text
stringlengths 1.79k
370k
| summary
stringlengths 34
23.5k
|
|---|---|
You are an expert at summarizing long articles. Proceed to summarize the following text:
the discovery of the top quark at fermilab fulfilled the three - family quark picture in the standard electroweak model . up to now , some knowledge on the mass spectra of @xmath2 and @xmath3 quarks has been accumulated through both experimental and theoretical ( or phenomenological ) attempts @xcite . the ratios of quark mass eigenvalues are obtainable after one renormalizes them to a common reference scale , e.g. , @xmath4 gev or @xmath5 . there exists a clear mass hierarchy in each quark sector : @xmath6 in comparison , the masses of three charged leptons manifest a similar hierarchical pattern @xcite . quark mass eigenstates are related to quark weak ( flavor ) eigenstates by the kobayashi - maskawa ( km ) matrix @xmath7 @xcite , which provides a quite natural description of flavor mixings and @xmath1 violation in the standard model . to date , many experimental constraints on the magnitudes of the km matrix elements have been achieved . the unitarity of @xmath7 together with current data requires a unique hierarchy among the nine matrix elements @xcite : @xmath8 here @xmath9 is a necessary condition for the presence of @xmath1 violation in the km matrix . how to understand the hierarchies of quark masses and flavor mixings is an important but unsolved problem in particle physics . a natural approach to the final solution of this problem is to look for the most favorable pattern of quark mass matrices ( see , e.g. , refs . @xcite ) , which can account for all low - energy phenomena of quark mixings and @xmath1 violation . the relevant symmetries hidden in such phenomenological schemes are possible to provide useful hints toward the dynamical details of fermion mass generation . it has been speculated by some authors that the realistic fermion mass matrices could arise from the flavor permutation symmetry and its spontaneous or explicit breaking @xcite . under exact @xmath10 symmetry the mass spectrum for either up or down quark sector consists of only two levels : one is of 2-fold degeneracy with vanishing mass eigenvalues , and the other is nondegenerate ( massive ) . an appropriate breakdown of the above symmetry may lead to the observed mass hierarchy and flavor mixings . although the way to introduce the minimum number of free parameters for permutation symmetry breaking is technically trivial , its consequences on quark mixings and @xmath1 violation may be physically instructive and may even shed some light on the proper relations between the km matrix elements and quark mass ratios . indeed there has not been a satisfactory symmetry breaking pattern with enough predictive power in the literature . in this work we first stress that some observed properties of the km matrix can be interpreted by the quark mass hierarchy without the assumption of specific mass matrices . in the quark mass limits such as @xmath11 , @xmath12 or @xmath13 , we find that simple but instructive relations between the km matrix elements and quark mass ratios are suggestible from current experimental data . then we present a new quark mass _ ansatz _ through the explicit breakdown of flavor permutation symmetry at the weak scale ( @xmath14 gev ) . this _ ansatz _ contains seven free parameters , thus it can give rise to three predictions for the phenomena of quark mixings and @xmath1 violation . the typical results are @xmath15 , @xmath16 and @xmath17 in the leading order approximation . prescribing the same _ ansatz _ at the supersymmetric grand unified theory ( gut ) scale ( @xmath18 gev ) , we derive the renormalized quark mass matrices at @xmath5 for small @xmath19 ( the ratio of higgs vacuum expectation values in the minimal supersymmetric model ) . we also renormalize some relations between the km matrix elements and quark mass ratios at @xmath5 for arbitrary @xmath19 , and find that the relevant results are in good agreement with experimental data . the scale - independent predictions of our _ ansatz _ for the characteristic measurables of @xmath1 asymmetries in weak @xmath20 decays , i.e. , @xmath21 , @xmath22 and @xmath23 , can be tested at the forthcoming kek and slac @xmath20-meson factories . the remaining part of this paper is organized as follows . some qualitative implications of the quark mass hierarchy on the km matrix elements , which are almost independent of the specific forms of quark mass matrices , are discussed in section 2 . in section 3 we suggest a new quark mass _ ansatz _ from the flavor permutation symmetry breaking at the weak scale , and study its various consequences on flavor mixings and @xmath1 violation . the same _ ansatz _ is prescribed at the supersymmetric gut scale in section 4 . by use of the one - loop renormalization group equations , we run the mass matrices from @xmath24 to @xmath5 and then discuss the renormalized relations between the km matrix elements and quark mass ratios . section 5 is devoted to a brief summary of this work . without loss of any generality , the up and down quark mass matrices ( denoted by @xmath25 and @xmath26 , respectively ) can be chosen to be hermitian . after the diagonalization of @xmath25 and @xmath26 through the unitary transformations @xmath27 one obtains the km matrix @xmath28 , which describes quark flavor mixings in the charged current . explicitly , the km matrix elements read @xmath29 depending upon the quark mass ratios @xmath30 , @xmath31 ( from @xmath32 ) and @xmath33 , @xmath34 ( from @xmath35 ) as well as other parameters of @xmath25 and @xmath26 ( e.g. , the non - trivial phase shifts between @xmath25 and @xmath26 ) . in view of the distinctive mass hierarchy in eq . ( 1.1 ) , we find that some interesting properties of @xmath7 can be interpreted without the assumption of specific forms of @xmath25 and @xmath26 . * a. @xmath36 and @xmath37 in the limits @xmath12 and @xmath13 * since the mass spectra of up and down quarks are absolutely dominated by @xmath38 and @xmath39 respectively , the limits @xmath12 and @xmath13 are expected to be very reliable when we discuss flavor mixings between @xmath40 and @xmath41 . in this case , the effective mass matrices turn out to be two @xmath42 matrices and the resultant flavor mixing matrix ( i.e. , the cabibbo matrix @xcite ) can not accommodate @xmath1 violation . the magnitudes of @xmath43 and @xmath44 can be obtained from eq . ( 2.2 ) , since @xmath45 holds for both sectors in the above - mentioned mass limits . we find that @xmath46 is a straightforward result guaranteed by the unitarity of @xmath32 and @xmath35 . the current experimental data , together with unitary conditions of the @xmath47 km matrix , have implied @xcite @xmath48 which is insensitive to allowed errors of the wolfenstein parameters @xmath49 , @xmath50 and @xmath51 @xcite . from the discussions above we realize that the near equality of @xmath36 and @xmath37 is in fact a natural consequence of @xmath52 and @xmath53 . the magnitude of @xmath43 ( or @xmath44 ) must be a function of the mass ratios @xmath30 and @xmath33 in the limits @xmath12 and @xmath13 , if @xmath25 and @xmath26 have parallel or quasi - parallel structures . considering the experimentally allowed regions of @xmath30 ( @xmath54 @xcite ) , @xmath55 ( @xmath56 @xcite ) and @xmath36 ( @xmath57 @xcite ) , one may guess that @xmath36 is dominated by @xmath58 but receives small correction from @xmath59 . indeed such an instructive result for @xmath36 or @xmath37 can be derived from @xmath42 hermitian mass matrices of the form @xcite @xmath60 where @xmath61 . denoting the phase difference between @xmath62 and @xmath63 as @xmath64 , we obtain @xmath65 although the @xmath42 flavor mixing matrix can not accommodate @xmath1 violation , the phase shift @xmath64 is non - trivial on the point that it sensitively determines the value of @xmath36 . for illustration , we calculate the allowed region of @xmath64 as a function of @xmath30 in fig . it is clear that the possibilities @xmath66 , @xmath67 and @xmath68 have all been ruled out by current data on @xmath43 and @xmath55 , since @xmath69 is expected to be true . we conclude that the presence of @xmath64 in the quark mass _ ansatz _ above is crucial for correct reproduction of @xmath36 and @xmath37 . such a non - trivial phase shift will definitely lead to @xmath1 violation , when the limits @xmath12 and @xmath13 are discarded . * b. @xmath70 and @xmath71 in the limit @xmath11 * considering the fact that @xmath72 and @xmath73 are negligibly small in the mass spectra of up and down quarks , one can take the reasonable limit @xmath11 to discuss flavor mixings between the second and third families . in this case , there is no mixing between @xmath40 and @xmath41 or between @xmath40 and @xmath74 . thus @xmath75 holds for both up and down mass matrices , and then we get @xmath76 . the relation @xmath77 is straightforwardly obtainable from eq . ( 2.2 ) by use of the unitary conditions of @xmath32 and @xmath35 . in contrast , the present data and unitarity of the km matrix requires @xcite @xmath78 we see that the near equality between @xmath70 and @xmath71 can be well understood , because the quark mass limit @xmath11 is a good approximation for @xmath25 and @xmath26 . we expect that @xmath70 and @xmath71 are functions of the mass ratios @xmath31 and @xmath34 in the limit @xmath11 . current experimental data give @xmath79 @xcite , while @xmath80 @xcite and @xmath81 @xcite are allowed . thus @xmath70 ( or @xmath71 ) should be dominated by @xmath34 , and it may get a little correction from @xmath31 . to obtain a linear relation among @xmath82 , @xmath34 and @xmath31 in the leading order approximation , one can investigate mass matrices of the following hermitian form : @xmath83 where @xmath84 and @xmath85 for both quark sectors . this generic pattern can also be regarded as a trivial generalization of the fritzsch _ ansatz _ , in which @xmath86 is assumed @xcite , but they have rather different consequences on the magnitudes of @xmath82 and @xmath87 . denoting @xmath88 , @xmath89 and @xmath90 , we find the approximate result @xmath91 one can see that @xmath92 holds approximately , if @xmath93 is comparable in magnitude with @xmath94 . here the phase shift @xmath95 plays an insignificant ( negligible ) role in confronting eq . ( 2.8 ) with the experimental data on @xmath70 , since the term proportional to @xmath31 is significantly suppressed . to determine the values of @xmath93 and @xmath94 , however , one has to rely on a more specific _ ansatz _ of quark mass matrices . * c. @xmath96 in @xmath13 and @xmath97 in @xmath12 * now let us take a look at the two smallest matrix elements of @xmath7 , @xmath98 and @xmath99 , in the quark mass limits . taking @xmath13 , we have @xmath100 , because @xmath26 turns out to be an effective @xmath42 matrix in this limit . then the ratio of @xmath98 to @xmath70 reads @xmath101 obtained from eq . contrary to common belief , @xmath96 is absolutely independent of the mass ratio @xmath33 in the limit @xmath102 ! therefore one expects that the left - handed side of eq . ( 2.9 ) is dominated by a simple function of the mass ratio @xmath30 , while the contribution from @xmath31 should be insignificant in most cases . the present numerical knowledge of @xmath96 ( @xmath103 @xcite ) and @xmath30 ( @xmath54 @xcite ) implies that @xmath16 is likely to be true . indeed such an approximate result can be reproduced from the fritzsch _ ansatz _ and a variety of its modified versions @xcite . in the mass limit @xmath12 , @xmath25 becomes an effective @xmath42 matrix , and then @xmath104 holds . the ratio of @xmath99 to @xmath71 is obtainable from eq . ( 2.2 ) as follows : @xmath105 here again we find that @xmath97 is independent of both @xmath30 and @xmath31 in the limit @xmath12 , thus it may be a simple function of the mass ratios @xmath33 and @xmath34 . the current data give @xmath106 @xcite , @xmath107 @xcite and @xmath108 @xcite . we expect that @xmath109 has a large chance to be true in the leading order approximation . note that this approximate relation can also be derived from the fritzsch _ ansatz _ or some of its revised versions @xcite . the qualitative discussions above have shown that some properties of the km matrix @xmath7 can be well understood just from the quark mass hierarchy . for example , @xmath110 and @xmath111 are natural consequences of arbitrary ( hermitian ) mass matrices with @xmath112 and @xmath113 respectively , where @xmath114 denote the mass eigenvalues of each quark sector . to a good degree of accuracy , @xmath36 and @xmath37 are expected to be independent of the mass ratios @xmath31 and @xmath34 , while @xmath70 and @xmath71 are independent of @xmath30 and @xmath33 . the ratios @xmath96 and @xmath97 may be simple functions of @xmath30 and @xmath33 , respectively , in the leading order approximations . these qualitative results should hold , in most cases and without fine tuning effects , for generic ( hermitian ) forms of @xmath25 and @xmath26 . they can be used as an enlightening clue for the construction of specific and predictive @xmath115 of quark mass matrices . we are now in a position to consider the realistic @xmath47 mass matrices in no assumption of the quark mass limits . such an _ ansatz _ should be able to yield the definite values of @xmath93 and @xmath94 in eq . ( 2.8 ) , and account for current experimental data on flavor mixings and @xmath1 violation at low - energy scales . * a. flavor permutation symmetry breaking * we start from the flavor permutation symmetry to construct quark mass matrices at the weak scale , so that the resultant km matrix can be directly confronted with the experimental data . the mass matrix with the @xmath10 symmetry reads @xmath116 where @xmath117 denotes the mass eigenvalue of the third - family quark ( @xmath118 or @xmath119 ) . note that @xmath120 is obtainable from another rank - one matrix @xmath121 through the unitary transformation @xmath122 , where @xmath123 to generate masses for the second- and first - family quarks , one has to break the permutation symmetry of @xmath120 to the @xmath124 and @xmath125 symmetries , respectively . here we assume that the up and down mass matrices have the parallel symmetry breaking patterns , corresponding to the parallel dynamical details of quark mass generation . we further assume that each symmetry breaking chain ( i.e. , @xmath126 or @xmath127 ) is induced by a single real parameter , and the possible phase shift between two quark sectors arises from an unknown dynamical mechanism . with the assumptions made above , a new _ ansatz _ for the up and down mass matrices can be given as follows : @xmath128 \ ; , \ ] ] where @xmath129 and @xmath130 are real ( dimensionless ) perturbation parameters responsible for the breakdowns of @xmath131 and @xmath124 symmetries of @xmath120 , respectively . in the basis of @xmath132 , the mass matrix @xmath133 takes the form @xmath134 which has three free parameters and three texture zeros . diagonalizing @xmath135 through the unitary transformation @xmath136 , one can determine @xmath137 , @xmath129 and @xmath130 in terms of the quark mass eigenvalues . in the next - to - leading order approximations , we get @xmath138 then the elements of @xmath139 are expressible in terms of the mass ratios @xmath140 and @xmath141 . the flavor mixing matrix can be given as @xmath142 , where @xmath143 is a diagonal phase matrix taking the form @xmath144 . here @xmath145 denotes the phase shift between up and down mass matrices , and its presence is necessary for the _ ansatz _ to correctly reproduce both @xmath36 ( or @xmath37 ) and @xmath1 violation . * b. flavor mixings and @xmath1 violation * calculating the km matrix elements @xmath36 and @xmath37 in the next - to - leading order approximation , we obtain @xmath146 this result is clearly consistent with that in eq . the allowed region of @xmath64 has been shown by fig . 1 with the inputs of @xmath55 and @xmath36 . we find @xmath147 for reasonable values of @xmath30 . in the leading order approximation of eq . ( 3.7 ) or eq . ( 2.5 ) , it is easy to check that @xmath37 , @xmath59 and @xmath58 form a triangle in the complex plane @xcite . in the next - to - leading order approximation , @xmath70 and @xmath71 can be given as @xmath148 \ ; . \ ] ] comparing between eqs . ( 3.8 ) and ( 2.8 ) , we get @xmath149 , determined by the quark mass _ ansatz _ in eq . ( 3.4 ) . by use of @xmath81 @xcite , we illustrate the allowed region of @xmath70 as a function of @xmath31 in fig . 2 , where the experimental constraint on @xmath70 ( @xmath150 @xcite ) has also been shown . we see that the result of @xmath70 obtained in eq . ( 3.8 ) is rather favored by current data . this implies that the pattern of permutation symmetry breaking ( i.e. , @xmath151 ) in eq . ( 3.4 ) may have a large chance to be true . the ratios @xmath96 and @xmath97 are found to be @xmath152 to a good degree of accuracy . with @xmath153 . the magnitude of @xmath154 may be as large as @xmath155 to @xmath156 for @xmath157 or @xmath68 , but it is only about @xmath158 for @xmath147 allowed by eq . ] by use of leutwyler s result @xmath107 @xcite , we get @xmath159 . in comparison , the current data together with unitarity of the @xmath47 km matrix yield @xmath160 @xcite . the allowed region of @xmath96 is constrained by that of @xmath30 , which has not been reliably determined . we find that @xmath161 is necessary for the quark mass _ ansatz _ in eq . ( 3.4 ) to accommodate the experimental result @xmath162 @xcite . in the leading order approximations , we have @xmath163 . small corrections to these diagonal elements are obtainable with the help of the unitary conditions of @xmath7 . if we rescale three sides of the unitarity triangle @xmath164 by @xmath165 , then the resultant triangle is approximately equivalent to that formed by @xmath44 , @xmath59 and @xmath58 in the complex plane @xcite . this interesting result can be easily shown by use of eqs . ( 3.7 ) , ( 3.8 ) and ( 3.9 ) . three inner angles of the unitarity triangle turn out to be @xmath166 in the approximations made above . at the forthcoming @xmath20-meson factories , these three angles will be determined from @xmath1 asymmetries in a variety of weak @xmath20 decays ( e.g. , @xmath167 , @xmath168 and @xmath169 ) . for illustration , we calculate @xmath170 , @xmath171 and @xmath172 by use of eq . ( 3.10 ) and plot their allowed regions in fig . 3 . clearly the quark mass _ ansatz _ under discussion favors @xmath173 , @xmath22 and @xmath23 . these results do not involve large errors , and they can be confronted with the relevant experiments of @xmath20 decays and @xmath1 violation in the near future . finally we point out that @xmath1 violation in the km matrix , measured by the jarlskog parameter @xmath174 @xcite , can also be estimated in terms of quark mass ratios . it is easy to obtain @xmath175 \sin\delta\phi \ ; . \ ] ] typically taking @xmath176 , @xmath177 , @xmath178 , @xmath179 and @xmath180 , we get @xmath181 . this result is of course consistent with current data on @xmath1 violation in the @xmath182 mixing system @xcite . it is interesting to speculate that the quark mass hierarchy and flavor mixings may arise from a certain symmetry breaking pattern in the context of supersymmetric guts @xcite . starting from the flavor permutation symmetry , here we prescribe the same _ ansatz _ for quark mass matrices as that proposed in eq . ( 3.4 ) at the supersymmetric gut scale @xmath24 . for simplicity we use @xmath183 and @xmath184 , which correspond to @xmath185 in eq . ( 3.4 ) and @xmath135 in eq . ( 3.5 ) , to denote the mass matrices at @xmath24 in two different bases . they are related to each other through the unitary transformation @xmath186 . the flavor mixing matrix derived from @xmath183 ( or @xmath184 ) is denoted by @xmath187 . the subsequent running effects of @xmath183 and @xmath187 from @xmath24 to @xmath5 can be calculated with the help of the renormalization group equations in the minimal supersymmetric standard model . * a. renormalized mass matrices at @xmath5 * the simplicity of @xmath183 ( or @xmath184 ) may be spoiled after it evolves from @xmath24 to @xmath5 . to illustrate this point , here we derive the renormalized mass matrices @xmath188 and @xmath189 at @xmath5 by use of the one - loop renormalization group equations for the yukawa matrices and gauge couplings @xcite . to get instructive analytical results , we constrain the ratio of higgs vacuum expectation values @xmath19 to be small enough ( @xmath190 ) , so that all non - leading terms in the yukawa couplings different from that of the top quark can be safely neglected @xcite . in this approximation , the evolution equations of @xmath188 and @xmath189 read @xmath191 \hat{m}^{\rm u}_0 \ ; , \nonumber \\ 16 \pi^2 \frac{{\rm d } \hat{m}^{\rm d}_0}{{\rm d } \chi } & = & \left [ \frac{1}{v^2 } \left ( \hat{m}^{\rm u}_0 \hat{m}^{{\rm u}^{\dagger}}_0 \right ) - g_{\rm d } \right ] \hat{m}^{\rm d}_0 \ ; , \ ] ] where @xmath192 , @xmath193 and @xmath194 are functions of the gauge couplings @xmath195 ( @xmath196 ) , and @xmath197 is the overall higgs vacuum expectation value normalized to 175 gev . for the charged lepton mass matrix @xmath198 , its evolution equation is dominated only by a linear term @xmath199 in the case of small @xmath19 . thus the hermitian structure of @xmath198 will be unchanged through the running from @xmath24 to @xmath5 ( in our discussions the neutrinos are assumed to be massless ) . the quantity @xmath200 ( n = u , d or e ) obeys the following equation : @xmath201 \ ; , \ ] ] where @xmath202 and @xmath203 are coefficients in the context of the minimal supersymmetric standard model . the values of @xmath204 , @xmath202 and @xmath203 are listed in table 1 . c|ccccc @xmath205 & @xmath206 & @xmath207 & @xmath208 & @xmath203 & @xmath209 + + 1 & 13/9 & 7/9 & 3 & 11 & 0.127 + + 2 & 3 & 3 & 3 & 1 & 0.42 + + 3 & 16/3 & 16/3 & 0 & @xmath2103 & 1.44 + + in order to solve eq . ( 4.1 ) , we diagonalize @xmath188 through the unitary transformation @xmath211 . making the same transformation for @xmath189 , i.e. , @xmath212 , we obtain the simplified evolution equations as follows : @xmath213 \hat{m}^{{\rm u}^{\prime}}_0 \ ; , \nonumber \\ 16 \pi^2 \frac{{\rm d}\hat{m}^{{\rm d}^{\prime}}_0}{{\rm d } \chi } & = & \left [ f^2_t \left ( \matrix { 0 & 0 & 0 \cr 0 & 0 & 0 \cr 0 & 0 & 1 } \right ) - g_{\rm u } \left ( \matrix { 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 } \right ) \right ] \hat{m}^{{\rm d}^{\prime}}_0 \ ; , \ ] ] where @xmath214 is the top quark yukawa coupling eigenvalue . for simplicity in presenting the results , we define @xmath215 \ ; , \nonumber \\ \xi_i & = & \exp \left [ - \frac{1}{16\pi^2 } \int^{\ln ( m_x / m_z)}_0 f^2_i ( \chi ) ~ { \rm d}\chi \right ] \ ] ] with @xmath216 ( or @xmath217 ) . by use of eq . ( 4.2 ) and the inputs listed in table 1 , one can explicitly calculate the magnitude of @xmath218 . we find @xmath219 , @xmath220 and @xmath221 for @xmath222 gev and @xmath223 gev . the size of @xmath224 depends upon the value of @xmath19 and will be estimated in the next subsection . solving eq . ( 4.3 ) and transforming @xmath225 back to @xmath226 , we get @xmath227 in the leading order approximation . since @xmath228 can be easily determined from @xmath229 and @xmath230 in the approximation of @xmath231 made above , we explicitly express @xmath232 and @xmath233 as follows : @xmath234 \ ; \ ] ] with @xmath235 , @xmath236 and @xmath237 ; and @xmath238 \ ; \ ] ] with @xmath239 , @xmath240 , @xmath241 and @xmath242 if one takes @xmath243 , which leads to @xmath244 , @xmath245 and in turn @xmath246 , @xmath247 , @xmath248 and @xmath249 , then eqs . ( 4.6 ) and ( 4.7 ) recover the form of @xmath250 as assumed in eq . ( 3.4 ) . to a good degree of accuracy , @xmath232 remains hermitian . the hermiticity of @xmath233 is violated by @xmath251 , which would vanish if the top and bottom quark masses were identical ( i.e. , @xmath252 ) . the presence of nonvanishing @xmath251 reflects the fact that @xmath38 dominates the mass spectra of both quark sectors @xcite . of course , one can transform the mass matrices obtained in eqs . ( 4.6 ) and ( 4.7 ) into the basis of @xmath184 . in doing so , we will find the inequality between ( 2,3 ) and ( 3,2 ) elements of @xmath253 , arising from @xmath251 . * b. renormalized flavor mixings at @xmath5 * calculating the magnitudes of flavor mixings from @xmath183 or @xmath184 at @xmath24 , we can obtain the same asymptotic relations between the km matrix elements and quark mass ratios as those given in eqs . ( 3.7 ) , ( 3.8 ) and ( 3.9 ) . now we renormalize such relations at the weak scale @xmath5 by means of the renormalization group equations . the quantities @xmath224 and @xmath254 defined in eq . ( 4.4 ) will be evaluated below for arbitrary @xmath19 , so that one can get some quantitative feeling about the running effects of quark mass matrices and flavor mixings from @xmath24 to @xmath5 . the one - loop renormalization group equations for quark mass ratios and elements of the km matrix @xmath187 have been explicitly presented by babu and shafi in ref . @xcite . in view of the hierarchy of yukawa couplings and quark mixing angles , one can make reliable analytical approximations for the relevant evolution equations by keeping only the leading terms . it has been found that ( 1 ) the running effects of @xmath30 and @xmath33 are negligibly small ; ( 2 ) the diagonal elements of the km matrix have negligible evolutions with energy ; ( 3 ) the evolutions of @xmath255 and @xmath256 involve the second - family yukawa couplings and thus they are negligible ; ( 4 ) the km matrix elements @xmath257 , @xmath258 , @xmath259 and @xmath260 have identical running behaviors . considering these points as well as the dominance of the third - family yukawa couplings ( i.e. , @xmath261 and @xmath262 ) , we get three key evolution equations in the minimal supersymmetric standard model : @xmath263 with @xmath264 , @xmath265 , @xmath266 or @xmath267 . in the same approximations , the renormalization group equations for the yukawa coupling eigenvalues @xmath261 , @xmath262 and @xmath268 read @xcite : @xmath269 where the quantities @xmath200 have been given in eq . ( 4.2 ) . with the typical inputs @xmath270 gev , @xmath271 gev , @xmath272 gev and those listed in table 1 , we calculate @xmath224 and @xmath254 for arbitrary @xmath19 by use of the above equations . our result is illustrated in fig . we see that @xmath273 for @xmath274 . this justifies our approximation made previously in deriving @xmath232 and @xmath233 . within the perturbatively allowed region of @xmath19 @xcite , @xmath254 may be comparable in magnitude with @xmath224 when @xmath275 . in this case , the evolution effects of quark mass matrices and flavor mixings are sensitive to both @xmath261 and @xmath262 . clearly the analytical results of @xmath255 , @xmath256 , @xmath276 and @xmath277 as those given in eqs . ( 3.7 ) and ( 3.9 ) are almost scale - independent , i.e. , they hold at both @xmath278 and @xmath279 . non - negligible running effects can only appear in the expression of @xmath258 or @xmath260 , which is a function of the mass ratios @xmath34 and @xmath31 ( see eq . ( 3.8 ) for illustration ) . with the help of eq . ( 4.9 ) , we find the renormalized relation between @xmath258 ( or @xmath260 ) and the quark mass ratios at the weak scale @xmath5 : @xmath280 \ ; . \ ] ] this result will recover that in eq . ( 3.8 ) if one takes @xmath243 ( i.e. , @xmath281 ) . using @xmath282 @xcite and taking @xmath283 typically , we confront eq . ( 4.11 ) with the experimental data on @xmath284 ( i.e. , @xmath285 @xcite ) . as shown in fig . 5 , our result is in good agreement with experiments for @xmath286 . this implies that the quark mass pattern @xmath183 or @xmath184 , proposed at the supersymmetric gut scale @xmath24 , may have a large chance to survive for reasonable values of @xmath19 . note that evolution of the @xmath1-violating parameter @xmath174 is dominated by that of @xmath287 . note also that three sides of the unitarity triangle @xmath288 have identical running effects from @xmath24 to @xmath5 , hence its three inner angles are scale - independent and take the same values as those given in eq . ( 3.10 ) or fig . 3 . as a result , measurements of @xmath289 , @xmath290 and @xmath291 in the forthcoming experiments of @xmath20 physics may check both the quark mass _ ansatz _ at the weak scale and that at the supersymmetric gut scale . without the assumption of specific mass matrices , we have pointed out that part of the observed properties of flavor mixings can be well understood just from the quark mass hierarchy . in the quark mass limits such as @xmath11 , @xmath12 or @xmath13 , a few instructive relations between the km matrix elements and quark mass ratios are suggestible from current experimental data . we stress that such _ ansatz_-independent results may serve as a useful guide in constructing the specific quark mass matrices at either low - energy scales or superheavy scales . starting from the flavor permutation symmetry and assuming an explicit pattern of symmetry breaking , we have proposed a new quark mass _ ansatz _ at the weak scale . we find that all experimental data on quark mixings and @xmath1 violation can be accounted for by our _ ansatz_. in particular , we obtain an instructive relation among @xmath70 , @xmath34 and @xmath31 in the next - to - leading approximation ( see eq . ( 3.8 ) ) . the scale - independent predictions of our quark mass pattern , such as @xmath21 , @xmath22 and @xmath23 , can be confronted with the forthcoming experiments at kek and slac @xmath20-meson factories . with the same _ ansatz _ prescribed at the supersymmetric gut scale @xmath24 , we have derived the renormalized quark mass matrices at the weak scale @xmath5 for small @xmath19 and calculated the renormalized flavor mixing matrix elements at @xmath5 for arbitrary @xmath19 . except @xmath258 and @xmath260 , the other asymptotic relations between the km matrix elements and quark mass ratios are almost scale - independent . we find that the renormalized result of @xmath258 ( or @xmath260 ) is in good agreement with the relevant experimental data for reasonable values of @xmath19 . in this work we neither assumed a specific form for the charged lepton mass matrix nor supposed its relation with the down quark mass matrix within the supersymmetric gut framework . of course , this can be done by following the strategy proposed in ref . @xcite . then one may obtain the relations between @xmath73 , @xmath292 , @xmath39 and @xmath293 , @xmath294 , @xmath295 . such an _ ansatz _ , based on the specific gut scheme and flavor permutation symmetry breaking , will be discussed somewhere else . the author would like to thank a.i . sanda for his warm hospitality and the japan society for the promotion of science for its financial support . he is also grateful to h. fritzsch , a.i . sanda and k. yamawaki for their useful comments on the topic of permutation symmetry breaking and on part of this work . xing , nucl . b ( proc . suppl . ) * 50 * , 24 ( 1996 ) ; nuovo cimento a * 109 * , 115 ( 1996 ) ; in the proceedings of the international europhysics conference on high energy physics , brussels , belgium , 1995 , edited by j. lemonne , c. vander velde , and f. verbeure ( world scientific , singapore , 1996 ) , p. 181 . h. harari , h. haut , and j. weyers , phys . b * 78 * , 459 ( 1978 ) ; y. koide , phys . rev . d * 28 * , 252 ( 1983 ) ; phys . d * 39 * , 1391 ( 1989 ) ; z. phys . c * 45 * , 39 ( 1989 ) ; h. fritzsch , in the proceedings of the europhysics topical conference on flavor mixing in weak interactions , erice , italy , 1984 , edited by l.l . chau ( plenum , new york , 1984 ) , p. 717 ; c. jarlskog , in the proceedings of the international symposium on production and decay of heavy hadrons , heidelberg , germany , 1986 , edited by k.r . schubert and r. waldi ( desy , hamburg , 1986 ) , p. 331 ; p. kaus and s. meshkov , ann . n. y. acad . sci . * 578 * , 353 ( 1989 ) ; phys . d * 42 * , 1863 ( 1990 ) ; m. tanimoto , phys . d * 41 * , 1586 ( 1990 ) ; g.c . branco , j.i . silva - marcos , and m.n . rebelo , phys . b * 237 * , 446 ( 1990 ) ; h. fritzsch and j. plankl , phys . b * 237 * , 451 ( 1990 ) ; c.e . lee , c. lin , and y.w . yang , phys . d * 42 * , 2355 ( 1990 ) ; p.f . harrison and w.g . scott , phys . b * 333 * , 471 ( 1994 ) ; g.c . branco and j.i . silva - marcos , phys . b * 359 * , 166 ( 1995 ) . for a brief review of other relevant references , see : s. meshkov , in the proceedings of the global foundation international conference on unified symmetry in the small and in the large , coral gables , florida , 1993 , edited by b.n . kursunoglu and a. perlmutter ( nova sci . publ . , new york , 1994 ) , p. 195 see , e.g. , l.j . hall and a. rasin , phys . b * 315 * , 164 ( 1993 ) ; d. du and z.z . xing , phys . d * 48 * , 2349 ( 1993 ) ; k. harayama , n. okamura , a.i . sanda , and z.z . xing , report no . dpnu-96 - 36 ( 1996 ) . s. dimopoulos , l.j . hall , and s. raby , phys . lett . * 68 * , 1984 ( 1992 ) ; g. anderson _ et al . _ , d * 47 * , 3702 ( 1993 ) ; p. ramond , r.g . roberts , and g.g . ross , nucl . b * 406 * , 19 ( 1993 ) ; a. kusenko and r. shrock , phys . d * 49 * , 4962 ( 1994 ) ; m. leurer , y. nir , and n. seiberg , nucl . b * 420 * , 468 ( 1994 ) ; l.e . ibanez and g.g . ross , phys . b * 332 * , 100 ( 1994 ) ; k.s . babu and r.n . mohapatra , phys . * 74 * , 2418 ( 1995 ) ; k.s . babu and s.b . barr , phys . lett . * 75 * , 2088 ( 1995 ) ; l.j . hall and s. raby , phys . d * 51 * , 6524 ( 1995 ) ; k.c . chou and y.l . wu , phys . d * 53 * , r3492 ( 1996 ) . for a brief review with more extensive references , see : c.d . froggatt , report no . gutpa/96/02/1 ( to be published in the proceedings of the fifth hellenic school and workshops on elementary particle physics , corfu , september , 1995 ) . | we stress that the observed pattern of flavor mixings can be partly interpreted by the quark mass hierarchy without the assumption of specific quark mass matrices .
the quantitatively proper relations between the kobayashi - maskawa matrix elements and quark mass ratios , such as @xmath0 \ ; , \ ] ] are obtainable from a simple _ ansatz _ of flavor permutation symmetry breaking at the weak scale .
we prescribe the same _ ansatz _ at the supersymmetric grand unified theory scale , and find that its all low - energy consequences on flavor mixings and @xmath1 violation are in good agreement with current experimental data .
= 17.1 cm = 24.71 cm = -15.mm = -11 mm plus 1pt minus 1pt * dpnu-96 - 39 + august 1996 * * implications of the quark mass hierarchy on flavor mixings * zhi - zhong xing _ department of physics , nagoya university , chikusa - ku , nagoya 464 - 01 , japan _
pacs number(s ) : 12.15.ff , 11.30.hv , 11.30.pb , 12.10.dm |
You are an expert at summarizing long articles. Proceed to summarize the following text:
this paper deals with combinatorial aspects in the representation theory of algebras . more precisely , for certain classes of algebras which are defined purely combinatorially by directed graphs and homogeneous relations we will characterize important representation - theoretic invariants in a combinatorial way . in particular , this leads to new explicit invariants for the derived module categories of the algebras involved . the starting point for this article is that the unimodular equivalence class of the cartan matrix of a finite dimensional algebra is invariant under derived equivalence . hence , being able to determine normal forms of cartan matrices yields invariants of the derived category . the class of algebras we study are the gentle algebras , and the related skewed - gentle algebras . gentle algebras are defined purely combinatorially in terms of a quiver with relations ( for details , see section [ sec - def ] ) ; the more general skewed - gentle algebras ( introduced in @xcite ) are then defined from gentle algebras by specifying special vertices which are split for the quiver of the skewed - gentle algebra ( see section [ sec - skewed ] ) . these algebras occur naturally in the representation theory of finite dimensional algebras , especially in the context of derived categories . for instance , the algebras which are derived equivalent to hereditary algebras of type @xmath2 are precisely the gentle algebras whose underlying undirected graph is a tree @xcite . the algebras which are derived equivalent to hereditary algebras of type @xmath3 are certain gentle algebras whose underlying graph has exactly one cycle @xcite . remarkably , the class of gentle algebras is closed under derived equivalence @xcite ; but note that the class of skewed - gentle algebras is not closed under derived equivalence . a fundamental distinction in the representation theory of algebras is given by the representation type , which can be either finite , tame or wild . in the modern context of derived categories , also derived representation types have been defined . again , gentle algebras occur naturally in this context . d. vossieck @xcite showed that an algebra @xmath4 has a discrete derived category if and only if either @xmath4 is derived equivalent to a hereditary algebra of type @xmath2 , @xmath5 , @xmath6 or @xmath4 is gentle with underlying quiver @xmath7 having exactly one ( undirected ) cycle and the number of clockwise and of counterclockwise paths of length 2 in the cycle that belong to @xmath8 are different . skewed - gentle algebras are known to be of derived tame representation type ( for a definition of derived tameness , see @xcite ) . it is a long - standing open problem to classify gentle algebras up to derived equivalence . a complete answer has only been obtained for the derived discrete case @xcite . the main problem is to find good invariants of the derived categories . in this paper we provide easy - to - compute invariants of the derived categories of skewed - gentle algebras which are of a purely combinatorial nature . our results are obtained from a detailed computation of the @xmath0-cartan matrices of gentle and skewed - gentle algebras , respectively . the following notion will be crucial throughout the paper . let @xmath7 be a ( gentle ) quiver with relations . an oriented path @xmath9 with arrows @xmath10 in @xmath11 is called an oriented @xmath12-cycle with full zero relations if @xmath13 has the same start and end point , and if @xmath14 for all @xmath15 and also @xmath16 . such a cycle is called minimal if the arrows @xmath17 on @xmath13 are pairwise different . we call two matrices @xmath18 with entries in a polynomial ring @xmath19 $ ] unimodularly equivalent ( over @xmath19 $ ] ) if there exist matrices @xmath20 over @xmath19 $ ] of determinant 1 such that @xmath21 . we can now state our main result on gentle algebras . * theorem 1 . * [ thm1 ] _ let @xmath7 be a gentle quiver , and @xmath22 the corresponding gentle algebra . denote by @xmath23 the number of minimal oriented @xmath12-cycles in @xmath11 with full zero relations . + then the @xmath0-cartan matrix @xmath24 is unimodularly equivalent ( over @xmath25 $ ] ) to a diagonal matrix with entries @xmath26 , with multiplicity @xmath27 , @xmath28 , and all further diagonal entries being @xmath29 . _ this theorem has the following direct consequences . * corollary 1 . * _ let @xmath7 be a gentle quiver , and @xmath30 the corresponding gentle algebra . denote by @xmath23 the number of minimal oriented @xmath12-cycles in @xmath11 with full zero relations . then the @xmath0-cartan matrix @xmath24 has determinant @xmath31 _ the following consequence of corollary 1 was first proved in @xcite . for a gentle quiver @xmath7 we denote by @xmath32 the number of minimal oriented cycles of odd length in @xmath11 having full zero relations , and by @xmath33 the number of analogous cycles of even length . * corollary 2 . * _ let @xmath7 be a gentle quiver , and @xmath30 the corresponding gentle algebra . then for the determinant of the cartan matrix @xmath34 the following holds . @xmath35 _ the most important application of theorem 1 is the following corollary which gives for gentle algebras easy - to - check combinatorial invariants of the derived category . * corollary 3 . * _ let @xmath7 and @xmath36 be gentle quivers , and let @xmath30 and @xmath37 be the corresponding gentle algebras . if @xmath4 and @xmath38 are derived equivalent , then @xmath39 and @xmath40 . _ as an illustration we give in section [ sec : gentles ] a complete derived equivalence classification of gentle algebras with two simple modules and of gentle algebras with three simple modules and cartan determinant 0 . our main result on skewed - gentle algebras determines the normal form of their @xmath0-cartan matrices . * theorem 2 . * _ let @xmath41 be a skewed - gentle algebra , arising from choosing a suitable set of special vertices in the gentle quiver @xmath7 . denote by @xmath23 the number of minimal oriented @xmath12-cycles in @xmath11 with full zero relations . + then the @xmath0-cartan matrix @xmath42 is unimodularly equivalent to a diagonal matrix with entries @xmath43 , with multiplicity @xmath27 , @xmath28 , and all further diagonal entries being @xmath29 . _ as an immediate consequence we obtain that the cartan determinant of any skewed - gentle algebra is the same as the cartan determinant for the underlying gentle algebra . * corollary 4 . * _ let @xmath41 be a skewed - gentle algebra , arising from choosing a suitable set of special vertices in the gentle quiver @xmath7 , with corresponding gentle algebra @xmath30 . then @xmath44 , and thus in particular , the determinants of the ordinary cartan matrices coincide , i.e. , @xmath45 . _ the paper is organized as follows . in section [ sec - def ] we collect the necessary background and definitions about quivers with relations and ( @xmath0-)cartan matrices . in section [ sec : gentles ] we prove all the main results about @xmath0-cartan matrices for gentle algebras . here we also give some extensive examples to illustrate our results . section [ sec - skewed ] contains the analogous main results for skewed - gentle algebras . * acknowledgement . * we thank the referee for very helpful and insightful comments . in particular , we are grateful for pointing out the importance of graded derived equivalences in our context of @xmath0-cartan matrices . algebras can be defined naturally from a combinatorial setting by using directed graphs . a finite directed graph @xmath11 is called a _ quiver_. for any arrow @xmath46 in @xmath11 we denote by @xmath47 its start vertex and by @xmath48 its end vertex . an oriented path @xmath13 in @xmath11 of length @xmath49 is a sequence @xmath50 of arrows @xmath51 such that @xmath52 for all @xmath53 . ( note that for each vertex @xmath54 in @xmath55 we allow a trivial path @xmath56 of length @xmath57 , having @xmath54 as its start and end vertex . ) for such a path @xmath13 we then denote by @xmath58 its start vertex and by @xmath59 its end vertex . the path algebra @xmath60 , where @xmath61 is any field , has as basis the set of all oriented paths in @xmath11 . the multiplication in the algebra @xmath60 is defined by concatenation of paths , i.e. , the product of two paths @xmath13 and @xmath0 is defined to be the concatenated path @xmath62 if @xmath63 , and zero otherwise . more general algebras can be obtained by introducing relations on a path algebra . an ideal @xmath64 is called admissible if @xmath65 where @xmath66 is the ideal of @xmath60 generated by the arrows of @xmath11 . the pair @xmath7 where @xmath11 is a quiver and @xmath64 is an admissible ideal is called a _ quiver with relations_. for any quiver with relations @xmath7 , we can consider the factor algebra @xmath30 , where @xmath61 is any field . we identify paths in the quiver @xmath11 with their cosets in @xmath4 let @xmath67 denote the set of vertices of @xmath11 . for any @xmath68 there is a path @xmath69 of length zero . these are primitive orthogonal idempotents in @xmath4 , the sum @xmath70 is the unit element in @xmath4 . in particular we get @xmath71 , hence the ( right ) @xmath4-modules @xmath72 are the indecomposable projective @xmath4-modules . the _ cartan matrix _ @xmath73 of an algebra @xmath30 is the @xmath74-matrix defined by setting @xmath75 . recall that when @xmath8 is generated by monomials , @xmath30 is called a monomial algebra . for monomial algebras , computing entries of the cartan matrix reduces to counting paths in the quiver @xmath11 which are nonzero in @xmath4 . in fact , any homomorphism @xmath76 of right @xmath4-modules is uniquely determined by @xmath77 , the @xmath61-vector space generated by all paths in @xmath11 from vertex @xmath78 to vertex @xmath79 , which are nonzero in @xmath30 . in particular , we have @xmath80 . this is the key viewpoint in this paper , enabling us to obtain results on the representation - theoretic cartan invariants by combinatorial methods . it allows to study a refined version of the cartan matrix , which we call the @xmath0-cartan matrix . ( it also occurred in the literature as filtered cartan matrix , see for instance @xcite . ) let @xmath11 be a quiver and assume that the relation ideal @xmath8 is generated by homogeneous relations , i.e. , by linear combinations of paths having the same length ( actually , for the algebras considered in this paper , the ideal @xmath8 will always be generated by monomials and commutativity ( mesh ) relations ) . the path algebra @xmath60 is a graded algebra , with grading given by path lengths . since @xmath8 is homogeneous , the factor algebra @xmath30 inherits this grading . so the morphism spaces @xmath81 become graded vector spaces . recall that the dimensions of these vector spaces are the entries of the ( ordinary ) cartan matrix . * definition . * let @xmath30 be a finite - dimensional algebra , and assume that the ideal @xmath8 is generated by homogeneous relations . for any vertices @xmath78 and @xmath79 in @xmath11 let @xmath82 be the graded components . let @xmath0 be an indeterminate . the @xmath0-cartan matrix @xmath83 of @xmath4 is defined as the matrix with entries @xmath84 $ ] . in other words , the entries of the @xmath0-cartan matrix are the poincar polynomials of the graded homomorphism spaces between projective modules . loosely speaking , when counting paths in the quiver of the algebra , each path is weighted by some power of @xmath0 according to its length . clearly , specializing @xmath1 gives back the usual cartan matrix @xmath34 ( i.e. , we forget the grading ) . even if we are mainly interested in the ordinary cartan matrix , the point of view of @xmath0-cartan matrices provides some new insights as we take a closer look at the invariants of the cartan matrix . [ brauer - ex ] _ we consider the following two quivers . _ let @xmath85 , where the ideal @xmath86 is generated by @xmath87 , @xmath88 and @xmath89 . the @xmath0-cartan matrix of @xmath4 has the form @xmath90 the second algebra @xmath91 is defined by the quiver @xmath92 , subject to the generating relations @xmath93 ( i.e. all paths of length four are zero ) . the @xmath0-cartan matrix of @xmath94 has the form @xmath95 cartan matrices provide invariants which are preserved under derived equivalences and thus improve our understanding of derived module categories ; this is our main motivation to study normal forms , invariant factors and determinants of cartan matrices in this paper . the following result is contained in the proof of ( * ? ? ? * proposition 1.5 ) . [ unimod ] let @xmath4 be a finite - dimensional algebra . the unimodular equivalence class of the cartan matrix @xmath34 is invariant under derived equivalence . in particular , the determinant of the cartan matrix is invariant under derived equivalence . we emphasize that the above theorem only deals with ordinary cartan matrices @xmath96 . the determinant of the @xmath0-cartan matrix is in general not invariant under derived equivalence . as an example , consider the algebras @xmath4 and @xmath94 from example [ brauer - ex ] , with @xmath97 and @xmath98 . but , in fact , the algebras @xmath4 and @xmath94 are derived equivalent ; they are brauer tree algebras for trees with the same number of edges and the same exceptional multiplicity @xcite . note that when specializing @xmath1 we indeed get the same determinants for the ordinary cartan matrices , as predicted by theorem [ unimod ] however , the natural setting when dealing with @xmath0-cartan matrices is that of graded derived categories . indeed , the determinant of the @xmath0-cartan matrix ( which is defined so as to take the grading into account ) is invariant under graded derived equivalences . we are very grateful to the referee for pointing this out to us . we do not discuss this aspect in this paper further , but shall address the topic of graded derived equivalences for gentle algebras in detail in a subsequent publication . for instance , the above algebra @xmath94 is graded derived equivalent to the algebra @xmath4 , where the grading on @xmath4 is chosen so that @xmath46 and @xmath99 are of degree 2 , and @xmath100 and @xmath101 of degree 1 . then rickard s derived equivalence @xcite lifts to a graded derived equivalence . in this section , we shall prove theorem [ thm1 ] on the unimodular equivalence class of the @xmath0-cartan matrix of an arbitrary gentle algebra . we first recall the definition of special biserial algebras and of gentle algebras , as these details will be crucial for what follows . let @xmath11 be a quiver and @xmath8 an admissible ideal in the path algebra @xmath60 . we call the pair @xmath7 a special biserial quiver ( with relations ) if it satisfies the following properties . \(i ) each vertex of @xmath11 is starting point of at most two arrows , and end point of at most two arrows . \(ii ) for each arrow @xmath46 in @xmath11 there is at most one arrow @xmath99 such that @xmath102 , and at most one arrow @xmath101 such that @xmath103 . a finite - dimensional algebra @xmath4 is called special biserial if it has a presentation as @xmath30 where @xmath7 is a special biserial quiver . gentle quivers form a subclass of the class of special biserial quivers . a pair ( @xmath104 as above is called a _ gentle quiver _ if it is special biserial and moreover the following holds . \(iii ) the ideal @xmath8 is generated by paths of length 2 . \(iv ) for each arrow @xmath46 in @xmath11 there is at most one arrow @xmath105 with @xmath106 such that @xmath107 , and there is at most one arrow @xmath108 with @xmath109 such that @xmath110 . a finite - dimensional algebra @xmath4 is called gentle if it has a presentation as @xmath30 where @xmath7 is a gentle quiver . the following lemma will turn out to be very useful . it does not only hold for gentle algebras but for those where we have dropped the final condition ( iv ) in the definition of gentle quivers . recall that two matrices @xmath111 , @xmath112 with entries in @xmath19 $ ] are called unimodularly equivalent ( over @xmath19 $ ] ) if there exist matrices @xmath113 , @xmath11 over @xmath19 $ ] of determinant 1 such that @xmath21 . [ lem : arrow - removal ] let @xmath7 be a special biserial quiver , and assume that @xmath8 is generated by paths of length 2 . let @xmath30 be the corresponding special biserial algebra . let @xmath46 be an arrow in @xmath11 , not a loop , such that there is no arrow @xmath114 with @xmath115 and @xmath116 , or there is no arrow @xmath117 with @xmath118 and @xmath119 . let @xmath120 be the quiver obtained from @xmath11 by removing the arrow @xmath46 , let @xmath121 be the corresponding relation ideal and @xmath37 . then the @xmath0-cartan matrices @xmath24 and @xmath122 are unimodularly equivalent ( over @xmath25 $ ] ) . * we consider the case where @xmath123 is an arrow in @xmath11 such that there is no arrow @xmath114 with @xmath115 and @xmath124 ; the second case is dual . + let @xmath125 . as @xmath7 is special biserial , there is a unique maximal non - zero path starting with @xmath126 , say @xmath127 , where @xmath128 , @xmath129 . as @xmath4 is finite - dimensional , the condition on @xmath130 guarantees that @xmath131 for all @xmath132 , but we may have @xmath133 for some @xmath134 . now any non - zero path of length @xmath79 , say , ending at @xmath135 can uniquely be extended to a non - zero path of length @xmath136 ending at @xmath137 , by concatenation with @xmath138 . conversely , any non - zero path ending at @xmath137 and involving @xmath126 arises in this way . + now denote the column corresponding to a vertex @xmath54 in the @xmath0-cartan matrix @xmath24 by @xmath139 . we perform column transformations on @xmath24 by replacing the columns @xmath140 by @xmath141 , for @xmath142 ( if @xmath133 for some @xmath143 , the column @xmath144 will then be replaced by @xmath145 ) . the resulting matrix @xmath146 is then exactly the cartan matrix @xmath122 to the algebra @xmath38 corresponding to the quiver @xmath120 where @xmath130 has been removed . @xmath147 + for any vertex in a quiver @xmath11 , its _ valency _ is defined as the number of arrows attached to it , i.e. , the number of incoming arrows plus the number of outgoing arrows ( note that in particular any loop contributes twice to the valency ) . [ thm : gentle ] let @xmath7 be a gentle quiver , and let @xmath30 be the corresponding gentle algebra . denote by @xmath23 the number of minimal oriented @xmath12-cycles in @xmath11 with full zero relations . + then the @xmath0-cartan matrix @xmath24 is unimodularly equivalent ( over @xmath25 $ ] ) to a diagonal matrix with entries @xmath26 , with multiplicity @xmath27 , @xmath28 , and all further diagonal entries being @xmath29 . * * we want to prove the claim by double induction on the number of vertices and the number of arrows . clearly the result holds if @xmath11 has no arrows or if it consists of one vertex with a loop . if @xmath11 has a vertex @xmath54 of valency 1 or 3 , or of valency 2 but with no zero relation at @xmath54 , then we can use lemma [ lem : arrow - removal ] to remove an arrow from @xmath11 ; note that by the conditions in lemma [ lem : arrow - removal ] the removed arrow is not involved in any oriented cycle with full zero relations . hence @xmath24 is unimodularly equivalent to @xmath122 , where the corresponding quiver has one arrow less but the same number of oriented cycles with full zero relations , and hence the result holds by induction . hence we may now assume that all vertices are of valency 0 , 2 or 4 , and if a vertex is of valency 2 , then there is a zero relation at the vertex . also , if @xmath11 is not connected , we may use induction on the number of vertices to have the result for the components and thus for the whole quiver ; hence we may assume that @xmath11 is connected . in particular , we now only have vertices @xmath54 of valency 2 with a non - loop zero relation at @xmath54 , and vertices of valency 4 . as we do not have paths of arbitrary lengths , not all vertices can be of valency 4 ( see also ( * ? ? ? * lemma 3 ) ) . now we take a vertex @xmath148 of valency 2 , with incoming arrow @xmath149 and outgoing arrow @xmath150 with @xmath151 ( here , @xmath152 ) . + as @xmath7 is gentle , there is a unique maximal path @xmath13 in @xmath11 with non - repeating arrows starting in @xmath135 with @xmath126 , such that the product of any two consecutive arrows is zero in @xmath4 ; in our present situation this path is an oriented cycle @xmath153 with full zero relations returning to @xmath135 . we denote the vertices on this path by @xmath154 , and the arrows by @xmath128 , @xmath155 ( also @xmath156 ) ; note that the arrows on @xmath13 are distinct , but the vertices are not necessarily distinct ( but we point out that @xmath157 for all @xmath158 ) . + denote by @xmath159 the row of the @xmath0-cartan matrix @xmath24 corresponding to the vertex @xmath160 . in @xmath24 , we now replace the row @xmath161 by the linear combination @xmath162 to obtain a new matrix @xmath146 ( note that this is a unimodular transformation over @xmath25 $ ] ) . + the careful choice of the coefficients is just made so that we can refine the argument in @xcite . we recall some of the notation there . for any arrow @xmath123 in @xmath11 let @xmath163 be the set of paths starting with @xmath123 which are non - zero in @xmath4 . at each vertex @xmath137 there is at most one outgoing arrow @xmath164 ; for this arrow we have @xmath165 , as @xmath7 is gentle . hence , cancelling @xmath166 induces a natural bijection @xmath167 , for @xmath168 , such that a path of @xmath0-weight @xmath169 is mapped to a path of @xmath0-weight @xmath170 ( if there is no arrow @xmath171 , we set @xmath172 ) . as @xmath54 is of valency 2 , with a zero - relation at @xmath54 , we also have the trivial bijection @xmath173 , again with a weight reduction by @xmath0 . now almost everything cancels in @xmath174 , apart from the one term @xmath175 that we obtain as the entry in the column corresponding to @xmath54 . + in the next step , we use the dual ( counter - clockwise ) operation on the columns labelled by the vertices on the cycle @xmath176 , i.e. , we set @xmath177 and replace the column @xmath178 by the linear combination @xmath179 ordering vertices so that @xmath54 corresponds to the first row and column of the cartan matrix , we have thus unimodularly transformed @xmath24 to a matrix of the form @xmath180 where @xmath181 is the @xmath0-cartan matrix of the gentle algebra @xmath38 for the quiver @xmath120 obtained from @xmath11 by removing @xmath54 and the arrows incident with @xmath54 . note that in comparison with @xmath11 , the quiver @xmath120 has one vertex less and one cycle with full zero relations of length @xmath182 less ; now by induction , the result holds for @xmath183 , and hence the result for @xmath24 follows immediately . @xmath147 + this result has several immediate nice consequences . [ q - det ] let @xmath7 be a gentle quiver , and let @xmath30 be the corresponding gentle algebra . denote by @xmath23 the number of minimal oriented @xmath12-cycles in @xmath11 with full zero relations . then the @xmath0-cartan matrix @xmath24 has determinant @xmath31 [ orient - cycles ] let @xmath7 be a gentle quiver , with set of vertices @xmath67 . then , as a direct consequence of theorem [ thm : gentle ] , there are at most @xmath184 minimal oriented cycles with full zero relations in the quiver ( this could also be proved directly by induction ) . note that the property of being gentle is invariant under derived equivalence @xcite , and we now have some invariants to distinguish the derived equivalence classes . for a gentle quiver @xmath7 , recall that @xmath33 and @xmath32 denote the number of minimal oriented cycles in @xmath11 with full zero relations of even and odd length , respectively . as an immediate consequence of corollary [ q - det ] we obtain the following formula for the cartan determinant which was the main result in @xcite : [ det ] let @xmath7 be a gentle quiver , and let @xmath30 be the corresponding gentle algebra . then for the determinant of the cartan matrix @xmath34 the following holds . @xmath185 note that in combination with remark [ orient - cycles ] this implies that the cartan determinant of a gentle algebra @xmath30 is at most @xmath186 , where @xmath187 is the number of simple modules of @xmath4 . the most important application of theorem [ thm : gentle ] is the following corollary which gives for gentle algebras new , combinatorial and easy - to - check invariants of the derived category . [ ec - oc ] let @xmath7 and @xmath36 be gentle quivers , and let @xmath30 and @xmath37 be the corresponding gentle algebras . if @xmath4 and @xmath38 are derived equivalent , then @xmath39 and @xmath40 . * since @xmath4 and @xmath38 are derived equivalent , their ( ordinary ) cartan matrices @xmath34 and @xmath188 are unimodularly equivalent over @xmath189 . by specializing to @xmath1 in theorem [ thm : gentle ] , representatives for the equivalence classes are given by diagonal matrices with entries @xmath190 for each minimal oriented cycle with full zero relations of odd length , an entry @xmath191 for each such cycle of even length , and remaining entries @xmath192 . these are precisely the elementary divisors over @xmath189 . the elementary divisors of an integer matrix are uniquely determined , and the diagonal matrices in theorem [ thm : gentle ] are actually the smith normal forms of @xmath34 and @xmath188 over @xmath189 . but by theorem [ unimod ] the unimodular equivalence class , and hence the smith normal form , is invariant under derived equivalence . hence , the diagonal entries in the above normal forms for @xmath34 and @xmath188 must occur with exactly the same multiplicities . thus we get the same number of minimal oriented cycles with full zero relations of even length and of odd length , respectively , i.e. , @xmath39 and @xmath40 . @xmath147 + we now illustrate our results and apply them to derived equivalence classifications of gentle algebras . [ derived - two ] * gentle algebras with two simple modules . * _ there are nine connected gentle quivers @xmath7 with two vertices , as given in the following list . the dotted lines indicate the zero relations generating the admissible ideal @xmath8 . _ in @xcite it was shown that these are precisely the basic connected algebras with two simple modules which are derived tame . as a direct illustration of our results we show how to classify these algebras up to derived equivalence . recall that the property of being gentle is invariant under derived equivalence @xcite . moreover , the number of simple modules of an algebra is a derived invariant @xcite . thus we will be able to describe the complete derived equivalence classes . we have shown above that the numbers @xmath32 and @xmath33 are derived invariants . in addition we look at two classical invariants , the center and the first hochschild cohomology group @xmath193 . recall that the center of an algebra ( and more generally the hochschild cohomology ring ) is invariant under derived equivalence @xcite . if the quiver contains a loop , then the dimension of @xmath193 depends on the characteristic being 2 or not . we indicate the dimension in characteristic 2 in parantheses in the table below . they can be computed using a method based on work of m. bardzell @xcite on minimal projective bimodule resolutions for monomial algebras ; a very nice explicit combinatorial description is given by c. strametz ( * ? ? ? * proposition 2.6 ) . @xmath194 the algebras @xmath195 are pairwise not derived equivalent . this can be deduced directly from the above table , since the dimensions of the first hochschild cohomology groups are different . the algebras @xmath196 and @xmath197 are derived equivalent . ( this can be shown by explicitly constructing a suitable tilting complex , similar to the detailed example given in the appendix . ) the algebras @xmath198 and @xmath199 with cartan determinant 0 are not derived equivalent , since their centers have different dimensions . in summary , there are exactly eight derived equivalence classes of connected gentle algebras with two simple modules . they are indicated by the double vertical lines in the above table . [ derived - three ] * gentle algebras with three simple modules . * let @xmath7 be a connected gentle quiver with three vertices , with corresponding gentle algebra @xmath30 . by corollary [ orient - cycles ] we deduce that @xmath200 . algebras with different cartan determinant can not be derived equivalent , by theorem [ unimod ] . as an illustration , we shall give a complete derived equivalence classification of those algebras with cartan determinant 0 . by corollary [ det ] , a gentle algebra has cartan determinant 0 if and only if the quiver contains an even oriented cycle with full zero relations . there are 18 connected gentle quivers with three vertices having cartan determinant 0 , as listed in the following figure . 0.5 cm the main tool will be corollary [ ec - oc ] which states that the numbers @xmath33 and @xmath32 of minimal oriented cycles with full zero relations of even ( resp . odd ) length are invariants of the derived category . this will already settle large parts of the classification . in addition we will need to look at the centers and at the first hochschild cohomology group . the following table collects all the necessary invariants . again , in the cases where the quiver has loops , the dimension of @xmath193 depends on the characteristic being 2 or not , and in these cases the dimension in characteristic 2 is given in parantheses . @xmath201 for the derived equivalence classification , it only remains to consider those algebras having the same invariants . in the cases where the algebras are in fact derived equivalent , we leave out the details of the construction of a suitable tilting complex ; in the appendix a detailed example is provided which serves to indicate the strategy which also works in all other cases . the algebras @xmath202 and @xmath203 are derived equivalent . moreover , the algebras @xmath204 and @xmath205 are derived equivalent . note that @xmath202 and @xmath204 represent different derived equivalence classes since their first hochschild cohomology groups have different dimensions . the algebras @xmath206 and @xmath207 are derived equivalent . ( the details for this case are provided in the appendix . ) similarly , the algebras @xmath208 , @xmath209 , @xmath210 and @xmath211 are derived equivalent , the algebras @xmath212 and @xmath213 are derived equivalent and moreover , the algebras @xmath214 and @xmath215 are derived equivalent . the case of @xmath216 is more subtle . this algebra has exactly the same invariants as the algebras @xmath204 and @xmath205 . however , we claim that @xmath216 is not derived equivalent to @xmath205 . in fact , the lie algebra structures on @xmath193 are not isomorphic . note that with the gerstenhaber bracket , the first hochschild cohomology becomes a lie algebra . by a result of b. keller @xcite , this lie algebra structure on @xmath193 is invariant under derived equivalence . as mentioned before , by work of m. bardzell @xcite there is an explicit way of computing @xmath193 for a gentle algebra , and a nice combinatorial version due to c. strametz ( * ? ? ? * proposition 2.6 ) ( for the additive structure ) and ( * ? ? ? * theorem 2.7 ) ( for the lie algebra structure ) . with this method one can compute that the four - dimensional lie algebras on @xmath217 and on @xmath218 are not isomorphic . in fact , the lie algebra center of @xmath217 is two - dimensional , whereas the lie algebra center of @xmath218 has dimension 1 . this completes the derived equivalence classification of connected gentle algebras with three simple modules and cartan determinant 0 . the ten derived equivalence classes are indicated in the above table by the horizontal double lines . skewed - gentle algebras were introduced in @xcite ; for the notation and definition we follow here mostly @xcite , but we try to explain how the construction works rather than repeating the technical definition from @xcite . we start with a gentle pair @xmath7 . a set @xmath219 of vertices of the quiver @xmath11 is an admissible set of _ special _ vertices if the quiver with relations obtained from @xmath11 by adding loops with square zero at these vertices is again gentle ; we denote this gentle pair by @xmath220 . the triple @xmath221 is then called _ skewed - gentle_. we want to point out that the admissibility of the set @xmath219 of special vertices is both a local as well as a global condition . let @xmath54 be a vertex in the gentle quiver @xmath7 ; then we can only add a loop at @xmath54 if @xmath54 is of valency 1 or 0 or if it is of valency 2 with a zero relation , but not one coming from a loop . hence only vertices of this type are potential special vertices . but for the choice of an admissible set of special vertices we also have to take care of the global condition that after adding all loops , the pair @xmath220 still does not have paths of arbitrary lengths . given a skewed - gentle triple @xmath221 , we now construct a new quiver with relations @xmath222 by doubling the special vertices , introducing arrows to and from these vertices corresponding to the previous such arrows and replacing a previous zero relation at the vertices by a mesh relation . + more precisely , we proceed as follows . the non - special ( or : ordinary ) vertices in @xmath11 are also vertices in the new quiver ; any arrow between non - special vertices as well as corresponding relations are also kept . any special vertex @xmath223 is replaced by two vertices @xmath224 and @xmath225 in the new quiver . an arrow @xmath226 in @xmath11 from a non - special vertex @xmath160 to @xmath54 ( or from @xmath54 to @xmath160 ) will be doubled to arrows @xmath227 ( or @xmath228 , resp . ) in the new quiver ; an arrow between two special vertices @xmath229 will correspondingly give four arrows between the pairs @xmath230 and @xmath231 . we say that these new arrows lie over the arrow @xmath226 . any relation @xmath232 where @xmath233 is non - special gives a corresponding zero relation for paths of length 2 with the same start and end points lying over @xmath234 . if @xmath54 is a special vertex of valency 2 in @xmath11 , then the corresponding zero relation at @xmath54 , say @xmath232 with @xmath235 , is replaced by mesh commutation relations saying that any two paths of length 2 lying over @xmath234 , having the same start and end points but running over @xmath224 and @xmath225 , respectively , coincide in the factor algebra to the new quiver with relations @xmath222 . + we will speak of @xmath222 as a skewed - gentle quiver _ covering _ the gentle pair @xmath7 . a @xmath61-algebra is then called _ skewed - gentle _ if it is morita equivalent to a factor algebra @xmath236 , where @xmath237 comes from a skewed - gentle triple @xmath221 as above . * remark . * let @xmath30 be gentle . in a gentle quiver , there is at most one non - zero cyclic path starting and ending at a given vertex ; hence the diagonal entries in the @xmath0-cartan matrix @xmath24 are @xmath29 or of the form @xmath238 , for some @xmath239 . + if a vertex @xmath54 in @xmath11 can be chosen as a special vertex for a covering skewed - gentle quiver @xmath240 , then the corresponding diagonal entry in @xmath24 is @xmath29 , as otherwise we have paths of arbitrary lengths in @xmath241 ; hence in the corresponding @xmath0-cartan matrix for the skewed - gentle algebra @xmath242 we have @xmath243 on the diagonal for the two split vertices @xmath230 in @xmath240 . + [ thm : skewed - gentle ] let @xmath7 be a gentle quiver , and @xmath237 a covering skewed - gentle quiver . let @xmath41 be the corresponding skewed - gentle algebra . denote by @xmath23 the number of oriented @xmath12-cycles in @xmath7 with full zero relations . + then the @xmath0-cartan matrix @xmath42 is unimodularly equivalent ( over @xmath19 $ ] ) to a diagonal matrix with entries @xmath43 , with multiplicity @xmath27 , @xmath28 , and all further diagonal entries being @xmath29 . * again , we argue by induction on the number of vertices and arrows . we let @xmath30 be the gentle algebra and @xmath24 the @xmath0-cartan matrix as before . + if @xmath11 has no arrows , then @xmath240 is just obtained by doubling the special vertices , and this still has no arrows , so the result clearly holds . + if @xmath11 has an arrow @xmath123 as in lemma [ lem : arrow - removal ] , with a non - special @xmath244 in the first case , and a non - special @xmath245 in the second case , respectively , then we can argue as in the proof of lemma [ lem : arrow - removal ] to remove @xmath123 . let us consider again the situation of the first case , so here @xmath246 is non - special . note that a maximal non - zero path @xmath13 starting from @xmath135 with @xmath123 or @xmath247 ( if @xmath248 is special ) will end on a non - special vertex ( and hence this maximal path is unique in @xmath242 ) ; in general , this path will be longer than the one taken in @xmath4 . in the column transformations , we only have to be careful at doubled vertices on the path @xmath13 ; here we replace both corresponding columns @xmath249 of @xmath42 by @xmath250 . this leads to the cartan matrix for the skewed - gentle algebra where @xmath123 or @xmath247 , respectively , has been removed from @xmath240 , which is a skewed - gentle cover for the quiver obtained from @xmath11 by deleting @xmath123 ; then the claim follows by induction . now assume @xmath11 has a source @xmath54 which is special ( w.l.o.g . the first vertex ) ; the case of a sink is dual . then the @xmath0-cartan matrix for @xmath242 has the form @xmath251 where @xmath252 is the @xmath0-cartan matrix of the skewed - gentle algebra @xmath253 for the quiver @xmath254 obtained from @xmath240 by removing @xmath255 and the arrows incident with @xmath230 . note that @xmath254 is the skewed - gentle cover for the quiver @xmath120 which is obtained from @xmath11 by removing @xmath54 and the arrow incident with @xmath54 , and the choice @xmath256 as the set of special vertices ; in short , we write this as @xmath257 . again , using induction the claim follows immediately . thus again , we may now assume that @xmath11 has only vertices of valency 2 with a ( non - loop ) zero relation or vertices of valency 4 ; note that any special vertex in @xmath11 has to be of valency 2 . as before , we may also assume that @xmath11 ( and hence also @xmath240 ) are connected . if there are no non - special vertices , or if all non - special vertices are of valency 4 , then @xmath241 is not gentle . hence @xmath11 has a non - special vertex @xmath54 of valency 2 with a zero relation at @xmath54 . let @xmath258 be the ( unique ) incoming arrow . + again we consider the unique maximal path @xmath13 in @xmath11 with non - repeating arrows starting in @xmath135 with @xmath126 , such that the product of any two consecutive arrows is zero in @xmath4 ; as before , we note that in our current situation @xmath13 has to be a cycle @xmath259 , where @xmath128 , @xmath155 , and @xmath260 . as in the previous situation , we note that the arrows are distinct , but vertices @xmath131 may be repeated . + for a vertex @xmath160 in @xmath11 we denote by @xmath159 the row of the @xmath0-cartan matrix @xmath24 corresponding to @xmath160 . if @xmath160 is non - special , we denote by @xmath261 the corresponding row in the @xmath0-cartan matrix @xmath262 . if @xmath160 is special , then for the two vertices @xmath231 we have two corresponding rows @xmath263 in the cartan matrix @xmath264 , and we then set @xmath265 . + before , we have transformed @xmath111 by replacing @xmath161 by @xmath266 and obtained a matrix @xmath146 . we now do a parallel transformation on @xmath264 , that is , we replace @xmath267 by @xmath268 and we obtain a matrix @xmath269 . we have to compare the differences and check that everything stays under control for the induction argument . + if a vertex @xmath137 , @xmath270 , is special , note that the doubled contribution in @xmath271 is needed on the one hand for the cancellation with the previous row , and on the other hand to continue around the cycle @xmath153 . as @xmath54 is non - special and of valency 2 with a zero - relation , we note that as before , in @xmath272 we only have the contribution @xmath175 at @xmath54 . following this by the parallel operation to the previous column operation we then replace the column @xmath273 by the linear combination @xmath274 where we use analogous conventions as before . with @xmath54 corresponding to the first row and column of the cartan matrix , we have thus unimodularly transformed @xmath264 to a matrix of the form @xmath275 where @xmath252 is the cartan matrix of the skewed - gentle algebra @xmath253 for the quiver @xmath254 obtained from @xmath240 by removing @xmath54 and the arrows incident with @xmath54 . note that in fact , @xmath276 in the notation of our previous proof , i.e. , as explained earlier , @xmath254 is the skewed - gentle cover for the quiver @xmath120 and the choice @xmath256 as the set of special vertices . thus the result follows by induction . @xmath147 + by comparing theorem [ thm : gentle ] and theorem [ thm : skewed - gentle ] we observe that the @xmath0-cartan matrix @xmath24 for the gentle algebra @xmath4 to @xmath7 , and the @xmath0-cartan matrix @xmath42 for a skewed - gentle cover @xmath242 are unimodularly equivalent to diagonal matrices which only differ by adding as many further @xmath29 s on the diagonal as there are special vertices chosen in @xmath11 . in particular , with notation as above , @xmath277 this observation has the following immediate consequence when specializing to @xmath1 . let @xmath7 be a gentle quiver , and @xmath237 a covering skewed - gentle quiver . then the determinant of the ordinary cartan matrix of the skewed - gentle algebra @xmath41 is the same as the one for the gentle algebra @xmath30 , i.e. , @xmath45 . a gentle algebra and a ( proper ) skewed - gentle algebra may have the same @xmath0-invariants but they can not be derived equivalent by @xcite , corollary 1.2 . this appendix is aimed at providing enough background on tilting complexes and explicit computations of their endomorphism rings so that the interested reader can fill in the details in the derived equivalence classifications of examples [ derived - two ] and [ derived - three ] . we explained there in detail how to distinguish derived equivalence classes ( since this is the main topic of this paper ) , but have been fairly short on indicating why certain algebras in the lists are actually derived equivalent . in this section we will go through one example in detail ; this will indicate the main strategy which also works in all other cases . two algebras @xmath4 and @xmath94 are called derived equivalent if @xmath278 and @xmath280 are equivalent as triangulated categories . by j. rickard s theorem @xcite , this happens if and only there exists a tilting complex @xmath281 for @xmath4 such that the endomorphism ring @xmath282 in the homotopy category is isomorphic to @xmath94 . a bounded complex @xmath281 of projective @xmath4-modules is called a tilting complex if the following conditions are satisfied . recall from section [ sec - def ] our conventions to deal with right modules and to read paths from left to right . in particular , left multiplication by a nonzero path from vertex @xmath79 to vertex @xmath78 gives a homomorphism @xmath287 . we define the following bounded complex @xmath288 of projective @xmath206-modules . let @xmath289 and @xmath290 be stalk complexes concentrated in degree 0 . moreover , let @xmath291 ( in degrees 0 and @xmath292 ) . we claim that @xmath281 is a tilting complex . property ( i ) above is obvious for all @xmath293 since we are dealing with two - term complexes . let @xmath294 , and consider possible maps @xmath295 $ ] where @xmath296 . this is given by a map of complexes as follows @xmath297 where @xmath11 could be either of @xmath298 , @xmath299 , or @xmath300 . but since we are dealing with gentle algebras , no nonzero map can be zero when composed with both @xmath100 and @xmath99 . so the only homomorphism of complexes @xmath295 $ ] is the zero map , as desired . directly from the definition we see that @xmath301)=0 $ ] and @xmath302)=0 $ ] ( since they are stalk complexes ) . now let @xmath304 . we have to consider maps @xmath305 $ ] ; these are given as follows @xmath306 where @xmath11 again can be either of @xmath298 , @xmath299 , or @xmath300 . now there certainly exist nonzero homomorphisms of complexes . but they are all homotopic to zero . in fact , every path in the quiver of @xmath206 from vertex 2 to vertex 1 or 3 either starts with @xmath100 or with @xmath99 . accordingly , every homomorphism @xmath307 can be factored through the map @xmath308 . it remains to show that the complex @xmath281 also satisfies property ( ii ) of the definition of a tilting complex . it suffices to show that the projective indecomposable modules @xmath298 , @xmath310 and @xmath299 , viewed as stalk complexes , can be generated by @xmath286 . this is clear for @xmath298 and @xmath299 since they occur as summands of @xmath281 . for @xmath310 , consider the map of complexes @xmath311 given by the identity map on @xmath300 in degree 0 . then the stalk complex @xmath312 $ ] with @xmath310 in degree 0 can be shown to be homotopy equivalent ( i.e. isomorphic in @xmath279 ) to the mapping cone of @xmath313 . thus we have a distinguished triangle @xmath314 \to \underbrace{t_2[1]}_{\in{\operatorname{add}}(t)}.\ ] ] by definition , @xmath286 is triangulated , so it follows that also the stalk complex @xmath312\in{\operatorname{add}}(t)$ ] , which proves ( ii ) . by rickard s theorem , the endomorphism ring of @xmath281 in the homotopy category is derived equivalent to @xmath206 . we need to show that @xmath315 is isomorphic to @xmath207 . note that the vertices of the quiver of @xmath316 correspond to the summands of @xmath281 . _ alternating sum formula . _ for a finite - dimensional algebra @xmath4 , let @xmath317 and @xmath318 be bounded complexes of projective @xmath4-modules . then @xmath319)= \sum_{r , s } ( -1)^{r - s } \dim{\operatorname{hom}}_a(q^r , r^s).\ ] ] in particular , if @xmath11 and @xmath320 are direct summands of a tilting complex then @xmath321 note that the cartan matrix of @xmath206 has the form . from that , using the alternating sum formula , we can compute the cartan matrix of @xmath316 to be . note that this is actually the cartan matrix of @xmath207 . now we have to define maps of complexes between the summands of @xmath281 , corresponding to the arrows of the quiver of @xmath207 . the final step then is to show that these maps satisfy the defining relations of @xmath207 , up to homotopy . we define @xmath322 by the map @xmath323 in degree 0 . note that this is indeed a homomorphism of complexes since @xmath324 in @xmath206 . moreover , we define @xmath325 and @xmath326 by the projection onto the first and second summand in degree 0 , respectively . finally , we define @xmath327 by @xmath328 . we now have to check the relations , up to homotopy . we write compositions from left to right ( as in the relations of the quiver of @xmath316 ) . clearly , @xmath329 . the composition @xmath330 is given in degree 0 by @xmath331 . so it is not the zero map , but is homotopic to zero via the homotopy map @xmath332 ( use that @xmath333 in @xmath206 ) . finally , consider @xmath334 on @xmath335 . it is given by @xmath336 in degree 0 and the zero map in degree @xmath292 . it is indeed homotopic to zero via the homotopy map @xmath337 . ( note that here we use that @xmath338 and @xmath324 in @xmath206 . ) thus , we have defined maps between the summands of @xmath281 , corresponding to the arrows of the quiver of @xmath207 . we have shown that they satisfy the defining relations of @xmath207 , and that the cartan matrices of @xmath316 and @xmath207 coincide . from this we can conclude that @xmath339 . hence , @xmath206 and @xmath207 are derived equivalent , as desired . all the other derived equivalences stated in examples [ derived - two ] and [ derived - three ] can be verified exactly along these lines . in particular , they can also be realized by tilting complexes with non - zero entries in only two degrees . | cartan matrices are of fundamental importance in representation theory . for algebras defined by quivers with monomial relations the computation of the entries of the cartan
matrix amounts to counting nonzero paths in the quivers , leading naturally to a combinatorial setting .
our main motivation are derived module categories and their invariants : the invariant factors , and hence the determinant , of the cartan matrix are preserved by derived equivalences .
the paper deals with the class of ( skewed- ) gentle algebras which occur naturally in representation theory , especially in the context of derived categories .
we study @xmath0-cartan matrices , where each nonzero path is weighted by a power of an indeterminate @xmath0 according to its length . specializing @xmath1
gives the classical cartan matrix .
we determine normal forms for the @xmath0-cartan matrices of skewed - gentle algebras .
in particular , we give explicit combinatorial formulae for the invariant factors and thus also for the determinant . as an application of our main results
we show how to use our formulae for the difficult problem of distinguishing derived equivalence classes .
msc - classification : 16g10 , 18e30 , 05e99 , 05c38 , 05c50 * @xmath0-cartan matrices and combinatorial invariants of derived categories for skewed - gentle algebras * * christine bessenrodt * fakultt fr mathematik und physik , universitt hannover , + welfengarten 1 , d-30167 hannover , germany * thorsten holm * department of pure mathematics , university of leeds , + leeds ls2 9jt , england |
You are an expert at summarizing long articles. Proceed to summarize the following text:
thermotropic liquid crystals exhibit exotic phase behavior upon temperature variation . the nematic phase is rich with a long - ranged orientational order but lacks translational order . the isotropic - nematic ( i - n ) phase transition , which is believed to be weakly first order in nature with certain characteristics of the continuous transition , has been a subject of immense attention in condensed matter physics and material sciences @xcite . in contrast , the dynamics of thermotropic liquid crystals have been much less studied , the focus being mostly on the long - time behavior of orientational relaxation near the i - n transition @xcite . a series of optical kerr effect ( oke ) measurements have , however , recently studied collective orientational relaxation in the isotropic phase near the i - n transition over a wide range of time scales @xcite . the dynamics have been found to be surprisingly rich , the most intriguing feature being the power law decay of the oke signal at short - to - intermediate times @xcite . the relaxation scenario appears to be strikingly similar to that of supercooled molecular liquids @xcite , even though the latter do not undergo any thermodynamic phase transition . although the analogous dynamics have been investigated in subsequent studies @xcite , a quantitative estimation of glassy dynamics of rodlike molecules near the i - n transition still eludes us . the prime objective of this paper is to provide a quantitative measure of glassy dynamics near the i - n transition . to this end , we have undertaken molecular dynamics simulations of a family of model systems consisting of rodlike molecules across the i - n transition in search of glassy behavior . given the involvement of the phase transition to an orientationally ordered mesophase upon lowering the temperature , we choose to probe the single - particle orientational dynamics . we have defined a fragility index and explored plausible correlation of the features of the underlying energy landscape with the observed fragility in analogy with supercooled liquids . the systems we have studied consist of ellipsoids of revolution . the gay - berne ( gb ) pair potential @xcite , that is well established to serve as a model potential for systems of thermotropic liquid crystals , has been employed . the gb pair potential , which uses a single - site representation for each ellipsoid of revolution , is an elegant generalization of the extensively used isotropic lennard - jones potential to incorporate anisotropy in both the attractive and the repulsive parts of the interaction @xcite.in the gb pair potential , @xmath0th ellipsoid of revolution is represented by the position @xmath1 of its center of mass and a unit vector @xmath2 along the long axis of the ellipsiod . the interaction potential between two ellipsoids of revolution @xmath0 and @xmath3 is given by @xmath4 where @xmath5 here @xmath6 defines the thickness or equivalently , the separation between the two ellipsoids of revolution in a side - by - side configuration , @xmath7 is the distance between the centers of mass of the ellipsoids of revolution @xmath0 and @xmath3 , and @xmath8 is a unit vector along the intermolecular separation vector @xmath9 . the molecular shape parameter @xmath10 and the energy parameter @xmath11 both depend on the unit vectors @xmath2 and @xmath12 as well as on @xmath13 as given by the following set of equations : @xmath14^{-1/2}\ ] ] with @xmath15 and @xmath16^{\nu } [ \epsilon_{2}({\bf \hat r}_{ij},{\bf e}_{i},{\bf e}_{j})]^{\mu}\ ] ] where the exponents @xmath17 and @xmath18 are adjustable parameter , and @xmath19^{-1/2}\ ] ] and @xmath20\ ] ] with @xmath21 . here @xmath22 is the aspect ratio of the ellipsoid of revolution with @xmath23 denoting the separation between two ellipsoids of revolution in a end - to - end configuration , and @xmath24 , and @xmath25 , where @xmath26 is the depth of the minimum of the potential for a pair of ellipsoids of revolution aligned in a side - by - side configuration , and @xmath27 is the corresponding depth for the end - to - end alignment . @xmath28 is the depth of the minimum of the pair potential between two ellipsoids of revolution alligned in cross configuration . the gb pair potential defines a family of models , each member of which is characterized by the values chosen for the set of four parameters @xmath29 and @xmath18 , and is represented by gb(@xmath30 ) @xcite . three systems , namely gb(3 , 5 , 2 , 1 ) , gb(3.4 , 5 , 2 , 1 ) , and gb(3.8 , 5 , 2 , 1 ) , that differ in the aspect ratio have been investigated . molecular dynamics simulations have been performed with each of these systems , consisting of 500 ellipsoids of revolution , in a cubic box with periodic boundary conditions . each of these systems has been studied along three isochors ( @xmath31 = 0.31 , 0.32 , and 0.33 for @xmath32 = 3.0 ; @xmath31 = 0.25 , 0.26 , and 0.27 for @xmath32 = 3.4 ; @xmath31 = 0.215 , 0.225 , and 0.235 for @xmath32 = 3.8 ) at several temperatures , starting from the high - temperature isotropic phase down to the nematic phase across the i - n phase boundary . all quantities are given in reduced units defined in terms of the gay - berne potential parameters @xmath28 and @xmath33 : length in units of @xmath33 , temperature in units of @xmath34 , and time in units of @xmath35 , m being the mass of the ellipsoids of revolution . the mass as well as the moment of inertia of each of the ellipsoids of revolution have been set equal to unity . the intermolecular potential is truncated at a distance @xmath36and shifted such that @xmath37 , @xmath7 being the separation between two ellipsoids of revolution i and j. the equations of motion have been integrated using the velocity - verlet algorithm with integration time step @xmath38 @xcite . equilibration has been done by periodic rescaling of linear and angular velocities of particles . this has been done for a time period of @xmath39 following which the system has been allowed to propagate with a constant energy for a time period of @xmath40 in order to ensure equilibration upon observation of no drift of temperature , pressure , and potential energy . the data collection has been executed in a microcanonical ensemble . at each state point , local potential energy minimization has been executed by the conjugate gradient technique for a subset of @xmath41 statistically independent configurations . the landscape analysis has been done with a system size of @xmath42 ellipsoids of revolution , which is big enough for having no qualitative change in the results due to the system size @xcite . minimization has been performed with three position coordinates and two euler angles for each particle , the third euler angle being redundant for ellipsoids of revolution . the single - particle second rank orientational time correlation function ( otcf ) @xmath43 is defined by @xmath44 where @xmath45 is the second rank legendre polynomial , @xmath46 is the unit vector along the long axis of @xmath0th ellipsoid of revolution , and the angular brackets stand for ensemble averaging . figure [ fig : power ] shows the time evolution of the single - particle second rank otcf for one of the three systems considered here as the temperature is lowered along an isochor from the high - temperature isotropic phase down to the nematic phase across the i - n phase boundary . in the inset , the average orientational order parameter @xmath47 is shown as a function of temperature along the isochor @xcite . the variation of @xmath47 with temperature serves to locate the i - n phase boundary . in the present study , the i - n transition temperature @xmath48 is taken as the temperature at which @xmath47 of the system is @xmath49 . for each aspect ratio , three isochors at different densities have been considered . the qualitative behavior has been found to be the same for all the three systems along all the isochors studied ( data not shown ) . the emergence of the power law decay in the isotropic phase near the i - n transition is evident in all the cases as a universal characteristic of i - n transition @xcite . as the i - n phase boundary is crossed upon cooling , the advent of two power law decay regimes separated by an intervening plateau at short - to - intermediate times imparts a step - like feature to the temporal behavior of the second rank otcf . such a feature bears remarkable similarity to what is observed for supercooled liquids as the glass transition is approached from the above @xcite . while for the supercooled liquid the emargence of step - like feature in the otcf is well understood as a consequence of @xmath50 relaxation , the origin of such a feature observed for liquid crystal defied of reliable explanation . we estimate the orientational correlation time @xmath51 as the time taken for @xmath43 to decay by @xmath52 , i.e. , @xmath53 . figure [ fig : frgl](a ) shows @xmath51 in the logarithmic scale as a function of the inverse temperature along the three isochors for each of the three systems considered . we have scaled the temperature by @xmath48 in the spirit of angell s plot , that displays the shear viscosity ( or the structural relaxation time , the inverse diffusivity , etc . ) of glass - forming liquids as a function of the inverse of the scaled temperature , the scaling being done in the latter case by the glass transition temperature @xmath54 @xcite . for all the three systems , two distinct features are common : ( i ) in the isotropic phase far away from the i - n transition , the orientational correlation time @xmath51 exhibits the arrhenius temperature dependence , i.e. , @xmath55 $ ] , where the activation energy @xmath56 and the pre - factor @xmath57 are both independent of temperature ; ( ii ) in the isotropic phase near the i - n transition , the temperature dependence of @xmath51 shows marked deviation from the arrhenius behavior and can be well described by the vogel - fulcher - tammann ( vft ) equation @xmath58 $ ] , where @xmath57 , @xmath59 , and @xmath60 are constants , independent of temperature . again these features bear remarkable similarity with those observed for fragile glass - forming liquid . a non - arrhenius temperature behavior is taken to be the signature of fragile liquids . for fragile liquids , the temperature dependence of the shear viscosity follows the arrhenius behavior far above @xmath54 and can be fitted to the vft functional form in the deeply supercooled regime near @xmath54 @xcite . the striking resemblance in the dynamical behavior described above between the isotropic phase of thermotropic liquid crystals near the i - n transition and supercooled liquids near the glass transition has prompted us to attempt a quantitative measure of glassy behavior near the i - n transition . for supercolled liquids , one quantifies the dynamics by a parameter called fragility index which measures the rapidity at which the liquid s propeties ( such as viscosity ) change as the glassy state is approached . in the same spirit @xcite that offers a quantitative estimation of the fragile behavior of supercooled liquids , we here define the fragility index @xmath61 of a thermotropic liquid crystalline system as @xmath62 figure [ fig : frgl](b ) shows the density dependence of the fragility index for the three systems with different aspect ratios . for a given aspect ratio , the fragility index increases with increasing density , the numerical values of the fragility index @xmath61 being comparable to those of supercooled liquids . the change in the fragility index for a given density difference ( @xmath63 ) increases with the decrease in the aspect ratio . another hallmark of fragile glass - forming liquids is spatially heterogeneous dynamics @xcite reflected in non - gaussian dynamical behavior @xcite . it is intuitive that the growth of the pseudo - nematic domains , characterized by local nematic order , in the isotropic phase near the i - n transition would result in heterogeneous dynamics in liquid crystals . we have , therefore , monitored the time evolution of the rotational non - gaussian parameter ( ngp ) @xcite , @xmath64 , which in the present case is defined as @xmath65 where @xmath66 here @xmath67 is the rotation vector like the position vector @xmath1 appears incase of translational ngp of @xmath0th ellipsoid of revolution , the change of which is defined by @xmath68 , @xmath69 being the corresponding angular velocity @xcite , and @xmath70 is the number of ellipsoids of revolution in the system . ngp will have value equal to zero when system dynamics is spatially homogeneous and will have a non - zero value when the system dynamics is spatially heterogeneous . as a typical behavior , fig . [ fig : rngp](a ) shows the time dependence of the rotational ngp for one of the systems at several temperatures across the i - n transition along an isochor . on approaching the i - n transition upon cooling , a bimodal feature starts appearing with the growth of a second peak , which eventually becomes the dominant one , at longer times . we further investgate the apearance of this bimodal feature in ngp plot . to this end we calculate mean square angular deviation ( msad ) of the system at different temperatures starting from high temperature isotropic phase to low temperature nematic phase . the appearance of the bimodal feature in the rotational ngp is accompanied by a signature of a sub - diffusive regime in the temporal evolution of the mean square angular deviation ( msad ) , the time scale of the short - time peak and that of the onset of the sub - diffusive regime being comparable , as shown in fig . [ fig : rngp](b ) . we further note that the dominant peak appears on a time scale which is comparable to that of onset of the diffusive motion in orientational degrees of freedom ( odof ) as evident in fig . [ fig : rngp](b ) and similar feature has been observed recently for glassy systems @xcite also . we also observe that the time scale at which long - time peak appears is also comparable to the onset of the plateau that is observed in the time evolution of @xmath43 , as shown in fig . [ fig : rngp](c ) . this striking similarities of the dynamics between liquid crystals near i - n transition and supercooled liquid near glass transition are also supported by the use of mode coupling theory ( mct ) to explain the dynamics of both the systems . while mct was used first for the supercooled liquid , recently it has been used for liquid crystals also . mct theory devloped by gottke et al . @xcite predicts that near i - n transition , the low frequency rotational memory kernel should diverge in a power law fashion . @xmath71 mean field treatment gives @xmath72 . invoking the rank ( @xmath73 ) dependence of the memory function , the single particle otcf can be written as @xcite @xmath74^{-1}\ ] ] the above equation can be laplace inverted to obtain a short - to - intermadiate power law decay in @xmath43 which is a universal characteristic of the i - n transition for several model liquid crystals @xcite . recently , @xcite have showed that it is also possible to formulate a schematic model that combines short - to - intermediate time relaxation with long time relaxation . in their model , they have expressed the total memory functon ( @xmath75 ) as the sum of mode coupling memory function ( @xmath76 ) and landau - de gennes memory function ( @xmath77 ) . @xmath78 where @xmath79 @xmath80 is the characteristic frequecny and @xmath81 . time dependence of @xmath82 can be written expressed in terms of the memory function @xmath83 @xcite and @xmath83 has the following form @xmath84 where @xmath85 is the autocorrelation function of the anisotropy of the polarizability and @xmath86 is the solution of a @xmath87 schematic model for what is referred to as the density correlator . @xmath32 being the coupling constant between them . now , @xmath77 can be written as @xmath88 here @xmath89 is the relaxation time ( @xmath90 ) and it diverges as @xmath91 as the critical temperature @xmath92 of the i - n transition is approached from the above . following the calculation of ref . @xcite , one can get two important relaxation equattions . @xmath93 with the initial conditions @xmath94 and @xmath95 and @xmath96 with the initial conditions @xmath97 and @xmath98 . @xmath99 and @xmath17 are the damping constants . @xmath100 is identical to one gets from the mct analysis of the supercooled liquids . difference between this scematic model and one applied for the supercooled liquid is in eq . @xmath101 . if @xmath102 is set equal to @xmath103 in eq . @xmath101 , the supercooled model is recovered . @xmath101 is the orientational correlation function coupled to the density correlation function with specific new terms that account for the long - time portion of the relaxation profile that has been previously described by ldg theory . several studies have attempted to interpret the fragility of glass - forming liquids in terms of the features of the underlying energy landscapes @xcite . energy landscape analysis gives the potential energy , which devoids of any kind of thermal motions , of inherent structures of the parent liquid and hence provide a better understanding of the structure and dynamics of the paprent liquid . a recent study on thermotropic liquid crystals has reported the temperature dependent exploration of the energy landscapes of a family of the gay - berne model systems across the mesophases @xcite . the average inherent structure ( is ) energy @xmath104 has been found to fall as @xmath47 grows across the i - n phase boundary and through the nematic phase in contrast to its insensitivity to the temperature in the high - temperature isotropic phase and the low - temperature smectic - b phase @xcite . such a fall in the average is energy is consistent with a gaussian form for the number density of inherent structures with energy @xmath105 , that predicts a linear variation of @xmath104 with the inverse temperature : @xmath106 , where @xmath107 and @xmath10 are parameters independent of temperature and @xmath108 is the boltzmann constant @xcite . note that this has been observed for a glassy system @xcite , where the average is energy also falls over a temperature range @xcite . in fig . [ fig : lscp](a ) , we demonstrate with the original and the most studied parameterization for the gb pair potential gb@xmath109 that the prediction holds good over the temperature range where @xmath104 is on a decline along all the three isochors studied . it then follows that a plot of @xmath110 versus @xmath111/@xmath112 would result in a collapse of the @xmath104 data for all densities onto a straight line with negative unit slope . this is indeed found to be true , as shown in fig . [ fig : lscp](b ) , implying the validity of the gaussian model in this case as well . it may be noted that when the distribution of is energy is gaussian , the fragility of glass - forming liquids has been shown to depend on the total number of inherent structures , the width of the gaussian , and the variation of the basin shape with the average is energy @xcite . the origin of the glassy orientational dynamics , both single particle and collective , of nematogens near the isotropic - nematic transition has been addressed in several publications in recent years @xcite . in these studies , the similarity between dynamics of supercooled glassy liquid and liquid crystals has been discussed in detail , but no quantitative measure of the similarity was provided . the fragility index introduced here serves to remove that lacuna . it is indeed surprising that even the values of the fragility parameter are in the range observed for glassy liquids as well . this is in agreement with the repeated observation by fayer and coworkers that the values of the power law exponents observed in the two systems are quite similar . further understanding of the relaxation mechanism has been obtained from a closure look on the heterogeneous dynamics . as figure 3 demonstrates , the rotational non - gaussian parameter shows a dramatic enhancement of hetrogeneous dynamics as the i - n phase boundary is approached . unlike what is found near the gas - liquid critical point @xcite , the single - particle dynamics near the i - n phase boundary are observed to be strongly affected by the approaching thermodynamic singularity . we have discussed mode coupling theory approaches introduced to understand anomalous dynamics observed in this problem . * acknowledgments * it is a pleasure to thank professor s. sastry for helpful discussions . this work was supported in parts by grants from dst and csir , india . b.j . acknowledges csir , india and d.c . acknowledges ugc , india for providing research fellowship | in an attempt to quantitatively characterize the recently observed slow dynamics in the isotropic and nematic phase of liquid crystals , we investigate the single - particle orientational dynamics of rodlike molecules across the isotropic - nematic transition in computer simulations of a family of model systems of thermotropic liquid crystals .
several remarkable features of glassy dynamics are on display including non - exponential relaxation , dynamical heterogeneity , and non - arrhenius temperature dependence of the orientational relaxation time . in order to obtain a _ quantitative _ measure of glassy dynamics in line with the estbalished methods in supercooled liquids ,
we construct a relaxation time versus scaled inverse temperature plot , and demonstrate that one can indeed define a fragility index for thermotropic liquid crystals , that depends on density and aspect ratio .
the values of the fragility parameter are surprisingly in the range one observed for glass forming liquids .
a plausible correlation between the energy landscape features and the observed fragility is discussed . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
physically meaningful interpretation of the measurement of relative motion from the point of view of an accelerated observer in a gravitational field requires the introduction of a special coordinate system ( i.e. fermi coordinates ) along the worldline of the observer . in this coordinate system , the equations of relative motion reveal tidal and inertial effects for ultrarelativistic motion ( with speed exceeding the critical value @xmath0 ) that are contrary to newtonian expectations . while there are general treatments of inertial and tidal effects in fermi coordinates @xcite and the special case of ultrarelativistic motion of particles has been discussed in our recent papers @xcite , the purpose of this work is to present a more systematic and complete description of motion beyond the critical speed @xmath0 . to this end , the equations of motion in fermi coordinates are discussed in section [ s0 ] and inertial effects are considered in sections [ s2 ] , [ s3 ] and appendix a. section [ s4 ] is devoted to tidal effects . a summary and brief discussion of our results is contained in section [ s5 ] . imagine an accelerated observer in a general relativistic spacetime following a worldline @xmath1 , where @xmath2 is the proper time along its trajectory . the local axes of the observer are given by an orthonormal tetrad frame @xmath3 that is carried along its path according to @xmath4 where @xmath5 is the antisymmetric acceleration tensor , @xmath6 is the local temporal direction and @xmath7 , @xmath8 , form the local spatial triad , and where ( here and throughout this paper ) greek indices run from 0 to 3 , latin indices run from 1 to 3 , the signature of the metric is @xmath9 and @xmath10 , unless specified otherwise . in analogy with the electromagnetic faraday tensor , we decompose the acceleration tensor into its `` electric '' and `` magnetic '' components @xmath11 ; these are given by the translational acceleration @xmath12 and the rotational frequency @xmath13 , respectively . that is , the acceleration of the reference trajectory is given by @xmath14 and @xmath15 is the frequency of rotation of the spatial triad with respect to a local nonrotating ( i.e. fermi - walker transported ) triad . let us next establish a fermi coordinate system @xmath16 in a neighborhood of the reference worldline @xcite . it turns out that fermi coordinates can be assigned uniquely only to spacetime events that are within a cylindrical region of finite radius along the observer s worldline . at each event @xmath1 on the observer s worldline , consider all spacelike geodesic curves that are normal to this reference worldline at this event . each event @xmath17 on the resulting hypersurface and within the cylindrical region under consideration is connected to @xmath1 by a unique spacelike geodesic curve that starts at @xmath1 and whose tangent vector @xmath18 at this event is normal to the reference worldline ( that is , @xmath19 ) . the event @xmath17 is assigned the fermi coordinates @xmath20 , where @xmath21 and @xmath22 here @xmath23 is the proper length of the normal spacelike geodesic segment connecting @xmath1 to @xmath17 . the fermi coordinate system is admissible in a cylindrical spacetime region around @xmath1 with radius @xmath24 , where @xmath25 is the minimum radius of curvature of spacetime along the reference worldline @xcite . the observer , at each instant @xmath26 of its proper time `` sees '' a locally euclidean three - dimensional space and `` instantaneously '' determines distances within it using the ( spacelike ) geodesic lengths beginning from its position at the spatial origin of the new fermi coordinates . a nearby particle worldline that punctures the sequence of three - dimensional spaces at @xmath27 is thus a graph over the reference worldline and each point on the graph is connected to the observer by a `` locally straight '' line ( i.e. a geodesic ) that is normal to the observer s worldline . from the viewpoint of the observer , the particle has relative coordinate velocity @xmath28 and relative coordinate acceleration @xmath29 . the general equation of motion for a test particle of mass @xmath30 in the fermi frame is @xmath31 where @xmath32 is the proper time along its worldline , @xmath33 , and @xmath34 is the external force acting on the particle . equation can be written as the system @xmath35 using the identity @xmath36 and the lorentz factor of the particle @xmath37 equation can be expressed as @xmath38 let @xmath39 . we note that @xmath40 and the physical content of the equation of motion is contained in equation together with @xmath41 and @xmath42 . these can be written respectively as @xmath43 and @xmath44 it is crucial to recognize that in fermi coordinates the velocity @xmath45 satisfies the condition @xmath46 only at @xmath47 ; indeed , away from the reference trajectory @xmath48 could in principle exceed unity in accordance with the equation of motion . it is natural to begin our discussion with an accelerated observer in minkowski spacetime . in this case , spacelike geodesic segments are straight lines ; therefore , the construction of fermi coordinates is simple . in fact , equation reduces to @xmath49 . differentiating this relation we find that @xmath50{\rm d}x^0+\lambda^\mu _ { \;\;(i ) } { \rm d}x^i,\ ] ] where @xmath51 and @xmath52 are given by @xmath53 the components @xmath54 of the minkowski metric tensor in fermi coordinates are given by @xmath55 moreover , @xmath56 and the inverse metric is given by @xmath57 the christoffel symbols can be evaluated using and ; the nonzero components are @xmath58_i,\end{aligned}\ ] ] where an overdot denotes differentiation with respect to @xmath59 and @xmath60 is defined by @xmath61 in minkowski spacetime , the fermi coordinates are admissible for @xmath62 . we note that the structure of the boundary region @xmath63 has been discussed in detail @xcite . in general , the boundary of the admissible region at a given time @xmath59 is a real quadric cone . if @xmath64 , this surface degenerates into coincident planes . moreover , if @xmath65 , but @xmath66 , the boundary surface is a hyperbolic cylinder for @xmath67 , a parabolic cylinder for @xmath68 , and an elliptic cylinder for @xmath69 . as is well known , for @xmath70 , the boundary surface is a circular cylinder of radius @xmath71 . we now consider the equations of motion in fermi coordinates . equations and are given by @xmath72 hence , equation has the form @xmath73 where @xmath74 represents the external force per unit mass @xmath75 .\ ] ] it is interesting to note that in equation the purely rotational inertial accelerations are essentially the same as in the nonrelativistic theory @xcite . inertial accelerations have been discussed by a number of authors @xcite . in particular , it has been shown @xcite that for @xmath76 and @xmath77 , the inertial acceleration experienced by the particle parallel to its motion is given to lowest order in @xmath78 by @xmath79 , where @xmath80 is the unit vector tangent to the spatial path of the particle ; therefore , there is a sign reversal for @xmath81 with consequences that are contrary to newtonian expectations . note that the acceleration @xmath82 and the inertial acceleration @xmath78 occur in rather different ways in equation . this difference is a consequence of the absolute character of acceleration in the theory of relativity ; that is , the fact that an observer is accelerated is independent of the choice of coordinates . in particular , while the critical speed associated with @xmath82 is @xmath83 ( on the basis of equation ) , the critical speed associated with @xmath78 in equation is @xmath0 . to obtain the limiting case of lightlike motion , we let @xmath84 , where @xmath85 is an affine parameter . in the limit as @xmath86 , we have that @xmath87 ; hence , equation is valid with @xmath88 ( see appendix a ) . for a rindler observer @xcite ( that is , an observer in hyperbolic motion in minkowski spacetime ) , consider an inertial frame in which the rindler observer is at rest at @xmath89 and has uniform translational acceleration @xmath90 along the @xmath91-axis . the observer s worldline is given by @xmath92 the nonrotating orthonormal tetrad frame along this worldline has the form @xmath93 where @xmath94 and @xmath95 , and the transformation from fermi to inertial coordinates is @xmath96 the rindler coordinates are admissible for @xmath97 and @xmath98 . we note that the manifold where @xmath99 is the rindler horizon . it is related to the null cone in the inertial frame , since equations and imply that @xmath100 . imagine a free particle moving along the @xmath91-axis with speed @xmath101 , @xmath102 , according to @xmath103 from the viewpoint of the rindler observer , this motion is given by @xmath104 for @xmath105 , the particle descends monotonically toward the horizon . for @xmath106 , its path crosses the path of the rindler observer @xmath107 at two events corresponding to @xmath108 and @xmath109 , where @xmath110 this value corresponds to inertial time @xmath111 , where @xmath112 is the lorentz factor associated with @xmath101 . at either event @xmath113 on the other hand , @xmath114 and @xmath115 . that is , the free particle ascends from @xmath116 to @xmath117 at @xmath118 such that @xmath119 , then descends back to @xmath116 at @xmath120 and finally approaches the horizon @xmath99 as @xmath121 . for @xmath122 , the particle has no acceleration at the crossing events . we note that for ultrarelativistic motion beyond the critical speed @xmath123 , the inertial acceleration of the free particle has the opposite sign at crossing events compared to intuitive expectations based on newtonian mechanics ; but , other aspects of the motion are not affected . for instance , the critical speed does not appear to play a role in the proper temporal intervals between the crossing events . the relevant proper time of the rindler observer @xmath120 given by equation is always less than the proper time of the free particle @xmath124 , since for @xmath125 , @xmath126 that is , the length of the geodesic segment is maximum . this circumstance , usually called `` the twin paradox , '' is a reflection of the absolute character of acceleration in the theory of relativity . for the limiting case of a null ray ( @xmath127 ) , @xmath128 and @xmath129 . the ray thus ascends monotonically toward infinity or descends monotonically toward the horizon always with positive acceleration . moreover , the fermi coordinate speed of the ray can range from zero to infinity . we now turn our attention to motion in a gravitational field @xcite . the spacetime metric in fermi coordinates then takes the form @xmath130 where @xmath131 is the projection of the riemann tensor on the tetrad frame of the observer . to simplify matters , we concentrate on tidal effects only and assume that the observer follows a geodesic along which a fermi coordinate system is constructed based on a parallel - propagated tetrad frame . moreover , we neglect external forces on the test particle . in this case , the equations of motion of the free particle in the fermi system , i.e. equations , reduce to @xmath132 and @xmath133 equality holds for null rays @xmath134 . in this case the right - hand side of equation is a first integral of the differential equation and higher - order tidal terms can not be neglected . the fermi coordinate system is admissible within a cylindrical region along the observer s worldline with @xmath135 , where @xmath136 is the supremum of @xmath137 . equation is the _ geodesic deviation equation _ in fermi coordinates , since it represents the relative motion of the free particle with respect to the fiducial observer that has been assumed here to follow a geodesic . to illustrate the tidal effects for test particles and null rays , we consider the gravitational field of a kerr black hole . the kerr metric , in boyer - lindquist coordinates @xmath138 , is given by @xmath139 where @xmath140 and @xmath141 . the kerr source has mass @xmath142 and angular momentum @xmath143 . for a black hole , the specific angular momentum @xmath144 is such that @xmath145 . the observer is taken to be on an escape trajectory along the axis of rotation : it starts at @xmath146 with @xmath147 on this axis and moves radially outward reaching infinity with zero kinetic energy . its geodesic worldline is given by @xmath148 to determine fermi coordinates , we choose a nonrotating orthonormal tetrad frame along the observer s worldline such that @xmath149 is parallel to the @xmath91-axis . in @xmath138 coordinates , we have @xmath150 where @xmath151 and @xmath152 are given in display . using the axial symmetry of the spacetime , @xmath153 and @xmath154 can be chosen uniquely up to a rotation about the @xmath91-axis . once a pair is chosen at @xmath146 , then @xmath155 and @xmath156 are parallel propagated along the reference worldline . we do not require the explicit transformation from boyer - lindquist to fermi coordinates to obtain the curvature components that appear in the approximate equations of motion . indeed , the symmetries of the riemann tensor make it possible to represent these quantities as elements of a symmetric @xmath157 matrix with indices that range over the set @xmath158 . the result is @xmath159,\ ] ] where @xmath160 and @xmath161 are @xmath162 matrices that are symmetric and traceless and correspond respectively to the electric and magnetic parts of the riemann tensor . moreover , @xmath163 and @xmath164 , where @xmath165 the curvature components are computed along the worldline of the observer , i.e. at @xmath166 in fermi coordinates . the electric and magnetic curvatures in and , respectively , only depend upon @xmath167 . therefore , it suffices to integrate the equation for @xmath168 in to obtain @xmath169 , which when substituted in and with @xmath170 results in @xmath171 and @xmath172 , respectively . the equation of motion can thus be written as the system @xmath173=0,\\ \label{eq45 } \ddot{y } & -\frac{1}{2}ky \left ( 1 - 2\dot{y}^2+\frac{4}{3 } \dot{x}^2- \frac{2}{3 } \dot{z}^2\right ) + \frac{1}{3 } k\dot{y } ( 5x\dot{x}-7z\dot{z})\nonumber \\ & -q [ \dot{x } \dot{y } \dot{z } y-\dot{z } x ( 1+\dot{y}^2 ) -2\dot{x } z]=0,\\ \label{eq46 } \ddot{z } & + kz \left [ 1 - 2\dot{z } ^2+\frac{1}{3 } ( \dot{x}^2 + \dot{y}^2)\right ] + \frac{2}{3 } k\dot{z } ( x\dot{x } + y\dot{y } ) \nonumber\\ & -q ( x\dot{y } -\dot{x } y)(1-\dot{z}^2)=0,\end{aligned}\ ] ] where we have neglected higher - order tidal terms for the sake of simplicity . these generalized jacobi equations reduce to the standard jacobi equations when the relative motion is so slow that the velocity terms can be neglected . consider , for instance , the one - dimensional motion of the test particle along the @xmath91-direction . equations - with @xmath174 for @xmath175 reduce to @xmath176 the behavior of the solutions of this equation , which contains the critical speed @xmath177 , has been studied in detail in our previous work @xcite . suppose that @xmath178 and @xmath179 ; then , the particle starts from the observer s location and decelerates along the @xmath91-axis toward the critical speed @xmath180 . in this case @xmath181 given by equation reduces to @xmath182 whose behavior is depicted in figure [ fig:1 ] . this extends our previous results @xcite and is consistent with the rapid decrease of curvature @xmath183 away from the black hole . a more interesting situation arises for motion normal to the direction of rotation of the black hole . equations with @xmath184 for @xmath175 reduce to @xmath185 which turns out to be the general equation for one - dimensional radial motion perpendicular to the rotation axis of the black hole as a consequence of the axial symmetry of the configuration under consideration here . the corresponding lorentz factor is given by as @xmath186 we note that for ultrarelativistic motion , the particles gain tidal energy and reach the speed of light as in figure [ fig:2 ] for an extreme kerr black hole @xmath187 and in figure [ fig:3 ] for a schwarzschild black hole @xmath188 . it follows from a comparison of figures [ fig:2 ] and [ fig:3 ] that the tidal acceleration mechanism does not depend sensitively upon the specific angular momentum of the black hole ; in fact , it appears that the time it takes for @xmath189 to diverge is in general slightly longer for @xmath190 as compared to the @xmath191 case . moreover , the divergence of @xmath189 indicates that our test - particle approach breaks down . in two recent papers @xcite , we approximated @xmath189 by @xmath192 and plotted its approach toward infinity as a consequence of the ultrarelativistic tidal acceleration mechanism . the present calculation of @xmath189 is based on an improved approximation scheme and goes beyond our previous work . to reach the speed of light or , equivalently , for the timelike particle worldline to become null the fermi speed @xmath193 should reach a value that is greater than unity as @xmath183 in equation is negative . we expect that this value of @xmath193 is reached at a later time @xmath59 as demonstrated by a comparison of figures [ fig:2 ] and [ fig:3 ] with our previous results @xcite . even the present calculations are approximate as they are based on dropping the higher - order tidal terms . it is clear that at a sufficiently long time @xmath194 , the fermi coordinate system breaks down . the kinematic breakdown of fermi coordinates , i.e. their inadmissibility , is logically independent of the divergence of @xmath189 , which indicates the failure of the dynamical theory presented here . our test - particle approximation ignores any back reaction ; furthermore , for charged particles the electromagnetic field configuration near the black hole could enhance or hinder the tidal acceleration mechanism . in any case , this breakdown of our dynamics nevertheless implies that sufficiently energetic particles emerge from the vicinity of the black hole after having experienced tidal acceleration by the gravitational field of the collapsed source . in this way , highly energetic particles may be created by microquasars in our galaxy . it is important to emphasize that the tidal acceleration mechanism is independent of the horizon structure of the black hole . these results are interesting in connection with the origin of the highest energy cosmic rays since cosmic rays with energies above the greisen - zatsepin - kuzmin limit @xmath195 are not expected to reach the earth from distant galaxies @xcite . fermi coordinates constitute a geodesic coordinate system that is a natural extension of the standard cartesian system and is indispensable for the theory of measurement in relativistic physics . we have discussed the general equation of motion of a pointlike test particle , as well as the limiting case of a null ray , in fermi coordinates . inertial and tidal effects of ultrarelativistic particles with speeds above the critical speed @xmath196 have been emphasized . this work goes beyond our previous work in this direction and strengthens the basis for our results and conclusions @xcite . let us compute the effective external force per unit mass given by equation for the electromagnetic case , namely , @xmath197 in accordance with the lorentz force law for the motion of a particle of charge @xmath198 in an external field @xmath199 . neglecting radiation reaction , we find that @xmath200 where @xmath85 is the affine parameter defined by @xmath201 . a massless charged particle does not exist ; therefore , we assume that @xmath202 as @xmath86 . with this assumption , it turns out that the massless limit of the trajectory of a charged particle is a null geodesic . there exist nongeodesic null curves in minkowski spacetime . these may be physically interpreted as follows : imagine the path of a null electromagnetic ray that is reflected from a collection of mirrors fixed in different places in space . the path consists of straight ( i.e. geodesic ) null segments separated by mirrors . next imagine a limiting situation involving an infinite number of such idealized pointlike mirrors . the resulting smooth curve is a nongeodesic null curve . an example of such a curve is @xmath203 where @xmath204 and @xmath205 are constants . the generalization to curved spacetime is straightforward . bm is grateful to donato bini and robert jantzen for helpful correspondence . xxxxx synge j l 1960 _ relativity : the general theory _ , north - holland , amsterdam marzlin k - p 1994 _ gen . rel . * 26 * 619 + marzlin k - p 1994 _ phys . d _ * 50 * 888 + marzlin k - p 1996 _ phys . lett a _ * 215 * 1 mashhoon b 1975 _ astrophys . j. _ * 197 * 705 + mashhoon b 1977 _ astrophys . j. _ * 216 * 591 ni w - t and zimmermann m 1978 _ phys . * 17 * 1473 + li w - q and ni w - t 1979 _ j. math . _ * 20 * 1473 + li w - q and ni w - t 1979 _ j. math . phys . _ * 20 * 1925 chicone c and mashhoon b 2002 _ class . quantum grav . _ * 19 * 4231 + chicone c and mashhoon b 2004 _ int . d _ * 13 * 945 chicone c and mashhoon b 2004 _ preprint _ astro - ph/0404170 + chicone c and mashhoon b 2004 _ preprint _ astro - ph/0406005 chicone c and mashhoon b 2004 _ class . quantum grav . _ , in press mashhoon b 2003 in : _ advances in general relativity and cosmology _ , edited by g. ferrarese , pitagora , bologna , pp . 323334 de felice f 1995 _ phys . rev . a _ * 52 * 3452 estabrook f b and wahlquist h d 1964 _ j. math . phys . _ * 5 * 1629 defacio b , dennis p w and retzloff d g 1978 _ phys . d _ * 18 * 2813 + defacio b , dennis p w and retzloff d g 1979 _ phys . d _ * 20 * 570 jaffe j and shapiro i i 1972 _ phys . d _ * 6 * 405 jantzen r t , carini p and bini d 1992 _ ann . phys . _ ( _ n y _ ) * 215 * 1 + bini d , carini p and jantzen r t 1995 _ class . quantum grav . _ * 12 * 2549 moliner i , portilla m and vives o 1995 _ phys . d _ * 52 * 1302 bunchaft f and carneiro s 1998 _ class . quantum grav . _ * 15 * 1557 rindler w 1969 _ am . j. phys . _ * 34 * 1174 + rindler w 2001 _ relativity : special , general and cosmological _ , oxford university press , oxford anchordoqui l a , dermer c d and ringwald a 2004 _ preprint _ hep - ph/0403001 | fermi coordinates are the natural generalization of inertial cartesian coordinates to accelerated systems and gravitational fields .
we study the motion of ultrarelativistic particles and light rays in fermi coordinates and investigate inertial and tidal effects beyond the critical speed @xmath0 .
in particular , we discuss the black - hole tidal acceleration mechanism for ultrarelativistic particles in connection with a possible origin for high - energy cosmic rays . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
input media for automatic summarization has varied from text @xcite to speech @xcite and video @xcite , but the application domain has been , in general , restricted to informative sources : news @xcite , meetings @xcite , or lectures @xcite . nevertheless , application areas within the entertainment industry are gaining attention : e.g. summarization of literary short stories @xcite , music summarization @xcite , summarization of books @xcite , or inclusion of character analyses in movie summaries @xcite . we follow this direction , creating extractive , text - driven video summaries for films and documentaries . documentaries started as cinematic portrayals of reality @xcite . today , they continue to portray historical events , argumentation , and research . they are commonly understood as capturing reality and therefore , seen as inherently non - fictional . films , in contrast , are usually associated with fiction . however , films and documentaries do not fundamentally differ : many of the strategies and narrative structures employed in films are also used in documentaries @xcite . in the context of our work , films ( fictional ) tell stories based on fictive events , whereas documentaries ( non - fictional ) address , mostly , scientific subjects . we study the parallelism between the information carried in subtitles and scripts of both films and documentaries . extractive summarization methods have been extensively explored for news documents @xcite . our main goal is to understand the quality of automatic summaries , produced for films and documentaries , using the well - known behavior of news articles as reference . generated summaries are evaluated against manual abstracts using rouge metrics , which correlate with human judgements @xcite . this article is organized as follows : section [ sec : generic - summarization ] presents the summarization algorithms ; section [ sec : datasets ] presents the collected datasets ; section [ sec : setup ] presents the evaluation setup ; section [ sec : results ] discusses our results ; section [ sec : conclusions ] presents conclusions and directions for future work . six text - based summarization approaches were used to summarize newspaper articles , subtitles , and scripts . they are described in the following sections . [ [ sub : mmr ] ] is a query - based summarization method @xcite . it iteratively selects sentences via equation [ eq : e1 ] ( @xmath0 is a query ; @xmath1 and @xmath2 are similarity metrics ; @xmath3 and @xmath4 are non - selected and previously selected sentences , respectively ) . @xmath5 balances relevance and novelty . can generate generic summaries by considering the input sentences centroid as a query @xcite . @xmath6\ ] ] lexrank @xcite is a centrality - based method based on google s pagerank @xcite . a graph is built using sentences , represented by tf - idf vectors , as vertexes . edges are created when the cosine similarity exceeds a threshold . equation [ eq : lexrank ] is computed at each vertex until the error rate between two successive iterations is lower than a certain value . in this equation , @xmath7 is a damping factor to ensure the method s convergence , @xmath8 is the number of vertexes , and @xmath9 is the score of the @xmath10th vertex . @xmath11}\frac{\operatorname{sim}\left(v_{i},v_{j}\right)}{\sum_{v_{k}\in \operatorname{adj}\left[v_{j}\right]}\operatorname{sim}\left(v_{j},v_{k}\right)}s\left(v_{j}\right)\ ] ] [ [ sub : lsa ] ] infers contextual usage of text based on word co - occurrence @xcite . important topics are determined without the need for external lexical resources @xcite : each word s occurrence context provides information concerning its meaning , producing relations between words and sentences that correlate with the way humans make associations . is applied to each document , represented by a @xmath12 term - by - sentences matrix @xmath13 , resulting in its decomposition @xmath14 . summarization consists of choosing the @xmath15 highest singular values from @xmath16 , giving @xmath17 . @xmath18 and @xmath19 are reduced to @xmath20 and @xmath21 , respectively , approximating @xmath13 by @xmath22 . the most important sentences are selected from @xmath21 . documents are typically composed by a mixture of subjects , involving a main and various minor themes . support sets are defined based on this observation @xcite . important content is determined by creating a support set for each passage , by comparing it with all others . the most semantically - related passages , determined via geometric proximity , are included in the support set . summaries are composed by selecting the most relevant passages , i.e. , the ones present in the largest number of support sets . for a segmented information source @xmath23 , support sets @xmath24 for each passage @xmath25 are defined by equation [ eq : e4 ] , where @xmath26 is a similarity function , and @xmath27 is a threshold . the most important passages are selected by equation [ eq : e5 ] . @xmath28 @xmath29 [ [ sub : support - sets_luis ] ] @xcite proposed an extension of the centrality algorithm described in section [ sub : support - sets ] , which uses a two - stage important passage retrieval method . the first stage consists of a feature - rich supervised key phrase extraction step , using the maui toolkit with additional semantic features : the detection of rhetorical signals , the number of named entities , tags , and 4 n - gram domain model probabilities @xcite . the second stage consists of the extraction of the most important passages , where key phrases are considered regular passages . [ [ sub : grasshopper ] ] @xcite is a re - ranking algorithm that maximizes diversity and minimizes redundancy . it takes a weighted graph @xmath30 ( @xmath31 : @xmath32 vertexes representing sentences ; weights are defined by a similarity measure ) , a probability distribution @xmath33 ( representing a prior ranking ) , and @xmath34 $ ] , that balances the relative importance of @xmath30 and @xmath33 . if there is no prior ranking , a uniform distribution can be used . sentences are ranked by applying the teleporting random walks method in an absorbing markov chain , based on the @xmath31 transition matrix @xmath35 ( calculated by normalizing the rows of @xmath30 ) , i.e. , @xmath36 . the first sentence to be scored is the one with the highest stationary probability @xmath37 according to the stationary distribution of @xmath38 : @xmath39 . already selected sentences may never be visited again , by defining @xmath40 and @xmath41 . the expected number of visits is given by matrix @xmath42 ( where @xmath43 is the expected number of visits to the sentence @xmath44 , if the random walker began at sentence @xmath10 ) . we obtain the average of all possible starting sentences to get the expected number of visits to the @xmath44th sentence , @xmath45 . the sentence to be selected is the one that satisfies @xmath46 . we use three datasets : newspaper articles ( baseline data ) , films , and documentaries . film data consists of subtitles and scripts , containing scene descriptions and dialog . documentary data consists of subtitles containing mostly monologue . reference data consists of manual abstracts ( for newspaper articles ) , plot summaries ( for films and documentaries ) , and synopses ( for films ) . plot summaries are concise descriptions , sufficient for the reader to get a sense of what happens in the film or documentary . synopses are much longer and may contain important details concerning the turn of events in the story . all datasets were normalized by removing punctuation inside sentences and timestamps from subtitles . temrio @xcite is composed by 100 newspaper articles in brazilian portuguese ( table [ tab : news_corpus ] ) , covering domains such as world " , politics " , and foreign affairs " . each article has a human - made reference summary ( abstract ) . .temrio corpus properties . [ cols="<,<,^,^,^",options="header " , ] ; support sets used cosine distance and threshold=@xmath47 ; kp - centrality used 50 key phrases . ] news articles intend to answer basic questions about a particular event : who , what , when , where , why , and often , how . their structure is sometimes referred to as inverted pyramid " , where the most essential information comes first . typically , the first sentences provide a good overview of the entire article and are more likely to be chosen when composing the final summary . although documentaries follow a narrative structure similar to films , they can be seen as more closely related to news than films , especially regarding their intrinsic informative nature . in spite of their different natures , however , summaries created by humans produce similar scores for all of them . it is possible to observe this behavior in figure [ fig : originais ] . note that documentaries achieve higher scores than news articles or films , when using the original subtitles documents against the corresponding manual plot summaries . figure [ fig : graphconcl1 ] presents an overview of the performance of each summarization algorithm across all domains . the results concerning news articles were the best out of all three datasets for all experiments . however , summaries for this dataset preserve , approximately , 31% of the original articles , in terms of sentences , which is significantly higher than for films and documentaries ( which preserve less than @xmath48 ) , necessarily leading to higher scores . nonetheless , we can observe the differences in behavior between these domains . notably , documentaries achieve the best results for plot summaries , in comparison with films , using scripts , subtitles , or the combination of both . the relative scores on the films dataset are influenced by two major aspects : the short sentences found in the films dialogs ; and , since the generated summaries are extracts from subtitles and scripts , they are not able to represent the film as a whole , in contrast with what happens with plot summaries or synopses . additionally , the experiments conducted for script+subtitles for films , in general , do not improve scores above those of scripts alone , except for support sets for r-1 . overall , lsa performed consistently better for news articles and documentaries . similar relatively good behavior had already been observed for meeting recordings , where the best summarizer was also lsa @xcite . one possible reason for these results is that lsa tries to capture the relation between words in sentences . by inferring contextual usage of text based on these relations , high scores , apart from r-1 , are produced for r-2 and r - su4 . for films , lexrank was the best performing algorithm for subtitles , scripts and the combination of both , using plot synopses , followed by lsa and support sets for plot summaries . mmr has the lowest scores for all metrics and all datasets . we observed that sentences closer to the centroid typically contain very few words , thus leading to shorter summaries and the corresponding low scores . interestingly , by observing the average of r-1 , r-2 , and r - su4 , it is possible to notice that it follows very closely the values of r - su4 . these results suggest that r - su4 adequately reflects the scores of both r-1 and r-2 , capturing the concepts derived from both unigrams and bigrams . overall , considering plot summaries , documentaries achieved higher results in comparison with films . however , in general , the highest score for these two domains is achieved using films scripts against plot synopses . note that synopses have a significant difference in terms of sentences in comparison with plot summaries . the average synopsis has 120 sentences , while plot summaries have , on average , 5 sentences for films , and 4 for documentaries . this gives synopses a clear advantage in terms of rouge ( recall - based ) scores , due to the high count of words . we analyzed the impact of the six summarization algorithms on three datasets . the newspaper articles dataset was used as a reference . the other two datasets , consisting of films and documentaries , were evaluated against plot summaries , for films and documentaries , and synopses , for films . despite the different nature of these domains , the abstractive summaries created by humans , used for evaluation , share similar scores across metrics . the best performing algorithms are lsa , for news and documentaries , and lexrank for films . moreover , we conducted experiments combining scripts and subtitles for films , in order to assess the performance of generic algorithms by inclusion of redundant content . our results suggest that this combination is unfavorable . additionally , it is possible to observe that all algorithms behave similarly for both subtitles and scripts . as previously mentioned , the average of the scores follows closely the values of r - su4 , suggesting that r - su4 is able to capture concepts derived from both unigrams and bigrams . we plan to use subtitles as a starting point to perform video summaries of films and documentaries . for films , the results from our experiments using plot summaries show that the summarization of scripts only marginally improved performance , in comparison with subtitles . this suggests that subtitles are a viable approach for text - driven film and documentary summarization . this positive aspect is compounded by their being broadly available , as opposed to scripts . this work was supported by national funds through fundao para a cincia e a tecnologia ( fct ) with reference uid / cec/50021/2013 . 40 natexlab#1#1[1]`#1 ` [ 2]#2 [ 1]#1 [ 1]http://dx.doi.org/#1 [ ] [ 1]pmid:#1 [ ] [ 2]#2 , , , , , . , in : . , pp . . , , , . . , , , . , in : , pp . . , , . , in : , pp . . , . , , , . , in : , pp . . , , , , . , in : , pp . . , , . , in : , pp . , , , , , , . , in : , pp . . , , , . . , , in : , pp . . , , . , in : , pp . . , , . , . , . . , . , , , , , . , in : , , , , , , , , ( eds . ) , , . , , , . , in : , . , in : , pp . . , , , , . , in : , pp . . , , , , , , , , , , . , in : , pp . . , , . , in : , pp . . , , , a. , in : , pp . . , , , b. , in : , pp . . ncleo interinstitucional de lingustica computacional ( nilc ) . , in : , pp . . , , , , . . , . , , , . . , . , , , , , , , , . , in : , pp . . , , . , in : , pp . . , , . , in : , pp . . , . . , . , , . , in : , pp . . , , , , , , , . , in : , pp . . | we assess the performance of generic text summarization algorithms applied to films and documentaries , using extracts from news articles produced by reference models of extractive summarization .
we use three datasets : ( i ) news articles , ( ii ) film scripts and subtitles , and ( iii ) documentary subtitles .
standard rouge metrics are used for comparing generated summaries against news abstracts , plot summaries , and synopses .
we show that the best performing algorithms are lsa , for news articles and documentaries , and lexrank and support sets , for films . despite the different nature of films and documentaries ,
their relative behavior is in accordance with that obtained for news articles .
automatic text summarization , generic summarization , summarization of films , summarization of documentaries |
You are an expert at summarizing long articles. Proceed to summarize the following text:
wormhole physics dates back to the formulation of general relativity ( gr ) . indeed , in 1916 , months after einstein presented his gravitational field equations , karl schwarzschild found the first solution of einstein s equations , which described the gravitational field of a vacuum non - rotating spherically symmetric solution . in the same year , ludwig flamm published a paper in which the geometry of the schwarzschild solution was studied more closely . he pointed out a `` tunnel - shaped '' nature of space near the schwarzschild radius , this being perhaps the first move towards the modern concept of the `` throat '' in wormholes @xcite . paging through the literature , one finds next that tunnel - like solutions were considered , in 1935 , by einstein and rosen , where they constructed an elementary particle model represented by a bridge connecting two identical sheets @xcite . they considered the possibility that fundamental particles such as the electron could be represented as microscopic spacetime tunnels that convey fluxes of the electric field . these tunnels were later denoted the _ einstein - rosen bridge_. in fact , the einstein - rosen bridge is a coordinate artifact arising from choosing a coordinate patch , which is defined to double - cover the asymptotically flat region exterior to the black hole event horizon . the field had lain dormant for about twenty years when , in 1955 , john wheeler , who was beginning to be interested in topological issues in gr , explored solutions of the coupled einstein maxwell equations , which he denoted _ geons _ ( gravitational - electromagnetic entities ) @xcite . these were considered to be objects of the quantum foam connecting different regions of spacetime at the planck scale . however , the term ` wormhole ' was only used for the first time in 1957 @xcite , where misner and wheeler presented a tour de force wherein riemannian geometry of manifolds of nontrivial topology was investigated with an ambitious view to explaining all of physics . the aim was to use the source - free maxwell equations , coupled to einstein gravity , with nontrivial topology , to build models for classical electrical charges and all other particle like entities in classical physics . subsequently to the geon concept , several wormhole solutions were obtained and discussed within different contexts @xcite . however , it was only in 1988 that the full - fledged renaissance of wormhole physics took place through the seminal morris - thorne paper @xcite , and the theme is still in full flight . morris and thorne , considered static and spherically symmetric traversable wormholes , and thoroughly analysed their fundamental properties . it was found that these traversable wormholes possess a stress - energy tensor that violates the null energy condition ( nec ) , a property that was denoted _ exotic matter_. beside being hypothetical short - cuts in spacetime and consequently useful for inter- and intra - universe travel , they were found to possess other intriguing applications , such as , the usage for time - travel @xcite and investigating the interior of a black hole @xcite , amongst others . thus , a fundamental ingredient for the morris - thorne wormhole , i.e. , for static and spherically symmetric wormhole solutions , is the violation of the nec @xcite . exotic matter is particularly troublesome for measurements made by observers traversing through the throat with a radial velocity close to the speed of light , as for sufficiently high velocities , @xmath0 , the observer will measure a negative energy density @xcite . although classical forms of matter are believed to obey these energy conditions , it is a well - known fact that they are violated by certain quantum fields , such as the casimir effect . in fact , the recent discovery that the universe is undergoing an accelerated expansion @xcite may be due to an exotic cosmic fluid that lies in the phantom regime . the realization of this fact has led to the study of wormhole solutions supported by different kinds of phantom fluids ( see , for instance @xcite ) . indeed the violation of the energy conditions is a subtle issue , as almost all known and physically possible forms of matter satisfy these energy conditions , and we recall that their imposition is one of the necessary assumptions for proving the hawking - penrose singularity theorems @xcite . thus , one may adopt the approach of minimizing the usage of exotic matter @xcite . in this context , a plethora of solutions have been investigated , using a wide variety of approaches diffpo , diffpo0,diffpo2,diffpo3,18 . more specifically , in rotating solutions it was found that the exotic matter lies in specific regions around the throat , so that it is possible for a certain class of infalling observers to move around the throat as to avoid the exotic matter supporting the wormhole @xcite . using the thin shell formalism , solutions where the exotic matter is concentrated at the throat have also been extensively investigated @xcite . in the context of modified gravity it was shown that one may impose that the matter threading the wormhole satisfies the energy conditions , so that it is the higher order curvature terms that sustain these exotic geometries @xcite . astrophysical signatures have also been explored in the literature harko:2008vy . for dynamic wormholes , the nec , or more precisely the averaged null energy condition , can be avoided in certain regions hochvisserprl98,hochvisserprd98,kar , kar - sahdev , kim - evolvwh , arellano:2006ex . a particularly interesting case is that of a wormhole in a time - dependent inflationary background @xcite , in which the primary goal was to use inflation to enlarge an initially small and possibly submicroscopic wormhole . it is also possible that the wormhole will continue to be enlarged by the subsequent frw phase of expansion . one could perform a similar analysis to @xcite by replacing the desitter scale factor by an frw scale factor @xcite . in particular , in kar , kar - sahdev specific examples for evolving wormholes that exist only for a finite time were considered , and a special class of scale factors that exhibit ` flashes ' of the wec violation were also analyzed . the present paper investigates the possibility and naturalness of expanding wormholes in higher dimensions which is an important ingredient of the modern theories of fundamental physics , such as string theory , supergravity , kaluza - klein , and others . one of our motivations for considering wormhole solutions in an expanding cosmological background refers to the inflation theory guth where the quantum fluctuations in the inflaton field are considered as the seed of large scale structures in the universe . as mentioned above , the non - trivial topological objects such as microscopic wormholes may have been formed during inflation and enlarged to macroscopic ones as the universe expanded @xcite . we also explore the possibility that these higher - dimensional wormholes satisfy the nec , and we explicitly show that this is indeed the case . this paper is organized in the following way : in section [ secii ] , we present the @xmath1dimensional field equations for the specific case of a spatially - independent curvature scalar . in section [ seciii ] , we analyse the two - way traversability conditions of the wormhole structure . in section [ seciv ] , we explore wormhole solutions in different expansionary regimes , and finally in section [ conclusion ] , we conclude . the action of gr in @xmath2dimensions is written as @xmath3where @xmath4 is the scalar curvature and @xmath5 is the matter lagrangian density ; we have considered @xmath6 . varying this action with respect to the metric , we obtain the @xmath2-dimensional einstein equations @xmath7 , where @xmath8 , and @xmath9 is the matter stress - energy tensor . since we are looking for expanding wormhole solutions in a cosmological background , we use the metric@xmath10 , \label{metn}\]]in which @xmath11 is the scale factor and @xmath12 is an unknown dimensionless function , defined as @xmath13 , where @xmath14 denotes the shape function @xcite . note that this metric is a generalization of the friedmann - robertson - walker ( frw ) metric , although being less symmetric than the latter . with this generalization , metric ( [ metn ] ) is still isotropic about the center of the symmetry , though not necessarily homogeneous . when the dimensionless shape function vanishes , @xmath15 the metric ( metn ) reduces to the flat frw metric ; and as @xmath16 it approaches the static wormhole metric . to see that the `` wormhole '' form of the metric is preserved with time , consider an embedding of @xmath17 and @xmath18 slices of the spacetime given by eq . ( [ metn ] ) , in a flat 3-dimensional euclidean space with metric @xmath19 in this context , the metric of the wormhole slice is @xmath20 now , comparing the coefficients of @xmath21 , one has @xmath22 it is important to keep in mind , in particular , when considering derivatives , that eqs . ( [ coef1:phi])-([coef2:phi ] ) do not represent a `` coordinate transformation '' , but rather a `` rescaling '' of the @xmath23 coordinate on each @xmath24 slice @xcite . with respect to the @xmath25 coordinates , the `` wormhole '' form of the metric will be preserved if the metric on the embedded slice has the form @xmath26 where @xmath27 , i.e. , @xmath28 has a minimum at some @xmath29 . equation ( [ slice ] ) can be rewritten in the form of eq . ( [ whslice ] ) by using eqs . ( [ coef1:phi])-(coef2:phi ) and @xmath30 the evolving wormhole will have the same overall size and shape relative to the @xmath25 coordinate system , as the initial wormhole had relative to the initial @xmath31 embedding space coordinate system . this is due to the fact that the embedding space corresponds to @xmath32 coordinates that `` scale '' with time ( each embedding space corresponds to a particular value of @xmath24 ) . following the embedding procedure @xcite , using eqs . ( [ barredslice ] ) and ( [ whslice ] ) , one deduces that @xmath33 which implies @xmath34 therefore , we see that the relation between the embedding space at any time @xmath35 and the initial embedding space at @xmath36 , from eqs . ( [ coef2:phi ] ) and ( [ embed : relation ] ) , is given by the following @xmath37\,.\ ] ] relative to the @xmath25 coordinate system the wormhole will always remain the same size , as the scaling of the embedding space compensates for the evolution of the wormhole . however , the wormhole will change size relative to the initial @xmath36 embedding space . writing the analog of the `` flaring out condition '' @xcite for the evolving wormhole we have @xmath38 , at or near the throat . from eqs . ( [ coef1:phi ] ) , ( [ coef2:phi ] ) , ( [ bar : b ] ) , and ( [ barredembedding ] ) , it follows that @xmath39 at or near the throat , where the prime denotes the derivative with respect to @xmath23 . note that this also implies @xmath40 , at or near the throat . taking into account eqs . ( [ coef1:phi ] ) , ( [ bar : b ] ) , and @xmath41 , one may rewrite the right - hand - side of eq . ( [ barred : flareout ] ) relative to the barred coordinates as @xmath42 or @xmath43 at or near the throat . one verifies that using the barred coordinates , the flaring out condition eq . ( [ barred : flareout2 ] ) , has the same form as for the static wormhole . thus , it can be shown that metric ( [ metn ] ) represents a traversable wormhole provided@xmath44where @xmath45 is the wormhole throat , which represents a minimum radius in the wormhole space - time @xcite . the second condition is imposed in order to avoid a change in the metric signature . the third condition is the flaring - out condition and plays a fundamental role in the analysis of the violation of the energy conditions . note that the comoving radial distance defined by @xmath46should be real and finite everywhere in spite of the fact that the @xmath47-component of the covariant metric diverges at the throat ; the @xmath48 signs denote the upper and lower parts of the wormhole . the energy - momentum tensor is @xmath49 @xmath50 @xmath51 @xmath52 @xmath53 , so that using the einstein field equations and eq . ( [ metn ] ) , the ( @xmath54)dimensional field equations are satisfied by the following stress - energy profile @xmath55where the overdot denotes a derivative with respect to time . the ricci scalar will play a fundamental role in our analysis , so we write it down explicitly as @xmath56since the ricci scalar is only a function of time in standard cosmological models , it provides a motivation to use this property as a simplifying assumption in our calculations , in the presence of a wormhole . in other words , we are looking for classes of solutions corresponding to the choice of a homogeneous ricci scalar , i.e. , @xmath57 , which implies @xmath58 the above differential equation yields the following solution @xmath59where the condition @xmath60 was used to eliminate the integration constant . we point out that although @xmath61 can , in principle , be a continuous variable , we have used the fact that the space - time is asymptotically frw and applied the normalization @xmath62 for the curvature constant . it is worthwhile to mention that it is common to consider static wormholes supported by radiation that have a traceless stress - energy tensor @xcite . in such a case , the ricci scalar vanishes if there is no cosmological constant within the framework of gr . our assumption leads to the same situation , if the scale factor is assumed to be independent of time , i.e. , for a static case . as mentioned before , the dimensionless shape function @xmath12 should satisfy the conditions ( [ 3 - 3 ] ) . it is easy to show that for @xmath63 and @xmath64 these conditions are satisfied whereas for @xmath65 they are not . therefore we continue our discussions using @xmath63 and @xmath64 which present flat and open universes , respectively . with @xmath12 in hand , given by eq . ( [ 3 ] ) , one can rewrite the field equations for the spatially flat background ( @xmath63 ) as : @xmath66where @xmath67 and @xmath68 are the respective flat background components given by @xmath69respectively . for the specific case of the open background ( @xmath70 ) , we have @xmath71where the @xmath72 and @xmath73 components correspond to the `` open background '' , and are given by @xmath74 since our solutions are in a spherically symmetric cosmological background , the components of the stress - energy tensor should be asymptotically independent of @xmath23 . it is easy to see this expected behaviour is obeyed by them . one of the most interesting properties of a wormhole as pointed out by morris and thorne @xcite is its two - way traversability . in this section , some proofs will be presented to show that the wormholes discussed in this paper are indeed two - way traversable . consider a radially moving light signal emitted from a co - moving source . we assume that the signal is emitted at @xmath75 ( @xmath76 is a co - moving coordinate ) and received by a distant co - moving observer at @xmath77 . using the metric ( [ metn ] ) and eq . ( [ 4 - 1 ] ) for a radial beam , we have @xmath78where @xmath79 is the comoving radial distance defined in eq . ( [ 4 - 1 ] ) . note that @xmath80 and @xmath76 can belong to either side of the throat . it is obvious that the rhs of eq . ( [ 42 ] ) is independent of time . therefore , the lhs should also be so and a signal which is emitted in an interval @xmath81 should be received in an interval @xmath82 such that ( @xmath83 ) @xmath84since @xmath82 and @xmath85 are very short time intervals , one deduces that @xmath86where @xmath87 is the scale factor at the time of observation , @xmath88 is the scale factor at the time of emission , and @xmath89 is the cosmological redshift . this shows that the redshift is the same as the cosmological redshift and no extra redshift is caused by the wormhole . it remains to examine whether the signal ever reaches the throat in a finite time or not . this will be addressed below . since @xmath23 is greater than @xmath90 on both sides of the throat , one can not trivially deduce whether the light signal passes through the throat or not using the @xmath23 coordinate . therefore , we need to transform from the radial coordinate @xmath23 to the comoving radial coordinate @xmath91 in order to analyse this behavior . using eq . ( [ 4 - 1 ] ) , the analysis is more transparent . consider radial motion so that the geodesic equation reads @xmath92and @xmath93equation ( [ 49 ] ) yields the first integral @xmath94equation ( [ 50 ] ) shows that @xmath95 does not undergo a sign change along the path and neither does it vanish . this shows that the particle or signal continues its path , passes the throat and goes to the other side of the wormhole and therefore the wormhole is two - way traversable . in order to prove that there is a finite proper distance between a specific point and the throat , eq . ( [ 4 - 1 ] ) should be solved explicitly . for arbitrary @xmath96 , there is no analytical solution for the integral ( [ 4 - 1 ] ) . therefore , one could obtain @xmath79 in the vicinity of the throat , which is sufficient for our purpose . using the approximate relation @xmath97 , one obtains @xmath98which are clearly finite distances . therefore , the throat is not located at spatial infinity and a finite time is required to reach it . we should mention that horizons are theoretical constructs that qualitatively have two main specific properties : first , they are one - way membranes , and second , the corresponding redshift ( as observed by a distant observer ) is infinite . more precise mathematical details can be found in @xcite . since it is common to consider the singularities of the metric as candidates of being horizons , one might ask whether the coordinate singularity at the throat forms a horizon . based on the above - mentioned general qualitative features of horizons and according to the results obtained in subsections ( [ sub-51 ] ) and ( [ sub-52 ] ) , it is justified that there is no horizon at or around the throat . the possibility of the existence of a cosmological horizon , however , depends on the behavior of the scale factor @xmath99 and is not essentially affected by the presence or absence of the wormhole . one of the properties of normal matter is that it satisfies the energy conditions , in particular , the null energy condition ( nec ) and the weak energy condition ( wec ) . it was mentioned in the introduction that the matter that supports the static wormhole geometry violates the nec and is therefore denoted ` exotic matter ' . the nec requires that @xmath100 , where @xmath101 is _ any _ null vector . in terms of the energy density , radial pressure and tangential pressure the nec becomes @xcite @xmath102note that the wec , in addition to the conditions considered above , also imposes a positive energy density , @xmath103 . in what follows , we investigate the nec for the wormhole solutions in different expansion regimes . in the case of a flat background , one can obtain two different solutions for the scale factor @xmath11 by applying the equation of state @xmath104 , given by @xmath105the specific case of @xmath106 represents the inflationary regime . the case of @xmath107 represents the radiation dominated and matter dominated expansion regimes , by considering @xmath108 and @xmath109 , respectively . consider first the inflationary expansion regime , where by using eqs . ( rhoflat)-([bgflat ] ) , one obtains @xmath110it is clear that @xmath111 is always negative while @xmath112 and @xmath113 are always positive . therefore , the nec is always violated . but @xmath114 tends to zero as @xmath115 increases and therefore the wormhole matter ranges from an exotic matter regime to normal matter over time . at the throat , assuming @xmath116 , with respect to time , with @xmath117 and @xmath118 for @xmath119 ( solid curve ) and @xmath120 ( dashed curve ) . the plot shows that although initially there is normal matter at throat , the throat matter tends to an exotic matter regime over time.,width=264 ] we continue our discussions with the second solution for the scale factor , for @xmath107 . in this case , we have @xmath121for @xmath122 , it is clear that the nec is violated due to @xmath123 . for @xmath124 , we have that @xmath112 and @xmath125 are always positive , while the quantity @xmath111 should be analysed in more detail . figure [ i1 ] shows the behaviour of @xmath111 with respect to time at the throat @xmath116 for @xmath118 , @xmath117 and @xmath109 and @xmath126 . it can be seen that although the wormhole matter at the throat initially satisfies the nec , the latter is violated as time passes . in fig . [ i2 ] , the quantity @xmath111 is plotted against @xmath23 and @xmath115 with @xmath116 , @xmath127 , @xmath117 for @xmath128 and @xmath126 . the figure shows that the region of exotic matter in the vicinity of the wormhole throat increases as time increases . as we are considering higher - dimensional wormholes in an expanding spacetime , an interesting scenario to examine is whether these dynamic wormholes could be constructed from normal matter for @xmath129 . this is indeed the case for the solutions discussed here , by choosing suitable values for the constants . figure [ ii1 ] plots @xmath111 against @xmath23 and @xmath115 for @xmath130 , @xmath131 , @xmath132 and @xmath133 . as depicted in the figure , @xmath134 is always positive for this choice of constants and therefore the nec ( and also wec ) is satisfied for the whole wormhole structure . with respect to @xmath23 and @xmath115 for @xmath132 , @xmath116 , @xmath135 and @xmath136 . this plot shows that for suitable choices of constants , there are wormhole structures constructed from matter that satisfy the null energy condition.,width=302 ] in the case of the open background , by applying @xmath137 , we consider the following analytical solutions @xmath138where the case @xmath106 corresponds to the inflationary regime . let us first investigate the solution corresponding to the inflationary expansion regime , @xmath139 . using eqs . ( [ rhoopen])-([bgopen ] ) , we have @xmath140as in the case of the flat background , @xmath111 is always negative , implying the violation of the nec throughout the spacetime , while @xmath112 and @xmath125 are always positive . however , @xmath111 tends to zero as @xmath115 increases and therefore during the inflationary era the wormhole matter tends from an exotic matter regime to a normal matter one , at temporal infinity . consider now the second case where @xmath141 and @xmath142 . in this case , one obtains the following relationships @xmath143 \frac{1}{{t}% ^{2 } } , \\ \rho + p_{t } & = & \bigg [ \frac{\left ( n-2\right ) \left ( { \mathit{r}}_{0}^{n}+{% \mathit{r}}_{0}^{n-2}\right ) } { 2{a}_{4}^{2}{r}^{n } } \notag \\ & & + \frac{(n-1)\left ( { a}_{4}^{2}-1\right ) } { { a}_{4}^{2}}\bigg ] \frac{1}{{t}% ^{2}}.\end{aligned}\]]for @xmath144 , it is clear that the nec is violated due to @xmath123 ; note also that @xmath145 . however , this case @xmath144 should be excluded , as @xmath112 coincides with the background energy density . this would imply that the energy density of the universe is negative , so it is not physically acceptable . for @xmath146 , it is obvious that @xmath147 and @xmath125 are always positive while @xmath111 should be investigated . figure [ ii2 ] depicts @xmath111 in terms of @xmath23 and @xmath115 for @xmath148 , @xmath149 and @xmath150 . this figure shows that by choosing suitable constants , there is a wormhole structure constructed from normal matter . with respect to @xmath23 and @xmath115 for @xmath132 , @xmath148 and @xmath149 . it shows that in the case of an open universe , it is possible to have wormholes constructed from normal matter.,width=302 ] the present paper investigates the possibility and naturalness of expanding wormholes in higher dimensions which is an important ingredient of modern theories of fundamental physics , for instance , string theory , supergravity and kaluza - klein , amongst others . one of our motivations for considering wormhole solutions in an expanding cosmological background refers to the inflationary theory where the quantum fluctuations in the inflaton field may have served as the seed for the large scale structures in the universe . non - trivial topological objects such as microscopic wormholes may have been formed through the quantum foam and enlarged to macroscopic size during inflation and in the subsequent expansion of the universe . indeed , if most of the wormholes in the quantum foam survived enlargement through inflation , then the universe might be far more inhomogeneous and topologically complicated than we observe . indeed , postulating higher - dimensional spacetimes is an important ingredient of modern theories of fundamental physics . in this context , the existence of higher dimensions may help construct wormhole solutions that respect energy conditions . in particular , in a cosmological set up , microscopic , dynamical wormholes produced in the early universe may be inflated to macroscopic scales and thus be at least in principle astrophysically observable . in this work , by assuming a homogeneous matter field ( i.e. energy density depending only on the time coordinate ) , which holds in the standard cosmology , we arrived at interestingly simple and exact solutions . more specifically , we considered a particular class of wormhole solutions corresponding to a spatially homogeneous ricci scalar . the possibility of obtaining solutions with normal and exotic matter was explored and we found new solutions including those that satisfy the nec in specific time intervals . in particular , in five dimensions , we found solutions that satisfy the nec everywhere . c. w. misner and j. a. wheeler , annals phys . * 2 * , 525 ( 1957 ) . h. g. ellis , j. math . phys . 14 ( 1973 ) 104 ; k. bronnikov , acta phys . pol . b 4 ( 1973 ) 251 ; k. a. bronnikov , v. n. melnikov , g. n. shikin , and k. p. staniukovich , scalar , ann . phys ( n.y ) 118 ( 1979 ) 84 ; g. clement , gen . rel . and grav . 16 ( 1984 ) 131 ; g. clement , gen . rel . and grav . 16 ( 1984 ) 477 ; g. clement , gen . rel . and grav . 16 ( 1984 ) 491 . d. hochberg and m. visser , phys . d 56 ( 1997 ) 4745 ; d. hochberg and m. visser , phys . rev . d 58 ( 1998 ) 044021 ; j. l. friedman , k. schliech , and d. m. witt , phys . 71 ( 1998 ) 1486 ; d. ida and s. a. heyward , phys . a 260 ( 1999 ) 175 ; m. parikh and j. p. van der schaar , arxiv:1406.5163 [ hep - th ] . s. v. sushkov , phys . d * 71 * 043520 ( 2005 ) ; f. s. n. lobo , phys . d * 71 * 084011 ( 2005 ) f. s. n. lobo , phys . d * 71 * , 124022 ( 2005 ) ; f. s. n. lobo , phys . rev . d * 73 * , 064028 ( 2006 ) ; f. s. n. lobo , class . quant . grav . * 23 * , 1525 ( 2006 ) a. debenedictis , r. garattini and f. s. n. lobo , phys . rev . d * 78 * , 104003 ( 2008 ) ; f. s. n. lobo , phys . d * 75 * , 024023 ( 2007 ) . h. b. g. casimir , proc . 51 ( 1948 ) 793 ; t. a. roman , phys . d 33 ( 1986 ) 3526 . m. visser , s. kar and n. dadhich , phys . * 90 * , 201102 ( 2003 ) b. bhawal and s. kar , phys . d * 46 * , 2464 ( 1992 ) ; m. h. dehghani and z. dayyani , phys . d * 79 * , 064010 ( 2009 ) ; s. h. mazharimousavi , m. halilsoy , and z. amirabi , phys . d * 81 * , 104002 ( 2010 ) ; s. h. mazharimousavi , m. halilsoy , and z. amirabi , class . gravity * 28 * , 025004 ( 2011 ) ; m. h. dehghani and m. r. mehdizadeh , phys d * 85 * , 024024 ( 2012 ) . f. s. n. lobo and m. visser , class . * 21 * , 5871 ( 2004 ) m. s. r. delgaty and r. b. mann , int . j. mod . phys . d 4 ( 1995 ) 231 ; j. p. s. lemos , f. s. n. lobo , and s. q. deoliveira , phys . d 68 ( 2003 ) 064004 ; j. p. s. lemos and f. s. n. lobo , phys . rev . d 69 ( 2004 ) 104007 . n. riazi , j. korean astron . ( 1996 ) 283 ; n. riazi , astrophys . space sci . 283 ( 2003 ) 235 ; j. m. maldacena and l. maoz , jhep 0402 ( 2004 ) 053 ; e. bergshoeff , a. collinucci , a. ploegh , s. vandoren , and t. van riet , jhep 0601 ( 2006 ) 061 ; n. arkani - hamed , j. orgera , and j. polchinski , jhep 0712 ( 2007 ) 018 ; v. sushkov and m. kozyrev , phys . d 84 ( 2011 ) 124026 ; f. s. n. lobo and m. a. oliveira , phys . d * 81 * , 067501 ( 2010 ) . s. w. kim and h. lee , phys . d 63 ( 2001 ) 064014 ; f. parisio , ibid . 63 , 087502 ( 2001 ) ; k. a. bronnikov , phys . rev . d 63 ( 2001 ) 044005 ; a. v. b. arellano and f. s. n. lobo , class . quant . grav . * 23 * , 7229 ( 2006 ) ; h. maeda , t. harada and b. j. carr , phys . d * 79 * , 044034 ( 2009 ) . e. teo , phys . rev . d * 58 * , 024014 ( 1998 ) m. visser , phys . d 39 ( 1989 ) 3182 ; m. visser , nucl . b 328 ( 1989 ) 203 ; e. poisson and m. visser , phys . rev . d * 52 * , 7318 ( 1995 ) ; f. s. n. lobo and p. crawford , class . quant . * 21 * , 391 ( 2004 ) e. f. eiroa and g. e. romero , gen . rel . grav . * 36 * , 651 ( 2004 ) f. s. n. lobo and p. crawford , class . * 22 * , 4869 ( 2005 ) j. p. s. lemos and f. s. n. lobo , phys . rev . d * 78 * , 044030 ( 2008 ) f. s. n. lobo , class . * 21 * , 4811 ( 2004 ) f. s. n. lobo and a. v. b. arellano , class . * 24 * , 1069 ( 2007 ) f. s. n. lobo , gen . * 37 * , 2023 ( 2005 ) . f. s. n. lobo , phys . d * 75 * , 064027 ( 2007 ) ; f. s. n. lobo , class . * 25 * , 175006 ( 2008 ) ; f. s. n. lobo and m. a. oliveira , phys . d * 80 * , 104012 ( 2009 ) ; n. m. garcia and f. s. n. lobo , phys . rev . d * 82 * , 104018 ( 2010 ) ; n. montelongo garcia and f. s. n. lobo , class . * 28 * , 085018 ( 2011 ) ; c. g. boehmer , t. harko and f. s. n. lobo , phys . d * 85 * , 044033 ( 2012 ) ; s. capozziello , t. harko , t. s. koivisto , f. s. n. lobo and g. j. olmo , phys . d * 86 * , 127504 ( 2012 ) ; t. harko , f. s. n. lobo , m. k. mak and s. v. sushkov , phys . rev . d * 87 * , no . 6 , 067504 ( 2013 ) . t. harko , z. kovacs and f. s. n. lobo , phys . d * 78 * , 084005 ( 2008 ) ; t. harko , z. kovacs and f. s. n. lobo , phys . d * 79 * , 064001 ( 2009 ) . d. hochberg and m. visser , phys . lett . * 81 * , 746 ( 1998 ) . d. hochberg and m. visser , phys . d * 58 * , 044021 ( 1998 ) . s. kar , phys . d * 49 * , 862 ( 1994 ) . s. kar and d. sahdev , phys . d * 52 * , 2030 ( 1995 ) ; n. dadhich , s. kar , s. mukherjee , and m. visser , phys . d 65 ( 2002 ) 064004 ; k.a . bronnikov and s.w . kim , phys . d 67 ( 2003 ) 064027 ; h. lu and j. mei , phys . b 666 ( 2008 ) 511 . | in this work , we consider the possibility of expanding wormholes in higher - dimensions , which is an important ingredient of modern theories of fundamental physics .
an important motivation is that non - trivial topological objects such as microscopic wormholes may have been enlarged to macroscopic sizes in an expanding inflationary cosmological background .
since the ricci scalar is only a function of time in standard cosmological models , we use this property as a simplifying assumption .
more specifically , we consider a particular class of wormhole solutions corresponding to the choice of a spatially homogeneous ricci scalar .
the possibility of obtaining solutions with normal and exotic matter is explored and we find a variety of solutions including those in four dimensions that satisfy the null energy condition ( nec ) in specific time intervals .
in particular , for five dimensions , we find solutions that satisfy the nec throughout the respective evolution . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
na@xmath6coo@xmath1 was originally studied as a kind of rechargeable battery material@xcite and thermoelectric material@xcite . more recently , the discovery of the superconductivity in its hydrated x=0.35 compound@xcite , where the effect of hydration is found to be limited to lattice expansion@xcite , makes it receive a renewed interest . na@xmath6coo@xmath1 has a crystal structure consisting of 2d triangular lattice co sheets , octahedrally coordinated with o above and below the co planes , and layers of na ions sandwiched between the coo@xmath1 sheets . the phase diagram of na@xmath6coo@xmath1 has been determined by changing the na content @xmath7 using a series of chemical reactions@xcite . their electronic and magnetic properties are found to be strongly dependent on the number of charge carriers introduced by deviations from stoichiometry in the na sublattice . theoretical studies on the electronic structure of na@xmath6coo@xmath1 are very active . the early work by singh @xcitefocused on the magnetic properties . he predicted a weak instabilities of itinerant ferromagnetic character in range @xmath8 to @xmath9 , with competing but weaker itinerant antiferromagnetic solution . he used the virtual crystal approximation , instead of supercell model , to describe the partially occupation of na sites . as a result , the possibility of sodium ion and co@xmath10/co@xmath4 charge ordering is not allowed . using pseudopotential plane wave method , ni _ _ et al.__@xcite have optimized the geometries for na@xmath6coo@xmath1 ( @xmath11 , 0.5 , 0.75 and @xmath12 ) with a @xmath13 supercell . they found that the itinerant ferromagnetism is strongly suppressed by the local distortions of the oxygen around the cobalt . with optimized geometries , they identified a phase transition from wide - band ferromagnetic to narrow band paramagnetic metals with @xmath7 increasing . by carefully examining the doped electron density , marianetti _ _ et al.__@xcite found that the rigid band model is not suitable for this na doped system . a rehybridization driven by a competition between the on - site coulomb interaction and the @xmath14-oxygen hybridization is identified . they have also used a modified hubbard model to study this kind of rehybridization . within the framework of density functional theory ( dft ) , the on - site coulomb correlation effects can be dealt with dft+u method@xcite . using this method , zou _ _ et al.__@xcite have studied the electronic structure of na@xmath6coo@xmath1 with varying @xmath15 ratio . they did not find critical change of the band structure near @xmath16 , except a notable decrease of the dos at @xmath16 . to consider the relations between correlation and charge ordering , pickett and coauthors@xcite have applied the dft+u method to a @xmath17 supercell of na@xmath6coo@xmath1 at @xmath18 and @xmath19 . they found that the parameter @xmath3 is critical to charge and spin orderings ( for small supercell , the so called charge disproportionation always leads to charge ordering ) . they suggested that the phase diagram of na@xmath6coo@xmath1 is characterized by a crossover from effective single - band character with @xmath20 for @xmath21 into a three - band regime for @xmath22 , where @xmath23 and correlation effects are substantially reduced . na@xmath0coo@xmath1 is of special interests among the na@xmath6coo@xmath1 family . in the recent experimental phase diagram of na@xmath6coo@xmath1 , a novel sodium ion ordered insulating state at @xmath24 was found@xcite , with the most strongly developed superstructure@xcite . this insulating state separates two kinds of metallic phase , _ i.e. _ curie - weiss metal for @xmath21 and paramagnetic metal for @xmath22 . it is suggested that the occurence of the insulating state indicates a strong interaction between the ions and holes even though they occupy separate layers , and a charge ordering is thought to accompany the sodium ion ordering@xcite . therefore it is interesting to compare the electronic structure based on this experimental sodium ion ordered structure with previous theoretical results to identify the effect of na ordering . for example , it will be interesting to see if the na ordering will produce a co charge ordering at dft level . on the other hand , the correlation effects of na@xmath0coo@xmath1 is considered to be a betweenness comparing to small values for @xmath21 and relatively large values for @xmath22@xcite , which demands a study at the dft+u level to identify the effect of correlation . unfortunately , there is no theoretical work in the literature for this important @xmath24 case with both na ordering and correlation effects being properly considered . in this paper , we report the electronic structure of na@xmath0coo@xmath1 focusing on the possibility of charge ordering and based on the experimental superstructure at both dft and dft+u level . the calculations in this work were performed with the vienna _ ab initio _ simulation package ( vasp)@xcite , which is a first - principles plane - wave code , treating the exchange and correlation in the dft scheme . in this study , the perdew - wang functional form@xcite of generalized gradient approximation ( gga ) was used . for spin - polarized calculations , the spin interpolation of vosko _ _ et al.__@xcite was also adopted . the projector augmented wave ( paw)@xcite method in its implementation of kresse and joubert@xcite was used to describe the electron - ion interaction . the kohn - sham equations were solved via a davidson - block iteration scheme@xcite . a @xmath25 supercell was introduced to describe the experimental sodium ion ordering . brillouin zone ( bz ) integrations were performed on a well converged grid of @xmath26 monkhorst - pack @xcite special points . the total energy and density of states ( dos ) were calculated using the linear tetrahedron method with blochl corrections@xcite . the plane wave kinetic energy cutoff was fixed to 500 ev . s approach@xcite was adopted for dft+u calculations . since this dft+u functional depends only on the difference of hubbard parameter @xmath3 and screened exchange parameter @xmath27 , @xmath27 was fixed to 1 ev during all dft+u calculations . the hubbard on - site coulomb interaction is applied to co @xmath2 orbitals only . geometry relaxation is performed at a spin unpolarized gga level . a @xmath25 superstructure of na@xmath0coo@xmath1 has been found by powder neutron diffraction experiment@xcite . this superstructure has the @xmath28 space group symmetry , with a cell formula : na@xmath29co@xmath30o@xmath31 . in this experimental structure model , the na1 and na2 sites ( @xmath32 and @xmath33 wyckoff sites for original @xmath34 space group , and @xmath32 and @xmath35 sites for this @xmath28 space group respectively ) are equally occupied , and the ordered na ions form one - dimensional zigzag chains . two types of co ions ( @xmath36 and @xmath37 ) , which differ subtly in their coordination by oxygen , are also in chains . the existence of two kinds of co sites allows the emergence of two inequivalent co ions , namely co@xmath10 and co@xmath4 , in the process of self - consistency . therefore this structure model can be used to study the possibility of charge ordering . the geometry optimization , starting from the experimental structure , reaches its convergence until all forces vanished within 0.01 ev / . the optimized geometry is shown in fig . [ geometry ] . during the optimization , the cell parameters are fixed to the experimental values ( @xmath38 , b=5.630 , and @xmath39 ) , which results in only a neglectable stress less than 20 kb . as listed in table [ tbl : geo ] , relaxations of atomic positions from the experimental ones are also found to be very small . the largest relaxation comes from the first two na atoms along @xmath7 direction ( 0.06 ) , typical relaxations are about [email protected] . the calculated two kinds of co - o distances ( 1.91 and 1.88 ) are very similar to experimental values ( 1.90 and 1.86 ) . in an ideal structure model of na@xmath6coo@xmath1 , co and o may form layers of edge - shared coo@xmath41 octahedra in a triangular lattice , but distortion of the octahedra with the variation of the o hight @xmath42 from its ideal value is always exists in real system . our optimized @xmath42 varies from 0.0845 @xmath43 to 0.0892 @xmath43 for different o sites , resulting in the co - o - co angles varying from 96.2@xmath44 to 97.0@xmath44 . comparing to the 90@xmath44 angle in the undistorted octahedra , the optimized structure shows a considerable distortion of the oxygen octahedra . for @xmath45 supercell , we get a slightly smaller distortion , but ni _ _ et al.__@xcite get a significantly larger one . we notice that they have used a relatively small kinetic energy cutoff in their study . we can clearly identify three well separated manifolds in the paramagnetic gga band structure and dos of na@xmath0coo@xmath1 , as shown in fig . [ ggaband ] . the lowest in energy is the o @xmath46 bands , with a bandwidth about 5.5 ev ( only part of them are shown in the figure ) . the other two manifolds come from the co @xmath2 orbitals , which are splitted into a lower lying @xmath47 manifold and an upper lying @xmath14 manifold , separated by approximately 2.5 ev according to the oxygen octahedral crystal field . although these three manifolds are well separated in energy , the hybridizations of o @xmath46 states with both the @xmath47 and @xmath14 states are notable . the fermi energy locates at the upper edge of the @xmath47 manifold , indicating a metallic gga ground state . the low energy properties of na@xmath0coo@xmath1 will be determined by the @xmath47 manifold , which has a bandwidth of 1.6 ev . the trigonal symmetry of the co sites further splits the @xmath47 states into one with @xmath48 symmetry and a doubly degenerate @xmath49 pair . in fig . [ ggaband ] , we also plot the @xmath50 ( co @xmath51 ) partial dos . as found by lee _ _ et al.__@xcite , the @xmath50 states have a bandwidth almost identical to that of the whole @xmath47 manifold , which indicates that the widely used single - band model may not be sufficient enough to describe the gga electronic structure of this material . fermi surface often plays an important role in the electronic structure of materials . our gga fermi surfaces for the five bands across fermi level are shown in fig . [ fs ] , which are obtained by a @xmath52 bz sampling and viewed by xcrysden package@xcite . lda / gga band structure is expected to give valuable and accurate information about the fermi surface even in system with strong correlations@xcite . in our supercell structure , there will be band folding from the first bz of the original hexagonal lattice ( black hexagon in the fig . [ fs]a ) , and the fermi surface thus becomes very complex . we notice that , on the other hand , the band structure of the superstructure geometry is far from simply band folding , for example , the fermi surface of the fourth band ( band 99 ) shows a prominent 3d character , which has not been reported in previous virtual crystal study@xcite . according to the complexity of the present fermi surfaces , nesting effect for na@xmath0coo@xmath1 may not be very strong , and we thus do nt expect to find corresponding spin density wave ( sdw ) state . spin - polarized calculations are carried out in order to address the possibility of magnetic ordering . when the ordering is constrained to be ferromagnetic ( fm ) , a stable low spin solution is obtained , with magnetic moments of 0.4 and 0.45 @xmath5 for co1 and co2 respectively . but the corresponding energy gain from ferromagnetic instability is only 26 mev / co . the moment values are somewhat smaller than that ( 0.5 @xmath5 ) reported by singh@xcite under virtual crystal approximation , which may due to the difference between their structural model and ours . in our structure , the triangular symmetry is slightly broken . antiferromagnetic ( afm ) ordering is frustrated on the 2d triangular lattice . but if the magnetic moments on a sublattice vanish , the left other sublattice may geometrically permit the presence of afm ordering . here we get an afm model ( afm1 ) by fixing the magnetic moments at co1 sites to be zero . as shown in fig . [ geometry ] , the co2 sites form a 1d chain in the @xmath7-@xmath53 plane , and it is easy to establish afm ordering within this chain . in a unit cell , the two afm chains are set to be out of phase . the afm1 model results in a magnetic moments of co2 ions about 0.39 @xmath5 , which is much larger the result ( 0.21 @xmath5 ) for the singh s afm model@xcite . we notice that in that model each co ion has also two nearest neighbors of like spin in addition to four of opposite spin . another afm model ( afm2 ) is constructed by alter the direction of magnetic moments of neighboring fm co layers . although the interlayer exchange interaction is expected to be small , we get lower energy for afm2 state than that of fm state . until now , we get a metallic ground state with intra - layer fm ordering and inter - layer afm ordering . comparing with the results of singh@xcite , we may conclude that na ordering alone will not qualitatively affect the gga electronic structure to generate the band gap and charge ordering observed in experiments . for the afm1 state , where charge ordering is enforced , the magnetic energy ( 7 mev / co ) is much smaller than those of afm2 and fm states , which indicates the enforced charge ordering under gga level is unfavorable in energy . this result is consistent with the calculations for na@xmath54coo@xmath1 , where attempts using lda to obtain self - consistent afm spin ordering always converge to the fm or nonmagnetic solution@xcite . this discrepancy between theory and experimental observation naturally leads to a further consideration of correlation effect . dft+u@xcite is a kind of method aiming to deal with the on - site coulomb interaction within the framework of dft . it treats the local electrons in a hartree - fock manner , which drives local orbital occupations to integral occupancy as @xmath3 increases . for transition metal oxide , one may find a larger split of @xmath55 orbitals for larger @xmath3 , as shown in fig . [ dosfm]-[dosafm2 ] , where partial dos of cobalt @xmath2 orbitals and oxygen @xmath46 orbitals are plotted for fm , afm1 and afm2 states respectively . the cobalt @xmath2 partial dos for co1 and co2 sites are separated . we do not present the results for spin restricted paramagnetic states in this section , because the energy differences between these states and corresponding magnetic states increase rapidly with @xmath3 ( 0.4 ev / co for @xmath56 ev ) . the paramagnetic states are still metallic even with @xmath3 as large as 7.0 ev . in fig . [ dosfm ] , the top panel is the gga result , and the three below panels represent three typical electronic structure behaviors for different @xmath3 values . for small @xmath3 ( see panel b ) , the electronic structure is similar to the above gga one , except for a strong trend towards gap opening and charge ordering . as shown in fig . [ moment ] , the moments on the two inequivalent co sites are nearly equal and also similar to the gga values ( which is the @xmath57 limit ) until @xmath583 ev . above @xmath59 , the unoccupied part of the @xmath48 states weight more and more for co2 states with the increase of @xmath3 . disproportionation from co@xmath60 into @xmath61 co@xmath4 and @xmath62 co@xmath10 ions is nearly complete at @xmath63 ev and is accompanied by a metal insulator ( mott - like ) transition from conducting to insulating . this insulating phase results from the splitting of the @xmath48 states , which lead to an unoccupied narrow band containing one hole for each co2 ions and an occupied band on the co1 ion , as shown in fig . [ dosfm]c . we judge the formation of charge ordering from local magnetic moments instead of from charge population directly , because the charge difference between co1 and co2 are relatively small ( only [email protected] electron ) . this is a result of the hybridization of the o @xmath46 and co @xmath2 orbitals . in fact , oxygen may directly contribute to the charge ordering in some oxides , as suggested by coey@xcite . we also notice that the charge ordering and gap opening with a moderate @xmath3 is far from obvious , since zou _ _ et al.__@xcite have gotten a metallic ground state with a relatively large coulomb @xmath3 ( 5 ev ) but without adopting any supercell structure . therefore the sodium ion ordering is critical to the insulating ground state . there is no clear boundary for moderate and large values of @xmath3 , but a relatively large @xmath3 value will feature several characters as shown in fig . [ dosfm]d . first , the empty @xmath48 band are pushed further higher , which make it strongly mix with the upper @xmath14 bands . secondly , the large splitting of the co @xmath2 orbitals even squeeze some co2 @xmath2 states to the bottom of the o @xmath46 manifold , and consequently the o @xmath46 character at the edge of valence band near fermi energy has exceeded the co @xmath2 characters . thirdly , the magnetic moments of cobalt are exceed 1 @xmath5 in this region ( up to 1.42 @xmath5 per co for @xmath56 ev ) , which seems curious at first sight . we now give two possible rationalizations for this observation . one interpretation comes from the population analysis process . we notice that oxygen is also spin polarized in our calculations for large @xmath3 , which makes the net magnetic moment of the whole supercell is still about 1.0 @xmath5 per co. therefore the nominal magnetic moment being larger than 1 @xmath5 per co may be just an artifact of paw population analysis as a result of the strong hybridization between co and o. another possibility is that the large on - site coulomb interaction may really promote a transition from low spin state to a intermediate spin state as demonstrated by korotin _ et al._@xcite for co ions in vertex sharing coo@xmath41 octahedra . this type of transition is consistent with the strong mixing of empty @xmath48 and @xmath14 bands . the evolution of electronic structures with @xmath3 for afm states is generally similar to that for fm state . the most significant difference is that the charge ordering occurs well before gap opening for afm1 state . as shown in fig . [ moment ] , the co@xmath4 moment grows immediately for afm1 state as @xmath3 increases from 1 ev , but the gap opens until @xmath64 ev , which is different from the results of lee _ _ et al.__@xcite for na@xmath6coo@xmath1 with @xmath18 and @xmath19 . the energy difference between the fm and afm phases as a function of @xmath3 is plotted in fig . [ moment]c . the general trend gives lower energy afm state for small @xmath3 and lower energy fm state for large @xmath3 , except the relatively higher afm1 energy for small @xmath3 , which is caused by the artificial charge ordering as previously discussed . for large @xmath3 , the lower energy of fm state is a natural result of the weakening of superexchange effect , and it is also consistent with the prominent dos of the majority spin component at deep energy as shown in fig . [ dosfm]d . the energy difference between fm and afm2 is generally small according to the relatively small exchange interaction between different layers . according to our result , the magnitude of on - site coulomb interaction @xmath3 in this material can be probed by experimental magnetic measurement . from fig . [ dosfm]-[dosafm2 ] , we can find an interesting splitting and ordering of the @xmath14 orbitals . with the increase of @xmath3 , the lower part of the @xmath14 orbitals becomes mainly from co1 sites and the upper part from co2 sites . we notice that this kind of orbital ordering of unoccupied @xmath14 orbitals will not affect the low energy properties of na@xmath0coo@xmath1 . although it has been found in lacoo@xmath65@xcite , orbital ordering of valence band orbitals is not very possible in this structure , since they are not doubly degenerated . a rehybridization process is proposed to describe the competition between the on - site coulomb interaction and the @xmath14-oxygen hybridization in doped coo layers@xcite . this process can be seen clearly from the partial dos of o @xmath46 orbitals as shown in fig . [ dosfm]-[dosafm2 ] . at gga level , hybridization of oxygen @xmath46 states and @xmath14 states are more significant than the @xmath47-oxygen hybridization . with the increase of @xmath3 , the on - site coulomb interaction becomes stronger , which can be minimized by unmixing the oxygen and @xmath14 orbitals in order to decrease the occupation of the @xmath14 orbitals . in fig . [ dosfm]-[dosafm2 ] , we can clearly see this kind of unmixing by the decrease of oxygen @xmath46 partial dos in the @xmath14 manifold . for large value of @xmath3 , the hybridization of oxygen @xmath46 states and @xmath47 states is somewhat enhanced , which is not included in the modified hubbard model study by marianetti _ _ et al.__@xcite . we have studied the effect of hubbard @xmath3 on the electronic structures of na@xmath0coo@xmath1 in the previous part of this section , but the actual value of @xmath3 in this material is not determined yet , of which very different values are proposed in the literature@xcite . based on the folowing two experimental observes , we estimate that the vaule of @xmath3 should be moderate , as also suggested by lee _ _ et al.__@xcite . first , the insulating ground state suggests that the value of @xmath3 should not be too small . secondly , na@xmath6coo@xmath1 with @xmath7 away from 0.5 is always metallic indicates a narrow energy gap for na@xmath0coo@xmath1 , which prefer a not too large value of @xmath3 . since there was experimental report@xcite on electron mass enhancement for small @xmath7 of na@xmath6coo@xmath1 , people have compared the experimental electronic specific heat coefficient @xmath66 with the band value to estimate the strength of correlation@xcite . but in the na@xmath0coo@xmath1 case , where hubbard type correlation drives system to a mott insulator , @xmath66 will reduced by the correlation effects . in fact , the experimental @xmath66 is only 3 mj / mol - k@xmath67,@xcite which is much smaller than our gga band value ( 10.5 mj / mol - k@xmath67 ) , but very near to the dft+u value ( zero ) . at a plausible value of @xmath3 , 4 ev for example , intralayer afm coupling is favored over fm and interlayer afm coupling ( ref fig . [ moment]c ) . we notice that the magnetic state in na@xmath6coo@xmath1 is far from being clearly determined in experiments . there are some experiments supporting afm coupling@xcite , while there are also some recent experiments supporting fm coupling@xcite . with this intermediate value of @xmath3 , the ground state ( afm1 ) magnetic moment for co2 is 1.0 @xmath5 , and the magnetic energy is 0.123 ev / co . based on dft+u method , we have studied the effects of na ordering and on - site coulomb correlation on the geometrical , electronic and magnetic properties of na@xmath0coo@xmath1 . without including on - site coulomb interaction @xmath3 , the experimental insulating phase can not be produced by dft . for small value of @xmath3 , ground state is still metallic . at this time , no charge ordering occur for fm coupling , and symmetry enforced charge ordering for intralayer afm coupling is unfavorable in energy . increasing parameter @xmath3 to a moderate value can open an energy gap and form charge ordering for both fm and afm states . large @xmath3 value will largely split the co @xmath2 orbitals , which leads to squeezing co @xmath2 states to the bottom of o @xmath46 manifold , mixing of empty @xmath48 and @xmath14 states , and large co magnetic moments . accompanying with the early work on na@xmath6coo@xmath1 ( @xmath18 and @xmath19)@xcite , our study will shed some insights into understanding of the complex interplay of na content , superstructure , correlation effects and tendency of various type of ordering in na@xmath6coo@xmath1 . however , a complete understanding on this issue , for example , why the @xmath19 case with stronger commensurability effect and maybe also stronger correlation is not a charge ordered insulator in experiment , will need much further work . this work is partially supported by the national project for the development of key fundamental sciences in china ( g1999075305 , g2001cb3095 ) , by the national natural science foundation of china ( 50121202 , 20025309 , 10074058 ) , by the foundation of ministry of education of china , and by icts , cas . 99 c. fouassier , c. delmas , and p. hagenmuller , mater . bull . * 10 * , 443 ( 1975 ) i. terasaki , y. sasago , and k. uchinokura , phys . b * 56 * , 12685 ( 1997 ) i. terasaki , physica b * 328 * , 63 ( 2003 ) y. wang , n. s. rogado , r. j. cava and n. p. ong , nature * 423 * , 425 ( 2003 ) k. takada , h. sakural , e. takayama - muromachi , f. izumi , r. a. dilanian , and t. sasaki , nature * 422 * , 53 ( 2003 ) c. a. marianetti , g. kotliar , and g. ceder , phys . rev . lett . * 92 * , 196405 ( 2004 ) m. d. johannes and d. j. singh , phys . rev . b * 70 * , 14507 ( 2004 ) m. l. foo , y. wang , s. watauchi , h. w. zandbergen , t. he , r. j. cava , and n. p. ong , phys lett . * 92 * , 247001 ( 2004 ) d. j. singh , phys . b * 61 * , 13397 ( 2000 ) d. j. singh , phys . rev . b * 68 * , 020503(r ) ( 2003 ) j. ni and g .- zhang , phys . b * 69 * , 214503 ( 2004 ) . v. i. anisimov , f. aryasetiawan and a. i. lichtenstein , j. phys . : condens . matter * 9 * 767 ( 1997 ) . zou , j .- l . wang and z. zeng , phys . b * 69 * , 132505 ( 2004 ) . j. kunes , k .- w . lee , and w. e. pickett , arxiv : cond - mat/0308388 . lee , j. kunes , and w. e. pickett , phys . b * 70 * , 45104 ( 2004 ) q. huang , m. l. foo , j. w. lynn , h. w. zandbergen , g. lawes , y. wang , b. h. toby , a. p. ramirez , n. p. ong , r. j. cava , arxiv : cond - mat/0402255 . h. w. zandbergen , m. l. foo , q. xu , v. kumar , r. j. cava , arxiv : cond - mat/0403206 . g. kresse and j. furthmuller , comput . mater . * 6 * , 15 ( 1996 ) g. kresse and j. furthmuller , phys . b * 54 * , 11169 ( 1996 ) . j. p. perdew , j. a. chevary , s. h. vosko , koblar a. jackson , mark r. pederson , d. j. singh , and c. fiolhais , phys . b * 46 * , 6671 ( 1992 ) . s. h. vosko , l. wilk and m. nusair , can . * 58 * , 1200 ( 1980 ) p. e. blochl , phys . b * 50 * , 17953 ( 1994 ) g. kresse and d. joubert , phys . rev . b * 59 * , 1758 ( 1996 ) h. j. monkhorst and j. d. pack , phys . rev . b * 13 * , 5188 ( 1976 ) . p. e. blochl , o. jepsen , and o. k. andersen , phys . b * 49 * , 16223 ( 1994 ) s. l. dudarev , g. a. botton , s. y. savrasov , c. j. humphreys , a. p. sutton , phys . b * 57 * , 1505 ( 1998 ) a. kokalj , j. mol . graphics mod . * 17 * , 176 ( 1999 ) . code available from http://www.xcrysden.org/. m. coey , nature , 430 , 155 ( 2004 ) m. a. korotin , s. yu . ezhov , i. v. solovyev , v. i. anisimov , d. i. khomskii and g. a. sawatzky , phys . b * 54 * , 5309 ( 1996 ) h. sakurai , k. takada , s. yoshii , t. sasaki , k. kindo , and e. takayama - muromachi , phys . rev . b * 68 * , 132507 ( 2003 ) t. fujimoto , g .- q . zheng , y. kitaoka , r. l. meng , j. cmaidalka , and c. w. chu , phys . * 92 * , 047004 ( 2003 ) a. chainani , t. yokoya , y. takata , k. tamasaku , m. taguchi , t. shimojima , n. kamakura , k. horiba , s. tsuda , s. shin , d. miwa , y. nishino , t. ishikawa , m. yabashi , k. kobayashi , h. namatame , m. taniguchi , k. takada , t. sasaki , h. sakurai , e. takayama - muromachi , arxiv : cond - mat/0312293 . f. c. chou , j. h. cho , p. a. lee , e.t . abel , k. matan , and y. s. lee , arxiv : cond - mat/0306659 . a. t. boothroyd , r. coldea , d. a. tennant , d. prabhakaran , l. m. helme , and c. d. frost , phys . lett . * 92 * , 197201 ( 2004 ) t. motohashi , r. ueda , e. naujalis , t. tojo , i. terasaki , t. atake , m. karppinen , and h. yamauchi , phys . b * 67 * , 64406 ( 2003 ) | we report a first - principles projector augmented wave ( paw ) study on na@xmath0coo@xmath1 . with the sodium ion ordered insulating phase being identified in experiments ,
pure density functional calculations fail to predict an insulating ground state , which indicates that na ordering alone can not produce accompanying co charge ordering , if additional correlation is not properly considered . at this level of theory ,
the most stable phase presents ferromagnetic ordering within the coo@xmath1 layer and antiferromagnetic coupling between these layers .
when the on - site coulomb interaction for co @xmath2 orbitals is included by an additional hubbard parameter @xmath3 , charge ordered insulating ground state can be obtained .
the effect of on - site interaction magnitude on electronic structure is studied . at a moderate value of @xmath3 ( 4.0 ev for example ) ,
the ground state is antiferromagnetic , with a co@xmath4 magnetic moment about 1.0 @xmath5 and a magnetic energy of 0.12 ev / co .
the rehybridization process is also studied in the dft+u point of view . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
short gamma - ray bursts ( sgrbs ) are usually defined by the prompt duration , i.e. , gamma - ray bursts ( grbs ) with @xmath15 @xcite . a significant fraction of the sgrbs is accompanied by longer duration ( up to @xmath1 ) extended emissions ( ees ) @xcite . the isotropic energy of the prompt spike is @xmath16 @xcite , while that of ee is @xmath17 @xcite . interestingly , in some cases , observed fluences of the ees are even larger than those of the prompt spikes , e.g. , @xmath18 for sgrb 050709 @xcite and @xmath19 for sgrb 080503 @xcite . @xcite found that @xmath20 of _ swift _ bat sgrbs have ees in the x - ray band ( see their table . 3 ) . on the other hand , @xcite searched ees in 296 batse sgrbs and found that the fraction of ees is @xmath21 , where they pointed out that this fraction should be regarded as the minimum value since dim and/or softer ees can not be detected by batse . in fact , batse measures fluence above @xmath22 while bat does down to @xmath23 , and the 5 kev lower threshold energy yields an 18% increase in the ee population . this suggests that a decrease in the threshold energy , say , as low as @xmath24 , might yield a dramatic increase in the ee population . here , we consider the possible origin of such ees based on the compact binary ( neutron star ( ns)-ns or ns - black hole ( bh ) ) merger scenario @xcite . if the maximum mass of a non - rotating ns is smaller than @xmath25 , the final outcome of such a merger will be a kerr bh with a mass of @xmath26 and a spin parameter of @xmath27 with an accretion disk with a mass of @xmath2 and possibly beyond , and a neutron - rich ejecta with a mass of @xmath6 with an expanding velocity of @xmath28 @xcite . here , we adopt this situation . if the maximum mass of a non - rotating ns is larger than @xmath25 , a rapidly rotating massive ns will be the final outcome and the magnetar activities may be responsible for the prompt spike and ees of the sgrbs as well as other electromagnetic counterparts @xcite . in this paper , we consider the huge rotational energy of a kerr bh to be the intrinsic energy budget of the ee ( see also * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the mass formula of the kerr bh with a gravitational mass of @xmath29 and an angular momentum @xmath30 is written as @xcite @xmath31 where @xmath32 is the irreducible mass of the kerr bh . writing @xmath33 , we have @xmath34 where @xmath35 is the well known kerr parameter and @xmath36 . ( [ eq : mass ] ) is rewritten as @xmath37 then , the energy available by the extraction of the angular momentum of the kerr bh is given by @xmath38 for @xmath39 , for example , @xmath40 is given by @xmath41 therefore , for an @xmath42 radiation efficiency , only @xmath43 of the rotational energy of the bh enables fueling of the ee even if the emission is isotropic . the problem is how to extract the rotational energy of the kerr bh on a timescale of a few 10 s to @xmath1 . one of the plausible mechanisms for extracting the rotation energy of the kerr bh is the blandford - znajek ( bz ) process @xcite . from the results of numerical simulations , @xcite found that within factors of order unity , the bz power is expressed in units of @xmath44 by @xmath45 where @xmath46 and @xmath47 is the magnetic flux threading the horizon with @xmath48 being the strength of the magnetic field formed by the disk around the kerr bh . recovering @xmath49 and @xmath50 , we have @xmath51 ^ 2 \left(\frac{gm_{bh}}{c^2 } \right)^2 c b^2.\ ] ] for @xmath39 , for example , @xmath52 is expressed as @xmath53 dividing eq . ( [ eq : del_e ] ) by eq . ( [ eq : l_bz ] ) , we have the characteristic time @xmath54 as @xmath55 this shows that if an accretion disk with @xmath56 and an accretion time @xmath1 exists , up to @xmath57 erg can be extracted from the kerr bh . this is just the timescale of the ees and only @xmath43 efficiency is enough to explain the emissions even if they are isotropic . this paper is organized as follows . in 2 , we analyze the time evolution of the neutrino - dominated accretion disk with finite mass and angular momentum around the bh , and estimate the resultant bz power and the duration . in 3 , we consider the interaction of the bz jets with the pre - ejected matter ( @xmath6 ) with an expanding velocity of @xmath28 , which produces mildly relativistic fireballs . we calculate the dissipative photospheric emissions from such fireballs . there , we also argue the detectability and the association with the observed ees . 4 is devoted to the discussion . we use @xmath58 in cgs units unless otherwise noted . @xcite calculated the steady - state solutions of a neutrino - dominated accretion disk around a kerr bh , which is also the case in our setup at the initial stage . they assumed an accretion rate of @xmath59 constant , but took into account the full neutrino process and the kerr geometry . their important conclusions are that ( i ) the pressure is dominated by baryons with @xmath60 , ( ii ) the disk is neutron dominated so that the electron fraction is as small as @xmath61 , ( iii ) the degeneracy of the electron is at most mild because , if the degeneracy is high , the neutrino cooling is lowered to increase the temperature , ( iv ) there is an ignition accretion rate for the neutrino cooling disk that is proportional to @xmath62 where @xmath63 is the parameter in the so - called @xmath63-disk model @xcite . @xcite performed simpler newtonian calculations of such disks both numerically and analytically . one of the important conclusions is that their analytical model fits well with the numerical ones by @xcite . therefore , here we adopt a simple newtonian analytical model to mimic the neutrino - dominated accretion disk around the kerr bh . one of the big differences from @xcite is that we take into account the time variation of the accretion rate for the finite disk mass and the finite disk angular momentum while they considered a constant accretion rate . the structure of the accretion disk can be derived from @xcite @xmath64 where @xmath65 and @xmath66 are the accretion rate , @xmath67 in cylindrical coordinates , the density , the half thickness of the disk , the infalling velocity , the @xmath63 parameter , the pressure , and the angular frequency of the disk , respectively . we can express @xmath68 by @xmath69 as @xmath70 @xmath71 @xmath72 where @xmath73 is the innermost stable circular orbit ( isco ) .. however we are using newtonian gravity so that the exact treatment of isco is not possible . one can take into account the change of isco by putting different value of @xmath74 in all the equations . in this case , various quantities are modified by powers of the above factor , but our results do not change qualitatively . ] the density @xmath69 can be determined by the energy equation and the equation of state . denoting the cooling rate @xmath75 , the energy balance is expressed as @xmath76 as for the energy loss rate , we consider two neutrino cooling processes relevant to the accretion disk we are interested in as @xcite @xmath77 where @xmath78 is the temperature in units of [ k ] . the urca process is @xmath79 and the pair neutrino process is @xmath80 . the urca process dominates over the pair process for @xmath81 . as for the pressure of the matter we should consider @xmath82 @xmath83 @xmath84 where @xmath85 is the radiation constant . the relativistically degenerate pressure dominates over the gas and the radiation pressure for @xmath86 while the gas pressure dominates over the radiation pressure for @xmath87 let us consider the case for @xmath88 , where the pressure is determined by gas pressure and the cooling process is dominated by the urca process so that eqs . ( [ eq : q_eq ] ) and ( [ eq : q_urca ] ) give @xmath89 with @xmath90 . from eqs . ( [ eq : p_gene ] ) , ( [ eq : eos ] ) , and ( [ eq : cool_eq ] ) , @xmath69 is expressed as @xmath91 we then have @xmath92 @xmath93 @xmath94 @xmath95 @xmath96 @xmath97 then , the surface density of the disk ( @xmath98 ) is given by @xmath99 let us introduce the coordinate @xmath100 by @xmath101 . we assume here that the disk has a minimum and a maximum @xmath100 as @xmath102 and @xmath103 , respectively . then for a given total mass @xmath104 and the total angular momentum @xmath105 , we have @xmath106 for a given value of @xmath104 , @xmath105 , @xmath63 , and @xmath29 , from eqs . ( [ eq : m_d ] ) and ( [ eq : j_t ] ) , we can determine @xmath107 and @xmath103 in general . let us define a new variable @xmath108 by @xmath109 , that is , the mean value of the angular momentum is @xmath110 times the minimum value of the specific angular momentum at the isco . then , @xmath103 is determined by @xmath111 it is easily shown that the left - hand side of eq . ( [ eq : x_max ] ) is a monotonically increasing function for @xmath112 and has a minimum value @xmath113 at @xmath114 so that for an arbitrary value of @xmath115 , there is an unique solution @xmath112 . in the accretion process , the total angular momentum of the system should be conserved in our case , since the kerr metric has a rotational killing vector , i.e. , a stationary axisymmetric system . some of the angular momentum is absorbed by the bh from the isco so that @xmath108 increases as a function of time . note that the spin up of the bh due to accretion is negligible in our case . if we denote the mass and the angular momentum of the accreted blob into the bh as @xmath116 and @xmath117 , we have @xmath118 the solution of eq . ( [ eq : beta ] ) is given by @xmath119 where @xmath120 and @xmath121 are the initial values of @xmath108 and @xmath104 , respectively . therefore , @xmath122 in the later phase of the accretion so that @xmath123 and the @xmath124 ^ 2 $ ] will be a good approximation . inserting this expression to eq . ( [ eq : m_d ] ) , we have @xmath125^{-8/5}\alpha_{-1}^{6/5}\left(\frac{m_{bh}}{3 \ m_\odot}\right)^{-6/5}.\ ] ] integration of eq . ( [ eq : d_m_d ] ) yields @xmath126 the method we adopted here to solve the evolution of the accretion disk is similar to the quasi - static evolution of the star where the nuclear timescale is much longer than the free fall time so that at each time , the star can be regarded in gravitational equilibrium ( see , e.g. , * ? ? ? * ) . in our case , from eq . ( [ eq : v_ratio ] ) , the accretion velocity is much smaller than the kepler velocity which determines the dynamical timescale so that we can regard the disk as stationary at each time . the decrease in total mass and angular momentum can be regarded as a decrease in total nuclear energy and a change of the composition in the stellar evolution case , which are very slowly changing in a dynamical timescale . let us assume that @xmath127 , @xmath128 , @xmath129 , and @xmath130 as suggested by numerical relativity calculations @xcite . solid lines in fig . [ mdot-1 ] show the accretion rates as a function of @xmath63 for the representative time such as 1 s , 3 s , 10 s , 30 s , 100 s , and 300 s. we are interested in the late time behavior ( @xmath131 s ) where the accretion rate decreases as a function of @xmath63 for a fixed time . this is because @xmath132 in eq . ( [ eq : a ] ) is smaller for larger @xmath63 so that the accretion rate is smaller . physically , the accreting velocity is larger for larger viscosity as is clear from eq . ( [ eq : v_r ] ) , where the consumption of the disk mass is faster . the dashed lines are the neutrino cooling ignition accretion rate obtained by @xcite for @xmath133 and @xmath134 ( eq . 42 of their paper ) . neutrino cooling is effective only above these dashed lines . in fig . [ l_bz ] , we show bz luminosity as a function of @xmath63 for typical time such as 1 s , 3 s , 10 s , 30 s , 100 s , and 300 s for @xmath127 , @xmath128 , @xmath129 , and @xmath130 . the magnetic field at the horizon is assumed to be determined by @xmath135 where @xmath136 is the pressure of the disk at the isco in eq . ( [ eq : p ] ) . the bz luminosity is then given by eq . ( [ eq : l_bz ] ) . we clearly see that the bz luminosity is higher for smaller @xmath63 for the same time . this comes from both the strong @xmath63 dependence of the @xmath137 in eq . ( [ eq : p ] ) and the accretion rate in fig . [ mdot-1 ] . physically , if the viscosity is low , the accretion rate decreases slowly and the pressure is high due to the accumulation of matter , which yields a strong magnetic field in our scenario . all these effects result in a larger bz luminosity for a smaller @xmath63 . one might suspect that our approximation ( @xmath138 ) to the solution of eq . ( [ eq : x_max ] ) for given @xmath108 affects the result . to check this , we show in fig . [ check_lc ] the time evolution of the luminosity for @xmath127 , @xmath128 , @xmath129 , @xmath130 , and @xmath139 . the red and blue solid lines show the numerical solution , which is obtained by integrating eqs . ( [ eq : m_d ] ) , ( [ eq : j_t ] ) , and ( [ eq : x_max ] ) directly over time , the analytic approximation ( eq . [ eq : m_d_ana ] ) . the dashed horizontal lines show the neutrino cooling ignition accretion rates for @xmath140 ( upper ) and @xmath141 ( lower ) , respectively . we see that the analytic approximation only slightly overestimates the luminosity . we can justify the use of the analytic approximate solution to eq . ( [ eq : x_max ] ) . it is suggested that the typical @xmath142 from various numerical simulations @xcite . to have possible isotropic ee , the typical luminosity of @xmath143 is needed at @xmath144 a few 10 s. from fig . [ mdot-1 ] , if @xmath145 , which is relatively smaller than that suggested by various numerical simulations , the accretion rate is above the ignition rate for neutrino cooling up to the observed time of @xmath146 s , and the luminosity can be above @xmath147 so that enough luminosity for ees seems to be obtained via the bz mechanism . however , from fig . [ mdot-1 ] , the accretion rate is above the ignition rate for neutrino cooling only up to @xmath148 s. in this case the duration of ee is at most @xmath149 s. to increase the duration of ee , we need smaller @xmath63 than 0.01 . when can the value of @xmath63 become smaller ? if the origin of viscosity is magnetic turbulence from the magneto rotational instability , there is a big difference from the usual situations , that is , the matter is neutron rich ( @xmath61 ) . since neutron does not directly couple to the magnetic field , and @xmath61 , only 10% of the mass feels the viscosity first . note that , for @xmath150 , @xmath151 , @xmath152 , which are typical values at the isco , the gyration radius of the proton @xmath153 cm is comparable to the mean free path of p - n collision with the cross section of @xmath154 . then , the effective @xmath155 might be reduced , in principle , 10% compared to the usual case of proton (= hydrogen ) dominated gas , i.e. , @xmath156 in the conventional definition could give @xmath157 . then , from fig . [ mdot-1 ] , the duration can be even longer than @xmath158 s and the bz luminosity can supply an enough energy for the isotropic ee . these arguments suggest that the rotational energy of the kerr bhs formed in compact binary mergers might supply the energy sources of the ees with the required duration even if the emissions are isotropic . in newtonian gravity , a numerical calculation of the accretion disk exists for the present problem , i.e. , the time evolution of an accretion disk of mass @xmath159 after the merger of an ns - ns binary @xcite . our treatment also uses newtonian gravity so that we can compare our analytic results with their numerical calculations to confirm the quantitative agreement . in their calculations , the mass of the central bh is @xmath160 and they solved the time evolution of a z - direction integrated quantity such as the surface density @xmath98 with neutrino and advection cooling . the initial surface density is given as @xmath161 with @xmath162 . since @xmath163 peaks at @xmath164 , the initial specific angular momentum is @xmath165 . in our crude model , we assumed the disk boundary is at @xmath166 while in their calculation inner edge (= disk boundary ) is @xmath167 so that we define the coordinate @xmath100 by @xmath168 in this paragraph . in the newtonian calculation , there is no isco so that the minimum specific angular momentum of their calculation is @xmath169 which is different from our model of @xmath170 . as for @xmath63 , they adopt 0.3 so that we need to rewrite eqs . ( [ eq : m_d ] ) and ( [ eq : j_t ] ) as @xmath171 defining @xmath108 by @xmath172 , we have the same equation as eq . ( [ eq : x_max ] ) . the argument for deriving equations corresponding to eqs . ( [ eq : beta ] ) and ( [ eq : beta_int ] ) is also the same by changing @xmath173 and @xmath174 , which gives @xmath175 integration of eq . ( [ eq : m_d_metzger ] ) yields @xmath176 now let us compare eq . ( [ eq : m_d_metzger_int ] ) with those of the numerical calculations by @xcite . their fig . 3 shows an accretion rate at @xmath177 for @xmath178 and @xmath179 are @xmath180 , @xmath181 and @xmath182 , respectively . ( [ eq : m_d_metzger_int ] ) yields @xmath183 , @xmath184 and @xmath185 . we can say that our analytic model agrees rather well with the numerical calculations at isco especially for later times , which is indispensable for the use of our analytic model to study ees . it has been shown that , in the late phase of the accretion of dense debris such as we consider here , the disk wind driven by energy injection via viscous heating and the recombination of nucleons into alpha - particles becomes relevant @xcite . although our calculation does not include this effect , since such outflows are predominantly triggered after the viscous timescale of the disk , our results can be still viable up to this timescale , e.g. , @xmath186 for @xmath145 ( see fig . in the previous section , we showed that the rotational energy of the kerr bh up to @xmath187 can be extracted as the poynting outflow via the bz process with a timescale of @xmath188 if the accretion of the debris @xmath189 occurs with @xmath145 . hereafter , we argue the resultant emissions from such outflows and their detectability . in the course of compact binary mergers , a fraction of baryons of mass @xmath6 can be ejected with an expansion velocity of @xmath190 @xcite . a certain duration after the merger , say @xmath191 , the hypermassive ns collapses into a bh due to the loss of rotational support by emitting gravitational waves ( gws ) and/or poynting fluxes . the poynting outflow by the bz process , which is relativistic , clashes with the pre - ejecta . the bz outflow or jet will be more or less beamed and drill through the pre - ejecta , forming a hot plasma cocoon surrounding the jet . recently , such a situation has been investigated numerically @xcite , although the jet injection timescale is set to be @xmath192 s , considering jets responsible for prompt emissions of sgrbs . these studies show that the jet dynamics are significantly affected by the pre - ejecta , especially for a jet luminosity of @xmath193 and a pre - ejecta mass ejection rate of @xmath194 , which we are interested in here . in particular , if the jet launch is delayed more than @xmath195 s from the pre - ejecta launch , such a jet will be choked in the ejecta and a significant fraction of its energy will be dissipated inside the ejecta , forming a cocoon fireball . in our case , the assumed duration of the outflow injection is @xmath196 s , thus , the bz outflow would more easily penetrate the pre - ejecta if the onset time of outflow injection is not delayed significantly . nevertheless , even after penetration , a fraction of the bz - outflow energy can dissipate at the interaction surface with the pre - ejecta or the cocoon , which typically occurs at @xmath197 where @xmath54 corresponds to the time @xmath198 in the previous section . note that the bz outflow is most likely magnetically dominated at the launching radius . in the case of magnetically dominated jets , the dynamics including the cocoon formation are different from those of hydrodynamical jets ( e.g. , * ? ? ? * ) , and the energy dissipation process is rather uncertain . hereafter , we simply parameterize such a dissipation process by the fraction of dissipated energy , @xmath199 and the beaming factor of the dissipation region , @xmath200 . after dissipation , the heated pre - ejecta can be regarded as a fireball . the temperature can be estimated as @xmath201 , or @xmath202 we note that the optical depth at around this radius is large , @xmath203 . here , @xmath204 is the opacity of the thomson scattering , @xmath205 is the mean density of the ejecta , and @xmath206 is the isotropic mass ejection . though the mass ejection , in general , is unisotropic ( e.g. * ? ? ? * ) , we can take into account this effect by changing @xmath200 and @xmath207 appropriately . the fireball is accelerated due to the large internal energy , and the lorentz factor saturates at @xmath208 , or @xmath209 which occurs at @xmath210 hereafter , we simply assume @xmath211 in the acceleration phase . , the fireball is still magnetically dominated , and the evolution of the lorentz factor is generally different from the above scaling ; @xmath212 with @xmath213 . ] for @xmath214 , the fireball moves as a shell with a shell width @xmath215 so that the temperature decreases as @xmath216 , while in the lateral direction @xmath217 , it expands as @xmath218 ( e.g. , * ? ? ? then , the fireball begins to expand almost spherically irrespective of the initial beaming angle beyond the radius given by @xmath219 the temperature decreases as @xmath220 for @xmath221 . the photospheric emission from the fireball can be expected around the photospheric radius , @xmath222 the temperature of the fireball in the comoving frame evolves as @xmath223 for @xmath224 , @xmath216 for @xmath225 , and @xmath220 for @xmath226 . from eqs . ( [ eq : r_exp ] ) and ( [ eq : r_ph ] ) , @xmath227 is satisfied if @xmath207 is larger than the critical mass given by @xmath228 in the case of _ dirty _ fireballs with @xmath229 , @xmath227 is expected . the temperature at the photospheric radius becomes @xmath230 , which is expressed as @xmath231 and the peak photon energy of the resultant photospheric emission can be estimated as @xmath232 , which is given by @xmath233 the peak intensity in the comoving frame can be approximated as @xmath234 , where @xmath235 with the planck constant @xmath236 . is not the half thickness of the disk but the planck constant . also we use @xmath108 as the spectral index in this section . ] in the observer frame , using the fact that @xmath237 is lorentz invariant ( e.g. , * ? ? ? * ) , the corresponding energy flux is given by @xmath238 with @xmath239 and @xmath240 being the luminosity distance and the relativistic beaming effect , respectively , as @xmath241 @xmath242 in general , the observed duration of the photospheric emission is given by @xmath243.\ ] ] in the case of dirty fireballs , @xmath244 ( see eqs . [ eq : r_o ] and [ eq : r_exp ] ) , and thus the first term on the right - hand side of the above equation always larger than the second one . then , @xmath245 we remark that ee durations of @xmath246 do not necessarily require durations of the bz jet of @xmath247 if the fireball is dirty and the duration is determined by eq . ( [ eq : t_dur_dirty ] ) . for a fiducial parameter set ( @xmath248 ) , the emission is characterized by @xmath249 , @xmath250 , and @xmath251 from @xmath252 , and @xmath253 , @xmath254 , and @xmath255 from @xmath256 . in general , the spectral shape is determined by additional dissipation processes , e.g. , internal shocks or magnetic reconnections occurring in @xmath257 , which may slightly boost the peak energy and most likely produce a quasi - thermal spectrum , @xmath258 for example , @xcite numerically calculated the photospheric emission from fireballs ( in their cases , hot - plasma cocoons ) including some dissipation processes at @xmath259 , and they showed that typically @xmath260 and @xmath261 . then , for our fiducial parameters , @xmath262 from @xmath252 so that the spectrum may range from @xmath263 to @xmath264 . we plot the possible @xmath265 spectra in fig . [ fig : ee_obs ] . although @xmath266 has parameter dependancies like eq . ( [ eq : ee_peak_dirty ] ) , it is most likely that @xmath267 is below @xmath23 for @xmath229 and @xmath268 . in this case , the emission energy is outside of the coverage of batse and _ swift _ bat so that such ees have not been detected so far . on the other hand , the soft ees can be detected by soft x - ray survey facilities like wide - field maxi , which has a @xmath269 angular resolution . let us estimate the possible detection rate of the soft ees discussed above , in particular , simultaneously with the gws from compact binary mergers . @xmath270 where @xmath271 is the fraction of the soft ees in all sgrbs , and @xmath14 represents the ns - ns merger rate within the detection horizon of advanced ligo , advanced virgo , and kagra , @xmath272 , which corresponds to a redshift of @xmath273 and a comoving volume of @xmath274 . we assume the standard @xmath275-cdm cosmology . note here that @xmath276 can be expected given that the fraction of an ee burst is significantly larger in softer energy bands ; @xmath20 in the _ swift _ bat samples ( @xmath277 ) and @xmath21 in the batse samples ( @xmath278 ) . we take into account the sky coverage of wide - field maxi @xmath279 and the anticipated duty cycle , @xmath280 for wide - field maxi and @xmath281 for the 2nd generation gw network . in the case of dirty fireballs , the beaming factor can be relatively large , e.g. , @xmath282 , and the above estimate gives @xmath283 for @xmath284 and @xmath285 . with a planned detection threshold flux of wide - field maxi @xmath286 , such soft ees can be detectable from @xmath287 , which corresponds to a luminosity distance of @xmath288 and a comoving volume of @xmath289 . the anticipated total detection rate can be estimated as @xmath290 next let us consider the observed ees in our scenario . as we argued above , the dissipative photospheric emissions from dirty fireballs are too soft and too dim for _ swift _ bat as far as @xmath291 and @xmath268 . importantly , the typical beaming angle of the observed ees can be estimated to be much smaller as follows . the ees are being detected predominantly by _ swift _ bat with the image trigger . here we set the trigger threshold as @xmath292 in 64 s @xcite . in this case , the trigger threshold by bat for a burst with @xmath293 can be calculated as @xmath294 . for an ee with a mean luminosity of @xmath295 ( like sgrb 061006 ) , the detection horizon also becomes @xmath296 , corresponding to @xmath297 and @xmath298 . for an effective total observation time for bat of @xmath299 , and a sky coverage of @xmath300 , the total number of detectable ees becomes @xmath301 . given that 14 ees have been identified in this interval @xcite , the fraction of ee bursts @xmath302 and the beaming factor @xmath303 can be constrained as @xmath304 , or @xmath305 which is small compared to the inferred beaming factor of the prompt spikes , @xmath306 @xcite . we should mention that candidates of orphan " ees without prompt spikes , i.e. , long grbs with @xmath307 whose redshifts and host galaxies are not identified , have been detected by _ swift _ bat . the detection rate of those candidates is roughly comparable to that of sgrbs with ees ] . given this fact , the constraint on the beaming factor ( eq . [ eq : omega_ee ] ) is a lower limit and can be larger by a factor . nevertheless , the possible simultaneous detection rate of such ees and gws from ns - ns mergers would be quite small , @xmath308 . note that the estimated value is independent of the relatively uncertain binary ns merger rate . in the context of our picture , a smaller @xmath200 corresponds to a relatively narrow fireball . we note that the mass ejection associated with the bh - ns merger is orders of magnitude smaller in the polar direction than the angle averaged one , which has been confirmed by numerical simulations @xcite where the bh spin is set to be parallel to the orbital angular momentum . in general , it is expected that they are misaligned due to the kick velocity in the formation process of nss . although more numerical simulations using general initial conditions are needed to know what happens in bh - ns mergers , the simulations by @xcite suggest that the matter along the bz jet axis is small compared with an ns - ns binary . as a result , one can expect that a smaller fraction of energy is dissipated in a smaller region of the pre - ejecta compared with the previous ns - ns cases where the outflows via the bz process clash with denser pre - ejecta . hereafter , we take @xmath309 , @xmath310 , @xmath311 as a fiducial value . such fireballs are _ clean _ in that @xmath312 , i.e. , @xmath313 the lorentz factor of such clean fireballs and the comoving temperature at the photospheric radius are @xmath314 and @xmath315 , respectively , and the peak energy of the photospheric emission @xmath316 can be estimated as @xmath317 from eq . ( [ eq : f_peak_dirty ] ) , the corresponding peak flux is @xmath318 in the case of clean fireballs , the emission duration is @xmath319 . for a fiducial parameter set ( @xmath320 ) , the emission is characterized by @xmath321 , @xmath322 , and @xmath323 from @xmath324 ( see fig . [ fig : ee_obs ] where we also plot an observed flux spectrum of ee associated with sgrb 061006 at @xmath324 ) . the observed ees can be consistently interpreted as the high energy tail of the photospheric emission from such a clean fireball . to test this scenario , simultaneous detections in the soft x - ray band ( @xmath325 ) are crucial . as for the ees from fireballs , we need to model , e.g. , the subphotospheric dissipation processes in detail to predict the spectra more precisely . nevertheless , a key message here is that the typical energy of ees is likely to be in soft x - ray bands . in our scenario , this is essentially due to the relatively large launching radii of the fireball , @xmath326 ( eq . [ eq : r_o ] ) compared with that of the conventional grb fireball , @xmath327 , which results in a lower initial temperature ( eq . [ eq : t_o ] ) . the importance of soft x - ray bands has been implied from the observed soft photon index of ees and the increase of the fraction of sgrb with ees in softer bands . we strongly encourage soft x - ray survey facilities like wide - field maxi , which can provide a useful electromagnetic counterpart to gws from compact binary mergers with an angular resolution of @xmath12 . such ee counterparts are also important in terms of time domain astronomy since they would be observed only @xmath328 after the mergers . if a larger detection rate as eq . ( [ eq : rate_soft_ee ] ) is realized , a statistical technique using a stacking approach might also be possible for the detection of gws , with the aid of soft ee counterparts . in our scenario , an ee duration of a few @xmath329 is attributed to a relatively small disk viscosity of @xmath145 and the effect of the disk spreading during the accretion . on the other hand , such a relatively long duration also may be realized if the disk accretion is suspended , but still the rotation energy of the bh is extracted via the interaction between the bh and the disk magnetosphere@xcite . so far , we have focused on the emission mechanism of ees , and not discussed that of the initial spike . in our picture , the initial spike can be provided by the initial inhomogeneity in the ejecta . if there is a direction with low column density , either the bz jet , the neutrino - antineutrino pair annihilation jet , or the magnetic tower jet would cause the initial spike . whatever origin of the initial spike is , our point here is that the major component of sgrbs can be the ees in terms of the energetics . if a half opening angle of the outflow responsible for the initial spike is @xmath330 , the total energy of the initial spike can be two to three orders of magnitude smaller than that of the ees . relatively soft ees without initial spikes might already have been detected by e.g. , swift , but misidentified as other types of events . one might think that soft ees should have already been detected by maxi , though the rapid sky sweeping ( @xmath331 ) makes it difficult to identify the @xmath1 emissions . our scenario can be more clearly tested by future soft x - ray observations . we note that the pressure in eq.(29 ) is a factor of five larger than that estimated by the general relativistic calculation by @xcite , which is due to our newtonian treatment . this causes a factor of five overestimate of the bz power through eq.(44 ) . to take into account this fact as well as the difference between our newtonian dynamics and the general relativistic one , we add a new phenomenological parameter @xmath332 in eq.(44 ) as @xmath333 fig . 5 shows the total energy of a bz jet as a function of @xmath63 for @xmath127 , @xmath128 , @xmath334 , and @xmath39 for three values of @xmath332 . the total energy is obtained by integrating the bz luminosity over time with a the mass accretion rate that is larger than the ignition rate of neutrino cooling , which is determined by @xcite for @xmath141 ( eq . 42 of their paper ) . the dashed lines show the rotational energy of a bh with @xmath39 and @xmath141 , respectively . for low @xmath63 , the figure shows that either the effect of back reaction is needed , @xmath332 is small or the disk accretion is suspended for a while as discussed in previous paragraphs in this paper . therefore it is urgent to undertake more precise analysis for more reliable quantitative predictions . nevertheless , qualitatively our newtonian model of ees presented in this paper is worthwhile . in the present paper , we treated the accretion disk in newtonian gravity for a constant accretion rate at first , then required a finite total disk mass and the total angular momentum . then , we solved the evolution of the size of the disk and the accretion rate . this treatment should be regarded as a crude approximation , and we should compare our results with the numerical simulations . unfortunately , we could not find numerical simulations of the evolution of a finite sized accretion disk either in kerr or schwarzschild metric in the literature . one of the basic problems here is the relativistic treatment of the @xmath63 viscosity . a simple generalization of the navier stokes equation as the one in the text book , for example , _ fluid mechanics _ ( chapter 127 ) by landau & lifshitz violates the causality @xcite , i.e. , the information propagates with a speed faster than the light velocity . the reason is simple and clear . let us consider the non - relativistic one - dimensional diffusion equation for some quantity @xmath335 , @xmath336 where @xmath337 is the diffusion constant . if we set the delta function source as @xmath338 the solution for @xmath339 is expressed as @xmath340 the above solution clearly shows that the initial disturbance at @xmath341 and @xmath342 propagates to any @xmath100 even for any very small value of @xmath343 , which means the causality is violated , i.e , the information propagates with infinite speed . the simple rule of changing the non - relativistic equation into a general relativistic one such as changing the derivative to the covariant derivative and using the projection of the tensor with @xmath344 does not help to guarantee the causality . we need to add several new terms with undetermined parameters to the basic equations such as done by @xcite . if we can start from the general relativistic boltzmann equation , there is no problem as for the causality in principle . however , in practice , we should treat the distribution function that depends on three coordinates and three momenta , but it is beyond the ability of the present computer power to simulate such a problem . one may think that the general relativistic resistive mhd simulations in three dimension are enough to solve this problem since the causality is not violated in such a system . however , the boltzmann equation should be solved anyway since the gyration radius of the proton is typically comparable to the mean free path of p - n collision at around the isco . the authors thank kunihito ioka , tsvi piran , norita kawanaka , kenta hotokezaka , hiroki nagakura , koutaro kyutoku , peter meszaros , peter veres , shuichiro inutsuka , makoto takamoto , and sanemichi takahashi for useful comments . this work is supported in part by the grant - in - aid from the ministry of education , culture , sports , science and technology ( mext ) of japan , no.23540305 ( t.n . ) , no.24103006 ( t.n . ) , no.23840023 ( y.s . ) and jsps fellowship ( k.k . ) . | we investigate the possible origin of extended emissions ( ees ) of short gamma - ray bursts with an isotropic energy of @xmath0 and a duration of a few 10 s to @xmath1 , based on a compact binary ( neutron star ( ns)-ns or ns - black hole ( bh ) ) merger scenario .
we analyze the evolution of magnetized neutrino - dominated accretion disks of mass @xmath2 around bhs formed after the mergers , and estimate the power of relativistic outflows via the blandford - znajek ( bz ) process .
we show that a rotation energy of the bh up to @xmath3 can be extracted with an observed time scale of @xmath4 with a relatively small disk viscosity parameter of @xmath5 .
such a bz power dissipates by clashing with non - relativistic pre - ejected matter of mass @xmath6 , and forms a mildly relativistic fireball .
we show that the dissipative photospheric emissions from such fireballs are likely in the soft x - ray band ( @xmath7 ) for @xmath8 possibly in ns - ns mergers , and in the bat band ( @xmath9 ) for @xmath10 possibly in ns - bh mergers . in the former case
, such soft ees can provide a good chance of @xmath11 for simultaneous detections of the gravitational waves with a @xmath12 angular resolution by soft x - ray survey facilities like wide - field maxi . here
, @xmath13 is the beaming factor of the soft ees and @xmath14 is the ns - ns merger rate detectable by advanced ligo , advanced virgo , and kagra . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
in a typical instance of a combinatorial optimization problem the underlying constraints model a static application frozen in one time step . in many applications however , one needs to solve instances of the combinatorial optimization problem that changes over time . while this is naturally handled by re - solving the optimization problem in each time step separately , changing the solution one holds from one time step to the next often incurs a transition cost . consider , for example , the problem faced by a vendor who needs to get supply of an item from @xmath10 different producers to meet her demand . on any given day , she could get prices from each of the producers and pick the @xmath10 cheapest ones to buy from . as prices change , this set of the @xmath10 cheapest producers may change . however , there is a fixed cost to starting and/or ending a relationship with any new producer . the goal of the vendor is to minimize the sum total of these two costs : an `` acquisition cost '' @xmath11 to be incurred each time she starts a new business relationship with a producer , and a per period cost @xmath12 of buying in period @xmath2 from the each of the @xmath10 producers that she picks in this period , summed over @xmath13 time periods . in this work we consider a generalization of this problem , where the constraint `` pick @xmath10 producers '' may be replaced by a more general combinatorial constraint . it is natural to ask whether simple combinatorial problems for which the one - shot problem is easy to solve , as the example above is , also admit good algorithms for the multistage version . the first problem we study is the _ multistage matroid maintenance _ problem ( ) , where the underlying combinatorial constraint is that of maintaining a base of a given matroid in each period . in the example above , the requirement the vendor buys from @xmath10 different producers could be expressed as optimizing over the @xmath14uniform matroid . in a more interesting case one may want to maintain a spanning tree of a given graph at each step , where the edge costs @xmath12 change over time , and an acquisition cost of @xmath11 has to paid every time a new edge enters the spanning tree . ( a formal definition of the problem appears in section [ sec : formal - defs ] . ) while our emphasis is on the online problem , we will mention results for the offline version as well , where the whole input is given in advance . a first observation we make is that if the matroid in question is allowed to be different in each time period , then the problem is hard to approximate to any non - trivial factor ( see section [ sec : time - varying ] ) even in the offline case . we therefore focus on the case where the same matroid is given at each time period . thus we restrict ourselves to the case when the matroid is the same for all time steps . to set the baseline , we first study the offline version of the problem ( in section [ sec : offline ] ) , where all the input parameters are known in advance . we show an lp - rounding algorithm which approximates the total cost up to a logarithmic factor . this approximation factor is no better than that using a simple greedy algorithm , but it will be useful to see the rounding algorithm , since we will use its extension in the online setting . we also show a matching hardness reduction , proving that the problem is hard to approximate to better than a logarithmic factor ; this hardness holds even for the special case of spanning trees in graphs . we then turn to the online version of the problem , where in each time period , we learn the costs @xmath12 of each element that is available at time @xmath2 , and we need to pick a base @xmath15 of the matroid for this period . we analyze the performance of our online algorithm in the competitive analysis framework : i.e. , we compare the cost of the online algorithm to that of the optimum solution to the offline instance thus generated . in section [ sec : online ] , we give an efficient randomized @xmath16-competitive algorithm for this problem against any oblivious adversary ( here @xmath17 is the universe for the matroid and @xmath6 is the rank of the matroid ) , and show that no polynomial - time online algorithm can do better . we also show that the requirement that the algorithm be randomized is necessary : any deterministic algorithm must incur an overhead of @xmath18 , even for the simplest of matroids . our results above crucially relied on the properties of matriods , and it is natural to ask if we can handle more general set systems , e.g. , @xmath19-systems . in section [ sec : matchings ] , we consider the case where the combinatorial object we need to find each time step is a perfect matching in a graph . somewhat surprisingly , the problem here is significantly harder than the matroid case , even in the offline case . in particular , we show that even when the number of periods is a constant , no polynomial time algorithm can achieve an approximation ratio better than @xmath20 for any constant @xmath21 . we first show that the problem , which is a packing - covering problem , can be reduced to the analogous problem of maintaining a spanning set of a matroid . we call the latter the _ multistage spanning set maintenance _ ( ) problem . while the reduction itself is fairly clean , it is surprisingly powerful and is what enables us to improve on previous works . the problem is a covering problem , so it admits better approximation ratios and allows for a much larger toolbox of techniques at our disposal . we note that this is the only place where we need the matroid to not change over time : our algorithms for work when the matroids change over time , and even when considering matroid intersections . the problem is then further reduced to the case where the holding cost of an element is in @xmath22 , this reduction simplifies the analysis . in the offline case , we present two algorithms . we first observe that a greedy algorithm easily gives an @xmath23-approximation . we then present a simple randomized rounding algorithm for the linear program . this is analyzed using recent results on contention resolution schemes @xcite , and gives an approximation of @xmath24 , which can be improved to @xmath25 when the acquisition costs are uniform . this lp - rounding algorithm will be an important constituent of our algorithm for the online case . for the online case we again use that the problem can be written as a covering problem , even though the natural lp formulation has both covering and packing constraints . phrasing it as a covering problem ( with box constraints ) enables us to use , as a black - box , results on online algorithms for the fractional problem @xcite . this formulation however has exponentially many constraints . we handle that by showing a method of adaptively picking violated constraints such that only a small number of constraints are ever picked . the crucial insight here is that if @xmath26 is such that @xmath27 is not feasible , then @xmath26 is at least @xmath28 away in @xmath29 distance from any feasible solution ; in fact there is a single constraint that is violated to an extent half . this insight allows us to make non - trivial progress ( using a natural potential function ) every time we bring in a constraint , and lets us bound the number of constraints we need to add until constraints are satisfied by @xmath27 . our work is related to several lines of research , and extends some of them . the paging problem is a special case of where the underlying matroid is a uniform one . our online algorithm generalizes the @xmath30-competitive algorithm for weighted caching @xcite , using existing online lp solvers in a black - box fashion . going from uniform to general matroids loses a logarithmic factor ( after rounding ) , we show such a loss is unavoidable unless we use exponential time . the problem is also a special case of classical metrical task systems @xcite ; see @xcite for more recent work . the best approximations for metrical task systems are poly - logarithmic in the size of the metric space . in our case the metric space is specified by the total number of bases of the matroid which is often exponential , so these algorithms only give a trivial approximation . in trying to unify online learning and competitive analysis , buchbinder et al . @xcite consider a problem on matroids very similar to ours . the salient differences are : ( a ) in their model all acquisition costs are the same , and ( b ) they work with fractional bases instead of integral ones . they give an @xmath31-competitive algorithm to solve the fractional online lp with uniform acquisition costs ( among other unrelated results ) . our online lp solving generalizes their result to arbitrary acquisition costs . they leave open the question of getting integer solutions online ( seffi naor , private communication ) , which we present in this work . in a more recent work , buchbinder , chen and naor @xcite use a regularization approach to solving a broader set of fractional problems , but once again can do not get integer solutions in a setting such as ours . shachnai et al . @xcite consider `` reoptimization '' problems : given a starting solution and a new instance , they want to balance the transition cost and the cost on the new instance . this is a two - timestep version of our problem , and the short time horizon raises a very different set of issues ( since the output solution does not need to itself hedge against possible subsequent futures ) . they consider a number of optimization / scheduling problems in their framework . cohen et al . @xcite consider several problems in the framework of the stability - versus - fit tradeoff ; e.g. , that of finding `` stable '' solutions which given the previous solution , like in reoptimization , is the current solution that maximizes the quality minus the transition costs . they show maintaining stable solutions for matroids becomes a repeated two - stage reoptimization problem ; their problem is poly - time solvable , whereas matroid problems in our model become np - hard . the reason is that the solution for two time steps does not necessarily lead to a base from which it is easy to move in subsequent time steps , as our hardness reduction shows . they consider a multistage offline version of their problem ( again maximizing fit minus stability ) which is very similar in spirit and form to our ( minimization ) problem , though the minus sign in the objective function makes it difficult to approximate in cases which are not in poly - time . in dynamic steiner tree maintenance @xcite where the goal is to maintain an approximately optimal steiner tree for a varying instance ( where terminals are added ) while changing few edges at each time step . in dynamic load balancing @xcite one has to maintain a good scheduling solution while moving a small number of jobs around . the work on lazy experts in the online prediction community @xcite also deals with similar concerns . there is also work on `` leasing '' problems @xcite : these are optimization problems where elements can be obtained for an interval of any length , where the cost is concave in the lengths ; the instance changes at each timestep . the main differences are that the solution only needs to be feasible at each timestep ( i.e. , the holding costs are @xmath32 ) , and that any element can be leased for any length @xmath33 of time starting at any timestep for a cost that depends only on @xmath33 , which gives these problems a lot of uniformity . in turn , these leasing problems are related to `` buy - at - bulk '' problems . given reals @xmath34 for elements @xmath35 , we will use @xmath36 for @xmath37 to denote @xmath38 . we denote @xmath39 by @xmath40 $ ] . we assume basic familiarity with matroids : see , e.g. , @xcite for a detailed treatment . given a matroid @xmath41 , a _ base _ is a maximum cardinality independent set , and a _ spanning set _ is a set @xmath42 such that @xmath43 ; equivalently , this set contains a base within it . the _ span _ of a set @xmath44 is @xmath45 . the _ matroid polytope _ @xmath46 is defined as @xmath47 . the _ base polytope _ @xmath48 . we will sometimes use @xmath5 to denote @xmath49 and @xmath6 to denote the rank of the matroid . an instance of the _ multistage matroid maintenance _ ( ) problem consists of a matroid @xmath41 , an _ acquisition cost _ @xmath50 for each @xmath35 , and for every timestep @xmath51 $ ] and element @xmath35 , a _ holding cost _ cost @xmath12 . the goal is to find bases @xmath52}$ ] to minimize @xmath53 where we define @xmath54 . a related problem is the _ multistage spanning set maintenance _ ( ) problem , where we want to maintain a spanning set @xmath55 at each time , and cost of the solution @xmath56}$ ] ( once again with @xmath57 ) is @xmath58 the following lemma shows the equivalence of maintaining bases and spanning sets . this enables us to significantly simplify the problem and avoid the difficulties faced by previous works on this problem . [ lem : pack - cover ] for matroids , the optimal solutions to and have the same costs . clearly , any solution to is also a solution to , since a base is also a spanning set . conversely , consider a solution @xmath59 to . set @xmath60 to any base in @xmath61 . given @xmath62 , start with @xmath63 , and extend it to any base @xmath64 of @xmath15 . this is the only step where we use the matroid properties indeed , since the matroid is the same at each time , the set @xmath63 remains independent at time @xmath2 , and by the matroid property this independent set can be extended to a base . observe that this process just requires us to know the base @xmath65 and the set @xmath15 , and hence can be performed in an online fashion . we claim that the cost of @xmath66 is no more than that of @xmath67 . indeed , @xmath68 , because @xmath69 . moreover , let @xmath70 , we pay @xmath71 for these elements we just added . to charge this , consider any such element @xmath72 , let @xmath73 be the time it was most recently added to the cover i.e . , @xmath74 for all @xmath75 $ ] , but @xmath76 . the solution paid for including @xmath77 at time @xmath78 , and we charge our acquisition of @xmath77 into @xmath64 to this pair @xmath79 . it suffices to now observe that we will not charge to this pair again , since the procedure to create @xmath80 ensures we do not drop @xmath77 from the base until it is dropped from @xmath15 itself the next time we pay an addition cost for element @xmath77 , it would have been dropped and added in @xmath81 as well . hence it suffices to give a good solution to the problem . we observe that the proof above uses the matroid property crucially and would not hold , e.g. , for matchings . it also requires that the _ same _ matroid be given at all time steps . also , as noted above , the reduction is online : the instance is the same , and given an solution it can be transformed online to a solution to . we will find it convenient to think of an instance of as being a matroid @xmath82 , where each element only has an acquisition cost @xmath83 , and it has a lifetime @xmath84 $ ] . there are no holding costs , but the element @xmath77 can be used in spanning sets only for timesteps @xmath85 . or one can equivalently think of holding costs being zero for @xmath86 and @xmath87 otherwise . _ an offline exact reduction . _ the translation is the natural one : given instance @xmath88 of , create elements @xmath89 for each @xmath90 and @xmath91 , with acquisition cost @xmath92 , and interval @xmath93 $ ] . ( the matroid is extended in the natural way , where all the elements @xmath89 associated with @xmath77 are parallel to each other . ) the equivalence of the original definition of and this interval view is easy to verify . _ an online approximate reduction . _ observe that the above reduction created at most @xmath94 copies of each element , and required knowledge of all the costs . if we are willing to lose a constant factor in the approximation , we can perform a reduction to the interval model in an _ online _ fashion as follows . for element @xmath35 , define @xmath95 , and create many parallel copies @xmath96 of this element ( modifying the matroid appropriately ) . now the @xmath97 interval for @xmath77 is @xmath98 $ ] , where @xmath99 is set to @xmath100 in case @xmath101 , else it is set to the _ largest _ time such that the total holding costs @xmath102 for this interval @xmath103 $ ] is at most @xmath11 . this interval @xmath104 is associated with element @xmath105 , which is only available for this interval , at cost @xmath106 . a few salient points about this reduction : the intervals for an original element @xmath77 now partition the entire time horizon @xmath40 $ ] . the number of elements in the modified matroid whose intervals contain any time @xmath2 is now only @xmath107 , the same as the original matroid ; each element of the modified matroid is only available for a single interval . moreover , the reduction can be done online : given the past history and the holding cost for the current time step @xmath2 , we can ascertain whether @xmath2 is the beginning of a new interval ( in which case the previous interval ended at @xmath108 ) and if so , we know the cost of acquiring a copy of @xmath77 for the new interval is @xmath109 . it is easy to check that the optimal cost in this interval model is within a constant factor of the optimal cost in the original acquisition / holding costs model . given the reductions of the previous section , we can focus on the problem . being a covering problem , is conceptually easier to solve : e.g. , we could use algorithms for submodular set cover @xcite with the submodular function being the sum of ranks at each of the timesteps , to get an @xmath23 approximation . in section [ sec : greedy ] , we give a dual - fitting proof of the performance of the greedy algorithm . here we give an lp - rounding algorithm which gives an @xmath110 approximation ; this can be improved to @xmath25 in the common case where all acquisition costs are unit . ( while the approximation guarantee is no better than that from submodular set cover , this lp - rounding algorithm will prove useful in the online case in section [ sec : online ] ) . finally , the hardness results of section [ sec : hardness - offline ] show that we can not hope to do much better than these logarithmic approximations . we now consider an lp - rounding algorithm for the problem ; this will generalize to the online setting , whereas it is unclear how to extend the greedy algorithm to that case . for the lp rounding , we use the standard definition of the problem to write the following lp relaxation . @xmath111 it remains to round the solution to get a feasible solution to ( i.e. , a spanning set @xmath15 for each time ) with expected cost at most @xmath31 times the lp value , since we can use lemma [ lem : pack - cover ] to convert this to a solution for at no extra cost . the following lemma is well - known ( see , e.g. @xcite ) . we give a proof for completeness . [ lem : alon ] for a fractional base @xmath112 , let @xmath113 be the set obtained by picking each element @xmath35 independently with probability @xmath114 . then @xmath115 \geq r(1 - 1/e)$ ] . we use the results of chekuri et al . @xcite ( extending those of chawla et al . @xcite ) on so - called contention resolution schemes . formally , for a matroid @xmath82 , they give a randomized procedure @xmath116 that takes the random set @xmath113 and outputs an independent set @xmath117 in @xmath82 , such that @xmath118 , and for each element @xmath77 in the support of @xmath119 , @xmath120 \geq ( 1 - 1/e)$ ] . ( they call this a @xmath121-balanced cr scheme . ) now , we get @xmath122 & \geq { { \mathbf{e } } } [ { \textsf{rank}}(\pi_z(r(z ) ) ) ] = \sum_{e \in \text{supp}(z ) } \pr [ e \in \pi_z(r(z ) ) ] \\ & = \sum_{e \in \text{supp}(z ) } \pr [ e \in \pi_z(r(z ) ) \mid e \in r(z ) ] \cdot \pr [ e \in r(z ) ] \\ & \geq \sum_{e \in \text{supp}(z ) } ( 1 - 1/e ) \cdot z_e = r(1 - 1/e ) . \end{aligned}\ ] ] the first inequality used the fact that @xmath117 is a subset of @xmath113 , the following equality used that @xmath117 is independent with probability 1 , the second inequality used the property of the cr scheme , and the final equality used the fact that @xmath119 was a fractional base . [ thm : lp - round ] any fractional solution can be randomly rounded to get solution to with cost @xmath24 times the fractional value , where @xmath6 is the rank of the matroid and @xmath13 the number of timesteps . set @xmath123 . for each element @xmath35 , choose a random threshold @xmath124 independently and uniformly from the interval @xmath125 $ ] . for each @xmath126 , define the set @xmath127 ; if @xmath128 does not have full rank , augment its rank using the cheapest elements according to @xmath129 to obtain a full rank set @xmath15 . since @xmath130 = \min\ { l\cdot z_t(e ) , 1\}$ ] , the cost @xmath131 . moreover , @xmath132 exactly when @xmath124 satisfies @xmath133 , which happens with probability at most @xmath134 hence the expected acquisition cost for the elements newly added to @xmath128 is at most @xmath135 . finally , we have to account for any elements added to extend @xmath128 to a full - rank set @xmath15 . [ lem : rand - round ] for any fixed @xmath51 $ ] , the set @xmath128 contains a basis of @xmath82 with probability at least @xmath136 . the set @xmath128 is obtained by threshold rounding of the fractional base @xmath137 as above . instead , consider taking @xmath138 different samples @xmath139 , where each sample is obtained by including each element @xmath35 independently with probability @xmath140 ; let @xmath141 . it is easy to check that @xmath142 \leq \pr [ { \textsf{rank}}(\widehat{s}_t ) = r]$ ] , so it suffices to give a lower bound on the former expression . for this , we use lemma [ lem : alon ] : the sample @xmath143 has expected rank @xmath144 , and using reverse markov , it has rank at least @xmath145 with probability at least @xmath146 . now focusing on the matroid @xmath147 obtained by contracting elements in @xmath148 ( which , say , has rank @xmath149 ) , the same argument says the set @xmath150 has rank @xmath151 with probability at least @xmath152 , etc . proceeding in this way , the probability that the rank of @xmath13 is less than @xmath6 is at most the probability that we see fewer than @xmath153 heads in @xmath154 flips of a coin of bias @xmath152 . by a chernoff bound , this is at most @xmath155 . now if the set @xmath128 does not have full rank , the elements we add have cost at most that of the min - cost base under the cost function @xmath156 , which is at most the optimum value for ( [ eq : lp2 ] ) . ( we use the fact that the lp is exact for a single matroid , and the global lp has cost at least the single timestep cost . ) this happens with probability at most @xmath157 , and hence the total expected cost of augmenting @xmath128 over all @xmath13 timesteps is at most @xmath158 times the lp value . this proves the main theorem . again , this algorithm for works with different matroids at each timestep , and also for intersections of matroids . to see this observe that the only requirements from the algorithm are that there is a separation oracle for the polytope and that the contention resolution scheme works . in the case of @xmath14matroid intersection , if we pay an extra @xmath30 penalty in the approximation ratio we have that the probability a rounded solution does not contain a base is @xmath159 so we can take a union bound over the multiple matroids . 0 we can replace the dependence on @xmath13 by a term that depends only on the variance in the acquisition costs . let us divide the period @xmath160 into `` epochs '' , where an epoch is an interval @xmath161 for @xmath162 such that the total fractional acquisition cost @xmath163 . we can afford to build a brand new tree at the beginning of each epoch and incur an acquisition cost of at most the rank @xmath6 , which we can charge to the lp s fractional acquisition cost in the epoch . by theorem [ thm : lp - round ] , naively applying the rounding algorithm to each epoch independently gives a guarantee of @xmath164 , where @xmath165 is the maximum length of an epoch . now we should be able to use the argument from the online section that says that we can ignore steps where the total movement is smaller than half . thus @xmath165 can be assumed to be @xmath166 . more details to be added once we consistentize notation . in fact , if we define epoch to be a period of acquisition cost @xmath167 , then the at least half means movement cost at least @xmath168 . thus the epoch only has @xmath169 relevant steps in it , so we get log of that . for the special case where all the acquisition costs @xmath11 are all the same , this implies we get rid of the @xmath13 term in the lp rounding , and get an @xmath25-approximation . when the ratio of the maximum to the minimum acquisition cost is small , we can improve the approximation factor above . more specifically , we show that essentially the same randomized rounding algorithm ( with a different choice of @xmath138 ) gives an approximation ratio of @xmath170 . we defer the argument to section [ sec : just - logr ] , as it needs some additional definitions and results that we present in the online section . [ [ an - improvement - avoiding - the - dependence - on - t.-1 ] ] an improvement : avoiding the dependence on @xmath13 . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + when the ratio of the maximum to the minimum acquisition cost is small , we can improve the approximation factor above . more specifically , we show that essentially the same randomized rounding algorithm ( with a different choice of @xmath138 ) gives an approximation ratio of @xmath170 . we defer the argument to section [ sec : just - logr ] , as it needs some additional definitions and results that we present in the online section . we defer the hardness proof to appendix [ app : offline ] , which shows that the and problems are np - hard to approximate better than @xmath171 even for graphical matroids . an integrality gap of @xmath172 appears in appendix [ sec : int - gap - matroids ] . [ thm : matr - hard ] the and problems are np - hard to approximate better than @xmath173 even for graphical matroids . we give a reduction from set cover to the problem for graphical matroids . given an instance @xmath174 of set cover , with @xmath175 sets and @xmath176 elements , we construct a graph as follows . there is a special vertex @xmath6 , and @xmath5 set vertices ( with vertices @xmath177 for each set @xmath178 ) . there are @xmath5 edges @xmath179 which all have inclusion weight @xmath180 and per - time cost @xmath181 for all @xmath2 . all other edges will be zero cost short - term edges as given below . in particular , there are @xmath182 timesteps . in timestep @xmath183 $ ] , define subset @xmath184 to be vertices corresponding to sets containing element @xmath185 . we have a set of edges @xmath186 for all @xmath187 , and all edges @xmath188 for @xmath189 . all these edges have zero inclusion weight @xmath11 , and are only alive at time @xmath190 . ( note this creates a graph with parallel edges , but this can be easily fixed by subdividing edges . ) in any solution to this problem , to connect the vertices in @xmath191 to @xmath6 , we must buy some edge @xmath192 for some @xmath193 . this is true for all @xmath190 , hence the root - set edges we buy correspond to a set cover . moreover , one can easily check that if we acquire edges @xmath194 such that the sets @xmath195 form a set cover , then we can always augment using zero cost edges to get a spanning tree . since the only edges we pay for are the @xmath194 edges , we should buy edges corresponding to a min - cardinality set cover , which is hard to approximate better than @xmath196 . finally , that the number of time periods is @xmath182 , and the rank of the matroid is @xmath197 for these hard instances . this gives us the claimed hardness . we now turn to solving in the online setting . in this setting , the acquisition costs @xmath11 are known up - front , but the holding costs @xmath12 for day @xmath2 are not known before day @xmath2 . since the equivalence given in lemma [ lem : pack - cover ] between and holds even in the online setting , we can just work on the problem . we show that the online problem admits an @xmath198-competitive ( oblivious ) randomized algorithm . to do this , we show that one can find an @xmath199-competitive fractional solution to the linear programming relaxation in section [ sec : offline ] , and then we round this lp relaxation online , losing another logarithmic factor . again , we work in the interval model outlined in section [ sec : intervals ] . recall that in this model , for each element @xmath77 there is a unique interval @xmath200 $ ] during which it is alive . the element @xmath77 has an acquisition cost @xmath11 , no holding costs . once an element has been acquired ( which can be done at any time during its interval ) , it can be used at all times in that interval , but not after that . in the online setting , at each time step @xmath2 we are told which intervals have ended ( and which have not ) ; also , which new elements @xmath77 are available starting at time @xmath2 , along with their acquisition costs @xmath11 . of course , we do not know when its interval @xmath201 will end ; this information is known only once the interval ends . we will work with the same lp as in section [ sec : lp - round ] , albeit now we have to solve it online . the variable @xmath202 is the indicator for whether we acquire element @xmath77 . @xmath203 \notag\end{aligned}\ ] ] note that this is not a packing or covering lp , which makes it more annoying to solve online . hence we consider a slight reformulation . let @xmath204 denote the _ spanning set polytope _ defined as the convex hull of the full - rank ( a.k.a . spanning ) sets @xmath205 . since each spanning set contains a base , we can write the constraints of ( [ eq:3 ] ) as : @xmath206 here we define @xmath207 to be the vector derived from @xmath208 by zeroing out the @xmath202 values for @xmath209 . it is known that the polytope @xmath204 can be written as a ( rather large ) set of covering constraints . indeed , @xmath210 , where @xmath211 is the dual matroid for @xmath82 . since the rank function of @xmath212 is given by @xmath213 , it follows that ( [ eq:4 ] ) can be written as @xmath214 thus we get a covering lp with `` box '' constraints over @xmath17 . the constraints can be presented one at a time : in timestep @xmath2 , we present all the covering constraints corresponding to @xmath215 . we remark that the newer machinery of @xcite may be applicable to [ eq : coveringconstraints ] . we next show that a simpler approach suffices will be useful in improving the rounding algorithm . ] . the general results of buchbinder and naor @xcite ( and its extension to row - sparse covering problems by @xcite ) imply a deterministic algorithm for fractionally solving this linear program online , with a competitive ratio of @xmath216 . however , this is not yet a polynomial - time algorithm , the number of constraints for each timestep being exponential . we next give an adaptive algorithm to generate a small yet sufficient set of constraints . [ [ solving - the - lp - online - in - polynomial - time.-1 ] ] solving the lp online in polynomial time . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + given a vector @xmath217^e$ ] , define @xmath218 as follows : @xmath219 clearly , @xmath220 and @xmath221^e$ ] . we next describe the algorithm for generating covering constraints in timestep @xmath2 . recall that @xcite give us an online algorithm @xmath222 for solving a fractional covering lp with box constraints ; we use this as a black - box . ( this lp solver only raises variables , a fact we will use . ) in timestep @xmath2 , we adaptively select a small subset of the covering constraints from ( [ eq : coveringconstraints ] ) , and present it to @xmath222 . moreover , given a fractional solution returned by @xmath222 , we will need to massage it at the end of timestep @xmath2 to get a solution satisfying all the constraints from ( [ eq : coveringconstraints ] ) corresponding to @xmath2 . let @xmath208 be the fractional solution to ( [ eq : coveringconstraints ] ) at the end of timestep @xmath108 . now given information about timestep @xmath2 , in particular the elements in @xmath215 and their acquisition costs , we do the following . given @xmath208 , we construct @xmath218 and check if @xmath223 , as one can separate for @xmath204 . if @xmath223 , then @xmath218 is feasible and we do not need to present any new constraints to @xmath222 , and we return @xmath218 . if not , our separation oracle presents an @xmath42 such that the constraint @xmath224 is violated . we present the constraint corresponding to @xmath42 to @xmath222 to get an updated @xmath208 , and repeat until @xmath218 is feasible for time @xmath2 . ( since @xmath222 only raises variables and we have a covering lp , the solution remains feasible for past timesteps . ) we next argue that we do not need to repeat this loop more than @xmath225 times . [ lem : farfromfeasible ] if for some @xmath208 and the corresponding @xmath218 , the constraint @xmath226 is violated . then @xmath227 let @xmath228 and let @xmath229 . let @xmath230 denote @xmath231 . thus @xmath232 since both @xmath233 and @xmath234 are integers , it follows that @xmath235 . on the other hand , for every @xmath236 , and thus @xmath237 . consequently @xmath238 finally , for any @xmath239 , @xmath240 , so the claim follows . the algorithm @xmath222 updates @xmath208 to satisfy the constraint given to it , and lemma [ lem : farfromfeasible ] implies that each constraint we give to it must increase @xmath241 by at least @xmath28 . the translation to the interval model ensures that the number of elements whose intervals contain @xmath2 is at most latexmath:[$|e_t| \leq time @xmath2 is at most @xmath243 . we summarize the discussion of this section in the following theorem . [ thm : lp - solve ] there is a polynomial - time online algorithm to compute an @xmath244-approximate solution to ( [ eq:3 ] ) . we observe that the solution to this linear program can be trivially transformed to one for the lp in section [ sec : lp - round ] . finally , the randomized rounding algorithm of section [ sec : lp - round ] can be implemented online by selecting a threshold @xmath245 $ ] the beginning of the algorithm , where @xmath246 and selecting element @xmath77 whenever @xmath247 exceeds @xmath248 : here we use the fact that the online algorithm only ever raises @xmath202 values , and this rounding algorithm is monotone . rerandomizing in case of failure gives us an expected cost of @xmath24 times the lp solution , and hence we get an @xmath249-competitive algorithm . the dependence on the time horizon @xmath13 is unsatisfactory in some settings , but we can do better using lemma [ lem : farfromfeasible ] . recall that the @xmath251-factor loss in the rounding follows from the naive union bound over the @xmath13 time steps . we can argue that when @xmath252 is small , we can afford for the rounding to fail occasionally , and charge it to the acquisition cost incurred by the linear program . the details appear in appendix [ sec : just - logr ] . the dependence on the time horizon @xmath13 is unsatisfactory in some settings , but we can do better using lemma [ lem : farfromfeasible ] . recall that the @xmath251-factor loss in the rounding follows from the naive union bound over the @xmath13 time steps . we now argue that when @xmath252 is small , we can afford for the rounding to fail occasionally , and charge it to the acquisition cost incurred by the linear program . let us divide the period @xmath253 $ ] into disjoint `` epochs '' , where an epoch ( except for the last ) is an interval @xmath161 for @xmath162 such that the total fractional acquisition cost @xmath254 . thus an epoch is a minimal interval where the linear program spends acquisition cost @xmath255 $ ] , so that we can afford to build a brand new tree once in each epoch and can charge it to the lp s fractional acquisition cost in the epoch . naively applying theorem [ thm : lp - round ] to each epoch independently gives us a guarantee of @xmath256 , where @xmath165 is the maximum length of an epoch . however , an epoch can be fairly long if the lp solution changes very slowly . we break up each epoch into phases , where each phase is a maximal subsequence such that the lp incurs acquisition cost at most @xmath257 ; clearly the epoch can be divided into at most @xmath258 disjoint phases . for a phase @xmath259 $ ] , let @xmath260}$ ] denote the solution defined as @xmath260}(e ) = \min_{t\in [ t_1,t_2 ] } z_t(e)$ ] . the definition of the phase implies that for any @xmath261 $ ] , the @xmath262 difference @xmath263 } - z_t\|_1 \leq \frac{1}{4}$ ] . now lemma [ lem : farfromfeasible ] implies that @xmath264}$ ] is in @xmath265 , where @xmath266 is defined as in ( [ eq:5 ] ) . suppose that in the randomized rounding algorithm , we pick the threshold @xmath267 $ ] for @xmath268 . let @xmath269}$ ] be the event that the rounding algorithm applied to @xmath260}$ ] gives a spanning set . since @xmath264 } \leq 2z_{[t_1,t_2]}$ ] is in @xmath270 for a phase @xmath259 $ ] , lemma [ lem : rand - round ] implies that the event @xmath269}$ ] occurs with probability @xmath271 . moreover , if @xmath269}$ ] occurs , it is easy to see that the randomized rounding solution is feasible for all @xmath272 $ ] . since there are @xmath273 phases within an epoch , the expected number of times that the randomized rounding fails any time during an epoch is @xmath274 . suppose that we rerandomize all thresholds whenever the randomized rounding fails . each rerandomization will cost us at most @xmath275 in expected acquisition cost . since the expected number of times we do this is less than once per epoch , we can charge this additional cost to the @xmath275 acquisition cost incurred by the lp during the epoch . thus we get an @xmath276-approximation . this argument also works for the online case ; hence for the common case where all the acquisition costs are the same , the loss due to randomized rounding is @xmath25 . we can show that any polynomial - time algorithm can not achieve better than an @xmath277 competitive ratio , via a reduction from online set cover . details appear in appendix [ app : sec : hardness - online ] . in the online set cover problem , one is given an instance @xmath278 of set cover , and in time step @xmath2 , the algorithm is presented an element @xmath279 , and is required to pick a set covering it . the competitive ratio of an algorithm on a sequence @xmath280}$ ] is the ratio of the number of sets picked by the algorithm to the optimum set - cover of the instance @xmath281\},{{\mathcal{f}}})$ ] . korman ( * ? ? ? * theorem 2.3.4 ) shows the following hardness for online set cover : there exists a constant @xmath282 such that if there is a ( possibly randomized ) polynomial time algorithm for online set cover with competitive ratio @xmath283 , then @xmath284 . recall that in the reduction in the proof of theorem [ thm : matr - hard ] , the set of long term edges depends only on @xmath285 . the short term edges alone depend on the elements to be covered . it can then we verified that the same approach gives a reduction from online set cover to online . it follows that the online problem does not admit an algorithm with competitive ratio better than @xmath286 unless @xmath284 . in fact this hardness holds even when the end time of each edge is known as soon as it appears , and the only non - zero costs are @xmath287 . we next consider the _ perfect matching maintenance _ ( ) problem where @xmath17 is the set of edges of a graph @xmath288 , and the at each step , we need to maintain a perfect matchings in @xmath289 . * integrality gap . * somewhat surprisingly , we show that the natural lp relaxation has an @xmath290 integrality gap , even for a constant number of timesteps . the lp and the ( very simple ) example appears in appendix [ sec : match - int - gap ] . the natural lp relaxation is : @xmath291 the polytope @xmath292 is now the perfect matching polytope for @xmath289 . [ lem : int - gap ] there is an @xmath290 integrality gap for the problem . consider the instance in the figure , and the following lp solution for 4 time steps . in @xmath293 , the edges of each of the two cycles has @xmath294 , and the cross - cycle edges have @xmath295 . in @xmath296 , we have @xmath297 and @xmath298 , and otherwise it is the same as @xmath293 . @xmath299 and @xmath300 are the same as @xmath293 . in @xmath301 , we have @xmath302 and @xmath303 , and otherwise it is the same as @xmath293 . for each time @xmath2 , the edges in the support of the solution @xmath304 have zero cost , and other edges have infinite cost . the only cost incurred by the lp is the movement cost , which is @xmath158 . consider the perfect matching found at time @xmath305 , which must consist of matchings on both the cycles . ( moreover , the matching in time 3 must be the same , else we would change @xmath290 edges . ) suppose this matching uses exactly one edge from @xmath306 and @xmath307 . then when we drop the edges @xmath308 and add in @xmath309 , we get a cycle on @xmath310 vertices , but to get a perfect matching on this in time @xmath311 we need to change @xmath290 edges . else the matching uses exactly one edge from @xmath306 and @xmath312 , in which case going from time @xmath313 to time @xmath314 requires @xmath290 changes . * hardness . * moreover , in appendix [ app : sec : match - hard ] we show that the perfect matching maintenance problem is very hard to approximate : for any @xmath315 it is np - hard to distinguish instances with cost @xmath316 from those with cost @xmath317 , where @xmath318 is the number of vertices in the graph . this holds even when the holding costs are in @xmath22 , acquisition costs are @xmath319 for all edges , and the number of time steps is a constant . in this section we prove the following hardness result : for any @xmath315 it is np - hard to distinguish instances with cost @xmath316 from those with cost @xmath317 , where @xmath318 is the number of vertices in the graph . this holds even when the holding costs are in @xmath22 , acquisition costs are @xmath319 for all edges , and the number of time steps is a constant . the proof is via reduction from @xmath313-coloring . we assume we are given an instance of @xmath313-coloring @xmath288 where the maximum degree of @xmath289 is constant . it is known that the @xmath313-coloring problem is still hard for graphs with bounded degree ( * ? ? ? * theorem 2 ) . we construct the following gadget @xmath320 for each vertex @xmath321 . ( a figure is given in figure [ fig : gadget ] . ) there are two cycles of length @xmath322 , where @xmath33 is odd . the first cycle ( say @xmath323 ) has three distinguished vertices @xmath324 at distance @xmath33 from each other . the second ( called @xmath325 ) has similar distinguished vertices @xmath326 at distance @xmath33 from each other . there are three more `` interface '' vertices @xmath327 . vertex @xmath328 is connected to @xmath329 and @xmath330 , similarly for @xmath331 and @xmath332 . there is a special `` switch '' vertex @xmath333 , which is connected to all three of @xmath334 . call these edges the _ switch _ edges . due to the two odd cycles , every perfect matching in @xmath320 has the structure that one of the interface vertices is matched to some vertex in @xmath323 , another to a vertex in @xmath325 and the third to the switch @xmath333 . we think of the subscript of the vertex matched to @xmath333 as the color assigned to the vertex @xmath335 . at every odd time step @xmath126 , the only allowed edges are those within the gadgets @xmath336 : i.e. , all the holding costs for edges within the gadgets is zero , and all edges between gadgets have holding costs @xmath87 . this is called the `` steady state '' . at every even time step @xmath2 , for some matching @xmath337 of the graph , we move into a `` test state '' , which intuitively tests whether the edges of a matching @xmath338 have been properly colored . we do this as follows . for every edge @xmath339 , the switch edges in @xmath340 become unavailable ( have infinite holding costs ) . moreover , now we allow some edges that go between @xmath320 and @xmath341 , namely the edge @xmath342 , and the edges @xmath343 for @xmath344 and @xmath345 . note that any perfect matching on the vertices of @xmath346 which only uses the available edges would have to match @xmath342 , and one interface vertex of @xmath320 must be matched to one interface vertex of @xmath341 . moreover , by the structure of the allowed edges , the colors of these vertices must differ . ( the other two interface vertices in each gadget must still be matched to their odd cycles to get a perfect matching . ) since the graph has bounded degree , we can partition the edges of @xmath289 into a constant number of matchings @xmath347 for some @xmath348 ( using vizing s theorem ) . hence , at time step @xmath349 , we test the edges of the matching @xmath350 . the number of timesteps is @xmath351 , which is a constant . . the test - state edges are on the right . ] suppose the graph @xmath289 was indeed @xmath313-colorable , say @xmath352 is the proper coloring . in the steady states , we choose a perfect matching within each gadget @xmath320 so that @xmath353 is matched . in the test state @xmath354 , if some edge @xmath355 is in the matching @xmath338 , we match @xmath342 and @xmath356 . since the coloring @xmath357 was a proper coloring , these edges are present and this is a valid perfect matching using only the edges allowed in this test state . note that the only changes are that for every test edge @xmath358 , the matching edges @xmath359 and @xmath360 are replaced by @xmath342 and @xmath361 . hence the total acquisition cost incurred at time @xmath354 is @xmath362 , and the same acquisition cost is incurred at time @xmath363 to revert to the steady state . hence the total acquisition cost , summed over all the timesteps , is @xmath364 . suppose @xmath289 is not @xmath313-colorable . we claim that there exists vertex @xmath365 such that the interface vertex not matched to the odd cycles is different in two different timesteps i.e . , there are times @xmath366 such that @xmath367 and @xmath185 ( for @xmath345 ) are the states . then the length of the augmenting path to get from the perfect matching at time @xmath368 to the perfect matching at @xmath369 is at least @xmath33 . now if we set @xmath370 , then we get a total acquisition cost of at least @xmath371 in this case . the size of the graph is @xmath372 , so the gap is between @xmath373 and @xmath374 . this proves the claim . in this paper we studied multistage optimization problems : an optimization problem ( think about finding a minimum - cost spanning tree in a graph ) needs to be solved repeatedly , each day a different set of element costs are presented , and there is a penalty for changing the elements picked as part of the solution . hence one has to hedge between sticking to a suboptimal solution and changing solutions too rapidly . we present online and offline algorithms when the optimization problem is maintaining a base in a matroid . we show that our results are optimal under standard complexity - theoretic assumptions . we also show that the problem of maintaining a perfect matching becomes impossibly hard . our work suggests several directions for future research . it is natural to study other combinatorial optimization problems , both polynomial time solvable ones such shortest path and min - cut , as well np - hard ones such as min - max load balancing and bin - packing in this multistage framework with acquisition costs . moreover , the approximability of the _ bipartite _ matching maintenance , as well as matroid intersection maintenance remains open . our hardness results for the matroid problem hold when edges have @xmath375 acquisition costs . the unweighted version where all acquisition costs are equal may be easier ; we currently know no hardness results , or sub - logarithmic approximations for this useful special case . an extension of /problems is to the case when the set of elements remain the same , but the matroids change over time . again the goal in is to maintain a matroid base at each time . [ thm : diff - matrs - wpb ] the problem with different matroids is np - hard to approximate better than a factor of @xmath376 , even for partition matroids , as long as @xmath377 . the reduction is from 3d - matching ( 3 dm ) . an instance of 3 dm has three sets @xmath378 of equal size @xmath379 , and a set of hyperedges @xmath380 . the goal is to choose a set of disjoint edges @xmath381 such that @xmath382 . first , consider the instance of with three timesteps @xmath383 . the universe elements correspond to the edges . for @xmath384 , create a partition with @xmath10 parts , with edges sharing a vertex in @xmath385 falling in the same part . the matroid @xmath386 is now to choose a set of elements with at most one element in each part . for @xmath387 , the partition now corresponds to edges that share a vertex in @xmath388 , and for @xmath389 , edges that share a vertex in @xmath390 . set the movement weights @xmath391 for all edges . if there exists a feasible solution to 3 dm with @xmath10 edges , choosing the corresponding elements form a solution with total weight @xmath10 . if the largest matching is of size @xmath392 , then we must pay @xmath393 extra over these three timesteps . this gives a @xmath10-vs-@xmath394 gap for three timesteps . to get a result for @xmath13 timesteps , we give the same matroids repeatedly , giving matroids @xmath395 at all times @xmath396 $ ] . in the `` yes '' case we would buy the edges corresponding to the 3d matching and pay nothing more than the initial @xmath10 , whereas in the `` no '' case we would pay @xmath397 every three timesteps . finally , the apx - hardness for 3 dm @xcite gives the claim . the time - varying problem does admit an @xmath24 approximation , as the randomized rounding ( or the greedy algorithm ) shows . however , the equivalence of and does not go through when the matroids change over time . the restriction that the matroids vary over time is essential for the np - hardness , since if the partition matroid is the same for all times , the complexity of the problem drops radically . [ thm : partition ] the problem with partition matroids can be solved in polynomial time . the problem can be solved using min - cost flow . indeed , consider the following reduction . create a node @xmath398 for each element @xmath77 and timestep @xmath2 . let the partition be @xmath399 . then for each @xmath400 $ ] and each @xmath401 , add an arc @xmath402 , with cost @xmath403 . add a cost of @xmath12 per unit flow through vertex @xmath398 . ( we could simulate this using edge - costs if needed . ) finally , add vertices @xmath404 and source @xmath405 . for each @xmath406 , add arcs from @xmath177 to all vertices @xmath407 with costs @xmath408 . all these arcs have infinite capacity . now add unit capacity edges from @xmath405 to each @xmath177 , and infinite capacity edges from all nodes @xmath409 to @xmath2 . since the flow polytope is integral for integral capacities , a flow of @xmath6 units will trace out @xmath6 paths from @xmath405 to @xmath2 , with the elements chosen at each time @xmath2 being independent in the partition matroid , and the cost being exactly the per - time costs and movement costs of the elements . observe that we could even have time - varying movement costs . whereas , for graphical matroids the problem is @xmath410 hard even when the movement costs for each element do not change over time , and even just lie in the set @xmath375 . moreover , the restriction in theorem [ thm : diff - matrs - wpb ] that @xmath411 is also necessary , as the following result shows . [ thm : two ] for the case of two rounds ( i.e. , @xmath412 ) the problem can be solved in polynomial time , even when the two matroids in the two rounds are different . the solution is simple , via matroid intersection . suppose the matroids in the two timesteps are @xmath413 and @xmath414 . create elements @xmath415 which corresponds to picking element @xmath77 and @xmath416 in the two time steps , with cost @xmath417 . lift the matroids @xmath386 and @xmath418 to these tuples in the natural way , and look for a common basis . we note that deterministic online algorithms can not get any non - trivial guarantee for the problem , even in the simple case of a @xmath319-uniform matroid . this is related to the lower bound for deterministic algorithms for paging . formally , we have the 1-uniform matroid on @xmath5 elements , and @xmath419 . all acquisition costs @xmath11 are 1 . in the first period , all holding costs are zero and the online algorithm picks an element , say @xmath420 . since we are in the non - oblivious model , the algorithm knows @xmath420 and can in the second time step , set @xmath421 , while leaving the other ones at zero . now the algorithm is forced to move to another edge , say @xmath422 , allowing the adversary to set @xmath423 and so on . at the end of @xmath419 rounds , the online algorithm is forced to incur a cost of 1 in each round , giving a total cost of @xmath13 . however , there is still an edge whose holding cost was zero throughout , so that the offline opt is 1 . thus against a non - oblivious adversary , any online algorithm must incur a @xmath424 overhead . in this section , we show that if the aspect ratio of the movement costs is not bounded , the linear program has a @xmath426 gap , even when @xmath13 is exponentially larger than @xmath5 . we present an instance where @xmath426 and @xmath427 are about @xmath6 with @xmath428 , and the linear program has a gap of @xmath425 . this shows that the @xmath429 term in our rounding algorithm is unavoidable . the instance is a graphical matroid , on a graph @xmath289 on @xmath430 , and @xmath431 . the edges @xmath432 for @xmath433 $ ] have acquisition cost @xmath434 and holding cost @xmath435 for all @xmath2 . the edges @xmath436 for @xmath437 $ ] have acquisition cost @xmath438 and have holding cost determined as follows : we find a bijection between the set @xmath40 $ ] and the set of partitions @xmath439 of @xmath440 with each of @xmath441 and @xmath442 having size @xmath443 ( by choice of @xmath13 such a bijection exists , and can be found e.g. by arranging the @xmath441 s in lexicographical order . ) . in time step @xmath2 , we set @xmath181 for @xmath444 , and @xmath445 for all @xmath446 . first observe that no feasible integral solution to this instance can pay acquisition cost less than @xmath443 on the @xmath432 edges . suppose that the solution picks edges @xmath447 for some set @xmath448 of size at most @xmath443 . then any time step @xmath2 such that @xmath449 , the solution has picked no edges connecting @xmath450 to @xmath442 , and all edges connecting @xmath441 to @xmath442 have infinite holding cost in this time step . this contradicts the feasibility of the solution . thus any integral solution has cost @xmath290 . finally , we show that on this instance , ( [ eq : lp2 ] ) from section [ sec : lp - round ] , has a feasible solution of cost @xmath158 . we set @xmath451 for all @xmath452 $ ] , and set @xmath453 for @xmath454 . it is easy to check that @xmath455 is in the spanning tree polytope for all time steps @xmath2 . finally , the total acquisition cost is at most @xmath456 for the edges incident on @xmath450 and at most @xmath457 for the other edges , both of which are @xmath158 . the holding costs paid by this solution is zero . thus the lp has a solution of cost @xmath158 the claim follows . the greedy algorithm for is the natural one . we consider the interval view of the problem ( as in section [ sec : intervals ] ) where each element only has acquisition costs @xmath11 , and can be used only in some interval @xmath201 . given a current subset @xmath458 , define @xmath459 . the benefit of adding an element @xmath77 to @xmath385 is @xmath460 and the greedy algorithm repeatedly picks an element @xmath77 maximizing @xmath461 and adds @xmath77 to @xmath385 . this is done until @xmath462 for all @xmath51 $ ] . phrased this way , an @xmath23 bound on the approximation ration follows from wolsey @xcite . we next give an alternate dual fitting proof . we do not know of an instance with uniform acquisition costs where greedy does not give a constant factor approximation . the dual fitting approach may be useful in proving a better approximation bound for this special case . using lagrangian variables @xmath465 for each @xmath77 and @xmath466 , we write a lower bound for @xmath467 by @xmath468 which using the integrality of the matroid polytope can be rewritten as : @xmath469 here , @xmath470 denotes the cost of the minimum weight base at time @xmath2 according to the element weights @xmath471 , where the available elements at time @xmath2 is @xmath472 . the best lower bound is : @xmath473 it is useful to maintain , for each time @xmath2 , a _ minimum weight base _ @xmath64 of the subset @xmath479 according to weights @xmath480 . hence the current dual value equals @xmath481 . we start with @xmath482 and @xmath483 for all @xmath2 , which satisfies the above properties . suppose we now pick @xmath77 maximizing @xmath461 and get new set @xmath484 . we use @xmath485 akin to our definition of @xmath486 . call a timestep @xmath2 `` interesting '' if @xmath487 ; there are @xmath488 interesting timesteps . how do we update the duals ? for @xmath489 , we set @xmath490 . note the element @xmath77 itself satisfies the condition of being in @xmath491 for precisely the interesting timesteps , and hence @xmath492 . for each interesting @xmath86 , define the base @xmath493 ; for all other times set @xmath494 . it is easy to verify that @xmath495 is a base in @xmath496 . but is it a min - weight base ? inductively assume that @xmath64 was a min - weight base of @xmath479 ; if @xmath2 is not interesting there is nothing to prove , so consider an interesting @xmath2 . all the elements in @xmath491 have just been assigned weight @xmath497 , which by the monotonicity properties of the greedy algorithm is at least as large as the weight of any element in @xmath479 . since @xmath77 lies in @xmath491 and is assigned value @xmath498 , it can not be swapped with any other element in @xmath496 to improve the weight of the base , and hence @xmath499 is an min - weight base of @xmath496 . it remains to show that the dual constraints are approximately satisfied . consider any element @xmath500 , and let @xmath501 . the first step where we update @xmath502 for some @xmath503 is when @xmath500 is in the span of @xmath486 for some time @xmath2 . we claim that @xmath504 . indeed , at this time @xmath500 is a potential element to be added to the solution and it would cause a rank increase for @xmath505 time steps . the greedy rule ensures that we must have picked an element @xmath77 with weight - to - coverage ratio at most as high . similarly , the next @xmath2 for which @xmath502 is updated will have @xmath506 , etc . hence we get the sum @xmath507 since each element can only be alive for all @xmath13 timesteps , we get the claimed @xmath23-approximation . note that the greedy algorithm would solve @xmath508 even if we had a different matroid @xmath509 at each time @xmath2 . however , the equivalence of and no longer holds in this setting , which is not surprising given the hardness of theorem [ thm : diff - matrs - wpb ] . | this paper is motivated by the fact that many systems need to be maintained continually while the underlying costs change over time .
the challenge then is to continually maintain near - optimal solutions to the underlying optimization problems , without creating too much churn in the solution itself .
we model this as a multistage combinatorial optimization problem where the input is a sequence of cost functions ( one for each time step ) ; while we can change the solution from step to step , we incur an additional cost for every such change .
we first study the multistage matroid maintenance problem , where we need to maintain a base of a matroid in each time step under the changing cost functions and acquisition costs for adding new elements .
the online version of this problem generalizes onine paging , and is a well - structured case of the metrical task systems .
e.g. , given a graph , we need to maintain a spanning tree @xmath0 at each step : we pay @xmath1 for the cost of the tree at time @xmath2 , and also @xmath3 for the number of edges changed at this step .
our main result is a polynomial time @xmath4-approximation to the online multistage matroid maintenance problem , where @xmath5 is the number of elements / edges and @xmath6 is the rank of the matroid .
this improves on results of buchbinder et al .
@xcite who addressed the _ fractional _ version of this problem under uniform acquisition costs , and buchbinder , chen and naor @xcite who studied the fractional version of a more general problem .
we also give an @xmath7 approximation for the offline version of the problem .
these bounds hold when the acquisition costs are non - uniform , in which case both these results are the best possible unless p = np .
we also study the perfect matching version of the problem , where we must maintain a perfect matching at each step under changing cost functions and costs for adding new elements .
surprisingly , the hardness drastically increases : for any constant @xmath8 , there is no @xmath9-approximation to the multistage matching maintenance problem , even in the offline case . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
bandwidth selection is a key issue in kernel density estimation that has deserved considerable attention during the last decades . the problem of selecting the most suitable bandwidth for the nonparametric kernel density estimator introduced by @xcite and @xcite is the main topic of the reviews of @xcite , @xcite and @xcite , among others . comprehensive references on kernel smoothing and bandwidth selection include the books by @xcite , @xcite and @xcite . bandwidth selection is still an active research field in density estimation , with some recent contributions like @xcite and @xcite in the last years . + kernel density estimation has been also adapted to directional data , that is , data in the unit hypersphere of dimension @xmath0 . due to the particular nature of directional data ( periodicity for @xmath1 and manifold structure for any @xmath0 ) , the usual multivariate techniques are not appropriate and specific methodology that accounts for their characteristics has to be considered . the classical references for the theory of directional statistics are the complete review of @xcite and the book by @xcite . the kernel density estimation with directional data was firstly proposed by @xcite , studying the properties of two types of kernel density estimators and providing cross - validatory bandwidth selectors . almost simultaneously , @xcite provided a similar definition of kernel estimator , establishing its pointwise and @xmath2 consistency . some of the results by @xcite were extended by @xcite , who studied the estimation of the laplacian of the density and other types of derivatives . whereas the framework for all these references is the general @xmath0-sphere , which comprises as particular case the circle ( @xmath1 ) , there exists a remarkable collection of works devoted to kernel density estimation and bandwidth selection for the circular scenario . specifically , @xcite presented the first plug - in bandwidth selector in this context and @xcite derived a selector based on mixtures and on the results of @xcite for the circular asymptotic mean integrated squared error ( amise ) . recently , @xcite proposed a product kernel density estimator on the @xmath0-dimensional torus and cross - validatory bandwidth selection methods for that situation . another nonparametric approximation for density estimation with circular data was given in @xcite and @xcite . in the general setting of spherical random fields @xcite derived an estimation method based on a needlet basis representation . + directional data arise in many applied fields . for the circular case ( @xmath1 ) a typical example is wind direction , studied among others in @xcite , @xcite and @xcite . the spherical case ( @xmath3 ) poses challenging applications in astronomy , for example in the study of stars position in the celestial sphere or in the study of the cosmic microwave background radiation @xcite . finally , a novel field where directional data is present for large @xmath0 is text mining @xcite , where documents are usually codified as high dimensional unit vectors . for all these situations , a reliable method for choosing the bandwidth parameter seems necessary to trust the density estimate . + the aim of this work is to introduce new bandwidth selectors for the kernel density estimator for directional data . the first one is a rule of thumb which assumes that the underlying density is a von mises and it is intended to be the directional analogue of the rule of thumb proposed by @xcite for data in the real line . this selector uses the amise expression that can be seen , among others , in @xcite . the novelty of the selector is that it is more general and robust than the previous proposal by @xcite , although both rules exhibit an unsatisfactory behaviour when the reference density spreads off from the von mises . to overcome this problem , two new selectors based on the use of mixtures of von mises for the reference density are proposed . one of them uses the aforementioned amise expression , whereas the other one uses the exact mise computation for mixtures of von mises densities given in @xcite . both of them use the expectation - maximization algorithm of @xcite to fit the mixtures and , to select the number of components , the bic criteria is employed . these selectors based on mixtures are inspired by the earlier ideas of @xcite , for the multivariate setting , and @xcite for the circular scenario . + this paper is organized as follows . section [ kdebwd : sec : kdedir ] presents some background on kernel density estimation for directional data and the available bandwidth selectors . the rule of thumb selector is introduced in section [ kdebwd : sec : ruleofthumb ] and the two selectors based on mixtures of von mises are presented in section [ kdebwd : sec : mixtures ] . section [ kdebwd : sec : comparative ] contains a simulation study comparing the proposed selectors with the ones available in the literature . finally , section [ kdebwd : sec : data ] illustrates a real data application and some conclusions are given in section [ kdebwd : sec : conclusions ] . supplementary materials with proofs , simulated models and extended tables are given in the appendix . denote by @xmath4 a directional random variable with density @xmath5 . the support of such variable is the @xmath0-dimensional sphere , namely @xmath6 , endowed with the lebesgue measure in @xmath7 , that will be denoted by @xmath8 . then , a directional density is a nonnegative function that satisfies @xmath9 . also , when there is no possible confusion , the area of @xmath7 will be denoted by @xmath10 where @xmath11 represents the gamma function defined as @xmath12 , @xmath13 . + among the directional distributions , the von mises - fisher distribution ( see @xcite ) is perhaps the most widely used . the von mises density , denoted by @xmath14 , is given by @xmath15 where @xmath16 is the directional mean , @xmath17 the concentration parameter around the mean , @xmath18 stands for the transpose operator and @xmath19 is the modified bessel function of order @xmath20 , @xmath21 this distribution is the main reference for directional models and , in that sense , plays the role of the normal distribution for directional data ( is also a multivariate normal @xmath22 conditioned on @xmath7 ; see @xcite ) . a particular case of this density sets @xmath23 , which corresponds to the uniform density that assigns probability @xmath24 to any direction in @xmath7 . + given a random sample @xmath25 from the directional random variable @xmath4 , the proposal of @xcite for the directional kernel density estimator at a point @xmath26 is @xmath27 where @xmath28 is a directional kernel ( a rapidly decaying function with nonnegative values and defined in @xmath29 ) , @xmath30 is the bandwidth parameter and @xmath31 is a normalizing constant . this constant is needed in order to ensure that the estimator is indeed a density and satisfies that @xmath32 as usual in kernel smoothing , the selection of the bandwidth is a crucial step that affects notably the final estimation : large values of @xmath33 result in a uniform density in the sphere , whereas small values of @xmath33 provide an undersmoothed estimator with high concentrations around the sample observations . on the other hand , the choice of the kernel is not seen as important for practical purposes and the most common choice is the so called von mises kernel @xmath34 . its name is due to the fact that the kernel estimator can be viewed as a mixture of von mises - fisher densities as follows : @xmath35 where , for each von mises component , the mean value is the @xmath36-th observation @xmath37 and the common concentration parameter is given by @xmath38 . + the classical error measurement in kernel density estimation is the @xmath39 distance between the estimator @xmath40 and the target density @xmath5 , the so called integrated squared error ( ise ) . as this is a random quantity depending on the sample , its expected value , the mean integrated squared error ( mise ) , is usually considered : @xmath41{\right]}}={\mathbb{e}{\left[}{\int_{\omega_{q } } { \left(\hat f_h({\mathbf{x}})-f({\mathbf{x}})\right)}^2\,\omega_{q}(d { \mathbf{x}})}{\right]}},\end{aligned}\ ] ] which depends on the bandwidth @xmath33 , the kernel @xmath28 , the sample size @xmath42 and the target density @xmath5 . whereas the two last elements are fixed when estimating a density from a random sample , the bandwidth has to be chosen ( also the kernel , although this does not present a big impact in the performance of the estimator ) . then , a possibility is to search for the bandwidth that minimizes the mise : @xmath43 to derive an easier form for the mise that allows to obtain @xmath44 , the following conditions on the elements of the estimator ( [ kdebwd : kernel_directional ] ) are required : 1 . extend @xmath5 from @xmath45 to @xmath46 by @xmath47 for all @xmath48 , where @xmath49 denotes the euclidean norm . assume that the gradient vector @xmath50 and the hessian matrix @xmath51 exist , are continuous and square integrable . [ kdebwd : cond : d1 ] 2 . assume that @xmath52 is a bounded and integrable function such that @xmath53 , @xmath54 , for @xmath55.[kdebwd : cond : d2 ] 3 . assume that @xmath56 is a positive sequence such that @xmath57 and @xmath58 as @xmath59.[kdebwd : cond : d3 ] the following result , available from @xcite , provides the mise expansion for the estimator ( [ kdebwd : kernel_directional ] ) . it is worth mentioning that , under similar conditions , @xcite and @xcite also derived analogous expressions . [ kdebwd : dir : prop:3 ] under conditions [ kdebwd : cond : d1][kdebwd : cond : d3 ] , the mise for the directional kernel density estimator ( [ kdebwd : kernel_directional ] ) is given by @xmath60 where @xmath61 , @xmath62 , @xmath63 and @xmath64 this results leads to the decomposition @xmath65 , where amise stands for the asymptotic mise . it is possible to derive an optimal bandwidth for the amise in this sense , @xmath66 , that will be close to @xmath44 when @xmath67 is small enough . [ kdebwd : dir : cor:1 ] the amise optimal bandwidth for the directional kernel density estimator ( [ kdebwd : kernel_directional ] ) is given by @xmath68}^{\frac{1}{4+q}},\label{kdebwd : dir : cor:1:1}\end{aligned}\ ] ] where @xmath69 . unfortunately , expression ( [ kdebwd : dir : cor:1:1 ] ) can not be used in practise since it depends on the curvature term @xmath70 of the unknown density @xmath5 . the first proposals for data - driven bandwidth selection with directional data are from @xcite , who provide cross - validatory selectors . specifically , least squares cross - validation ( lscv ) and likelihood cross - validation ( lcv ) selectors are introduced , arising as the minimizers of the cross - validated estimates of the squared error loss and the kullback - leibler loss , respectively . the selectors have the following expressions : @xmath71 where @xmath72 represents the kernel estimator computed without the @xmath36-th observation . see remark [ kdebwd : rem:3 ] for an efficient computation of @xmath73 . + recently , @xcite proposed a plug - in selector for the case of circular data ( @xmath1 ) for the estimator with the von mises kernel . the selector of @xcite uses from the beginning the assumption that the reference density is a von mises to construct the amise . this contrasts with the classic rule of thumb selector of @xcite , which supposes at the end ( _ i.e. _ , after deriving the amise expression ) that the reference density is a normal . the bandwidth parameter is chosen by first obtaining an estimation @xmath74 of the concentration parameter @xmath75 in the reference density ( for example , by maximum likelihood ) and using the formula @xmath76}^\frac{1}{5}.\end{aligned}\ ] ] note that the parametrization of @xcite has been adapted to the context of the estimator ( [ kdebwd : kernel_directional ] ) by denoting by @xmath33 the inverse of the squared concentration parameter employed in his paper . + more recently , @xcite proposed a selector that improves the performance of @xcite allowing for more flexibility in the reference density , considering a mixture of von mises . this selector is also devoted to the circular case and is mainly based on two elements . first , the amise expansion that @xcite derived for the circular kernel density estimator by the use of fourier expansions of the circular kernels . this expression has the following form when the kernel is a circular von mises ( the estimator is equivalent to consider @xmath34 , @xmath1 and @xmath33 as the inverse of the squared concentration parameter in ( [ kdebwd : kernel_directional ] ) ) : @xmath77}^2\int_0^{2\pi } f''(\theta)^2\,d\theta+\frac{{\mathcal{i}}_0\big(2h^{-1/2}\big)}{2n\pi{\mathcal{i}}_0{\left(h^{-1/2}\right)}^2}. \label{kdebwd : dimarzio}\end{aligned}\ ] ] the second element is the expectation - maximization ( em ) algorithm of @xcite for fitting mixtures of directional von mises . the selector , that will be denoted by @xmath78 , proceeds as follows : 1 . use the em algorithm to fit mixtures from a determined range of components . choose the fitted mixture with minimum aic . 3 . compute the curvature term in ( [ kdebwd : dimarzio ] ) using the fitted mixture and seek for the @xmath33 that minimizes this expression , that will be @xmath78 . using the properties of the von mises density it is possible to derive a directional analogue to the rule of thumb of @xcite , which is the optimal amise bandwidth for normal reference density and normal kernel . the rule is resumed in the following result . [ kdebwd : prop : rot ] the curvature term for a von mises density @xmath14 is @xmath79}.\end{aligned}\ ] ] if @xmath74 is a suitable estimator for @xmath75 , then the rule of thumb selector for the kernel estimator ( [ kdebwd : kernel_directional ] ) with a directional kernel @xmath28 is @xmath80}^{\frac{1}{4+q}}.\end{aligned}\ ] ] if @xmath28 is the von mises kernel , then : @xmath81}n}\right]}^\frac{1}{5 } , & q=1,\\ \displaystyle{\left[\frac{8\sinh^2(\hat\kappa)}{\hat\kappa{\left[(1 + 4\hat\kappa^2)\sinh(2\hat\kappa)-2\hat\kappa\cosh(2\hat\kappa)\right]}n}\right]}^\frac{1}{6 } , & q=2,\\ \displaystyle{\left[\frac{4\pi^\frac{1}{2}{\mathcal{i}}_{\frac{q-1}{2}}(\hat\kappa)^2}{\hat\kappa^{\frac{q+1}{2}}{\left[2q{\mathcal{i}}_{\frac{q+1}{2}}(2\hat\kappa)+(2+q)\hat\kappa{\mathcal{i}}_{\frac{q+3}{2}}(2\hat\kappa)\right]}n}\right]}^\frac{1}{4+q } , & q\geq3 . \end{array}{\right.}\label{kdebwd : rot}\end{aligned}\ ] ] the parameter @xmath75 can be estimated by maximum likelihood . in view of the expression for @xmath82 in ( [ kdebwd : rot ] ) , it is interesting to compare it with @xmath83 when @xmath1 . as it can be seen , both selectors coincide except for one difference : the term @xmath84 in the sum in the denominator of @xmath82 . this `` extra term '' can be explained by examining the way that both selectors are derived . whereas the selector @xmath82 derives the bandwidth supposing that the reference density is a von mises when the amise is already derived in a general way , the selector @xmath83 uses the von mises assumption to compute it . therefore , it is expected that the selector @xmath85 will be more robust against deviations from the von mises density . + figure [ kdebwd : fig : vs ] collects two graphs exposing these comments , that are also corroborated in section [ kdebwd : sec : comparative ] . the left plot shows the mise for @xmath86 and @xmath82 for the density @xmath87 , where @xmath88 $ ] . this model represents two equally concentrated von mises densities that spread off from being the same to being antipodal . as it can be seen , the @xmath85 selector is slightly more accurate when the von mises model holds ( @xmath89 ) and when the deviation is large ( @xmath90 $ ] ) . when @xmath91 $ ] , both selectors perform similar . this graph also illustrates the main problem of these selectors : the von mises density is not flexible enough to capture densities with multimodality and it approximates them by the flat uniform density . + when the density is a @xmath14 , the right plot of figure [ kdebwd : fig : vs ] shows the output of @xmath83 , @xmath85 , @xmath92 and their corresponding errors with respect to @xmath75 . the effect of the `` extra term '' is visible for low values of @xmath75 , where @xmath93 presents a local maxima . this corresponds with higher values of @xmath83 with respect to @xmath85 and @xmath92 , which means that the former produces oversmoothed estimations of the density ( _ i.e. _ tend to the uniform case faster ) . despite the worse behaviour of @xmath83 , when the concentration parameter increases the effect of the `` extra term '' is mitigated and both selectors are almost the same . . left plot : logarithm of the curves of @xmath93 , @xmath94 and @xmath95 for sample size @xmath96 . the curves are computed by @xmath97 monte carlo samples and @xmath92 is obtained exactly . the abscissae axis represents the variation of the parameter @xmath98}$ ] , which indexes the reference density @xmath87 . right plot : logarithm of @xmath83 , @xmath85 , @xmath92 and their corresponding mise for different values of @xmath75 , with @xmath96.[kdebwd : fig : vs],title="fig : " ] . left plot : logarithm of the curves of @xmath93 , @xmath94 and @xmath95 for sample size @xmath96 . the curves are computed by @xmath97 monte carlo samples and @xmath92 is obtained exactly . the abscissae axis represents the variation of the parameter @xmath98}$ ] , which indexes the reference density @xmath87 . right plot : logarithm of @xmath83 , @xmath85 , @xmath92 and their corresponding mise for different values of @xmath75 , with @xmath96.[kdebwd : fig : vs],title="fig : " ] the results of the previous section show that , although the rule of thumb presents a significant improvement with respect to the @xcite selector in terms of generality and robustness , it also shares the same drawbacks when the underlying density is not the von mises model ( see figure [ kdebwd : fig : vs ] ) . to overcome these problems , two alternatives for improving @xmath85 will be considered . + the first one is related with improving the reference density to plug - in into the curvature term . the von mises density has been proved to be not flexible enough to estimate properly the curvature term in ( [ kdebwd : dir : cor:1:1 ] ) . this is specially visible when the underlying model is a mixture of antipodal von mises , but the estimated curvature term is close to zero ( the curvature of a uniform density ) . a modification in this direction is to consider a suitable mixture of von mises for the reference density , that will be able to capture the curvature of rather complex underlying densities . this idea was employed first by @xcite considering mixtures of multivariate normals and by @xcite in the circular setting . + the second improvement is concerned with the error criterion for the choice of the bandwidth . until now , the error criterion considered was the amise , which is the usual in the literature of kernel smoothing . however , as @xcite showed for the linear case and @xcite did for the directional situation , the amise and mise may differ significantly for moderate and even large sample sizes , with a potential significative misfit between @xmath99 and @xmath92 . then , a substantial decreasing of the error of the estimator ( [ kdebwd : kernel_directional ] ) is likely to happen if the bandwidth is obtained from the exact mise , instead of the asymptotic version . obviously , the problem of this new approach is how to compute exactly the mise , but this can be done if the reference density is a mixture of von mises . + the previous two considerations , improve the reference density and the error criterion , will lead to the bandwidth selectors of asymptotic mixtures ( ami ) , denoted by @xmath100 , and exact mixtures ( emi ) , denoted by @xmath101 . before explaining in detail the two proposed selectors , it is required to introduce some notation on mixtures of von mises . + an @xmath102-mixture of von mises densities with means @xmath103 , concentration parameters @xmath104 and weights @xmath105,with @xmath106 , is denoted by @xmath107 when dealing with mixtures , the tuning parameter is the number of components , @xmath102 , which can be estimated from the sample . the notation @xmath108 will be employed to represent the mixture of @xmath109 components where the parameters are estimated and @xmath109 is obtained from the sample . the details of this fitting are explained later in algorithm [ kdebwd : algo : nm ] . + then , the ami selector follows from modifying the rule of thumb selector to allow fitted mixtures of von mises . it is stated in the next procedure . [ kdebwd : algo : ami ] let @xmath25 be a random sample of a directional variable@xmath4 . 1 . compute a suitable estimation @xmath108 using algorithm [ kdebwd : algo : nm ] . 2 . for a directional kernel @xmath28 , set @xmath110}^{\frac{1}{4+q}}\end{aligned}\ ] ] and for the von mises kernel , @xmath111}^{-\frac{1}{4+q}}.\end{aligned}\ ] ] [ kdebwd : rem:1 ] unfortunately , the curvature term @xmath112 does not admit a simple closed expression , unless for the case where @xmath113 , _ i.e. _ , when @xmath100 is equivalent to @xmath85 . this is due to the cross - product terms between the derivatives of the mixtures that appear in the integrand . however , this issue can be bypassed by using either numerical integration in @xmath0-spherical coordinates or monte carlo integration to compute @xmath112 for any @xmath114 . the emi selector relies on the exact expression of the mise for densities of the type ( [ kdebwd : mise : mvm ] ) , that will be denoted by @xmath115}}.\end{aligned}\ ] ] similarly to what @xcite did for the linear case , @xcite derived the closed expression of @xmath116 when the directional kernel is the von mises one . the calculations are based on the convolution properties of the von mises , which unfortunately are not so straightforward as the ones for the normal , resulting in more complex expressions . [ kdebwd : mise : th:1 ] let @xmath117 be the density of an @xmath102-mixture of directional von mises ( [ kdebwd : mise : mvm ] ) . the exact mise of the directional kernel estimator ( [ kdebwd : kernel_directional ] ) with von mises kernel and obtained from a random sample of size @xmath42 is @xmath118}\mathbf{p } , \label{kdebwd : mise : mvm:1}\end{aligned}\ ] ] where @xmath119 and @xmath120 . the matrices @xmath121 , @xmath122 have entries : @xmath123 where @xmath124 is defined in equation ( [ kdebwd : dir : cq ] ) . [ kdebwd : rem:2 ] a more efficient way to implement ( [ kdebwd : mise : mvm:1 ] ) , specially for large sample sizes and higher dimensions , is the following expression : @xmath125}}-f_m({\mathbf{x}})\right)}^2-{\mathbb{e}{\left[}\hat f_h({\mathbf{x}}){\right]}}^2\right\}}\,\omega_{q}(d { \mathbf{x}})},\end{aligned}\ ] ] where the integral is either evaluated numerically using @xmath0-spherical coordinates or monte carlo integration and @xmath126}}$ ] is computed using @xmath127}}&=\sum_{j=1}^m p_j \frac{c_q(\kappa_j)c_q{\left(1/h^2\right)}}{c_q\big(||{\mathbf{x}}/h^2+\kappa_j{\boldsymbol\mu}_j||\big)}.\end{aligned}\ ] ] [ kdebwd : rem:3 ] by the use of similar techniques , when the kernel is von mises , the lscv selector admits an easier expression for the cv@xmath128 loss that avoids the calculation of the integral of @xmath129 : @xmath130}-\frac{c_q(1/h^2)^2}{nc_q(2/h^2)}.\end{aligned}\ ] ] based on the previous result , the philosophy of the emi selector is the following : using a suitable pilot parametric estimation of the unknown density ( given by algorithm [ kdebwd : algo : nm ] ) , build the exact mise and obtain the bandwidth that minimizes it . this is summarized in the following procedure . [ kdebwd : algo : emi ] consider the von mises kernel and let @xmath25 be a random sample of a directional variable @xmath4 . 1 . compute a suitable estimation @xmath108 using algorithm [ kdebwd : algo : nm ] . 2 . obtain @xmath131 . the em algorithm of @xcite , implemented in the ` r ` package ` movmf ` ( see @xcite ) , provides a complete solution to the problem of estimation of the parameters in a mixture of directional von mises of dimension @xmath0 . however , the issue of selecting the number of components of the mixture in an automatic and optimal way is still an open problem . + the propose considered in this work is an heuristic approach based on the bayesian information criterion ( bic ) , defined as @xmath132 , where @xmath133 is the log - likelihood of the model and @xmath134 is the number of parameters . the procedure looks for the fitted mixture with a number of components @xmath102 that minimizes the bic . this problem can be summarized as the global minimization of a function ( bic ) defined on the naturals ( number of components ) . + the heuristic procedure starts by fitting mixtures from @xmath135 to @xmath136 , computing their bic and providing @xmath114 , the number of components with minimum bic . then , in order to ensure that @xmath114 is a global minimum and not a local one , @xmath137 neighbours next to @xmath114 are explored ( _ i.e. _ fit mixture , compute bic and update @xmath114 ) , if they were not previously explored . this procedure continues until @xmath114 has at least @xmath137 neighbours at each side with larger bics . a reasonable compromise for @xmath138 and @xmath137 , checked by simulations , is to set @xmath139 and @xmath140 . in order to avoid spurious solutions , fitted mixtures with any @xmath141 are removed . the procedure is detailed as follows . [ kdebwd : algo : nm ] let @xmath25 be a random sample of a directional variable @xmath4 with density @xmath5 . 1 . set @xmath139 and @xmath137 as the user supplies , usually @xmath140 . [ kdebwd : algo : nm:1 ] 2 . for @xmath102 varying from @xmath142 to @xmath138 , [ kdebwd : algo : nm:2 ] 1 . estimate the @xmath102-mixture with the em algorithm of @xcite and 2 . compute the bic of the fitted mixture . 3 . set @xmath114 as the number of components of the mixture with lower bic . [ kdebwd : algo : nm:3 ] 4 . if @xmath143 , set @xmath144 and turn back to step [ kdebwd : algo : nm:2 ] . otherwise , end with the final estimation @xmath108 . [ kdebwd : algo : nm:4 ] other informative criteria , such as the akaike information criterion ( aic ) and its corrected version , aicc , were checked in the simulation study together with bic . the bic turned out to be the best choice to use with the ami and emi selectors , as it yielded the minimum errors . along this section , the three new bandwidth selectors will be compared with the already proposed selectors described in subsection [ kdebwd : subsec : bwsels ] . a collection of directional models , with their corresponding simulation schemes , are considered . subsection [ kdebwd : subsec : dirmods ] is devoted to comment the directional models used in the simulation study ( all of them are defined for any arbitrary dimension @xmath0 , not just for the circular or spherical case ) . these models are also described in the appendix . + for each of the different combinations of dimension , sample size and model , the mise of each selector was estimated empirically by @xmath97 monte carlo samples , with the same seed for the different selectors . this is used in the computation of @xmath95 , where @xmath92 is obtained as a numerical minimization of the estimated mise . the calculus of the ise was done by : simpson quadrature rule with @xmath145 discretization points for @xmath1 ; @xcite rule with @xmath146 nodes for @xmath3 and monte carlo integration with @xmath147 sampling points for @xmath148 ( same seed for all the integrations ) . finally , the kernel considered in the study is the von mises . the first models considered are the uniform density in @xmath7 and the von mises density given in ( [ kdebwd : dir : cq ] ) . the analogous of the von mises for axial data ( _ i.e. _ , directional data where @xmath149 ) is the watson distribution @xmath150 @xcite : @xmath151 where @xmath152 . this density has two antipodal modes : @xmath153 and @xmath154 , both of them with concentration parameter @xmath17 . a further extension of this density is the called small circle distribution @xmath155 @xcite : @xmath156 where @xmath157 , @xmath158 and @xmath159 . for the case @xmath160 , this density has a kind of modal strip along the @xmath161-sphere @xmath162 . + a common feature of all these densities is that they are rotationally symmetric , that is , their contourlines are @xmath161-spheres orthogonal to a particular direction . this characteristic can be exploited by means of the so called tangent - normal decomposition ( see @xcite ) , that leads to the change of variables @xmath163 where @xmath164 is a fixed vector , @xmath165 ( measures the distance of @xmath166 from @xmath153 ) , @xmath167 and @xmath168 is the semi - orthonormal matrix ( @xmath169 and @xmath170 , with @xmath171 the @xmath0-identity matrix ) resulting from the completion of @xmath153 to the orthonormal basis @xmath172 . the family of rotationally symmetric densities can be parametrized as @xmath173 where @xmath174 is a function depending on a vector parameter @xmath175 and such that @xmath176 is a density in @xmath177 , for all @xmath178 . using this property , it is easy to simulate from ( [ kdebwd : rotsym ] ) . 1 . sample @xmath179 from the density @xmath176.[kdebwd : algo : rotsym:1 ] 2 . sample @xmath180 from a uniform in @xmath181 @xmath182.[kdebwd : algo : rotsym:2 ] 3 . @xmath183 is a sample from @xmath184.[kdebwd : algo : rotsym:3 ] step [ kdebwd : algo : rotsym:1 ] can always be performed using the inversion method @xcite . this approach can be computationally expensive : it involves solving the root of the distribution function , which is computed from an integral evaluated numerically if no closed expression is available . a reasonable solution to this ( for a fixed choice of @xmath174 and @xmath153 ) is to evaluate once the quantile function in a dense grid ( for example , @xmath145 points equispaced in @xmath185 ) , save the grid and use it to interpolate using cubic splines the new evaluations , which is computationally fast . extending these ideas for rotationally symmetric models , two new directional densities are proposed . the first one is the directional cauchy density @xmath186 , defined as an analogy with the usual cauchy distribution as @xmath187 where @xmath153 is the mode direction and @xmath17 the concentration parameter around it ( @xmath23 gives the uniform density ) . this density shares also some of the characteristics of the usual cauchy distribution : high concentration around a peaked mode and a power decay of the density . the other proposed density is the skew normal directional density @xmath188 , @xmath189 where @xmath190 is the skew normal density of @xcite with location @xmath191 , scale @xmath192 and shape @xmath193 that is truncated to the interval @xmath177 . the density is inspired by the wrapped skew normal distribution of @xcite , although it is based on the rotationally symmetry rather than in wrapping techniques . a particular form of this density is an homogeneous `` cap '' in a neighbourhood of @xmath153 that decreases very fast outside of it . + non rotationally symmetric densities can be created by mixtures of rotationally symmetric . however , it is interesting to introduce a purely non rotationally symmetric density : the projected normal distribution of @xcite . denoted by @xmath194 , the corresponding densityis @xmath195 where @xmath196 , @xmath197 , @xmath198 and @xmath199 . sampling from this distribution is extremely easy : just sample @xmath200 and then project @xmath4 to @xmath7 by @xmath201 . + the whole collection of models , with @xmath202 densities in total , are detailed in table [ kdebwd : tab : models ] in appendix [ kdebwd : ap : models ] . figures [ kdebwd : fig : circ ] and [ kdebwd : fig : sph ] show the plots of these densities for the circular and spherical cases . for the circular case , the comparative study has been done for the @xmath202 models described in figure [ kdebwd : fig : circ ] ( see table [ kdebwd : tab : models ] to see their densities ) , for the circular selectors @xmath203 , @xmath204 , @xmath86 , @xmath78 , @xmath82 , @xmath205 and @xmath206 and for the sample sizes @xmath207 , @xmath208 , @xmath209 and @xmath97 . due to space limitations , only the results for sample size @xmath209 are shown in table [ kdebwd : tab : cir ] , and the rest of them are relegated to appendix [ kdebwd : ap : tables ] . + in addition , to help summarizing the results a ranking similar to ranking b of @xcite will be constructed . the ranking will be computed according to the following criteria : for each model , the @xmath191 bandwidth selectors @xmath210 considered are sorted from the best performance ( lowest error ) to the worst performance ( largest error ) . the best bandwidth receives @xmath191 points , the second @xmath211 and so on . these points , denoted by @xmath212 , are standardized by @xmath191 and multiplied by the relative performance of each selector compared with the best one . in other words , the points of the selector @xmath213 , if @xmath214 is the best one , are @xmath215 . the final score for each selector is the sum of the ranks obtained in all the twenty models ( thus , a selector which is the best in all models will have @xmath202 points ) . with this ranking , it is easy to group the results in a single and easy to read table . + in view of the results , the following conclusions can be extracted . firstly , @xmath85 performs well in certain unimodal models such as m3 ( von mises ) and m6 ( skew normal directional ) , but its performance is very poor with multimodal models like m15 ( watson ) . in its particular comparison with @xmath83 , it can be observed that both selectors share the same order of error , but being @xmath85 better in all the situations except for one : the uniform model ( m1 ) . this is due to the `` extra term '' commented in section [ kdebwd : sec : ruleofthumb ] : its absence in the denominator makes that @xmath216 faster than @xmath85 when the concentration parameter @xmath217 and , what is a disadvantage for @xmath218 , turns out in an advantage for the uniform case . with respect to @xmath100 and @xmath101 , although their performance becomes more similar when the sample size increases , something expected , @xmath101 seems to be on average a step ahead from @xmath100 , specially for low sample sizes . among the cross - validated selectors , @xmath219 performs better than @xmath73 , a fact that was previously noted by simulation studies carried out by @xcite and @xcite . finally , @xmath220 presents the most competitive behaviour among the previous proposals in the literature when the sample size is reasonably large ( see table [ kdebwd : tab : rankcirsph ] ) . + the comparison between the circular selectors is summarized in the scores of table [ kdebwd : tab : rankcirsph ] . for all the sample sizes considered , @xmath101 is the most competitive selector , followed by @xmath100 for all the sample sizes except @xmath221 , where @xmath219 is the second . the effect of the sample effect is also interesting to comment . for @xmath221 , @xmath219 and @xmath85 perform surprisingly well , in contrast with @xmath220 , which is the second worst selector for this case . when the sample size increases , @xmath85 and @xmath83 have a decreasing performance and @xmath220 stretches differences with @xmath100 , showing a similar behaviour . this was something expected as both selectors are based on error criteria that are asymptotically equivalent . the cross - validated selectors show a stable performance for sample sizes larger than@xmath221 . [ kdebwd : fig : circ],title="fig:",scaledwidth=22.5% ] is drawn . [ kdebwd : fig : circ],title="fig:",scaledwidth=22.5% ] is drawn . [ kdebwd : fig : circ],title="fig:",scaledwidth=22.5% ] is drawn . [ kdebwd : fig : circ],title="fig:",scaledwidth=22.5% ] + is drawn . [ kdebwd : fig : circ],title="fig:",scaledwidth=22.5% ] is drawn . [ kdebwd : fig : circ],title="fig:",scaledwidth=22.5% ] is drawn . [ kdebwd : fig : circ],title="fig:",scaledwidth=22.5% ] is drawn . [ kdebwd : fig : circ],title="fig:",scaledwidth=22.5% ] + is drawn . [ kdebwd : fig : circ],title="fig:",scaledwidth=22.5% ] is drawn . [ kdebwd : fig : circ],title="fig:",scaledwidth=22.5% ] is drawn . [ kdebwd : fig : circ],title="fig:",scaledwidth=22.5% ] is drawn . [ kdebwd : fig : circ],title="fig:",scaledwidth=22.5% ] + is drawn . [ kdebwd : fig : circ],title="fig:",scaledwidth=22.5% ] is drawn . [ kdebwd : fig : circ],title="fig:",scaledwidth=22.5% ] is drawn . [ kdebwd : fig : circ],title="fig:",scaledwidth=22.5% ] is drawn . [ kdebwd : fig : circ],title="fig:",scaledwidth=22.5% ] + is drawn . [ kdebwd : fig : circ],title="fig:",scaledwidth=22.5% ] is drawn . [ kdebwd : fig : circ],title="fig:",scaledwidth=22.5% ] is drawn . [ kdebwd : fig : circ],title="fig:",scaledwidth=22.5% ] is drawn . [ kdebwd : fig : circ],title="fig:",scaledwidth=22.5% ] .comparative study for the circular case , with sample size @xmath222 . columns of the selector @xmath223 represent the @xmath224 , with bold type for the minimum of the errors . the standard deviation of the @xmath225 is given between parentheses.[kdebwd : tab : cir ] [ cols=">,>,>,>,>,>,>,>,>,>",options="header " , ] | new bandwidth selectors for kernel density estimation with directional data are presented in this work .
these selectors are based on asymptotic and exact error expressions for the kernel density estimator combined with mixtures of von mises distributions .
the performance of the proposed selectors is investigated in a simulation study and compared with other existing rules for a large variety of directional scenarios , sample sizes and dimensions .
the selector based on the exact error expression turns out to have the best behaviour of the studied selectors for almost all the situations .
this selector is illustrated with real data for the circular and spherical cases .
* keywords : * bandwidth selection ; directional data ; mixtures ; kernel density estimator ; von mises . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
the invention of the scanning tunneling microscope@xcite in 1981 and a consequent development in the beginning of the nineties of the remarkably simple experimental technique known as mechanically controllable break junction ( mcbj)@xcite led to the possibility of fabrication of metallic point contacts approaching the atomic scale . the recent review article ( ref . ) summarizes the numerous achievements in this field . in the experiments the conductance measured as a function of the elongation of the nanocontacts decreases in a stepwise fashion @xcite with steps of order of the conductance quantum @xmath0 . such behavior of the conductance is attributed to atomic rearrangements that entails a discrete variation of the contact diameter . @xcite the electron transport in metallic nanocontacts is purely ballistic and phase - coherent because their size is much smaller than all scattering lengths of the system . according to landauer,@xcite conductance is understood as transport through nonmixing channels , @xmath1 where @xmath2 s are transmission probabilities . they are defined as eigenvalues of the transmission matrix @xmath3 . here the matrix element @xmath4 gives the probability amplitude for an incoming electron wave in the transverse mode ( channel ) @xmath5 on the left from the contact to be transmitted to the outgoing wave in the mode @xmath6 on the right . consequently , the eigenvectors of @xmath3 are usually called eigenchannels . it was shown in the pioneering work by scheer _ et al._@xcite that a study of the current - voltage relation for the superconducting atomic - sized contacts allowed to obtain transmission probabilities @xmath2 s for particular atomic configurations realized in mcbj experiments . the @xmath2 s are found by fitting theoretical and experimental @xmath7 curve which has a peculiar nonlinear behavior for superconducting contacts at voltages @xmath8 smaller than the energy gap @xmath9 of a superconductor@xcite . the origin of such effect is explained in terms of multiple andreev reflections.@xcite the analysis of mcbj experiments within the tight - binding ( tb ) model suggested by cuevas _ et al._@xcite gave a strong evidence to the relation between the number of conducting modes and the number of valence orbitals of a contact atom . to describe the electronic and transport properties of nanocontacts , quite a big number of different methods which supplemented each other were developed during the last 15 years . early models employed a free - electron - like approximation.@xcite further approaches based on density functional theory ( dft ) used psuedopotentials to describe atomic chains suspended between jellium electrodes.@xcite the tb models were applied to the problem of the conduction eigenchannels@xcite and to the study of the breaking processes of nanowires.@xcite the up - to - date fully self - consistent _ ab initio _ methods@xcite allowed to treat both the leads and the constriction region on the same footing and to evaluate the non - equilibrium transport properties as well@xcite . the scattering waves , underlying a concept of eigenchannels introduced by landauer and bttiker,@xcite do not form an appropriate basis for the most of _ ab initio _ methods . instead , one considers conduction channels as eigenvectors of some hermitian transmission matrix written in terms of local , atom centered basis set.@xcite one of the goals of the present paper is to establish a missing link between these approaches . below we introduce a formalism for the evaluation of conduction eigenchannels , which combines an _ ab initio _ korringa - kohn - rostoker ( kkr ) green s function method@xcite for the electronic structure calculations and the baranger and stone formulation of the ballistic transport@xcite . in recent publications,@xcite we have successfully applied this method to the study of the electron transport through atomic contacts contaminated by impurities . in the present paper , mathematical aspects of the problem are considered , followed by some applications . in particular , we analyze the symmetry of channels and relate our approach to the orbital classification of eigenmodes introduced by cuevas _ et al._@xcite the paper is organized as follows . a short description of the kkr method is given in sec . we proceed in sec . iii with a formal definition of eigenchannels for the case of realistic crystalline leads attached to atomic constriction . iv supplemented by appendices a and b contains mathematical formulation of the method . briefly , using the equivalence of the kubo and landauer approaches for the conductance , @xcite we build the transmission matrix @xmath10 in the scattering wave representation . the angular momentum expansion of the scattering bloch states within each cell is used further to find an equivalent , kkr representation of the transmission operator for which the eigenvalue problem can be solved . applications of the method are presented in sec . v. in particular , we focus on transition metal contacts ( such as ni , co and pd ) , since experimental@xcite and theoretical studies @xcite of their transport properties have been attracting much attention during the last years . experiments @xcite regarding ballistic magnetoresistance ( bmr ) effect in ferromagnetic contacts are commented . a summary of our results is given in sec . the systems under consideration consist of two semi - infinite crystalline leads , left ( l ) and right ( r ) , coupled through a cluster of atoms which models an atomic constriction . in fig . 1 a typical configuration used in the calculations is shown the two fcc ( 001 ) pyramids attached to the electrodes are joined via the vertex atoms . we employed the _ ab initio _ screened kkr green s function method to calculate the electronic structure of the systems . since details of the approach can be found elsewhere , @xcite only a brief description is given below . in the kkr formalism one divides the whole space into non - overlapping , space - filling cells , with the atoms ( and empty spheres ) positioned at the sites @xmath11 , so that the crystal potential @xmath12 is expressed in each cell as @xmath13 . the one - electron retarded green s function is expressed in terms of local functions centered at sites @xmath11 : @xmath14 where @xmath15 , @xmath16 are restricted to the cells @xmath5 and @xmath17 ; @xmath18 , @xmath19 denote one of the two vectors @xmath15 or @xmath16 with the smaller or the larger absolute value , and local functions @xmath20 and @xmath21 are the regular and irregular solutions of the schrdinger equation for the single potential @xmath22 . here the index @xmath23 stands for the angular momentum quantum numbers and atomic units are used : @xmath24 , @xmath25 , @xmath26 . the structural green s function @xmath27 ( structure constants ) in eq . ( [ grr ] ) is related to the known structure constants of the appropriately chosen reference system by the algebraic dyson equation which includes the difference @xmath28 between local @xmath29-matrices of the physical and a reference system . in the screened kkr method@xcite we use a lattice of strongly repulsive , constant muffin - tin potentials ( typically , @xmath30ry height ) as reference system that leads to structure constants which decay exponentially in real space . within the screened kkr method both a constriction region and the leads are treated on the same footing . this is achieved by using the hierarchy of green s functions connected by a dyson equation , so that we perform the self - consistent electronic structure calculations of complicated systems in a step - like manner . first , using the concept of principal layers together with the decimation technique,@xcite we calculate the structural green s function of the auxiliary system consisting of semi - infinite leads separated by a vacuum barrier . at the second step , the self - consistent solution of the impurity problem is found by embedding a cluster with perturbed potentials caused by the atomic contact into the auxiliary system . due to effective screening of perturbation , the algebraic dyson equation for the structure constants is solved in real space.@xcite the concept of eigenchannels is introduced in the landauer approach to ballistic transport , where the problem of the conductance evaluation is considered from viewpoint of scattering theory . following landauer,@xcite we look at the system shown in fig . 1 as consisting of two semi - infinite leads ( electrodes ) attached to a scattering region ( atomic - sized constriction ) . far away from the scattering region the propagating states are the unperturbed bloch waves @xmath31 of the left ( l , @xmath32 ) and right ( r , @xmath33 ) leads , where @xmath34 belongs to the isoenergetic surface @xmath35 ( fermi surface , @xmath36 , in case of conductance ) and a common notation @xmath37 is used to denote bloch vector @xmath38 and band index @xmath39 . for the eigenchannel problem one considers in - coming and out - going states in the l and r leads normalized to a unit flux . the in - states in l and out - states in r are @xmath40 with positive velocity @xmath41 along @xmath42-axis . the conjugated states @xmath43 are the out - states in l and in - states in r with negative velocity @xmath44 . here @xmath45 is a @xmath42-component of the group velocity @xmath46 ; a proportionality factor between @xmath47 and @xmath48 related to a particular choice of normalization of the bloch waves is introduced further in sec.iv.b . the potential @xmath49 describing the constriction introduces a perturbation to the perfect conductor . let @xmath50 be a perturbed state which is a solution of the lippmann - schwinger equation for an in - coming state in l : @xmath51 where the integral goes over all space , and @xmath52 is the retarded green s function of the perfect conductor . asymptotic behavior of @xmath50 is @xmath53 where @xmath54 and @xmath55 are transmission and reflection amplitudes , assuming elastic scattering ( @xmath56 ) . according to the landauer - bttiker formula,@xcite conductance is given by @xmath57 , where trace goes over in - coming states ( @xmath58 ) in the left electrode and @xmath36 . an equivalent formulation with respect to in - coming states ( @xmath59 ) from the right electrode reads as @xmath60 . eigenchannels appear from a unitary transformation of in- and out - states . let @xmath61 be a unitary transform of in - states in l : @xmath62 . the corresponding solution @xmath63 of eq . ( [ ls ] ) for an arbitrary * r * is @xmath64 the unitary transform @xmath61 is defined such way that the transmission matrix @xmath65 is diagonal in the basis @xmath66 : @xmath67 and the conductance reads as @xmath68 , where the @xmath69 s are transmission probabilities of eigenchannels . the matrix @xmath70 , however , is not diagonal in basis @xmath66 . following ref . one can introduce a unitary matrix @xmath71 which satisfies the equation : @xmath72 where all quantities are energy dependent . the solution is @xmath73 . the following properties of @xmath71 can be checked : @xmath74,@xmath75 , thus @xmath71 is indeed the unitary matrix . it diagonalizes @xmath76 : @xmath77 matrix @xmath71 performs a unitary transform of out - states in r , so that the linear combination @xmath66 of in - coming states in l scatters into the linear combination @xmath66 of the out - states in r , @xmath78 with the transmission amplitude @xmath79 , namely : @xmath80 . one can show@xcite that for the bloch states at the same energy ( @xmath81 ) in the ideal leads the following relations hold for the current matrix elements : @xmath82 \ = \ \frac{v_{\boldsymbol{k}}}{2\pi } \ , \delta_{\boldsymbol{kk'}},\ ] ] where the bloch waves are either left ( @xmath83 ) or right - travelling ( @xmath84 ) , the operator @xmath85 is defined as @xmath86 , and the integral goes over infinite plane @xmath87 ( cross - section of the lead ) which is perpendicular to the current direction @xmath42 . in case of the perturbed system the orthogonality relation holds for current matrix elements in the basis of eigenchannels . using eqs . ( [ phi_k_limit ] ) , ( [ omega ] ) and ( [ curr_bloch1 ] ) we can compute it in the asymptotic region of the right ( r ) lead:@xcite @xmath88_{z\to + \infty } = \frac{t_{\nu}(e)}{2\pi}\,\delta_{\nu\mu}.\ ] ] we note , that eq . ( [ curr_right ] ) holds for _ any _ position @xmath42 of the plane @xmath87 . because the wave functions of channels are solutions of the schrdinger equation with a real potential corresponding to the same energy , the flux through arbitrary plane @xmath87 is conserved ( a proof is similar to that of appendix a of ref . ) . to calculate the ballistic conductance , we employ the kubo linear response theory as formulated by baranger and stone : @xcite @xmath89 where @xmath90 and @xmath91 are retarded and advanced green s functions , respectively , and @xmath92 . the integration is performed over left ( @xmath93 ) and right ( @xmath94 ) planes which connect the leads with the scattering region ( fig . 1 ) . the current flows in @xmath42 direction . the implementation of eq . ( [ cond_g ] ) within the kkr method , related convergence tests and further details were discussed in recent publication.@xcite in this paper we present a further extension of ref . , a method for the evaluation of conduction eigenchannels . for that , we will follow closely the analysis of refs . where the equivalence of the kubo and landauer approaches to conductance problem was shown . we proceed in four steps : ( i ) we remind the kkr representation of the bloch functions ; ( ii ) we build up the asymptotic expansion of the green s function in terms of unperturbed bloch states of the leads ; ( iii ) we construct the transmission matrix @xmath95 in @xmath58-space ; ( iv ) finally , we find an equivalent representation of the transmission matrix within the local kkr basis . further solution of the eigenvalue problem leads us to conduction channels . we mention here one aspect of the problem : for realistic calculations the planes @xmath93 and @xmath94 ( see fig . 2 for details ) are usually placed in finite distance from the atomic constriction . nevertheless , we first focus on the asymptotic limit before we discuss the realistic situation which is considered in sec . iv.e and appendix a. let @xmath96 be arbitrary point in the asymptotic region of the lead ( fig . 2 ) . in the kkr method the local , energy dependent basis of atomic orbitals is defined at each unit cell @xmath5 : @xmath97 where @xmath98 s are real regular solutions @xcite of the schrdinger equation for the potential @xmath99 at cell @xmath5 , and @xmath100function is 1 inside cell @xmath5 and is 0 outside it . the unperturbed bloch function is given by expansion over atomic orbitals at all sites @xmath5 in the born - von krmn supercell : @xmath101 with @xmath102 . here the common notation @xmath37 for the bloch vector @xmath38 of the 1st brillouin zone ( bz ) and the band index @xmath39 is used . the @xmath103 are solutions of the kkr band structure equations @xcite with energy @xmath104 . considering a waveguide geometry , we will assume the bloch functions to be normalized per cross section of the born - von krmn supercell ( see fig . 2 ) with open boundary conditions along @xmath42-axis . thus , @xmath105 with @xmath106 being number of atoms per cross - section , and orthogonality condition for bloch waves takes a form @xcite : @xmath107 here @xmath108 is a volume of the supercell , @xmath109 is an area of the @xmath110-unit cell in the electrode , @xmath111 are signs of @xmath112 and @xmath113 , relation @xmath114 has been used , and velocity @xmath47 along current flow is defined as @xmath115 . since the bloch waves form a complete set , a back transform of eq . ( [ psik_nl ] ) exists : @xmath116 where @xmath117 sum runs over all @xmath118-points in the 1st bz and over all bands @xmath119 . the @xmath120 symbol means hermitian conjugate . the expression for @xmath121 can be obtained @xcite from known matrix @xmath122 . one can prove further , that @xmath122 and @xmath121 obey the following orthogonality relations : @xmath123 where in the second equation a sum over @xmath58 is restricted to states with @xmath124 . starting from the site angular momentum representation ( [ grr ] ) of the retarded green s function within the kkr method and using eqs . ( [ psik_nl ] ) and ( [ phi_nl ] ) we obtain the asymptotic expansion for @xmath125 over the unperturbed bloch waves : @xcite @xmath126 with @xmath127 , @xmath128 ( see fig . 2 ) , and @xmath129 or in a matrix form : @xmath130 . formally , the @xmath58-sums in eq . ( [ g_knl ] ) are performed over all @xmath38-states in the 1st bz and over all bands @xmath39 . however , since the green s function for @xmath131 is a solution of the schrdinger equation without a source term , only states @xmath132 at the isoenergetic surface of energy @xmath104 contribute to the sum in the asymptotic expansion.@xcite therefore , @xmath133 where @xmath134 . here @xmath135 is number of atom sites in born - von krmn supercell along @xmath42 axis , @xmath136 is volume of the 1st bz , @xmath137 is the area of the isoenergetic surface corresponding to band @xmath39 , @xmath138 . for the discrete @xmath58-points , the function @xmath139 equals 1 if @xmath140 , and is 0 otherwise . in addition , boundary conditions for the green s function@xcite constrain matrix elements @xmath141 to be non - zeros only if @xmath58 and @xmath117-states are right - travelling waves ( with positive velocity along @xmath42-axis ) that corresponds to the in - states @xmath58 in the left lead and out - states @xmath117 in the right one . we proceed further , and use the asymptotic representation ( [ g_knl ] ) of the green s function to evaluate conductance according to eq . ( [ cond_g ] ) . assuming the integration planes to be placed within the leads infinitely far from the scattering region , we obtain : @xmath142,\ ] ] where the diagonal operators of velocities @xmath143 and @xmath144 ( related to the left and right planes ) acting in the @xmath58-space were introduced : @xmath145_{\boldsymbol{kk ' } } = \left({v_{\boldsymbol{k}}}/{2\pi}\right ) \delta_{\boldsymbol{kk'}}$ ] . formally , the trace ( tr ) in eq . ( [ g_tr_vava ] ) goes over all @xmath58-states and the fermi surface is taken into account by means of eq . ( [ a0_kk ] ) where @xmath146 . the velocity operators can be decomposed into sum of two operators related to the bloch states with positive and negative velocities along @xmath42 : @xmath147 where @xmath148 is nonzero for right - travelling waves only , while @xmath149 is nonzero for left - travelling ones . in the asymptotic limit , only in - coming and out - going @xmath58-states with positive velocities contribute to the sums in eq.([g_tr_vava ] ) . using the relation between expansion coefficients @xmath150 and the transmission amplitudes @xmath151 derived in refs . , @xmath152 we obtain : @xmath153 \\ \nonumber { } & = & g_0 \mathrm{tr}_{(\boldsymbol{k})}\,[t(e_f)],\end{aligned}\ ] ] where a representation of @xmath65 in @xmath58-space is given by @xmath154 with a positive definite operator under square - roots . the @xmath58-representation is formal but not suitable for implementation . to solve the eigenvalue problem for @xmath65 the mapping on the site - angular momentum @xmath155-space of the kkr method should be presented . such mapping is realized through the expansion ( [ psik_nl ] ) of the bloch functions over atomic orbitals , so that velocity operators in @xmath58-space take a form : @xmath156_{\boldsymbol{kk ' } } = \sum_{nn'\in s^0_{\mathrm{l } } } \sum_{ll ' } c_{\boldsymbol{k},nl}\ , \bigl[d_{\mathrm{l(r)}}\bigr]^{nn'}_{ll'}\ , c^{\dagger}_{n'l',\boldsymbol{k'}},\ ] ] where site - diagonal operators @xmath157 and @xmath158 defined on atomic orbitals are the kkr - analogue of velocities : @xcite @xmath159^{nn'}_{ll'}\ = } \\ \nonumber\displaystyle & & \pm\ , \delta_{nn'}\ , \int_{s^n_{\mathrm{l(r ) } } } ds \left [ r^n_l({\bf r},e)\ , i\stackrel{\leftrightarrow}{\partial_z } r^{n}_{l'}({\bf r},e ) \right]\end{aligned}\ ] ] here integral is restricted to the cross - section of the unit cell around site @xmath5 . now we can evaluate conductance according to eq.([g_tr_vava ] ) . taking into account that @xmath160 [ eq.([bc ] ) ] , we obtain : @xmath161 here ( and further ) all matrices are assumed to be taken at the fermi energy , @xmath162 stands for matrix notation of the structural green s function introduced in eq . ( [ grr ] ) , and trace ( tr ) involves sites and orbitals related to the atomic plane @xmath163 ( fig . 2 ) . the operators @xmath158 , @xmath157 are anti - symmetric and hermitian , thus their spectrum consists of pairs of positive and negative eigenvalues : @xmath164 . let @xmath165 be the unitary transform of matrix @xmath166 ( either @xmath158 or @xmath157 ) to a diagonal form : @xmath167 decomposition of operator @xmath168 into two terms , @xmath169 , related to the right- and left - travelling waves is naturally given in the basis of eigenvectors : @xmath170 with @xmath171 where @xmath172 , @xmath173 are positive ( non - negative ) diagonal matrices . the analogue of eq . ( [ g_vp_a_vp_a ] ) reads as @xmath174 now we are ready to build the @xmath155-representation for transmission matrix @xmath65 . for that , one should extract a square - root from the positive definite operator @xmath175 defined on @xmath58-space with help of @xmath155-space . namely , because the @xmath176 is positive definite , it can be represented in the following form : @xcite @xmath177 . here an operator @xmath178 maps the @xmath58-space on the @xmath155-space , @xmath179 . the solution for @xmath180 is @xmath181 where @xmath182 is an arbitrary unitary matrix ( @xmath183 ) and @xmath184^{1/2 } u^{\dagger}_l $ ] with @xmath185^{1/2}$ ] being square - root of the positive definite diagonal matrix [ eq.([d_ud0u ] ) ] . to find @xmath65 , we start from eq . ( [ g_vp_a_vp_a ] ) and proceed as follows : @xmath186\end{aligned}\ ] ] where eq.([a_kk ] ) was used . because of @xmath187 [ eq . ( [ bc ] ) ] , we obtain : @xmath188 where , in the asymptotic limit , the transmission matrix @xmath65 in @xmath155-representation is given by @xmath189 the trace in eq.([g_tr_nl ] ) goes over all sites @xmath5 and orbitals @xmath190 of the atomic plane @xmath163 in the left lead ( fig . 2 ) . equivalent formula can be written for the right lead . to conclude , one can prove that spectrum of the obtained matrix coincides with spectrum of transmission matrix @xmath191 [ eq . ( [ tke ] ) ] defined in @xmath58-space ( see appendix b for details ) . therefore , solution of the eigenvalue problem for @xmath192 gives us required transmission probabilities . in practical calculations of conductance with the use of the kubo formula , the left ( l ) and right ( r ) planes are positioned _ somewhere _ in the leads ( fig . 2 ) . expression ( [ g_tracedgdg ] ) is valid in general case and result is exactly the same as in ref . . operator @xmath158 in eq.([g_tracedgdg ] ) is sum of two terms : @xmath193 . therefore , we can write down @xmath194 where @xmath195 and @xmath196 denote two contributions . in a formal theory , when the atomic plane @xmath93 is placed in the asymptotic region of the left lead far from the atomic constriction the second term @xmath196 in eq.([2_terms ] ) is equal to zero . in practice , the real space summation of current contributions includes only a finite number of sites at both atomic planes , because the current flow along @xmath42 direction is localized in the vicinity of the contact . due to numerical effort we are forced to take integration planes closer to the constriction in order to obtain convergent value for the conductance with respect to number of atoms included in summation . in addition , even better convergence for matrix elements is required to solve the eigenvalue problem . a compromise can be usually achieved but positions of the atomic planes @xmath93 and @xmath94 do not meet the asymptotic limit criterion . however , since the electron current through the structure is conserved , any position of the planes is suitable for the calculation of conductance . if @xmath93 is placed somewhere in the scattering region we have to sum up all multiple scattering contributions . we show in appendix a , that all multiple scattering contributions in direction of the current cause @xmath195 , whereas all scattering contributions in opposite direction give rise to @xmath196 . thus , the first term , @xmath195 , in eq . ( [ 2_terms ] ) is always positive , while the second one , @xmath196 , is always negative . to make this statement clear , an illustration of scattering events is shown in fig . 3 assuming a simple free - electron model . in the region of the lead where the potential is a small perturbation with respect to the bulk potential the contribution to the conductance due to @xmath196 is one order of magnitude smaller than @xmath195 . to find transmission probabilities of eigenchannels , one has to apply the procedure introduced in the previous section independently for both terms , @xmath195 and @xmath196 . we refer to appendix a for a mathematical justification . expression for conductance takes a form : @xmath197 with @xmath198 we show in appendix a that all eigenvalues of @xmath199 are either positive or zeros whereas all eigenvalues of @xmath200 are negative or zeros . to identify transmission probabilities @xmath2 of channels the spectra of operators @xmath199 and @xmath200 have to be arranged in a proper way . then transmission of the @xmath5-th channel is given by @xmath201 , where @xmath202 are positive and negative eigenvalues of the operators @xmath199 and @xmath200 , respectively . the @xmath2 does not depend on the positions of the left ( @xmath93 ) and right ( @xmath94 ) planes , while @xmath203 are @xmath42-dependent . in the asymptotic limit @xmath204 and @xmath205 , so that the landauer picture is restored . in general case the direct way to find the pairs of eigenvalues is not evident without a back transform to the @xmath58-space . however , from the point of view of applications to the extremely small symmetric atomic contacts , as the ones we are studying in this work , the problem is easy to handle . since the number of contributing eigenmodes is limited , the pairs of eigenvalues can be found by symmetry analysis of the eigenvectors of @xmath199 and @xmath200 . namely , using the symmetry properties of the structural green s function @xmath206 and current matrix elements @xmath207 one can show that channel s transmission @xmath2 bounded between 0 and 1 is defined by eigenvalues @xmath208 and @xmath209 which belong to the same irreducible representation of the symmetry point group . in recent papers @xcite we have verified the method described here by studying systematic changes in the conductance of metallic constrictions in the presence of defect atoms . illustrative examples presented below focus on single - atom contacts made of pure metals such as cu , ni , co and pd . copper serves mainly for test purposes . it is a representative of the noble metals and has electronic properties similar to ag and au , for which a lot of experimental results @xcite as well as dft based calculations @xcite are available . in particular , a large number of experiments for alkali metals ( li , na , k ) @xcite and noble metals ( au , ag , cu ) , @xcite employing different techniques under room an liquid - he temperatures , show that conductance histograms have a dominant peak very close to one conductance quantum @xmath0 , and smaller peaks close to integer values . however , for transition metal contacts ( examples of which are ni , co and pd ) the situation differs significantly.@xcite only one broad maximum centered somewhere between @xmath210 and @xmath211 is usually observed in conductance histograms.@xcite that is a signature of the nontrivial decomposition of conductance consisting of more than one perfectly transmitting channel , @xcite since for transition metal atoms @xmath212 states of different symmetry are available at the fermi level . the question on half - integer conductance quantization has been addressed.@xcite however , recent experiments @xcite do not confirm this hypothesis thus pointing to the conclusion that the electron transport through ferromagnetic contacts can never be fully spin - polarized . another issue is a large magnetoresistance ( mr ) effect @xcite observed for metallic point contacts made of different magnetic materials . this field is known to be full of controversy . there is still a continuing discussion on whether or not the enormous mr values found experimentally could be of electronic origin @xcite or the effect is just due to atomic rearrangements in the neck region of a contact as a response to the applied magnetic field.@xcite extensive discussion on this topic can be found in the recent review paper by marrows . @xcite we will comment further on the above issues . an atomic configuration of a constriction used in our calculations is presented in fig . 1 . the single atom contact was modeled by a small cluster attached to the semi - infinite fcc ( 001 ) leads . the cluster consists of two pyramids joined via the vertex atoms separated by a distance @xmath213 . here @xmath214 is the experimental lattice constant of fcc metals : 6.83 a.u . for cu , 6.66 a.u . for ni , 6.70 a.u . for co , and 7.35 a.u . for pd . the metals under consideration do not have a tendency to form chains.@xcite an atomic bridge is most likely to be broken just after the single atom limit is achieved . thus , a configuration shown in fig . 1 resembles one of limiting configurations of point contacts which could appear in the mcbj experiments . our calculations are based on dft within the local density approximation . the parametrization of vosko , wilk , and nusair @xcite for the exchange and correlation energy was used . the potentials were assumed to be spherically symmetric around each atom ( atomic sphere approximation , asa ) . however , the full charge density , rather than its spherically symmetric part , was taken into account . to achieve well converged results the angular momentum cut - off for the wave functions and the green s function was chosen to be @xmath215 that imposed a natural cut - off @xmath216 for the charge density expansion . in case of heavy element pd the scalar relativistic approximation@xcite was employed . for the conductance calculation the surface green s function was computed using a small imaginary part @xmath217 and about 250.000 were taken in the 2d brillouin zone . instead of integration over planes , current matrix elements ( [ d_lr ] ) were averaged over atomic layers as described in detail in ref . . a typical error in the calculation of conductance was @xmath218 . to understand the relation between the electronic structure and the transport properties of atomic contacts we consider the energy - dependent total transmission , @xmath219 , and its decomposition to the conduction eigenchannels , @xmath220 . results are shown in figs . 4,6,8,9 for the case of cu , ni , co and pd point contacts , respectively . the investigated structure ( fig . 1 ) has a @xmath221 symmetry . further we denote individual channels by the indices of irreducible representations of this group using notations of ref . , common in band theory . in addition , each channel can be classified according to the angular momentum contributions when the channel wave function is projected on the contact atom of the constriction . this is very helpful since the channel transmission can be related to the states of the contact atom.@xcite for example , the identity representation @xmath222 of the @xmath221 group is compatible with the @xmath223 , @xmath224 and @xmath225 orbitals ( here the @xmath42 is perpendicular to the surface and passes through the contact atom ) , while the two - dimensional representation @xmath226 is compatible with the @xmath227 , @xmath228 , @xmath229 , @xmath230 orbitals . the basis functions of @xmath231 and @xmath232 are @xmath233 and @xmath234 harmonics , respectively . the energy - dependent transmission of cu atomic contact ( shown in fig . 1 ) is presented in fig.4 together with the eigenchannel decomposition . at the fermi energy the calculated conductance value is @xmath235 . it mainly consists of one open channel of @xmath222 symmetry which arises locally from @xmath223 , @xmath224 and @xmath225 orbitals when the wave function is projected on the contact atom . this result is in a good agreement with a lot of experiments@xcite mentioned previously as well as with other calculations involving different approaches.@xcite the additional twofold degenerate channel has @xmath236 symmetry . transmission of this channel increases at energies above the fermi level ( @xmath237 ) together with an increase of the @xmath238 contribution to the local density of states ( ldos ) at the contact atom . however , at @xmath239 ev below the @xmath237 the @xmath236 channel is built mainly from the @xmath240 orbitals of the cu atom . we would like to point out that in case of noble metals the conductance of single - atom contact is not necessarily restricted to one channel . an example of a configuration which has more channels ( but still has only one cu atom at the central position ) is presented on the top of fig . 5 . because of the larger opening angle for incoming waves as compared with the preceding case , conductance of such system is @xmath241 with major contribution from four channels ( fig . 5 and its caption ) . the value @xmath242 correlates with a position of the third peak in the conductance histogram of cu , which is shifted from @xmath243 to smaller values.@xcite as it is seen from the presented example , conductance quantization does not occur for the metallic atomic - sized contacts . in general , even for noble metals , conduction channels are only partially open@xcite in contrary to the case of quantum point contacts realized in the two - dimensional electron gas where a clear conductance quantization was observed.@xcite for illustration , we present in fig . 5 probability amplitudes of the eigenchannles @xmath244 in real space resolved with respect to atoms of the 2nd plane ( s-2 ) below the surface . we see that the wave functions of the 1st and the 4th channel with the highest symmetry ( @xmath222 ) obey all eight symmetry transformations of the @xmath221 group , while two different wave functions of the double degenerate channel ( @xmath236 ) are transformed to each other after some symmetry operations . we turn to transition metals , and consider the ferromagnetic ni assuming a uniform magnetization of the sample . transmission @xmath219 split per spin of a ni contact is shown in fig . a shift about @xmath245 ev along the energy axis between transmission curves is seen that is in agreement with exchange - splitting of the ni @xmath212 states . similar computational results regarding transmission of ni constrictions were reported by solanki _ et al_.,@xcite rocha _ et al._@xcite and smogunov _ et al._.@xcite exchange splitting estimated from their works varies from 0.8 ev till 1.0 ev , but fine details differ because of different atomic configurations and employed methodologies . in this regard , exchange splitting about 2.0 ev observed in transport calculations of jacob _ et al._@xcite in case of ni contact seems to be overestimated . the shift in energy due to different spins is observed as well for the transmissions of individual channels ( fig . we see from fig . 7 and table i that at the fermi energy the spin - up ( majority ) conductance of ni contact is mainly determined by one open @xmath222 channel ( similar to the case of cu ) , while three partially open channels , of @xmath222 and @xmath236 symmetry , contribute to the spin - down ( minority ) conductance . the minority @xmath236 channel arises locally from @xmath229 and @xmath230 states , rather than from @xmath246 states whose contribution to the spin - down ldos at the fermi energy is much smaller ( fig . the calculated conductance , @xmath247 , correlates with a position of the wide peak in the conductance histogram of ni centered between @xmath210 and @xmath248.@xcite [ cols=">,^,^,^,^,^,^ " , ] coming to the experimental situation on ballistic magnetoresistance ( bmr ) effect in ferromagnetic contacts , we point out that large mr values @xcite were usually measured for much thicker constrictions ( as compared with atomic - sized contacts ) with resistance in the range of hundreds of ohms . it is believed,@xcite that such experiments suffer from many unavoidable artifacts induced by magnetomechanical effects that mimics the real mr signal which would come from the spin - polarized transport alone . however , recent studies by sullivan _ et al_.@xcite and chopra _ et al._@xcite on ni and co atomic - sized contacts report bmr values in the range of @xmath249 with discussion on the electronic origin of the effect due to domain wall scattering . nevertheless attempts to minimize magnetostrictive effects were undertaken , we just can repeat@xcite that a natural explanation of these@xcite and similar experiments @xcite is that , due to magnetization reversal processes , unstable in time atomic constriction changes its contact area when magnetic field is applied . characteristic steps and jumps in the measured field - dependent conductance ( fig . 4 of ref . ) or resistance ( fig . 3a of ref . , fig . 3 of ref . ) are distinct evidence of atomic reconstructions and fractional changes of the contact cross section . for example , just eliminating one contact atom from the configuration shown in fig . 1 changes conductance of a ni constriction from @xmath250 ( chain of two atoms , see table i ) up to @xmath251 ( one contact atom only , see ref . ) , thus producing @xmath252% mr . further increase of a contact area can give arbitrary high mr values , that supports hypothesis on mechanical nature of the effect . to summarize , we have presented a formalism for the evaluation of conduction eigenchannels of metallic atomic - sized contacts from first - principles . we have combined the _ ab initio _ kkr green s function approach with the kubo linear response theory . starting from the scattering wave formulation of the conductance problem , we have built a special representation of the transmission matrix in terms of local , energy and angular momentum dependent basis inherent to the kkr method . we have proven that solutions of the eigenvalue problem for the obtained matrix are identical to conduction eigenchannels introduced by landauer and bttiker . applications of the method have been presented by studying ballistic electron transport through cu , pd , ni and co single - atom contacts . the symmetry analysis of eigenchannels and its connection to the orbital classification known from the tight - binding approach were discussed in detail . experiments on the electron transport through magnetic contacts were commented . we acknowledge a financial support through the deutsche forschungsgemeinschaft ( dfg ) , priority programme 1165 : `` nanowires and nanotubes '' . we consider a practical implementation of the kubo formula ( [ cond_g ] ) : positions @xmath253 and @xmath254 of the @xmath255 and @xmath256 planes are chosen within the semi - infinite electrodes , where @xmath257-matrix describing a scattering region is sufficiently small ( for details , see fig . 2 ) . a sketch as how to find the conduction eigenchannels in this case has been given in sec . below we complete that discussion by presenting a mathematical justification of the method . we proceed in three steps : ( i ) since a position of @xmath255 and @xmath256 planes in eq . ( [ cond_g ] ) does not meet an asymptotic limit criterion , we expand the green s function in terms of scattering bloch states in contrary to expansion ( [ g_knl ] ) involving unperturbed states ; ( ii ) we use the kubo formula ( [ cond_g ] ) and express conductance as a trace of the appropriately defined current operator in @xmath58-space ; the obtained operator is identified with transmission matrix @xmath76 ; ( iii ) finally , we build up the equivalent site - angular momentum representation of the transmission matrix , that leads us to the required formulae [ eqs . ( [ g_2terms ] ) , ( [ tt ] ) of sec . iv.e ] . to begin the proof , we note that the asymptotic representation ( [ g_knl ] ) of the green s function can be rewritten as @xmath258_{\boldsymbol{kk'}}}\ \phi_{\boldsymbol{k'}}(\mathbf{r'},e)\end{aligned}\ ] ] where an expansion is performed over perturbed bloch waves @xmath259 and @xmath260 which were introduced in sec . we remind , that function @xmath261 is the solution of the lippmann - schwinger equation ( [ ls ] ) associated with the in - coming unperturbed bloch wave @xmath262 in the left lead propagating towards atomic constriction ( @xmath263 ) , while the second function , @xmath264 , is the solution of the equation associated with the in - coming unperturbed bloch wave @xmath265 in the right lead ( @xmath266 ) . consider now a general case , when @xmath253 and @xmath254 points related to planes @xmath255 and @xmath256 ( see fig . 2 ) are not taken infinitely far from atomic constriction . we note , that for the conductance evaluation one needs only the back - scattering term @xmath267 in eq . ( [ grr ] ) , which is the solution of the schrdinger equation without a source term . therefore , we can expand @xmath268 over the eigenfunctions of the whole system corresponding to energy @xmath104 . these are the propagating perturbed bloch states , @xmath269 and the evanescent states , @xmath270 where , in turn , both type of functions are expanded over atomic orbitals , and @xmath271 and @xmath272 are expansion coefficients . in particular , matrix @xmath273 is related to the matrix @xmath274 corresponding to the unperturbed bloch waves ( for details , see ref . ): @xmath275^{nn'}_{ll ' } { c}_{\boldsymbol{k},n'l'}(e)\ ] ] where the @xmath257-matrix describes the whole scattering region ( a vacuum barrier plus an atomic constriction ) and @xmath276 is the structural green s function of the three - dimensional periodic bulk crystal . both equations ( [ psik_pert ] ) and ( [ x_a ] ) can be joined to one matrix equation : @xmath277 \phi(\mathbf{r},e),\ ] ] where ( ) stands for a column - vector , while [ ] denotes a matrix . we also expand the conjugated states as @xmath278 \phi(\mathbf{r},e).\ ] ] we expand the back - scattering term @xmath267 over eigenfunctions @xmath279 and @xmath280 with energy @xmath104 : @xmath281 \left ( \begin{array}{c } \psi(\mathbf{r'},e ) \\ { \cal{x}}(\mathbf{r'},e ) \end{array } \right ) \\ \nonumber & = & \phantom{+\ } \sum_{\boldsymbol{k k ' } } \overline\psi_{\boldsymbol{k}}(\mathbf{r},e)\ , f^{00}_{\boldsymbol{kk'}}(e)\ , \psi_{\boldsymbol{k'}}(\mathbf{r'},e ) \\ \nonumber & & + \ \sum_{\boldsymbol{k}\beta } \overline\psi_{\boldsymbol{k}}(\mathbf{r},e)\ , f^{01}_{\boldsymbol{k}\beta}(e)\ , { \cal{x}}_{\beta}(\mathbf{r'},e ) \\ \nonumber & & + \ \sum_{\alpha\boldsymbol{k ' } } { \cal{x}}^{*}_{\alpha}(\mathbf{r},e)\ , f^{10}_{\alpha\boldsymbol{k'}}(e)\ , \psi_{\boldsymbol{k'}}(\mathbf{r'},e ) \\ \nonumber & & + \ \sum_{\alpha\beta } { \cal{x}}^{*}_{\alpha}(\mathbf{r},e)\ , f^{11}_{\alpha\beta}(e)\ , { \cal{x}}_{\beta}(\mathbf{r},e ) \\ \label{g_rr_e_fexp } & = & \delta g_{+}^{(1)}(\mathbf{r},\mathbf{r'},e ) + \delta g_{+}^{(2)}(\mathbf{r},\mathbf{r'},e).\end{aligned}\ ] ] only the first term @xmath282 exists in the asymptotic case , while other three terms , denoted as @xmath283 , contain contributions of the evanescent states which decay within the leads . the unknown matrix @xmath284 in eq . ( [ g_rr_e_fexp ] ) satisfies the equation ( below we will skip for simplicity the obvious energy dependence ) : @xmath285 \times \left [ \begin{array}{cc } f^{00 } & f^{01 } \\ f^{10 } & f^{11 } \end{array } \right ] \times \left [ \begin{array}{c } { \mathscr{c } } \\ \gamma \end{array } \right ] = g_{\mathrm{lr}},\ ] ] where symbol @xmath191 denotes transpose , and matrix @xmath286 is introduced which contains selected matrix elements of the structural green s function @xmath287 with @xmath288 and @xmath289 . to find matrices @xmath290 , we introduce the block - matrix @xmath291 $ ] which is a solution of the following equation ( here symbol @xmath120 denotes hermitian conjugate ) : @xmath292 \times \left [ { \cal{b}}^{\dagger}\ \delta^{\dagger } \right ] = \left [ \begin{array}{cc } { \mathscr{c } } { \cal{b}}^{\dagger } & { \mathscr{c}}\delta^{\dagger } \\ \gamma { \cal{b}}^{\dagger } & \gamma \delta^{\dagger } \end{array } \right ] = \left [ \begin{array}{cc } 1 & 0 \\ 0 & 1 \end{array } \right].\ ] ] this means , that matrix @xmath293 $ ] is the inverse or ( in general case ) the pseudoinverse @xcite matrix to the matrix with @xmath294- and @xmath295-blocks.@xcite . the equation similar to eq . ( [ eq_cg_bd ] ) exists for other matrices : @xmath296 \times \left [ \widetilde{\mathscr{c}}^t\ \widetilde\gamma^t \right ] = \left [ \begin{array}{cc } \widetilde{\cal{b}}^{*}\ , \widetilde{\mathscr{c}}^t & \widetilde{\cal{b}}^{*}\ , \widetilde{\gamma}^t \\ \widetilde{\delta}^{*}\ , \widetilde{\mathscr{c}}^t & \widetilde{\delta}^{*}\ , \widetilde{\gamma}^t \end{array } \right ] = \left [ \begin{array}{cc } 1 & 0 \\ 0 & 1 \end{array } \right],\ ] ] where symbol @xmath297 denotes complex conjugate of the matrix elements only . with the use of eqs . ( [ eq_cg_bd ] ) , ( [ eq_cg_bdx ] ) we find solution for @xmath284 : @xmath298 = \left [ \begin{array}{cc } \widetilde{\cal{b}}^ { * } g_{\mathrm{lr } } { \cal{b}}^{\dagger } & \widetilde{\cal{b}}^ { * } g_{\mathrm{lr } } { \delta}^{\dagger } \\ \widetilde{\delta}^ { * } g_{\mathrm{lr } } { \cal{b}}^{\dagger } & \widetilde{\delta}^ { * } g_{\mathrm{lr } } { \delta}^{\dagger } \end{array } \right],\ ] ] then expansion coefficients of the @xmath299 term read as @xmath300^{nn'}_{ll ' } { \cal{b}}^{\dagger}_{n'l',\boldsymbol{k'}}(e).\end{aligned}\ ] ] with help of matrix @xmath301 , @xmath302_{\boldsymbol{kk ' } } = -\sqrt{v_{\boldsymbol{\kappa}}}\ \frac{f^{00}_{\boldsymbol{kk'}}(e)}{2\pi i } \ \sqrt{v_{\boldsymbol{\kappa'}}}\ ] ] ( with positive velocities under square - roots ) we write down eq . ( [ g1_f ] ) as @xmath303_{\boldsymbol{kk ' } } \phi_{\boldsymbol{k'}}(\mathbf{r'},e).\end{aligned}\ ] ] comparing obtained equation with one for the asymptotic limit [ eq . ( [ g_pert_wf ] ) ] and taking into account that in fact @xmath301 is independent on the positions of the l and r planes , we obtain : @xmath304_{\boldsymbol{kk ' } } = \left[\tau^{-1}(e)\right]_{\boldsymbol{kk'}}.\ ] ] when conductance is evaluated , only @xmath305 term in expansion ( [ g_rr_e_fexp ] ) contributes . other term , @xmath306 , contains the evanescent states which decay within the leads and , therefore , have zero velocities along the current flow . when eq . ( [ g1_tau ] ) is inserted into the kubo formula ( [ cond_g ] ) we obtain : @xmath307_{\boldsymbol { k k ' } } { \cal{j}}^{\mathrm{r}}_{\boldsymbol{k'k'_1 } } \left[\frac{1}{\tau^{\dagger}_{\mathrm{rl } } } \right]_{\boldsymbol{k'_1 k_1}},\ ] ] where all matrices are evaluated at the fermi energy ( @xmath308 ) . here we have introduced the current operators related to the left ( l ) lead , @xmath309 \ = \ \bigl [ \tau^{\dagger}\tau \bigr]_{\boldsymbol{kk'}},\end{aligned}\ ] ] and to the right ( r ) one : @xmath310 \ = \ \bigl [ \tau\tau^{\dagger}\bigr]_{\boldsymbol{kk'}},\end{aligned}\ ] ] where basis @xmath66 of eigenchannels ( see sec . iii ) was used to evaluate the matrix elements . with help of ( [ jl_def ] ) and ( [ jr_def ] ) , equation ( [ g_jtjt ] ) for conductance @xmath311 takes a form : @xmath312 where @xmath313 is the unitary operator in the @xmath58-space : @xmath314 and eqs . ( [ tau_lr_tau ] ) and ( [ jr_def ] ) were used . when the perturbed bloch waves are expanded over atomic orbitals , @xmath315 current operators @xmath316 and @xmath317 take a form ( energy dependence is skipped ) : @xmath318^{nn'}_{ll ' } \widetilde{x}^{*\dagger}_{n'l',\boldsymbol{k ' } } , \\ \nonumber { \cal{j}}^{\mathrm{r}}_{\boldsymbol{kk ' } } & = & \sum_{nn'\in s_r}\sum_{ll ' } { x}_{\boldsymbol{k},nl } \left[d_{\mathrm{r } } \right]^{nn'}_{ll ' } { x}^{\dagger}_{n'l',\boldsymbol{k'}},\end{aligned}\ ] ] where operators @xmath319 and @xmath320 are defined as @xmath321 and operators @xmath158 and @xmath157 were introduced in eq . ( [ d_lr ] ) . according to eq . ( [ g_trjl ] ) conductance is determined by operator @xmath322 . the formal representation ( [ jlr_kk ] ) of the current operator @xmath323 in @xmath58-space is not suitable for practical implementation . it can be used , however , to find an equivalent representation in @xmath155-space . using ( [ jlr_kk ] ) together with representation of operator @xmath158 [ eqs . ( 23 ) , ( 24 ) ] we can decompose @xmath323 in two terms : @xmath324 where @xmath325^{\dagger}\ ] ] has only positive ( non - negative ) eigenvalues , while @xmath326^{\dagger}\ ] ] has only negative ( or zero- ) eigenvalues , and @xmath327 are diagonal matrices with nonnegative elements [ eqs . ( 23 ) , ( 24 ) ] . the advantage of operators @xmath328 over operator @xmath329 is that matrices @xmath330 are positive definite , while anti - symmetric matrix @xmath158 is not . as we show below in appendix b , that makes possible to find for operators @xmath331 an equivalent , site - angular momentum @xmath155 representation , while it is not the case for full operator @xmath332 . we mention also , that in the asymptotic limit the contribution due to @xmath333 vanishes . however , in general case , spectra of both operators should be found . to clarify these ideas , let us elucidate a physical meaning of eq . ( [ jl_2tems ] ) . consider new basis functions in @xmath155-space which are constructed with help of unitary matrix @xmath334 , @xmath335_{nl,\alpha},\ ] ] so that the velocity operator @xmath336 in @xmath155-space introduced in eq . ( 21 ) is diagonal in the new basis : @xmath337_{\alpha\beta } } \\ \nonumber & = & \int_{s_{\mathrm{l } } } ds \left [ \chi^{*}_{\alpha}(\mathbf{r},e)\ , i\!\stackrel{\leftrightarrow}{\partial_{z } } \chi_{\beta}(\mathbf{r},e)\right ] \ = \ \delta^{\circ}_{\alpha}(e ) \ , \delta_{\alpha\beta}.\end{aligned}\ ] ] let us examine further the wave function @xmath264 which defines the current operator @xmath316 [ eq . ( [ jl_def ] ) ] . this function is the solution of the lippmann - schwinger equation associated with the in - coming unperturbed bloch wave @xmath338 ( initial channel ) propagating from the right ( r ) lead towards the nanocontact and being scattered on it . within the left ( l ) lead ( @xmath339 ) we have : @xmath340 where @xmath341 here the ( @xmath342)-sums run over the half of the indices of the basis functions @xmath343 , corresponding either to the `` positive '' or to the `` negative '' window of the spectrum thus , the wave function @xmath264 in the l lead is a linear combination of two functions . the first one , @xmath344 carries the flux in the initial direction of the in - coming wave with momentum @xmath345 , from the l electrode to the r one . the second function , @xmath346 , carries the flux in the direction being opposite to the propagation direction of the in - coming wave . an example illustrating this idea for free electrons is shown in fig . one can check , that operator @xmath347 is defined on functions @xmath344 only , while @xmath348 is defined on @xmath349 : @xmath350_{\boldsymbol{kk ' } } = 2\pi \int\limits_{s_l } ds \left [ \overline\phi^{(\pm)\,*}_{\boldsymbol{k}}(\mathbf{r},e)\ , i\!\stackrel{\leftrightarrow}{\partial_{z } } \overline\phi^{(\pm)}_{\boldsymbol{k'}}(\mathbf{r},e)\right],\ ] ] because , according to eqs . ( [ dl_delta ] ) and ( [ phi_k_pm ] ) , the cross - terms involving both functions , @xmath351 and @xmath352 , vanish . obviously , that if the @xmath93 plane is chosen way behind the scattering region , the @xmath346 turns to zero and the contribution due to @xmath353 vanishes . in general case , both operators , @xmath354 and @xmath355 , should be considered . in this section we show that transmission probability of the conduction eigenchannel @xmath66 is given by @xmath356 , where @xmath357 and @xmath358 are positive and negative eigenvalues of operators @xmath359 and @xmath333 , respectively . assume , that we know matrix @xmath71 ( introduced in sec.iii ) which is the unitary transform to the basis of eigenchannels . according to ref . we have : @xmath360 thus , we have decomposed the wave function of each eigenchannel in two terms : @xmath361 . consider further the transmission matrix @xmath191 in the basis of eigenchannels : @xmath362_{\nu\mu } \\ & = & 2\pi \int_{s_{\mathrm{l } } } ds\ \left[\overline\phi_{\nu}^{\,*}(\mathbf{r},e)\ i\!\stackrel{\leftrightarrow}{\partial_{z } } \overline\phi_{\mu}(\mathbf{r},e ) \right ] \\ & = & t^{(+)}_{\nu\mu } + t^{(-)}_{\nu\mu},\end{aligned}\ ] ] with @xmath363.\ ] ] operators @xmath364 are related to current operators @xmath331 via the unitary transform @xmath71 : @xmath365 . the unitary transform @xmath71 diagonalizes the full operator @xmath366 . using the representation ( [ jlr_kk ] ) of operators @xmath331 we can check , that @xmath359 and @xmath333 do not commutate . thus , the @xmath364 operators do not have a diagonal form in the basis of eigenchannels . therefore , we represent each of these operators as sum of diagonal and off - diagonal terms : @xmath367 here @xmath368 , and off - diagonal contributions are of different signs because the sum of two matrices is the diagonal matrix @xmath369 . the diagonal terms are @xmath370 where @xmath371.\ ] ] and transmission probability of the eigenchannel is given by @xmath372 . here the first term arises due to all multiple scattering contributions in the direction of the current , while the second term is due to scattering contributions in the opposite direction . in spite of the fact that operators @xmath373 and @xmath374 can not be diagonalized within the unique unitary transform , one can show that off - diagonal terms @xmath375 are small . they are determined by functions @xmath376 with negative velocities with respect to the initial velocity of the in - coming channel @xmath66 incident from the right ( r ) lead . the function @xmath376 in the left ( l ) lead is small as compared with @xmath377 . when the @xmath93 plane is placed far enough from the surface , the @xmath376 collects all multiple scattering events in the opposite direction to the current flow . however , such scattering processes are possible only due to small inhomogeneities of the potential which is not exactly the bulk one around @xmath378 ( see fig . 3 ) . thus , we can write down : @xmath379 where parameter @xmath380 has a meaning of reflection amplitude ( see caption of fig . 3 ) . the @xmath381 is of the order of reflection probability , which ( in realistic calculations ) is @xmath382 . thus , off - diagonal terms , @xmath383 , contain small parameter @xmath384 . since @xmath385 , according to the perturbation theory @xcite the difference between eigenvalues of operators @xmath386 [ eq.([tpm_full ] ) ] and @xmath387 [ eq.([t0_diag ] ) ] _ appears only in the second order _ , which is @xmath388 . therefore , the operators @xmath331 provide a feasible way to find @xmath389 with good enough precision . to find the spectrum of positive definite operators @xmath390 we represent them in the form @xmath391 , where @xmath392 maps the site - orbital space on the @xmath58-space . we can prove ( see appendix b ) that spectrum of @xmath393 is the same as for operator @xmath394 which is defined in @xmath155-space and reads as @xmath395^{\dagger } { \cal{o}}^{\,\mathrm{lr}}\widetilde{x}^ { * } \left(\pm d^{\pm}_{\mathrm{l}}\right)^{1/2}.\ ] ] here we have used the unit operator in @xmath58-space @xmath396 introduced in eq . ( [ o_kk ] ) . by using equations ( [ tau1_kk ] ) , ( [ o_kk ] ) , ( [ jlr_kk ] ) , ( [ xc_1 ] ) , we obtain expression for operator @xmath397 : @xmath398^{\dagger } \left [ \tau^{-1}_{\mathrm{lr}}\ , { \cal{j}}_{\mathrm{r } } \bigl(\tau^{\dagger}_{\mathrm{rl}}\bigr)^{-1 } \right ] \widetilde{x}^ { * } \left(\pm d^{\pm}_{\mathrm{l}}\right)^{1/2 } \\ & = & \pm \left(\pm d^{+}_{\mathrm{l}}\right)^{1/2 } \bigl[\widetilde{\mathscr{c}}^{*}\bigr]^{\dagger } \left(\widetilde{\cal{b}}^{*}\ , g_{\mathrm{lr}}\ , { \cal{b}}^{\dagger } \right ) \mathscr{c}\ , d_{\mathrm{r}}\ , { \mathscr{c}}^{\dagger } \\ & & \times \left({\cal{b}}\ , g^{\dagger}_{\mathrm{rl}}\ , [ \widetilde{\cal{b}}^{*}]^{\dagger } \right ) \widetilde{\mathscr{c}}^ { * } \left(\pm d^{\pm}_{\mathrm{l } } \right)^{1/2 } \\ & = & \pm \left(\pm d^{\pm}_{\mathrm{l}}\right)^{1/2}\ , g_{\mathrm{lr}}\ , d_{\mathrm{r } } g^{\dagger}_{\mathrm{rl } } \left(\pm d^{\pm}_{\mathrm{l}}\right)^{1/2},\end{aligned}\ ] ] where orthogonality relation@xcite @xmath399^{\dagger } \widetilde{\cal{c}}^ { * } = 1 $ ] was used . thus , we come to eq . ( 30 ) of sec . let us consider operator @xmath400 acting in @xmath58-space of a following kind : @xmath401 where @xmath402 is a positive definite hermitian operator acting in the site - orbital @xmath155-space . let @xmath403 be a unitary transform which diagonalizes operator @xmath404 , @xmath405 our goal is to find a spectrum of @xmath404 with help of @xmath155-space . following sec.iii , we represent @xmath404 in the form : @xmath406 with @xmath407 and @xmath408 . a square - root from the positive definite hermitian operator is defined as @xmath409 where the unitary matrix @xmath165 transforms @xmath402 to the diagonal form @xmath410 . let us show that a spectrum of operator @xmath411 acting in the @xmath58-space , is the same as a spectrum of operator @xmath412 acting in the @xmath155-space . let @xmath413 be such matrix that @xmath414 solution for @xmath413 is @xmath415 . matrix @xmath413 diagonalizes @xmath416 : @xmath417 to finish the proof , one has to check that @xmath413 is indeed the unitary transform , i.e. the following properties should hold : ( i ) @xmath418 , and ( ii ) @xmath419 . first , we check property ( i ) : @xmath420 next , we check property ( ii ) : @xmath421^{-1 } \sigma^{\dagger } = 1,\end{aligned}\ ] ] so the proof is complete . ya . m. blanter and m. bttiker , physics reports * 336 * , 1 ( 2000 ) ; m. bttiker , phys . rev . b * 46 * , 12485 ( 1992 ) ; m. bttiker , phys . lett . * 57 * , 1761 ( 1986 ) ; m. bttiker , y. imry , r. landauer , and s. pinhas , phys . b * 31 * , 6207 ( 1985 ) . j. a. torres , j. i. pascual , and j. j. senz , phys . b * 49 * , 16581 ( 1994 ) ; j. a. torres , and j. j. senz , phys . lett . * 77 * , 2245 ( 1996 ) ; a. m. bratkovsky and s. n. rashkeev , phys . b * 53 * , 13074 ( 1996 ) . n. kobayashi , m. brandbyge and m. tsukada , jpn . . phys . * 38 * , 336 ( 1999 ) ; n. kobayashi , m. brandbyge and m. tsukada , phys . b * 62 * , 8430 ( 2000 ) ; n. kobayashi , m. aono , and m. tsukada , phys . b * 64 * , 121402(r ) ( 2001 ) ; k. hirose , n. kobayashi , and m. tsukada , phys . b * 69 * , 245412 ( 2004 ) . a. nakamura , m. brandbyge , l. b. hansen , and k. w. jacobsen , phys . * 82 * , 1538 ( 1999 ) ; k. s. thygesen , m. v. bollinger , and k. w. jacobsen , phys . rev . b * 67 * , 115404 ( 2003 ) ; p. jelnek , r. prez , j. ortega , and f. flores , phys . b * 68 * , 085403 ( 2003 ) ; y. fujimoto and k. hirose , phys . b * 67 * , 195315 ( 2003 ) ; k. palots , b. lazarovits , l. szunyogh , and p. weinberger , phys . b * 70 * , 134421 ( 2004 ) ; p. a. khomyakov and g. brocks , phys . b * 70 * , 195402 ( 2004 ) . m. brandbyge , j .- l . mozos , p. ordejn , j. taylor , and k. stokbro , phys . b * 65 * , 165401 ( 2002 ) ; j. taylor , h. guo , and j. wang , phys . b * 63 * , 245407 ( 2001 ) ; h. mehrez , a. wlasenko , b. larade , j. taylor , and p. grtter , h. guo , phys . rev . b * 65 * , 195419 ( 2002 ) . a. pecchia and a. di carlo , rep . phys . * 67 * , 1497 ( 2004 ) ; g. c. solomon , a. gagliardi , a. pecchia , th . frauenheim , a. di carlo , j. r. reimersa , and n. s. hush , j. chem . * 125 * , 184702 ( 2006 ) ; s. kurth , g. stefanucci , c .- o . almbladh , a. rubio , and e. k. u. gross , phys . b * 72 * , 035308 ( 2005 ) ; c. verdozzi , g. stefanucci , and c .- o . almbladh , phys . lett . * 97 * , 046603 ( 2006 ) . r. zeller , p. h. dederichs , b. ujfalussy , l. szunyogh and p. weinberger , phys . b * 52 * , 8807 ( 1995 ) ; r. zeller , phys . rev . b * 55 * , 9400 ( 1997 ) . n. papanikolaou , r. zeller , and p. h. dederichs , j. phys . : condens . matter * 14 * , 2799 ( 2002 ) . we note , that a similar result holds for the eigenfunctions @xmath422 of the system defined as superposition of scattered states coming from the right lead : @xmath423 , where the perturbed bloch state @xmath424 is the solution of eq . ( [ ls ] ) corresponding to the initial in - coming state @xmath425 in r ( @xmath33 ) . the notation with bar , @xmath422 , is used to distinguish from @xmath426 . taking into account the property of microscopic reversibility for the transmission amplitudes , @xmath427 , and eqs . ( [ theta ] ) and ( [ curr_bloch1 ] ) , we obtain : @xmath428_{z\to -\infty } = -\ , \frac{t_{\nu}(e)}{2\pi}\ , \delta_{\nu\mu}.\ ] ] since cell potential @xmath99 is real , both regular @xmath429 and irregular @xmath430 solutions of the radial schrdinger equation can be found as real valued functions . however , in practical implementation of the kkr method , we find @xmath431 as solution of the lippmann - schwinger equation for the incoming spherical bessel function @xmath432 , where @xmath433 is a real spherical harmonic . for example , in case of spherical potentials ( asa ) employed in this work , such defined functions @xmath431 carry multipliers @xmath434 $ ] , where @xmath435 is a scattering phase shift . without loss of generality , these phase factors can be ascribed to structure constants @xmath436 [ see eq . ( [ grr ] ) ] and solutions @xmath429 can be considered as real valued functions . this expression reads @xmath437 @xmath438 $ ] , where a matrix @xmath404 is introduced : @xmath439 $ ] , here @xmath440 is a volume of a unit cell , whereas @xmath441 is number of atoms in a supercell . because of orthogonality of the basis functions , @xmath442 \sim \delta_{ll'}\ , \delta(e - e')$ ] , the major contribution to eq . ( [ phi_nl ] ) comes from @xmath443-states with @xmath444 . this property is used to prove the second relation in eq . ( [ bc ] ) . w. f. egelhoff , jr . , l. gan , h. ettedgui , y. kadmon , c. j. powell , p. j. chen , a. j. shapiro , r. d. mcmichael , j. j. mallett , t. p. moffat , m. d. stiles , and e. b. svedberg , j. appl . phys . * 95 * , 7554 ( 2004 ) . recent _ ab intio _ calculations by burton _ ( see above ref . ) predict 340% domain wall mr in case of one - dimensional , fcc @xmath445 , ni wire . this result follows from the quantized conductance being @xmath446 in case of uniformly magnetized wire reduced down to @xmath447 in the presence of domain wall . however , one has to take into account that transmission probabilities of many wire channels even without domain wall will be significantly reduced ( especially those which are built from @xmath212 electrons ) when realistic geometry of a contact is considered . that can diminish mr to quite moderate values comparable to ones estimated in the present work . without a loss of generality , the number @xmath449 of @xmath58-points on the isoenergetic surface @xmath450 plus number of ( possibly existing ) evanescent states ( with energy @xmath104 ) @xmath451 can be chosen to be @xmath452 where @xmath453 is number of atoms in the l ( or r ) atomic planes , so that the inverse matrix operation is well defined . even if @xmath454 is not exactly equal to @xmath455 the rectangular matrix @xmath291 $ ] can be found as the pseudoinverse matrix ( ref . ) to the matrix with @xmath294- and @xmath295-blocks . from eq . ( [ eq_cg_bd ] ) we obtain : @xmath456 , and @xmath457 . in order the @xmath5-sum be converged , matrix elements of @xmath294 , @xmath458 , @xmath295 and @xmath459 should carry coefficients @xmath460 ( where @xmath461 is number of atoms per cross - section of the born - von krmn supercell ) . on the other hand , since matrix @xmath462 $ ] is the inverse to the one with @xmath294 and @xmath295 blocks , the following relation holds : @xmath463 . the second term in this equation is zero , because the number of evanescent ( surface ) states @xmath464 at given energy @xmath104 is much smaller than the total number of surface states @xmath465 , so that @xmath466 . finally , we obtain : @xmath467 . | we develop a formalism for the evaluation of conduction eigenchannels of atomic - sized contacts from first - principles .
the multiple scattering korringa - kohn - rostoker ( kkr ) green s function method is combined with the kubo linear response theory .
solutions of the eigenvalue problem for the transmission matrix are proven to be identical to eigenchannels introduced by landauer and bttiker .
applications of the method are presented by studying ballistic electron transport through cu , pd , ni and co single - atom contacts .
we show in detail how the eigenchannels are classified in terms of irreducible representations of the symmetry group of the system as well as by orbital contributions when the channels wave functions are projected on the contact atom . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
* the background and the problem . * broadcasting and gossiping are fundamental communication tasks in networks . in broadcasting , one node of a network , called the _ source _ , has a message that must be learned by all other nodes . in gossiping , every node has a ( possibly different ) input message , and all messages must be learned by all nodes . we study these well - researched tasks in a very weak communication model , called the _ beeping model_. communication proceeds in synchronous rounds . in each round , a node can either listen , i.e. , stay silent , or beep , i.e. , emit a signal . a node hears a beep in a round , if it listens in this round and if one or more adjacent nodes beep in this round . the beeping model has been introduced in @xcite for vertex coloring and used in @xcite to solve the mis problem . the beeping model is widely applicable , as it makes small demands on communicating devices , relying only on carrier sensing . in fact , as mentioned in @xcite , beeps are an even weaker way of communicating than using one - bit messages , as the latter ones allow three different states ( 0,1 and no message ) , while beeps permit to differentiate only between a signal and its absence . the network is modeled as a simple connected undirected graph . initially all nodes are dormant . the adversary wakes up the source in the case of broadcasting and some nonempty subset of nodes , at possibly different times , in the case of gossiping . a woken up node starts executing the algorithm . a dormant node is woken up by a beep of any neighbor . our aim is to provide fast deterministic algorithms for broadcasting and gossiping in the beeping model . the time of broadcasting is defined as the number of rounds between the wakeup of the source and the round in which all nodes of the network acquire the source message . the time of gossiping is defined as the number of rounds between the wakeup of the first node and the round in which all nodes acquire all messages . messages are considered as binary strings and the size of a message is the length of this string . in the case of broadcasting , our algorithm does not assume any information about the network , and it does not require any labeling of nodes . in the case of gossiping , we assume that all nodes have different labels from the set @xmath0 and that they know @xmath7 . moreover , we assume that all nodes know the same upper bound @xmath1 on the size of the network and the same upper bound @xmath4 on the size of all input messages . without loss of generality we may assume that @xmath8 . indeed , the parameter @xmath7 known to nodes is an upper bound on the size of the network , as all nodes have different labels . let @xmath2 be the diameter of the network , initially unknown to the nodes . * our results . * for the task of broadcasting we give an algorithm working in time @xmath5 for arbitrary networks , where @xmath3 is the size of the source message . this complexity is optimal . for the task of gossiping we give an algorithm working in time @xmath6 for arbitrary networks . due to space restrictions several proofs are moved to the appendix . * related work . * broadcasting and gossiping have been studied in various models for over four decades . early work focused on the telephone model , where in each round communication proceeds between pairs of nodes forming a matching , and nodes that communicate exchange all previously acquired information . deterministic broadcasting in this model has been studied , e.g. , in @xcite and deterministic gossiping in @xcite . in @xcite the authors studied randomized broadcasting . in the telephone model studies focused on the time of the communication task and on the number of messages it uses . early literature on communication in the telephone and related models is surveyed in @xcite . fault tolerant aspects of broadcasting and gossiping are surveyed in @xcite . more recently , broadcasting and gossiping have been studied in the radio model . while radio networks model wireless communication , similarly as the beeping model , in radio networks nodes send entire messages of some bounded , or even unbounded size in a single round , which makes communication drastically different from that in the beeping model . the focus in the literature on radio networks was usually on the time of communication . deterministic broadcasting in the radio model was studied , e.g. , in @xcite and deterministic gossiping in @xcite . randomized broadcasting was studied in @xcite and randomized gossiping in @xcite . the book @xcite is devoted to algorithmic aspects of communication in radio networks . randomized leader election in the radio and in the beeping model was studied in @xcite . deterministic leader election in the beeping model was studied in @xcite . the authors showed an algorithm working in time @xmath9 in networks of diameter @xmath2 with labels polynomial in the size @xmath10 of the network . in this section we consider the simpler of our two communication tasks , that of broadcasting . even for this easier task , the restrictions of the beeping model require the solution of the basic problem of detecting the beginning and the end of the transmitted message . the naive idea would be to adapt the method of beeping waves , used in @xcite in a different context , and transmit a message by coding bit 1 by a beep and bit 0 by silence , other nodes relaying these signals after getting them . however , in this coding there is no difference between message @xmath11 and message @xmath12 because both these messages are coded by a single beep . hence we need to reserve some sequence of beeps to mark the beginning and end of a message , and code bits by some other sequences of beeps and silent rounds . one way of defining such a coding is the following . consider the message @xmath13 that has to be transmitted by some node @xmath14 . let @xmath15 denote a round in which @xmath14 beeps , and let @xmath16 denote a round in which @xmath14 is silent . the beginning and end of the message are marked by the sequence @xmath17 , bit @xmath18 is coded by @xmath19 , and bit @xmath20 is coded by @xmath21 . hence message @xmath22 is transmitted as the sequence @xmath23 , where + @xmath24 , for @xmath25 , and + @xmath26 and @xmath27 , if @xmath28 , for @xmath29 , + @xmath30 and @xmath31 , if @xmath32 , for @xmath29 . a node @xmath33 hearing the sequence @xmath23 of beeps and silent rounds can correctly decode message @xmath22 as follows . upon hearing two beeps in the first two rounds it divides the successive rounds into segments of length 2 and records all beeps and silent rounds until a segment with two consecutive beeps . each segment between the two segments @xmath17 is either of the from @xmath19 or of the form @xmath21 . the node @xmath33 decodes each segment @xmath19 as 1 and each segment @xmath21 as 0 . in this way , message @xmath22 is correctly reconstructed . note that the time of transmitting this message is @xmath34 and hence linear in its size . we will call the above sequence of beeps and silent rounds , chosen by a node @xmath14 for the message @xmath22 , the _ canonical sequence _ for @xmath22 transmitted by node @xmath14 . the way of reconstructing message @xmath22 by node @xmath33 is called the _ canonical decoding_. using canonical sequences we formulate our broadcasting algorithm that works for arbitrary networks . * algorithm broadcast * let @xmath35 be the round in which the source is woken up . given a source message @xmath22 , the source transmits the canonical sequence for @xmath22 in rounds @xmath36 , for @xmath37 , and stops . more precisely , if @xmath38 , then the source beeps in round @xmath36 , and if @xmath39 , then the source is silent in this round . in all other rounds the source is silent . every other node @xmath33 that is woken up by a beep in round @xmath40 , beeps in round @xmath41 . if it hears a beep in a round @xmath42 , for some positive integer @xmath43 , it beeps in round @xmath44 . in all other rounds node @xmath33 is silent . it divides all rounds @xmath42 , for @xmath45 , into segments of length 2 . after the second segment when it hears @xmath17 , the node decodes the source message using the canonical decoding of the sequence received in rounds @xmath42 , for @xmath45 . then it beeps in the next round and stops . @xmath46 [ broad - trees ] algorithm broadcast is a correct broadcasting algorithm working in any network of diameter @xmath2 in time @xmath5 , where @xmath3 is the size of the source message . the time complexity of this algorithm is optimal . consider any network and define its @xmath47th layer as the set of nodes at distance @xmath47 from the source . consider any node @xmath33 in the @xmath47th layer of this network , other than the source , and suppose that @xmath33 is woken up in round @xmath40 . hence nodes in layer @xmath48 beep only in rounds @xmath42 , nodes in layer @xmath47 beep only in rounds @xmath44 , and nodes in layer @xmath49 beep only in rounds @xmath50 , for some integer @xmath43 . consequently , in rounds @xmath42 the node @xmath33 hears a beep in exactly these rounds in which nodes from the layer @xmath48 beep . by induction on the distance of @xmath33 from the source we get that @xmath33 gets correctly the canonical sequence for the source message and hence decodes the source message correctly . if the source is woken up in round @xmath35 and @xmath33 is at distance @xmath47 from the source , it starts receiving transmissions in round @xmath51 and stops in round @xmath52 . since @xmath53 , it follows that the algorithm works in time @xmath5 . in order to prove that this time complexity is optimal , it is enough to show that every algorithm requires at least time @xmath54 and at least time @xmath55 . the first fact is immediate because the farthest node from the source must be at distance at least @xmath56 from it , and no signal can get to this node from the source faster than after @xmath56 rounds . the second fact holds even in the two - node network . suppose that some broadcasting algorithm transmits every message of size @xmath3 from one node of this network ( the source ) to the other , in time @xmath57 . the number of sequences of length @xmath58 with terms from the set @xmath59 is @xmath60 . since the number of possible source messages of size @xmath3 is @xmath61 , it follows that for two distinct source messages the source must behave identically , and hence the input of the other node is identical . hence the other node must decode the same message in both cases , which contradicts the correctness of the algorithm . @xmath62 in this section we investigate the more complex task of gossiping . one way to accomplish this task is to have each node broadcast its input message . however , in the highly contrived beeping model , periods of broadcasting by different nodes should be disjoint , otherwise messages , transmitted as series of beeps and silent rounds , risk to become damaged , when a node receives simultaneously a beep being part of the transmission of one message and should hear a silent round being part of the transmission of another message . the receiving node will then just hear the beep and the transmission of the message requiring this round to be silent becomes scrambled . in order to broadcast in disjoint time intervals , nodes must establish an order between them and reserve the @xmath47th time interval to the broadcast of the @xmath47th node in this order . this yields the following high - level idea of a gossiping algorithm . first we establish a procedure that finds the node with the largest label . this is done in such a way that all nodes learn the largest label . ( notice that we can not use the leader election algorithm from @xcite because this algorithm works only under an additional strong assumption that all nodes that are woken by the adversary and not by hearing a beep are woken simultaneously in the first round of the algorithm execution . ) next , using this elected leader , all nodes are synchronized : they agree on a common round , and hence can simultaneously start the rest of the algorithm execution . then the procedure of finding the largest - labeled node is repeated at most @xmath1 times , where @xmath1 is an upper bound on the size of the network , each time the currently found largest - labeled node withdrawing from the competition . in this way , after at most @xmath1 repetitions all nodes know the order between them , and subsequently they broadcast their values in disjoint time intervals , in this order . we assume that all nodes know the size @xmath7 of the label space . let @xmath63 and let @xmath64 be the binary representation of the integer @xmath65 . we assume that all sequences @xmath64 are of length @xmath66 , the representations being padded by a prefix of 0 s , if necessary . hence the representation of a smaller integer is lexicographically smaller than the representation of a larger integer . we also assume that all nodes know a common upper bound @xmath1 on the number of nodes in the network . our first procedure finds the largest label among a set @xmath67 called the set of _ participating nodes_. * procedure find max * when a node is woken up in round @xmath68 , either by the adversary or by a beep , it defines the following round numbers : @xmath69 , for @xmath70 . then , for @xmath71 , the node defines the time interval @xmath72 $ ] . note that these intervals are pairwise disjoint . the node beeps in round @xmath73 . the rest of the procedure is divided into @xmath66 stages , corresponding to time intervals @xmath74 , for @xmath71 . first assume that the node is participating . let @xmath75 be the binary representation of the node s label , of length @xmath66 . in the beginning of the first stage the node is _ active_. if the node is still active at the beginning of the @xmath43th stage , then it behaves as follows . if @xmath76 , the node listens in all rounds of the time interval @xmath74 until it hears a beep . if it does not hear any beep , it remains active and proceeds to stage @xmath77 . if it hears a beep for the first time in some round @xmath35 , then it beeps in round @xmath78 and becomes non - active . if @xmath79 , the node listens until it hears a beep or until round @xmath80 , whichever comes earlier . if it hears a beep in some round @xmath81 , it beeps in round @xmath78 , listens till the end of time interval @xmath74 and remains active . otherwise , it beeps in round @xmath80 , listens till the end of time interval @xmath74 and remains active . if the node is non - active at the beginning of the @xmath43th stage , then it listens in all rounds of the time interval @xmath74 until it hears a beep . if it does not hear any beep , it remains non - active and proceeds to stage @xmath77 . if it hears a beep for the first time in some round @xmath35 , then it beeps in round @xmath78 , listens till the end of time interval @xmath74 and remains non - active . a non - participating node is never active . it listens in all rounds of the time interval @xmath74 until it hears a beep . if it does not hear any beep , it remains non - participating and proceeds to stage @xmath77 . if it hears a beep for the first time in some round @xmath35 , then it beeps in round @xmath78 , listens till the end of the stage and remains non - participating . at the end of stage @xmath66 , the ( unique ) participating node that remained active till the end of this stage is the node with the largest label among participating nodes . @xmath46 [ find - max ] at the end of the execution of procedure find max , there is exactly one active participating node . this node has the largest label among participating nodes . all nodes know the label of this node . procedure find max takes time @xmath82 . the goal of our next procedure is synchronizing all processors . the procedure will be used upon completion of procedure find max , and hence we assume that the largest label is known to all nodes . let @xmath83 be the node with the largest label . upon completion of procedure synchronization , each node declares a specific round to be _ red _ , and this round is the same for all nodes . * procedure synchronization * each node other than @xmath83 has an integer variable @xmath84 initially set to 0 . let @xmath85 . for every integer @xmath86 , let @xmath87 be the binary representation of @xmath47 of length @xmath88 , padded by a prefix of 0 s , if necessary . a string @xmath87 will be transmitted by nodes of the network , using the canonical sequence , as it was done for broadcasting in section 2 . we briefly recall this coding . let @xmath15 denote a round in which @xmath14 beeps , and let @xmath16 denote a round in which @xmath14 is silent . in @xmath89 rounds , node @xmath14 transmits the following message @xmath90 . the beginning and end of the message are marked by the sequence @xmath17 , every bit @xmath18 of @xmath87 is coded by @xmath19 , and every bit @xmath20 of @xmath87 is coded by @xmath21 . at the beginning of the procedure node @xmath83 transmits @xmath91 starting in round @xmath78 , where @xmath35 is the round in which @xmath83 completed the execution of procedure find max . after completing this transmission , node @xmath83 waits till round @xmath92 and declares it to be _ red_. every node other than @xmath83 that is at @xmath84 0 waits until it hears two consecutive beeps . then it partitions the following rounds into consecutive segments of length 2 , and decodes each segment of the form @xmath19 as 1 and each segment of the form @xmath21 as 0 . as soon as it hears a segment @xmath93 consisting of two beeps , it considers the previously decoded string of bits as the binary representation of an integer @xmath43 . it changes the value of its variable @xmath84 to @xmath77 , transmits @xmath94 in @xmath89 rounds , starting in the round @xmath40 following the segment @xmath93 , then waits till round @xmath95 and declares it to be _ red_. @xmath46 [ synch ] all nodes declare the same round to be _ red_. procedure synchronization takes time @xmath96 . we prove the following invariant by induction on @xmath97 . 1 . in time interval @xmath98 $ ] the only message transmitted is @xmath99 and it is transmitted by all nodes at distance @xmath97 from node @xmath83 and only by these nodes ; 2 . a node at distance @xmath97 from the node @xmath83 declares round @xmath92 as _ red_. the invariant is clearly satisfied for @xmath100 . suppose that it holds for some @xmath101 . the only nodes that hear the beeps transmitted in time interval @xmath98 $ ] are those at distance at most @xmath102 from node @xmath83 . the only nodes among them that have value of @xmath84 0 are nodes at distance exactly @xmath102 from node @xmath83 . since all the nodes at distance @xmath97 from @xmath83 transmit the same message @xmath99 in this time interval , they all beep exactly in the same rounds of the interval . hence the value of @xmath97 is correctly decoded by all nodes at distance @xmath102 from node @xmath83 . these nodes , and only these nodes , transmit @xmath103 in the time interval @xmath104 $ ] . this proves the first part of the invariant . all these nodes set their value of @xmath84 to @xmath102 and declare as _ red _ the round @xmath105 , where @xmath40 is the last round of the preceding time interval , i.e. , @xmath106 . hence the declared round is @xmath107 , which proves the second part of the invariant . this implies , in particular , that part 2 of the invariant is true for nodes at any distance @xmath97 from node @xmath83 , and hence all nodes of the network declare the same round as _ red_. since there are at most @xmath1 time intervals used in the procedure and each of them has length @xmath108 , the entire procedure takes time @xmath96 . @xmath62 as we mentioned at the beginning of this section , we want to use our broadcasting algorithm many times , each time starting from a different node . in order to take advantage of the efficiency of broadcasting , which depends on the diameter and not on the size of the network , all nodes need to have a linear upper bound on the diameter of the network . note , that in order to accomplish one execution of this algorithm , from one source node , no such upper bound was needed . it becomes needed for multiple broadcasts , as we want to execute each of them in a separate time interval , and thus we need a good estimate of the time of each execution . recall , that we assume knowledge of a bound @xmath1 on the size of the network but no knowledge of any such bound on the diameter . clearly @xmath1 is an upper bound on the diameter as well , but may be vastly larger than the diameter . the following procedure is devoted to obtaining a linear upper bound on the diameter @xmath2 of a network . it will be executed after the execution of procedure find max and procedure synchronization . hence we may assume that the largest of all labels is known to all nodes . let @xmath83 be the node with this label . we may also assume that all nodes declared the same round @xmath40 as _ red_. moreover , each node other than @xmath83 has its variable @xmath84 set to its distance from @xmath83 ( this is done in procedure synchronization ) . * procedure diameter estimation * each node defines consecutive time intervals @xmath109 $ ] , for positive integers @xmath43 . in time interval @xmath110 each node at level @xmath111 beeps in round @xmath111 of this interval . for @xmath112 , if a node at @xmath84 @xmath111 heard a beep in round @xmath113 of interval @xmath114 , then it beeps in round @xmath111 of interval @xmath115 . in the first round @xmath16 of the form @xmath116 in which @xmath83 does not hear a beep , it sets @xmath117 . let @xmath22 be the binary representation of the integer @xmath118 and let @xmath3 be the size of message @xmath22 . all nodes execute algorithm broadcast with node @xmath83 as the source and message @xmath22 as the source message . in this execution the role of round @xmath35 in which @xmath83 is woken up is played by round @xmath119 . every node decodes the integer @xmath120 as an upper bound on the diameter @xmath2 of the network . all nodes declare round @xmath121 as _ blue_. @xmath46 [ diamest ] upon completion of procedure diameter estimation all nodes have the same linear upper bound @xmath122 on the diameter of the graph . they all declare the same round as _ blue _ , and procedure diameter estimation is completed by this round . procedure diameter estimation takes time @xmath123 . if nodes know a linear upper bound @xmath122 on the diameter of the network , procedure find max can be modified to work faster . the modifications are detailed below . * procedure modified find max * in procedure find max replace the wakeup round @xmath68 by some round @xmath124 , given as input in its call . round @xmath125 will be called the _ starting round _ of the procedure . let @xmath126 instead of @xmath69 , for @xmath70 . let @xmath127 $ ] instead of @xmath72 $ ] , for @xmath71 . the rest of procedure find max remains unchanged . @xmath46 the proof of the following proposition is the same as that of lemma [ find - max ] , using the above modifications . [ prop1 ] at the end of the execution of procedure modified find max , there is exactly one active participating node . this node has the largest label among participating nodes . all nodes know the label of this node . procedure modified find max takes time @xmath128 . using procedure modified find max we now establish the order between all nodes as follows . the procedure will be called after the execution of procedure find max , procedure synchronization and procedure diameter estimation . hence we assume that @xmath83 is the node with the largest label , found by procedure find max , and that @xmath15 is the common _ blue _ round found by all nodes in procedure diameter estimation . all nodes start procedure ordering in round @xmath129 . let @xmath130 be an upper bound on the duration of procedure modified find max , established in proposition [ prop1 ] . * procedure ordering * @xmath131 the set of all nodes except node @xmath83 node @xmath83 assigns itself rank 0 * for * @xmath132 * to * @xmath1 * do * + execute procedure modified find max in the time interval + @xmath133 $ ] , with the set @xmath134 of participating nodes ; + the node found in the current execution of procedure modified find max + removes itself from @xmath134 and assigns itself @xmath135 . @xmath46 [ ranks ] ranks assigned to nodes in the execution of procedure ordering define a strictly decreasing function of the node labels . procedure ordering is completed in round @xmath136 and takes time @xmath137 . since after each execution of procedure modified find max , the node with the largest label among participating nodes is found , and this node stops participating in the following executions , the rank assigned to the @xmath43th largest node is @xmath138 . since there are @xmath1 time intervals , each of length @xmath139 , the time estimate follows . @xmath62 we are now ready to formulate a gossiping algorithm working for arbitrary networks . let @xmath4 be the upper bound on the size of all input messages , known to all nodes . let @xmath140 be the upper bound on the duration of algorithm broadcast established in theorem [ broad - trees ] , for the value @xmath122 of the diameter and for the size @xmath4 of the message . * algorithm gossiping * 1 . execute procedure find max ; let @xmath83 be the node with the largest label ; 2 . execute procedure synchronization ; let @xmath40 be the _ red _ round found in this procedure ; 3 . execute procedure diameter estimation starting in round @xmath41 ; let @xmath122 be the upper bound on the diameter of the network found in this procedure ; let @xmath15 be the _ blue _ round found in procedure diameter estimation ; 4 . execute procedure ordering starting in round @xmath129 ; 5 . let @xmath74 be the time interval @xmath141 $ ] ; 6 . in time interval @xmath74 execute algorithm broadcast with node of rank @xmath43 , found in procedure ordering , as the source , and the input message of this node as the source message . ( for each @xmath43 , the role of the round @xmath35 when the source is woken up is played by the first round of interval @xmath74 . ) @xmath46 * remark . * note that , in the first step of the algorithm , we have to use procedure find max instead of the more efficient procedure modified find max because at this point the only estimate on the diameter , known to nodes , is @xmath142 . however , since this procedure is executed only once , it does not increase the time complexity of the entire algorithm . algorithm gossiping is a correct gossiping algorithm working in any network of diameter @xmath2 with at most @xmath1 nodes in time @xmath6 , where @xmath4 is an upper bound on the size of all input messages , known to all nodes . by lemma [ find - max ] , procedure find max correctly finds the node @xmath83 with the largest label . by lemma [ synch ] , all nodes compute the same round @xmath40 , and hence start procedure diameter estimation in the same round @xmath41 . by lemma [ ranks ] , there is at most one node with any rank @xmath143 . by lemma [ diamest ] , @xmath122 is a linear upper bound on the diameter of the network . hence , in view of theorem [ broad - trees ] , the upper bound @xmath144 is indeed an upper bound on the time of execution of algorithm broadcast starting from any source node . this implies that all nodes broadcast their messages in pairwise disjoint time intervals , and hence all broadcasts are correct , in view of theorem [ broad - trees ] . this proves the correctness of algorithm gossiping . it remains to estimate the execution time of the algorithm . procedure find max takes time @xmath82 . procedure synchronization takes time @xmath96 . procedure diameter estimation takes time @xmath145 . procedure ordering takes time @xmath137 . at most @xmath1 executions of algorithm broadcast in step 6 of algorithm gossiping take time @xmath146 . hence algorithm gossiping takes time @xmath6 . @xmath62 we conclude with the following lower bound on the time of gossiping that holds even for the class of trees of bounded diameter . [ lb ] assume that all input messages have size @xmath147 for some constant @xmath148 . then there exist bounded diameter trees of size @xmath149 for which every gossiping algorithm takes time @xmath150 . we considered two basic communication tasks , broadcasting and gossiping , in the arguably weakest communication model , in which in every round each node has only the choice to beep or to listen . for the task of broadcasting , we proposed an optimal algorithm working in time @xmath5 for arbitrary networks of diameter @xmath2 , where @xmath3 is the message size . for the task of gossiping , we presented an algorithm working in time @xmath6 for arbitrary networks of diameter @xmath2 with at most @xmath1 nodes . here @xmath4 denotes an upper bound on the size of all input messages , known to all nodes . it remains open if this complexity can be improved in general . note however , that our gossiping algorithm has optimal time for networks of diameter bounded by a constant , if the following two assumptions are satisfied : the size of all input messages is the same , it is known to all nodes , and satisfies @xmath147 for some constant @xmath148 , and the size @xmath7 of the label space is polynomial in @xmath1 . indeed , in this case @xmath151 and @xmath152 . since for bounded @xmath2 , we have @xmath153 , algorithm gossiping works in time @xmath154 in this case , which matches the lower bound @xmath150 , shown in proposition [ lb ] . the two above assumptions do not seem to be overly strong . indeed , in most applications , we want messages exchanged by gossiping nodes to be large enough to hold at least the node s label and some other useful information , which justifies the assumption @xmath147 for some constant @xmath148 . on the other hand , labels of nodes are often assumed to be polynomial in the size of the network , as there is usually no need of larger labels . notice , moreover , that if these assumptions are satisfied , the complexity of our gossiping algorithm is @xmath155 , i.e. , it exceeds the lower bound @xmath150 only by a factor of @xmath2 , for any value of the diameter @xmath2 . thus , our gossiping algorithm is not only optimal for networks of bounded diameter , but it is close to optimal for `` shallow '' networks , e.g. , networks whose diameter is logarithmic in their size , such as the hypercube . let @xmath156 be the earliest round in which some node is woken up by the adversary . ( there may be several nodes woken up in round @xmath156 ) . since each node beeps in the round following its wakeup ( either by the adversary or by a beep ) , all nodes are woken up until round @xmath157 . in order to prove the claim , first note that stage @xmath43 of node @xmath14 starts after round @xmath158 . for any node @xmath159 and any non - negative integer @xmath47 , let @xmath160 denote the round @xmath161 computed by node @xmath159 ( relative to its wakeup round ) . since non - active nodes beep in some round @xmath162 only if they heard a beep in round @xmath163 , it follows that if @xmath14 heard a beep in some round @xmath35 of its stage @xmath43 , then an active node @xmath33 must have beeped in some round @xmath164 , such that no node beeped in round @xmath165 . according to the procedure , the only reason for such a beep is that , for some @xmath166 , @xmath167 , node @xmath33 is active in its stage @xmath166 , and that @xmath168 , where @xmath75 is the binary representation of the label of @xmath33 . suppose that @xmath169 . this implies @xmath170 , hence @xmath171 . hence the wakeup rounds of nodes @xmath14 and @xmath33 differ by more than @xmath1 , which is a contradiction . thus @xmath166 can not be smaller than @xmath43 . for similar reasons , @xmath166 can not be larger than @xmath43 . this leaves the only possibility of @xmath172 , which proves the claim . next , we prove that the node @xmath83 with the largest label among participating nodes remains active till the end of its stage @xmath66 . let @xmath173 be the binary representation of this label . suppose , for contradiction , that @xmath83 becomes passive in some stage @xmath174 . according to the procedure , this implies that it heard a beep in stage @xmath43 and that @xmath175 . in view of the claim , there is a node @xmath33 active in stage @xmath43 , with the binary representation @xmath176 of its label , such that @xmath79 . consider any index @xmath169 . if @xmath177 then also @xmath178 . otherwise , since node @xmath83 is active in stage @xmath166 , it beeps in stage @xmath166 and hence node @xmath33 would become passive in stage @xmath166 , which contradicts the fact that it became passive only in stage @xmath43 . this proves that the sequence @xmath173 is lexicographically smaller than the sequence @xmath176 , which contradicts the assumption that @xmath83 has the largest label among participating nodes . further , we prove that no node other than the node @xmath83 with the largest label among participating nodes remains active till the end of its stage @xmath66 . let @xmath33 be such a node with the binary representation @xmath176 of its label , and let @xmath16 be the first index where @xmath179 . since @xmath83 is active in stage @xmath16 and @xmath180 , node @xmath83 beeps in stage @xmath16 , and hence @xmath33 hears it and becomes passive ( at the latest ) in stage @xmath16 . it follows that @xmath83 is the only participating node that is active at the end of the execution of procedure find max . node @xmath83 knows that it remained active at the end , so it knows that it has the largest label . every other node @xmath33 ( it is passive at the end of the procedure execution ) deduces the binary representation @xmath173 of the label of @xmath83 as follows : @xmath181 , if and only if @xmath33 heard a beep in stage @xmath47 . indeed , if @xmath181 then it beeped in stage @xmath47 , because it was active in this stage , and hence @xmath33 heard a beep at most @xmath1 rounds later , hence still in its stage @xmath47 . if @xmath182 , then no node beeped in stage @xmath47 because all nodes that have 1 as the @xmath47th term of the binary representation of their label must be already passive in stage @xmath47 , as this representation is lexicographically smaller than @xmath173 . let @xmath184 be the largest distance of any node from @xmath83 . for every node other than @xmath83 , the value of the variable @xmath84 is its distance from @xmath83 . hence in time interval @xmath115 exactly nodes at distance at most @xmath185 from @xmath83 beep . it follows that @xmath186 is the first round of the form @xmath116 in which @xmath83 does not hear a beep . consequently @xmath187 . by the correctness of algorithm broadcast , all nodes correctly decode the integer @xmath120 . since @xmath156 is the largest distance of any node from @xmath83 , the diameter @xmath2 of the network satisfies inequalities @xmath188 . thus @xmath122 is a linear upper bound on the diameter of the network . after decoding the integer @xmath122 , all nodes know @xmath1 , @xmath40 , @xmath156 and @xmath3 . hence the round declared as _ blue _ is the same for all nodes . it was follows from the proof of theorem [ broad - trees ] that algorithm broadcast takes time at most @xmath189 . since @xmath83 starts broadcasting in round @xmath119 , by round _ blue _ the procedure is completed . it takes time @xmath190 . consider the star @xmath67 with @xmath191 leaves , i.e. , a tree with one node @xmath14 of degree @xmath191 and with @xmath191 nodes of degree 1 . @xmath67 is a tree of diameter 2 . let @xmath33 be any leaf . consider any algorithm @xmath192 accomplishing gossiping in @xmath67 in time @xmath35 . the leaf @xmath33 can obtain information only from node @xmath14 . in time @xmath35 node @xmath14 can transmit @xmath193 sequences with terms in the set @xmath59 , where @xmath15 denotes a beep and @xmath16 denotes silence . consider the family @xmath194 of possible sets of input messages initially held by the @xmath191 nodes of @xmath67 other than @xmath33 , assuming that each node has a different message of size @xmath4 . if @xmath195 , then node @xmath14 executing algorithm @xmath192 must generate the same sequence with terms in the set @xmath59 for two distinct sets of messages initially held by the @xmath191 nodes of @xmath67 other than @xmath33 , and consequently @xmath33 can not correctly deduce the set of messages held by other nodes of @xmath67 . this implies that @xmath196 . | broadcasting and gossiping are fundamental communication tasks in networks . in broadcasting ,
one node of a network has a message that must be learned by all other nodes . in gossiping
, every node has a ( possibly different ) message , and all messages must be learned by all nodes .
we study these well - researched tasks in a very weak communication model , called the _ beeping model_. communication proceeds in synchronous rounds . in each round
, a node can either listen , i.e. , stay silent , or beep , i.e. , emit a signal .
a node hears a beep in a round , if it listens in this round and if one or more adjacent nodes beep in this round .
all nodes have different labels from the set @xmath0 .
our aim is to provide fast deterministic algorithms for broadcasting and gossiping in the beeping model .
let @xmath1 be an upper bound on the size of the network and @xmath2 its diameter .
let @xmath3 be the size of the message in broadcasting , and @xmath4 an upper bound on the size of all input messages in gossiping .
for the task of broadcasting we give an algorithm working in time @xmath5 for arbitrary networks , which is optimal .
for the task of gossiping we give an algorithm working in time @xmath6 for arbitrary networks .
* keywords : * algorithm , broadcasting , gossiping , deterministic , graph , network , beep . at the time of writing this paper
we were unaware of the paper + a. czumaj , p. davis , communicating with beeps , arxiv:1505.06107 [ cs.dc ] which contains the same results for broadcasting and a stronger upper bound for gossiping in a slightly different model . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
since the first announcement of a planet orbiting a sun - like star , 51 peg , by @xcite , around 140 stars with planets ( swps ) have been reported . the main properties of swps have been investigated by many works . among these , spectroscopic observations of most swps based on high - resolution and high signal - to - noise ratio spectra have been carried out . all of these works indicate a high mean metallicity of swps relative to normal field stars . are there any other chemical signatures for swps ? there is no conclusive result on this issue , and only a few works have reported abundance anomalies in some elements . for example , @xcite suggested a high [ c / fe ] ratio for swps as compared with the normal stars from @xcite . however , they later reported that there is no difference in abundance for c , na , mg , and al elements between swps and normal stars @xcite . present knowledge on the abundances of 25 elements ( except for li and be ) available in the literature shows no special feature for swps except for the high overall metallicity . for lithium abundance , @xcite claimed that @xmath0 was detected in the atmosphere of hd 82943 , while @xcite did not confirm this detection . @xcite reported no @xmath1 presence in several lithium - poor stars . for @xmath2 , @xcite first found several swps with low li abundances , and they related them with the presence of a planetary companion . @xcite concluded that smaller li abundances are found for swps after correcting for trends with temperature , metallicity , and age . however , @xcite suggested that there is no significant difference in lithium abundance for swps , and later work by @xcite agreed with ryan s conclusion . with a large sample , @xcite suggested again that in the temperature range 56005850k , swps show additional depletion of li compared to normal stars . they also showed that for higher temperatures , the li abundances were similar for swps and for normal stars . recently , measurements of be abundances by @xcite indicated further evidence on the composition difference between stars with and without planets . lithium is a special element because it is easily destroyed in stars during even pre main sequence and main sequence evolution , which is not the case for other elements . therefore , li abundance reflects the mixing history of the star . for swps , li is further related with the accretion of material and the angular momentum evolution of the system , and thus li abundances have been used to distinguish the current hypotheses on planet formation mechanisms . in this respect , it is important to verify whether li abundance anomalies for swps at the temperature range of 56005850k reported by @xcite are real , which will provide new information on the planet formation process and its influence on host stars . in this work we conduct a parallel abundance analysis of lithium abundances for 16 swps and 20 comparison stars in the same metallicity of @xmath3}}\sim 0.1 $ ] dex at the temperature range of 56005900k in order to investigate whether any anomalies persist . furthermore , this study is quite interesting for our understanding of the lithium behavior of solar metallicity stars , which show a large scatter at @xmath4 k and which the standard model of stellar evolution can not explain . with a consistent analysis procedure , this work aims to investigate the effect of planet presence on stellar evolution . this will provide new information on the chemical evolution of lithium in the galaxy . swps are selected from a catalog in the temperature range of 56005900k , and their spectra are available from our observations with the 2.16 m telescope of the national astronomical observatories ( xinglong , china ) with a resolving power of 37000 , or from @xcite , who observed with the 1.93 m telescope of the observatoire de haute provence with r@xmath542000 . @xcite present a database of high resolution spectra of 709 stars covering a large range of atmospheric parameters , which enables us to select a comparison sample of stars spanning the same ranges in temperature , gravity and metallicity as the above sample of planet stars but not having been reported to harbor planets . that is , both samples cover the atmospheric parameters of @xmath4 k , @xmath6 , and @xmath3 } } > -0.3 $ ] . in the selection , strmgren @xmath7 indices from @xcite are adopted and the metallicity is estimated based on the calibrations by @xcite . along with the initial metallicity , temperature and gravity are derived in the same way as described below . all spectra have signal - to - noise ratios above 150 pixel@xmath8 at the @xmath96707 line . the spectra were reduced using standard midas ( for xinglong data ) and iraf ( for elodie data ) routines for order definition , background correction , flat - fielding , extraction of echelle orders , wavelength calibration , and continuum fitting . the lines suitable for measurement were carefully selected , and the equivalent widths ( ews ) were obtained by the fitting of a gaussian function . atomic data for lines were the same as those in @xcite . the wavelengths and oscillator strengths of the @xmath96707 line are taken from @xcite . crccrrrrrrrr star & @xmath10 & @xmath11 & @xmath12 & @xmath3}}$ ] & li & age & ew(1 ) & @xmath13 & ew(2 ) & ew(3 ) & ew(4 ) + hd & k & & @xmath14 & & & gyr & ma & ma & ma & ma & ma + + sun & 5780 & 4.44 & 1.15 & @xmath150.03&@xmath16 & 4.6 & ... & 1.2 & ... & ... & ... + 12661 & 5682 & 4.26 & 1.50 & 0.30 & 1.74 & 9.6&16.6 & 0.8 & ... & ... & ... + 16141 & 5752 & 4.13 & 1.40 & 0.08 & @xmath17 & 7.6 & ... & 1.9 & @xmath18 & ... & ... + 23596 & 5906 & 4.09 & 1.60 & 0.31 & 2.78 & 5.9&86.7 & 1.6 & ... & ... & ... + 33636 & 5817 & 4.41 & 1.30 & @xmath150.03 & 2.29 & 5.8&40.3 & 1.5 & 49.0 & ... & ... + 72659 & 5846 & 4.16 & 1.15 & 0.04 & 2.25 & 6.3&35.4 & 0.6 & ... & ... & ... + 82943 & 5858 & 4.32 & 1.40 & 0.20 & 2.37 & 6.0&44.7 & 2.2&44.1 & ... & ... + 92788 & 5683 & 4.36 & 1.40 & 0.18&@xmath19 & 6.6 & ... & 1.7 & ... & ... & ... + 95128 & 5755 & 4.23 & 1.30 & @xmath150.00 & 1.68 & 9.3 & 10.5 & 2.0&12.5 & ... & 18.0 + 106252 & 5769 & 4.32 & 1.00 & 0.01 & 1.72 & 9.6&13.5 & 2.1 & ... & ... & ... + 134987 & 5735 & 4.27 & 1.15 & 0.24 & @xmath20 & 8.3 & ... & 1.2&@xmath18 & ... & ... + 143761 & 5701 & 4.26 & 1.20 & @xmath150.21 & 1.37 & 10.8&6.2 & 1.1 & 5.9 & ... & 6.0 + 150706 & 5764 & 4.43 & 1.20 & 0.01&2.40 & 7.5 & 53.3 & 1.4 & 55.4 & ... & ... + 187123 & 5717 & 4.26 & 1.15 & 0.11&@xmath21 & 8.8 & ... & 1.8 & ... & ... & ... + 195019 & 5729 & 4.07 & 1.15 & 0.03 & 1.55 & 7.3 & 9.9 & 1.8 & 6.9 & ... & ... + 217014 & 5654 & 4.27 & 1.15 & 0.20&@xmath22 & 10.4 & ... & 0.8&4.4 & ... & ... + + 4307 & 5748 & 3.95 & 1.30 & @xmath150.20 & 2.38 & 6.9&53.9 & 1.9 & 52.3 & ... & ... + 4614 & 5818 & 4.31 & 1.20 & @xmath150.23 & 2.08 & 10.3&21.7 & 1.0 & 26.3 & 24.0 & 21.0 + 9562 & 5786 & 3.99 & 1.20 & 0.18 & 2.51 & 4.6&61.0 & 1.7&61.2 & ... & ... + 15335 & 5797 & 3.93 & 1.40 & @xmath150.15 & 2.55 & 7.0&65.7 & 0.7 & ... & 57.9 & 52.0 + 34411 & 5800 & 4.24 & 1.20 & 0.07 & 2.07 & 7.2&24.3 & 1.1 & 25.7 & 27.4 & ... + 39587 & 5833 & 4.41 & 1.40 & @xmath150.00 & 2.85 & 5.8&103.9 & 1.1&101.3 & 105.6 & 95.0 + 52711 & 5778 & 4.31 & 1.30 & @xmath150.10 & 1.84 & 9.2&17.2 & 1.5&17.5 & ... & ... + 70110 & 5880 & 3.93 & 1.40 & 0.10 & 2.49 & 3.9&53.2 & 1.9&52.0 & 55.8 & ... + 76151 & 5702 & 4.39 & 1.20 & 0.05 & 1.80 & 6.0&18.2 & 1.6&18.7 & ... & ... + 79028 & 5818 & 4.05 & 1.50 & 0.03 & 2.65 & 7.2&76.1 & 1.4&74.1 & 76.5 & 11.0 + 84737 & 5822 & 4.11 & 1.30 & 0.17 & 2.32 & 5.7&42.9 & 1.5&41.8 & ... + 88986 & 5750 & 4.09 & 1.30 & @xmath150.02 & 1.97 & 7.2&23.6 & 1.5&26.2 & ... & ... + 109358 & 5742 & 4.28 & 1.20 & @xmath150.26 & 1.59 & 12.0&10.7 & 0.8 & 9.3 & ... & 9.0 + 114710 & 5920 & 4.39 & 1.30 & 0.13 & 2.56 & 2.3 & 63.3 & 1.0&57.8 & ... & ... + 115383 & 5918 & 4.19 & 1.30 & 0.19 & 2.78 & 5.2 & 79.2 & 1.3&83.1 & ... & ... + 141004 & 5806 & 4.17 & 1.40 & @xmath150.02 & 1.79 & 8.2 & 14.7 & 1.5&16.0 & ... & 20.0 + 182572 & 5648 & 4.12 & 1.40 & 0.15&@xmath23 & 8.8 & ... & 2.4&@xmath24 & ... & ... + 190406 & 5797 & 4.38 & 1.40 & 0.04 & 2.26 & 5.6 & 39.4 & 1.8 & 37.6 & ... & ... + 193664 & 5795 & 4.39 & 1.20 & @xmath150.11 & 2.27 & 6.1 & 39.9 & 1.2&34.0 & ... & ... + 196755 & 5675 & 3.66 & 1.40 & @xmath150.04 & 1.28 & 3.0 & 6.6 & 1.3&10.8 & ... & ... + [ fig : plot1 ] [ fig : plot2 ] [ fig : plot3 ] [ fig : plot2 ] [ fig : plot3 ] effective temperatures were determined from the strmgren @xmath25 color index @xcite using the calibration of @xcite . the gravities were determined via _ hipparcos _ parallaxes @xcite , and the microturbulences were obtained by forcing lines with different strengths to give the same abundances . * here the oscillator strengths of the lines are taken from @xcite . * the new metallicity derived from spectroscopic analysis was updated , and the procedure of atmospheric parameter determination was iterated for consistency , even though there is no systematic difference between the initial metallicity and the new value , with a deviation of @xmath26 dex . the internal errors of the atmospheric parameters are around 70k in temperature , 0.1 dex in @xmath11 , 0.1 dex in @xmath3}}$ ] , and 0.3 @xmath14 in microturbulence . but the absolute uncertainties of the stellar parameters may be slightly larger . we have found that the photometry - based temperatures in the work are 130 k lower than the spectroscopically derived values in @xcite for swps in common . however , our temperatures are 83 k higher than those in @xcite , who derive temperatures by spectroscopic method for 29 stars in common . it seems that spectroscopically derived temperatures have significant uncertainties , and the values depend on stellar model atmosphere , the selection of lines , the adopted atomic data , and the way the temperatures are determined . in addition , as already noted by @xcite , surface gravities determined from the balance of and abundances are systematically higher than those derived from parallaxes ( adopted in our work ) by the order of 0.150.20 dex . however , iron abundances in this work and @xcite are similar since we adopted iron abundances from lines and they forced and lines to give the same iron abundances when they derived gravities . for the comparison stars , there are 15 stars in common with @xcite , and our temperatures are 87k lower than their values . this is due to different temperature calibrations , and our values , based on the infrared flux method temperatures by @xcite , may be more reliable . actually , the absolute temperature and gravity are not very important , since we are carrying out a differential analysis , and it is crucial that the stellar parameters all be in the same scale between the target and the comparison sample . the model atmospheres were interpolated from a grid of plane - parallel , lte models provided by @xcite in which convective overshoot is switched off . the abontest8 program developed by p. magain in the lige group was used to carry out the calculations of theoretical line ews , and abundances were derived by matching the theoretical ews to the observed values . abundance errors , estimated from the uncertainties of atmospheric parameters and ews , are 0.07 dex for iron abundances , and less than 0.1 dex for li abundances . here the contribution of the @xmath27 isotope to the @xmath96707 line was assumed to be negligible . finally , non - lte corrections were applied to the derived li abundances based on the work by @xcite , who studied non - lte formation of the @xmath28 line as a function of effective temperature , gravity , metallicity , and li abundance . stellar parameters and li and fe abundances for swps and the comparison stars , as well as the ews of the lines and their errors are shown in table 1 . li abundances as a function of metallicity for all stars in the present survey are shown in fig . 1 , in which stars with downward directed arrows have an undetected lithium line , and the symbol size corresponds to stellar age . the upper limits of the li abundances are estimated by assuming ews of 3.0 m , which is the estimated error in ew from the comparison used in @xcite . based on table 1 , the scatter of the ew comparison between this work and @xcite , @xcite , and @xcite is around 4.0 m , which indicates an error of @xmath29 massuming the same uncertainties for the ews from the other works . note that the statistical uncertainties based on the formula by @xcite , presented in table 1 , are smaller . since experimental errors by different authors during spectrum normalization and ew measurements are not considered in cayrel s ( 1988 ) formula , we believe that an upper limit to the ews of 3.0 m for stars with an undetected li line may be more reasonable . an interesting result from fig . 1 is that there is a lack of normal stars with @xmath3 } } > + 0.2 $ ] . this suggestion is supported by large samples of stars in the literature . there are 22% normal stars with @xmath3 } } > + 0.2 $ ] in @xcite s sample at the temperature range of 56005900k , but note that their temperatures , derived from the fitting of synthetic spectra with observed spectra , are 107k higher than our values for stars in common , and the sample is too small for statistics . with photometry - based temperatures and a larger sample of stars from @xcite , this fraction is reduced to 2% , indicating a low probability of super metal rich stars in the solar neighbourhood . this is probably due to the fact that most metal rich stars tend to have formed planetary systems , and thus it is difficult to obtain a comparison sample that consists of super metal rich stars not harboring a planet . this is consistent with the current knowledge that swps have higher mean metallicities than normal stars . the second feature from fig . 1 is that six of the 16 swps versus two of the 20 normal stars have undetected lithium lines . all of the six swps with undetected li lines are main sequence stars while the two normal stars with depleted li abundances are subgiants , as shown below ( fig . 5 ) . it seems that main sequence swps in the temperature range of 56005900k destroy their li much more easily than do normal stars . a kolmogorov - smirnov test is applied to stars with and without planets , and the cumulative distribution functions are shown in fig . it shows that the maximum value of the absolute difference between the two distribution functions is 0.54 , and the probability for the two samples being the same group is 0.007 . this is also evident from @xcite , as shown in fig . 3 , in which eight of 16 main sequence swps at @xmath30 k only have upper limits for li abundances , while three of the 14 normal main sequence stars from @xcite are depleted in li . note that the two comparison samples , from this work and @xcite , are independently selected with six common stars . thus , this property could be intrinsic , but further investigations with large samples of both swps and normal stars are needed . if this suggestion that main sequence swps have a higher probability of depleting their lithium abundances is true , it will provide new information on stellar evolution for swps . we suspect that the stellar evolution history between swps and normal stars could be somewhat different due to the presence of a planet , which appears to affect only li abundances , with no other elements showing any difference . as suggested by @xcite , the presence of planets or associated circumstellar disks may affect a parent star s initial angular momentum and/or subsequent evolution . specifically , the conservation of angular momentum in the protoplanetary disk may induce increased mixing by causing rotational breaking in the host stars during the pre main sequence evolution . this effect , once it happens , will destroy the li completely . considering this suggestion as one possibility , @xcite proposed a second hypothesis : that migration triggers tidal forces and creates a shear instability that leads to a strong depletion of li abundance . both mechanisms occur during the pre main sequence evolution , and thus they can be used to explain our result . further theoretical and observational studies are very desirable to investigate whether the two mechanisms are reasonable . 4 shows li abundances versus temperatures for all stars . it seems that li abundance generally increases with increasing @xmath10 , and there is no significant gap between swps and normal stars at a given temperature . 5 shows the stellar positions of the two samples of stars in the hr diagram as compared with the evolutionary tracks of @xcite with @xmath3}}\sim + 0.12 $ ] dex . stellar ages are also estimated from this comparison , and they are presented in table 1 . the error of the ages due to the uncertainties of @xmath10 , @xmath31 , and @xmath3}}$ ] is about 15% . it is clear that our two samples of stars are generally located in the same regions of the hr diagram , and during the selection all these stars are classified as main sequence stars with similar parameters . however , the two stars from the comparison sample showing depleted li abundances , as indicated by downward directed arrows in fig . 5 , are actually subgiants . other possible subgiant stars are not evolved far away from the main sequence , and there is no difference in li abundance between swps and normal stars . this observation that subgiants tend to have depleted li abundances reminds us to investigate the stellar positions of swps at @xmath30 k in @xcite . here we redetermined the temperatures for swps with @xmath30 k from @xcite in the same way as @xcite . in fig . 3 , the stellar positions of these swps with new temperatures are plotted against normal stars from @xcite in this temperature range with @xmath3 } } > -0.3 $ ] . it is clear that a few swps in israelian et al.s ( 2004 ) sample based on the updated temperatures are actually evolved subgiants , and downward directed arrows in fig . 3 indicate stars with undetected @xmath96708 lines . the fact that evolved subgiants tend to have depleted li abundances is consistent with the knowledge of stellar evolution theory . it has been suggested in @xcite that the li abundances of subgiants can be somewhat depleted due to the dilution proces without complete destruction , as opposed to main sequence stars . in this scenario , once a star evolves onto the subgiant branch , it dilutes surface li when convection brings li - poor material from deep layers to the surface . this dilution is not obvious for slightly evolved subgiants in the present study , and a few subgiants with 1.2 @xmath32 show higher li abundances than main sequence stars . moreover , there is no difference in li abundance between planet hosts and normal stars on the subgiant branch . actually , the dilution process mentioned above is difficult to detect , and its effect is masked by the large scatter of li abundances for even slightly evolved subgiants . it is impossible to investigate how the presence of a planet will affect lithium evolution for evolved subgiants in the present work , since their li abundances have already been depleted due to their low temperatures . as shown in fig . 4 , there is a substantial scatter of li abundances at @xmath33 k. considering the uncertainty in temperature , it is difficult to investigate whether the scatter is real . there seems to be a tendency that stars in the upper envelope of the li versus @xmath10 diagram seem to be slightly younger than stars with the lowest li abundances in the @xmath34 k range . however , since there is a lack of young stars in the low temperature range and of old stars at the high temperature edge in our sample , we refrain from drawing any firm conclusions . a larger sample of stars with different ages at a given temperature is needed to clarify this issue . the excess of li depletion in swps in the temperature range of @xmath30 k suggested by @xcite is based on the fact that li abundances for swps are lower than for comparison stars . a closer comparison shows that the average of li abundances for 24 swps at this temperature range in @xcite is @xmath35 , while this value is @xmath36 for the comparison stars from @xcite . in our work the average li abundance is @xmath37 for swps versus @xmath38 for the comparison stars . as presented above , this anomaly in li abundance of swps might reflect the effect of planet presence on stellar evolution and can not be used to provide constraints on planet formation . in addition , this effect is difficult to detect from stars in the subgiant branch or later stages if their li abundances have already been depleted by other mechanisms . moreover , the deviations of the average li abundance between stars with and without planets are 0.27 dex for @xmath39 k and 0.58 dex for @xmath40k . it seems that li anomalies concentrate in the range of @xmath40k , and thus they do not relate to the li dip in the hyades , which happens in the temperature range of 63006600k as first found by @xcite . in connection with planet formation mechanism , our result does not support the accretion scenario of planet formation . theoretically , @xcite predicted quantitative estimates of the main sequence evolution of stellar surface lithium after planet ingestion . they found that the preservation of @xmath41 occurs in a large mass range of 0.91.3 @xmath32 at solar metallicity . for swps with a typically super - solar metallicity , the mass range is slightly reduced . according to their theory , if a 1 @xmath42 planet were dissolved in the convective envelope of an swp at a sufficiently old age for extra mixing to be inefficient , it would produce an enhancement of 0.3 dex in li abundance , which is unfortunately within the scatter of li abundance at the temperature range we investigated . but li abundances for stars with @xmath43 k in @xcite do not show such a large scatter , and the similar abundance between swps and the comparison stars argues against the accretion scenario . furthermore , most swps known so far have giant planets with masses larger than @xmath44 , and if @xmath45 were engulfed the enhancement would reach 0.9 dex , which is easily detected . therefore , we favor the scenario that the surface composition of swps is not polluted by the accretion of planetary material , and the high mean metallicity of swps is primordial . we have made a parallel abundance analysis for 16 swps and 20 normal stars at the metallicity of @xmath46 } } < + 0.4 $ ] and the temperature range of @xmath4 k. our results show that there is a higher probability for swps to have depleted li abundances than for comparison stars , which confirms the possible li abundance anomalies for swps at this temperature range reported by @xcite after excluding evolved subgiants . we proposed that the presence of planets around host stars may affect stellar evolution by inducing additional mixing or shear instability , which leads to a high probability of destroying surface lithium while other elements are not affected . this effect can only be detected for main sequence swps since other depletion mechanisms would not have started to act on main sequence stars . in view of this , the effect of planet presence should be considered when the large scatters of the galactic lithium in field disk stars at the temperature range of @xmath4 k are investigated . this work is supported by the national natural science foundation of china under grants 10433010 and 10203002 . alonso , a. , arribas , s. & martnez - roger , c. 1996 , a&as , 117 , 227 * boesgaard , a.m. , & tripicco , m.j . 1986 , apj , 302 , l49 * carlsson , m. , rutten , r.j . , bruls , j.h.m.j . , & shchukina , n.g . 1994 , a&a , 288 , 860 cayrel , r. 1988 , iau symp . , 132 , 345 chen , y.q . , nissen , p.e . , zhao , g. et al . , 2000 , a&as , 141 , 491 chen , y.q . , nissen , p.e . , benoni , t. , & zhao , g. 2001 , , 371 , 943 chen , y.q . , nissen , p.e . , zhao , g. , & asplund m. 2002 , , 390 , 925 dantona , f. , & mazzitelli , 1994 , apjs , 90 , 467 edvardsson , b. , anderson , j. , gustafsson b. , lambert , d.l . , nissen , p.e . , tomkin , j. , 1993 , , 275 , 101 gonzalez , g. , & laws , c. 2000 , , 119 , 390 gonzalez , g. , lawsi , c. , tyagii , s. , & reddy , b.e . 2001 , aj , 285 , 403 gustafsson , b. , karlsson , t. , olsson , e. , edvardsson , b. , & ryde , n. 1999 , a&a , 342 , 426 israelian , g. , santos , n. , mayor , m , & rebolo , r. 2001 , nature , 411 , 163 israelian , g. , santos , n. , mayor , m , & rebolo , r. 2004 , , 414 , 211 king , j.r . , deliysnnis , c.p . , hiltgen , d.d . et al . , 1997 , aj , 113 , 1871 kurucz , r.l . , 1995 , cdrom , no . 23 lambert , d.l . , heath , j.e . , & edvardsson , b. , 1991 , mnras , 253 , 610 mandell , a.m. , & ge , j. 2004 , aj , 127 , 1147 mayor , m. , & queloz , d. 1995 , nature , 378 , 357 montalbn , j. , & rebolo , r. , 386 , 1043 nordstrm , b. , mayor , m. , andersen , j. , et al . , 2004 , a&a , 418 , 989 olsen e.h . , 1983 , a&as 54 , 55 olsen e.h . , 1993 , a&a 102 , 89 perryman , m.a.c . , 1997 , the _ hipparcos _ and tycho catalogs ( esa sp-1200 , noordwijk : esa ) prugniel , p. , & soubiran , c. 2001 , a&a , 369 , 1048 reddy , b. , lambert , d.l . , laws , c. , gonzalez , g. , & covery , k. 2002 , , 572 , 1012 ryan , r.g . 2000 , , 316 , l35 santos , n.c . , israelian , g. , garca lpex , g.j et al . , 2004 , , 427 , 1085 schuster , b.w . , & nissen , p.e . 1989 , a&a , 221 , 65 smith , v.v . , lambert , d.l . , & nissen , p.e . 1998 , , 506 , 405 takeda , y. , & kawanomoto , s. , 2005 , pasj , 57 , 45 valenti , j.a , & fischer , d.a . , 2005 , apjs , 159 , 141 vandenberg , d.a . , swenson , f.j . , rogers , f.j . , iglesias , c.a . , & alexander , d.r . 2000 , , 532 , 430 | we have investigated the abundance anomalies of lithium for stars with planets in the temperature range of 56005900k reported by israelian and coworkers , as compared to 20 normal stars at the same temperature and metallicity ranges .
our result indicates a higher probability of lithium depletion for stars with planets in the main sequence stage .
it seems that stellar photospheric abundances of lithium in stars with planets may be somewhat affected by the presence of planets .
two possible mechanisms are considered to account for the lower li abundances of stars with planets .
one is related to the rotation - induced mixing due to the conservation of angular momentum by the protoplanetary disk , and the other is a shear instability triggered by planet migration .
these results provide new information on stellar evolution and the lithium evolution of the galaxy . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
the nuclear rotation is one of the most typical collective motions in nuclei @xcite . its semiclassical nature makes it possible to introduce the classical concepts like the rotational frequency and the transformation to the rotating frame , which are very useful to analyze the high - spin rotational bands @xcite and nowadays provide a standard method called the cranked shell model @xcite . however , the nucleus is a quantum many - body system and the basis of such a semiclassical treatment of the rotational motion is the symmetry breaking caused by the deformed mean - field , see e.g. @xcite . in fact , the rotational motion emerges as a symmetry restoring collective motion of atomic nucleus as a whole and can be described full - quantum mechanically by the angular - momentum - projection method @xcite . although a nice rotational spectrum can be obtained by the angular - momentum - projection from the deformed mean - field state , it has been known that the level spacing of the obtained rotational spectrum tends to be larger than that of the experimental data ; i.e. the moment of inertia is quite often underestimated . inclusion of the time - odd components into the deformed mean - field , from which the projection performed , improves this problem and it can be easily realized by the cranking procedure @xcite . it has been demonstrated that the small cranking frequency is enough to increase the moment of inertia and the result of projection does not depend on the actual value of the frequency ; we call it `` infinitesimal cranking '' for the angular - momentum - projection @xcite . recently we have extended the study of the rotational motion by the angular - momentum - projection method @xcite . namely the cranking procedure is combined with the projection by employing the configuration - mixing with respect to the finite rotational frequency ; we call it angular - momentum - projected multi - cranked configuration - mixing . this method was originally proposed by peierls and thouless long time ago @xcite , but has not been taken seriously . we have applied it to a few examples to show that it gives a reliable description of the rotational motion at high - spin states @xcite . the angular - momentum - projected configuration - mixing with respect to a few cranked mean - field states has recently been performed also in ref . @xcite . as for the application of the angular - momentum - projection method to the nuclear rotational motion , many pioneering works have been done by the projected shell model , see e.g. ref . @xcite . while the basic idea is the same , much larger but simple configurations , like the zero- , two- , four- , ... , quasiparticle excited bands are mixed in the sense of the shell model . we believe the multi - cranked configuration - mixing @xcite is an alternative , which incorporates a relatively small number of mean - field configurations with the help of the cranking procedure . the main purpose of the present work is to demonstrate that the multi - cranked configuration - mixing is indeed a reliable method to describe the rotational band with the angular - momentum - projection method . we first apply the method to the ground - state rotational bands for a number of selected nuclei in the rare earth region . at high - spin states , it is well - known that the band crossing ( back - bending ) phenomenon between the ground - state ( g- ) band and the stockholm ( s- ) band , i.e. , the two - neutron aligned band , occurs . therefore , we try to study the s - band in a typical nucleus @xmath0er with the same multi - cranked configuration - mixing method ; we are able to study the g- and s - bands separately without the inter - band mixing . the cranked mean - field states are determined selfconsistently by the hartree - fock - bogoliubov ( hfb ) method for given rotational frequencies employing the finite - range gogny interaction @xcite with the d1s parameter set @xcite . after briefly explaining the theoretical framework in sec . [ sec : multi ] , we show the results of calculations in sec . [ sec : results ] . the conclusion is drawn in sec . [ sec : concls ] our basic approach to study the high - spin states of the nuclear collective rotation is the angular - momentum - projected configuration - mixing , or the projected generator coordinate method ( gcm ) , where the cranking frequency @xmath1 is employed as a generator coordinate . it was first proposed by peierls - thouless @xcite , and the wave function is calculated by @xmath2 where the operator @xmath3 is the angular momentum projector , and the mean - field wave function , @xmath4 , is obtained by the selfconsistent cranking procedure with the cranked hamiltonian , @xmath5 , @xmath6 in the present work , the ground - state mean - field states are axially deformed and the cranking axis is chosen to be the @xmath7-axis perpendicular to the symmetry axis ( @xmath8-axis ) . practically we discretize the generator coordinate , i.e. , the cranking frequency , as ( @xmath9 ; @xmath10 ) in eq . ( [ eq : ptanz ] ) , @xmath11 and solve the configuration - mixing amplitude , @xmath12 , with the so - called hill - wheeler equation , @xmath13 where the hamiltonian and norm kernels are defined as usual , @xmath14 we do not perform the number projection in the present work , and treat the number conservation approximately by replacing @xmath15 , where @xmath16 and @xmath17 are the neutron and proton numbers to be fixed . as for the neutron and proton chemical potentials @xmath18 and @xmath19 we use those obtained for the hfb ground - state . we have recently developed an efficient method for the angular - momentum - projection and the configuration - mixing @xcite . this method is fully utilized also in the present work . to solve the hfb equation and to perform the projection calculation the harmonic oscillator basis expansion is employed . more details of our theoretical framework can be found in refs . as it is mentioned we employ the gogny force with the d1s parameter set @xcite as an effective interaction . therefore there is no ambiguity for the hamiltonian . the hfb equation is solved in the space generated by the isotropic harmonic oscillator potential with the frequency @xmath20 mev . the size of the space is controlled by the oscillator quantum number @xmath21 ; all the basis states satisfying @xmath22 are included . we use @xmath23 for the following systematic calculations of the rotational spectra . the main target of the present work is the most basic rotational band , i.e. , the g - band . therefore , we selected typical deformed nuclei in the rare earth region , i.e. , three isotopes with the neutron number in @xmath24100 in each gd , dy , er , and yb nuclide , as they are tabulated in table [ tab : mf_rare ] . in this table we show the nuclear radii , the deformation parameters , and the average pairing gaps obtained by the hfb calculations in the ground - states of these selected nuclei . the ground - states are axially symmetric in all nuclei and the deformation parameter @xmath25 is defined by @xcite @xmath26 where the @xmath27-pole moment @xmath28 and the radius @xmath29 are calculated by the expectation value with respect to the hfb state , @xmath30^{1/2}. \label{eq : qmom}\ ] ] the average pairing gap is defined by @xmath31 \left[\sum_{a>0}\kappa^*_{a\tilde{a}}\right]^{-1},\qquad \delta_{ab}= \sum_{c > d}\bar{v}_{ab , cd}\,\kappa_{cd } , \label{eq : avgap}\ ] ] where the quantities @xmath32 and @xmath33 are the anti - symmetrized matrix element of the two - body interaction and the abnormal density matrix ( pairing tensor ) , respectively @xcite , and @xmath34 means the time - reversal conjugate state of @xmath35 . here we corrected the misprinted expression of denominator in the definition of @xmath36 in refs.@xcite . the even - odd mass differences calculated by the 4th - order difference formula based on the 2003 mass table @xcite are also included in table [ tab : mf_rare ] . [ cols="^,<,^,<,^,<,^,<,^,^,<,^,^,<,^ " , ] the result of the deformation parameter @xmath37 roughly corresponds to experimental data deduced by the measured @xmath38 values @xcite , but the calculated @xmath37 are slightly smaller . this is merely due to the differences in the definitions of @xmath37 in ref . @xcite and in the present work . the calculated radius @xmath29 is 2.42.9% larger than the empirical value @xmath39 [ fm ] , which is mostly due to the effect of deformation . in these stable nuclei the difference of the radius from the empirical value is not so large , but for the unstable nuclei the difference is expected to be larger . therefore , it is important to measure the nuclear radius to reliably extract the deformation parameters @xcite . as for the average pairing gaps selfconsistently calculated values are smaller than the even - odd mass differences in most cases . especially the neutron pairing gap in @xmath40gd is only 65% of the even - odd mass difference , and in @xmath41yb the calculated average gaps for both the neutron and proton are about 27% smaller . on the other hand , for @xmath42gd , @xmath43dy and @xmath0dy , the calculated gaps relatively well correspond to the even - odd mass differences , and their differences are less than 14% . as it is discussed in the following , the agreement of the calculated gaps with the even - odd mass differences is crucial to reproduce the moments of inertia of the g - band . if the deformed superconducting state is obtained for the ground - state , we generate the cranked hfb states in eq . ( [ eq : crank ] ) for a given set of rotational frequencies ( @xmath9 ; @xmath10 ) . it was demonstrated @xcite that the result of configuration - mixing does not depend on the choice of a set of frequencies , if the number of frequencies , @xmath44 , is five . therefore , we take @xmath45 and choose them ( almost ) equidistantly . since the ground - state rotational band is studied , only the frequencies before the g - s crossing should be selected ; we choose @xmath46 mev as in ref . @xcite , for most of nuclei in table [ tab : mf_rare ] . however , it is known that there is no sharp g - s crossing observed in n=98 isotopes , so that we choose @xmath47 mev for @xmath0dy and @xmath48yb . once the cranked hfb states are prepared we perform the angular - momentum - projected multi - cranked configuration - mixing , see eq . ( [ eq : proj ] ) , to obtain the rotational spectrum . in order for the efficient calculation of projection , the cut - off of the quasiparticle basis is employed @xcite , namely the canonical basis states of the hfb wave function whose occupation numbers are larger than @xmath49 are only retained . as for the integration mesh points for the euler angles ( @xmath50 ) in the angular - momentum - projector , we take @xmath51 and @xmath52 . for the present systematic calculation of the ground - state band , @xmath53 and @xmath54 are chosen ; the smaller value of @xmath55 is enough because the hfb wave function is nearly axially symmetric ( @xmath56 mixing is mainly induced by the cranking procedure ) . in the configuration - mixing in eq . ( [ eq : proj ] ) , the superposition of the states with respect to @xmath57 for given @xmath58 is overcomplete and there are vanishingly small norm states , which causes numerical problems @xcite . therefore , the norm cut - off should be done ; namely , the eigenvalues of the norm kernel are first calculated and the small norm states should be excluded when solving the hill - wheeler eq . ( [ eq : hw ] ) . the value for the norm cut - off is better to be as small as possible not to miss important contributions . we start from the value @xmath59 and increase it to avoid the numerical problems in each case . the actual values used in the following calculation are also denoted in table [ tab : mf_rare ] ; they are in the range , @xmath60 . figures [ fig : eir1 ] and [ fig : eir2 ] show the resultant rotational spectra of the configuration - mixing for the ground - state bands in nuclei in table [ tab : mf_rare ] . the results of the simple projection from one cranked hfb state with small frequency @xmath61 mev , i.e. , those of the infinitesimal cranking , are also included . the experimental data are taken from the table of isotope homepage @xcite . to show the detail we subtract the reference rotational energy , @xmath62 mev , in each spectrum . by comparing the results of the configuration - mixing and of the simple projection from one infinitesimally cranked hfb state in these figures , it is clear that the infinitesimal cranking gives a good description of the low - spin states . however , the deviations become non - negligible quickly at higher spins , @xmath63 . with the multi - cranked configuration - mixing the energy gain at @xmath64 from the simple projection is about @xmath65 mev , and therefore the effect of configuration - mixing is crucial for the description of the high - spin states . in comparison with the experimental spectra , the results of the configuration - mixing reproduce the bending - down behaviors of the data at higher spins , which reflects the increase of moments of inertia . in contrast , the spectra of the simple projection with the infinitesimal cranking do not change or even increase as spin increases , which means that the moments of inertia obtained by the simple projection are rather constant . the deviation of the calculated spectrum from the measured spectrum is rather large in @xmath40gd , @xmath66dy , @xmath42er , and @xmath41yb , for which the average pairing gaps are considerably smaller than the even - odd mass differences , especially for neutron , see table [ tab : mf_rare ] . on the other hand , the agreements are almost perfect for @xmath42gd , @xmath43dy , @xmath0dy , and @xmath0yb , in which the calculated average pairing gaps for both neutron and proton well correspond to the even - odd mass differences . thus the reproduction of the pairing properties is very important to achieve a good description of the ground - state rotational band , which is a rather well - known fact . it should be emphasized that the agreements for other nuclei are rather satisfactory considering the fact that we have no room for adjustment in the present calculations . the relative difference between the isotopes , e.g. , between @xmath43er and @xmath0er , or between @xmath0yb and @xmath48yb , is also reproduced . to study the rotational property in more detail , we show the first ( or kinematic ) moment of inertia in figs . [ fig : emo1 ] and [ fig : emo2 ] , which is defined by @xmath67 as a reference , the result of the cranked hfb calculation is also included in these figures , which is calculated by @xmath68 and is plotted as a function of @xmath69 we do not try to search the minimum energy at the fixed spin value in the cranked hfb calculation , and therefore the back - bending behavior of the moment of inertia is not obtained . it should be noticed that the cranked hfb inertia in eq . ( [ eq : j1chfb ] ) at high - spin states after the alignment of two quasineutrons should be considered to be that of the s - band and the inertia is unphysical in the band crossing region . the irregularities seen in the experimental moment of inertia are due to the effect of the g - s band crossing . it can be seen in figs . [ fig : emo1 ] and [ fig : emo2 ] that the moments of inertia for the ground - state band increase gradually as functions of spin . this behavior is quite nicely reproduced by the multi - cranked configuration - mixing calculations . the values of moment of inertia are considerably overestimated at low - spins in @xmath40gd , @xmath66dy , @xmath42er , and @xmath41yb ; again this is mainly because the calculated pairing gaps are markedly smaller than the even - odd mass differences . in these nuclei the amounts of increase for the moment of inertia are also much smaller than the experimental data . this indicates that the increase of the moment of inertia is mainly related to the reduction of the pairing correlations at higher spins . the agreements between the calculated and measured inertias are excellent for @xmath42gd , @xmath43dy , @xmath0dy , and @xmath0yb in the whole spin range shown in the figures . for the other cases , @xmath66gd , @xmath43er , @xmath0er , and @xmath48yb , the deviations of the calculated moments of inertia from the experimental data are less than 20% , which is quite non trivial . the different spin - dependence observed between the isotopes , e.g. , between @xmath43er and @xmath0er , is also nicely reproduced by the configuration - mixing calculations . the inertia calculated by the cranked hfb at low - spin is slightly larger than the result of configuration - mixing ; the large increase of the cranked hfb inertia is caused by the effect of the two - neutron alignment mentioned above . it should be emphasized that the moment of inertia obtained by the simple projection from the non - cranked hfb state ( not shown in the present work ) is about 3040% smaller than the result of the infinitesimal cranking @xcite . the inclusion of the time - odd components of the wave functions with @xmath70 and the subsequent @xmath56-mixing is very important to reproduce the correct magnitude of the moments of inertia . in figures [ fig : emm1 ] and [ fig : emm2 ] we show how the moments of inertia is changed by the multi - cranked configuration - mixing . namely , the inertias calculated by the simple projection from one intrinsic hfb state with five frequencies are compared with the result of the configuration - mixing . the calculated inertias with higher cranking frequencies are generally larger because of the coriolis anti - pairing effect . however , those calculated by the simple projection from one hfb states are almost constant as spin increases or even decrease in several cases . in contrast , the resultant inertias of the configuration - mixing always increase as functions of spin in accordance with experimental data . in this way the configuration - mixing is important to obtain gradually increasing behavior , which is general for the g - band in the rare earth region . note , however , that the amount of increase is quite different in each nucleus ; there is a trend that if the difference between the inertias calculated with different cranking frequencies at high - spin is large , then the amount of increase is larger . it is interesting to note that the behaviors of calculated inertias by the simple projection with finite frequencies are rather different in each nucleus . for example , all five results of the simple projection are similar in @xmath0er , while they are considerably different in @xmath41yb . however , the results of the configuration - mixing make the behaviors of moment of inertia in all nuclei rather similar , i.e. , gradually increasing as functions of spin . it is worthwhile mentioning that the result of the infinitesimal cranking , i.e. , the calculated inertia with @xmath61 mev coincides with that of the configuration - mixing at low - spins in most cases . this clearly shows that the infinitesimal cranking is enough for a good description of the rotational band at low - spin states , although the configuration - mixing is crucial at high - spin states . in the previous section only the g - band is considered . however , it is well - known that a different rotational band intersects with the g - band and becomes the yrast state at higher spin values . this band is called the s - band , in which two quasineutrons are excited to align their angular momenta to the axis of collective rotation . the band crossing between the g- and s - bands is the origin of the back - bending phenomenon first observed in ref . @xcite , where the rotational frequency decreases when spin increases along the yrast line . how this alignment of the two quasineutron occurs can be nicely understand by the semiclassical cranking model , see e.g. refs . we have shown that the g - band can be nicely described by our multi - cranked configuration - mixing method . therefore it is natural to study the s - band with the same method . the reference rotational energy , @xmath62 mev , is subtracted . the result of the projected multi - cranked configuration - mixing ( mixed ) is compared with the experimental data ( exp . ) . , width=302 ] er obtained by the simple projection from one intrinsic hfb state with five values of the cranking frequencies , @xmath710.01 , 0.05 , 0.10 , 0.15 , 0.20 mev , compared with the result of the projected configuration - mixing employing those five cranked hfb states . the energy origin is taken as the energy of the ground state of the configuration - mixing calculation . , width=302 ] er obtained by the simple projection from one intrinsic hfb state with four values of the cranking frequencies , @xmath710.25 , 0.30 , 0.35 , 0.40 mev , compared with the result of the projected configuration - mixing employing those four cranked hfb states . the energy origin is taken as the energy of the ground state of the configuration - mixing calculation . , width=302 ] we have already investigated the g - band in @xmath0er in our previous work @xcite with @xmath72 . therefore , we use the same model space for the s - band in this section . the method of the calculation is the same as in ref . @xcite except that the set of cranking frequencies is suitably chosen for the description of the s - band . by the cranked hfb calculation , the alignment of the two quasineutrons occurs at @xmath73 mev in @xmath0er with the gogny d1s parameter set , see fig . 5 of ref . we use the four cranked hfb states with frequencies , @xmath710.25 , 0.30 , 0.35 , 0.40 mev , for the configuration - mixing calculation of the s - band . the value of the norm cut - off is taken to be @xmath74 , with which the result is stabilized , and @xmath75 and @xmath76 are used . the resultant spectrum for the s - band is depicted in fig . [ fig : er164gsbr ] ; for completeness the calculated result of the g - band in ref . @xcite is also included , which was obtained with the set of five frequencies @xmath710.01 , 0.05 , 0.10 , 0.15 , 0.20 mev , as in the previous section . again , the reference rotational energy , @xmath62 mev , is subtracted . as it can be seen in the figure , we have successfully obtained the band crossing between the g- and s - bands , although the crossing occurs at slightly higher spin compared with the observation ; at @xmath77 in the calculation , while between @xmath78 and 16 in the experimental data . this is non trivial because we have no kind of adjustment in the present calculation . it should be emphasized that the g- and s - bands are calculated independently without the inter - band mixing ; the effect of the inter - band mixing is not taken into account in the present work . the reason why the calculated crossing is delayed is that the calculated excitation energies of the s - band are higher than the experimental data ; the excitation energy of @xmath79 state is 2.69 mev , which is about 170 kev higher than the experimentally measured one . in figs . [ fig : er164gb5 m ] and [ fig : er164sb4 m ] , we show how the resultant spectrum is obtained by the configuration - mixing calculation for the g- and s - bands , respectively , where the five ( four ) spectra calculated by the projection from one cranked hfb state with different frequencies are depicted in addition to the result of the configuration - mixing for the g - band ( s - band ) . from fig . [ fig : er164gb5 m ] it can be seen that the five spectra for the g - band , each of which is obtained from the one cranked state with @xmath80 mev , are rather similar , and the energy gain by the configuration - mixing is larger at higher spin , which leads to the increase of the moment of inertia as a function of spin . this suggests that the @xmath56-mixing and configuration - mixing induced by the cranking is more effective at higher spins ; the hill - wheeler equation ( [ eq : hw ] ) should be solved even for the case of projection from a single hfb state and its dimension of increases as spin increases . in contrast , the four spectra for the s - band obtained from the one cranked state with @xmath81 mev are rather different as is shown in fig . [ fig : er164sb4 m ] . non of the spectra looks like the observed one and the configuration - mixing is very important for the s - band to obtain the correct spectrum . each spectrum in figs . [ fig : er164sb4 m ] has a minimum energy at a finite spin value , and the spin value that gives a minimum is larger for the spectrum obtained from the cranked hfb with larger cranking frequency . moreover , the energy gain by the configuration - mixing is considerable at lower spin , while it is much smaller at @xmath82 and the resultant configuration - mixed spectrum looks more like the envelope curve of the four spectra . note that the aligned angular momentum of the two quasineutrons is estimated to be about 10 @xmath83 in @xmath0er . thus , the role played by the configuration - mixing seems to be somewhat different in the g- and s - bands , and the configuration - mixing is much more crucial for the s - band than for the g - band . er obtained by the projected configuration - mixing calculations in comparison with the experimental data . the result of the cranked hfb ( chfb ) is also included . , width=340 ] although the nice band crossing is obtained by the calculation , the agreement of the moment of inertia for the s - band is not as good as that for the g - band , which is shown in fig . [ fig : er164moim ] . especially for the s - band , the inertia is considerably overestimated in @xmath84 , and the experimentally observed inertia decreases as spin increases , while the calculated one is almost constant in the range @xmath85 and increases afterward . the overestimation of the moment of inertia for the s - band is mainly due to the fact that the neutron pairing vanishes by the alignment of two quasineutrons , namely , the cranked hfb states used for the configuration - mixing calculation for the s - band are in fact the unpaired states ( the slater determinants ) for neutrons . the proton pairing is non vanishing but reduces considerably ; i.e. , the average pairing gap of proton is @xmath86 mev at @xmath87 mev and @xmath88 mev at @xmath89 mev . this fact is reflected also in the resultant inertia of the cranked hfb calculation , which is considerably larger at @xmath84 . because of this problem we do not discuss the higher spin part at @xmath90 in the present work . but the calculations with employing five values of the cranking frequencies , @xmath710.250 , 0.275 , 0.300 , 0.325 , 0.350 mev . , width=302 ] er obtained by the configuration - mixing calculations with the two different sets of rotational frequencies , @xmath710.25 , 0.30 , 0.35 , 0.40 mev ( mixed1 ) , and @xmath710.250 , 0.275 , 0.300 , 0.325 , 0.350 mev ( mixed2 ) in comparison with the experimental data . the scale of the ordinate is enlarged from that of fig . [ fig : er164moim ] . , width=302 ] in the case of the g - band the result of the configuration - mixing does not depend on the choice of a set of frequencies for the cranked hfb states @xcite . however , it turns out that the result for the s - band is not completely independent of the choice of frequencies . we show an example of another set of the cranking frequencies , @xmath710.250 , 0.275 , 0.300 , 0.325 , 0.350 mev in fig . [ fig : er164sb5 m ] , where the range of the frequencies are reduced from @xmath91 $ ] to @xmath92 $ ] and they are chosen more densely in the given interval . comparing the resultant configuration - mixed spectra obtained with the two sets of cranking frequencies in figs . [ fig : er164sb4 m ] and [ fig : er164sb5 m ] , one can see they are slightly different . to see the difference more clearly , we show the moments of inertia for the s - band calculated with these two sets of frequencies in fig . [ fig : er164moid ] in comparison with the experimental data . note that the scale of the ordinate is enlarged . the difference is non - negligible especially in the lower spin range @xmath93 . moreover , the inertia obtained with the first set , @xmath710.25 , 0.30 , 0.35 , 0.40 mev ( mixed1 ) , is monotonically increasing as spin increases , while the one with the second set , @xmath710.250 , 0.275 , 0.300 , 0.325 , 0.350 mev , decreases first in @xmath94 and then turns to increase . we think that the result with the second set ( mixed2 ) is more reliable in the considered spin - range because a larger number of the cranked hfb states are employed in the relevant frequency interval . the expectation values of spin in eq . ( [ eq : iavchfb ] ) for the cranked hfb states are @xmath95 19.3 , 23.6 , 29.1 , 35.1 at @xmath71 0.25 , 0.30 , 0.35 , 0.40 mev , respectively ; see also fig . 5 of ref . the cranked hfb state is most suitable for describing the states with spin around its expectation value . in fact , the difference of moments of inertia obtained with the two sets of frequencies is small in the spin range @xmath84 ; the results of configuration - mixing mainly differs in the lower spin region , @xmath96 . it may be necessary to include the cranked hfb states with lower spin - expectation values for the configuration - mixing of the s - band in order to obtain the result that is independent of the detailed choice of the frequencies . thus , the description of the s - band is not as simple as in the case of the g - band , especially in the low - spin region , @xmath96 . it may not , however , be a serious problem because the s - bands for such low - spin parts have not been observed in experiments and it is difficult to compare with experimental data . in any case , we need to study further for more satisfactory description of the s - band . we have investigated the rotational bands in the rare earth nuclei by employing our recently developed microscopic framework , the angular - momentum - projected multi - cranked configuration - mixing method @xcite . in this method several cranked hfb states are utilized with a suitably chosen set of rotational frequencies . we use the gogny force with the d1s parameter set as an effective interaction , and there is no ambiguity for the hamiltonian . we first apply our method to the g - band of various selected nuclei in the rare earth region . reasonably good overall agreements are obtained for the energy spectra and the moments of inertia up to about @xmath97 . in a few cases the moments of inertia at low spin are considerably overestimated and the increase of the inertia as a function of spin is not enough compared with the experimental data . it is found that the selfconsistently calculated pairing correlations are too weak for such nuclei ; the average pairing gaps for both neutrons and protons are only about 70% of or even less than the even - odd mass differences . if the pairing properties are nicely reproduced , the agreements of the moments of inertia are found to be excellent . in this way we have confirmed that our method is capable to reliably describe the nuclear rotational motion near the ground state . next we apply our approach to the study of the s - band in the nucleus @xmath0er for the first time . the method of calculation is the same for the s - band ; the only difference is that the cranked hfb states with higher rotational frequencies are employed for the configuration - mixing , in which the two quasineutrons align their angular momenta . thus the g- and s - bands can be calculated separately without the inter - band mixing between them . the band crossing between the g- and s - bands can be reproduced , although the spin value , at which the two bands cross , is slightly larger than the observed one . the calculated moment of inertia of the s - band is overestimated especially at high - spin states . this is mainly because the selfconsistently calculated pairing correlation for neutrons vanishes in the cranked hfb states due to the alignment of two quasineutrons , which can not be avoided as long as the gogny d1s force is employed . it is found that the result of configuration - mixing weakly depends on the choice of the set of cranking frequencies for the s - band , especially in the lower spin region , in contrast to the case of the g - band , where the result is independent of the choice of frequencies . thus , further investigation is necessary for the proper description of the s - band , which is an important future work . this work is supported in part by grant - in - aid for scientific research ( c ) no . 25@xmath98949 from japan society for the promotion of science . | recently we have proposed a reliable method to describe the rotational band in a fully microscopic manner .
the method has recourse to the configuration - mixing of several cranked mean - field wave functions after the angular - momentum - projection . by applying the method with the gogny d1s force as an effective interaction
, we investigate the moments of inertia of the ground state rotational bands in a number of selected nuclei in the rare earth region . as another application
we try to describe , for the first time , the two - neutron aligned band in @xmath0er , which crosses the ground state band and becomes the yrast states at higher spins .
fairly good overall agreements with the experimental data are achieved ; for nuclei , where the pairing correlations are properly described , the agreements are excellent .
this confirms that the previously proposed method is really useful for study of the nuclear rotational motion . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
magnetoelectric ( me ) effects in multiferroic ( mf ) or ferromagnetic ( metallic ) films have brought remarkable interest since promising technological applications in spintronics and ultrafast electric field control on magnetic data storage are seen as imminent@xcite,@xcite . characterization of the relative strength for the me coupling can be obtained by implementing terahertz spectroscopy in rare earth manganites of the type rmno@xmath2 ( r = tb , gd , dy , eu : y ) @xcite,@xcite,@xcite,@xcite demonstrating that the generated electromagnons ( mixed spin - waves and photon states ) represent , among others , the signature of the me effect for an approximate range of frequencies between 10 @xmath3 and 40 @xmath3 at temperatures where antiferromagnetic resonance modes ( afmr ) coexist , or more recently , the key mechanism for controllable magnetochromism in ba@xmath4mg@xmath4fe@xmath5o@xmath6 hexaferrites @xcite . the magnetoelectric effect emerges when a magnetic field @xmath7 can induce a polarization vector @xmath8 at zero applied electric field ( @xmath9 ) . likewise , the magnetization of the substance @xmath10 can be generated for an electric field @xmath11 with @xmath12 . the minimal coupling for describing the thermodynamic potential associated with this effect is given by @xmath13 , where @xmath14 is an unsymmetrical magnetoelectric tensor , whose components depend on the magnetic symmetry class@xcite . the primary origin for the me coupling is commonly associated with the dzyaloshinskii - moriya relativistic exchange - interaction @xcite,@xcite which is appropriate for the description of asymmetric spin wave dispersion on double layer fe - films@xcite as well as for those materials where weak ferromagnetism emerges , namely the ilmenite fetio@xmath2 , tbmno@xmath2 , eu@xmath15y@xmath16mno@xmath2 ( @xmath17 at @xmath18 k@xcite ) or the widely studied pyroelectric ferromagnet bamnf@xmath1 @xcite . weak ferromagnetism on this compound is generated by canting effects between antiferromagnetic sub - lattices , leading to a spontaneous polarization @xmath8 perpendicular to the resulting magnetization @xmath10@xcite . considerations in the symmetry change of the static polarization and magnetization fields have brought interesting unconventional optical phenomena labeled as non - reciprocal dichroism associated with the sign reversal of @xmath19 , recently reported in the perovskite eu@xmath20y@xmath21mno@xmath22 , with magnetoelectric activity for photon energies around 0.8 mev ( sub thz regime ) in the cycloidal phase at 4 k@xcite . intense activity in the last decade has also been dedicated to achieve possible optical and photonic band gap control via surface plasmon ( sp ) propagation in periodic arrays@xcite , since modern lithographic techniques allow to design functional objects with almost any desirable geometrical pattern at a sub - wavelength scale@xcite . plasmon localization and its coupling with incident light depend on the dielectric properties of the metal in conjunction of its surrounding environment , enlightening an alternative route for engineering highly efficient sp photonic devices via externally applied fields , rare earth doping or electron charge transference from the modified metal@xcite . in this communication , we study an electrodynamic - based model for estimating the optical response generated by the contact between a material exhibiting weak ferromagnetism in contact with a 2d metallic film . it is found that a specific strength of the me interaction might couple with localized charge - sheet modes for electron carrier densities about @xmath23 @xmath24 and incident frequencies around 18 @xmath3 , leading to a change in the reflectance from the metallic film . applied magnetic field effects on relative reflective are discussed in section iii . localized charge - sheet modes in a 2d conducting medium in the framework of drude approximation is obtained from the nakayama result @xcite,@xcite,@xcite : @xmath25 where @xmath26 corresponds to the quasiwavevector in the @xmath27 direction , @xmath28 is defined as @xmath29 and @xmath30 denotes the electron density concentration in a two dimensional space . @xmath31 is related with the wavevector along @xmath32-direction through @xmath33 , ( @xmath34 ) . the term @xmath35 represents the relative dielectric function value for @xmath36-th medium , with @xmath37 for vacuum . in the range of wavelengths behind the far infrared radiation ( @xmath38 mm ) , the dielectric function approaches to the well recognized lyddane - sachs - teller ( lst ) relationship : @xmath39 , where @xmath40 corresponds to the dielectric permittivity of the medium @xmath41 and @xmath42 represents the longitudinal ( transverse)-optical phonon frequency . for numerical purposes , we have set @xmath43 , which coincides with the relationship for the @xmath44-axis normal phonon modes in bamnf@xmath1 . the permittivity @xmath40 is a functional depending on mechanical strain deformations and polarization field depletion in the proximities between the multiferroic slab and metal film@xcite , and is taken as constant for zero applied ( electric ) field and fixed temperature . formula ( [ nakay ] ) is derived by solving the complete set of maxwell equations with normal ( tm wave ) incidence for @xmath45 , and boundary conditions on the plane @xmath46 with the _ ansatz _ for propagating fields @xmath47 in the region @xmath46 . magnetoelectric effects are taken into consideration throughout the transverse susceptibility @xmath48 and the electric displacement vector @xmath49 is written into the constitutive equation like @xmath50 . after inserting the additional term @xmath51 $ ] , the expression ( [ nakay ] ) shall be modified under @xmath52 . in the plane @xmath46 , and in agreement with the geometrical configuration shown in figure ( [ r0 ] ) , the non - zero surface current density component is defined as @xmath53 , where @xmath54 corresponds to the @xmath55-element of the generalized conductivity tensor@xcite , and @xmath56 is the electrical field propagating on the @xmath32 direction . [ fig1 ] the generic expression for the transverse susceptibility @xmath48 is obtained from first principles by minimizing the free - energy density functional @xmath57 , which contains the two sublattice magnetizations , the polarization as well as external fields . it can be summarized as : @xmath58 $ ] , where @xmath59 is a coupling parameter which is an involved function of the canting angle between two adjacent ( antiferromagnetic ) sublattices , the spontaneous magnetization @xmath10 and the polarization vector @xmath8 , as well as the parameters @xmath60 . factor @xmath61 is defined in terms of the characteristic magnetoelectric frequency @xmath62 as @xmath63 , given in units of mm@xmath64 all throughout this paper@xcite , in concordance with the spectral weight intrinsically associated with the fitting procedure for the transmittance spectra via lorentzian model in various multiferroic species , namely rmn@xmath65o@xmath66(r : y , tb ) , tbmno@xmath22 or lumno@xmath2@xcite , and its dependence with the externally applied magnetic field has been neglected for small canting angles ( see for instance eqs . ( 38 ) and ( 47 ) in reference [ 14 ] . two main poles are clearly identified for @xmath48 : the optical antiferromagnetic resonance mode ( afmr ) @xmath67 and the soft - phonon along @xmath10 with resonance frequency @xmath68 , with @xmath69 . classical plasmon excitations in low 2d carrier electron density are experimentally detected and theoretically estimated for wavevectors @xmath70 @xmath3 and energies @xmath71 mev@xcite , @xcite,@xcite , therefore the condition @xmath72 remains valid in the range of interest , and the dispersion relationship for the coupled magnetoelectric plasma mode is obtained by solving the modified equation ( [ nakay ] ) : @xmath73,\ ] ] with @xmath74 , @xmath75 and @xmath76 . for @xmath77 , i.e. , no magnetoelectric effects taken under consideration , we reproduce the expression for the localized plasmon mode @xcite : @xmath78 where ( @xmath79 ) sign in equation ( [ cuasiwv ] ) has been selected . complex index of refraction @xmath80 is directly estimated from the wavenumber @xcite @xmath81 @xmath82 . the lowest - order reflectance coefficient @xmath83 for normal incidence is defined as @xmath84 and its numerical profile discussed on the next section . @xmath80 can be considered as the _ effective _ index of refraction for the composite 2d metallic foil in contact with a multiferroic ( ferroelectric ) system under normal incidence of a electromagnetic wave oscillating in the thz regime . applied magnetic field @xmath85 along @xmath86-direction enters into the formalism by taking symmetry considerations upon the dependence of the electrical conductivity as a function of @xmath85 under the transformation @xmath87 , with @xmath88 . expression ( [ cuasiwv ] ) may be reconstructed as : @xmath89 @xmath90,\ ] ] with @xmath91 . the classical localized magnetoplasmon mode ( [ qy ] ) is rewritten for @xmath92 and under @xmath93 like@xcite : @xmath94 in similarity with the result ( [ qy ] ) . in this particular case the antireflective condition ( @xmath95 ) depends on the external magnetic field intensity @xmath96 , which leads to a quadratic correlation @xmath97 for @xmath98 . for an arbitrary orientation of @xmath0 , equation ( [ nakay ] ) shall be modified on its right side accordingly @xmath99 , where @xmath100 is a function of the directors @xmath101@xcite . optical reflectivity response for this structure might also be verified by adapting the rouard method@xcite,@xcite : @xmath102 where @xmath103 corresponds to the internal reflectivity between media labeled @xmath104 ( @xmath36 ) and @xmath105 is the phase difference on the second medium with thickness @xmath106 , defined as @xmath107 . the index of refraction @xmath108 is a function of the components for the conductivity tensor @xmath109 $ ] depending on the incoming electromagnetic field polarization . in this particular case , it is calculated as : @xmath110 while @xmath55 is explicitly given by @xmath111 where @xmath112 represents the electronic plasma frequency for the _ bulk _ system , which is related to @xmath113 through @xmath114 where @xmath115 being the volumetric electron density concentration . reference values for plasma frequencies were taken as @xmath116 hz and @xmath117 hz for gold ( au ) in the framework of the drude model fitting@xcite . factors @xmath103 in formula ( [ rou ] ) are given explicitly by @xmath118 , and @xmath119 , with @xmath120 . indeces of refraction are directly obtained by reconstructing the set of maxwell equations on each material media . in the general case , taking into account the me effect in the formalism by inserting the tensor @xmath121 $ ] , the propagating electric field @xmath11 must satisfy : @xmath122\mathbf{e}\right)_{m - mf}\\ \nonumber + \omega^{2}\mu_{0}\mathbf{d}_{m - mf}+4\pi\omega\nabla\times\left(\left[\chi\right]\mathbf{e}\right)_{m - mf},\end{aligned}\ ] ] where @xmath123 $ ] is the conductivity tensor , and @xmath49 previously defined as the electric displacement vector , and subscript @xmath124 indicates the region where fields propagation are evaluated , namely the metal ( m ) or multiferroic ( mf ) slab . figure ( [ r1 ] ) exhibits the zero field reflectance response as a function of the 2d electronic carrier concentration @xmath30 , for different wavelengths and the magneto - electric coupling parameter @xmath61 fixed at @xmath125 mm@xmath64 , the dielectric permittivity values have been taken as @xmath126 and @xmath127 for the pyroelectric ferromagnet bamnf@xmath1 , which correspond to the values measured along its @xmath128 and @xmath44 crystallographic axes , respectively . dotted curves ( a ) and ( b ) are set as reference for @xmath92 . comparative results are shown for rouard s method ( rm ) and the modified nakayama ( n ) expression ( eq . [ nakay ] ) , indicating the change in the reflectivity spectra under the me effect and different values for the dielectric constant @xmath129 . the reflectance response increases from @xmath130 ( @xmath131 ) to @xmath132 ( @xmath133 ) for electronic densities lower than @xmath134 @xmath24 , while it augments monotonically to @xmath135 for electronic concentrations greater than @xmath136 @xmath24 regardless of the value of @xmath61 , in the framework of the rm approach . one of the discrepancies with the nakayama results is due to the difference between the 2d intrinsic plasma frequency @xmath113 and those associated with the plasma frequency in the _ bulk _ system @xmath137 . variation in the electronic carrier density in the former case has been simulated by inserting the thickness film dependence @xmath106 on @xmath137 , providing good agreement for @xmath138 nm ( @xmath139 @xmath24 ) as proven in fig.([comp ] ) . minima of reflectivity obtained from eq . ( [ cuasiwv ] ) , are located at @xmath140 , or @xmath141 , indicating that the critical wavelength for _ bare _ plasmon excitations is larger as the electronic concentration decreases . afmr mode lies in the range thz range , with @xmath142 thz , while the transverse phonon frequency is taken as 7.53 thz for the bamnf@xmath1 compound@xcite . metallic behavior predominates for concentrations higher than @xmath143 @xmath24 and smaller than @xmath144 @xmath24 and selected wavelengths between 0.5 mm and 0.6 mm . resonant plasmon modes ( i.e. , collective electronic excitations under me interaction ) are important for carrier densities around @xmath145 @xmath24 , where radiative absorption or antireflective phenomena become strong and the reflectance spectrum is therefore significantly modified by diminishing the percentage of absorbed radiation only when the external frequency approaches the characteristic mode @xmath67 , and @xmath146 . figure ( [ r2 ] ) depicts the shifting of the minimum of reflectance in the @xmath147 plane for the nakayama approach . the me effect becomes relevant by decreasing the _ critical _ carrier density @xmath148 as @xmath61 increases , and it remains essentially unmodified for those frequencies away from the afmr characteristic mode as indicated in line ( d ) . dotted vertical line is tagged at @xmath133 mm@xmath64 as a eye guide for identifying the critical density change as the incident wavelength varies around @xmath149 . critical density @xmath150 shall be understood as the electron carrier concentration which maximizes antireflective effects for the composite metal / multiferroic system . figure ( [ bparallel0 ] ) shows the reflectance response under applied magnetic field with magnitude @xmath151 t for different directions on the @xmath152 plane . afmr resonance at @xmath153 is not essentially affected by the orientation of the external field , but it becomes sensitive with the azimuthal angle for frequencies between the edge of the thz range and the microwave ( shf ) band . highly reflective effects are more intense for external magnetic fields which are applied in the opposite direction with respect to the weak ferromagnetic state @xmath10 , favoring the metallic behavior for long wavelengths and shielding the resulting me interaction . _ in - plane _ applied field @xmath0 effects on the reflectance as a function of carrier density @xmath30 are illustrated in fig . ( [ bparallel ] ) . @xmath154 tends to increase for @xmath0 parallel to @xmath155-axis and decreases for @xmath0 along @xmath156 axis . curve ( b ) for null @xmath0 overlaps the outcome of @xmath157 at @xmath158 t , @xmath159 and @xmath160 ( i.e. , parallel to @xmath32 axis ) , indicating no substantial variation in the optical reflectance for applied fields in the same direction of the plasmonic wavevector @xmath161 for carrier densities smaller than @xmath162 @xmath24 . equation ( [ max ] ) has also been treated by implementing finite element method ( fem ) and standard boundary conditions for @xmath49 and @xmath163\mathbf{e}$ ] fields in order to calculate the reflectance response as a function of incident wavelengths . comparative results on the calculated response of the reflectance are shown in figures ( [ comp ] ) and ( [ 3cc ] ) . under nakayama s formalism , the metallic medium is treated as a 2d system , while rouard and fem methods converge with the first one for a film thickness around @xmath138 nm , which roughly corresponds to an electronic carrier density of 147.42 @xmath24 after calculating the correlation between two intrinsic plasma frequencies @xmath137 and @xmath113 . iso - reflective lines for @xmath164@xcite close to @xmath153 and the externally applied magnetic field ( in @xmath86 direction ) are shown in figure ( [ control ] ) . projected lines preserve symmetrical distribution under magnetic field inversion nearby @xmath165 although strong fluctuations and a sign flip on @xmath166 are present for wavelengths slightly different from @xmath165 and magnetic fields greater than @xmath167 t , indicating that interacting me and plasmonic activity might increase the reflectance outcome from systems with low electronic density and without applied field . we have developed a model for studying the magnetoelectric interactions on 2d plasmonic modes in the thz range for a metal / multiferroic composite device . the multiferroic medium exhibits weak ferromagnetism and the metallic behavior enters into the formalism in the framework of the classical drude - lorentz model . relative reflectance response for normal incidence is numerically calculated for a particular me coupling strength @xmath61 and wavelengths near to the optical antiferromagnetic resonance frequency @xmath168 by using three different approaches : nakayama s formalism , rouard s method and finite elements ( fem ) . characteristic soft phonon and afmr frequencies were taken for the pyroelectric ferromagnet bamnf@xmath1 , showing that a particular condition for reflectivity might be adjustable by varying the intensity of the applied field , its orientability , film thickness or incident frequency of radiation , mainly in a range @xmath169 . spectra of reflectance demonstrate that the magnetoelectric interaction predominates for metallic film thicknesses smaller than 25 nm in the thz regime , while for thicker films ( 50 - 100 nm ) the optical outcomes are not significantly affected by this interaction ; instead , total reflectance from the film is observed along a wide range of frequencies up to the cut - off bulk value @xmath170 phz , in which reflectivity decays abruptly to zero and exhibits oscillatory behavior for greater frequencies . the chosen value of @xmath112 is into the typical order of magnitude for good conductors like gold , silver or copper , despite that the calculations and comparison with the strictly 2d system were made just for the first one . there is not a clear signature of the plasmonic cut - off for intermediate film thicknesses ( 25 - 50 nm ) and the reflectivity curve does not breach abruptly as for wider ones ; rather , it reaches its maximum value in a broad interval of @xmath171 hz , suggesting a variation of the effective dielectric response associated with the metal under me interaction . further analysis shall be proposed for other metals or semiconducting materials , since optical control experiments on the thz range have recently been achieved on gaas wafers via stimulated photocarriers generated by interband light absorption . the resulting reflectivity spectrum is tuned from antireflective ( @xmath172 ) to high reflective ( @xmath173 ) limits under controlled power illumination@xcite,@xcite . although all numerical simulations were conducted for @xmath174 , ( @xmath175 being taken as @xmath176 and @xmath177 in the range of interest ) , simultaneous electric field control @xmath178 on optical properties for the composite device might also be achieved under the dielectric function dependence for a multiferroic material @xmath179 $ ] , the polarization @xmath180 and temperature , issue that shall be addressed in future investigations . h.v . wants to thank computing accessibility at pom group . c. v .- h . acknowledges financial support provided by dima , _ direccin de investigacin sede manizales _ , universidad nacional de colombia . h.v . declares no competing financial interest . 00 c - g duan , j. p. velev , r. f. sabirianov , z. zhu , j. chu , s. s. jaswal and e. y. tsymbal , phys . * 101 * , 137201 ( 2008 ) . l. gerhard , t. k. yamada , t. balashov , a. f. takcs , r. j. h. wesselink , m. dne , m. fechner , s. ostanin , a. ernst , i. mertig and w. wulfhekel , nature nanotechnology * 5 * , 792 ( 2010 ) . doi:10.1038/nnano.2010.214 a. pimenov , a. m. shuvaev , a. a. mukhin and a. loidl , j. phys . : condens . matter * 20 * , 434209 ( 2008 ) . d. talbayev , s. a. trugman , a. v. balatsky , t. kimura , a. j. taylor and r. d. averitt , physical review letters * 101 * , 097603 ( 2008 ) . h. n@xmath181mec , f. kadlec , p. ku@xmath182el , l. duvillaret and j .- l . coutaz , optics communications * 260 * , 175 ( 2006 ) . a. pimenov , a. a. mukhin , v. yu ivanov , v. d. travkin , a. m. balbashov and a. loidl , nature physics * 2 * , 97 - 100 ( 2006 ) . n. kida and y. yokura , journal of magnetism and magnetic materials * 324 * , 3512 ( 2012 ) . l. d. landau , e. m. lifshitz and l. p. pitaevskii , _ electrodynamics of continuous media _ , 2nd ed . , ch.*v * , butterworth - heinemann ( 2006 ) . h. katsura , n. nagaosa and a. v. balatsky , physical review letters * 95 * , 057205 ( 2005 ) . h. katsura , a. v. balatsky and n. nagaosa , physical review letters * 98 * , 027203 ( 2007 ) . zakeri , y. zhang , j. prokop , t .- h . chuang , n. sakr , w. x. tang and j. kirschner , physical review letters * 104 * , 137203 ( 2010 ) . a. a. mukhin , v. yu ivanov , v. d. travkin , a. s. prokhorov , a. a. volkov , a. v. pimenov , a. m. shuvaev and a. loidl , phys .- usp * 52 * , 851 ( 2009 ) . doi : 10.3367/ufne.0179.200908j.0904 j. f. scott , rep . prog . phys . * 42 * , 1055 ( 1979 ) . v. gunawan and r. l. stamp , j. phys . : condensed matter * 23 * , 105901 ( 2011 ) . y. takahashi , r. shimano , y. kaneko , h. murukawa and y. tokura , nature physics * 8 * , 121 ( 2012 ) , doi:10.1038/nphys2161 w. l. barnes , a. dereux and t. w. ebbensen , nature * 424 * , 824 ( 2003 ) . s. li , m. jadidi , t. e. murphy and g. kumar , optical express * 21 * no . 6 , 7041 ( 2013 ) . h. j. freund , surface science * 601 * , 1438 ( 2007 ) . m. nakayama j. phys . soc . jpn.*36 * , 393 ( 1974 ) . m. cottam and d. r. tilley , _ introduction to surface and superlattice excitations _ , cambridge university press ( 1989 ) . j. m. pitarke , v. m. silkin , e. v. chulkov and p. m. echenique , rep . . phys . * 70 * , 1 - 87 ( 2007 ) . s. zhong , s. alpay , v , nagarajan , j. mater . res . * 21 * ( 6 ) ( 2006 ) 1600 . j. slyom , _ fundamentals of the physics of solids _ , vol . ii , springer - verlag ( 2009 ) . d. r. tilley and j. f. scott , physical review b * 25 * , 3251 ( 1982 ) . k. l. livesey and r. l. stamps , physical review b * 81 * , 094405 ( 2010 ) . j .- rivera , eur . j. b. * 71 * , 299 ( 2009 ) . a. b. sushkov , r. valds aguilar , s. park , s - w . cheong and h. d. drew , physical review letters * 98 * , 027202 ( 2007 ) . e. h. hwang and s. das sarma , physical review b * 64 * , 165409 ( 2001 ) . i. a. nechaev , v. m. silkin and e. v. chulkov , journal of experimental and theoretical physics * 112 * , issue 1 , pp 134 - 139 ( 2011 ) . f. kanjouri , a. h. esmailian and m. molayem , eur . j. b * 79 * , 429 ( 2011 ) . g. r. fowles , _ introduction to modern optics _ , second edition , dover ( 1989 ) ; j. b. marion and m. a. heald , _ classical electromagnetic radiation _ , second edition , chapter * 6 * , academic press ( 1980 ) . @xmath183 , with @xmath184 , @xmath185 and @xmath186 , @xmath187 and @xmath188 . p. lecaruyer , e. maillart , m. canva and j. rolland , applied optics * 45 * , 8419 ( 2006 ) . o. s. heavens , _ optical properties of thin solid films _ , chapter * 4 * , dover publications ( 1991 ) . e. j. zeman and g. c. schatz , j. phys . 91(3 ) , 634 - 643 ( 1987 ) . j. barna , journal of magnetism and magnetic materials * 62 * , 381 ( 1986 ) . m. a. eriksson , a. pinczuk , b. s. dennis , c. f. hirjibehedin , s. h. simon , l. n. pfeiffer , k.w . west , physica e * 6 * , 165 ( 2000 ) . j. d. e. mcintyre and d. e. aspnes , surface science * 24 * , 417 ( 1970 ) . l. fekete , j. y. hlinka , f. kadlec , p. ku@xmath182el and p. mounaix , optical letters , * 30 * , no . 15 , 1992 ( 2005 ) . _ recent developments in terahertz optoelectronics _ , edited by jean - louis coutaz , comptes rendus physique , * 9 * , issue 2 , pages 127 - 284 ( 2008 ) . | the rle of the magnetoelectric effect upon optical reflectivity is studied by adapting an electrodynamic - based model for a system composed by a 2d metallic film in contact with an extended multiferroic material exhibiting weak ferromagnetism .
the well - known _ nakayama s _ boundary condition is reformulated by taking into account the magnetoelectric coupling as well as an externally applied magnetic field @xmath0 in an arbitrary direction . it is found that the reflectance shows strong fluctuations for incident radiation close to the characteristic antiferromagnetic resonance frequency associated with the multiferroic material in the thz regime . these results were verified for a 10 nm metallic foil by using a finite element method ( fem ) and the rouard s approach , for a wide range of wavelengths ( 0.1 - 5 mm ) , showing good agreement with respect to nakayama s outcome , for the particular material bamnf@xmath1 .
+ + _ keywords : _ multiferroics , magnetoelectric effect , surface plasmon , reflectance . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
physicists have recently shown that network analysis is a powerful tool to study the statistical properties of complex biological , technological and social systems of diverse kinds@xcite . many networks exhibit a scale - free degree distribution in which the probability @xmath0 that a vertex is connected to @xmath1 other vertices falls as a power @xmath2 . this property is not sufficient to completely describe natural networks because such systems also exhibit degree correlations the degrees of the vertices at the end points of any given edge are not independent @xcite . it is not surprising that natural systems depend on properties that do not appear explicitly in degree distributions . in particular , protein interaction networks depend on the availability of sufficient binding free energy@xcite to cause interactions to occur ( links between vertices to exist ) . caldarelli _ et al . _ @xcite and sderberg @xcite proposed models in which vertices are characterized by a fitness parameter assigned according to a chosen probability distribution . then , pairs of vertices are independently joined by an undirected edge with a probability depending on the fitnesses of the end points . @xcite generalized these models as a class of models with hidden variables and presented a detailed formalism showing how to compute network properties using the conditional probability ( propagator ) that a vertex with a given value of a hidden variable is connected to other @xmath1 vertices . this formalism , valid for any markovian ( binary ) network , provides the generating function for the propagator , but not the propagator itself . the purpose of this paper is twofold . we first use a mean field approximation to derive a general analytic formula for the propagator , therefore finding a general approximate solution to to the inversion problem . this enables one to compute network properties without the use of a simulation procedure , thereby simplifying the computational procedure and potentially broadening the ability of scientists from all fields to use network theory . the validity of the method is assessed by comparing the results of using our approximation with published results . we then use this method to compute clustering coefficients of a specific hidden variable model for protein - protein interaction networks ( pin ) from several organisms developed by us@xcite that previously had obtained degree distributions in agreement with measured data . we show that two models with the same degree distribution have very different clustering coefficients . we outline this in more detail . [ sec : formalism ] reviews the hidden variable formalism and our approximate solution to the inversion problem . we distinguish between sparse ( which have been solved in ref . @xcite ) and non - sparse networks which are solved here . the next section [ sec : models ] studies the models of refs . @xcite and @xcite . our averaging procedure is found to work well for most situations . our own model@xcite is presented in [ sec : pin ] . we present an analytic result for the average connection probability and extend the results of @xcite to computing the clustering coefficients . the final section [ sec : summary ] is reserved for a brief summary and discussion . we present the formalism for hidden variable models @xcite . the probability that a node has a hidden continuous variable @xmath3 is given by @xmath4 , normalized so that its integral over its domain is unity . this function is chosen to be an exponential in @xcite and a gaussian in @xcite . the connection probability for two nodes of @xmath5 is defined to be @xmath6 . this is taken as a step function in @xcite , and a fermi function in @xcite . the two functions @xmath4 and @xmath6 can be chosen in a wide variety of ways to capture the properties of a given network . reference @xcite presents the probability generating function , @xmath7 , that determines @xmath0 in terms of the generating function for the propagator , @xmath8 , as g_0(z)= dg(g ) _ 0(z , g),[bog1]where _ 0(z , g)= ndg ( g)(1-(1-z)p(g , g ) ) . [ gbog]the propagator @xmath9 giving the conditional probability that a vertex of hidden variable @xmath3 is connected to @xmath1 other vertices is given implicitly by _ 0(z , g)=_k=0^z^kg_0(k , g ) . [ g0 kg ] knowledge of @xmath9 determines the conditional probability @xmath10 that a node of degree @xmath1 is connected to a node of degree @xmath11 , @xcite ( as well as @xmath0 ) , and those two functions completely define a markovian network . once @xmath9 is the determined , all of the properties of the given network are determined . the most well - known example is the degree distribution @xmath0 : p_k=_0^dg _ ( g)g_0(k , g ) . it would seem that determining @xmath9 from eq . ( [ gbog ] ) is a simple technical matter , but this is not the case@xcite . the purpose of the present section is to provide a simple , analytic and accurate method to determine @xmath9 . we obtain @xmath9 from eq . ( [ gbog ] ) by using the tautology p(g , g)= |p(g ) + ( p(g , g)-|p(g)[exp ] ) in eq . ( [ gbog ] ) , choosing @xmath12 so as to eliminate the effects of the second term , and then treating the remaining higher powers of @xmath13 as an expansion parameter . using eq . ( [ exp ] ) in eq . ( [ gbog ] ) yields & & _ 0(z , g)= _ 0(z , g)=(1-(1-z)|p(g))^n- n(1-z)dg(g)(|p(g)-p(g , g))1-(1-z)|p(g ) & & -n_n=2^dg(g)(p(g , g)-|p(g))1-(1-z)|p(g))^n . [ gbog01 ] in analogy with the mean - field ( hartree ) approximation of atomic and nuclear physics , we find that the second term of eq . ( [ gbog01 ] ) vanishes if we choose @xmath12 to be the average of @xmath6 over @xmath14 : |p(g)=dg(g)p(g , g).[pave]with eq . ( [ pave ] ) the effects of the term of first order in @xmath13 vanish . we therefore obtain the result : _ 0(z , g)=(1-(1-z)|p(g))^n - n_n=2^dg(g)(p(g , g)-|p(g))1-(1-z)|p(g))^n , [ gbog1 ] with the putative term with @xmath15 vanishing by virtue of eq . ( [ pave ] ) . we treat the first term of eq . ( [ gbog1 ] ) as the leading order ( @xmath16 ) term and regard the remainder as a correction . the validity of this approach can be checked by comparison with simulations , or ( in certain cases ) with analytic results . numerical results for the pin of current interest @xcite indicate that the corrections to the _ lo _ terms induce errors in @xmath0 of no more than a few percent and that the approximation becomes more accurate for large values of @xmath1 . therefore we use the _ lo _ approximation . using exponentiation and the binomial theorem in the first term of eq . ( [ gbog1 ] ) leads to the result ^(lo)_0(k , g)= ( cn + k ) ( 1-|p(g))^n - k|p(g)^k,[glo]which is of the form of a random binomial distribution in which the connection probability depends on the hidden variable @xmath3 . ( [ glo ] ) is our central new general result that can be used for any hidden variable network . this binomial distribution has both the normal gaussian and poisson @xmath17 distributions as limiting cases . ref . @xcite explained the difference between sparse and nonsparse networks . sparse networks have a well - defined thermodynamic limit for the average degree , while this quantity diverges as the network size @xmath18 approaches infinity . @xcite defines criteria for sparseness by pointing out the relevance of @xmath19 of eq . ( [ pave ] ) in determining whether or not a network is sparse . given this quantity the average degree is k = dg ( g)|p(g)= dgdg ( g)p(g , g)(g).if the @xmath4 is independent of @xmath18 the only way to obtain a non - divergent value @xmath20 is for the connection probability @xcite to scale as @xmath21 : p^sparse(g , g)=c(g , g)n , .[sparse ] under the specific assumption that eq . ( [ sparse ] ) holds , ref . @xcite finds a very interesting result . in our notation , this amounts to using eq . ( [ sparse ] ) in eq . ( [ gbog ] ) and taking the limit that @xmath18 approaches infinity . then g_0^sparse(z , g)=(z-1)dg(g)c(g , g ) . this shows that the poisson limit of eq . ( [ glo ] ) is obtained for the very special case of sparse networks in which the connection probability scales as @xmath21 . none of the models of interest here @xcite are sparse , so it is our present result ( [ glo ] ) that is widely applicable . turning to the use of the use of the propagator , we obtain the degree distribution as p_k = dg ( g ) g_0(k , g)dg ( g ) g_0^(lo)(k , g ) . [ ours]this expression can be thought of as averaging a binomial distribution over the hidden variable and is a natural generalization of classical graph theory . a similar expression for @xmath0 has been obtained , in the poisson limit , in ref . @xcite . in that work , @xmath0 is presented as an integral of the poisson distribution for @xmath22 multiplied by the `` @xmath23 representation '' of a density matrix . comparing eq . ( [ glo ] ) with the result ( 3 ) of @xcite shows that our propagator is proportional to the @xmath23 representation , essentially our @xmath4 . @xcite shows , how under certain assumptions , to use @xmath24 to determine the @xmath23 representation . our method allows underlying network properties , denoted by @xmath4 and @xmath6 , to predict various network properties . the clustering coefficient which measures transitivity @xcite : if vertex @xmath25 is connected to vertex @xmath26 and vertex @xmath26 to vertex @xmath27 , there is an increased probability that vertices @xmath25 and @xmath27 are connected . in graph theory , the clustering coefficient @xmath28 is the ratio of the number of triangles to the number of pairs , computed for nodes of degree @xmath1 . @xcite shows that & & c(k)= dg ( g)g_0(k , g)c(g ) [ eq:21 ] + & & c(g)= dg dg(g)p(g , g)|p(g ) p(g , g)(g)p(g,g)|p(g).[cofg ] our calculations replace @xmath29 by @xmath30 of eq . ( [ glo ] ) . one way to verify the _ lo _ approximation is to show that it reproduces analytic results for previously published models . we consider the models of @xcite and @xcite in this section . in both of these models @xmath6 is taken as a step function ( the 0 temperature limit of our model ) : p(g , g)= ( g+g-).[sharp]the two models differ in their choice of @xmath4 , but the use of eq . ( [ sharp ] ) allows one to obtain compact general expressions for the generating functions @xmath31 and @xmath28 . we present these first and discuss specific details of the individual models in separate sub - sections . the use of eq . ( [ sharp ] ) in eq . ( [ gbog ] ) yields _ 0(z , g)= n(z)=n|p(g)(z ) , [ gbogsharp ] so that _ 0(z , g)=z^n|p(g).[gsharp ] it is interesting to observe that eq . ( [ gbog1 ] ) reduces to the above result . this is because powers of @xmath32 for eq . ( [ sharp ] ) , so that the integration appearing in eq . ( [ gbog1 ] ) leads to an expression that is a function of @xmath33 then the use of the binomial theorem allows the second term of eq . ( [ gbog1 ] ) to be expressed as a summable power series in @xmath19 which ultimately leads to the result eq . ( [ gsharp ] ) . if we follow @xcite and treat @xmath1 as a continuous variable ( which requires large values of @xmath1 ) we find _ 0(k , g)=(k - n|p(g ) ) = , , [ gdelta ] + where @xmath34 is the solution of the equation k = n|p(g).[gn]note that for @xmath35 , @xmath34 can take on any value greater than @xmath36 . the result eq . ( [ gdelta ] ) is the same as eq.(34 ) of @xcite , but written in a more compact form . the use of eq . ( [ gdelta ] ) in eq . ( [ ours ] ) and eq . ( [ eq:21 ] ) yields the results p_k=(g_n(k))n|p(g_n(k ) ) + this model is defined by using @xmath37 , but we generalize to take the form _ ( g)=(-g).ref . @xcite works out this model using their green s function formalism . our purpose here is to compare the results of our averaging approximation with their results . for this model the average interaction probability @xmath12 is given by |p(g)=_0^dg ( g+g-)= ( g- ) + ( -g).[pbarss]then our approximation eq . ( [ ours ] ) for the degree distribution @xmath0 is given by p_k= ( cn + k ) _ 0^dg ( 1- ) ^n - kdefine the integration variable @xmath38}$ ] so that & & p_k= ( cn + k ) e^-_t_0 ^ 1 dtt^2t^k(1-t)^n - k , t_0e^- + & & p_k>1= ( cn + k ) e^-((n+1-k)(k-1)(n)-b_t_0(k-1,n+1-k)),[pkif ] + & & p_k=1=n e^-(1-t_0)^nn_2f_1(1,n;n+1,1-t_0)where @xmath39 is the confluent hypergeometric function and @xmath40 is the incomplete beta function ( and with @xmath41 the beta function ) : b_z(a , b)_0^z dt t^a-1 ( 1-t)^b-1,b_1(a , b)=b(a , b).consider the case 1<k,10,(the latter is typical of our biological model ) so that the second term of eq . ( [ pkif ] ) can be neglected . evaluating the remaining gamma functions gives p_k = e^-nk(k-1).[cutus]ref . @xcite computes the degree distribution for this model in analytic manner , using the approximation eq . ( [ gdelta ] ) in which @xmath1 is treated as a continuous variable and therefore `` is expected to perform poorly for small values of @xmath1 '' . the result of @xcite @xmath42 is p_k^bps = e^-nk^2 + e^-(k- n)[cutum]which corresponds to agreement ( for @xmath43 ) within the stated domain of accuracy of ref . the confluence of eq . ( [ cutus ] ) and eq . ( [ cutum ] ) provides a verification of the accuracy of the averaging approximation . the results for @xmath35 seem to disagree , so we examine this more closely . use eq . ( [ gsharp ] ) directly to obtain the generating function @xmath44 as @xmath45 . one obtains a result @xmath46 for all values of @xmath3 ( @xmath47 ) such that @xmath48 . using this generating function yields the result p_k = n = dg(g ) ( g-).the specific value of the integral depends on the choice of @xmath4 , but the result is a finite number for any choice of @xmath4 that satisfies the normalization condition that its integral over its domain is unity . thus we believe that the correct result of using the propagator ( eq(34 ) of @xcite in their eq(11 ) ) is p_k^bps = e^-nk^2[cutum1]instead of eq . ( [ cutum ] ) , which is in agreement with our result . our approximation works very well in reproducing the computed clustering coefficient of @xcite . in particular , we evaluate @xmath49 of eq . ( [ cofg ] ) to find that ( 2g-+1)).numerical evaluation of this approximate expression accurately reproduces the result of fig . 3 of ref . thus our mean field approximation is accurate for both our model@xcite and the model of ref . our principal application is to the the pin of ref . this model is based on the concept of free energy of association . for a given pair of proteins the association free energy ( in units of @xmath50 ) is assumed to deviate from an average value a number contributed by both proteins additively as @xmath51 . this is a unique approximation to first - order in @xmath3 and @xmath52 . thermodynamics and the assumption that the interaction probability is independent of concentration allows us to write p(g , g)=1/ ( 1+e^-g - g),[pdef]which reduces to a step function in the zero temperature limit , but otherwise provides a smooth function . increasing the value of @xmath36 weakens the strength of interactions , and previous results @xcite showed the existence of an evolutionary trend to weaker interactions in more complex organisms . the probability that a protein has a value of @xmath3 is given by the probability distribution _ ( g)=ee^-g , -1g+,[rhodef]where the positive real value of @xmath53 governs the fluctuations of @xmath3 . we previously chose the species - dependent values of @xmath53 and @xmath36 so as to reproduce measured degree distributions obtained using the yeast two - hybrid method ( y2h ) that reports binary results for protein - protein binding under a controlled setting@xcite . those parameters are displayed in table i. the impact of the parameters @xmath53 and @xmath36 are explained in ref . @xcite and displayed in fig . 3 of that reference . increasing the value of @xmath53 increases the causes a more rapid decrease of @xmath0the slope of @xmath0 increases in magnitude . increasing the value of @xmath36 decreases the magnitude of @xmath0 without altering the slope much for values of @xmath1 greater than about 10 . the ability to vary both the slope and magnitude of @xmath0 gives this model flexibility that allows us to describe the available degree distributions for different species . [ table1 ] .parameters obtained in ref . @xcite [ cols="^,^,^,^",options="header " , ] we obtain an analytic form for the for @xmath12 eq . ( [ pave ] ) of this model . given eq . ( [ rhodef ] ) and eq . ( [ pdef ] ) we find an analytic result : |p(g,)=_2f_1(1,;+1;-),[analpbar]where @xmath39 is the confluent hypergeometric function . the special case @xmath54 yields a closed form expression : obtained in contrast with the result of the sharp cutoff model eq . ( [ pbarss ] ) . this shown in fig . [ fig : pbar ] . .95 cm ( 10,10)(5,0 ) ( color online ) average connection probability @xmath54 , @xmath55 . solid ( red ) : result of eq . ( [ special ] ) ; dashed ( blue ) ( containing the step function ) result of eq . ( [ pbarss ] ) . the approach to unity is smooth for eq . ( [ special]).,title="fig:",width=10,height=377 ] it is useful to define the variable > 0 , and note that an integral representation@xcite _ 2f_1(n,;+1;-)=_0 ^ 1dtt^-1(1+t)^-n,[alg]is convenient for numerical evaluations . knowledge of the propagator eq . ( [ glo ] ) allows us to compute the clustering coefficients of diverse species . the resulting degree distributions of @xmath0 ( shown for the sake of completeness ) and the newly computed clustering coefficients @xmath56 for yeast _ s. cerevisiae _ @xcite , worm _ c. elegans _ @xcite and fruit fly _ d. melanogaster _ @xcite are shown in fig . the parameters @xmath53 and @xmath36 are those of @xcite , so the calculations of the clustering coefficients represent an independent major new prediction of our model . results of numerical simulations and our analytic procedure are presented . the excellent agreement between the two methods verifies the @xmath16 approximation . more importantly , the agreement between our calculations and the measured clustering coefficients is generally very good , so our model survives a very significant test . this bolsters the notion that the properties of a pin are determined by a distribution of free energy . the clustering coefficient for yeast drops rapidly for large values of @xmath1 ( where statistics are poor ) , a feature not contained in our model . it is worthwhile to compare our model with that of @xcite . that work chooses a gaussian form of @xmath4 , based on hydrophobicity , a step function form of @xmath6 , and is applied only to yeast . we found @xcite that @xmath0 of @xcite is scale free only for a narrow range of parameters , and we could not reproduce the data for diverse species using that model . .95 cm ( 10,10)(2,-1 . ) ( color online ) degree distributions @xmath0 and clustering coefficients @xmath57 of diverse species . degree distributions @xmath0 : the solid ( red ) curves are derived from the @xmath16 theory . the black dots are the results of experimental data as referenced in the text . the small ( blue ) circles are the results of a numerical simulation using the procedure of @xcite . clustering coefficients @xmath57 : the solid ( red ) curves are derived from the @xmath16 theory . the small ( blue ) dots are the results of a numerical simulation using the procedure of @xcite and the heavy ( black ) dots represent the measured data . , title="fig:",width=10,height=566 ] the human interactome is of special interest . [ hdegc]a shows the human degree distributions computed with two sets of parameters , one from ref . @xcite ( table i ) and the other using values of @xmath58 shown in the caption . the degree distributions are essentially identical , so only one curve can be shown . each is approximately of a power law form and each describes the measured degree distribution very well@xcite . calculations of degree correlations allows one to distinguish the two parameter sets . figure [ hdegc]b shows that the cluster coefficients differ by a factor of two . we find that @xmath56 decreases substantially as @xmath53 increases . the increase in @xmath53 reduces the allowed spread in the value of @xmath3 and reduces the value of integrand of eq . ( [ eq:21 ] ) . it is interesting to note that the two existing measurements of the human @xmath56 differ by a factor of about an order of magnitude with the measurements of ref.@xcite obtaining much smaller values than those of @xcite . the results of @xcite are closer to our computed @xmath56 results for @xmath59 . in contrast with the results for other species , our @xmath56 lie significantly above the data . however , the two data sets disagree substantially ( by a factor of as much as 100 for certain values of @xmath1 ) and both show a clustering coefficient that is generally significantly smaller than that of the other species . several possibilities may account for the discrepancies between these two measurements of @xmath56 in humans and also for the differences between our model predictions and the experimental results . i ) the human studies sample a limited subset of links of the complete network and this could bias the results . ii ) the human protein subsets used in the two studies differ . iii ) the human interactome is truly less connected than that of other species . this demonstrates the importance of measuring degree correlations to determine the underlying properties of the network . the current model and these considerations suggest the need for better design of future pin studies that will not only include other species , but also comparisons between the pins of different organs of a given species . furthermore , comparisons between normal and malignant tissues could also be very fruitful . .9 cm ( color online ) human degree distribution @xmath0 ; the solid ( red ) curve is obtained using both set a @xmath60 and set b @xmath59 . the black dots represent the experimental data . the data set is that of @xcite , but nearly identical data is obtained from @xcite . human cluster coefficient @xmath56 : the solid ( red ) curve is computed using set a @xmath60 and the dashed ( green ) using set b @xmath59 . measured human clustering coefficients are from @xcite triangles ( blue ) and @xcite heavy dots ( pink ) . , title="fig:",width=9 ] in summary , this work provides a method to obtain the properties of hidden variable network models . the use of the approximation eq . ( [ pave ] ) , used to obtain the propagator eq . ( [ glo ] ) , provides an excellent numerical approximation to exact results for the models considered here . if necessary , the method can be systematically improved through the calculation of higher order corrections . our principal example is the pin of ref . @xcite . not only does the use of eq . ( [ glo ] ) provide an accurate numerical result , but the model correctly predicts the clustering coefficients of most species . for the human interactome , two different parameter sets yield nearly the same degree distribution but very different clustering coefficients , showing the importance of measuring degree correlations to determine the underlying nature of the network . 99 s. h. strogatz , nature ( london ) * 410*,268 ( 2001 ) . r. albert and a .- l.barabsi , rev . mod . phys . * 74*,47 ( 2002 ) , siam rev . * 45 * , 167 ( 2003 ) . pastor - satorras , a. vzquez , and a. vespignani , phys . lett . * 87 * , 258701 ( 2001 ) . b. alberts _ et al . _ , _ the cell _ , ( garland science , new york 2002 ) . g. caldarelli , a. capocci , p.delosrios , and m. a. muoz , phys . * 89 * , 258702 ( 2002 ) . b. sderberg , phys . e * 66 * , 066121 ( 2002 ) . m. bogu and r. pastor - satorras , phys . e * 68 * , 036112 ( 2003 ) . yi y. shi , g.a . miller , h. qian , and k. bomsztyk , proc . sci . * 103 * , 11527 ( 2006 ) . deeds , o. ashenberg , and e.i . shakhonovich , proc . sci . * 103 * , 311 ( 2006 ) . m. abramowitz and i. a. stegun , _ handbook of mathematical functions _ , ( dover , new york 1970 ) . s. abe and s. thurner , phys . e * 72 * , 036102 ( 2005 ) ; s. abe and s. thurner , int . j. mod c * 17 * , 1303 ( 2006 ) . s. fields and s. song , nature * 340 * , 245 ( 1989 ) . networks / resources / protein / bo.dat.qz s. li , _ et al . _ , science * 303 * , 540 ( 2004 ) . l. giot _ et al . _ , science * 302 * , 1727 ( 2003 ) . the value of @xmath61 is a testable result of our model , even though experimentalists do not measure this quantity . the predicted number of proteins with no interactions is @xmath62 , where the value of @xmath18 is given in table i. the experimentalists conventionally normalize their distributions as @xmath63 , so we multiply our computed @xmath0 by a factor of @xmath64 so that the computed sum @xmath65 is unity . rual , _ et al . _ nature * 437 * , 1173 ( 2005 ) u. stelzl , _ el al . _ cell * 122 * , 957 ( 2005 ) | the properties of certain networks are determined by hidden variables that are not explicitly measured .
the conditional probability ( propagator ) that a vertex with a given value of the hidden variable is connected to k of other vertices determines all measurable properties .
we study hidden variable models and find an averaging approximation that enables us to obtain a general analytical result for the propagator .
analytic results showing the validity of the approximation are obtained .
we apply hidden variable models to protein - protein interaction networks ( pins ) in which the hidden variable is the association free - energy , determined by distributions that depend on biochemistry and evolution .
we compute degree distributions as well as clustering coefficients of several pins of different species ; good agreement with measured data is obtained . for the human interactome
two different parameter sets give the same degree distributions , but the computed clustering coefficients differ by a factor of about two .
this shows that degree distributions are not sufficient to determine the properties of pins .
= 10000 0.5 cm |
You are an expert at summarizing long articles. Proceed to summarize the following text:
when matter is exposed to intense laser fields , high harmonics ( hhs ) of the incident radiation may be produced . usually , only odd harmonics are obtained even when the laser pulses are short ( for theoretical and experimental work which demonstrates this see @xcite and @xcite , respectively ) . since the duration of the pulse in time is inversely proportional to its width in energy space , one may find this result surprising , as one may expect to obtain also a large distribution of frequencies in the scattered field . why are only odd harmonics obtained even when the laser pulses are short ? for cw lasers ( and symmetric field - free potential ) using the non - hermitian floquet theory it was proved that only odd harmonics are obtained when the dynamics is controlled by a single resonance floquet quasienergy ( qe ) state @xcite,@xcite . when laser pulses are used it was argued that this proof holds since usually the populated resonance states are associated with very different lifetimes and the dynamics is controlled by the resonance state which has the longest lifetime . however , this argument may hold only when the duration of the laser pulses is large enough to enable the decay of the short lived resonances . indeed numerical simulations showed that the harmonic generation spectra ( hgs ) as obtained from a single non - hermitian ( complex scaled ) resonance floquet state is in a remarkable agreement with the results obtained from conventional ( i.e. , hermitian ) time dependent simulation @xcite . the question that is addressed in this work is weather an analytical criteria for the shape and duration of the laser pulse for which the system is controlled by a single resonance floquet state can be given . it is obvious that the question regarding the possibility of population of a single resonance state is connected with the question regarding the degree of adiabaticity of the process . the question is therefore under which conditions can a short laser pulse be defined as adiabatic one . the answer to this question is important not only to harmonic generation ( hg ) studies but also for other , more general studies where lasers are used to control the dynamics , for example adiabatic stirap procedures @xcite . in order to answer this question we use the ( t , t ) formalism @xcite together with the non - hermitian quantum mechanics ( nhqm ) formalism . the use of nhqm formalism to describe the dynamics of atoms / molecules subjected to cw laser fields is essential , since only then can the dynamics be described in terms physical , square integrable , resonance floquet states . otherwise , the description of the dynamics in terms of hermitian floquet states results in very little physical insight on the problem , as well as numerical problems , not to mention that it is limited to the description of bound systems only . in the hermitian case the spectrum is continuous and becomes discrete only due to the use of finite box quantization ; moreover , a single floquet state in hermitian qm can not describe neither the resonance phenomena nor the field ionization phenomena @xcite . our strategy is as follows . in section 2 we give a brief review of the formalism used in our derivation [ namely the ( t , t ) formalism for hermitian and non - hermitian hamiltonians ] . in section 3 we introduce the new derivation of the adiabatic theorem for open quantum systems in strong laser pulses . in section 4 we apply the adiabatic theorem as derived in section 3 to a test - case model hamiltonian which describes a 1d xe atom subjected to a sin - square pulse of monochromatic laser . in section 4 we conclude . the ( t , t ) formalism enables one to obtain analytical solutions for any time - dependent schrdinger equation ( tdse ) with time - dependent hamiltonians . the formalism rests on lifting of the tdse to an extended hilbert space , propagation of the wavefunction there and finally projecting back to the physical hilbert space . the solution of a general tdse is given as usual by : @xmath0 with the initial condition @xmath1 and the vector operator @xmath2 describes the internal degrees of freedom . it has been shown by peskin and moiseyev that by regarding time as an extra coordinate @xmath3 , one can obtain another schrdinger equation with a time independent hamiltonian in the extended hilbert space @xmath4 , whose solution @xmath5 has an analytical time dependence ( given by the analytical time evolution operator associated with time - independent hamiltonians ) @xcite . our desired wavfunction @xmath6 can be deduced from this wavfunction by a simple operation ( that will be shown later ) . the advantage is that efficient propagation schemes designed for solving the tdse with time - independent hamiltonians could then be used also for time dependent hamiltonians , thus releasing one from the difficulties associated with time ordering . let us define the following floquet - type operator @xmath7 where @xmath3 should be regarded as a * coordinate * and @xmath8 is the final time of propagation . when the hamiltonian @xmath9 is time periodic with period @xmath10 , the operator is the floquet operator and @xmath11 . provided that correct boundary conditions are chosen for the @xmath3 coordinate , this operator is hermitian since it is the sum of two hermitian operators . this operator has eigenstates and eigenvalues which are given by the eigenvalue equation : @xmath12 and the set of eigenstates is complete in the extended hilbert space @xmath4 with respect to the inner product @xmath13 say the following tdse with time - independent hamiltonian need to be solved with the initial state @xmath14 : @xmath15 using the definition of @xmath16 this equation reads @xmath17 by setting the cut @xmath18 on eq.[eq6 ] one gets : @xmath19|_{t'=t}=h(\mathbf{r},t ) [ \overline{\psi}(\mathbf{r},t',t)|_{t'=t}]=i\hbar \frac{\partial}{\partial t } [ \overline{\psi}(\mathbf{r},t',t)|_{t'=t}]\label{eq7}\ ] ] where the following property @xmath20= [ ( \frac{\partial}{\partial t}+\frac{\partial}{\partial t ' } ) \overline{\psi}(\mathbf{r},t',t)]|_{t'=t}\label{eq8}\ ] ] ( which holds true for * any * function of @xmath3 and @xmath21 . the relation between the solution of eq.[eq7 ] and the solution of the original tdse ( eq.[eq1 ] ) is given by @xmath22 provided that these two differential equations have the same initial condition . hence , @xmath23 it is seen from eq.[eq10 ] that apparently the initial condition is nt unique . while this holds true in case that one is interested only in the physical wavefunction @xmath6 , it should be noted that if one wishes to calculate physical quantities in the extended hilbert space using the function @xmath5 and then go back to the original hilbert space , and get the correct results , the initial condition in the extended hilbert space should behaves as a delta - function in @xmath3 . the correct initial condition will therefore be @xmath24 . the main advantage of the ( t , t ) formalism from a numerical point of view is that it enables the use of an analytical expression for the time evolution operator , without the necessity of time ordering , even when the hamiltonian is strongly time - dependent . any tdse with time - dependent hamiltonian could be replaced by a different tdse , with time - independent hamiltonian , for which a greater number of accurate integration schemes exist and the solution is given formally by @xmath25 ) . the price one pays however is that the new tdse need to be integrated over one more dimension . the main advantage of the _ ( t , t ) _ formalism from a conceptional point of view is that it enables to describe any time - dependent dynamics in terms of stationary eigenstates and eigenvalues . so far the derivation has been carried out within the framework of the conventional ( i.e. , hermitian ) quantum mechanics . using the complex - scaling ( cs ) transformation @xcite @xmath26 the quasienergy spectrum of the floquet hamiltonian becomes complex and square - integrable resonance states , that were embedded in the continuum in the unscaled problem , are uncovered . for sake of simplicity we drop the index @xmath27 in all @xmath27-dependent expressions ( operators , eigenvalues , eigenvectors etc . ) in the proceeding text . the floquet operator @xmath28 can be represented with the orthogonal fourier basis set @xmath29 as a square matrix @xmath30_{n',n}=\frac{1}{t}\int_{0}^{t}dt'e^{-i\omega n ' t ' } h_f(\mathbf{r},t')e^{-i\omega n t } \label{eq11}\ ] ] where @xmath31 . the left and right eigenvectors ( bi - orthogonal set of eigenvectors , @xcite ) of this floquet matrix are the fourier components of the floquet states as defined in eq.[eq3 ] , that is @xmath30^{t}\overrightarrow{\varphi}^{l}_{\alpha}(\textbf{r})=e_{\alpha}\overrightarrow{\varphi}^{l}_{\alpha}(\textbf{r } ) \label{eq11a}\ ] ] @xmath30\overrightarrow{\varphi}^{r}_{\alpha}(\textbf{r})=e_{\alpha}\overrightarrow{\varphi}^{r}_{\alpha}(\textbf{r } ) \label{eq11b}\ ] ] where @xmath32_{n}\equiv\varphi^{l / r}_{\alpha , n}(\textbf{r } ) . \label{eq11c}\ ] ] since in our case @xmath33^{t}=[\underline{\underline{h_f(\mathbf{r})}}]$ ] then @xmath34 . there are two sets of eigenfunctions of the floquet operator @xmath35 @xmath36 and @xmath37 as pointed out in @xcite the c - product , which is associated with the non usual inner - product in linear algebra ( see for example wilkinson s text book @xcite ) , reads @xmath38 where @xmath39 . however , as proposed recently by moiseyev and lein ( @xcite ) , the inner - product should be modified even further when time - dependent functions are used as basis set due to the time - asymmetry problem in nhqm . if we define @xmath40 and @xmath41 to be the floquet eigenfunctions and eigenvalues respectively , then the time - dependent basis functions could be @xmath42 ( which are solutions of the tdse ) and also the left functions @xmath43 following the modified definition of the inner product ( `` finite - range '' product , `` f - product '' @xcite ) @xmath44 it implies that the @xmath45 quasienergy state decays exponentially in time . for more detailed discussion see @xcite,@xcite . the adiabatic theorem for time - dependent * bound * systems was derived in 1997 by kohn _ _ @xcite and in 1999 by holthaus _ @xcite who used the ( t , t ) formalism to describe the evolution of a system subjected to chirped laser pulses . as discussed by kohn and co - workers , the adiabatic approach is * not * applicable for open systems since the quasienergy level spacing reduce to zero as the number of basis functions used in the numerical calculation is increased . to avoid this difficulty baer et al . @xcite applied the adiabatic theorem to time - dependent * open * systems in the high frequency regime where the system was stabilized and the resonances ( which were embedded in the continuum part of the floquet spectra ) became so narrow that they could be practically treated as bound states . the purpose of our work is to derive the adiabatic theorem for general time - dependent open systems where there are no bound states and the resonances are nt necessarily narrow . we are using the non - hermitian floquet formalism ( through the cs formalism ) which allow us to describe the dynamics in term of non - hermitian resonance states ( see for example @xcite-@xcite , and also the work of day et al . @xcite who used the nh floquet multistate method to study the applicability of the single floquet resonance approximation in the description of the dynamics of h atom subjected to intense laser fields of various strengths ) . below we derive the adiabatic theorem for time - dependent open systems using the extended ( t , t ) formalism . by the term `` extended '' _ ( t , t ) _ formalism we mean that in the same manner presented , one may add any number of time `` coordinates '' to the schrdinger equation as one wishes , if by this a better understanding or easier solution of the problem is achieved . here we found that by addition of 2 time `` coordiantes '' to the tdse , we simplified the derivation of the adiabaticity criteria for photo - induced dynamical systems . in this sense , we are using a _ ( t , t,t ) _ formalism . we would like to study the dynamics of a single active electron in an atom or molecule , subjected to a pulse of strong monochromatic linearly polarized laser radiation . in the dipole approximation the tdse which describes this process is : @xmath46 where , @xmath47 and , @xmath48 here @xmath49 is the function which describes the envelope of the laser pulse and @xmath50 is a vector as defined in eq.[eq18 ] ; @xmath51 is the laser s amplitude , @xmath52 is a unit vector in the direction of the electric component of the laser field , @xmath53 is the laser s frequency , with @xmath54 the optical period . @xmath55 is the field - free hamiltonian and the vector operator @xmath2 describes the internal degrees of freedom ( the coordinates are complex - scaled throughout ) . in the same spirit of subsection ( 2.1 ) , we define the following operator @xmath56 where @xmath57 and @xmath58 should be regarded as additional * coordinates*. upon complex - scaling @xcite @xmath59 becomes non - hermitian . therefore the inner c - product should be used as mentioned before . the quasi - energy solutions of this complex - scaled floquet - type operator are : @xmath60 @xmath61 where the symbol @xmath62 does nt stand for an operator but for the transpose of the matrix representing the operator @xmath63 . the eigenfunctions form a complete set in the extended hilbert space @xmath64 . say we want to solve the following tdse with the initial state @xmath65 : @xmath66 the solution to this equation is @xmath67 and a function @xmath68 , which is * not * a solution of any schrdinger equation , is defined as @xmath69 where by taking the cut @xmath70 on eq.[eq24 ] it is easily seen that the expansion coefficients are @xmath71 . [ @xmath72 is a solution of the original tdse ( apply the cut t=t=t on eq.[eq21 ] and compare the result to the result obtained when the function @xmath72 is substituted in the original tdse ) ; therefore , any linear combination of these solutions is also a solution ] . let us now return to the main purpose of this article , the derivation of the adiabatic theorem for non - hermitian open systems . we would like to treat @xmath73 as an adiabatic coordinate ( this is the `` coordinate '' associated with the pulse envelope ) in the same way that the electronic motion is separated from the nuclear one in the treatment of molecules within the born - oppenheimer approximation . first we define the following operator @xmath74 where @xmath73 should be regarded as a parameter now . this means that this hamiltonian is a floquet hamiltonian describing the interaction of the atom with * cw * laser of strength @xmath75 where following eq.[eq18 ] @xmath76 the eigenstates of this operator form a complete basis ( in the @xmath77 space ) , for every value of the parameter @xmath73 : @xmath78 notice that due to the complex scaling , @xmath79 and @xmath80 get complex values . we can expand each eigenstate of the complete problem ( eq.[eq21 ] ) in this basis : @xmath81 substituting eq.[eq28 ] into eq.[eq21 ] , multiplying the obtained equation from the left hand side by @xmath82 and integrating over @xmath2 and @xmath3 one gets , in matrix notation , the equality : @xmath83\overrightarrow{\chi}_{k}(t'')=\lambda_{k}\overrightarrow{\chi}_{k}(t '' ) \label{eq29}\ ] ] where @xmath84_{\alpha,\alpha'}=\varepsilon^{ad}_{\alpha}(t'')\delta_{\alpha,\alpha'},\ ] ] @xmath85_{\alpha,\alpha'}=((\psi^{ad , l}_{\alpha}(\mathbf{r},t',t'')|-i\hbar \frac{\partial}{\partial t '' } @xmath86_{\alpha}=\chi_{\alpha , k}(t '' ) . \label{eq30}\ ] ] notice that in case that the matrix on the left hand side of eq.[eq29 ] is diagonal , a homogeneous systems of uncoupled equations is obtained . in such a case one should solve each equation separately . therefore , the sum in eq.[eq28 ] reduces to a single product . this is exactly the adiabatic approximation as appears in the born - oppenheimer context . the next step in our derivation is to represent the matrix @xmath87 by its spectral decomposition @xmath88\underline{\underline{d}}^{r}(t'')=\underline{\underline{d}}^{r}(t'')\underline{\underline{w}}(t '' ) \label{eq31}\ ] ] @xmath88^{t}\underline{\underline{d}}^{l}(t'')=\underline{\underline{d}}^{l}(t'')\underline{\underline{w}}(t '' ) \label{eq32}\ ] ] the matrix of eigenvalues @xmath89 is diagonal and the right and left eigenvectors are normalized with respect to each other in order to maintain the correct inner product : @xmath90^{t}\underline{\underline{d}}^{r}(t'')=\underline{\underline{i } } \label{eq33}\ ] ] in the case that the matrix @xmath87 is not strictly diagonal we can use first - order perturbation theory to get the first - order deviation from diagonal . if we treat the matrix @xmath91 as perturbation , we get @xmath92_{\alpha',\alpha}=\delta_{\alpha',\alpha}+\frac{[\underline{\underline{v}}(t'')]_{\alpha',\alpha}}{\varepsilon^{ad}_{\alpha}(t'')-\varepsilon^{ad}_{\alpha'}(t '' ) } \label{eq34}\ ] ] the matrix will be diagonal to a good approximation if @xmath93_{\alpha',\alpha}|\ll 1 \label{eq35}\ ] ] which produces the following adiabaticity criteria , @xmath94 using the specific form of the hamiltonian of the problem given in eq.[eq17 ] and the hellman - feynman theorem one gets the adiabatic condition for time - dependent open systems : @xmath95 where @xmath96 the index @xmath97 is a super - index ; since ( eq.[eq27 ] ) it is easily seen that not only is @xmath98 a solution of the eigenvalue equation , with eigenvalue @xmath80 , but also @xmath99 is a solution , with the eigenvalue @xmath100 , for any integer @xmath101 . let us take all the states whose corresponding eigenvalues lie in the interval @xmath102 $ ] ( the first brillouin zone ) and define them to have index @xmath103 ; we call these states @xmath104 , @xmath105 and the corresponding eigenvalues @xmath106 and get : @xmath107 @xmath108 @xmath109 where @xmath110 thus , the index @xmath97 actually counts both the position of the quasienergy within the first brillouin zone ( the index @xmath111 ) and the brillouin zone itself ( the index @xmath112 ) . with respect to the generalized inner product , two states with one or more of the indices ( @xmath111,@xmath112 ) different are orthogonal . going back to eq.[eq37 ] now , it is seen that the probability to couple an initial adiabatic state @xmath113 to any other adiabatic state @xmath114 is given by , @xmath115 where @xmath116 @xmath117 and the functions @xmath118 , that will be termed as _ `` adiabatic cross terms '' _ from now on , are given by @xmath119 since the energies @xmath120 are complex ( the hamiltonian is non - hermitian ) then for @xmath121 it is most unlikely that @xmath122 . it is clear that the denominator hardly ever vanishes even when @xmath103 . this holds true even when @xmath123 but @xmath124 . _ therefore , the criteria for a pulse to be considered adiabatic is that the condition _ @xmath125 _ be fulfilled . _ who is the adiabatic state @xmath113 whose couplings to all other states @xmath114 should remain small in the adiabatic limit ? assuming that the system is in a stationary state of the field - free problem @xmath126 before the action of the field ( relaying on the superposition principle of the solutions of the tdse generality is not lost by this assumption ) and provided that the field is switched adiabatically , this floquet resonanace state @xmath113 is the state which is `` born '' from the stationary state @xmath126 as the field is switched on . if the process is not done adiabatically many floquet resonanace states will be populated , resulting in considerable couplings of @xmath113 to them and collapse of the adiabatic condition in eq.[eq46 ] . the only adiabatic check which is physically meaningful is one in which _ @xmath97 denotes a resonance state _ ( which is associated with square - integrable function ) . @xmath127 however _ stands for both resonances and rotated continuum states_. one should notice that the derivation of the adiabatic theorem presented above holds for many electron systems . in order to avoid complicated notation the symbol @xmath2 can stand for many electrons . it also holds for polychromatic radiation , where the cw field is a collection of monochromatic fields with frequencies @xmath128 and phases @xmath129 . notice that for a given problem ( given spectral profile of the cw field and given field - free potential ) , the sum over absolute value of the adiabatic cross terms should be calculated as function of the effective cw - field intensity ( which is symbolized here through @xmath73 ) * only once*. the adiabatic cross terms should then be converted to be functions of time , through the explicit time dependence of the pulse envelope and then the sum of their absolute values should be multiplied by the time derivative of the pulse envelope and by the maximal field intensity to obtain the final expression which indicates whether the adiabatic criteria is fulfilled or not . it is easily seen in eq.[eq43 ] that for a given system , the shape and intensity of the laser pulse determines its adiabaticity since these parameters influence the shape - derivative term . a short pulse which is switched on or off abruptly and has a high maximal intensity , will most likely not be adiabatic . in the case that the adiabaticity criteria is fulfilled , the sum in eq.[eq28 ] could be reduced to a single product : @xmath130 and the adiabatic states are assigned with two good quantum numbers @xmath131 and @xmath111 . the solution to the eigenvalue equation eq.[eq29 ] is now ( @xmath132_{\alpha,\alpha'}\approx 0 $ ] ) @xmath133 by using eq.[eq24],[eq51],[eq52 ] we get that the adiabatic solution , in the ( t , t,t ) formalism , is given by , @xmath134\phi^{ad , r}_{j}(\mathbf{r},t',t'')e^{-\frac{i}{\hbar}\int^{t''}d\tau e^{ad}_{j}(\tau)}\label{eq53}\ ] ] now , applying the cut @xmath135 in order to obtain the physical solution of eq.[eq16 ] one gets ( eliminating the phase factor @xmath136 ) : @xmath137 this is the adiabatic solution of the tdse associated with initial state @xmath126 . let us summarize and clarify the procedure that need to be made in order to determine if a pulse is adiabatic or not ; the determination is carried out through the calculation of the expression @xmath138 ( it is t now , not t ! ): \(1 ) perform non - hermitian adiabatic floquet simulations ( eq.[eq27 ] ) with * cw * field , for a range of intensities @xmath75 which covers all intensities between zero and the maximal intensity @xmath51 that the studied laser pulse reaches . the adiabatic floquet hamiltonian is therefore @xmath139 and the eigenvalue equation to be solved is @xmath140 obtain the quasienergy spectrum @xmath141 and all the adiabatic cross terms @xmath142 as defined in eq.[eq45],[eq49 ] as function of the intensities . this stage is done only once , for a given system . \(2 ) for a given laser pulse @xmath49 with a maximal intensity @xmath51 , evaluate the effective cw - field intensity as function of time @xmath143 ( eq.[eq18 ] ) . then , convert the adiabatic cross terms to be functions of time via the equality ( eq.[eq26a ] ) @xmath144 using @xmath142 as calculated in step 1 @xmath145 where @xmath146 is the transformation which fulfills @xmath147=t$ ] . \(3 ) for each given resonance state @xmath148 calculate @xmath149 using @xmath150 from step 2 where notice that here @xmath73 in eqs.([eq42]-[eq45 ] ) is replaced by @xmath21 . if for a given resonance state @xmath148 the corresponding expression @xmath138 is smaller then unity at every instant , it is guaranteed that the system initially at the bound state which corresponds to this resonance , will evolve adiabatically to that resonance . in this case the hgs spectra will show only odd harmonics and the ionization probability as function of time will have a simple form that will be shown . we studied a single - electron 1d xe atom subjected to a single sin - square pulse of strong monochromatic laser field in two approaches . in the first one hermitian simulations were carried out whereas in the second non - hermitian floquet simulations based on the complex scaling method were carried out . the hermitian simulations were carried out by solving the following tdse @xmath151\psi(x , t)=i\hbar \frac{\partial}{\partial t}\psi(x , t ) \label{eq55}\ ] ] with a sine - square envelope @xmath152 and the field - free effective potential @xmath153 was an inverse gaussian @xmath154 which supports two bound states that mimic the two lowest electronic states of xe , with energies @xmath155 , @xmath156 and a third weakly bound state with energy @xmath157 the wave function was taken initially at the ground state ( g.s . ) of the field - free hamiltonian of the system : @xmath158 and was calculated for times @xmath159 ( single sine - square pulse , @xmath160 was the number of optical cycles that entered the pulse ) . using this wavefunction the ionization probability at times @xmath161 was calculated ( at these times the potential felt by the electron was the field - free potential @xmath153 ) @xmath162 where @xmath129 were the bound states of the field - free problem ( 3 bound states over which we summed in this example ) . also hgs was calculated , which following the classical - quantum correspondence principle ( larmor formula , ) equals the modulus - square of the fourier - transformed time - dependent acceleration expectation value . this is actually the intensity of the radiation emitted by the oscillating electron as presented in energy space . @xmath163 the non - hermitian simulations were floquet simulations which were carried out for different field intensities . the quasienergy spectrum of complex energies @xmath164 and the adiabatic cross terms @xmath142 were calculated for each intensity . then , these quantities were expressed as function of time through eq.[eq26a ] , i.e. @xmath165 . it was verified that in the cases where the adiabatic criteria was fulfilled , the hgs obtained from the hermitian propagation simulation contained only odd harmonics . a more quantitative measure of the existence of the adiabatic criteria was obtained through the comparison of ionization probabilities as obtained from the hermitian propagation simulation and the non hermitian simulation . in order to get non - hermitian floquet hamiltonian , the complex coordinate method was used . the floquet hamiltonian ( eq.[eq48 ] ) @xmath166 was diagonalized . provided that the scaling parameter @xmath27 was sufficiently large , the resonance quasienergy states were @xmath27 independent : @xmath167 @xmath168 being the position of the state , @xmath169 being the width of the state and @xmath170 its lifetime . since the resonance states had finite lifetimes , and since the resonances are the states which are associated with the dynamics , these resonance lifetimes should have fingerprints in the hermitian propagation simulation . indeed it was found to be so when the ionization probabilities , as computed in the two simulations , were compared ; the ionization probability at each instant was given by the following expression , which was obtained using eq.[eq54 ] and the f - product definition for the inner product @xcite : @xmath171 where @xmath172 was associated with the floquet resonanace state that was `` born '' from the ground stationary state @xmath173 as the field was turned on . the resonances that were `` born '' from the field - free hamiltonian bound states were identified by plotting the quasienergy spectrum as function of the effective field intensity @xmath75 . the resonance complex quasienergy trajectories started from the field - free hamiltonian bound states real energies and formed continuous trajectories in the complex energy plane as function of the intensity . how was this expression for the ionization probability obtained ? according to the f - product definition when the complex energy given in eq.[eq62 ] is substituted in eq.[eq54 ] , the `` ket '' ( right ) solution of the non - hermitian tdse is obtained : @xmath174 and it is easily seen that this function decays with time . the `` bra '' ( left ) solution is not a solution of a schrdinger equation but is derived from the `` ket '' solution ( the explanation of this point is beyond the scope of this work ; for an explanation see @xcite ) @xmath175 and also this function decays with time . when the f - product of @xmath176 and @xmath177 is calculated ( this is an overlap integral without complex conjugation of the left state ; in the usual dirac - product notation this reads @xmath178 ) the terms containing the real part of the energy cancel each other and the overlap integral ( c - product ) of the adiabatic floquet states gives unity ( remember the completeness property of floquet states also in coordinate space alone ) . the expression in eq.[eq63 ] is obtained for the specific case that only the floquet resonance state which is `` born '' from the field - free bound state is populated . when several floquet states are populated the same type of calculation could be repeated , where this time @xmath176 and @xmath177 are given by liner combination of terms as in eq.[eq64 ] with the proper coefficients . since the adiabatic floquet states are orthogonal to each other , a generalization of the result of eq.[eq63 ] is obtained : @xmath179 notice that in the non - hermitian formalism the norm is not conserved but decays with time ; it contains the knowledge about decay processes inherently . the two functions as given in eq.[eq59],[eq63 ] were compared and it was found that the resemblance between the functions increased as the adiabatic limit was increasingly reached by the laser pulse parameters . the numerical method used to solve the tdse with hermitian hamiltonian was the split operator forest - ruth algorithm with 7 points @xcite ; the grid size , time step and/or grid step were adjusted as required to achieve convergence . for the non - hermitian floquet simulation , the ( t , t ) formalism was used , with the complex coordinate method . the number of basis functions , box length and scaling angle were adjusted as required to achieve convergence . in fig.[fig1 ] the hgs as obtained for pulse strength of @xmath180 ( corresponding to intensity of @xmath181 ) , pulse - durations of @xmath182 optical cycles , laser frequencies of @xmath183 ( corresponding to energies of @xmath184 and wavelengths of @xmath185 , @xmath186 , respectively ) and initial state @xmath187 are shown . the appearance of odd harmonics and the absence of even or non - integer harmonics , is in general the main feature which appears for long pulses , regardless of the frequency . the odd harmonics are obtained even for the shortest laser pulses although the existence of odd - symmetry selection rules is sensitive to the frequency : some deviation appears at frequency of @xmath188 and the obtained spectra is more complicated . in fig.[fig2 ] the complex quasienergies of the 3 resonances which are `` born '' from the 3 bound states of the field - free hamiltonian ( will be given indices 1 - 3 from now on ) , as obtained from adiabatic floquet simulations ( eq.[eq49 ] ) for @xmath189 are shown as function of the cw - field strength @xmath75 . notice that the lifetimes of these resonances are not monotonic functions of the field intensity . it should be noted that these were not the only resonances that appeared in the quasienergy spectrum ; there were also other resonances which did nt emerge from the field - free bound states . however , as will be shown in fig.[fig4 ] , in this case for not too large field intensities resonances 1 - 3 were the only important resonances and they alone determined the dynamical behavior of the system . in fig.[fig3 ] the ionization probability as obtained from the hermitian simulation ( eq.[eq59 ] ) with pulse strength of @xmath180 , pulse - durations of @xmath182 optical cycles , laser frequency of @xmath189 and initial state @xmath187 , is compared to the ionization probability as obtained from the expression given in eq.[eq63 ] , which is derived from the f - product formalism together with the resonances quasienergies obtained from the non - hermitian simulation . it is seen that as the pulse becomes longer the results obtained from the two simulations become identical . both results of hgs and ionization probabilities showed that the time - dependent wavefunction of the studied systems could be well approximated by the adiabatic expression given in eq.[eq54 ] , even for short pulses . the values of the terms which check the adiabatic criteria , as seen in fig.[fig6 ] , gave the explanation why this was so . in fig.[fig4 ] the expressions @xmath190 ( eq.[eq44 ] ) which describe the couplings between every two resonances from the set of 3 tracked resonances is shown as function of the cw - field strength @xmath75 , for the case @xmath189 . it can be seen that the coupling between resonances 1 and 2 is strong for field strength of @xmath191 , @xmath192 and @xmath193 . this could be partially explained on the basis of the values of the quasienergies , as seen in fig.[fig2 ] , at least for two field strengths out of the three . it is seen that for field strength of @xmath194 and @xmath192 the real parts of the quasienergies cross , resulting in small value of the denominator in the expression given in eq.[eq45 ] for @xmath195 ( at least for one @xmath101 term ) . in the same way the strong couplings between resonances 1 and 3 at field strength of @xmath196 , and between resonances 2 and 3 at field strength of @xmath197 could be explained on the basis of the quasienergy values at these field - strengths . in particular it should be noticed that the couplings between resonances 1 and 3 at @xmath196 are the strongest among all 3 resonances due to the close values of both real and imaginary parts of the quasienergies . however , it should be noted that crossings in the quaisenergy plot are not always indications of large couplings since also the overlap between the wavefunctions ( the nominator of the expression for @xmath198 ) is important . in fig.[fig5 ] the sum @xmath199 which describes the couplings between * all * quasistates of the system @xmath200 to resonance 1 @xmath201 is shown as function of the cw - field strength @xmath75 , for the case @xmath189 . in addition , also the partial sum @xmath202 which describes the couplings between resonances 2 and 3 to the resonance state @xmath201 is shown . it can be shown that up to a moderate intensity of @xmath203 the first resonance is mainly coupled only to the other 2 resonances and not to other , higher resonances or continuum states . the entire dynamics is governed almost solely by the 3 resonances which are `` born '' from the 3 bound states of the field - free hamiltonian . the sums described above , which are functions of a cw - field strength @xmath75 , are converted to be explicit functions of time for the specific sine - square pulse with maximal intensity @xmath180 used in the simulation ( fig.[fig5 ] , upper part ) . for this purpose for each time @xmath21 @xmath159 the effective cw - field strength @xmath204 is calculated and the values of the 2 functions shown in fig.[fig5 ] which fit this effective cw - field strength are taken . hence the 2 sum functions are converted to be explicit functions of time . in the middle part of fig.[fig6 ] the full term @xmath205 , which represents the degree of adiabaticity in the process of shining a 1d xe atom initially at the ground state with a sine - square laser pulse supporting @xmath206 optical cycles of monochromatic radiation with frequency @xmath189 and strength @xmath180 , is shown as function of time . it is seen that the term @xmath205 is bounded by the value of @xmath207 for all times , whether it is calculated by coupling of the first resonance to other 2 resonances only or to all other states . the structure of the hhs for @xmath206 seen in fig.[fig1 ] implies that this value is indeed small and the process is adiabatic . in the lower part of fig.[fig6 ] the same quantity is shown , but for @xmath188 . here it is seen the terms @xmath205 are bounded by a much larger value of @xmath208 for all times and the more complex structure of the hhs for @xmath206 seen in fig.[fig1 ] implies that this value is not small enough to indicate the appearance of an adiabatic process . it should be noted that for a given system with field - strength @xmath51 , frequency @xmath53 and sine - square pulse envelope , we have @xmath209 . therefore , as the number of optical cycles the pulse supports increases the general shape of the terms @xmath205 is kept the same but is attenuated . therefore , for @xmath188 as @xmath160 gets bigger , the pulse s envelope varies more slowly and the process becomes more and more adiabatic , as seen in the hhs spectra for @xmath210 for example . with the help of the ( t , t ) formalism we derive an adiabatic theorem for open systems . the use of the complex scaling transformation plays a key role in our derivation . for example , the spectrum of the floquet hamiltonian of an open system is changed dramatically . rather than a continuous spectrum that is responsible for the absence of an adiabatic limit for @xmath160(number of basis functions)@xmath211 in the conventional qm , the resonances are associated with a point spectrum and are separated from the continuum which is rotated into the lower half of the complex energy plane . an interesting important numerical result of our derivation is that the calculation of the effect of the pulses s shape on the dynamics does not require heavy computations . the entire effect of the laser pulse is embedded in a multiplication factor of @xmath212 where @xmath213 is the variation of the maximum field amplitude as function of time . as a numerical example we applied the adiabatic theorem we derived to a model hamiltonian of xe atom ( with symmetric field - free potential ) which interacts with strong , monochromatic laser pulses . we have shown that the generation of odd - order harmonics and the absence of even - order harmonics , even when the pulses are extremely short , can be explained with the help of the adiabatic theorem we derived . the use of a single - electron 1d model to describe a realistic atom is justified since it has been shown before in many cases that all the main strong field effects are reproduced . therefore the conclusions obtained with this model are also valid for a realistic atom . this work was supported in part by the by the israel science foundation and by the fund of promotion of research at the technion . milan indelka is acknowledged for most helpful and fruitful discussions . s. x. hu and z. z. xu , _ phys . rev . a _ * 56 * , 3916 ( 1997 ) ; s. dionissopoulou , t. mercouris and c. a. nicolaides , _ ibid . _ * 61 * , 063402 ( 2000 ) ; a. di piazza and e. fiordilino , _ ibid . _ * 64 * , 013802 ( 2001 ) ; v. vniard , r. taeb and a. maquet , _ ibid . _ * 65 * , 013202 ( 2001 ) . j. a. fleck , j. r. morris and m. d. feit , _ appl . * 10 * , 129 ( 1976 ) ; m. d. feit , j. a. fleck and a. steiger , _ j. comput . phys . _ * 47 * , 412 ( 1982 ) ; e. forest and r. d. ruth , _ physica d _ * 43 * , 105 ( 1990 ) ; m. campostrini and p. rossy , _ nucl . b _ * 329 * , 753 ( 1990 ) ; j. candy and w. rozmus , _ j. comput . phys . _ * 92 * , 230 ( 1991 ) . | in the conventional quantum mechanics ( i.e. , hermitian qm ) the adiabatic theorem for systems subjected to time periodic fields holds only for bound systems and not for open ones ( where ionization and dissociation take place ) [ d. w. hone , r. ketzmerik , and w. kohn , phys .
rev .
a * 56 * , 4045 ( 1997 ) ] . here with the help of the ( t , t ) formalism combined with the complex scaling method we derive an adiabatic theorem for open systems and provide an analytical criteria for the validity of the adiabatic limit .
the use of the complex scaling transformation plays a key role in our derivation . as a numerical example
we apply the adiabatic theorem we derived to a 1d model hamiltonian of xe atom which interacts with strong , monochromatic sine - square laser pulses .
we show that the generation of odd - order harmonics and the absence of hyper - raman lines , even when the pulses are extremely short , can be explained with the help of the adiabatic theorem we derived .
department of chemistry and minerva center of nonlinear physics in complex systems technion israel institute of technology haifa 32000 , israel .
+ 03.65.-w , 42.50.hz , 42.65.-ky , 32.80.rm |
You are an expert at summarizing long articles. Proceed to summarize the following text:
third harmonic ( th ) generation and supercontinuum ( sc ) emission are two phenomena which have attracted broad interest in the past years @xcite . an evident reason is their direct application in atmospheric remote sensing measurements based on lidar ( light detection and ranging ) femtosecond laser setups @xcite . in this context , spectral broadening originates from complex mechanisms that drive the long - range propagation of ultrashort pulses , when they form narrow filaments in optically - transparent media . the physics of isolated femtosecond filaments in air is nowadays rather well understood ( see , e.g. , @xcite and references therein ) . it involves the competition between kerr self - focusing and plasma defocusing , triggered whenever the input pulse power exceeds the critical power for self - focusing @xmath0 . here , @xmath1 is the central laser wavelength , @xmath2 and @xmath3 are the linear and nonlinear refraction indices in air , respectively . for high enough powers , multiple filaments nucleated after an early stage of modulational instability have also been widely investigated @xcite . they produce spectral patterns mostly analogous to those generated by a single filament , as filamentary cells emerge in phase from the background field and possess the same phase link @xcite . by comparing terawatt ( tw ) multifilamented beams with gigawatt ( gw ) single filaments in air , this property was again verified in the uv - visible region ( 230 - 500 nm ) , where femtosecond self - focusing pulses centered at 800 nm generically produce a tremendous plateau of wavelengths @xcite . this latter phenomenon has recently become a subject of inspiration for several researchers . two scenarios have been proposed for justifying the build - up of new wavelengths in the uv - visible range . on the one hand , temporal steepening phenomena undergone by the pump were shown to deeply modify the filament spectrum @xcite . full chromatic dispersion included in the optical field wave number @xmath4 affects both the diffraction operator and the nonlinearities . this induces shock - like dynamics at the back edge of the pulse through space - time focusing and self - steepening effects , which strongly blueshift the spectra . on the other hand , spectral broadening becomes enhanced by harmonic generation . the coupling of th with an infrared ( ir ) pump produces a two - colored filament from pump intensities above 10 tw/@xmath5 @xcite . the amount of pump energy transferred into th radiation depends on the linear wave vector mismatch parameter @xmath6^{-1}$ ] fixing the coherence length @xmath7 . the smaller the coherence length , the weaker th fields . along meter - range distances , the th component can stabilize the pump wave with about @xmath8 conversion efficiency @xcite . experimental and numerical data reported ring structures embarking most of the th energy and having a half - divergence angle of about 0.5 mrad @xcite . this process contributes to create a continuous spectral band of uv - visible wavelengths @xcite . resembling spectral dynamics have also been reported from 1-mj infrared pulses propagating in argon at atmospheric pressure , after subsequent compression by chirped mirrors @xcite . simulations of these experiments @xcite , discarding th emission , revealed that temporal gradients inherent to the steepening operators are sufficient to amplify uv shifts and cover the th bandwidth down to 250 and 210 nm for initial pulse durations of 10 and 6 fs , respectively . very recently , numerical simulations @xcite refound this tendency for atmospheric propagation , i.e. , th generation , while it affects the pump dynamics to some extent over long ranges , does not change significantly sc spectra , whose variations are mostly induced by the fundamental field in air . despite these last results , we are still missing a detailed understanding of the key parameters which are supposed to drive sc generation . a first important parameter is , of course , the laser wavelength itself : how does the supercontinuum evolve when @xmath1 is varied ? this question was addressed in ref . @xcite for various laser wavelengths , at which some spectral components were seen to merge . however , the model used a two - envelope approximation ( for the pump and th fields , separately ) . as emphasized in @xcite , splitting into th and sc pump within envelopes becomes problematic when their respective spectra overlap inside a wide frequency interval where the basic validity condition @xmath9 ( @xmath10 ) may no longer be fulfilled . actually , th radiation produced through the nonlinear polarization needs to be described self - consistently from a single equation governing the total real optical field . this model was missing in refs . @xcite , which made the role of th overestimated in the white light emission . another important parameter is the length of the self - guiding range : successive cycles of focusing and defocusing events promote the creation of shorter peaks in the pulse temporal profile and lead to a maximal extension of the spectrum . a third potential player is the input pulse duration . in @xcite , this was shown to affect the spectra in noble gases for pulses containing a few optical cycles mainly . clearing this aspect requires several simulations using distinct pulse durations and exploiting different propagation ranges . in connection , we demonstrate that spectral enlargements are directly linked to the level of maximum intensity : steepening operators as well as th radiation broaden all the more the spectra as the intensity in the filament is high . the paper is organized as follows : sec . [ model_equations ] presents the model equations , namely , a unidirectional propagation equation for the total electric field that generates higher - order harmonics ( mostly th ) through kerr nonlinearities . results from this equation will be compared with those inferred from the standard nonlinear evolution equation ( nee ) for the pump wave . the major difference between these two models lies in the production of the th field and its coupling with the pump wave . [ long - range ] is devoted to the long - range propagation of 127-fs pulses in air described by the previous models . emphasis is put on the influence of the central wavelength @xmath1 ( 248 , 800 , 1550 nm ) . we discuss spectral modifications versus the height of @xmath11 , the input duration , together with the temporal steepening dynamics and merging between th and pump spectral bands . [ short - range ] revisits sc for short - range ( focused ) propagations . it is shown that @xmath11 becomes closer to analytical evaluations when the beam develops few focusing / defocusing cycles . in this configuration , a lesser broadening may be achieved . [ conclusion ] finally summarizes the generic features resulting from our analysis . our unidirectional pulse propagation equation ( uppe ) assumes scalar and radially - symmetric approximations . it also supposes negligible backscattering . these hypotheses hold as long as the beam keeps transverse extensions larger than the central laser wavelength and as the nonlinear responses ( together with their longitudinal variations ) are small compared with the linear refraction index . straightforward manipulations of maxwell equations allow us to establish the equation for the spectral amplitude of the optical electric field in the forward direction as @xcite @xmath12 where @xmath13 is the fourier transform of the forward electric field component , @xmath14 is the propagation variable , @xmath15 ( @xmath16 ) is the diffraction operator , @xmath17 , @xmath18 is the wavenumber of the optical field depending on the linear susceptibility tensor @xmath19 defined at frequency @xmath20 . in eq . ( [ uppe_1 ] ) , @xmath21 is the fourier transform of the nonlinearities that include the nonlinear optical polarization @xmath22 and the current density @xmath23 created by charged particles . ( [ uppe_1 ] ) restores the earlier uppe formulation proposed by kolesik _ et al . _ @xcite in the limit @xmath24 ( @xmath25 ) . for practical use , it is convenient to introduce the complex version of the electric field @xmath26 where @xmath27 employs the central wavenumber and frequency of the pump wave ( @xmath28 ) and @xmath29 denotes the heaviside function . because @xmath30 satisfies @xmath31 ( @xmath32 means complex conjugate ) , it is then sufficient to treat the uppe model ( [ uppe_1 ] ) in the frequency domain @xmath33 only . the field intensity can be defined by @xmath34 averaged over an optical period at least . expressed in w/@xmath5 , it is simply given by the classical relation @xmath35 . concerning the nonlinearities , we assume a linearly polarized field . we consider a cubic susceptibility tensor @xmath36 keeping a constant value around @xmath37 , so that @xmath22 contains the instantaneous cubic polarization expressed as @xmath38 . in addition , the phenomenon of raman scattering comes into play when the laser field interacts with anisotropic molecules , in which vibrational and rotational states are excited . depending on the transition frequency @xmath39 in three - level molecular systems and related dipole matrix element @xmath40 @xcite , the raman response takes the form @xmath41 where @xmath42 is the inverse of the fundamental rotational frequency and @xmath43 is the dipole dephasing time . expressed in terms of the rescaled complex field @xmath30 [ eq . ( [ complex ] ) ] and with appropriate normalizations @xcite , eq . ( [ praman ] ) completes the cubic polarization as [ raman_123 ] @xmath44 where @xmath45 is the kerr nonlinear index . expression ( [ raman_1 ] ) possesses both retarded and instantaneous components in the ratio @xmath46 . the instantaneous part @xmath47 of eq . ( [ raman_3 ] ) describes the response from the bound electrons . the retarded part @xmath48 accounts for the raman contribution , in which fast oscillations in @xmath34 give negligible contributions , as @xmath49 fs are assumed to exceed the optical period @xmath50 . when free electrons are created , they induce a current density @xmath51 , which depends on the electron charge @xmath52 , the electron density @xmath53 and the electron velocity @xmath54 . @xmath23 is computed from fluid equations involving external plasma sources and the electron collision frequency @xmath55 . at moderate intensities ( @xmath56 w/@xmath5 ) , the current density obeys @xmath57 assuming electrons born at rest , the growth of the electron density is only governed by external source terms , i.e. , @xmath58 that include photo - ionization processes with rate @xmath59 and collisional ionization with cross - section @xmath60 here , @xmath61 and @xmath62 are the density of neutral species and the ionization potential , respectively . electron recombination in gases is efficient over long ( ns ) time scales , and therefore we omit it . in eq . ( [ rho1 ] ) , the rate for photo - ionization @xmath59 follows from the perelomov , popov and terentev ( ppt ) s theory @xcite incorporating ammosov , delone and krainov ( adk ) coefficients @xcite ( see also ref . optical field ionization theories stress two major limits bounded by the keldysh parameter , @xmath63 namely , the limit for multi - photon ionization ( mpi , @xmath64 ) concerned with rather low intensities and the tunnel limit ( @xmath65 ) concerned with high intensities , from which the coulomb barrier becomes low enough to let the electron tunnel out . here , @xmath66 denotes the peak optical amplitude . for laser intensities @xmath67 w/@xmath5 , mpi characterized by the limit @xmath68 dominates , where @xmath69 is the number of photons necessary to liberate one electron . the level of clamped intensity , @xmath11 , depends on the selected ionization rate . energy lost by the pulse through single ionization processes is determined by a local version of the poynting theorem , yielding the loss current @xmath70 , such that @xmath71 . as a result , our uppe model reads in fourier space as @xmath72 \widehat{\cal e } + \frac{i \mu_0 \omega^2}{2 k(\omega ) \sqrt{c_1 } } \theta(\omega ) \widehat{p}_{\rm nl } \\ & - \frac{i k_0 ^ 2 \theta(\omega)}{2 \epsilon(\omega_0 ) k(\omega)(1+\frac{\nu_e^2}{\omega^2 } ) } \left(\widehat{\frac{\rho \mathcal{e}}{\rho_c}}\right ) - \frac{\theta(\omega)}{2 } \sqrt{\frac{\epsilon(\omega_0)}{\epsilon(\omega ) } } { \cal l}(\omega ) , \end{split}\ ] ] where @xmath73 \mbox{e}^{i \omega t } dt.\ ] ] in eq . ( [ finaluppe1 ] ) , @xmath22 and the expression containing the electron density @xmath74 [ eq . ( [ rho1 ] ) ] must be transformed to fourier space , from which we retain only positive frequencies for the symmetry reasons given above . alternatively , when a central frequency @xmath37 is imposed , eq . ( [ uppe_1 ] ) restitutes the nonlinear envelope equation ( nee ) , earlier derived by brabec and krausz @xcite . we can make use of the taylor expansion @xmath75 where @xmath76 , @xmath77 , @xmath78 , and take the inverse fourier transform of eq . ( [ uppe_1 ] ) in which terms with @xmath4 in their denominator are expanded up to first order in @xmath79 only . after introducing the complex - field representation @xmath80 , the new time variable @xmath81 can be utilized to replace the pulse into the frame moving with the group velocity @xmath82 . furthermore assuming @xmath83 , @xmath84 and ignoring the th component , the nonlinear envelope equation for the forward pump envelope @xmath85 expands as follows : @xmath86 u \\ & \quad - i \frac{k_0}{2n_0 ^ 2 \rho_c } { t}^{-1 } \rho u - \frac{\sigma}{2 } \rho u - ( \rho_{\rm nt } - \rho)\frac{u_i w(i)}{2|u|^2 } u , \end{split } % \end{align}\ ] ] where @xmath87 , @xmath88 and @xmath89 . the first term of the operator @xmath90 corresponds to group - velocity dispersion with coefficient @xmath91 . equation ( [ 1 ] ) describes wave diffraction , kerr focusing response , plasma generation , chromatic dispersion with self - consistent deviations from the classical slowly - varying envelope approximation through the space - time focusing and self - steepening operators [ @xmath92 and @xmath93 , respectively ] . this model will be integrated numerically by using initially gaussian pulses , @xmath94 which involves the input power @xmath95 , the beam waist @xmath96 and @xmath97 pulse half - width @xmath98 . input pulses can be focused through a lens of focal length @xmath99 and they linearly diffract over the distance @xmath100 where @xmath101 is the rayleigh range of the collimated beam ( @xmath102 ) . in the coming analysis , we shall employ the nonlinear refractive indices @xmath103 , @xmath104 , and @xmath105 @xmath5/w for the wavelengths @xmath106 , 800 and 1550 nm , respectively . at 800 nm , we consider a fitted mpi formulation for the ionization rate , @xmath107 , where @xmath108 and @xmath109 s@xmath110@xmath111/w@xmath112 . this approximation is known to reproduce experimental data at @xmath113 nm rather faithfully @xcite . for the two other wavelengths , we lack well established formulations , so we employ ppt ionization rates . we consider @xmath114 molecules having the lowest gap potential ( @xmath115 ev ) as the main specy undergoing ionization with an effective residual charge @xmath116 @xcite . all ionization rates used in the present paper are illustrated in fig . they yield saturation intensity below the threshold of 100 tw/@xmath5 currently claimed in the literature @xcite . the dispersion relation for air has been parametrized as in ref . @xcite . ionization rates for the different wavelengths used throughout this paper : @xmath117 nm ( solid line ) , @xmath118 nm ( dotted line ) , and @xmath119 nm ( dashed line ) . the dashed - dotted line shows the overestimated ionization rate chosen in sec . [ imax ] for @xmath119 nm . ] since our outlook is to understand spectral variations versus the propagation dynamics at different wavelengths , we find it instructive to fix the same ratio of input power over critical , e. g. , @xmath120 , at all wavelengths . to locate the kerr - driven filamentation onset upon comparable @xmath14 scales , we also adjust the ratio of the nonlinear focus @xmath121 and rayleigh range @xmath122 between 2 and 4 , by adapting suitably the beam waist @xmath96 between 1 and 4 mm . in air , dispersion is weak with @xmath123 fs@xmath124/cm in the uv as well as in the mid - ir domains . self - channeling then mainly relies on the dynamical balance between kerr self - focusing and plasma defocusing , so that estimates for peak intensities ( @xmath11 ) , electron densities ( @xmath125 ) and filament radius ( @xmath126 ) can be deduced from equating diffraction , kerr and ionization responses in eq . ( [ 1 ] ) . this yields the simple relations [ estimate13 ] @xmath127 where @xmath128 represents the maximal effective kerr index over the initial pulse profile . for practical use , @xmath129 can be simplified to @xmath130 in mpi - like formulation . the magnitude of @xmath11 directly impacts spectral broadening , which is basically driven by self - phase modulation ( spm ) . because the frequency spectrum is expanded by the nonlinearity , spm leads to sc , as the wave intensity strongly increases through the self - focusing process . noting by @xmath131 the phase of the field envelope , frequency variations are dictated in the limit @xmath132 by @xmath133 which varies with the superimposed actions of the kerr and plasma responses . near the focus point @xmath121 , only the front edge of the pulse survives from this interplay and a redshift is enhanced by plasma generation . at later distances , second focusing / defocusing sequences attenuate this first tendency . in contrast , when accounting for temporal steepening ( @xmath134 ) , shock edges in the back of the pulse are created and a blue shoulder appears in the spectrum , to the detriment of the early redshift @xcite . in addition , the cubic polarization generates third - order harmonics , modeled by the last term of eq . ( [ raman_1 ] ) . in self - focusing regimes , the third - harmonic intensity usually contributes by a little percentage to the overall beam fluence @xcite . despite its smallness , this component may act as a saturable nonlinearity for the carrier wave . it lowers the peak intensity of the pump and contributes to enhance the blue side of the spectrum after the th and pump bandwidths increase and overlap @xcite . we numerically analyze supercontinuum generation for the three laser wavelengths of 248 nm , 800 nm and 1550 nm . results of the nonlinear schrdinger - like equation ( [ 1 ] ) for the pump envelope involving or not space - time focusing and self - steepening operators are compared with those of the unidirectional propagation equation ( [ finaluppe1 ] ) avoiding any taylor expansion in the dispersion relation . special attention is here given to the long - range propagation , which rather favors several cycles of focusing - defocusing events . before proceeding with the above parameters specifically , we perform three different series of simulations showing sc at 800 nm , whose results are summarized in fig . the first one concerns direct integrations of eq . ( [ finaluppe1 ] ) ; the second refers to the same pulse described by eq . ( [ 1 ] ) , which eludes th production ; the third approach relies on eq . ( [ 1 ] ) , in which temporal steepening is omitted , i.e. , @xmath135 . peak intensities ( solid curves , left - hand side scales ) and peak electron densities ( dashed curves , right - hand side scales ) of 127-fs pulses with ratio of input power over critical equal to 4 and waist @xmath136 at @xmath117 nm . the insets show on - axis spectra : dotted curves represent the initial spectra ; the solid curves the spectra at the propagation distance @xmath137 m where maximal broadening is observed : ( a ) uppe model eq . ( [ finaluppe1 ] ) ; ( b ) nee eq . ( [ 1 ] ) applied to the pump wave ; ( c ) nee eq . ( [ 1 ] ) modified with setting @xmath138.,title="fig : " ] peak intensities ( solid curves , left - hand side scales ) and peak electron densities ( dashed curves , right - hand side scales ) of 127-fs pulses with ratio of input power over critical equal to 4 and waist @xmath136 at @xmath117 nm . the insets show on - axis spectra : dotted curves represent the initial spectra ; the solid curves the spectra at the propagation distance @xmath137 m where maximal broadening is observed : ( a ) uppe model eq . ( [ finaluppe1 ] ) ; ( b ) nee eq . ( [ 1 ] ) applied to the pump wave ; ( c ) nee eq . ( [ 1 ] ) modified with setting @xmath138.,title="fig : " ] peak intensities ( solid curves , left - hand side scales ) and peak electron densities ( dashed curves , right - hand side scales ) of 127-fs pulses with ratio of input power over critical equal to 4 and waist @xmath136 at @xmath117 nm . the insets show on - axis spectra : dotted curves represent the initial spectra ; the solid curves the spectra at the propagation distance @xmath137 m where maximal broadening is observed : ( a ) uppe model eq . ( [ finaluppe1 ] ) ; ( b ) nee eq . ( [ 1 ] ) applied to the pump wave ; ( c ) nee eq . ( [ 1 ] ) modified with setting @xmath138.,title="fig : " ] the insets in fig . [ fig1 ] show the on - axis spectra when sc is maximal . following the uppe description , th , which emerges from @xmath139 m , develops a limited redshift , whereas sc of the fundamental is widely extending towards the blue / uv wavelengths [ fig . [ fig1](a ) ] . note that , although a broad plateau occurs in this domain , the th bandwidth still appears separated from the pump spectrum . following the nee description , there is no th generation . however , sc is so amplified in the blue region by temporal steepening effects , that it overlaps the th zone and simply hides it [ fig . [ fig1](b ) ] . finally , when neglecting temporal steepening , the pump instead develops a wide redshift ( overestimated by plasma coupling ) and a much narrower blueshift [ fig . [ fig1](c ) ] . a first observation can be drawn from fig . [ fig1 ] : since th is responsible for lowering the saturation intensity of the pump @xcite , @xmath11 reached in the uppe model is lower and the frequency variations @xmath140 are diminished compared with nee spectra for the pump wave alone . apart from this difference , no significant other change was detected between both these models , so that nee seems to be nothing else but the uppe description subtracted by the self - generated harmonics @xcite . importantly , omitting temporal derivatives of the operators @xmath141 imply more serious discrepancies , as can be seen from fig . [ fig1 ] . we now examine gaussian pulses at different wavelengths ( 248 , 800 and 1550 nm ) with @xmath142 fs and @xmath143 as initial conditions for the uppe model in parallel geometry @xmath144 . figures [ fig1](a ) and [ fig2 ] show snapshots of spectra at maximal extent , together with associated peak intensities and electron densities . we here specify that no th generation was included for @xmath106 nm , because no reliable data of the dispersion relation was available for this wavelength . we believe , instead , that spectral components below 90 nm should be rapidly absorbed by the medium . peak intensities ( solid curves , left - hand side scales ) and peak electron densities ( dashed curves , right - hand side scales ) of 127-fs pulses with ratio of input power over critical equal to 4 at different wavelengths @xmath1 . the insets show on - axis spectra : dotted curves represent the initial spectra ; the solid curves the spectra at the propagation distance @xmath145 where maximal broadening is observed : ( a ) @xmath118 nm , @xmath146 mm , @xmath147 m ; ( b ) @xmath119 nm , @xmath148 mm , @xmath149 m.,title="fig : " ] peak intensities ( solid curves , left - hand side scales ) and peak electron densities ( dashed curves , right - hand side scales ) of 127-fs pulses with ratio of input power over critical equal to 4 at different wavelengths @xmath1 . the insets show on - axis spectra : dotted curves represent the initial spectra ; the solid curves the spectra at the propagation distance @xmath145 where maximal broadening is observed : ( a ) @xmath118 nm , @xmath146 mm , @xmath147 m ; ( b ) @xmath119 nm , @xmath148 mm , @xmath149 m.,title="fig : " ] temporal evolutions of the pulses shown in figs . [ fig1](a ) and [ fig2 ] in the @xmath150 plane : ( a ) @xmath118 nm ; ( b ) @xmath117 nm ; ( c ) @xmath119 nm.,title="fig : " ] temporal evolutions of the pulses shown in figs . [ fig1](a ) and [ fig2 ] in the @xmath150 plane : ( a ) @xmath118 nm ; ( b ) @xmath117 nm ; ( c ) @xmath119 nm.,title="fig : " ] temporal evolutions of the pulses shown in figs . [ fig1](a ) and [ fig2 ] in the @xmath150 plane : ( a ) @xmath118 nm ; ( b ) @xmath117 nm ; ( c ) @xmath119 nm.,title="fig : " ] by comparing figs . [ fig1](a ) and [ fig2 ] , it is seen right away that supercontinuum generation increases with the wavelength . to quantify this observation we introduce @xmath151 as the total extension of the on - axis spectra over wavelengths at @xmath152 times the maximal spectral intensity . then a measurement for the effective broadening is the ratio @xmath153 , which we find close to @xmath154 at 248 nm , @xmath155 at 800 nm and @xmath156 at 1550 nm . a look at the propagation dynamics reveals that the self - guiding range is noticeably augmented at longer wavelengths . this can be explained by the transverse size of the filament . equation ( [ estimate3 ] ) gives an estimate for the beam waist in filamentation regime . if we assume comparable @xmath11 for all wavelengths , we deduce that the filament diameter at 1550 nm is about one order of magnitude larger than that at 248 nm , which is compatible with our numerical data . hence , the larger the wavelength , the slower the filament is expected to diffract . moreover , by virtue of the formula for the critical power @xmath157 and since a filament conveys a few @xmath158 @xcite , it is obvious that ir filaments contain much more energy than their uv counterparts . thus , nonlinear losses along the filamentation range are less dramatic in the ir domain . due to the longer propagation range , more focusing / defocusing events participate in enlarging the spectra at longer wavelengths . figure 4 details the evolution of the filaments in the plane @xmath150 . it is seen that the time window in which the pulses disperse occupies the length of the input pulse duration . although shorter temporal peaks arise through self - focusing / defocusing events , multi - peaked profiles mostly develop patterns having a whole extent close to @xmath98 . before going on , we find it worth investigating sc at 1550 nm more thoroughly , in relationship with third - harmonic generation . figure [ fig3 ] plots three of the sc development stages , first when th and pump components are clearly separated ( @xmath159 m ) , second when they start to merge ( @xmath160 m ) . at the last propagation distance ( @xmath161 m ) , we can observe that , unlike fig . [ fig1 ] , th and pump spectra overlap and make the th bandwidth not distinguishable . for comparison , results from the nee model for the pump wave alone have also been plotted . on - axis spectra for the pulse used in fig . [ fig2](b ) ( @xmath162 nm ) for the propagation distances ( a ) @xmath163 m ( solid line ) and @xmath164 m ( dash - dotted line ) and ( b ) @xmath165 m ( solid line ) . the dashed line in ( b ) refers to a spectrum at @xmath165 m computed from the nee model.,title="fig : " ] on - axis spectra for the pulse used in fig . [ fig2](b ) ( @xmath162 nm ) for the propagation distances ( a ) @xmath163 m ( solid line ) and @xmath164 m ( dash - dotted line ) and ( b ) @xmath165 m ( solid line ) . the dashed line in ( b ) refers to a spectrum at @xmath165 m computed from the nee model.,title="fig : " ] the impact of the saturation intensity onto sc is investigated by simply changing the ionization model : by decreasing the photo - ionization rate artificially it is possible to increase @xmath11 and the maximal plasma level @xmath125 accordingly . reversely , increasing @xmath59 reduces these two quantities , which can have a direct influence on the spectral broadening , as inferred from eq . ( [ spmchin ] ) . to study this point , we concentrate on the wavelength of 1550 nm only , because it yields the broadest spectra explored so far . figure [ fig4 ] shows the maximal intensity and peak electron density at this wavelength , when the ionization rate is artificially increased ( see fig . [ fig0 ] ) . all other parameters are unchanged , compared with the simulation shown in fig . [ fig2](b ) . with the original ionization rate , @xmath11 reaches the value of 80 tw/@xmath5 ; with the artificial one , @xmath11 stays below 13 tw/@xmath5 . the inset in fig . [ fig4 ] details the corresponding spectrum at @xmath166 m. with low @xmath11 , the th component is reduced to some extent as the pump intensity barely exceeds the th conversion threshold @xcite . meanwhile , sc of the pump driven by the @xmath141 operators in the blue side decreases , i.e. , a lower @xmath11 for analogous pulse compression implies less sharp temporal gradients and smoother optical shocks , which weakens blueshifted frequency variations . these features are visible in fig . [ fig4 ] , where th and pump broadbands remain separated . in figs . [ fig2](b ) and [ fig3](b ) , in contrast , sc extends beyond the th wavelength and increases more the pulse spectrum . similar features were observed at the two other wavelengths , when the ionization rate was changed . peak intensity ( solid curve , left - hand side scale ) and peak electron density ( dashed curve , right - hand side scale ) for the same pulse as in fig . [ fig2](b ) ( @xmath162 nm ) computed from the uppe model with an overestimated ionization rate ( see fig . [ fig0 ] ) . the inset shows maximal spectral broadening attained at @xmath166 m. ] we investigate the influence of the initial pulse duration on the propagation dynamics and sc generation . since we consider transform - limited pulses , the value of @xmath98 is directly linked to the initial spectral width . moreover , @xmath11 comes into play in sc generation and is expected to scale as @xmath167 [ see eq . ( [ estimate1 ] ) ] . thus , the initial pulse duration should play a significant role in spectral broadening . to check this assessment , we performed several simulations using the uppe model , by varying @xmath98 from 20 fs up to 500 fs . because group - velocity dispersion becomes very efficient at short pulse durations and may even stop the kerr self - focusing at powers too close to critical @xcite , we increased the input power up to 20 @xmath158 for @xmath168 fs . with this , we ensure to trigger a filamentation regime even for this short input duration . to illustrate the dependency of @xmath11 upon @xmath98 , short wavelengths are preferable because the number of photons for ionization is small . figure [ fig6 ] shows maximum intensity , peak electron density and maximal spectral extent of a 20-fs pulse at 248 nm . at this wavelength , @xmath169 and , following eq . ( [ estimate1 ] ) , @xmath11 and @xmath125 should increase by a factor @xmath170 compared to the 127-pulse shown in fig . [ fig2](a ) . indeed , both quantities are increased by a factor of @xmath171 in the simulation . spectral broadening is augmented from 0.5 to 0.8 in terms of @xmath172 , especially to the blue side . as explained in sec . [ imax ] , this results from the action of the steepening operators . the overall propagation dynamics , characterized by the filamentation length and number of focusing / defocusing cycles are , however , comparable for both the 20-fs and the 127-fs pulses . peak intensity ( solid curve , left - hand side scale ) and peak electron density ( dashed curve , right - hand side scale ) of a 20-fs pulse with ratio of input power over critical equal to 20 , @xmath118 nm , @xmath146 mm . the inset shows on - axis spectra : the dotted curve represents the initial spectrum ; the solid curve the spectrum at the distance @xmath173 m where maximal broadening is observed . ] on the other hand , if we increase the initial duration @xmath98 towards the ps time scale , the propagation dynamics changes drastically . as an example , figure [ fig5bis2 ] shows the temporal evolution of a 500-fs pulse at 800 nm . compared with fig . [ fig2bis](b ) employing @xmath142 fs , the obvious difference is the huge number of focusing / defocusing cycles . the action of the generated plasma breaks the pulse profile into a larger number of shorter peaks . with a longer pulse duration , time slices are available for feeding successive focusing events . the filamentation range is increased and @xmath174 is maintained over several meters . inspection of the simulations , however , reveals maximal spectral extent comparable with that dispayed in fig . [ fig1](a ) . temporal evolution of a 500-fs pulse at 800 nm with 4 @xmath158 and waist @xmath136 in the @xmath150 plane . ] at 1550 nm the influence of @xmath98 on the maximal intensity is much less pronounced , since we have @xmath175 [ see figs . [ fig2](b ) and [ fig5 ] ] . for all pulse durations , we indeed observe @xmath176 tw/@xmath5 . this can explain why the maximal spectral extension @xmath172 is always found between 1.5 and 2 , regardless @xmath98 may be . the major difference lies in the filamentation range , which increases with the initial pulse duration . so , there is no significant change in sc generation between short and long pulses over large enough propagation scales . peak intensities ( solid curves , left - hand side scales ) and peak electron densities ( dashed curves , right - hand side scales ) of 4 mm waisted pulses with different durations and ratios @xmath177 at @xmath119 nm . the insets show on - axis spectra : dotted curves represent the initial spectra ; the solid curves the spectra at the propagation distance @xmath145 where maximal broadening is observed : ( a ) @xmath178 fs , @xmath179 , @xmath180 m ; ( b ) @xmath181 fs , @xmath182 , @xmath166 m.,title="fig : " ] peak intensities ( solid curves , left - hand side scales ) and peak electron densities ( dashed curves , right - hand side scales ) of 4 mm waisted pulses with different durations and ratios @xmath177 at @xmath119 nm . the insets show on - axis spectra : dotted curves represent the initial spectra ; the solid curves the spectra at the propagation distance @xmath145 where maximal broadening is observed : ( a ) @xmath178 fs , @xmath179 , @xmath180 m ; ( b ) @xmath181 fs , @xmath182 , @xmath166 m.,title="fig : " ] this last observation invites us to look at the temporal profiles upon propagation . if the spectral extent is comparable , we might also find similar temporal patterns . indeed , the on - axis temporal profiles shown in fig . [ fig5bis ] all exhibit structures with duration of 10 - 15 fs . it seems that the initial pulse length @xmath98 just determines how many of these peaks appear , or , in other words , how many focusing / defocusing cycles the pulse is able to develop upon propagation . another indication for the change in the effective pulse duration upon propagation is provided by the curve of the intensity maximum in fig . [ fig5](b ) : the first focusing cycle is halted at slightly lower intensities @xmath183 60 tw/@xmath5 , because in this early stage the peak duration remains of the order of @xmath181 fs . at later stages , @xmath11 increases as the pulse undergoes temporal compression . on - axis temporal profiles of pulses with different duration at @xmath145 : @xmath178 fs [ solid line , parameters used in fig . [ fig5](a ) ] ; @xmath184 fs [ dashed line , parameters used in fig . [ fig2](b ) ] ; @xmath181 fs [ dotted line , parameters used in fig . [ fig5](b ) ] . ] so far , we have analyzed free propagation dynamics where long filaments achieve temporal gradients and sc extents similar to those produced by initially much shorter pulses . now , we force all pulses to cover the same short filamentation range through a focusing optics ( @xmath185 m ) . results are shown in fig . [ fig7 ] for @xmath168 and 500 fs at 1550 nm . we can check that the maximum intensity @xmath11 follows the theoretical expectations ( [ estimate13 ] ) involving @xmath98 . the short filamentation range prevents the occurrence of several focusing / defocusing cycles , especially for the 500-fs pulse . hence , although the pulse self - focuses , its temporal extent remains comparable with @xmath98 , as we can see in figs . [ fig7](c ) and ( d ) . the 20-fs pulse shows significant spectral broadening with a visible , broadened th peak . the 500-fs pulse is spectrally too narrow to generate supercontinuum over roughly 1-m filamentation range . therefore , the fundamental and harmonic peaks clearly stay separated ( note the occurrence of the fifth harmonics at 310 nm , which is self - consistently described by the uppe model ) . thus , the initial pulse duration strongly influences spectral broadening in configurations of short filamentation range mainly , which is consistent with the numerical results of ref . @xcite . peak intensities ( solid curves , left - hand side scales ) and peak electron densities ( dashed curves , right - hand side scales ) of 4 mm waisted pulses with different duration and ratio of input power over critical equal to 4 at @xmath119 nm propagating in focused geometry ( @xmath186 m ) . the insets show on - axis spectra : dotted curves represent the initial spectra ; the solid curves the spectra at the distance @xmath187 m where maximal broadening is observed : ( a ) @xmath178 fs ; ( b ) @xmath181 fs . ( c ) and ( d ) show the respective temporal evolution for the pulses of ( a ) and ( b ) in the @xmath150 plane.,title="fig : " ] peak intensities ( solid curves , left - hand side scales ) and peak electron densities ( dashed curves , right - hand side scales ) of 4 mm waisted pulses with different duration and ratio of input power over critical equal to 4 at @xmath119 nm propagating in focused geometry ( @xmath186 m ) . the insets show on - axis spectra : dotted curves represent the initial spectra ; the solid curves the spectra at the distance @xmath187 m where maximal broadening is observed : ( a ) @xmath178 fs ; ( b ) @xmath181 fs . ( c ) and ( d ) show the respective temporal evolution for the pulses of ( a ) and ( b ) in the @xmath150 plane.,title="fig : " ] peak intensities ( solid curves , left - hand side scales ) and peak electron densities ( dashed curves , right - hand side scales ) of 4 mm waisted pulses with different duration and ratio of input power over critical equal to 4 at @xmath119 nm propagating in focused geometry ( @xmath186 m ) . the insets show on - axis spectra : dotted curves represent the initial spectra ; the solid curves the spectra at the distance @xmath187 m where maximal broadening is observed : ( a ) @xmath178 fs ; ( b ) @xmath181 fs . ( c ) and ( d ) show the respective temporal evolution for the pulses of ( a ) and ( b ) in the @xmath150 plane.,title="fig : " ] in summary , we have revisited recent works on sc generation versus third harmonic emission , by showing from a complete uppe model that spectral enlargements of femtosecond pulses in self - guiding regime are mostly driven by space - time focusing and self - steepening . th generation , although changing the pump dynamics , affects the spectra to a limited extent only . this conclusion corrects some previous statements @xcite , based on a propagation model in which temporal steepening terms were analyzed separately from an envelope description for th generation . going one step beyond , we have demonstrated the important role of the saturation intensity in the frequency variations enlarging both th and pump broadbands . the input pulse duration becomes a significant player in the spectral extents as long as pulses do not propagate too far , i.e. , they do not let the temporal profiles of the pulse fluctuate so much that many peaks and sharp gradients can develop along the optical path . this property is mainly enlightened in short - range focused geometry and lost in long - range parallel geometry . finally , the role of the central wavelength is preeminent : our numerical simulations displayed evidence that sc clearly augments with the laser wavelength . s. l. chin , a. brodeur , s. petit , o. g. kosareva , v. p. kandidov , filamentation and supercontinuum generation during the propagation of powerful ultrashort laser pulses in optical media ( white light laser ) , j. nonlinear opt . ( 1999 ) 121 . a. b. fedotov , n. i. koroteev , m. m. t. loy , x. xiao , a. m. zheltikov , saturation of third - harmonic generation in a plasma of self - induced optical breakdown due to the self - action of 80-fs light pulses , opt . commun . 133 ( 1997 ) 587 . j. kasparian , m. rodriguez , g. mjean , j. yu , e. salmon , h. wille , r. bourayou , s. frey , y. b. andr , a. mysyrowicz , r. sauerbrey , j. p. wolf , l. wste , white - light filaments for atmospheric analysis , science 301 ( 2003 ) 61 . l. berg , s. skupin , r. nuter , j. kasparian , j. p. wolf , optical ultrashort filaments in weakly - ionized , optically - transparent media , http://arxiv.org/abs/physics/0612063 , submitted to rev . ( 2006 ) . s. l. chin , s. petit , w. liu , a. iwasaki , m .- c . nadeau , v. p. kandidov , o. g. kosareva , k. y. andrianov , interference of transverse rings in multifilamentation of powerful femtosecond laser pulses in air , opt . commun . 210 ( 2002 ) 329 . n. akzbek , a. becker , m. scalora , s. l. chin , c. m. bowden , continuum generation of the third - harmonic pulse generated by an intense femtosecond ir laser pulse in air , appl . b : lasers & optics 77 ( 2003 ) 177 . l. berg , s. skupin , g. mjean , j. kasparian , j. yu , s. frey , e. salmon , j. p. wolf , supercontinuum emission and enhanced self - guiding of infrared femtosecond filaments sustained by third - harmonic generation in air , phys . rev . e 71 ( 2005 ) 016602 . h. yang , j. zhang , j. zhang , l. z. zhao , y. j. li , h. teng , y. t. li , z. h. wang , z. l. chen , z. y. wei , j. x. ma , w. yu , z. m. sheng , third - order harmonic generation by self - guided femtosecond pulses in air , phys . e 67 ( 2003 ) 015401(r ) . n. akzbek , a. iwasaki , a. becker , m. scalora , s. l. chin , c. m. bowden , third - harmonic generation and self - channeling in air using high - power femtosecond laser pulses , phys . ( 2002 ) 143901 . i. alexeev , a. c. ting , d. f. gordon , e. briscoe , b. hafizi , p. sprangle , characterization of the third - harmonic radiation generated by intense laser self - formed filaments in air , opt . ( 2005 ) 1503 . g. mjean , j. kasparian , j. yu , s. frey , e. salmon , r. ackermann , j .- wolf , l. berg , s. skupin , uv - supercontinuum generated by femtosecond pulse filamentation in air : meter - range experiments versus numerical simulations , appl . b : lasers & optics 82 ( 2006 ) 341 . s. a. trushin , s. panja , k. kosma , w. e. schmid , w. fuss , supercontinuum extending from @xmath188 to 250 nm , generated by focusing ten - fs laser pulses at 805 nm into ar , appl . b : lasers & optics 80 ( 2005 ) 399 . n. akzbek , s. a. trushin , a. baltuka , w. fuss , e. goulielmakis , k. kosma , f. krausz , s. panja , m. uiberacker , w. e. schmid , a. becker , m. scalora , m. bloemer , extending the supercontinuum spectrum down to 200 nm with few - cycle pulses , new j. phys . 8 ( 2006 ) 177 . s. skupin , l. berg , u. peschel , f. lederer , g. mjean , j. yu , j. kasparian , e. salmon , j .- wolf , m. rodriguez , l. wste , r. bourayou , r. sauerbrey , filamentation of femtosecond light pulses in the air : turbulent cells versus long - range clusters , phys . e 70 ( 2004 ) 046602 . | supercontinuum generation by femtosecond filaments in air is investigated for different laser wavelengths ranging from ultraviolet to infrared .
particular attention is paid on the role of third - harmonic generation and temporal steepening effects , which enlarge the blue part of the spectrum . a unidirectional pulse propagation model and nonlinear evolution equations are numerically integrated and their results are compared .
apart from the choice of the central wavelength , we emphasize the importance of the saturation intensity reached by self - guided pulses , together with their temporal duration and propagation length as key players acting on both supercontinuum generation of the pump wave and emergence of the third harmonics .
maximal broadening is observed for large wavelengths and long filamentation ranges . , supercontinuum generation , femtosecond filaments , nonlinear schrdinger equation , self - phase modulation 42.65.tg , 52.38.hb , 42.65.jx , 42.68.ay |
You are an expert at summarizing long articles. Proceed to summarize the following text:
perturbation theory is a powerful tool to obtain reliable predictions for physical observables within the standard model of particle physics or its extensions . due to the high precision of experimental results , e.g. , at the cern large hadron collider ( lhc ) or at the @xmath2 factories , it is on the one hand mandatory to advance the development of tools , which can be used for higher order calculations . on the other hand it is necessary to improve the understanding of the perturbative structure of quantum field theories . form factors are ideal objects to obtain deeper insight into the latter . especially in the context of qcd they are indispensable tools to investigate the infrared structure of scattering amplitudes to high orders in perturbation theory . moreover , from the pole part it is possible to extract universal process - independent quantities @xcite like the cusp anomalous dimension which can be extracted from the @xmath3 pole of the form factor . the finite parts of the form factors serve as building blocks for a variety of physical processes . for example , the quark - anti - quark - photon form factor enters the virtual corrections of the drell - yan process for the production of lepton pairs at hadron colliders . in this paper we consider the quark - anti - quark - photon form factor which is conveniently obtained from the photon - quark vertex function @xmath4 by applying an appropriate projector . in @xmath5 space - time dimensions we have @xmath6 with @xmath7 where @xmath8 and @xmath9 are the incoming quark and anti - quark momenta and @xmath10 is the momentum of the photon . we perform our calculation in the framework of qcd keeping the number of colours , @xmath1 , generic . in the limit of large @xmath1 the calculation of @xmath0 is simplified since only planar feynman diagrams contribute . this is the limit we consider in this paper . besides @xmath1 we also keep the number of active quark flavours , @xmath11 as a parameter and thus have at four - loop order the colour structures @xmath12 , @xmath13 , @xmath14 , @xmath15 where each factor of @xmath11 counts the number of closed fermion loops . two- and three - loop corrections to @xmath0 have been computed in refs . @xcite . to obtain the four - loop corrections two obstacles need to be overcome : ( i ) the reduction to a set of basis integrals and ( ii ) the ( if possible ) analytic calculation of the latter . recently , the first steps towards four loops have been initiated by computing the fermionic contributions to @xmath0 in the planar limit @xcite . the @xmath16 terms have been confirmed in ref . @xcite using different methods both for the reduction and the computation of the master integrals . let us mention that the four - loop corrections to the cusp anomalous dimension with two and three closed fermion loops have also been obtained in ref . @xcite . in ref . @xcite one finds a discussion of non - planar master intergrals relevant for the four - loop form factor in a @xmath17 supersymmetric yang - mills theory . in the present paper , we evaluate the @xmath18 contribution and therefore complete the evaluation of the form factors and anomalous dimensions in the limit of large @xmath1 . in the next section we provide some technical details , in particular to the calculation of the most complicated master integral , and we discuss our results in section [ sec::res ] . we provide explicit expressions for the four - loop cusp and collinear anomalous dimensions in the planar limit . furthermore , we provide results for the finite part of @xmath19 . sample four - loop feynman diagrams contributing to @xmath0 . solid , curly and wavy lines represent quarks , gluons and photons , respectively . all particles are massless.,title="fig:",scaledwidth=30.0% ] sample four - loop feynman diagrams contributing to @xmath0 . solid , curly and wavy lines represent quarks , gluons and photons , respectively . all particles are massless.,title="fig:",scaledwidth=30.0% ] sample four - loop feynman diagrams contributing to @xmath0 . solid , curly and wavy lines represent quarks , gluons and photons , respectively . all particles are massless.,title="fig:",scaledwidth=30.0% ] for the calculation of @xmath0 we use a well - tested automated chain of programs which work hand - in - hand . the feynman amplitudes are generated with qgraf @xcite . since there is no possibility to select already at this point the planar diagrams also non - planar amplitudes are generated and we obtain in total 1 , 15 , 337 and 9784 diagrams at one , two , three and four loops . sample feynman diagrams at four loops can be found in fig . [ fig::diags ] . next , we transform the output to form @xcite notation using q2e and exp @xcite . the program exp furthermore maps each feynman diagram to predefined integral families for massless four - loop vertices with two different non - vanishing external momenta ; 68 of them are of planar type . at this point we perform the dirac algebra and decompose the numerator into terms which appear in the denominator . this allows us to express each feynman integral as a linear combination of scalar functions which belong to the corresponding family . after exploiting the symmetries connected to the exchange of the external momenta we can reduce the number of families , for which integral tables have to be generated , from 68 to 38 . note that for the fermionic contributions , which have been considered in ref . @xcite , only 24 families are needed . for the reduction to master integrals we use the program fire @xcite which we apply in combination with litered @xcite . we observe that the non - fermionic diagrams lead to more complex integrals for which the reduction time significantly increases . let us remark that we have adopted feynman gauge for the calculation of the non - fermionic parts . once reduction tables for each family are available we apply tsort , which is part of the latest fire version @xcite . it is based on ideas presented in ref . @xcite to establish relations between between primary master integrals and thus minimize their number . in this way we arrive at 99 master integrals . in the remainder of this section we describe in detail the calculation of the most complicated integral corresponding to the graph of fig . [ fig::i99 ] , @xmath20 . massless four - loop vertex diagram with two external ( incoming ) momenta @xmath10 and @xmath21 with @xmath22 and @xmath23 . the master integral @xmath20 corresponds to @xmath24.,scaledwidth=50.0% ] to have the possibility to apply differential equations for the evaluation of the form factor master integral corresponding to the graph of fig . [ fig::i99 ] we introduce a second mass scale by considering @xmath25 and derive differential equations with respect to the ratio of the two scales , @xmath26 . this strategy was advocated in @xcite and used for the four - loop form - factor integrals in @xcite . we derive differential equations for the corresponding 332 master integrals using litered @xcite . to solve our differential equations we use an important observation made in ref . it has been suggested to turn from the basis of primary master integrals to a so - called canonical basis where the corresponding integrals satisfy a system differential equations which has a particular structure : the dependence on @xmath27 appears as a linear prefactor and the matrix in front of the vector of master integrals has only simple poles in @xmath26 , i.e. has only so - called fuchsian singularities . such a system can then easily be solved in terms of iterated integrals . in ref . @xcite , it was proposed that choosing integrals with constant leading singularities provides a canonical basis of the differential equations . in subsequent work , this was used in a variety of cases , see e.g. refs . @xcite . as was motivated in ref . @xcite , this choice can be done in a systematic way at the level of the loop integrand . the first algorithm to convert a given differential system to a canonical form at the level of the differential equations has been provided in the case of one variable in ref . @xcite ( therein called @xmath27-form , see also ref . recently a public implementation @xcite of the algorithm of ref . @xcite in a computer code fuchsia became available . in this paper we follow @xcite to construct a canonical basis . an independent evaluation of @xmath20 can be found in ref . @xcite where the canonical basis was chosen at the level of the loop integrand . the size of the system is large but , as it concerns the diagonal blocks , their size is at most @xmath28 . consequently , the reduction of diagonal blocks within the approach of ref . @xcite is simple . note that already after this step one might claim that the solution is expressible in terms of harmonic polylogarithms ( and also construct this solution ) . nevertheless , we follow the prescription of section 7 of ref . @xcite to reduce the whole system to an @xmath27-form . similar to ref . @xcite we find that the differential equation - based hierarchy of the set of the master integrals is too restrictive and the use of a sector - based hierarchy when factoring @xmath27 out of the whole matrix is necessary . the family of one - scale feynman integrals ( with @xmath24 and with @xmath29 ) corresponding to fig . [ fig::i99 ] contains 76 master integrals . after introducing @xmath30 we define @xmath31 and obtain a family of feynman integrals with 332 master integrals . our strategy is to turn from a primary basis to a canonical basis , solve differential equations for the canonical basis , evaluate the naive values of the elements of the canonical basis at @xmath32 ( i.e. , setting @xmath32 under the integral sign ) from which it will be straightforward to obtain analytical results for the primary master integrals of our one - scale family , in particular , @xmath20 . following ref . @xcite we arrive at a canonical basis @xmath33 which is obtained from the primary basis @xmath34 by a linear transformation with a matrix @xmath35 , @xmath36 the vectors @xmath34 and @xmath33 have @xmath37 entries and @xmath35 is a @xmath38 matrix . the dependence of @xmath34 , @xmath33 and @xmath35 on @xmath26 and @xmath27 has been suppressed . it is convenient @xcite to normalize the canonical master integrals such that they have uniformly transcendental @xmath27-expansion which starts from @xmath39 . in our four - loop case , one has to compute expansion terms including @xmath40 terms and we have @xmath41 the canonical basis @xmath33 satisfies ( by definition ) a differential equation of the form @xmath42 where @xmath43 with constant matrices @xmath44 and @xmath45 . it is straightforward to construct the generic solution of ( [ eq::dgl ] ) order - by - order in @xmath27 in terms of harmonic polylogarithms ( hpls ) @xcite with letters 0 and 1 , with @xmath46 unknown constants . to fix these constants we use boundary conditions at the point @xmath47 . the primary master integrals are regular at this point where the integrals become propagator - type integrals which are well known @xcite . in particular , all the corresponding 28 master integrals are known analytically @xcite in an @xmath27-expansion up to weight 12 and have been cross checked numerically @xcite . we obtain the boundary values of the elements of the canonical basis @xmath33 at @xmath47 by inverting eq . ( [ eq::t ] ) and considering the limit @xmath48 . the matrix @xmath49 involves elements which develop poles up to order @xmath50 . this requires that the first seven expansion terms for @xmath48 of the primary master integrals @xmath34 have to be computed . the corresponding reduction tables are again generated with fire . note that the resulting two - point integrals with external momentum @xmath10 have scalar products in the numerator involving the momentum @xmath21 which complicates the calculation . it is an important cross check of the calculation that all poles in @xmath51 cancel in the combination @xmath52 and we obtain the values for @xmath33 at @xmath47 , in an @xmath27-expansion up to @xmath40 . this fixes the solution of the system of differential equations in eq . ( [ eq::dgl ] ) . in a next step we analyze the leading asymptotic behaviour of @xmath33 near @xmath32 . on the one hand , we obtain it from the differential equations ( [ eq::dgl ] ) where the term @xmath53 on the right - hand side can be neglected and the solution has the form @xmath54 . the quantity @xmath55 can be evaluated using the mathematica command matrixexp [ ] which leads to a @xmath38 matrix where each element is a linear combination of terms @xmath56 with integer @xmath57 . in general both non - positive @xmath57 and positive @xmath57 might appear . however , in the case of feynman integrals only terms with non - positive @xmath57 can be present which we use as a check . on the other hand , the leading asymptotic behaviour in the limit @xmath58 can also be obtained with the help of the mathematica package hpl @xcite from our analytic expression for the canonical basis . matching the two expressions provides values for the vector @xmath59 in an @xmath27-expansion up to @xmath40 and terms @xmath56 with @xmath60 . note that this step involves powers of @xmath61 terms ; their cancellation in the matching provides a welcome check for our calculation . finally , the naive value for @xmath62 at @xmath32 is obtained by setting all terms @xmath56 with @xmath63 to zero in the expression for @xmath64 . in the last step , using eq . ( [ eq::t ] ) , we compute the naive values of the elements of the primary basis @xmath34 from the naive expansion of the canonical basis @xmath65 near @xmath58 . note that some of the matrix elements of @xmath35 involve singularities up to order @xmath66 . thus , the naive expansion of @xmath65 up to order @xmath67 is needed . it is obtained following the prescription outlined in ref . @xcite where the expansion terms can be computed from the leading order asymptotics at @xmath68 after recursively solving matrix equations with @xmath38 entries . after inserting the expansion of @xmath33 in eq . ( [ eq::t ] ) the poles cancel and the naive values of the primary master integrals at @xmath32 are obtained . the naive value of one of the elements of our primary basis @xmath34 is nothing but the one - scale master integral @xmath20 ( cf . [ fig::i99 ] ) for which we obtain the following analytic result @xmath69 + \frac{1}{\epsilon^6 } \bigg [ \frac{13}{576 } \bigg ] + \frac{1}{\epsilon^5 } \bigg [ -\frac{101}{576}-\frac{\pi ^2}{48 } \bigg ] \nonumber\\&&\mbox { } + \frac{1}{\epsilon^4 } \bigg [ -\frac{17 \zeta _ 3}{54}+\frac{5 \pi ^2}{36}+\frac{145}{96 } \bigg ] + \frac{1}{\epsilon^3 } \bigg [ \frac{1775 \zeta _ 3}{432}-\frac{767 \pi ^4}{17280}-\frac{5 \pi ^2}{8}-\frac{1669}{144 } \bigg ] \nonumber\\&&\mbox { } + \frac{1}{\epsilon^2 } \bigg [ -\frac{83}{72 } \pi ^2 \zeta _ 3-\frac{21899 \zeta _ 3}{864}-\frac{3659 \zeta _ 5}{360}+\frac{31333 \pi ^4}{103680}+\frac{659 \pi ^2}{288}+\frac{11243}{144 } \bigg ] \nonumber\\&&\mbox { } + \frac{1}{\epsilon } \bigg [ -\frac{40231 \zeta _ 3 ^ 2}{1296}+\frac{745 \pi ^2 \zeta _ 3}{288}+\frac{18751 \zeta _ 3}{144}+\frac{50191 \zeta _ 5}{360}-\frac{277703 \pi ^6}{2177280}-\frac{14015 \pi ^4}{10368 } \nonumber\\&&\mbox { } -\frac{149 \pi ^2}{24}-\frac{22757}{48 } \bigg ] \nonumber\\&&\mbox { } + \bigg [ \frac{39173 \zeta _ 3 ^ 2}{324}-\frac{77399 \pi ^4 \zeta _ 3}{25920}+\frac{4013 \pi ^2 \zeta _ 3}{432}-\frac{259559 \zeta _ 3}{432}-\frac{568 \pi ^2 \zeta _ 5}{45}-\frac{1123223 \zeta _ 5}{1440 } \nonumber\\&&\mbox { } -\frac{2778103 \zeta _ 7}{4032}+\frac{3129533 \pi ^6}{4354560}+\frac{28201 \pi ^4}{5760}+\frac{173 \pi ^2}{36}+\frac{382375}{144 } \bigg ] \nonumber\\&&\mbox { } + \epsilon \bigg [ \frac{4931 s_{8a}}{30}+\frac{2615}{144 } \pi ^2 \zeta _ 3 ^ 2-\frac{276671 \zeta _ 3 ^ 2}{2592}-\frac{2702413 \zeta _ 5 \zeta _ 3}{1080}+\frac{154037 \pi ^4 \zeta _ 3}{31104 } \nonumber\\&&\mbox { } -\frac{55327 \pi ^2 \zeta _ 3}{432}+\frac{1100461 \zeta _ 3}{432}+\frac{205 \pi ^2 \zeta _ 5}{9}+\frac{155029 \zeta _ 5}{48}+\frac{2732549 \zeta _ 7}{1008}-\frac{665217829 \pi ^8}{1306368000 } \nonumber\\&&\mbox { } -\frac{131003 \pi ^6}{45360}-\frac{747929 \pi ^4}{51840}+\frac{2995 \pi ^2}{36}-\frac{2005247}{144 } \bigg ] \bigg\}\,,\end{aligned}\ ] ] where @xmath70 is riemann s zeta function evaluated at @xmath71 and @xmath72 @xmath73 are multiple zeta values given by @xmath74 as by - product we also obtain analytic results for the remaining 75 one - scale master integrals and we find agreement with the results obtained in ref . this constitutes a further cross check for our procedure . we want to stress that the calculation which is outlined in this section is largely independent from the one performed in ref . @xcite . this section is devoted to the analytic results of the cusp and collinear anomalous dimensions and the finite part of @xmath0 . generic formulae where the pole part of @xmath0 is parametrized in terms of the cusp and collinear anomalous dimensions and the qcd beta function can , e.g. , be found in refs . in what follows we use eq . ( 2.3 ) of ref . @xcite which displays the pole parts of @xmath19 up to four - loop order . in this formula it is assumed that the one - loop coefficient of the beta function is given by @xmath75 with @xmath11 being the number of active quarks and the coefficients of the anomalous dimensions are defined through @xmath76 with @xmath77 and @xmath78 is the renormalized coupling constant with @xmath11 active flavours . from eq . ( 2.3 ) of ref . @xcite one observes that the four - loop corrections of @xmath79 follows from the @xmath3 term of @xmath19 and @xmath80 from the linear pole terms . in the following we start with explicit results for the cusp and collinear anomalous dimensions . the four - loop corrections to @xmath79 reads @xmath81 we note that , after taking into account the difference between fundamental and adjoint representation of the external fields , the leading transcendental piece of this expression agrees with the result for planar @xmath17 super yang - mills @xcite , in agreement with expectations from @xcite . note that @xmath82 entering the @xmath3 pole of @xmath19 is multiplied by @xmath83 . for this reason there is only a @xmath84 factor in front of the @xmath11-independent term in eq . ( [ eq::gamma_cusp3 ] ) . the one- , two- and three - loop corrections in the large-@xmath1 limit can be found in eq . ( 2.6 ) of ref . @xcite where also the fermionic part of @xmath82 is shown . the four - loop coefficient of the collinear anomalous dimension is given by @xmath85+\left(-\frac{8 \pi ^4}{1215 } -\frac{356 \zeta _ 3}{243}-\frac{2 \pi ^2}{81}+\frac{18691}{13122}\right ) n_c n_f^3+\left(-\frac{2}{3 } \pi ^2 \zeta _ 3 \right.\nonumber\\&&\left.\mbox { } + \frac{166 \zeta _ 5}{9}+\frac{331 \pi ^4}{2430 } -\frac{2131 \zeta _ 3}{243 } -\frac{68201 \pi ^2}{17496}-\frac{82181}{69984}\right ) n_c^2 n_f^2 \nonumber\\&&\mbox { } + n_c^4 \left(\frac{1175 \zeta _ 3 ^ 2}{9}+\frac{82 \pi ^4 \zeta _ 3}{45}-\frac{377 \pi ^2 \zeta _ 3}{6}+\frac{867397 \zeta _ 3}{972}+24 \pi ^2 \zeta _ 5 - 1489 \zeta _ 5 + 705 \zeta _ 7 \right.\nonumber\\&&\left.\mbox { } + \frac{114967 \pi ^6}{204120}-\frac{59509 \pi ^4}{9720}-\frac{120659 \pi ^2}{17496}-\frac{187905439}{839808}\right ) \ , . \label{eq::gamma_q}\end{aligned}\ ] ] the one- , two- and three - loop corrections and the fermionic four - loop terms to @xmath86 are listed in eq . ( 2.7 ) of ref . the @xmath12 term is new . finally , we also present the finite part of the form factor . we parametrize the perturbative expansion in terms of the renormalized coupling constant and set @xmath87 . furthermore , it is convenient to consider @xmath19 which leads to more compact expressions . thus , we have the following parametrization @xmath88 in the large-@xmath1 limit the four - loop term reads @xmath89 where the ellipses refer to the fermionic contributions which are given in eq . ( 2.8 ) of ref . @xcite . for convenience of the reader we provide the results for the form factor @xmath0 expanded in the bare strong coupling constant in an ancillary file which can be downloaded from ` https://www.ttp.kit.edu/preprints/2016/ttp16-055/ ` . this file also contains the lower - loop results expanded to higher order in @xmath27 . furthermore , it contains the dependence of the renormalization scale @xmath90 . we compute the photon - quark form factor to four loops up to the finite term in @xmath27 in the large-@xmath1 limit which is obtained from the planar feynman diagrams . from the pole parts we extract the cusp and collinear anomalous dimensions . we discuss in detail the calculation of the most complicated master integral ( see fig . [ fig::i99 ] ) and present analytic results expanded in @xmath27 up to transcendental weight eight . an independent calculation of this integral , together with a discussion of the remaining master integrals can be found in refs . @xcite to the required order in @xmath27 . we want to remark that the same master integrals enter the higgs gluon form factor in the planar limit . however , the corresponding reduction is significantly more complicated . the logical next step is the calculation of the non - planar contributions . note that also here both the reduction and the computation of the master integrals turn out to be much more complicated . we are grateful to gang yang for checking numerically the highest poles of our result for @xmath20 . is supported in part by a gfk fellowship and by the prisma cluster of excellence at mainz university . this work is supported by the deutsche forschungsgemeinschaft through the project `` infrared and threshold effects in qcd '' . * note added * : + after submission of our mansucript to arxiv.org we were contacted by colleagues drawing our attention to a talk by ben ruijl at the university of zurich where the result for @xmath82 has been shown . the result agrees with our eq . ( 11 ) . a. h. mueller , phys . d * 20 * ( 1979 ) 2037 . doi:10.1103/physrevd.20.2037 j. c. collins , phys . rev . d * 22 * ( 1980 ) 1478 . doi:10.1103/physrevd.22.1478 a. sen , phys . d * 24 * ( 1981 ) 3281 . doi:10.1103/physrevd.24.3281 l. magnea and g. f. sterman , phys . rev . d * 42 * ( 1990 ) 4222 . doi:10.1103/physrevd.42.4222 g. p. korchemsky and a. v. radyushkin , phys . b * 279 * ( 1992 ) 359 doi:10.1016/0370 - 2693(92)90405-s [ hep - ph/9203222 ] . i. a. korchemskaya and g. p. korchemsky , phys . b * 287 * ( 1992 ) 169 . doi:10.1016/0370 - 2693(92)91895-g g. f. sterman and m. e. tejeda - yeomans , phys . b * 552 * ( 2003 ) 48 doi:10.1016/s0370 - 2693(02)03100 - 3 [ hep - ph/0210130 ] . g. kramer and b. lampe , z. phys . c * 34 * ( 1987 ) 497 [ erratum - ibid . c * 42 * ( 1989 ) 504 ] . t. matsuura and w. l. van neerven , z. phys . c * 38 * ( 1988 ) 623 . t. matsuura , s. c. van der marck and w. l. van neerven , nucl . b * 319 * ( 1989 ) 570 . t. gehrmann , t. huber and d. maitre , phys . b * 622 * ( 2005 ) 295 [ arxiv : hep - ph/0507061 ] . a. baikov , k. g. chetyrkin , a. v. smirnov , v. a. smirnov and m. steinhauser , phys . * 102 * ( 2009 ) 212002 doi:10.1103/physrevlett.102.212002 [ arxiv:0902.3519 [ hep - ph ] ] . t. gehrmann , e. w. n. glover , t. huber , n. ikizlerli and c. studerus , jhep * 1006 * ( 2010 ) 094 doi:10.1007/jhep06(2010)094 [ arxiv:1004.3653 [ hep - ph ] ] . r. n. lee and v. a. smirnov , jhep * 1102 * ( 2011 ) 102 doi:10.1007/jhep02(2011)102 [ arxiv:1010.1334 [ hep - ph ] ] . t. gehrmann , e. w. n. glover , t. huber , n. ikizlerli and c. studerus , jhep * 1011 * ( 2010 ) 102 doi:10.1007/jhep11(2010)102 [ arxiv:1010.4478 [ hep - ph ] ] . j. m. henn , a. v. smirnov , v. a. smirnov and m. steinhauser , jhep * 1605 * ( 2016 ) 066 doi:10.1007/jhep05(2016)066 [ arxiv:1604.03126 [ hep - ph ] ] . a. von manteuffel and r. m. schabinger , arxiv:1611.00795 [ hep - ph ] . j. davies , a. vogt , b. ruijl , t. ueda and j. a. m. vermaseren , arxiv:1610.07477 [ hep - ph ] . r. boels , b. a. kniehl and g. yang , nucl . b * 902 * ( 2016 ) 387 doi:10.1016/j.nuclphysb.2015.11.016 [ arxiv:1508.03717 [ hep - th ] ] . p. nogueira , j. comput . phys . * 105 * ( 1993 ) 279 . j. kuipers , t. ueda , j. a. m. vermaseren and j. vollinga , comput . phys . commun . * 184 * ( 2013 ) 1453 doi:10.1016/j.cpc.2012.12.028 [ arxiv:1203.6543 [ cs.sc ] ] . r. harlander , t. seidensticker and m. steinhauser , phys . b * 426 * ( 1998 ) 125 [ hep - ph/9712228 ] . t. seidensticker , hep - ph/9905298 . a. v. smirnov , jhep * 0810 * ( 2008 ) 107 doi:10.1088/1126 - 6708/2008/10/107 [ arxiv:0807.3243 [ hep - ph ] ] . a. v. smirnov and v. a. smirnov , comput . commun . * 184 * ( 2013 ) 2820 doi:10.1016/j.cpc.2013.06.016 [ arxiv:1302.5885 [ hep - ph ] ] . a. v. smirnov , comput . commun . * 189 * ( 2015 ) 182 doi:10.1016/j.cpc.2014.11.024 [ arxiv:1408.2372 [ hep - ph ] ] . r. n. lee , arxiv:1212.2685 [ hep - ph ] . r. n. lee , j. phys . * 523 * ( 2014 ) 012059 doi:10.1088/1742 - 6596/523/1/012059 [ arxiv:1310.1145 [ hep - ph ] ] . j. m. henn , a. v. smirnov and v. a. smirnov , jhep * 1403 * ( 2014 ) 088 doi:10.1007/jhep03(2014)088 [ arxiv:1312.2588 [ hep - th ] ] . j. m. henn , a. v. smirnov and v. a. smirnov , to appear . j. m. henn , phys . * 110 * ( 2013 ) 251601 doi:10.1103/physrevlett.110.251601 [ arxiv:1304.1806 [ hep - th ] ] . j. m. henn , j. phys . a * 48 * ( 2015 ) 153001 doi:10.1088/1751 - 8113/48/15/153001 [ arxiv:1412.2296 [ hep - ph ] ] . t. gehrmann , a. von manteuffel , l. tancredi and e. weihs , jhep * 1406 * ( 2014 ) 032 doi:10.1007/jhep06(2014)032 [ arxiv:1404.4853 [ hep - ph ] ] . p. mastrolia , m. argeri , s. di vita , e. mirabella , j. schlenk , u. schubert and l. tancredi , pos ll * 2014 * ( 2014 ) 007 . c. meyer , arxiv:1611.01087 [ hep - ph ] . r. n. lee , jhep * 1504 * ( 2015 ) 108 doi:10.1007/jhep04(2015)108 [ arxiv:1411.0911 [ hep - ph ] ] . o. gituliar and v. magerya , pos ll * 2016 * ( 2016 ) 030 [ arxiv:1607.00759 [ hep - ph ] ] . r. n. lee and v. a. smirnov , jhep * 1610 * ( 2016 ) 089 doi:10.1007/jhep10(2016)089 [ arxiv:1608.02605 [ hep - ph ] ] . e. remiddi and j. a. m. vermaseren , int . j. mod . phys . a * 15 * ( 2000 ) 725 doi:10.1142/s0217751x00000367 [ hep - ph/9905237 ] . p. a. baikov and k. g. chetyrkin , nucl . b * 837 * ( 2010 ) 186 doi:10.1016/j.nuclphysb.2010.05.004 [ arxiv:1004.1153 [ hep - ph ] ] . r. n. lee , a. v. smirnov and v. a. smirnov , nucl . b * 856 * ( 2012 ) 95 doi:10.1016/j.nuclphysb.2011.11.005 [ arxiv:1108.0732 [ hep - th ] ] . a. v. smirnov and m. tentyukov , nucl . b * 837 * ( 2010 ) 40 doi:10.1016/j.nuclphysb.2010.04.020 [ arxiv:1004.1149 [ hep - ph ] ] . d. maitre , comput . commun . * 174 * ( 2006 ) 222 [ arxiv : hep - ph/0507152 ] . w. wasow , `` asymptotic expansions for ordinary differential equations '' , interscience publishers john wiley & sons , inc . , new york - london - sydney , pure and applied mathematics , vol . xiv , 1965 . t. becher and m. neubert , jhep * 0906 * ( 2009 ) 081 erratum : [ jhep * 1311 * ( 2013 ) 024 ] doi:10.1088/1126 - 6708/2009/06/081 , 10.1007/jhep11(2013)024 [ arxiv:0903.1126 [ hep - ph ] ] . z. bern , m. czakon , l. j. dixon , d. a. kosower and v. a. smirnov , phys . d * 75 * ( 2007 ) 085010 doi:10.1103/physrevd.75.085010 [ hep - th/0610248 ] . n. beisert , b. eden and m. staudacher , j. stat . * 0701 * ( 2007 ) p01021 doi:10.1088/1742 - 5468/2007/01/p01021 [ hep - th/0610251 ] . a. v. kotikov , l. n. lipatov , a. i. onishchenko and v. n. velizhanin , phys . b * 595 * ( 2004 ) 521 erratum : [ phys . b * 632 * ( 2006 ) 754 ] doi:10.1016/j.physletb.2004.05.078 , 10.1016/j.physletb.2005.11.002 [ hep - th/0404092 ] . | we compute the four - loop qcd corrections to the massless quark - anti - quark - photon form factor @xmath0 in the large-@xmath1 limit . from the pole part
we extract analytic expressions for the corresponding cusp and collinear anomalous dimensions .
pacs numbers : 11.15.bt , 12.38.bx , 12.38.cy |
You are an expert at summarizing long articles. Proceed to summarize the following text:
the many - body problem can not be solved exactly for realistic interactions and systems . numerous approximations have been formulated and applied over the years . one of the problems is that many particles require a large hilbert space to allow for the many possible correlations . in fact , the space typically grows exponentially with the particle number . the necessary reduction of the hilbert space to obtain a tractable problem has to be accompanied by a correponding transformation of the interaction which in turn has to be used in the smaller space . this is the method of effective interactions in reduced spaces . a different approach starts by an approximation applied to the interaction itself . in many physical fields , the two - body interaction is a complicated entity , and therefore it is desirable to reduce this complexity . one such direction is to perform a harmonic approximation of the interaction hamiltonian which leaves all terms ( one- and two - body ) as a polynomial of at most second order in the coordinates of all particles . this yields an exactly solvable model for the many - body system @xcite . however , to utilize such an approach , the input parameters must be adjusted to reproduce properties of realistic systems for low particle numbers . this could be two - body binding and scattering information and structural expectation values . this sort of approach has been pursued in nuclear and condensed - matter physics for many years @xcite as a source of exact insight into many - body problems @xcite when numerics are either intractable or hard to interpret . more recently , cold atomic gases have emerged as an arena and testing ground for various models due to the large degree of controllability in experiments @xcite . in particular , interaction can be controlled through interatomic feshbach resonances @xcite . however , these systems are almost exclusively studied in the presence of an external confining potential which can often be modelled by a ( possibly deformed ) harmonic oscillator . for experimental setups with very deformed traps , the geometry can in fact be manipulated to study effective one- or two - dimensional dynamics . very recently , it has even become possible to explore the few - body limit with trapping of just a few atoms near the degenerate regime @xcite . at the moment , the physics of ultracold atomic gases is , however , dominated by short - range interacting neutral atoms @xcite . in addition , since the samples are often extremely dilute , the details of the potential at short - range does not matter , and only low - energy scattering information is important @xcite . the replacement of the potential by a regularized zero - range potential is therefore often justified . in the presence of a harmonic confinement , the solution of the two - body problem with a zero - range potential was presented by busch _ this solution was subsequently confirmed in experiments @xcite . more recently , this model has been used as a starting point for addressing the few- and many - body problem in a controlled manner in both nuclear and cold atom physics @xcite . the zero - range approximation scheme is one approach to effective interactions . here we pursue a different one via harmonic expansion of the interaction hamiltonian . system that can be described in some regimes of parameter space by harmonic hamiltonians are extremely useful due to their solvability . of course , as mentioned above , the parameters have to be meningfully related to realistic physical systems . this can be done through fits to two - body properties as described previously in ref . @xcite . here we take a parametric approach and study the details of the harmonic approximation for identical quantum particles as function of the two relevant frequencies for such a system ; the ( internal ) interaction frequency and the ( external ) trapping frequencies , both of which are described by ( isotropic ) harmonic terms . deformation can be easily implemented and will not be considered here . the objective of the present paper is to describe a method for calculating thermodynamical quantities for systems of identical bosons and fermions , a problem of relevance for any subfield of physics concerned with quantum mechanical behavior of multiparticle system in equilibrium . this requires full access to the partition function and the thermodynamic quantities that can be derived from it . the quantum statistics of ensemble of particles is needed here , and we present a general method for obtaining such information for harmonic hamiltonians that works for any number of particles in principle . there is great interest in the universal thermodynamics of strongly interacting fermi gases @xcite , and measurements have determined the equation of state of the system @xcite . those studies are mostly concerned with trap - averaged quantities that are less sensitive to sudden changes in thermodynamic behavior as expected near phase transitions . however , recently it has become possible to determine local properties of the gas by density fluctuation measurements @xcite . quantities that involve derivatives of the energy or pressure like heat capacity and compressibility can thus be used to study for instance superfluid properties of both fermionic @xcite and bosonic systems @xcite . in addition , the same experiments are also able to explore effects of changing the dimensionality of the system through optical lattice potentials . in the current presentation we focus on extending th harmonic approximation into the statictical regime , and more particularly on the technicalities of the method used to achieve this . to demonstrate our method we consider the case of a two - dimensional system of identical bosons or fermions for different ratios of interaction to external trapping frequencies . the paper is organized as follows . in sec . [ secmethod ] we briefly recap the harmonic method and then discuss the partition function in the harmonic approximation . the most important part of the method is the counting of the degeneracies in the partition function which we discuss in extended detail . the thermodynamic quantities are briefly outlined and a discussion of how to related the parameters in the harmonic hamiltonian to the model by busch _ @xcite is given . practical numerical results are obtained and discussed in sec . [ secthermo ] for a system of cold atoms confined to two spatial dimensions . both bosons and fermions are explicitely investigated . finally , give a summary and an outlook in sec . [ seccon ] . we consider a system of @xmath0 identical quantum particles , fermions or bosons , that have two - body interactions , all of which are confined by an external field . all interactions are substituted by carefully chosen harmonic oscillators . the solutions of the hamiltonian can then be found explicitly and treatment of thermodynamic properties is possible . we first present the necessary ingredients of the model , and continue to develope the method for calculating the partition function . this includes the key discussion of how to obtain the degeneracies of the many - body states . the @xmath0-body hamiltonian contains kinetic energy , one- and two - body interaction terms , all of which we assume can be written in terms of second order polynomials in the particle coordinates when proper choice of effective interaction parameters have been made . the resulting harmonic oscillator structure can be separated into terms related to each of the cartesian coordinates and the hamiltonian in one , two , and three dimensions is a sum of terms from each coordinate . for example , in the @xmath1-direction , the effective hamiltonian is @xmath2 where @xmath3 is the mass of the identical particles and @xmath4 is the reduced mass . the one - body ( external ) harmonic trap frequency is @xmath5 , while the two - body interaction frequency between all pairs is denoted @xmath6 ( internal ) . @xmath7 is a constant energy shift which is included for generality . double counting of interactions account for the factor @xmath8 . for two and three dimensions , the total hamiltonian is @xmath9 and @xmath10 . we will assume that the external harmonic trap is isotropic , extension to the deformed case are straightforward and will be addressed in future work . all particles are identical and interact in the same way which means that @xmath6 is independent of @xmath11 and @xmath12 . also the one - body frequency , @xmath5 , is independent of @xmath12 . the schrdinger equation for the hamiltonian in eq . can be solved analytically for any particle number ( details can be found in @xcite ) the pertinent properties in the present context are that the energies , @xmath13 , of the excited states have oscillator character . here @xmath13 denotes the @xmath14th _ many - body _ excited state , i.e. the energy of all @xmath0 particles when they are distributed in a certain way in the oscillator levels that come out of the diagonalization of the full hamiltonian . in this paper we will concentrate mostly on the case of a two - dimensional system , and we therefore specialize to this case now . the energy of a many - body state , @xmath13 , can be written @xmath15 where @xmath16 and @xmath17 are quanta in the @xmath1-direction related to the relative and center - of - mass excitations with the former carrying the index @xmath12 running over all @xmath0 particles ( and similar terms for the @xmath18-direction ) . the total number of relative and center - of - mass quanta in the many - body state @xmath14 are denoted @xmath19 and @xmath20 respectively . note that @xmath16 , @xmath17 , @xmath21 , and @xmath22 must all be specified to completely characterize the many - body state of energy @xmath13 . however , more than one set of individual quantum numbers can lead to the same energy @xmath13 and this determines the degeneracy of the state , which we denote @xmath23 and discuss at length below . @xmath13 and @xmath23 are all that is needed to compute the partition function and thus the thermodynamic properties of the system . the constants , @xmath24 and @xmath25 in the formula for @xmath13 , are from the two - dimensional zero - point energy of intrinsic and center of mass parts , respectively . the one - body frequency , @xmath5 , correspondings to center of mass motion . the other frequency , @xmath26 , is @xmath27 times degenerate for identical particles . it is given by the relation @xcite @xmath28 if the two - body interaction goes to zero ( @xmath29 ) , @xmath30 , and the resulting eigenvalues and eigenfunctions converge to those of the external one - body trapping potential . the solutions are @xmath0 non - interacting identical particles occupying @xmath0 states in a harmonic oscillator . for interacting particles , we obtain instead @xmath27 identical frequencies with value given by eq . . the extension of the above discuss to three dimensions involves also the @xmath31-direction and is straightforward , although the extra quantum numbers makes the degeneracy larger and the counting of many - body states more involved . with the harmonic hamiltonian , the exact spectrum of excited many - body states , @xmath13 , are thus available for a given number of particles . for the thermodynamic calculations , the natural choice is to use the canonical ensemble applicable for a definite number of particles . the canonical partition function is @xmath32 where @xmath23 is the degeneracy of the @xmath14th many - body state and @xmath33 is the temperature measured in energy units ( the boltzmann constant , @xmath34 ) . once @xmath35 is obtained , all thermodynamic quantities can be derived from it . the most involved question is how to obtain @xmath23 for a given energy of the many - body system . this will now be discussed . the only quantities in the partition function are the energies , @xmath13 , and their degeneracies , @xmath23 . the energies are appropriate multiples of the harmonic oscillator frequencies introduced above . the degeneracies depend strongly on which symmetry restriction is imposed . problems arise when all , or a group of , particles can not be distinguished because they are identical and allow occupation of the same states . we then have to count the number of ( anti-)symmetric states for the indistinguishable particles of a given excitation energy , @xmath13 , and the corresponding degeneracies , @xmath23 , of each _ many - body _ state of energy @xmath13 . we now describe our procedure to obtain @xmath23 . it is based on knowledge of the non - interacting system , a recursive procedure , and the fact that when the two - body interaction vanishes , the degeneracies remain the same as we will now explain . we start with @xmath0 identical particles without any two - body interaction ( @xmath29 ) , and all in the same external trap with frequency @xmath5 . this hamiltonian will have @xmath0 degenerate solutions of frequency @xmath5 per spatial dimension ( so @xmath36 for the two dimensional case ) . the many - body solutions are the products of @xmath0 combinations of the single - particle harmonic oscillator wave functions . for identical fermions , the pauli principle , or equivalently the antisymmetrization , requires that each of the single - particle states at most is occupied by one particle . in addition , if this condition of occupancy by at most one particle for each state is fulfilled , then one and only one antisymmetric state is uniquely constructed as the slater determinant of the @xmath0 singly occupied single - particle states . importantly , the energy can be characterized by the sum , @xmath37 , of single - particle oscillator quantum numbers for all particles in the state @xmath13 , as seen in eqs . and . assume that we have computed the number , @xmath38 , of properly symmetrized states of non - interacting identical particles in the external potential . the quantity @xmath38 can be computed in a simple manner by trail - and - error , i.e. taking all possible permutations and testing which are completely symmetric and which are completely antisymmetric . this is , however , notoriously difficult since the computational effort is exponentially increasing with particle number . also , it does not distinguish between center of mass motion and relative motion and this needs to be disentangled . this can be done by recalling the symmetry of the center of mass as we now discuss . the center of mass motion is a symmetric mode in all permutations of the particle coordinates , simply because the center of mass coordinate is the sum of the individual particle coordinates . this motion is always symmetric , independent of the number of quanta in this mode , @xmath20 . therefore , the relative motion determines the symmetry of the total wave function . the excited states consist of a combination of two distinctly different types of excitations . they are characterized by the number of quanta in the center of mass motion , @xmath20 , and the number of quanta in the relative motion , @xmath19 . we denote the number of properly ( anti)symmetric @xmath0-body states by @xmath39 , which is the number of states that have the proper symmetry in terms of the _ relative _ coordinates . it can be computed from knowledge of the total number of non - interacting many - body states , @xmath40 , for all values of @xmath37 . to do so , one must subtract the number of states of different center of mass quanta from the total number of non - interacting many - body states in a recursive manner , i.e. @xmath41 where the degeneracy , @xmath42 , of the center of mass state depends on the spatial dimension of the oscillator ; @xmath43 for one , @xmath44 for two , and @xmath45 for three dimensions , respectively . the formula comes from the fact that we know only the total number of quanta , @xmath46 , but _ not _ @xmath19 and @xmath20 individually . we need to recursively subtract the states that correspond to @xmath47 , 2 , @xmath48 , @xmath46 as the formula prescribes . the factor @xmath42 is due to the degeneracy of the center of mass motion itself for given number of quanta , @xmath20 . note that @xmath49 is well - defined and corresponds to a state with no quanta of excitation , i.e. the ground state . it can be zero or non - zero depending on @xmath0 and whether the particles are bosons or fermions . . determines the degeneracy of an @xmath0-body state in a harmonic system and is one of the main results of this work . the degeneracy can now be calculated iteratively from eq . for any number of quanta , @xmath46 from a starting point based on a non - interacting system of identical particles in a trap . these states must then be either completely symmetric or antisymmetric with respect to interchange of all coordinates of any pair of particles to obey either bosonic or fermionic statistics . now we add the two - body interaction on top of the external one - body potential . the external frequency , @xmath5 , remains in the spectrum corresponding to the center of mass motion , and an @xmath27 times degenerate frequency , @xmath26 , appears corresponding to relative motion . the absolutely crucial point is that the degeneracies do _ not _ depend on the two - body interaction strength which will only influence the value of the degenerate frequency . the counting remains completely unchanged , because the number of states of a given symmetry is a discrete number which does not change with continuous variation of the interaction . mathematically , this corresponds to a continuous map ( the scheme for counting degeneracies ) from a continuous interval ( the interaction frequency , @xmath26 ) to a discrete set ( number of states , @xmath23 , with given energy ) . such a map must necessarily be a constant map . the underlying point here is that our splitting into @xmath20 and @xmath19 is done in a way that moves the states that are related to the relative motion when @xmath26 is increased from zero to its full value , while the states that are related to center of mass motion remain constant in energy since @xmath5 remains a solutions . we confirmed this simple counting method by an elaborate and computationally very slow brute force procedure which is much worse than computing @xmath38 above . the ( anti-)symmetrization of each of the computed wave functions is performed by permutations of all the coordinates followed by extraction of a basis with a size equal to the smallest number of linearly independent states . for sufficiently small @xmath0 this can be computed in reasonable time and comparison can be made . in all cases considered below we found perfect agreement between the numerics and the procedure outlined above . the procedure above is , however , far superior since it needs only counting of states with given energy , _ not _ the full wave functions . we can now use eq . to get the number of center of mass states and the number of relative states as functions of the sum of corresponding quanta . along with the frequencies , this is all we need to calculate the partition function in eq . . this simplification occurs since the structure of the wave functions do not enter in the partition function . the center of mass mode of frequency @xmath5 separates in eq . because both the exponential function and the degeneracy factorizes , that is @xmath50 , where the center of mass contribution is analytical and given below eq . . we have @xmath51 where @xmath52 is the dimension , @xmath53 , and @xmath54 . in practice , the infinite sum over relative quanta has to be limited by a cut - off value which is chosen sufficiently high to achieve the required accuracy . certain quantities are more sensitive to the cut - off than others , such as those with more derivatives of the partition function ( heat capacity , compressibility , etc . ) . for all the results presented below we have checked convergence by increasing the cut - off and identifying the value of @xmath33 below which the relevant quantities remain unchanged . from the canonical partition function , eqs . or , we get immediately the basic quantities of energy , @xmath55 and free energy , @xmath56 , that is @xmath57 where @xmath58 is the entropy , and @xmath59 is the heat capacity @xcite . in thermodynamic formulations of macroscopic systems , both temperature and volume is usually external parameters . both the energy , @xmath55 , and heat capacity , @xmath59 , defined above , are in principle the derivatives for a fixed volume , @xmath60 ( and also fixed @xmath33 and @xmath0 ) . in our case with an externally confined @xmath0-body system the volume is not the fixed quantity , but rather it is the external trapping frequency , @xmath5 , that is fixed . the derivatives above are thus taken for fixed @xmath5 . usually , variation of the volume allows information about pressure and compressibility . definitions of these quantities and connection to their thermodynamic counterparts require a precise translation between @xmath5 and volume . here we employ the simplest choice and define the volume via the length parameter related to the external trap , @xmath61 , which is defined by @xmath62 . we now have @xmath63 , where @xmath64 and @xmath65 are the two and three dimensional angular surface areas . derivatives of any quantity , @xmath66 , with respect to volume can now be written @xmath67 for the pressure , @xmath68 , we find @xmath69 which , from the definition in eq . , gives @xmath70 the derivatives are easily worked out since they only contain @xmath71 . continuation to the second derivative with respect to the volume produce the isothermal bulk modulus , @xmath72 , defined as @xmath73 which is the reciprocal of the isothermal compressibility , @xmath74 . again two terms arise related to center of mass and relative degrees of freedom . several terms appear by carrying out the two derivatives . this kind of compressibility , that of a response of a system to an applied pressure , is different to the compressibility of a self - bound system such as a nucleus @xcite . for self - bound systems , the compressibility refers to resistance to density fluctuations . this can be written as the second derivate of the energy per particle with respect to the fermi momentum for fermionic systems . from eq . and the external trap frequency , @xmath5 , as function of the scattering length , @xmath75 , in two dimensions for the busch model @xcite with @xmath76-wave interactions for bosons ( upper ) and for the @xmath77-wave interaction identical ( i.e. spin - polarized ) fermions ( lower ) . in both plots , the different curves correspond to different particle numbers . from the bottom to top , the curves are for 3 , 4 , 5 , 6 , 10 , 20 , and 30 particles . the length parameter is the trapping length , @xmath78 , introduces in the text.,scaledwidth=48.0% ] before we present results for degeneracies and various thermodynamic properties , we discuss how one can fix the two - body interaction term to capture the properties of the realistic system which have interactions that are not harmonic . in the case of identical bosons , the procedure was already presented in ref . @xcite but we repeat the arguments here for completeness . we assume that we are in the situation that the real two - body interaction is of much shorter range than the external trap length @xmath61 above . in this case , we can use the zero - range model of busch _ et al . _ @xcite to obtain the solutions in the trap . the effective two - body interaction that we want to use is a harmonic oscillator , which in general contains two parts ; an oscillator frequency , @xmath6 , and a shift , @xmath79 . to fix these parameters we employ the conditions that the two - body harmonic interaction should reproduce the correct two - body binding energy of the ground state in the busch model , and also the spatial extend of the wave function . the latter condition is implemented by calculating the root - mean - square radius , @xmath80 , in the exact solution and fixing the oscillator parameter to reproduce value . the energy shift is suqsequently tuned so as to also reproduce the two - body binding energy . clearly , since the busch model uses a zero - range interaction , the only scale left is the external trap frequency , @xmath5 . therefore what we obtain from this is the quantity @xmath81 . we ignore the shift from now on since it merely provides an overall shift in the @xmath0-body harmonic hamiltonian which is not of interest in the current paper . the next step is to insert this value into eq . to obtain @xmath82 , which will now naturally depend on @xmath0 . in the upper panel in fig . [ busch ] we show the result of this procedure for the case of bosons in two dimensions ( more detail for bosons in both two and three dimensions can be found in ref . the results are plotted as function of the two - dimensional scattering length @xmath75 . note that when @xmath75 is small , the ground state becomes strongly bound , while for large @xmath75 it goes to the non - interacting limit . this is clearly reflected in the behavior of @xmath82 . the result of this procedure agrees rather well with predictions from the well - studied problem of bosons interacting through a zero - range interaction @xcite in the strong and weak coupling limits @xcite , also in the case where higher - order interaction terms are included @xcite . recently , a very similar procedure has been used to study polar molecules in layered system @xcite where excellent agreement with exact methods has been found both for isotropic @xcite and anisotropic potentials @xcite which have reasonably large potential pockets . incidentally , one - dimensional dipolar system also turn out to have such potential pockets @xcite and applying the harmonic approximation to these systems is an interesting direction for future research . for identical ( spin - polarized ) fermions a complication arises from the fact that the original busch model applies to particles interacting through an isotropic @xmath76-wave interaction only . the pauli principle dictates that @xmath76-wave interactions are zero for identical fermions , and therefore the interactions must have @xmath77-wave ( or higher odd partial wave ) character . in two dimensions , the equivalent of the busch model with @xmath77-waves can be solved and the spectrum turns out to be very similar to the @xmath76-wave case except for ( unimportant ) shifts @xcite . however , a further complication arises since the corresponding ground state wave function is singular at the origin in a manner that does not allow it to be normalized @xcite . this can be fixed by a properly defined scalar product as discussed for the three - dimensional case in ref . we will not pursue this approach here but instead we note that any higher order structural average of the type @xmath83 with @xmath84 an integer is still perfectly convergent . we thus consider a higher order average , @xmath85 , in order to fix the oscillator parameter for identical fermions . we are thus tacitly using that the @xmath77-wave spectrum is very similar to the @xmath76-wave one and assume that the wave function should be so as well . of course , it must be kept in mind that this is _ not _ at odds with the pauli principle since the two - body wave function in the @xmath77-wave channel is still zero at the origin . in any case , the pauli principle is exactly enforced on the @xmath0-body system as discussed above . the results for @xmath82 are shown in the lower panel of fig . [ busch ] . the similarity to @xmath76-waves and bosons is quite clear , and the only difference seems to be slightly lower overall values for the fermionic case . below we will illustrate our method by calculating thermodynamic quantities for both fermions and bosons . here the only thing that matters is the ratio @xmath82 . one can then reverse the process and use fig . [ busch ] to obtain a corresponding value of the scattering length and thus compare to a realistic system with trapped bosons or single - component fermions . since this is mainly a discussion of method , we will not dwell on this any further . an important thing to note , however , is that since the ratios @xmath82 are similar for bosons and fermions , the results below will truly isolate the statistical behavior coming from the exchange requirements in the different cases . we now proceed to demonstrate our methods by computing the density of states and the thermodynamic quantities that have been introduced above . they depend on particle mass , dimension , number of particles and their quantum statistics , temperature , one- and two - body frequencies . if we measure all energies in units of @xmath86 , all results for given particles in @xmath52 dimensions , depend only on the two ratios , @xmath87 and @xmath82 . this implies that any kind of interaction model used to define the effective harmonic hamiltonian only has to provide these two quantities as pointed out above . , while it is @xmath88 for fermions.,title="fig:",scaledwidth=48.0% ] , while it is @xmath88 for fermions.,title="fig:",scaledwidth=48.0% ] first we discuss the degeneracies themselves which are found by the procedure discussed above , essentially through the recursive formula eq . . while the number of non - interacting ( anti-)symmetric states increases exponentially as function of excitation energy , the numbers can , however , be tremendously reduced by the symmetry requirements . a simple example indicating this tendency is that for bosons all particles are allowed to occupy the lowest level with zero quanta , @xmath89 . in contrast , a given number of fermions has a minimum sum of quanta , @xmath90 , in the ground state , that is @xmath91 . we show the numbers of given symmetry as functions of excitation energy in fig . [ fermbos ] . since we are dealing with harmonic oscillators , a given energy corresponds to a given total number of quanta . the starting point for the ground state is naturally @xmath92 and @xmath93 , which is larger for fermions than for bosons due to the requirement that at most one fermion can occupy each state . the corresponding higher degeneracy implies therefore that the number of states of given excitation energy is larger for fermions than for bosons because the latter do not need to obey the pauli principle . for bosons , the lowest excited state consisting of one quantum added to zero quanta in the ground state can only be a center of mass excitation , since one quantum in the relative motion would be antisymmetric in one pair of coordinates ( it would have to be a state of @xmath77-wave / parity - odd symmetry ) . the lowest completely symmetric excited state of the relative motion appear for @xmath94 . in conclusion , there is always a gap in the excitation spectrum for bosons corresponding to @xmath95 where fermions only have a gap of @xmath96 . this will be reflected in the temperature dependence below . for bosons ( upper part of fig . [ fermbos ] ) the same degeneracy is found for low energy ( small total number of quanta ) for all particle numbers . in the upper plot of fig . [ fermbos ] the horizontal axis is @xmath97 since we have found that this is a good measure for the excitation energy in the system . for fermions , one can give a qualitative argument for the exponential behavior which we discuss below . the similarity of degeneracies at low energy for bosons can be understood by the direct counting procedure where a given number of particles , @xmath0 , has to be distributed to add up to the total number of quanta , @xmath19 . first , @xmath27 particles are placed in the lowest oscillator level , and the one remaining particle then has to be placed in the level with total number of quanta equal to @xmath19 . then we move the single particle one step down to @xmath98 , and simultaneously compensating by moving one particle one step up from the lowest level . we continue with these combinations until we have @xmath19 particles in the second oscillator level and all others in the lowest level . adding one particle and repeating the counting process we realize that their is a one - to - one correspondence between the configurations of @xmath0 and @xmath99 particles . going from one to the other is simply by removing or adding one particle in the lowest oscillator level with zero contribution to the total number of quanta . for @xmath0 bosons , the deviation from this universal curve starts for @xmath100 . the reason is again found by following the counting procedure . the configurations with all particles in the lowest two levels are only possible when @xmath101 . this implies that we find fewer states when @xmath102 for @xmath0 than for @xmath99 particles . the curves break away from the universal curve for increasing @xmath0 when @xmath100 . the degeneracies for fermions have very different behavior , as seen in the lower part of fig . [ fermbos ] . notice that the horizontal axis is different in the two plots . we notice a regime of linear dependence which has the same origin as the exponential square root dependence of excitation energy of the free fermi gas level density @xcite . for two dimensions , the particle number dependence is roughly @xmath103 as reflected on the axis in the fermion plot . this holds for intemediate excitation energies and for relatively large particle numbers . a qualitative understanding of the behavior can be obtained in a manner following ref . the density of _ single - partile _ states at the fermi level in a two - dimensional harmonic trap is roughly @xmath104 , while the typical excitation energy in the system is @xmath105 . the _ many - body _ level density in this situation is then proportional to @xmath106 $ ] . this explains the choice of horizontal axis for fermions in fig . [ fermbos ] . the suggested linear dependence is not very clear but the assumptions are not well fulfilled . the requirements is that the excitation energy has to be sufficiently large to allow statistcial treatment and sufficiently small not to exhaust particles at the bottom of the potential . this is not true for the relatively small particle numbers that we must necessarily work with to make the problem tractable by both brute force symmetrization and the counting scheme developed here . , for several boson ( upper ) and fermion ( lower ) systems , identified by the number of particles and frequency ratio , @xmath107 . the ground state energy is subtracted , and the energy unit is @xmath108 for both energies and temperatures . the ground state energy , @xmath109 , has been substracted.,scaledwidth=50.0% ] we are now in a position to explore the thermodynamics and to compare bosonic and fermionic behavior in detail . the free energies per particle for a two - dimensional system are shown in fig . [ bosf ] as function of temperature for different interactions and particle numbers , both for bosons ( upper panel ) and fermions ( lower panel ) . note that the ground state energy has been subtracted as it is not of interest here . the curves start out very flat at zero showing that the low - temperature dependence is at least quadratic . the decrease with increasing temperature is actually rather similar to a quadratic behavior with @xmath33 . the behavoir is modified when @xmath33 exceeds @xmath25 and the center of mass mode becomes active . notice that the dependence on particle number is relatively weak . this is in contrast to the interaction dependence where the free energy varies substantially more for the smallest two - body interactions . in addition , we note that fermion free energies are always smaller than boson free energies . this demonstrates that the free energy is entropy driven , i.e. the entropy term grows faster than the energy with temperature . again the larger degeneracy of fermionic systems is seen by the lower free energies . , for several boson ( upper ) and fermion ( lower ) systems , identified by the number of particles and frequency ratios , @xmath107 . the ground state energy , @xmath109 , has been subtracted from each curve.,title="fig:",scaledwidth=48.0% ] , for several boson ( upper ) and fermion ( lower ) systems , identified by the number of particles and frequency ratios , @xmath107 . the ground state energy , @xmath109 , has been subtracted from each curve.,title="fig:",scaledwidth=48.0% ] fig . [ bosu ] displays the energy per particle for bosons and fermions . we measure energy in units of @xmath110 to isolate the high temperature behavior which is @xmath111 for two dimensionsal harmonic system by the equipartition theorem . again we subtract the uninteresting constant ground state energy . the increase from zero at zero temperature is rather steep for both types of particles . all curves continue to increase with temperature but much slower after a few units of @xmath25 . the largest energy per @xmath112 is for the system with the least particles and the smallest frequency ratio . this can be understood from the fact that for large @xmath82 , only the center of mass modes ( twice degenerate in two dimensions ) are active . however , since we divide by @xmath0 , the values become smaller for larger @xmath0 , but the behavior remains the same ( self - similar lines for @xmath113 in fig . [ bosu ] ) . this is similar for both bosons and fermions . we also observe that the equipartition limit at high temperature is reached for substantially larger temperatures , a clear sign of the interaction effects . the low - temperature behavior is , however , different for bosons and fermions . the low - energy @xmath95 gap in the boson spectra arises due to a ground state with all particles with zero oscillator quanta of excitation . then there are no states with one quanta of excitation . for fermions , there are only @xmath114 gaps . therefore the energy exhibits flat regions at relatively low temperature , but these features are observed first for fermions and later for bosons due to the difference in energy gap which is related to the activation energy . this demonstrates a clear signature of quantum statistics in the harmonic model , but which should be expected in generic interacting systems with identical particles . , for boson ( upper ) and fermion ( lower ) systems , identified by the number of particles and frequency ratios , @xmath107 . , scaledwidth=48.0% ] next , we show the entropy , @xmath58 , in fig . we note the increase of available states per particle as function of temperature goes from zero to values around a few times the temperature in units of @xmath25 . for lower ratios @xmath82 , the increase with temperature is faster since less energy is required to excite the internal modes of the system and more states are available , correspondingly increasing @xmath58 . again , for larger values of @xmath82 , the center of mass is the only active mode at low temperature and the division by @xmath0 in the plots explains the lower value of @xmath58 for larger @xmath0 . only at substantially larger temperatures will both internal and center of mass modes become active . the effect of symmetry is not very pronounced , although it is noticeable that the entropy is larger for fermionic than for bosonic systems due to the larger degeneracy . an interesting feature of the fermion plot is that for @xmath115 the limit of @xmath116 for @xmath117 is non - vanishing . the number of available states is finite reflecting that the ground state itself is degenerate . this is seen by simply counting of oscillator degeneracy for a two - dimensional harmonic oscillator . the lowest three quantum levels can hold 6 identical fermions ( 1 , 2 , and 3 , respectively ) , and the fourth can hold additionally four particles . this means that eight particles only occupy half of the last level , leaving the ground state as six times degenerate . this beavior at zero temperature then nicely indicate the presence of shell structure . in two dimensions for boson ( upper ) and fermions ( lower ) systems , identified by the number of particles and frequency ratios , @xmath107 . the black line shows the unscaled value for the mean field center of mass mode.,title="fig:",scaledwidth=48.0% ] in two dimensions for boson ( upper ) and fermions ( lower ) systems , identified by the number of particles and frequency ratios , @xmath107 . the black line shows the unscaled value for the mean field center of mass mode.,title="fig:",scaledwidth=48.0% ] we now consider some derived thermodynamic quantities that are of experimental interest in many fields of physics . the heat capacity and the compressiblity are two such quantities that can be obtained from second derivatives of the partition function as discussed above . in fig . [ boscv ] , we show the heat capacity per particle , @xmath118 , defined in eq . as function of temperature for both bosons and fermions . they both start with an initial activation of the external trap mode , since the degenerate relative degrees of freedom all require higher temperature to be excited . after a delay , these internal modes are activated , and the heat capacity increases with temperature . the delay and the rate of increase depend strongly on @xmath82 with a slower increase of @xmath59 for larger interaction ratios . the tendency to increase slower and in steps is related to gaps in the energy spectrum . at high excitation energy , the spectrum becomes denser , and gap sizes larger than the temperature can not appear . however , the presence of a gap in the low - energy spectrum is important for the heat capacity in general . in fact , this is clearly seen by the fact that bosons have a flat profile for a region of low temperature , while the fermions have two flat plateaus . fermions rise faster initially , and always approach the equipartition heat capacity from below ( @xmath119 for two dimensions ) . the larger gap causes a delay in the boson systems . the heat capacity slightly overshoots the equipartition value , oscillate back below the equipartion value ( at a temperature outside the scale in fig . [ boscv ] ) , and eventually approach the limit from below . the curve marked com shows the heat capacity when assuming that only the center of mass mode is excited . all the other curves will approach this at very large temperatures when the internal structure is washed out . however , as the plots clearly show , the approach is very different for different particle numbers and interaction strengths . ) divided by @xmath0 for boson ( upper ) and fermions ( lower ) systems , identified by the number of particles and frequency ratios , @xmath107 . , scaledwidth=48.0% ] in fig . [ boskt ] , we show the isothermal compressibility per particle , @xmath120 , as defined in eq .. the high temperature behavior of a harmonic system is @xmath121 . the compressiblity shows a small increase through a maximum at very small temperature followed by steady decrease towards zero at large temperature with a @xmath121 slope . the compressiblity indicates how easy it is to squeeze the system and a large value indicates that the system is very susceptible to compression . we see that the more strongly interacting systems ( larger @xmath82 ) have larger @xmath74 . this comes from the fact that the attraction in these systems will make it energetically favourable to contract , due to the larger degeneracy of the interaction frequency , @xmath26 . this also explains why @xmath74 increases with @xmath0 . comparing bosons and fermions we find the same qualitative behavior . the fermions are slightly less compressible , with a sharper dependence on particle number and interaction strength . this is a common feature of fermi systems and usually attributed to the pauli principle . for comparison , [ 3dferkt ] shows the compressibility for fermions in three spatial dimensions , which shows the same qualitative behavior . the results for bosons are similar and we do not show them here . notice that the peak features in the compressibility are sharper in the three - dimensional case . our system is similar to an attractively interacting fermi gas where superfluidity is expected to show a signature in the compressibility @xcite . our results are consistent with the fact that phase transitions are less pronounced in lower dimensions . ) divided by @xmath0 for fermionic systems in three dimensions , identified by the number of particles and frequency ratios , @xmath107 . , scaledwidth=48.0% ] the harmonic approximation is extremely useful because its simplicity allows transparent calculations of otherwise complicated properties . the only approximation lies in the choice of harmonic potentials acting on each particle and between pairs of particles . one can therefore think of the harmonic approximation as an effective interaction scheme and subsequently investiagate the behavior of its predictions under changes in the input parameters . the latter should naturally be connected to whatever realistic physical system one is interested in studying . in this paper , we explore the harmonic approximation scheme by considering the hamiltonian for a many - body system consisting of a given number of identical particles . in particular , we explore the consequences of symmetry requirements on the properties of the system . this is done with thermodynamic applications in mind since this is a venue where the quantum statistics plays a decisive role at low temperatures . we develop a new method for counting the number of correctly symmetrized _ many - body _ states that reduces the complications that arise from this fundamental problem of statistical mechanics and thermodynamics . the advantage of the harmonic approach is obviously that the energy spectrum is analytically known . however , the degeneracy of each many - body state of given total energy still remains to be determined before the partition function is fully defined and possible to compute numerically . we design a novel procedure to obtain the degeneracy of each state , subject to requirements of symmetry and antisymmetry appropriate for bosons and fermions , respectively . we separate the completely permutation symmetric center of mass motion from the relative motion , which then has to carry the symmetry corresponding to bosons or fermions . we count by subtracting the number of states of different quanta in the center of mass motion from the total number of non - interacting states of given symmetry . to demonstrate the method , we consider the case of a two - dimensional system with identical bosons or fermions ( with no internal degrees of freedom ) . within the canonical ensemble , we compute the partition function , and from it the free energy , entropy , heat capacity , and compressibility of the system . this is done for a relatively small number of particles ( up to 20 ) . the method is a considerable improvement over the brute force method where one explicitly checks for symmetry properties by exchanging all pairs of particles one by one . however , it is still computationally involved when going beyond the particle numbers considered here . however , as is known from for instance the virial expansion @xcite , it is often enough to consider small particle numbers and then extrapolate to large system sizes from this information . the effective harmonic interaction can be related to quantities in realistic systems and we discuss a case of great usefulness within the realm of ultracold atomic gases , that of two particles interaction in a harmonic trap through a two - body interaction of zero - range . while bosons can interact in the @xmath76-wave channel originally considered by busch _ @xcite , identical fermions must have an antisymmetric relative wave function and thus an @xmath76-wave interaction of short - range will vanish . we therefore have to consider the @xmath77-wave channel , but we find that the effective harmonic interaction frequencies are very similar to the @xmath76-wave case in the two - dimensional setup that we consider here . therefore we have chosen to parametrize the discussion of the thermodynamic quantities by the harmonic oscillator frequency of the two - body interaction itself . one can then make the connection to a realistic system by working backwards through the model of busch _ et al._. our numerical results show that the low - temperature behavior reflects shell structure for the particle numbers we study . the large - temperature equipartition limits are recovered for energy and heat capacity , and the transition from small to large temperature is fastest for smaller two - body interaction strength . the qualitative behavior is rather similar for bosons and fermions . however , bosons always have a gap in the low - energy spectrum since one quantum of excitation of the relative motion must be antisymmetric and therefore forbidden . this leads to a slower variation with temperature , since this gap has to be overcome before the number of available states goes up . our results for the density of states turn out to scale with the number of relative excitation quanta in a manner that is very similar to the treatment of the many - body density of states for a uniform fermi gas . surprisingly , the bosonic many - body level density scales similarly to the fermionic one , although the particle number enters differently in our interpolation formulas . in future studies it will be interesting to consider also multi - component systems , something which is simply done within the harmonic approach since the level counting can be factorized in the different components . also , a study of the virial expansion based on the harmonic approach is currently on - going , both in two and three dimensions . in addition , we note that one dimensional system have attracted a lot of attention recently due to their realization in ultracold atomic physics @xcite . it would be interesting to test our prediction against some of the models that are being explored in the experiments and for which a number of exactly solvable many - body models are known @xcite . furthermore , recent experiments studying ultracold few - body two - component fermi systems ( particle numbers of ten or less ) @xcite would be an interesting comparison for the harmonic approximation . m. a. zaluska - kotur , m. gajda , a. orlowski , and j. mostowski , phys . a * 61 * , 033613 ( 2000 ) ; j. yan , j. stat . phys . * 113 * , 623 ( 2003 ) ; m. gajda , phys . rev . a * 73 * , 023603 ( 2006 ) ; j. r. armstrong , n. t. zinner , d. v. fedorov , and a. s. jensen , _ j. phys . * 44 * ( 2011)055303 . b. sutherland : _ beautiful models _ , ( world scientific publishing co. , singapore , 2004 ) ; r. j. baxter : _ exactly solved models in statistical mechanics _ , ( academic press , new york , 1982 ) ; v. e. korepin : _ exactly solvable models of strongly correlated electrons _ , ( world scientific publishing co. , singapore , 1994 ) . t. stferle , h. moritz , k. gnter , m. khl , and t. esslinger , phys . * 96 * , 030401 ( 2006 ) ; t. volz _ et al . _ , nature phys . * 2 * , 692 ( 2006 ) ; g. thalhammer _ et al . _ , lett . * 96 * , 050402 ( 2006 ) ; c. ospelkaus _ et al . lett . * 97 * , 120402 ( 2006 ) . w. c. haxton and t. luu , phys . lett . * 89 * , 182503 ( 2002 ) . i. stetcu , b. r. barrett , and u. van kolck , phys . b * 653 * , 358 ( 2007 ) ; i. stetcu , b. r. barrett , u. van kolck , and j. p. vary , phys . rev . a * 76 * , 063613 ( 2007 ) ; y. alhassid , g. f. bertsch , and l. fang , phys . rev . lett . * 100 * , 230401 ( 2008 ) ; n. t. zinner , k. mlmer , c. zen , d. j. dean , and k. langanke , phys . rev . a * 80 * , 013613 ( 2009 ) ; i. stetcu , j. rotureau , b. r. barrett , and u. van kolck , ann . phys . * 325 * , 1644 ( 2010 ) ; t. luu , m. j. savage , a. schwenk , and j. p. vary , phys . rev . c * 82 * , 034003 ( 2010 ) ; j. rotureau , i. stetcu , b. r. barrett , m. c. birse , and u. van kolck , phys . rev . a * 82 * , 032711 ( 2010 ) . h. hu , p. d. drummond , and x .- j . liu , nature phys . * 3 * , 469 ( 2007 ) . m. horikoshi , s. nakajima , m. ueda , and t. makaiyama , science * 327 * , 442 ( 2010 ) ; s. nascimbne _ et al . _ , nature * 463 * , 1057 ( 2010 ) ; n. navon , s. nascimbne , f. chevy , and c. salomon , science * 328 * , 729 ( 2010 ) ; c. cao _ et al . _ , science * 331 * , 58 ( 2011 ) . c. sanner _ et al . _ , 105 * , 040402 ( 2010 ) ; t. mller _ lett . * 105 * , 040401 ( 2010 ) . m. j. h. ku , a. t. sommer , l. w. clark , and m. w. zwierlein , arxiv:1110.3309v1 . c. l. hung , x. zhang , n. gemelke , and c. chin , nature * 470 * , 236 ( 2011 ) . d. a. mcquarrie : _ statistical mechanics _ , ( harpercollins , new york , 1976 ) . k. huang : _ statistical mechanics _ ( john wiley & sons , new york , 1987 ) . s. s. m. wong : _ introductory nuclear physics _ , ( wiley interscience , new york , 1998 ) . r. v. e. lovelace and t. j. tommila , phys . a * 35 * , 3597 ( 1987 ) ; g. baym and c. j. pethick , phys . rev . lett . * 76 * , 6 ( 1996 ) . h. fu , y. wang , and b. gao , phys . a * 67 * , 053612 ( 2003 ) ; n. t. zinner and m. thgersen , phys . a * 80 * , 023607 ( 2009 ) ; m. thgersen , n. t. zinner , and a. s. jensen , phys . rev . a * 80 * , 043625 ( 2009 ) . j. r. armstrong , n. t. zinner , d. v. fedorov , and a. s. jensen , europhys . * 91 * , 16001 ( 2010 ) ; n. t. zinner , b. wunsch , d. pekker , and d .- w . wang , arxiv:1009.2030v3 ; n. t. zinner _ _ , arxiv:1105.6264v1 . a. g. volosniev _ et al . _ , j. phys . b : at . mol . 44 * , 125301 ( 2011 ) ; a. g. volosniev , d. v. fedorov , a. s. jensen , and n. t. zinner , phys . * 106 * , 250401 ( 2011 ) ; a. g. volosniev _ _ , arxiv:1112.2541v1 . ho and e. j. mueller , phys . rev . * 92 * , 160404 ( 2004 ) ; x .- j . liu , h. hu , and p. d. drummond , phys . rev . lett . * 102 * , 160401 ( 2009 ) ; phys . rev . a * 82 * , 023619 ( 2010 ) ; s .- g . peng , s .- q . li , p. d. drummond , and x .- j . liu , phys . a * 83 * , 063618 ( 2011 ) . | we describe a method to compute thermodynamic quantities in the harmonic approximation for identical bosons and fermions in an external confining field .
we use the canonical partition function where only energies and their degeneracies enter .
the number of states of given energy and symmetry is found by separating the center of mass motion , and counting the remaining states of given symmetry and excitation energy of the relative motion .
the oscillator frequencies that enter the harmonic hamiltonian can be derived from realistic model parameters and the method corresponds to an effective interaction approach based on harmonic interactions . to demonstrate the method
, we apply it to systems in two dimensions .
numerical calculations are compared to a brute force method that is considerably more computationally intensive . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
according to increasing astrophysical indicia , the evolution of the universe seems to be largely governed by dark energy with negative pressure together with pressureless cold dark matter ( see @xcite for the latest review ) in a two to one proportion . however , little is know about the origin of either component , which in the standard cosmological model would play very different roles : dark matter would be responsible for matter clustering , whereas dark energy @xcite would account for accelerated expansion . several candidates for dark energy haven proposed and confronted with observations : a purely cosmological constant , quintessence with a single field ( see @xcite for earliest papers ) or two coupled fields @xcite , k - essence scalar fields , and phantom energy @xcite . interestingly , a bolder alternative presented recently suggests that an effective dark energy - like equation of state could be due to averaged quantum effects @xcite . the lack of information regarding the provenance of dark matter and dark energy allows for speculation with the economical and aesthetic idea that a single component acted in fact as both dark matter and dark energy . the unification of those two components has risen a considerable theoretical interest , because on the one hand model building becomes considerable simpler , and on the other hand such unification implies the existence of an era during which the energy densities of dark matter and dark energy are strikingly similar . one possible way to achieve that unification is through a particular k - essence fluid , the generalized chaplygin gas @xcite , with the exotic equation of state @xmath1 where constants @xmath2 and @xmath3 satisfy respectively @xmath4 and @xmath5 . using the energy conservation equation and the einstein equation @xmath6 one obtains the evolution @xmath7 where as usual @xmath8 is the scale factor , @xmath9 and @xmath10 is an integration constant . this model interpolates between a @xmath11 evolution law at early times and @xmath12 at late times ( i.e. the model is dominated by dust in its early stages and by vacuum energy in its late history ) . in the intermediate regime the matter content of the model can be approximated by the sum of a cosmological constant an a fluid with a soft equation of state @xmath13 . the traditional chaplygin gas @xcite corresponds to @xmath14 ( stiff equation of state ) . another possibility which has emerged recently is the modified chaplygin gas ( mcg ) @xcite . it is characterized by @xmath15 with @xmath16 a constant . alternatively , such evolution can be seen as coming from a modified gravity approach , along the lines of the dvali - gabadadze - porrati @xcite , cardassian @xcite and dvali - turner @xcite models . in those works the present acceleration of the universe is not attributed to an exotic component in the universe , but to modifications in gravitational physics at subhorizon scales . following the proposal by @xcite , an evolution like that arising from ( [ eqstate ] ) in standard gravity , could alternatively be obtained in the modified gravity picture for a pure dust dust configuration under the modification @xmath17 where @xmath18 . modified chaplygin cosmologies with @xmath19 are transient models which interpolate between a @xmath20 evolution law at early times and a de sitter phase at late times , but interestingly the matter content at the intermediate stage is a mixture of dust and a cosmological constant . the sound speed for the modified chaplygin gas @xcite becomes [ cs ] c_s^2=. the observational tests of traditional and generalized chaplygin models are numerous . several teams have analyzed the compatibility of those models with the cosmic microwave background radiation ( cmbr ) peak location and amplitude @xcite , supernovae data @xcite and gravitational lensing statistics @xcite . the main results of those papers can be summarized as follows : models with @xmath21 and some small curvature ( positive or negative ) are favored over the @xmath0cdm model , and chaplygin cosmologies are much likelier as dark energy models than as unified dark matter models . in what regards the modified chaplygin gas , it has only been tested observationally in @xcite , using the most updated and reliable compilation of supernovae data so far : the gold dataset by riess et al . @xcite . by means of a statistical test which depends no only on @xmath22 ( as in usual procedures ) bu also on the number of parameters of the parametrization of the hubble factor as a function of redshift , it was concluded that the modified chaplygin gas cosmologies give better fits than usual and generalized chaplygin cosmologies . in this paper we shall be concerned with the evolution of large - scale inhomogeneities in modified chaplygin cosmologies . this is an issue of interest because candidates for the dark matter and dark matter unification will only be valid if they ensure that initial perturbations can evolve into a deeply nonlinear regime to form a gravitational condensate of super - particles that can act like cold dark matter . here we will follow the covariant and sufficiently general zeldovich - like non - perturbative approach given in @xcite , because it can be adapted to any balometric or parametric equation of state . our results indicate that our model fits well in the standard structure formation scenarios , and we find , in general , a fairly similar behavior to generalized chaplygin models @xcite for the modified chaplygin gas described by eqs . ( [ eqstate ] ) and ( [ effecden])the effective equation of state in the intermediate regime between the dust dominated phase and the de sitter phase can be obtained expanding eqs . ( [ effecden ] ) and ( [ eqstate ] ) in powers of @xmath23 , we get @xmath24 which corresponds to a mixture of vacuum energy density @xmath25 , presureless dust and other perfect fluids which dominate at the very beginning of the universe . in the intermediate regime the modified chaplygin gas behaves as dust at the time where the energy density satisfies the condition @xmath26 . at very early times the equation of state parameter @xmath27 becomes @xmath28 so that for very large @xmath29 the dust - like behavior is recovered . the next step is to investigate what sort of cosmological model arises when we consider a slight inhomogeneous modified chaplygin cosmologies . for a general metric @xmath30 , the proper time @xmath31 , and @xmath32 as the determinant of the induced @xmath33-metric , one has _ i j = g_i0g_j0 g_00 - g_ij . [ inducedmetric ] in the first approximation it will be interesting to investigate the contribution of inhomogeneities introduced in the modified chaplygin gas through the expression = ( a + ) ^ /(-1 ) . [ genevoldeninh ] the latter result suggests that the evolution of inhomogeneities can be studied using the zeldovich method through the deformation tensor @xcite : d_i^j = a(t ) ( _ i^j - b(t ) ^2 ( ) q^i q^j ) , [ defromationt ] where @xmath34 parametrizes the time evolution of the inhomogeneities and @xmath35 are generalized lagrangian coordinates so that _ i j = _ mnd_i^md_j^n , [ inducedmetric1 ] and @xmath36 is a perturbation h = 2 b(t ) _ , [ hevol ] hence , using the equations above and eqs . ( [ rhoapprox ] ) and ( [ papprox ] ) , it follows that @xmath37 where @xmath38 is given by eq . ( [ eqstate ] ) and the density contrast @xmath39 is related to @xmath36 through = h 2 ( 1 + w ) , [ delta ] where @xmath40 . finally , after some algebra we get |p=|(w + ) . [ omega ] now , the metric ( [ inducedmetric1 ] ) leads to the following @xmath41 component of the einstein equations : - 3 a + 1 2 + h = 4 g |(1 + 3w + ( 1 + ) ) , [ einstein ] where the unperturbed part of this equation corresponds to the raychaudhuri equation - 3 a = 4 g |(1 + 3 w ) . [ raychaudhuri ] using the friedmann equation for a flat spacetime @xmath42 , one can rewrite eq . ( [ einstein ] ) as a differential equation for @xmath43 : a^2 b + ( 1 - w ) a b - ( 1 + w ) ( 1 + ) b = 0 , [ bprimes ] where the primes denote derivatives with respect to the scale - factor , @xmath8 . an expression for @xmath44 as a function of the scale - factor can be derived from eqs . ( [ eqstate ] ) and ( [ effecden ] ) : @xmath45 the latter must be conveniently recast in terms of the fractional vacuum and matter energy densities . this can be done by using @xmath46 combined with @xmath47 where @xmath48 and @xmath49 are , respectively , the current value of the hubble and scale factor , and @xmath50 and @xmath51 are , respectively , the fractional vacuum and matter energy densities today . setting @xmath52 we obtain @xmath53 and consistently @xmath54 for the modified chaplygin gas for @xmath55 ( continuous lines ) as compared with @xmath0cdm ( dashed line ) and cdm ( dashed - dotted line ) . lower curves correspond to higher values of @xmath29.,title="fig:",width=264,height=170 ] ( -222,62)@xmath56 ( -90,-6)@xmath8 we have used this expression to integrate eq . ( [ bprimes ] ) numerically , for different values of @xmath29 , and taking @xmath57 and @xmath58 @xcite . we have set @xmath59 for matter - radiation equilibrium ( while keeping @xmath60 at present ) , and our initial condition is @xmath61 . our results are shown in figures [ evbsmall ] and [ evblarge ] . we find that modified chaplygin scenarios start differing from the @xmath0cdm only recently ( @xmath62 ) and that , in any case , they yield a density contrast that closely resembles , for any value of @xmath63 , the standard cdm before the present . notice that @xmath0cdm corresponds effectively to using eq . ( [ limit ] ) and removing the factor @xmath64 in eq . ( [ bprimes ] ) . figures [ evbsmall ] and [ evblarge ] show also that , for any value of @xmath29 , @xmath43 saturates as in the @xmath0cdm case . for the modified chaplygin gas for @xmath65 ( continuous lines ) as compared with @xmath0cdm ( dashed line ) and cdm ( dashed - dotted line ) . lower curves correspond to higher values of @xmath29.,title="fig:",width=264,height=170 ] ( -224,62)@xmath56 ( -90,-6)@xmath8 for the modified chaplygin gas for @xmath66 ( continuous lines ) as compared with @xmath0cdm ( dashed line ) . lower curves correspond to higher values of @xmath29.,title="fig:",width=264,height=170 ] ( -228,62)@xmath67 ( -90,-6)@xmath8 in what regards the density contrast , @xmath39 , using eqs . ( [ hevol ] ) , ( [ delta ] ) and ( [ wparam ] ) one can deduce that the ratio between this quantity in the modified chaplygin and the @xmath0cdm scenarios is simply given by = b_mchap b_cdm , [ ratio ] and its behavior is depicted in figure [ ratiofig ] . we find that it asymptotically evolves to a constant value . now , in figure [ contrast ] , we have plotted @xmath39 as a function of @xmath8 for different values of @xmath29 . as happens in the the traditional @xcite and generalized chaplygin models , in our models the density contrast decays for large @xmath8 also . for the modified chaplygin gas for @xmath68 ( continuous lines ) as compared with @xmath0cdm ( dashed line ) . lower curves correspond to higher values of @xmath29.,title="fig:",width=264,height=170 ] ( -228,62)@xmath69 ( -90,-6)@xmath8 using a zeldovich - like approximation , we have studied the evolution of large - scale perturbations in a recently proposed theoretical framework for the unification of dark matter and dark energy : the so - called modified chaplygin cosmologies @xcite , with equation of state @xmath70 with @xmath71 . this model evolves from a phase that is initially dominated by non - relativistic matter to a phase that is asymptotically de sitter . the intermediate regime corresponds to a phase where the effective equation of state is given by @xmath72 plus a cosmological constant . we have estimated the fate of the inhomogeneities admitted in the model and shown that these evolve consistently with the observations as the density contrast they introduce is smaller than the one typical of cdm scenarios . on general grounds , the pattern of evolution of perturbations follows is similar to the one in the @xmath0cdm models and in generalized chaplygin cosmologies , and therefore our represent plausible alternatives alternatives as usual , in modified chaplygin cosmologies , the equation of state parameter @xmath44 can be expressed in terms of the scale factor and a free parameter @xmath29 , and the value of the latter can be chosen so that the model resembles as much as desired the @xmath0cdm model . it would be very interesting to deepen in the comparison between modified and generalized chaplygin models , particularly from the observational point of view ( as already done in @xcite ) . it would also be worth generalizing our study by going beyond the zeldovich approximation , to incorporate the effects of finite sound speed . this can be done by generalizing the spherical model to incorporate the jeans length as in @xcite . alternatively , following @xcite one could investigate whether the modified chaplygin admits an unique decomposition into dark energy and dark matter , and if that were the case then study structure formation and show that difficulties associated to unphysical oscillations or blow - up in the matter power spectrum can be circumvented . we hope this will be addressed in future works . is partially funded by the university of buenos aires under project x224 , and the consejo nacional de investigaciones cientficas y tcnicas under proyect 02205 . is supported by the university of the basque country through research grant upv00172.310 - 14456/2002 and by the spanish ministry of education and culture through research grant fis2004 - 01626 . b. ratra and p.j.e . peebles , ( 1988 ) 3406 ; ( 1988 ) 117 ; c. wetterich , ( 1988 ) 668 ; j. a. frieman , c. t. hill , a. stebbins and i. waga , phys . rev . * 75 * ( 1995 ) 2077 ; k. coble , s. dodelson and j. a. frieman , phys . d * 55 * ( 1997 ) 1851 ; m. s. turner and m. j. white , phys . d * 56 * ( 1997 ) 4439 ; p. g. ferreira and m. joyce , phys . * 79 * ( 1997 ) 4740 ; p. g. ferreira and m. joyce , phys . d * 58 * ( 1998 ) 023503 ; e. j. copeland , a. r. liddle and d. wands , phys . d * 57 * ( 1998 ) 4686 ; r.r . caldwell and r. dave , p.j . steinhardt , ( 1998 ) 1582 ; p.g . ferreira and m. joyce , ( 1998 ) 023503 ; l. m. wang and p. j. steinhardt , astrophys . j. * 508 * ( 1998 ) 483 ; i. zlatev , l. wang , and p.j . steinhardt , ( 1999 ) 986 ; p. j. steinhardt , l. m. wang and i. zlatev , phys . d * 59 * ( 1999 ) 123504 ; i. zlatev and p. j. steinhardt , phys . b * 459 * ( 1999 ) 570 ; p. bintruy , ( 1999 ) 063502 ; j.e . kim , 9905 ( 1999 ) 022 ; m.c . bento and o. bertolami , ( 1999 ) 1461 ; j.p . uzan , ( 1999 ) 123510 ; t. chiba , ( 1999 ) 083508 ; l. amendola , ( 1999 ) 043501 . y. fujii , ( 2000 ) 023504 ; a. masiero , m. pietroni and f. rosati , ( 2000 ) 023504 ; m.c . bento , o. bertolami , and n. santos , ( 2002 ) 067301 . r. r. caldwell , phys . b * 545 * ( 2002 ) 23 ; a. e. schulz and m. j. white , phys . d * 64 * ( 2001 ) 043514 ; b. mcinnes , jhep * 0208 * ( 2002 ) 029 ; r. vaas , bild . * 2003n8 * ( 2003 ) 52 ; r. r. caldwell , m. kamionkowski and n. n. weinberg , phys . * 91 * ( 2003 ) 071301 ; j. g. hao and x. z. li , phys . d * 67 * ( 2003 ) 107303 ; g. w. gibbons , hep - th/0302199 ; x. z. li and j. g. hao , phys . d * 69 * ( 2004 ) 107303 ; s. nojiri and s. d. odintsov , phys . b * 562 * ( 2003 ) 147 ; s. nojiri and s. d. odintsov , phys . b * 565 * ( 2003 ) 1 ; a. yurov , astro - ph/0305019 ; p. singh , m. sami and n. dadhich , phys . d * 68 * ( 2003 ) 023522 ; j. g. hao and x. z. li , phys . d * 68 * ( 2003 ) 043501 ; p. f. gonzlez - daz , phys . d * 68 * ( 2003 ) 021303 ; s. nojiri and s. d. odintsov , phys . b * 571 * ( 2003 ) 1 ; m. p. dabrowski , t. stachowiak and m. szydlowski , phys . d * 68 * ( 2003 ) 103519 ; d. j. liu and x. z. li , phys . d * 68 * ( 2003 ) 067301 ; l. p. chimento and r. lazkoz , * 91 * ( 2003 ) 211301 ; j. g. hao and x. z. li , phys . d * 70 * ( 2004 ) 043529 ; e. elizalde and j. q. hurtado , mod . a * 19 * ( 2004 ) 29 ; h. stefancic , phys . b * 586 * ( 2004 ) 5 ; v. b. johri , phys . d * 70 * ( 2004 ) 041303 ; j. m. cline , s. y. jeon and g. d. moore , phys . d * 70 * ( 2004 ) 043543 ; m. sami and a. toporensky , mod . . lett . a * 19 * ( 2004 ) 1509 ; h. q. lu , hep - th/0312082 ; h. stefancic , eur . j. c * 36 * ( 2004 ) 523 ; p. f. gonzlez - daz , phys . b * 586 * ( 2004 ) 1 ; y. s. piao and y. z. zhang , astro - ph/0401231 ; p. f. gonzlez - daz , phys . d * 69 * ( 2004 ) 063522 ; m. szydlowski , w. czaja and a. krawiec , astro - ph/0401293 ; e. babichev , v. dokuchaev and y. eroshenko , phys . * 93 * ( 2004 ) 021102 ; j. m. aguirregabiria , l. p. chimento and r. lazkoz , phys . rev . d * 70 * ( 2004 ) 023509 ; j. cepa , astro - ph/0403616 ; j. g. hao and x. z. li , phys.lett . b*606 * ( 2005 ) 7 ; z. k. guo , y. s. piao and y. z. zhang , phys . b * 594 * ( 2004 ) 247 ; m. bouhmadi - lopez and j. a. jimnez madrid , astro - ph/0404540 ; v. faraoni , phys . rev . d * 69 * ( 2004 ) 123520 ; e. elizalde , s. nojiri and s. d. odintsov , phys . d * 70 * ( 2004 ) 043539 ; j. e. lidsey , phys . d * 70 * ( 2004 ) 041302 ; m. g. brown , k. freese and w. h. kinney , astro - ph/0405353 ; l. p. chimento and r. lazkoz , astro - ph/0405518 ; p. f. gonzlez - daz and j. a. jimnez - madrid , phys . b * 596 * ( 2004 ) 16 ; p. f. gonzlez - daz , hep - th/0408225 ; a. vikman , phys.rev . d * 71 * ( 2005 ) 023515 ; r. koley and s. kar , mod . a * 20 * ( 2005 ) 363 ; p. f. gonzalez - daz and c. l. siguenza , nucl . b * 697 * ( 2004 ) 363 ; p. x. wu and h. w. yu , astro - ph/0407424 ; w. fang , h. q. lu , z. g. huang and k. f. zhang , hep - th/0409080 ; l. amendola , phys.rev.lett . * 93 * ( 2004 ) 181102 ; y. h. wei , gr - qc/0410050 ; s. nesseris and l. perivolaropoulos , phys.rev . d * 70 * ( 2004 ) 123529 ; r. j. scherrer , phys . rev . d * 71 * ( 2005 ) 063519 ; w. hu , phys . d * 71 * ( 2005 ) 047301 ; e. majerotto , d. sapone and l. amendola , astro - ph/0410543 ; z. k. guo , y. s. piao , x. zhang and y. z. zhang , phys . b * 608 * ( 2005 ) 177 . v. k. onemli and r. p. woodard , class . * 19 * ( 2002 ) 4607 ; v. k. onemli and r. p. woodard , phys . d * 70 * ( 2004 ) 107301 . m. c. bento , o. bertolami and a. a. sen , phys . d * 66 * ( 2002 ) 043507 . a. kamenshchik , u. moschella , v. pasquier , ( 2001 ) 265 . l. p. chimento , phys . d * 69 * ( 2004 ) 123517 . @xcite t. barreiro and a. a. sen , phys . d * 70 * ( 2004 ) 124013 . g. r. dvali , g. gabadadze and m. porrati , phys . b * 485 * ( 2000 ) 208 ; c. deffayet , phys . b * 502 * ( 2001 ) 199 . k. freese and m. lewis , phys . b * 540 * ( 2002 ) 1 ; p. gondolo and k. freese , phys . d * 68 * ( 2003 ) 063509 . g. dvali and m. s. turner , astro - ph/0301510 . t. barreiro and a. a. sen , phys . d * 70 * ( 2004 ) 124013 . m. c. bento , o. bertolami and a. a. sen , phys . d * 67 * ( 2003 ) 063003 ; phys . lett . b * 575 * ( 2003 ) 172 ; gen . * 35 * ( 2003 ) 2063 ; d. carturan and f. finelli , phys . d * 68 * ( 2003 ) 103501 ; l. amendola , f. finelli , c. burigana and d. carturan , jcap * 0307 * ( 2003 ) 005 . m. c. bento , o. bertolami , n. m. c. santos and a. a. sen , phys . d * 71 * ( 2005 ) 063501 ; o. bertolami , a. a. sen , s. sen and p. t. silva , mon . not . * 353 * ( 2004 ) 329 ; r. j. colistete and j. c. fabris , astro - ph/0501519 ; r. j. colistete , j. c. fabris and s. v. b. goncalves , astro - ph/0409245 ; r. j. colistete , j. c. fabris , s. v. b. goncalves and p. e. de souza , int . j. mod . d * 13 * ( 2004 ) 669 ; m. biesiada , w. godlowski and m. szydlowski , astro - ph/0403305 ; m. makler , s. quinet de oliveira and i. waga , phys . lett . b * 555 * ( 2003 ) 1 ; j. c. fabris , s. v. b. goncalves and p. e. d. souza , astro - ph/0207430 ; a. dev , d. jain and j. s. alcaniz , phys . d * 67 * ( 2003 ) 023515 ; v. gorini , a. kamenshchik and u. moschella , phys . d * 67 * ( 2003 ) 063509 j. s. alcaniz , d. jain and a. dev , phys . d * 67 * ( 2003 ) 043514 ; r. bean and o. dore , phys . d * 68 * ( 2003 ) 023515 . p. t. silva and o. bertolami , astrophys . j. * 599 * ( 2003 ) 829 ; a. dev , d. jain and j. s. alcaniz , astron . astrophys . * 417 * ( 2004 ) 847 . | we extend the homogeneous modified chaplygin cosmologies to large - scale perturbations by formulating a zeldovich - like approximation .
we show that the model interpolates between an epoch with a soft equation of state and a de sitter phase , and that in the intermediate regime its matter content is simply the sum of dust and a cosmological constant .
we then study how the large - scale inhomogeneities evolve and compare the results with cold dark matter ( cdm ) , @xmath0cdm and generalized chaplygin scenarios .
interestingly , we find that like the latter , our models resemble @xmath0cdm . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
the minimal supersymmetric standard model ( mssm ) may be described by a lagrangian containing interactions consistent with invariance under the gauge group @xmath3 and global supersymmetry plus a lagrangian containing a restricted set of soft supersymmetry breaking terms @xcite . these terms break supersymmetry while maintaining a useful property of a supersymmetric theory , namely the cancellation of quadratic divergences @xcite . the absence of these divergences is necessary in order to define the renormalized mass of a fundamental scalar , such as the higgs boson , without a fine - tuning of the cancellation between the bare mass and the scalar self - energy @xcite . the presence of fundamental scalar fields in the mssm , besides the higgs bosons , leads to the possibility that these fields may acquire non - zero vacuum expectation values ( vevs ) . since this would violate the conservation of color and/or electric charge symmetry , this leads to forbidden regions of the parameter space of the theory . we will calculate numerical estimates of the boundary of the allowed region of soft - breaking parameters using both the tree - level potential and the one - loop effective potential . many studies of the mssm mass spectrum neglect these charge / color breaking , or ccb , bounds in their analyses . previously , ccb bounds were obtained for various supersymmetric models , however no systematic numerical study of ccb constraints for a realistic approximation to the mssm using the one - loop effective potential has been done @xcite . one may assume that there are relations among the soft breaking terms , such as in the minimal supergravity model in which all scalar masses and scalar trilinear couplings are the same at the unification scale , of order @xmath4 @xcite . however we will find constraints on the soft - breaking parameters at a low - energy scale , @xmath5 , with @xmath6 . this is an indeterminate upper limit on particle masses if the mssm is to explain the gauge hierarchy problem . we will not make any assumptions about the theory near the gut scale nor the particle spectrum above @xmath5 . we will use an approximation to the mssm that includes only the top flavor supermultiplets . this follows from evidence that the top quark mass @xmath7 @xcite . we use the conventional definition @xmath8 , with @xmath9 , @xmath10 the vevs for the higgs scalar fields , @xmath11 and @xmath12 , respectively . assuming a small value for @xmath1 , near @xmath13 , gives the top quark yukawa coupling , @xmath14 . the contributions from the bottom supermultiplets may then be ignored . there are various reasons to choose these particular values of @xmath1 and to consider only the top squarks as acquiring a non - zero vev . first of all , there is an infrared quasi - fixed point in the renormalization group equation for @xmath15 which corresponds to a value @xmath16 @xcite . the mass relation m_t(m_t ) = h_t^fp(m_t)(2)vgives @xmath17 if one uses the relation between the top quark mass defined by the pole in its propagator and its running mass to first - order in the qcd gauge coupling @xcite . m_pole = m_t(m_t ) . therefore a value of @xmath1 at @xmath5 in the range @xmath18 results from a large range of @xmath19 values at the gut scale . although @xmath1 is not required to be in this range , it indicates that this is a natural choice . one motivation for considering only the top sector comes from assuming common soft - breaking parameters at the gut scale . a large value of @xmath19 causes the third generation parameters to undergo the largest change as they are evolved from @xmath20 down to @xmath5 . for this same reason , @xmath19 also gives the largest contribution to the radiative gauge symmetry breaking of @xmath21 @xcite . therefore , if one assumes that the minimum of the effective potential at energy scales @xmath22 gives zero vevs for the scalar fields , such as in the case of universality at the gut scale , as one evolves to @xmath5 the third - generation parameters undergo the largest change and the ccb constraints from third generation scalar fields will be the most restrictive . finally , as discussed in ref . @xcite , the potential barrier height for tunneling from the the symmetric vacuum at a high temperature ( @xmath23 ) , early in the expansion of the universe , to a lower ccb minimum is proportional to @xmath24 where @xmath25 is the smallest of the yukawa couplings for the slepton and squark fields that have non - zero vevs at the ccb minimum . this implies that one should consider ccb vacua in which only the higgs fields and the top squarks have non - zero vevs in order for the tunneling from the symmetric to the ccb vacuum to have occurred in a time less than the present age of the universe ( @xmath26 years ) . we use a consistent approximation to the mssm with @xmath27 as a small value near the fixed point value and interactions with the bottom quark superfields are ignored . we use all mssm interactions between the following fields @xmath11 , @xmath12 , @xmath28 , @xmath29 , @xmath30 , @xmath31 , @xmath32 , @xmath33 , @xmath34 , and @xmath35 . @xmath11 and @xmath12 are respectively the hypercharge @xmath36 and @xmath37 higgs boson doublets . the corresponding field variables with a tilde are the higgsino doublets . @xmath30 and @xmath31 are the left - handed component of the top quark and the right - handed component of the charge conjugate top quark field respectively . again , the corresponding field variables with tildes are the top squarks . @xmath34 is the gluon field and @xmath35 is the gluino field . notice that the field content in this approximation is supersymmetric . this arises from including all interactions with the top quark supermultiplet involving the parameters @xmath19 and @xmath38 . the potential in this approximation as well as the definitions of the parameters appearing in it are shown in the appendix . we use the values of the gauge couplings at the weak scale @xmath39 : @xmath40 , @xmath41 , and @xmath42 for all calculations @xcite . since we will take the renormalization scale to be @xmath43 , the running of the gauge couplings from @xmath39 to this scale is negligible . we also omit the couplings @xmath44 and @xmath45 in the calculation of the one - loop effective potential . however , we retain the terms with these couplings at tree - level since they include the quartic higgs scalar interactions responsible for the non - zero higgs vev of the standard model ( sm ) . quantum corrections may affect whether spontaneous symmetry breaking occurs in a field theory . the vevs of the scalar fields are the values of the classical fields at the minimum of the effective potential . we use the one - loop correction to the effective potential in the dimensional reduction , @xmath46 , renormalization scheme @xcite [ one - loop veff ] v_eff^1-loop = 1str , where the supertrace , @xmath47 , is over all color and spin degrees of freedom with a minus sign for fermions . @xmath48 is the square of the renormalization scale , which we take to be equal to @xmath49 . @xmath50 is the field dependent mass matrix ; for instance , for a theory with scalar fields @xmath51 _ ij = .^2v |_=_c . @xmath52 as well as @xmath53 are function of the classical fields @xmath54 . since the top yukawa coupling , @xmath19 , and the strong gauge coupling , @xmath55 , are large one may expect significant contributions to the effective potential from one - loop corrections . the one - loop effective potential is also more stable under a change of renormalization scale @xcite . since the parameters appearing in the equation for the one - loop correction to @xmath53 are the renormalized parameters we consider the resulting ccb constraints as being limits on these renormalized parameters in the @xmath56 scheme at momentum scale @xmath5 . if large logs of ratios of massive parameters appear in the effective potential , such as @xmath57 or @xmath58 , then one may , in principle , sum these using the renormalization group equation @xcite . however , this is difficult when there are multiple scalar fields and masses . we only consider masses , @xmath59 , and the renormalization scale , @xmath5 , which differ by less than two orders of magnitude . furthermore , since we are only interested in the effective potential near its minimum where @xmath60 , i.e. , the classical field near the minimum of @xmath53 is of the same order of magnitude as the masses , the logarithms appearing in @xmath53 are small . renormalization group improvement of the effective potential in this case is unnecessary . we also assume that the lagrangian parameters are real . in a field basis where the higgs fields are real and positive and @xmath19 and the scalar masses , @xmath61 , real , only one of the parameters @xmath0 , @xmath38 , or @xmath62 may be made real by redefining the fields . however , it was shown in ref . @xcite that the presence of complex phases in these parameters , greater than about @xmath63 , gives too large of a contribution to the neutron s electric dipole moment due to gluino loops . we evaluate the one - loop effective potential using eq . ( [ one - loop veff ] ) by diagonalizing the mass matrix @xmath52 . if one chooses the landau gauge for the gluons , in which the propagator is - , the generalized mass matrix @xmath52 is block diagonal in the gluon fields , bosonic fields , and fermionic fields . the top squark fields may also be rotated by a global @xmath64 transformation so that they are of the form _ i = ( c q_1r + 0 + 0 ) , _ i = ( c t_1r + t_1i + t_2r + 0 ) , with @xmath65 , @xmath66 , @xmath67 , and @xmath68 are real . we then use only the real part of the one - loop effective potential ; the imaginary part is related to the decay rate of the vacuum as shown in ref . @xcite . the global minimum of the effective potential is found by calculating the local minimum using a standard algorithm from ref . @xcite starting from a set of field values on a rectangular grid . if this global minimum occurs for non - zero values of the squark fields then there is a ccb vacuum . this process is repeated for a set of soft - breaking parameter values on a rectangular grid in parameter space . a quadratic surface is then fit to the parameter points on the boundary between those that give a symmetric vacuum and those that lead to a ccb vacuum . the surface is fit to the boundary points by varying the coefficients in the equation of the surface , @xmath69 , until the average of the distance squared from the boundary points to the nearest point on the surface , @xmath70 , is minimized . the equation of the surface is [ quadratic_surface ] a_t^4+p_1a_t^2+p_2m_2 ^ 4+p_3m_2 ^ 2 + p_4m_^4+p_5m_^2+p_6m_^4+p_7m_^2 -p_8=0 with @xmath71 a point in parameter space . these soft - breaking parameters are defined in the appendix . this particular parameterization of the surface is chosen solely because of its simplicity , but as will be seen later it gives an accurate characterization of the ccb boundary . because of the large dimension of the parameter space in which we want to find ccb bounds , it is necessary to initially constrain some of the parameters values in order to reduce the calculation time . two of the soft - breaking parameters , @xmath72 and @xmath73 , are constrained using two different methods . in method s the values of these parameters are chosen such that there is a standard model minimum in the effective potential with @xmath74 and @xmath75 , @xmath76 with @xmath77 . this is done by solving the analytical expression for the one - loop effective potential with zero vevs for the top squarks in terms of these parameters . these relations are shown in eq . ( [ one_loop_m1m3 ] ) . in the other procedure , method f , we simply fix @xmath72 and @xmath73 at constant values , i.e. , independent of the other soft - breaking parameters . we begin with a numerical analysis of the model with three real scalar fields @xmath78 , @xmath79 , and @xmath80 of ref . the potential in this model is @xmath81 \nonumber\end{aligned}\ ] ] this is a simplified version of the potential described in the appendix with @xmath82 , @xmath83 , and @xmath84 . the ccb bound [ analytical_bound ] a^2 < 3(m_2 ^ 2 + m_q^2 + m_t^2 ) , derived in ref . @xcite follows from minimizing the potential only in the equal - field direction , _ i.e. _ , @xmath85 . this is valid , in general , only if the d - term dominates , or if @xmath86 with @xmath87 the smallest gauge coupling . we perform a numerical analysis of the potential in eq . ( [ ghs_pot ] ) using the parameter value ranges shown in table [ ghs_range ] and with @xmath88 . the gauge couplings are fixed at their running values at the weak scale @xmath39 as stated earlier . the resulting best fit to the ccb boundary is shown in table [ ghs_fits ] . clearly , this is close to the analytical result of eq . ( [ analytical_bound ] ) , as it should be with a small value for @xmath89 . next we repeat the analysis for the potential of eq . ( [ ghs_pot ] ) with the same soft - breaking parameter ranges of table [ ghs_range ] , but we set @xmath90 . this is the proper value for the top quark yukawa coupling for small @xmath1 . the corresponding ccb bound is given in table [ ghs_fits ] . the ccb bound for @xmath90 is more stringent than the bound for @xmath88 , i.e. , for a given set of parameter values within the range given in table [ ghs_range ] , @xmath91 , the value of the @xmath92 parameter at which the vacuum becomes ccb is lower . the fractional difference in the @xmath92 parameter bound between these cases is largest when the other parameters , @xmath93 , are near the lower end of their range in table [ ghs_range ] , where @xmath94 . also the value of @xmath92 at the ccb bound in this case is small . the average fractional difference over the entire parameter ranges shown in table [ ghs_range ] for the two bounds is @xmath95 next we examine the ccb bounds obtained from of our approximation to the mssm with the lagrangian given in the appendix . in all of the following analyses the mass units are @xmath96 . we also fix @xmath97 and @xmath98 . the renormalization scale for the one - loop calculation of the effective potential is set at @xmath99 . we first examine the bound for method f with @xmath100 and @xmath101 . these masses result in a value of @xmath102 if a s.m . minimum were required , i.e. , zero squark vevs . these values for @xmath72 and @xmath73 were chosen since they are consistent with a s.m . minimum for @xmath97 and they are close to the renormalization scale , @xmath103 . the ranges of parameter values for which the effective potential was calculated are shown in table [ mssm_range ] . the ccb bounds for the analyses of both the tree - level and the one - loop effective potential are shown in table [ method_f_fits ] . parameter values which give a potential that is unbounded below are not used in determining the best fit surface for the ccb bound . this includes values with @xmath104 for the tree - level potential since in this case the potential is unbounded for @xmath105 and zero squark vevs . we do not examine negative @xmath106 values since in this case , for the small range of @xmath107 and @xmath108 values for which the potential is bounded below , the vevs for the c.c.b . vacuum are too small for the numerical methods to distinguish this minimum from the minimum with all vevs vanishing . however , according to eq . ( [ higgs_mass_def ] ) , the higgs soft - breaking mass , @xmath109 , is negative over part of the parameter range examined . next we calculate the ccb bounds using method s , in which @xmath72 and @xmath73 are fixed by requiring a s.m . the parameter values listed in table [ mssm_range ] were used in determining the bounds . since there is no standard model minimum for @xmath110 , we do not examine @xmath106 values in this range . the ccb bounds calculated by finding the global minimum of the both the tree - level potential and the one - loop effective potential are given in table [ method_s_fits ] . the quadratic surface of eq . ( [ quadratic_surface ] ) is sufficient to provide an accurate characterization of the numerical ccb bound . all of the least - square fits give the average distance squared , @xmath111 . the one - loop ccb bound calculations yield a larger value for @xmath112 because the longer calculation time requires using less parameter grid points than for the tree - level calculation . typically it takes around @xmath113 days of cpu time on a digital alpha workstation to calculate the one - loop ccb boundary points and perform a least - squares fit so we are limited by the computation time . as stated in section [ simplified model ] , the numerical ccb bound for the simplified lagrangian of ref . @xcite with yukawa coupling @xmath114 is significantly different from the numerical bound with @xmath115 , and hence also different from the analytical bound of eq . ( [ analytical_bound ] ) , when the other soft - breaking parameters are small . however , for the remainder of the parameter values tested , the numerical bound with @xmath114 is quite close to the analytical bound . one would not expect the ccb bound of eq . ( [ analytical_bound ] ) to be correct for large @xmath89 . one possible explanation is that the @xmath116 d - term is large enough at least to insure that the minimum is in the direction @xmath117 . also for @xmath114 the numerical bound does give a more stringent ccb bound than that of eq . ( [ analytical_bound ] ) over the entire range of parameters tested . we present some contour plots of the ccb bounds with @xmath118 for comparison . the contours show the value of @xmath0 on the ccb boundary , i.e. , lower values of @xmath0 result in a symmetric vacuum whereas higher values result in a ccb vacuum with nonzero squark vevs . figs . [ tree_f_fig ] and [ loop_f_fig ] show the bounds for method f using the tree - level and one - loop effective potential respectively [ tree_s_fig ] and [ loop_s_fig ] show the tree - level and one - loop method s ccb bounds . fig . [ analytical_fig ] shows the analytical bound of eq . ( [ analytical_bound ] ) with @xmath118 . since there is not the additional constraint of requiring a standard model minimum for the higgs field for both the method f ccb bounds and for the analytical bound , one may compare these ccb bounds . the tree - level ccb bound on the @xmath0 parameter for method f is lower than the analytical bound of eq . ( [ analytical_bound ] ) for the entire range of parameter values , @xmath91 , shown in table [ mssm_range ] . the one - loop correction to the effective potential raises the value of @xmath0 for the ccb bound over about @xmath119 of the parameter range considered . however , even with the one - loop corrections , the ccb bound for the mssm potential is more stringent , i.e. , gives a lower value for @xmath0 , than the bound of eq . ( [ analytical_bound ] ) for @xmath120 of the parameter range . the one - loop corrections for the ccb bound calculated using method s give a lower @xmath0 value than the tree - level bound over the entire range of parameters examined . for values of the parameters , @xmath121 , that give a small value for the @xmath0 parameter ccb bound , the one - loop corrections to the effective potential makes the ccb bound significantly stricter . in conclusion , ccb bounds on the soft - breaking parameters of the higgs and top quark / squark sectors of the mssm provide important constraints for these parameters . these constraints may be expressed as a maximum value of the @xmath0 parameter for given values of the remaining soft - breaking parameters . the numerical ccb constraints that we calculated give more stringent ccb bounds than the analytical constraint of eq . ( [ analytical_bound ] ) for most of the ranges of parameter values considered . because of the large top yukawa coupling , one - loop corrections to the effective potential may result in significantly different ccb bounds than those for the tree - level potential the author wishes to thank y. okada for suggesting this topic and b. wright for many useful discussions . this work was partially supported by the japan society for the promotion of science . the potential may be divided into several parts . a sum over group and spinor indices , where applicable , is implied . the supersymmetric d - terms are @xmath122 where @xmath44 , @xmath45 , and @xmath55 are respectively the @xmath123 , @xmath124 , and @xmath64 couplings , @xmath125 are the pauli matrices and @xmath126 are the antihermitian generators of @xmath64 . using the relations _ ij^a_kl^a = 2_il_jk - _ ij_kl and t_ij^at_kl^a = ( _ il_jk -_ij_kl ) the @xmath124 contribution becomes @xmath127 \nonumber\end{aligned}\ ] ] and the @xmath64 one is v_su(3 ) = g_3 ^ 2 . the superpotential or f term is v_f = h_t^2(||^2|h_2 ^ 0|^2 + ||^2|h_2 ^ 0|^2 + ||^2 ) + h_t(h_1 ^ 0 * ) + h.c . , with @xmath128 denoting the hermitian conjugate and @xmath129 and @xmath130 are the neutral components of the higgs scalar doublets . the higgs scalar and fermion doublet components are h_1 = ( ) . the quark - squark - gluino interaction terms are v_q = g_3(p_l^(a)t^a^ * -t^a^(a ) p_rt + ^*t^ap_lt - t^ap_r^(a ) ) . @xmath131 are the projection operators for left- and right - handed chiral spinors , @xmath132 , @xmath133 is the four component spinor field for the top quark , and @xmath134 are the majorana gluino fields . the quark - squark - higgsino terms are v_q = h_t(|_2 ^ 0p_lt + ^*p_r_2 ^ 0 - p_l_2 ^ 0 - ^*|_2 ^ 0p_rt ) . the higgsino interaction terms are v _ = ( _ 1 ^ 0_2 ^ 0 - _ 1 ^ -_2^+ ) + h.c . finally the susy soft - breaking terms are @xmath135 with the addition of the supersymmetric higgs interactions , the masses for @xmath11 and @xmath12 become @xmath136 respectively . if the squark vevs are zero the one - loop contributions to the higgs effective potential from top squark and quark loops may be written in an analytical form @xcite . after requiring that the minimum of the effective potential be at @xmath137 and @xmath138 and solving for @xmath72 and @xmath73 we obtain @xmath139 , \nonumber \\ m_{1}^{2 } & = & m_{3}^{2}\tan\beta - m_{z}^{2}\cos 2\beta \nonumber \\ & + & { 3\over{16\pi^{2}}}h_{t}^{2}\mu(a_{t}\tan\beta + \mu ) { { f(m_{1}^{'2})-f(m_{2}^{'2})}\over{m_{2}^{'2}-m_{1}^{'2 } } } , \nonumber\end{aligned}\ ] ] with @xmath140 . the definition @xmath141 and the tree - level z boson mass , @xmath142 , were used . the tree - level relation follows by including only the first term in the above equations for @xmath72 and @xmath73 . p. fayet , phys . b * 69 * ( 1977 ) 489 ; j. rosiek , phys . rev . d * 41 * ( 1990 ) 3464 . l. girardello , and m.t . grisaru , nucl . * b194 * ( 1982 ) 65 . nilles , phys . rep . * 110 * ( 1984 ) 1 . frere , d.r.t . jones , s. raby , nucl . * 222 * ( 1983 ) 11 ; l. alvarez - gaum , j. polchinski , m.b . wise , _ ibid . _ * 221 * ( 1983 ) 495 ; c. kounnas , a.b . lahanas , d.v . nanopoulos , m. quiros , _ ibid . _ * 236 * ( 1984 ) 438 ; j.p . derendinger and c.a . savoy , _ ibid . _ * 237 * ( 1984 ) 307 ; h. komatsu , phys . b * 215 * ( 1988 ) 323 ; p. langacker and n. polonsky , phys . d * 50 * ( 1994 ) 2199 . gunion , h.e . haber , and m. sher , nucl . b306 * ( 1988 ) 1 . l. alvarez - gaum and j. polchinski , nucl . phys . * b221 * ( 1983 ) 495 . cdf collaboration , phys . * 73 * ( 1994 ) 225 ; _ ibid . _ , `` observation of top quark production in pp collisions with the cdf detector at fermilab '' , preprint fermilab - pub-95/022-e , hep - ex/9503002 . b. pendleton and g.g . ross , phys . b * 98 * ( 1981 ) 291 ; c.t . hill , phys . b * 98 * ( 1981 ) 291 ; v. barger , m. berger , p. ohmann , and r.j.n . phillips , phys . lett . b * 314 * ( 1993 ) 351 . r. tarrach , nucl . phys . * b183 * ( 1981 ) 384 . ibez and g.g . ross , phys . b * 110 * ( 1982 ) 215 ; k. inoue , a. kakuto , h. komatsu , and s. takeshita , prog . * 68 * ( 1982 ) 96 ; l. alvarez - gaum , m. claudson , and m.b . wise , nucl . b207 * ( 1982 ) 96 . g. gamberini , g. ridolfi , and f. zwirner , nucl . b331 * ( 1990 ) 331 . m. claudson , j. hall , and i. hinchliffe , nucl . b228 * ( 1983 ) 501 . particle data group , phys . d * 50 * ( 1994 ) 1173 . w. siegel , phys . b * 84 * ( 1979 ) 193 ; d.m . capper , d.r.t . jones , and p. nieuwenhuizen , nucl b167 * ( 1980 ) 479 . m.b . einhorn and d.r.t . jones , nucl . b230 * ( 1984 ) 261 ; h. nakano and y. yoshida , phys . d * 49 * ( 1994 ) 5393 ; c. ford , _ ibid . _ 50 ( 1994 ) 7531 . j. polchinski and m. wise , phys . b * 125 * ( 1983 ) 393 . e. weinberg and a. wu , phys . d * 36 * ( 1987 ) 2474 . w. press , b. flannery , s. teukolsky , and w. vetterling , _ numerical recipes - the art of scientific computing _ , ( cambridge university press , cambridge , 1986 ) . j. ellis , g. ridolfi , and f. zwirner , phys . b * 257 * ( 1991 ) 83 ; j. ellis _ _ , _ ibid . _ * 262 * ( 1991 ) 477 ; a. brignole , j. ellis , g. ridolfi , f. zwirner , _ ibid . _ * 271 * ( 1991 ) 123 . c|dd parameter & & + @xmath145 & 0.7246 & 2.723 + @xmath146 & 0.1608 & 3.527 + @xmath147 & -3.808 & -8.310 + @xmath148 & 0.1991 & 3.370 + @xmath149 & -3.830 & -8.228 + @xmath150 & 0.1985 & 3.452 + @xmath151 & -3.836 & -8.310 + @xmath152 & 0.4911 & 0.5877 + @xmath153 & 2.@xmath154 & 4.@xmath155 cdddd parameter & min . value & max . value & tree - level grid spacing & one - loop grid spacing + @xmath0 & 0.0 & 3.0 & 0.1 & 0.4 + @xmath106 & 2.@xmath156 & 0.7525 & 0.125 & 0.25 + @xmath107 & 2.@xmath156 & 0.7525 & 0.125 & 0.25 + @xmath108 & 2.@xmath156 & 0.7525 & 0.125 & 0.25 c|dd parameter & & + @xmath145 & 6.@xmath157 & 5.@xmath158 + @xmath146 & 3.@xmath159 & 1.@xmath160 + @xmath147 & -3.@xmath161 & -4.@xmath162 + @xmath148 & 6.@xmath163 & 6.@xmath164 + @xmath149 & -1.@xmath165 & -1.@xmath166 + @xmath150 & 6.@xmath167 & 7.@xmath168 + @xmath151 & -1.@xmath169 & -1.@xmath170 + @xmath152 & -1.@xmath171 & -1.@xmath172 + @xmath153 & 1.@xmath173 & 4.@xmath174 c|dd parameter & & + @xmath145 & 4.@xmath175 & 3.@xmath176 + @xmath146 & 9.@xmath177 & -1.@xmath178 + @xmath147 & -1.@xmath179 & 5.@xmath180 + @xmath148 & 3.@xmath181 & 7.@xmath182 + @xmath149 & -6.@xmath183 & -9.@xmath184 + @xmath150 & 3.@xmath185 & -7.@xmath186 + @xmath151 & -6.@xmath187 & 3.@xmath188 + @xmath152 & -2.@xmath189 & -2.@xmath190 + @xmath153 & 1.@xmath191 & 9.@xmath192 | we calculate limits on the trilinear soft - breaking parameter , @xmath0 , in the minimal supersymmetric standard model by requiring the absence of nonzero top squark vacuum expectation values . assuming a low @xmath1 , which implies a large top yukawa coupling
, we also calculate one - loop corrections to the effective potential .
the resulting numerical calculations of the charge / color breaking limits are presented as best - fit surfaces .
we compare these results with the analytical limit , @xmath2 , and find that although this is a good estimate of the charge / color breaking bounds for a simplified model of the top sector , stricter bounds are found by a numerical minimization of the minimal supersymmetric standard model potential . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
for a range of scientific studies including quantum computation , quantum information and quantum simulation , an important task is to learn and engineer quantum systems [ @xcite , @xcite ( @xcite , @xcite ) , @xcite , @xcite,@xcite , @xcite , and wang ( @xcite , @xcite ) ] . a quantum system is described by its state characterized by a density matrix , which is a positive semidefinite hermitian matrix with unit trace . determining a quantum state , often referred to as quantum state tomography , is an important but difficult task [ @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , and @xcite ] . it is often inferred by performing measurements on a large number of identically prepared quantum systems . more specifically , we describe a quantum spin system by the @xmath0-dimensional complex space @xmath1 and its quantum state by a complex matrix on @xmath1 . when measuring the quantum system by performing measurements on some observables which can be represented by hermitian matrices , we obtain the measurement outcomes for each observable , where the measurements take values at random from all eigenvalues of the observable , with the probability of observing a particular eigenvalue equal to the trace of the product of the density matrix and the projection matrix onto the eigenspace corresponding to the eigenvalue . to handle the up and down states of particles in a quantum spin system , a common approach is to employ the well - known pauli matrices as observables to perform measurements and obtain the so - called pauli measurements [ @xcite , @xcite , @xcite , @xcite , @xcite , and wang ( @xcite , @xcite ) ] . since all pauli matrices have @xmath2 eigenvalues , pauli measurements takes discrete values @xmath3 and @xmath4 , and the resulted measurement distributions can be characterized by binomial distributions . the goal is to estimate the density matrix based on the pauli measurements . traditional quantum tomography employs classical statistical models and methods to deduce quantum states from quantum measurements . these approaches are designed for the setting where the size of a density matrix is greatly exceeded by the number of quantum measurements , which is almost never the case even for moderate quantum systems in practice because the dimension of the density matrix grows exponentially in the size of the quantum system . for example , the density matrix for @xmath5 spin-@xmath6 quantum systems is of size @xmath7 . in this paper , we aim to effectively and efficiently reconstruct the density matrix for a large - scale quantum system with a relatively limited number of quantum measurements . quantum state tomography is fundamentally connected to the problem of recovering a high - dimensional matrix based on noisy observations [ @xcite ] . the latter problem arises naturally in many applications in statistics and machine learning and has attracted considerable recent attention . when assuming that the unknown matrix of interest is of ( approximately ) low - rank , many regularization techniques have been developed . examples include @xcite , @xcite , @xcite ( @xcite , @xcite ) , @xcite , @xcite , bunea , she and wegkamp ( @xcite , @xcite ) , klopp ( @xcite , @xcite ) , @xcite , @xcite , @xcite , @xcite , @xcite , and @xcite , among many others . taking advantage of the low - rank structure of the unknown matrix , these approaches can often be applied to estimate unknown matrices of high dimensions . yet these methods do not fully account for the specific structure of quantum state tomography . as demonstrated in a pioneering article , @xcite argued that , when considering quantum measurements characterized by the pauli matrices , the density matrix can often be characterized by the sparsity with respect to the pauli basis . built upon this connection , they suggested a compressed sensing [ @xcite ] strategy for quantum state tomography [ @xcite and @xcite ] . although promising , their proposal assumes exact measurements , which is rarely the case in practice , and adopts the constrained nuclear norm minimization method , which may not be an appropriate matrix completion approach for estimating a density matrix with unit trace ( or unit nuclear norm ) . we specifically address such challenges in the present paper . in particular , we establish the minimax optimal rates of convergence for the density matrix estimation under both the spectral and frobenius norm losses when assuming that the true density matrix is approximately sparse under the pauli basis . furthermore , we show that these rates could be achieved by carefully thresholding the coefficients with respect to the pauli basis . because the quantum pauli measurements are characterized by the binomial distributions , the convergence rates and minimax lower bounds are derived by asymptotic analysis with manipulations of binomial distributions instead of the usual normal distribution based calculations . the rest of paper proceeds as follows . section [ sec2 ] gives some background on quantum state tomography and introduces a thresholding based density matrix estimator . section [ sec3 ] develops theoretical properties for the density matrix estimation problem . in particular , the convergence rates of the proposed density matrix estimator and its minimax optimality with respect to both the spectral and frobenius norm losses are established . section [ sec4 ] features a simulation study to illustrate finite sample performance of the proposed estimators . all technical proofs are collected in section [ proofs ] . in this section , we first review the quantum state and density matrix and introduce pauli matrices and pauli measurements . we also develop results to describe density matrix representations through pauli matrices and characterize the distributions of pauli measurements via binomial distribution before introducing a thresholding based density matrix estimator . for a @xmath0-dimensional quantum system , we describe its quantum state by a density matrix @xmath8 on @xmath0 dimensional complex space @xmath1 , where density matrix @xmath8 is a @xmath0 by @xmath0 complex matrix satisfying ( 1 ) hermitian , that is , @xmath8 is equal to its conjugate transpose ; ( 2 ) positive semidefinite ; ( 3 ) unit trace , that is , @xmath9 . for a quantum system , it is important but difficult to know its quantum state . experiments are conducted to perform measurements on the quantum system and obtain data for studying the quantum system and estimating its density matrix . in physics literature , quantum state tomography refers to reconstruction of a quantum state based on measurements for the quantum systems . statistically , it is the problem of estimating the density matrix from the measurements . common quantum measurements are on observable @xmath10 , which is defined as a hermitian matrix on @xmath1 . assume that the observable @xmath10 has the following spectral decomposition : @xmath11 where @xmath12 are @xmath13 different real eigenvalues of @xmath10 , and @xmath14 are projections onto the eigenspaces corresponding to @xmath12 . for the quantum system prepared in state @xmath8 , we need a probability space @xmath15 to describe measurement outcomes when performing measurements on the observable @xmath10 . denote by @xmath16 the measurement outcome of @xmath10 . according to the theory of quantum mechanics , @xmath16 is a random variable on @xmath15 taking values in @xmath17 , with probability distribution given by @xmath18 we may perform measurements on an observable for a quantum system that is identically prepared under the state and obtain independent and identically distributed observations . see @xcite , @xcite , and @xcite . the pauli matrices as observables are widely used in quantum physics and quantum information science to perform quantum measurements . let @xmath19 where @xmath20 , @xmath21 and @xmath22 are called the two - dimensional pauli matrices . tensor products are used to define high - dimensional pauli matrices . let @xmath23 for some integer @xmath5 . we form @xmath5-fold tensor products of @xmath24 , @xmath20 , @xmath21 and @xmath22 to obtain @xmath0 dimensional pauli matrices @xmath25 we identify index @xmath26 with @xmath27 . for example , @xmath28 corresponds to @xmath29 . with the index identification we denote by @xmath30 the pauli matrix @xmath31 , with @xmath32 . we have the following theorem to describe pauli matrices and represent a density matrix by pauli matrices . [ thm-1 ] pauli matrices @xmath33 are of full rank and have eigenvalues @xmath2 . denote by @xmath34 the projections onto the eigen - spaces of @xmath30 corresponding to eigenvalues @xmath2 , respectively . then for @xmath35 , @xmath36 denote by @xmath37 the space of all @xmath0 by @xmath0 complex matrices equipped with the frobenius norm . all pauli matrices defined by ( [ pauli - matrix ] ) form an orthogonal basis for all complex hermitian matrices . given a density matrix @xmath8 , we can expand it under the pauli basis as follows : @xmath38 where @xmath39 are coefficients . for @xmath40 , @xmath41 suppose that an experiment is conducted to perform measurements on pauli observable @xmath30 independently for @xmath42 quantum systems which are identically prepared in the same quantum state @xmath8 . as @xmath30 has eigenvalues @xmath2 , the pauli measurements take values @xmath3 and @xmath4 , and thus the average of the @xmath42 measurements for each @xmath30 is a sufficient statistic . denote by @xmath43 the average of the @xmath42 measurement outcomes obtained from measuring @xmath30 , @xmath44 . our goal is to estimate @xmath8 based on @xmath45 . the following proposition provides a simple binomial characterization for the distributions of @xmath43 . [ thm-2 ] suppose that @xmath8 is given by ( [ beta - representation ] ) . then @xmath45 are independent with @xmath46 and @xmath47 follows a binomial distribution with @xmath42 trials and cell probabilities @xmath48 , where @xmath49 denotes the projection onto the eigenspace of @xmath30 corresponding to eigenvalue @xmath3 , and @xmath39 is the coefficient of @xmath30 in the expansion of @xmath8 in ( [ beta - representation ] ) . since the dimension of a quantum system grows exponentially with its components such as the number of particles in the system , the matrix size of @xmath8 tends to be very large even for a moderate quantum system . we need to impose some structure such as sparsity on @xmath8 in order to make it consistently estimable . suppose that @xmath8 has a sparse representation under the pauli basis , following wavelet shrinkage estimation we construct a density matrix estimator of @xmath8 . assume that representation ( [ beta - representation ] ) is sparse in a sense that there is only a relatively small number of coefficients @xmath50 with large magnitudes . formally , we specify sparsity by assuming that coefficients @xmath51 satisfy @xmath52 where @xmath53 , and @xmath54 is a deterministic function with slow growth in @xmath0 such as @xmath55 . pauli matrices are used to describe the spins of spin-@xmath6 particles along different directions , and density matrix @xmath8 in ( [ beta - representation ] ) represents a mixture of quantum states with spins along many directions . sparsity assumption ( [ csparsity ] ) with @xmath56 indicates the mixed state involving spins along a relatively small number of directions corresponding to those pauli matrices with nonzero @xmath50 . the sparsity reduces the complexity of mixed states . sparse density matrices often occur in quantum systems where particles have sparse interactions such as location interactions . examples include many quantum systems in quantum information and quantum computation [ @xcite , @xcite , @xcite , @xcite , @xcite , and wang ( @xcite , @xcite ) ] . since @xmath57 are independent , and @xmath58 . we naturally estimate @xmath50 by @xmath57 and threshold @xmath57 to estimate large @xmath50 , ignoring small @xmath59 , and obtain @xmath60 \label{threshold1 } \\[-8pt ] \nonumber \hat{\beta } _ k & = & \operatorname{sign}(n_k ) \bigl(|n_k|- \varpi\bigr)_+ , \qquad k = 2 , \ldots , d^2,\end{aligned}\ ] ] and then we use @xmath61 to construct the following estimator of @xmath8 , @xmath62 where the two estimation methods in ( [ threshold1 ] ) are called hard and soft thresholding rules , and @xmath63 is a threshold value which , we reason below , can be chosen to be @xmath64 for some constant @xmath65 . the threshold value is designed such that for small @xmath50 , @xmath57 must be bounded by threshold @xmath63 with overwhelming probability , and the hard and soft thresholding rules select only those @xmath57 with large signal components @xmath50 . as @xmath66 , an application of bernstein s inequality leads to that for any @xmath67 , @xmath68 and @xmath69^{d^2 - 1 } = \bigl [ 1 - 2 d^{-2 \hbar/(1+o(1 ) ) } \bigr]^{d^2 - 1 } \rightarrow1 , \ ] ] as @xmath70 and @xmath71 , that is , with probability tending to one , @xmath72 uniformly for @xmath73 . thus , we can select @xmath64 to threshold @xmath57 and obtain @xmath61 in ( [ threshold1 ] ) . we fix matrix norm notation for our asymptotic analysis . let @xmath74 be a @xmath0-dimensional vector and @xmath75 be a @xmath0 by @xmath0 matrix , and define their @xmath76 norms @xmath77 denote by @xmath78 the frobenius norm of @xmath79 . for the case of matrix , the @xmath80 norm is called the matrix spectral norm or operator norm . @xmath81 is equal to the square root of the largest eigenvalue of @xmath82 , @xmath83 and @xmath84 for a real symmetric or complex hermitian matrix @xmath79 , @xmath85 is equal to the largest absolute eigenvalue of @xmath79 , @xmath86 is the square root of the sum of squared eigenvalues , @xmath87 , and ( [ ell-1infty - norm])([ell-12infty - norm ] ) imply that @xmath88 . the following theorem gives the convergence rates for @xmath89 under the spectral and frobenius norms . [ thm-3 ] denote by @xmath90 the class of density matrices satisfying the sparsity condition ( [ csparsity ] ) . assume @xmath91 for some constants @xmath92 and @xmath93 . for density matrix estimator @xmath89 defined by ( [ threshold1])([threshold - estimator1 ] ) with threshold @xmath94 for some constant @xmath65 , we have @xmath95 & \leq & c_2 \pi^2_n(d ) \frac{1}{d^2 } \biggl ( \frac { \log d } { n } \biggr)^{1-q } , \\ \sup_{{\bolds{\rho}}\in\theta } e\bigl[\| \hat{{\bolds{\rho } } } - { \bolds{\rho}}\|_f^2 \bigr ] & \leq & c_3 \pi_n(d ) \frac{1}{d } \biggl ( \frac { \log d } { n } \biggr)^{1-q/2},\end{aligned}\ ] ] where @xmath96 and @xmath97 are constants free of @xmath42 and @xmath0 . [ rem1 ] theorem [ thm-3 ] shows that @xmath89 achieves the convergence rate @xmath98 under the squared frobenius norm loss and the convergence rate @xmath99 under the squared spectral norm loss . both rates will be shown to be optimal in the next section . similar to the optimal convergence rates for large covariance and volatility matrix estimation [ @xcite and @xcite ] , the optimal convergence rates here have factors involving @xmath54 and @xmath100 . however , unlike the covariance and volatility matrix estimation case , the convergence rates in theorem [ thm-3 ] have factors @xmath101 and @xmath102 for the squared spectral and frobenius norms , respectively , and go to zero as @xmath0 approaches to infinity . in particular , the result implies that mses of the proposed estimator get smaller for large @xmath0 . this is quite contrary to large covariance and volatility matrix estimation where the traces are typically diverge , the optimal convergence rates grow with the logarithm of matrix size , and the corresponding mses increase in matrix size . the new phenomenon may be due to the unit trace constraint on density matrix and that the density matrix representation ( [ beta - representation ] ) needs a scaling factor @xmath101 to satisfy the constraint . also for finite sample @xmath89 may not be positive semidefinite , we may project @xmath89 onto the cone formed by all density matrices under a given matrix norm @xmath103 , and obtain a positive semidefinite density matrix estimator @xmath104 . since the underlying true density matrix @xmath8 is positive semidefinite with unit trace , and the representation ( [ threshold - estimator1 ] ) ensures that @xmath89 has unit trace , the projection implies @xmath105 . thus , @xmath106 . taking @xmath103 as the spectral norm or the frobenius norm and using theorem [ thm-3 ] , we conclude that @xmath104 has the same convergence rates as @xmath89 . the following theorem establishes a minimax lower bound for estimating @xmath8 under the spectral norm . [ thm-4 ] we assume that @xmath54 in the sparsity condition ( [ csparsity ] ) satisfies @xmath107 for some constant @xmath108 and @xmath109 . then @xmath110 \geq c_4 \pi^2_n(d ) \frac { 1}{d^2 } \biggl ( \frac { \log d } { n } \biggr)^{1-q},\ ] ] where @xmath111 denotes any estimator of @xmath8 based on measurement data @xmath45 , and @xmath112 is a constant free of @xmath42 and @xmath0 . [ rem2 ] the lower bound in theorem [ thm-4 ] matches the convergence rate of @xmath89 under the spectral norm in theorem [ thm-3 ] , so we conclude that @xmath89 achieves the optimal convergence rate under the spectral norm . to establish the minimax lower bound in theorem [ thm-4 ] , we construct a special subclass of density matrices and then apply le cam s lemma . assumption ( [ cond - pi ] ) is needed to guarantee the positive definiteness of the constructed matrices as density matrix candidates and to ensure the boundedness below from zero for the total variation of related probability distributions in le cam s lemma . assumption ( [ cond - pi ] ) is reasonable in a sense that if the right - hand side of ( [ cond - pi ] ) is large enough , ( [ cond - pi ] ) will not impose very restrictive condition on @xmath54 . we evaluate the dominating factor @xmath113 on the right - hand side of ( [ cond - pi ] ) for various scenarios . first , consider @xmath56 , the assumption becomes @xmath114 , @xmath115 , and so assumption ( [ cond - pi ] ) essentially requires @xmath54 grows in @xmath0 not faster than @xmath116 , which is not restrictive at all as @xmath54 usually grows slowly in @xmath0 . the asymptotic analysis of high - dimensional statistics usually allows both @xmath0 and @xmath42 go to infinity . typically , we may assume @xmath0 grows polynomially or exponentially in @xmath42 . if @xmath0 grows exponentially in @xmath42 , that is , @xmath117 for some @xmath118 , then @xmath119 is negligible in comparison with @xmath120 , and @xmath113 behavior like @xmath120 . the assumption in this case is not very restrictive . for the case of polynomial growth , that is , @xmath121 for some @xmath122 , then @xmath123 . if @xmath124 , @xmath125 grows in @xmath0 like some positive power of @xmath0 . since we may take @xmath126 arbitrarily close to @xmath127 , the positiveness of @xmath128 essentially requires @xmath129 , which can often be quite realistic given that @xmath130 is usually very small . the theorem below provides a minimax lower bound for estimating @xmath8 under the frobenius norm . [ thm-5 ] we assume that @xmath54 in the sparsity condition ( [ csparsity ] ) satisfies @xmath131 for some constants @xmath132 and @xmath133 . then @xmath134 \geq c_5 \pi_n(d ) \frac{1}{d } \biggl ( \frac { \log d } { n } \biggr)^{1-q/2},\ ] ] where @xmath111 denotes any estimator of @xmath8 based on measurement data @xmath45 , and @xmath135 is a constant free of @xmath42 and @xmath0 . [ rem3 ] the lower bound in theorem [ thm-5 ] matches the convergence rate of @xmath89 under the frobenius norm in theorem [ thm-3 ] , so we conclude that @xmath89 achieves the optimal convergence rate under the frobenius norm . similar to the remark [ rem2 ] after theorem [ thm-4 ] , we need to apply assouad s lemma to establish the minimax lower bound in theorem [ thm-5 ] , and assumption ( [ cond - pi-0 ] ) is used to guarantee the positive definiteness of the constructed matrices as density matrix candidates and to ensure the boundedness below from zero for the total variation of related probability distributions in assouad s lemma . also the appropriateness of ( [ cond - pi-0 ] ) is more relaxed than ( [ cond - pi ] ) , as @xmath136 and the right - hand side of ( [ cond - pi-0 ] ) has main powers more than the square of that of ( [ cond - pi ] ) . it is interesting to consider density matrix estimation under a schatten norm , where given a matrix @xmath79 of size @xmath0 , we define its schatten @xmath137-norm by @xmath138 are the eigenvalues of the square root of @xmath139 . spectral norm and frobenius norm are two special cases of the schatten @xmath137-norm with @xmath140 and , respectively , and the nuclear norm corresponds to the schatten @xmath137-norm with @xmath141 . the following result provides the convergence rate for the proposed thresholding estimator under the schatten @xmath137-norm loss for @xmath142 . [ schatten.prop ] under the assumptions of theorem [ thm-3 ] , the density matrix estimator @xmath89 defined by ( [ threshold1])([threshold - estimator1 ] ) with threshold @xmath94 for some constant @xmath65 satisfies @xmath143 \leq c \bigl[\pi_n(d)\bigr ] ^{2 - 2/\max(s,2 ) } \frac{1}{d^{2 - 2/s } } \biggl(\frac { \log d}{n } \biggr ) ^{1-q+q/\max(s,2)}\ ] ] for @xmath142 , where @xmath144 is a constant not depending on @xmath42 and @xmath0 . the upper bound in ( [ schatten - s1 ] ) matches the minimax convergence rates for both the spectral norm and frobenius norm . moreover , for the case of the nuclear norm corresponding to the schatten @xmath137-norm with @xmath141 , ( [ schatten - s1 ] ) leads to an upper bound with the convergence rate @xmath145 . we conjecture that the upper bounds in ( [ schatten - s1 ] ) are rate - optimal under the schatten @xmath137-norm loss for all @xmath146 . however , establishing a matching lower bound for the general schatten norm loss is a difficult task , and we believe that a new approach is needed for studying minimax density matrix estimation under the schatten @xmath137-norm , particularly the nuclear norm . [ rem4 ] the pauli basis expansion ( [ beta - representation ] ) is orthogonal with respect to the usual euclidean inner product , and as in the proof of lemma [ lem - norm ] we have @xmath147 where @xmath148 and @xmath89 are threshold estimators of @xmath149 and @xmath8 , respectively . the sparse vector estimation problem is well studied under the gaussian or sub - gaussian noise case [ @xcite and @xcite ] and can be used to recover the minimax result for density matrix estimation under the frobenius norm loss , because of orthogonality . in fact , our relatively simple proof of the minimax results for the frobenius norm loss is essentially the same as the sparse vector estimation approach . however , such an equivalence between sparse density matrix estimation and sparse vector estimation breaks down for the general schatten norm loss such as the commonly used spectral norm and nuclear norm losses . for the spectral norm loss , lemma [ lem - norm ] in section [ proofs ] provides a sharp upper bound for @xmath150 $ ] through the @xmath151-norm of @xmath152 , and the proof of the minimax lower bound in theorem [ thm-4 ] relies on the property that the spectral norm is determined by the largest eigenvalue only . such a special property allows us to reduce the problem to a simple subproblem and establish the lower bound under the spectral norm loss . the arguments can not be applied to the case of the general schatten norm loss in particular the nuclear norm loss . moreover , instead of directly applying lemma [ lem - norm ] and remark [ rem5 ] in section [ proofs ] to derive upper bounds for the general schatten norm loss , we use the obtained sharp upper bounds for the spectral norm and frobenius norm losses together with moment inequalities to derive sharper upper bounds in proposition [ schatten.prop ] . however , similar lower bounds are not available . our analysis leads us to believe that it is not possible to use sparse vector estimation to recover minimax lower bound results for the general schatten norm loss in particular for the spectral norm loss . a simulation study was conducted to investigate the performance of the proposed density matrix estimator for the finite sample . we took @xmath153 and generated a true density matrix @xmath8 for each case as follows . @xmath8 has an expansion over the pauli basis @xmath154 where @xmath155 , @xmath40 . from @xmath51 , we randomly selected @xmath156 $ ] coefficients @xmath39 and set the rest of @xmath39 to be zero . we simulated @xmath157 $ ] values independently from a uniform distribution on @xmath158 $ ] , assigned the simulated values at random to the selected @xmath39 , and then constructed @xmath8 from ( [ rho - simulation ] ) . the constructed @xmath8 always has unit trace but may not be positive semi - definite . the procedure was repeated until we generated a positive semi - definite @xmath8 . we took it as the true density matrix . the simulation procedure guarantees the obtained @xmath8 is a density matrix and has a sparse representation under the pauli basis . for each true density matrix @xmath8 , as described in section [ sec-2 - 2 ] we simulated data @xmath43 from a binomial distribution with cell probability @xmath39 and the number of cells @xmath159 . we constructed coefficient estimators @xmath160 by ( [ threshold1 ] ) and obtained density matrix estimator @xmath89 using ( [ threshold - estimator1 ] ) . the whole estimation procedure is repeated @xmath161 times . the density matrix estimator is measured by the mean squared errors ( mse ) , @xmath162 and @xmath163 , that are evaluated by the average of @xmath164 and @xmath165 over @xmath161 repetitions , respectively . three thresholds were used in the simulation study : the universal threshold @xmath166 for all @xmath39 , the individual threshold @xmath167 for each @xmath39 , and the optimal threshold for all @xmath39 , which minimizes the computed mse for each corresponding hard or soft threshold method . the individual threshold takes into account the fact in theorem [ thm-2 ] that the mean and variance of @xmath43 are @xmath39 and @xmath168 , respectively , and the variance of @xmath43 is estimated by @xmath169 . . are plots of mses based on the spectral norm for @xmath153 , respectively , and are plots of mses based on the frobenius norm for @xmath170 , respectively . ] [ figure1 ] . are plots of mses based on the spectral norm for @xmath153 , respectively , and are plots of mses based on the frobenius norm for @xmath171 , respectively . ] . are plots of mses based on the spectral norm for @xmath172 , respectively , and are plots of mses based on the frobenius norm for @xmath173 , respectively . ] or @xmath174 against matrix size @xmath0 for the proposed density estimator with hard and soft threshold rules for @xmath175 . are plots of @xmath174 times of mses based on the spectral norm for @xmath175 , respectively , and are plots of @xmath0 times of mses based on the frobenius norm for @xmath175 , respectively . ] figures [ figure1 ] and [ figure2 ] plot the mses of the density matrix estimators with hard and soft threshold rules and its corresponding density matrix estimator without thresholding [ i.e. , @xmath39 are estimated by @xmath43 in ( [ threshold - estimator1 ] ) ] against the sample size @xmath42 for different matrix size @xmath0 , and figures [ figure3 ] and [ figure4 ] plot their mses against matrix size @xmath0 for different sample size . the numerical values of the mses are reported in table [ table1 ] . figures [ figure1 ] and [ figure2 ] show that the mses usually decrease in sample size @xmath42 , and the thresholding density matrix estimators enjoy superior performances than that the density matrix estimator without thresholding even for @xmath176 ; while all threshold rules and threshold values yield thresholding density matrix estimators with very close mses , the soft threshold rule with individual and universal threshold values produce larger mses than others for larger sample size such as @xmath177 and the soft threshold rule tends to give somewhat better performance than the hard threshold rule for smaller sample size like @xmath178 . figures [ figure3 ] and [ figure4 ] demonstrate that while the mses of all thresholding density matrix estimators decrease in the matrix size @xmath0 , but if we rescale the mses by multiplying it with @xmath174 for the spectral norm case and @xmath0 for the frobenius norm case , the rescaled mses slowly increase in matrix size @xmath0 . the simulation results largely confirm the theoretical findings discussed in remark [ rem1 ] . let @xmath179 . denote by @xmath180 s generic constants whose values are free of @xmath42 and @xmath181 and may change from appearance to appearance . let @xmath182 and @xmath183 be the maximum and minimum of @xmath184 and @xmath126 , respectively . for two sequences @xmath185 and @xmath186 , we write @xmath187 if @xmath188 as @xmath189 , and write @xmath190 if there exist positive constants @xmath191 and @xmath192 free of @xmath42 and @xmath181 such that @xmath193 . without confusion we may write @xmath54 as @xmath194 . proof of proposition [ thm-1 ] in two dimensions , pauli matrices satisfy @xmath195 , and @xmath196 , @xmath197 have eigenvalues @xmath2 , the square of a pauli matrix is equal to the identity matrix , and the multiplications of any two pauli matrices are equal to the third pauli matrix multiplying by @xmath198 , for example , @xmath199 , @xmath200 , and @xmath201 . for @xmath206 , @xmath207 , @xmath208 and @xmath209 , @xmath210\otimes[{\bolds{\sigma}}_{\ell_2}{\bolds{\sigma}}_{\ell^\prime_2 } ] \otimes \cdots\otimes[{\bolds{\sigma}}_{\ell_b } { \bolds{\sigma}}_{\ell^\prime_b}],\ ] ] is equal to a @xmath0 dimensional pauli matrix multiplying by @xmath211 , which has zero trace . thus , @xmath212 , that is , @xmath30 and @xmath213 are orthogonal , and @xmath214 form an orthogonal basis . @xmath215 . in particular @xmath32 , and @xmath216 . @lcd4.3d1.3d1.3d1.3d1.3d1.3d1.3d2.3d2.3d2.3@ & & & + & & & + & & & & & + & & & & & & & & & + & & & & & & & & & + & & & & & & & + & & & & & & & + & & & & & & & & & & & + 32 & 100 & 348.544 & 4.816 & 4.648 & 5.468 & 4.790 & 6.104 & 4.762 & 24.782 & 15.180 & 0.619 + & 200 & 175.034 & 4.449 & 4.257 & 5.043 & 4.708 & 5.293 & 4.667 & 17.524 & 7.739 & 0.562 + & 500 & 70.069 & 2.831 & 3.054 & 3.344 & 4.130 & 3.260 & 4.071 & 11.083 & 2.397 & 0.373 + & 1000 & 35.028 & 1.537 & 1.974 & 1.875 & 3.201 & 1.875 & 3.155 & 7.837 & 1.099 & 0.212 + & 2000 & 17.307 & 0.785 & 1.195 & 1.001 & 2.230 & 0.989 & 2.200 & 5.541 & 0.551 & 0.116 + 64 & 100 & 368.842 & 1.583 & 1.572 & 1.744 & 1.583 & 1.954 & 1.586 & 27.148 & 16.660 & 0.395 + & 200 & 183.050 & 1.565 & 1.534 & 1.669 & 1.575 & 1.833 & 1.571 & 19.196 & 9.252 & 0.376 + & 500 & 73.399 & 1.175 & 1.228 & 1.367 & 1.490 & 1.347 & 1.476 & 12.141 & 2.900 & 0.307 + & 1000 & 36.692 & 0.566 & 0.807 & 0.747 & 1.249 & 0.722 & 1.233 & 8.585 & 1.308 & 0.177 + & 2000 & 18.402 & 0.186 & 0.443 & 0.255 & 0.832 & 0.251 & 0.820 & 6.070 & 0.657 & 0.061 + 128 & 100 & 381.032 & 0.543 & 0.542 & 0.574 & 0.543 & 0.705 & 0.545 & 29.323 & 17.500 & 0.237 + & 200 & 190.113 & 0.541 & 0.539 & 0.570 & 0.542 & 0.594 & 0.542 & 20.734 & 10.246 & 0.235 + & 500 & 75.824 & 0.471 & 0.480 & 0.514 & 0.525 & 0.509 & 0.522 & 13.114 & 3.547 & 0.213 + & 1000 & 38.010 & 0.309 & 0.350 & 0.355 & 0.470 & 0.354 & 0.466 & 9.273 & 1.613 & 0.146 + & 2000 & 18.907 & 0.142 & 0.216 & 0.194 & 0.359 & 0.194 & 0.356 & 6.557 & 0.725 & 0.080 + @lcd4.3d1.3d1.3d1.3d1.3d1.3d1.3d2.3d2.3d2.3@ & & & + & & & + & & & & & + & & & & & & & & & + & & & & & & & & & + & & & & & & & + & & & & & & & + & & & & & & & & & & & + 32 & 100 & 317.873 & 6.052 & 5.119 & 6.195 & 5.274 & 7.050 & 5.246 & 24.782 & 11.004 & 9.936 + & 200 & 159.679 & 5.217 & 4.629 & 5.616 & 5.187 & 5.874 & 5.143 & 17.524 & 5.681 & 3.771 + & 500 & 63.823 & 3.165 & 3.229 & 3.732 & 4.575 & 3.642 & 4.512 & 11.083 & 2.286 & 0.954 + & 1000 & 31.856 & 1.722 & 2.053 & 2.119 & 3.540 & 2.119 & 3.492 & 7.837 & 1.100 & 0.401 + & 2000 & 15.967 & 0.894 & 1.219 & 1.155 & 2.424 & 1.141 & 2.394 & 5.541 & 0.546 & 0.182 + 64 & 100 & 641.437 & 3.909 & 3.528 & 3.951 & 3.563 & 4.463 & 3.562 & 27.148 & 13.719 & 13.234 + & 200 & 319.720 & 3.706 & 3.401 & 3.755 & 3.548 & 4.082 & 3.536 & 19.196 & 7.042 & 5.515 + & 500 & 127.958 & 2.691 & 2.551 & 3.069 & 3.342 & 3.023 & 3.309 & 12.141 & 2.800 & 1.275 + & 1000 & 63.845 & 1.335 & 1.628 & 1.765 & 2.791 & 1.717 & 2.756 & 8.585 & 1.277 & 0.548 + & 2000 & 31.952 & 0.433 & 0.882 & 0.610 & 1.842 & 0.596 & 1.817 & 6.070 & 0.647 & 0.258 + 128 & 100 & 1283.182 & 2.370 & 2.240 & 2.370 & 2.242 & 2.924 & 2.245 & 29.323 & 15.989 & 16.128 + & 200 & 639.556 & 2.349 & 2.219 & 2.354 & 2.238 & 2.444 & 2.238 & 20.734 & 8.218 & 7.799 + & 500 & 255.954 & 1.990 & 1.906 & 2.125 & 2.172 & 2.102 & 2.160 & 13.114 & 3.355 & 1.773 + & 1000 & 127.714 & 1.221 & 1.341 & 1.463 & 1.943 & 1.448 & 1.924 & 9.273 & 1.546 & 0.729 + & 2000 & 63.921 & 0.581 & 0.815 & 0.798 & 1.471 & 0.798 & 1.456 & 6.557 & 0.719 & 0.327 + denote by @xmath34 the projections onto the eigenspaces corresponding to eigenvalues @xmath2 , respectively . then for @xmath217 , @xmath218 and solving the equations we get @xmath219 for @xmath220 , @xmath221 , @xmath222 and @xmath213 are orthogonal , @xmath223 and @xmath224 which imply @xmath225 for a density matrix @xmath8 with representation ( [ beta - representation ] ) under the pauli basis ( [ pauli - matrix ] ) , from ( [ eq1 ] ) we have @xmath226 and @xmath227 , and thus @xmath228 \label{tr - qrho2 } \\[-8pt ] \nonumber & = & \frac{1}{2 } + \frac{\beta_k}{d } \operatorname{tr}({\mathbf{b}}_{k } { \mathbf{q}}_{k \pm } ) = \frac{1 \pm\beta_k}{2}.\end{aligned}\ ] ] proof of proposition [ thm-2 ] we perform measurements on each pauli observable @xmath229 independently for @xmath42 quantum systems that are identically prepared under state @xmath8 . denote by @xmath230 the @xmath42 measurement outcomes for measuring @xmath229 , @xmath231 . @xmath232 @xmath233 , @xmath234 , @xmath235 , are independent , and take values @xmath2 , with distributions given by @xmath236 as random variables @xmath230 are i.i.d . and take eigenvalues @xmath2 , @xmath237 is equal to the total number of random variables @xmath230 taking eigenvalue @xmath3 , and thus @xmath238 follows a binomial distribution with @xmath42 trials and cell probability @xmath239 . from ( [ tomography3])([tomography4 ] ) and proposition [ thm-1 ] , we have for @xmath234 , @xmath240 from proposition [ thm-2 ] and ( [ tomography3])([tomography4 ] ) , we have that @xmath43 is the average of @xmath249 , which are i.i.d . random variables taking values @xmath2 , @xmath250 , @xmath251 and @xmath252 . applying bernstein s inequality , we obtain for any @xmath67 , @xmath253 both @xmath254 and @xmath255 are less than @xmath256 , which is bounded by @xmath257 [ lem - norm ] @xmath258 + \biggl\ { \sum _ { j=2}^p e\bigl[|\hat{\beta}_j - \beta_j|\bigr ] \biggr\}^2 \nonumber \\[-8pt ] \label{e - spectral - norm - square } \\[-8pt ] \nonumber & & { } - \sum_{j=2}^p \bigl\{e\bigl(|\hat { \beta}_j - \beta _ j|\bigr)\bigr\}^2.\end{aligned}\ ] ] since pauli matrices @xmath30 are orthogonal with respect to the usual euclidean inner product , with @xmath259 , and @xmath260 , we have @xmath261 \label{trace - norm } \\[-8pt ] \nonumber & = & \sum_{j=2}^p |\hat { \beta}_j - \beta_j|^2/d , \\ p^{1/2 } \| \hat{{\bolds{\rho } } } - { \bolds{\rho}}\|_2 & = & \biggl\| \sum _ { j=2}^p ( \hat{\beta } _ j - \beta_j ) { \mathbf{b}}_j \biggr\|_2 \leq\sum _ { j=2}^p |\hat{\beta}_j - \beta _ j| \| { \mathbf{b}}_j \|_2 \nonumber \\[-8pt ] \label{spectral - norm } \\[-8pt ] \nonumber & = & \sum_{j=2}^p |\hat { \beta}_j - \beta_j| , \\ p \| \hat{{\bolds{\rho } } } - { \bolds{\rho}}\|_2 ^ 2 & = & \biggl\| \sum _ { j=2}^p ( \hat{\beta}_j - \beta_j ) { \mathbf{b}}_j \biggr\|_2 ^ 2 \nonumber\\ & \leq & \sum_{j=2}^p |\hat { \beta}_j - \beta_j|^2 \| { \mathbf{b}}_j \|_2 ^ 2 + 2 \sum_{i < j } ^p \bigl|(\hat{\beta}_i - \beta_i ) ( \hat{\beta}_j - \beta_j)\bigr| \| { \mathbf{b}}_i { \mathbf{b}}_j \|_2 \nonumber \\[-8pt ] \label{spectral - norm - square } \\[-8pt ] \nonumber & \leq & \sum_{j=2}^p |\hat { \beta}_j - \beta_j|^2 \| { \mathbf{b}}_j \|_2 ^ 2 + 2 \sum_{i < j } ^p \bigl|(\hat{\beta}_i - \beta_i ) ( \hat{\beta}_j - \beta_j)\bigr| \| { \mathbf{b}}_i\|_2 \| { \mathbf{b}}_j \|_2 \nonumber \\ & = & \sum_{j=2}^p |\hat { \beta}_j - \beta_j|^2 + 2 \sum _ { i < j } ^p\bigl|(\hat{\beta}_i - \beta_i ) ( \hat{\beta}_j - \beta_j)\bigr| . \nonumber\end{aligned}\ ] ] as @xmath262 are independent , @xmath263 are independent . thus , from ( [ trace - norm])([spectral - norm - square ] ) we obtain ( [ e - trace - norm])([e - spectral - norm ] ) , and @xmath264 + \biggl\ { \sum _ { j=2}^p e\bigl[|\hat{\beta}_j - \beta_j|\bigr ] \biggr\}^2 - \sum_{j=2}^p \bigl\{e\bigl(|\hat{\beta}_j - \beta_j|\bigr)\bigr \}^2.\end{aligned}\ ] ] [ rem5 ] since pauli matrices @xmath30 have eigenvalues @xmath2 , the schatten @xmath137-norm @xmath265 . similar to ( [ spectral - norm])([spectral - norm - square ] ) , we obtain that @xmath266 \label{s - norm - square } \\[-8pt ] \nonumber & = & d^{2/s } \biggl [ \sum_{j=2}^p | \hat{\beta}_j - \beta_j|^2 + 2 \sum _ { i < j } ^p \bigl|(\hat{\beta}_i - \beta_i ) ( \hat{\beta}_j - \beta_j)\bigr| \biggr].\hspace*{-12pt } \nonumber\end{aligned}\ ] ] using ( [ threshold1 ] ) , we have @xmath268 + |\beta_j| p\bigl(|n_j| \leq\varpi \bigr ) \nonumber \\ & & \qquad \leq \bigl[e| n_j - \beta_j|^2 p\bigl(|n_j| \geq\varpi\bigr ) \bigr]^{1/2 } + \varpi p\bigl(|n_j| \geq\varpi\bigr ) + |\beta_j| p\bigl(|n_j| \leq\varpi\bigr ) \nonumber \\ & & \qquad \leq \bigl [ n^{-1 } \bigl(1-\beta_j^2\bigr ) p\bigl(|n_j| \geq\varpi\bigr ) \bigr]^{1/2 } + \varpi p\bigl(|n_j| \geq\varpi\bigr ) + |\beta_j| p\bigl(|n_j| \leq\varpi \bigr ) \nonumber \\ & & \qquad\leq 2 \varpi \bigl[p\bigl(|n_j| \geq\varpi\bigr ) \bigr]^{1/2 } + | \beta _ j| p\bigl(|n_j| \leq\varpi\bigr ) \nonumber \\ & & \qquad= 2 \varpi \bigl[p\bigl(|n_j| \geq\varpi\bigr ) \bigr]^{1/2 } \bigl\{1\bigl(|\beta _ j| > a_1 \varpi\bigr)+ 1\bigl(|\beta_j| \leq a_1 \varpi\bigr)\bigr\ } \nonumber \\ & & \qquad\quad{}+ |\beta_j| p\bigl(|n_j| \leq\varpi\bigr ) \bigl\{1\bigl(| \beta_j| > a_2 \varpi\bigr)+1\bigl(|\beta_j| \leq a_2 \varpi\bigr)\bigr\ } \nonumber \\ & & \qquad\leq 2\varpi1\bigl(|\beta_j| > a_1 \varpi\bigr ) + 2 \varpi \bigl[p\bigl(|n_j| \geq\varpi\bigr ) \bigr]^{1/2 } 1\bigl(| \beta_j| \leq a_1 \varpi\bigr ) \nonumber \\ & & \qquad\quad{}+ p\bigl(|n_j|\leq\varpi\bigr ) 1\bigl(|\beta_j| > a_2 \varpi\bigr ) + |\beta _ j| 1\bigl(|\beta_j| \leq a_2 \varpi\bigr),\end{aligned}\ ] ] where @xmath269 and @xmath270 are two constants satisfying @xmath271 whose values will be chosen later , and @xmath272^{1/2 } 1\bigl(| \beta_j| \leq a_1 \varpi\bigr)\label{beta-1 } \\ & & { } + \sum_{j=2}^p p\bigl(|n_j| \leq\varpi\bigr ) 1\bigl(|\beta _ \varpi\bigr ) + \sum _ { j=2}^p |\beta_j| 1\bigl(| \beta_j| \leq a_2 \varpi\bigr ) . \nonumber\end{aligned}\ ] ] similarly , @xmath273 ^ 2\\ & & \qquad \leq e\bigl[|\hat{\beta}_j - \beta _ j|^2\bigr ] \nonumber \\ & & \qquad\leq e\bigl [ 2\bigl(| n_j - \beta_j|^2 + \varpi^2 \bigr ) 1\bigl(|n_j| \geq\varpi\bigr)\bigr ] + | \beta_j|^2 p\bigl(|n_j| \leq\varpi\bigr ) \nonumber \\ & & \qquad \leq 2 \bigl[e| n_j - \beta_j|^4 p\bigl(|n_j| \geq\varpi\bigr ) \bigr]^{1/2}\\ & & \qquad\quad { } + 2 \varpi ^2 p\bigl(|n_j| \geq\varpi\bigr ) + |\beta_j|^2 p\bigl(|n_j| \leq\varpi\bigr ) \nonumber \\ & & \qquad\leq c \varpi^2 \bigl[p\bigl(|n_j| \geq\varpi\bigr ) \bigr]^{1/2 } + |\beta_j|^2 p\bigl(|n_j| \leq\varpi\bigr ) \nonumber \\ & & \qquad = c \varpi^2 \bigl[p\bigl(|n_j| \geq\varpi\bigr ) \bigr]^{1/2 } \bigl\{1\bigl(|\beta_j| > a_1 \varpi\bigr)+ 1\bigl(|\beta_j| \leq a_1 \varpi\bigr)\bigr\ } \nonumber \\ & & \qquad\quad{}+ |\beta_j|^2 p\bigl(|n_j| \leq\varpi\bigr ) \bigl[1\bigl(|\beta_j| > a_2 \varpi\bigr ) + 1\bigl(| \beta_j| \leq a_2 \varpi\bigr)\bigr ] \nonumber \\ & & \qquad \leq c \varpi^2 1\bigl(|\beta_j| > a_1 \varpi\bigr ) + c \varpi^2 \bigl[p\bigl(|n_j| \geq\varpi\bigr ) \bigr]^{1/2 } 1\bigl(|\beta_j| \leq a_1 \varpi\bigr ) \nonumber \\ & & \qquad\quad{}+ p\bigl(|n_j|\leq\varpi\bigr ) 1\bigl(|\beta_j| > a_2 \varpi\bigr ) + |\beta_j|^2 1\bigl(|\beta_j| \leq a_2 \varpi\bigr),\end{aligned}\ ] ] and @xmath274\nonumber\\ & & \qquad \leq c \varpi^2 \sum_{j=2}^p 1\bigl(| \beta_j| > a_1 \varpi\bigr)\nonumber \\[-8pt ] \label{beta-2 } \\[-8pt ] \nonumber & & \qquad\quad { } + c \varpi^2 \sum_{j=2}^p \bigl[p\bigl(|n_j| \geq\varpi \bigr ) \bigr]^{1/2 } 1\bigl(| \beta_j| \leq a_1 \varpi\bigr ) \\ & & \qquad\quad{}+ \sum_{j=2}^p p\bigl(|n_j| \leq\varpi\bigr ) 1\bigl(|\beta_j| > a_2 \varpi\bigr ) + \sum _ { j=2}^p |\beta_j|^2 1\bigl(| \beta_j| \leq a_2 \varpi \bigr ) . \nonumber\end{aligned}\ ] ] by lemma [ lem - sparse ] , we have @xmath275 \label{beta - sparse-11 } \\[-8pt ] \nonumber & & \qquad \leq ( a _ 2 \varpi)^{2-q } \sum _ { j=2}^{p } |\beta_j|^q 1\bigl(| \beta_j| \leq a_2 \varpi\bigr ) \leq a_2^{2-q } \pi_n(d ) \varpi^{2-q } , \\ \label{beta - sparse-2 } & & \varpi \sum_{j=2}^p 1\bigl(| \beta_j| \geq a_1 \varpi\bigr ) \leq a_1^{-q}\pi_n(d ) \varpi^{1-q}.\end{aligned}\ ] ] on the other hand , @xmath276 \\ & & \qquad\leq4 p^{1- \vafrac{\hbar^2 |a_2 - 1|^2}{1+o(1 ) } } = 4 p^{-1 - ( 2-q)/(2 c_0 ) } \leq4 p^{-1 } n^{-(q-2)/2}\nonumber\\ & & \qquad = o\bigl(\pi_n(d ) \varpi^{2-q}\bigr ) , \nonumber\end{aligned}\ ] ] where the third inequality is from lemma [ lem - tail ] , the first equality is due the fact that we take @xmath277 so that @xmath278 , and @xmath279 is the constant in assumption @xmath280 . finally , we can show @xmath281^{1/2 } 1\bigl(| \beta_j| \leq a_1 \varpi\bigr ) \\ & & \qquad\leq\varpi \sum_{j=2}^p \bigl[p ( n_j - \beta_j \leq- \varpi- \beta _ j ) \nonumber \\ \label{beta - sparse-4 } & & \quad\qquad{}+ p(n_j - \beta_j \geq\varpi- \beta_j ) \bigr]^{1/2 } 1\bigl(|\beta_j| \leq a_1 \varpi\bigr ) \\ & & \qquad\leq\varpi \sum_{j=2}^p \bigl[p\bigl ( n_j - \beta_j \leq- |1-a_1| \varpi \bigr ) + p\bigl(n_j - \beta_j \geq|1-a_1| \varpi\bigr ) \bigr]^{1/2 } \nonumber \\ & & \qquad \leq2\varpi p^{1- \hbar^2 ( 1-a_1)^2/(2(1+o(1 ) ) ) } = 2\varpi p^{-1 } = o\bigl ( \pi_n(d ) \varpi^{1-q}\bigr ) , \nonumber\end{aligned}\ ] ] where the third inequality is from lemma [ lem - tail ] , and the first equality is due to the fact that we take @xmath282 so that @xmath283 . plugging ( [ beta - sparse-1])([beta - sparse-4 ] ) into ( [ beta-1 ] ) and ( [ beta-2 ] ) , we prove the lemma . proof of theorem [ thm-3 ] combining lemma [ lem - thm3 ] and ( [ e - trace - norm])([e - spectral - norm ] ) in lemma [ lem - norm ] , we easily obtain @xmath284 & \leq & c_1 \frac{\pi_n(d)}{p^{1/2 } } \biggl ( \frac { \log p } { n } \biggr)^{\vfrac{1-q}{2 } } , \\ e\bigl[\| \hat{{\bolds{\rho } } } - { \bolds{\rho}}\|_f^2\bigr ] & \leq & c_0 \pi_n(d ) \frac{1}{d } \biggl ( \frac { \log p } { n } \biggr)^{1-q/2}.\end{aligned}\ ] ] using lemma [ lem - thm3 ] and ( [ e - spectral - norm - square ] ) in lemma [ lem - norm ] , we conclude @xmath285 & \leq & c_2 \biggl [ \pi^2_n(d ) \frac{1}{p } \biggl ( \frac { \log p } { n } \biggr)^{1-q } + \pi_n(d ) \frac{1}{p } \biggl ( \frac { \log p } { n } \biggr)^{1-q/2 } \biggr ] \nonumber \\[-8pt ] \label{equation - spectral - norm - square } \\[-8pt ] \nonumber & \leq & c \frac{\pi^2_n(d)}{d^2 } \biggl ( \frac { \log p } { n } \biggr)^{1-q},\end{aligned}\ ] ] where the last inequality is due to the fact that the first term on the right - hand side of ( [ equation - spectral - norm - square ] ) dominates its second term . proof of proposition [ schatten.prop ] applying the lyapunov s moment inequality to the schatten @xmath137-norm , we have for @xmath286 $ ] and @xmath287 , @xmath288 & \leq & d^{-1 + 2/s } e \bigl [ \| \hat{{\bolds{\rho } } } -{\bolds{\rho}}\|_{*2 } ^2 \bigr ] \\ & = & d^{-1 + 2/s } e \bigl [ \| \hat{{\bolds{\rho } } } -{\bolds{\rho}}\|_{f } ^2 \bigr ] \\ & \leq & c_1 \pi_n ( d ) d^{-2 + 2/s } \biggl(\frac{\log d}{n } \biggr ) ^{1-q/2},\end{aligned}\ ] ] where the last inequality is due to theorem [ thm-3 ] . on the other hand , applying hlder s inequality by interpolating between schatten @xmath137-norms with @xmath289 and @xmath290 , we obtain for @xmath291 $ ] and @xmath287 , @xmath292 & \leq & e \bigl [ \| \hat{{\bolds{\rho}}}- { \bolds{\rho}}\|_{*2}^{4/s } \| \hat{{\bolds{\rho}}}- { \bolds{\rho}}\| _ { * \infty } ^{2 - 4/s } \bigr ] \\ & \leq & \bigl [ e \| \hat { { \bolds{\rho}}}- { \bolds{\rho}}\|_{*2}^{2 } \bigr ] ^{2/s } \bigl[e \| \hat{{\bolds{\rho}}}- { \bolds{\rho}}\|_{*\infty } ^{2 } \bigr ] ^{1 - 2/s } \\ & \leq & c_7 \pi_n ^{2 - 2/s}(d ) d^{-2 + 2/s } \biggl(\frac{\log d}{n } \biggr ) ^{1-q+q / s},\end{aligned}\ ] ] where the last inequality is due to theorem [ thm-3 ] , and @xmath293 . the result follows by combining the above two inequalities together . proof of theorem [ thm-4 ] we first define a subset of the parameter space @xmath90 . it will be shown later that the risk upper bound under the spectral norm is sharp up to a constant factor , when the parameter space is sufficiently sparse . consider a subset of the pauli basis , @xmath294 , where @xmath295 or @xmath296 . its cardinality is @xmath297 . denote each element of the subset by @xmath298 , @xmath299 , and let @xmath300 . we will define each element of @xmath90 as a linear combination of @xmath298 . let @xmath301 , @xmath302 , and denote @xmath303 . the value of @xmath304 is either @xmath305 or @xmath306 , where @xmath306 is the largest integer less than or equal to @xmath307 . by assumption ( [ cond - pi ] ) , we have @xmath308 let @xmath309 and set @xmath310 . now we are ready to define @xmath90 , @xmath311 note that @xmath90 is a subset of the parameter space , since @xmath312 and its cardinality is @xmath313 . we need to show that @xmath314 note that for each element in @xmath90 , its first entry @xmath315 may take the form @xmath316 . it can be shown that @xmath317 it is then enough to show that @xmath318 which immediately implies @xmath319 we prove equation ( [ klowerbd ] ) by applying le cam s lemma . from observations @xmath320 , @xmath321 , we define @xmath322 , which is @xmath323 . let @xmath324 be the joint distribution of independent random variables @xmath325 . the cardinality of @xmath326 is @xmath313 . for two probability measures @xmath327 and @xmath328 with density @xmath329 and @xmath330 with respect to any common dominating measure @xmath331 , write the total variation affinity @xmath332 , and the chi - square distance @xmath333 . define @xmath334 the following lemma is a direct consequence of le cam s lemma [ cf . @xcite and @xcite ] . it is enough to show that @xmath342 which implies @xmath343 , then we have @xmath344 . let @xmath345 denote the number of overlapping nonzero coordinates between @xmath346 and @xmath347 . note that @xmath348 when @xmath349 , we have @xmath350 \biggr ) ^{j } \\ & = & \biggl ( \sum_{l=0}^{n } \biggl [ \pmatrix{n \cr l } \biggl ( \frac { ( 1+a ) ^{2}}{2 } \biggr ) ^{l } \biggl ( \frac { ( 1-a ) ^{2}}{2 } \biggr ) ^{n - l } \biggr ] \biggr ) ^{j } \\ & = & \biggl ( \frac { ( 1+a ) ^{2}}{2}+\frac { ( 1-a ) ^{2}}{2 } \biggr ) ^{nj } \\ & = & \bigl ( 1+a^{2 } \bigr ) ^{nj}\leq\exp \bigl ( na^{2}j \bigr),\end{aligned}\ ] ] which implies @xmath351 since @xmath352 ^{2}\cdot ( d-1-k ) \cdot \ldots\cdot ( d-2k+j ) } { j!\cdot ( d-1 ) \cdot \ldots \cdot ( d - k ) } \\ & \leq&\frac{k^{2j } ( d-1-k ) ^{k - j } } { ( d - k ) ^{k}}\leq \biggl ( \frac{k^{2}}{d - k } \biggr ) ^{j},\end{aligned}\ ] ] and @xmath309 , we then have @xmath353 ^{j } \leq\sum_{1\leq j\leq k } \biggl [ \frac{d^{2v+ ( 1 - 2v ) /2}}{d - k } \biggr ] ^{j}\rightarrow0.\end{aligned}\ ] ] apply assouad s lemma , and we show below that @xmath134 \geq c \pi_n(p ) \frac{1}{d } \biggl ( \frac { \log p } { n } \biggr)^{1-q/2},\ ] ] where @xmath111 denotes any estimator of @xmath8 based on measurement data @xmath262 , and @xmath180 is a constant free of @xmath42 and @xmath181 . to this end , it suffices to construct a collection of @xmath356 density matrices @xmath357 such that ( i ) for any distinct @xmath358 and @xmath359 , @xmath360 where @xmath191 is a constant ; ( ii ) there exists a constant @xmath361 such that @xmath362 where @xmath363 denotes the kullback leibler divergence . by the gilbert varshamov bound [ cf . @xcite ] , we have that for any @xmath364 , there exist @xmath365 binary vectors @xmath366 , @xmath367 , such that ( i ) @xmath368 , ( ii ) @xmath369 , and ( iii ) @xmath370 . let @xmath371 where @xmath372 since @xmath373 , @xmath374 whenever @xmath375 . moreover , @xmath376 on the other hand , @xmath377 now the lower bound can be established by taking @xmath378 and then @xmath379 which are allowed by the assumption @xmath380 for @xmath381 . | quantum state tomography aims to determine the state of a quantum system as represented by a density matrix .
it is a fundamental task in modern scientific studies involving quantum systems . in this paper , we study estimation of high - dimensional density matrices based on pauli measurements . in particular , under appropriate notion of sparsity , we establish the minimax optimal rates of convergence for estimation of the density matrix under both the spectral and frobenius norm losses ; and show how these rates can be achieved by a common thresholding approach .
numerical performance of the proposed estimator is also investigated .
./style / arxiv - general.cfg , , , |
You are an expert at summarizing long articles. Proceed to summarize the following text:
quantum systems when placed in low dimensional lattices typically exhibit strongly correlated effects driving them towards regimes with no classical analog . many properties of these regimes or quantum phases @xcite depend in turn on the properties of their ground state and low lying energy excitations @xcite . a problem of particular interest in the field of strongly correlated systems is the emergence of critical phases in a system where the generic behaviour as coupling constants are varied is to be a gapped system , although those gapped phases may be of different nature . in this paper we address this problem by selecting a system of quantum spins that allows us to perform a detailed study of critical and non - critical phases on equal footing , i.e. , without any bias towards an a priori preferred phase . for reasons explained in sect.[sect_model ] , the quantum spins are arranged in a 2-leg ladder lattice @xcite with anti - ferromagnetic heisenberg couplings along the legs while rung couplings are ferromagnetic . in addition , we also introduce an explicit dimerization coupling in the hamiltonian along the leg directions , which can be varied from zero to strong values . this coupling plays a major role in order to create the aforementioned critical phases out of a system with only gapped phases . this particular type of 2-leg ladder system has a number of open problems such as the precise location of critical phases in the phase diagram of the coupling constants , and the nature of the gapped phases it exhibits . our study is complete enough so as to be able to solve for these problems in a very precise manner . the understanding of these purely quantum effects is usually a hard problem . perturbative and variational methods in quasi - one dimensional systems like chains and ladders are not well suited to uncover the physics in the whole range of coupling constants involved in the description of the interactions in the system . on the contrary the dmrg method @xcite , @xcite , @xcite , @xcite , @xcite allows us to identify the critical phases clearly and without any bias . this is so because the method is non - perturbative and allows a controllable management of errors . our studies are also of interest since experiments on ladder materials have revealed a very complex behaviour , such as an interplay between a spin - gapped normal states and superconductivity @xcite . moreover , a new field of study for these complex effects has been opened by the simulation of strongly correlated systems in optical lattices @xcite , in particular quantum spin chains and ladders @xcite . this paper is organized as follows : in sect.[sect_model ] we introduce the model hamiltonian describing a 2-leg ladder lattice of spins @xmath3 with columnar bond - alternating antiferromagnetic couplings in the horizontal direction and ferromagnetic couplings in the vertical direction , see fig.[ladderpic ] . we can identify some particular behaviours in appropriate weak and strong coupling limits , but not for generic values of the couplings . in sect.[sect_critical ] we point out the rich physical effects posed by open boundary conditions in these 2-leg ladders with finite length , although it also implies an a priori analysis in order to find out which low - lying states contribute to the gap of the system in the thermodynamic limit . this we can be done with the dmrg method by targeting several states and measuring their magnetization properties in the bulk and at the ends . then , we compute numerically the gap and we establish the existence of a critical line in the quantum phase diagram of the model . a numerical fit of this critical curve is also given . in sect.[sect_haldanedimer ] we determine the structure of the phase diagram by identifying the type of gapped phases occurring at each side of the critical line found in the previous section . they correspond to haldane and dimer phases . they are identified by measuring the string order parameter and the dimerization parameter with the dmrg method . we complete our study of these phases measuring different observables . sect.[sect_conclusions ] is devoted to conclusions . competing ferromagnetic versus antiferromagnetic spin interactions may give rise to critical phases if they are appropriately arranged in certain quasi - one dimensional lattices . one emblematic example of this phenomenon is a lattice of quantum spins with the shape of a 2-leg ladder such that there are antiferromagnetic couplings along the legs and ferromagnetic interactions along the rungs connecting both legs . in addition , the antiferromagnetic couplings are bond - alternating in a columnar fashion . dimerization interactions in the hamiltonian are also known as staggered interactions . this configuration is shown in fig.[ladderpic ] . more precisely , this configuration of heisenberg - like interactions is associated with the following quantum hamiltonian . in the horizontal direction ( legs ) , we picture the bond alternation with strong links @xmath4 and weak links @xmath5 . in the vertical direction ( rungs ) , the system is arranged in the form of a columnar dimerization : strong links are parallel to one another , and similarly for weak links in the lattice . ] @xmath6 where @xmath7 are quantum spin @xmath3 operators located at site @xmath8 of the leg @xmath9 , and @xmath10 , @xmath11 , @xmath12 $ ] are the antiferromagnetic , ferromagnetic and staggering couplings , respectively , as mentioned above . notice that several known regimes can be reached by tuning the coupling constants towards particular values . in the weak coupling limit , making @xmath13 we end up with a system consisting on two effectively decoupled @xmath14 heisenberg chains with bond alternation ( bahc ) , which are known to be gapped for every value of the dimerization parameter @xmath15 @xcite , except for the point @xmath16 . in the strong coupling limit , making @xmath17 the system can be effectively described by a @xmath18 spin chain with bond alternation , which is predicted to be gapped for all values of @xmath15 except for a critical point at a non - zero value @xmath19 @xcite . these predictions are based on an approximate mapping onto the o(3 ) @xmath20 model @xcite at topological angle @xmath21 . this yields a critical value of @xmath22 when @xmath23 , and similarly another symmetric critical value at @xmath24 . thus , we shall always concentrate in the region @xmath25 , due to the symmetry @xmath26 in the hamiltonian . this non - linear sigma model ( nl@xmath20 m ) prediction misses the correct location of the critical point due to the approximations involved in that mapping . the exact location of this point has been widely studied @xcite and results slightly varied depending on the approach , however modern studies place it at @xmath27 @xcite@xcite , also compatible with fig.[l500vsl2x140](_lower _ ) which gives @xmath28 for a chain of @xmath29 sites . these studies also conclude that the region @xmath30 corresponds to a haldane phase while for @xmath31 we move to a dimer phase . the emergence of a dimerized @xmath18 spin chain in the strong coupling limit can be explained by noting that as the rung coupling is ferromagnetic and strong @xmath32 , the two spins @xmath3 in each rung find energetically favorable to form a spin triplet . ladder at the point @xmath33 . _ up _ : mean magnetization @xmath34 in the states with @xmath35 and @xmath36 , as explained in the text . _ inset _ : cumulative sum of the magnetization over the whole extent of the ladder and the states with @xmath37 and @xmath38 . the order of the sites in the @xmath39-axis corresponds in this case to the path used to traverse the ladder in a dmrg sweep . _ down _ : difference of energy density ( doe ) of the excited states with @xmath37 and @xmath38 : @xmath40 . .the scale on the right axis corresponds to the cumulative sum . see text for more explanations . , title="fig : " ] ladder at the point @xmath33 . _ up _ : mean magnetization @xmath34 in the states with @xmath35 and @xmath36 , as explained in the text . _ inset _ : cumulative sum of the magnetization over the whole extent of the ladder and the states with @xmath37 and @xmath38 . the order of the sites in the @xmath39-axis corresponds in this case to the path used to traverse the ladder in a dmrg sweep . _ down _ : difference of energy density ( doe ) of the excited states with @xmath37 and @xmath38 : @xmath40 . .the scale on the right axis corresponds to the cumulative sum . see text for more explanations . , title="fig : " ] for generic values of the coupling constants in the hamiltonian , this model has been the subject of a series of conjectures based on exact diagonalization numerical studies @xcite in the absence of dimerization @xmath16 and analytical studies using bosonization and nl@xmath20 m mapping @xcite in the presence of dimerization @xmath41 . those numerical methods only allowed to reach ladder lengths typically of @xmath42 or so , which prevents from reaching any definitive conclusion on the bulk properties of the system in the thermodynamic limit . as for the analytical studies , they conjectured the existence of a possible critical region , but due to the nature of the methods it is not possible to give its location in terms of the original coupling constants in the model hamiltonian . computed on a @xmath43 ladder for different values of the parameter @xmath44 . each minimum in the gap belongs to the critical line . ] one of the main issues in this model is whether it exhibits a critical line in the quantum phase diagram of @xmath44 vs. @xmath15 . we solve this open problem in the positive by using the dmrg method in finite version algorithm which provides us with better accuracy values than the infinite method version , although at the expense of more demanding time computing resources . the performance of the finite dmrg algorithm is characterized by the following parameters : the number of states @xmath45 retained in the truncation process of the rg method , the weight of the discarded states @xmath46 which is a measure of the dmrg error , the number of sweeps @xmath47 or iterations of the method after the initial warm - up process and the tolerance @xmath48 of the target state energy which controls the average number of iterations that will need the diagonalization algorithm ( lanczos in our case ) to compute the target state . we shall provide values of these parameters in our numerical computations below . before applying the finite - size dmrg method , two important remarks are in order : i/ as we shall always work with a fixed value of @xmath49 the length of the lattice , the gap @xmath50 is always finite and only in the thermodynamic limit @xmath51 it may vanish for certain values of @xmath44 and @xmath15 which define the critical line we are searching for . thus , the signature of a gap in @xmath50 for fixed @xmath44 and varying @xmath15 will show up as a minimum in the dimerization parameter . upon increasing the value of @xmath49 , we shall obtain more robust estimations of the critical value @xmath52 from the minima @xmath53 . this is a finite - size scaling analysis of the dmrg numerical data . . the points of the critical line correspond to the coordinates that minimize @xmath54 . the solid line is only a guide for the eye._bottom _ : the value of the minimum gap for the ladder with @xmath43 sites is very similar to the corresponding @xmath18 bahc with @xmath55 but the value of @xmath19 that minimizes this gap is still a bit shifted , which constitutes a signal that @xmath56 is still a low value to accurately mimic the limit bahc behaviour . the computations for the @xmath57 bahc were performed storing @xmath58 eigenvectors of the density matrix . , title="fig : " ] . the points of the critical line correspond to the coordinates that minimize @xmath54 . the solid line is only a guide for the eye._bottom _ : the value of the minimum gap for the ladder with @xmath43 sites is very similar to the corresponding @xmath18 bahc with @xmath55 but the value of @xmath19 that minimizes this gap is still a bit shifted , which constitutes a signal that @xmath56 is still a low value to accurately mimic the limit bahc behaviour . the computations for the @xmath57 bahc were performed storing @xmath58 eigenvectors of the density matrix . , title="fig : " ] ii/ the physics of this 2-leg ladder is richer when the lattice has open boundary conditions . moreover , the numerical performance of the dmrg method is also better under these conditions . however , open boundary conditions must be handled with care in order to identify the gap @xmath50 we are after . we shall provide ways to do this identification by targeting appropriate low - lying states and measuring convenient observables with them . in particular , using open boundary conditions we have found that the first excited state lies within the sector with total z - spin angular momentum @xmath37 , but in the haldane phase it converges to the ground state that has @xmath59 as we take larger sizes of the system . we have then to consider the next excited state in the sector with @xmath38 to compute the gap of the spectrum as @xmath60 fits very well to a potential function of the form @xmath61 , with @xmath62 and @xmath63 . ] the reason for considering @xmath64 instead of @xmath65 as the gap of the system , can be justified as follows : in the complete dimerized limit @xmath66 , it is clear that the difference in energy between two arbitrary consecutive levels is the same , and corresponds exactly to the energy needed to promote one singlet bond to a triplet . the argument for the limit @xmath16 makes use of the properties of haldane phases , where it is known to appear a non bulk excitation due to the existence of virtual spins at the end of the chain . our conclusion is that the lowest lying state with @xmath67 consists on a superposition of two kind of excitations , namely the haldane non - bulk triplet mentioned before and the bulk itself , also giving a triplet . considering this scheme , in order to obtain the gap related to the bulk excitations we have to substract the non bulk excitations present in the lowest lying states of sectors @xmath68 and @xmath67 . in fig.[obcexcitations ] we show rigourous comparations of these two states in the haldane limit @xmath16 . computations have been done on a @xmath69 ladder at the point @xmath33 . on the _ up _ part of the figure , we plot the mean magnetization @xmath34 in the states with @xmath35 and @xmath36 , computed in one leg of the ladder , since due to the symmetry of the hamiltonian , the magnetization is the same in both legs . as a check of the accuracy of our computations we observed that the results in both legs are the same up to the fifth or sixth decimal digit . in the _ inset _ of that figure , we show the cumulative sum of the magnetization over the whole length of the ladder and the states with @xmath37 and @xmath38 . the order of the sites in the @xmath39-axis corresponds in this case to the path used to traverse the ladder in a dmrg sweep . in the _ down _ part of this figure , we plot the difference of energy density of the excited states with @xmath37 and @xmath38 : @xmath40 the difference has been divided into three contributions : the contribution labelled with _ even _ stands for links involving sites in the same leg and the even sublattice ( @xmath70 ) , _ odd _ involves links joining sites in the same leg and the odd sublattice ( @xmath71 ) , and _ perpendicular _ denotes links among legs ( @xmath72 ) . the cumulative sum of the difference of the various contribution , measured in the right axis scale , is also shown . interestingly enough , we can observe the magnetization pattern at the ends being almost identical in the states with @xmath73 and @xmath74 . the contribution to the @xmath75-axis projection of the spin coming from the ends is equal to 1 in both cases . notice also that the difference of the density of energy between these states is close to zero at the ends , while it becomes clearly apreciable in the bulk . all these facts strongly support the picture of a non - bulk triplet excitation with the same nature in both states , that leaves the bulk of the chain with a neat value of the projection equal to @xmath76 and @xmath77 and gives a strong hint on the equivalence of @xmath78 with periodic boundary conditions and @xmath64 for open systems . sites . the sop has been computed forming the triplets with adjacent @xmath14 spins located in different legs . clearly the value of this parameter is non - vanishing in the low region of @xmath15 , where the system is in the haldane phase . the dimer phase is nonetheless characterized by a vanishing sop._inset _ : sop computed for a @xmath18 bahc . the resemblance between both systems is evident in the region with @xmath79 . ] after this previous analysis to identify the states needed to target the gap of the system , we present in fig.[gap21 ] some values of the gap @xmath64 for a ladder consisting on @xmath43 sites , at different regions of the parameter space . computations have been performed retaining @xmath80 states of the density matrix and the grid used to explore the phase diagram is @xmath81 $ ] , and @xmath82 . the existence of a set of minima in the function @xmath54 is clear in this graph , although they shall become more distinguishable as we move to higher values of @xmath83 . as an instance of the accuracy of our results , we point out that a systematic examination of the error in each of the truncations of our dmrg computations reveals that the highest values in the whole process are of the order of @xmath84 , and mostly they are of order @xmath85 . to obtain a suitable acuracy in the results we have set the number of sweeps @xmath86 and the tolerance to @xmath87 . to compute with enough precision the critical value @xmath52 that minimizes the gap it becomes necessary to use large amount of data . on this regard , we have used interpolated values resulting from the dmrg computations . computed in the middle of a ladder of @xmath88 sites . since we are explicitly introducing some staggering in the hamiltonian , the dimerization parameter is non - vanishing even in the haldane phase . however , the shape of the graphs seem to have an inflexion at the critical point . the graph corresponding to the @xmath18 bahc , has been scaled down by a factor @xmath89 due to the effective coupling constant of the ladder , which is known to be half the constant corresponding to the bahc . ] now , we can detect the presence of a critical line in the quantum phase diagram separating gapped phases . in fig.[l500vsl2x140 ] we plot the critical region consisting of the coordinates for each minimum in fig.[gap21 ] . in earlier studies @xcite , we placed the critical point of the @xmath18 bond - alternating heisenberg chain ( bahc ) at @xmath27 . the curve shown in fig.[l500vsl2x140 ] shows a vertical asymptota that is still a bit off from this limiting value corresponding to the region @xmath79 , but this is simply because we have chosen a value of @xmath90 which is still not big enough and also due to finite - size effects on the 2-leg ladder . in the lower plots of fig.[l500vsl2x140 ] we address these possibilities by comparing our ladder in the strong ferromagnetic limit with a pure @xmath18 bahcs with different sizes . two parameters are important in this discussion , namely , the value @xmath52 that minimizes the gap , and the value of the gap itself at this point @xmath91 . as we can observe in fig.[gap21 ] , the value of @xmath91 does not strongly depend on the particular choice of the coupling constant ratio @xmath44 , while it is definitely influenced by the size of the system . in fig.[l500vsl2x140](lower ) it is shown that the shift of @xmath19 computed for two @xmath18 bahcs with different sizes , but still large enough both , is less noticeable than the difference in their value of @xmath91 . it is clear the similarity of this magnitude in the case of the ladder and the corresponding bahc , as well as the shift in the value of @xmath19 . all this make us conclude that in order to attain a better convergence with the @xmath18 bahc and a better estimate of the critical asymptota @xmath27 , we shall increase the strength of the ferromagnetic coupling rather than the size of the system . as we have a set of numerical data from the finite - size analysis of the critical line , we can also make a numerical estimation of the criticality curve . in fig . [ fit ] we present a fit of the critical curve in the region close to @xmath92 . we choose as trial function for this fitting an inverse power law with some coefficients and exponents that are fixed by our numerics , namely , @xmath93 the fitting yields the following estimations for the values of the parameters @xmath94 and @xmath95 that best fit the data : @xmath62 and @xmath96 , and for simplicity the value of @xmath27 is taken for fixed . in the ground state of a @xmath69 sites ladder and @xmath56 . the plot on the left shows the correlation value for some arbitrary values of @xmath15 . the plot on the right shows the value of this magnitude in the whole region of the parameter @xmath15 . ] once we have established the existence of a critical line in the quantum phase diagram of the model , it is natural to wonder about the two gapped phases that this line separates . more specifically , whether they are different or not and their identification as quantum phases in the framework of strongly correlated systems . the possible nature of those phases can be guessed from the strong ferromagnetic limit @xmath97 of the ladder , effectively leading to the @xmath18 bahc . the phases of this chain are known to be the massive haldane phase , separated by a critical point from the also massive dimer phase . to test the nature of each phase , we will resort to two different order parameters . the haldane phase is known to exhibit a particular hidden order that can be measured by the string order parameter ( sop ) @xcite , @xcite . the definition of this operator for a spin-1 chain is as follows : @xmath98 this operator acting on our ground state measures how far it is from a spin liquid nel state consisting on a sequence of @xmath18 spins such that every spin with projection @xmath99 is followed by @xmath100 and @xmath101 for every @xmath102 . when we deal with @xmath14 particles , to compute the sop we have to define the pairs of particles which are most likely to couple to give a triplet and compute the sop along the path connecting them . in our case , the existence of a ferromagnetic coupling clearly suggests that the triplets will result via this coupling . it is also worth recalling that the sop is a parameter suited to work with translational invariant systems . in order to correctly estimate the sop in open systems , we must restrict the computation to a region shorter than the whole length of the chain where end - effects are negligible and only bulk physics is relevant . in fig.[sop ] we show the sop computed traversing the path shown in fig.[laddersop ] . we can observe a non - vanishing sop in the haldane region , while it rapidly decays to zero in the dimer phase . the inset shows the sop computed for a @xmath18 bahc and the resemblance between both systems is apparent . therefore , the phase below the critical line in the numerical phase diagram of fig.[l500vsl2x140 ] is a gapped haldane phase . in the first lying excited state in the sector with @xmath37 of a @xmath69 ladder and @xmath56 . ] in the first lying excited state in the sector with @xmath38 of a ladder consisting on @xmath88 sites and @xmath56 . ] as for the region above the critical line in fig.[l500vsl2x140 ] , we have guessed from the strong coupling limit that it may be a dimer phase . the structure of a dimer phase is such that full translational invariance symmetry of the system is broken by one unit cell of the lattice . this situation can be detected by means of the dimerization parameter , which can be defined for our particular 2-leg ladder as @xmath103 ladder for the lowest lying states in the sectors @xmath37 ( _ top _ ) , and @xmath38 ( _ bottom _ ) . the curves are separated in the different sublattices consisting on the sites ocuppying odd or even positions . notice that both states present a peaked magnetization at the ends for @xmath104 well into the haldane phase , while it vanishes in the dimer phase with @xmath105 . , title="fig : " ] ladder for the lowest lying states in the sectors @xmath37 ( _ top _ ) , and @xmath38 ( _ bottom _ ) . the curves are separated in the different sublattices consisting on the sites ocuppying odd or even positions . notice that both states present a peaked magnetization at the ends for @xmath104 well into the haldane phase , while it vanishes in the dimer phase with @xmath105 . , title="fig : " ] the subindex @xmath8 is necessary since open systems are intrinsically not translationally invariant . in fig.[dimerization ] it is plotted the dimerization parameter of the ladder measured in the middle of the chain . since staggering is explicitly introduced into the hamiltonian , the order parameter vanishes only at @xmath16 , but is finite even in the haldane phase . nevertheless , our plots clearly exhibit different behaviours related to the convexity of the parameter at each phase . this observation indicates that an accurate estimation of the point of inflexion in the dimerization parameter could be used as a measure of the critical point separating both phases . we have also performed some measurements in the ladder to give more hints to understand the nature of both phases . figures [ colormap0 ] , [ colormap1 ] , and [ colormap2 ] show the correlation @xmath106 between sites in the perpendicular rungs . the pattern of the correlation can be understood by noticing that the correlation between two isolated @xmath14 spins coupled to give a singlet is @xmath107 while it equals @xmath108 if the spins form a triplet . from these values we observe that the perpendicular rungs in the ground state are forming triplets and the distribution is uniform all along the ladder . in the excited states however , the triplet strength of some rungs is weakened , signaling the presence of magnon - like excitations , also apparent in fig.[szgraph ] . the nature of the non - bulk excitation present in the haldane phase is not magnon - like and that explains the different number of kinks in the haldane and dimer phase in fig.[colormap1 ] and fig.[colormap2 ] . we have determined the existence of a critical line in the quantum phase diagram of a 2-leg ladder with columnar dimerization and ferromagnetic vs. antiferromagnetic couplings . in this study , we use the finite - size system dmrg method which allows us to give the location of that critical curve . moreover , we have clearly identified the two phases separated by the critical line to be a haldane phase and a dimer phase . this identification is carried out by measuring the string order parameter and the dimerization order parameter in the whole range of values of the coupling constant ratio @xmath44 and dimerization parameter @xmath15 . as a byproduct , we have introduced a systematic analysis of the spins at the borders of the open 2-leg ladder lattice . our model is based on @xmath3 spins , then these end - chain spins exhibit physical effects of their own . they are real spins unlike the virtual spins appearing in integer spin chains or ladders . their physics is specially interesting when the system size is finite , and even during the process of reaching the thermodynamic limit they produce non - trivial finite - size effects along the way . these facts difficult the technical analysis of the opening or closing of a gap in the low - lying spectrum of a 2-leg ladder with open boundary conditions . we have solved these difficulties by analyzing the ground state and low - lying energy excitations with respect to their bulk and boundary properties such as local magnetization and the like . with this information , it is possible to identify which states contribute to the gap in thermodynamic limit . these low - lying states have a definite total spin @xmath109 and they can be targeted with the dmrg method . in this fashion , we have been able to identify the gapped or gapless behaviour of the model within the framework of the dmrg with open boundary conditions . m. kohno , m. takahashi , m. hagiwara , `` low - temperature properties of the spin-1 antiferromagnetic heisenberg chain with bond alternation '' , physical review b , * 57 * , 1046 - 1051 ( 1998 ) . h. watanabe , `` numerical diagonalization study of an s=1/2 ladder model with open boundary conditions '' ; phys . b * 50 , ( 13442 - 13448 1994 ) . | we obtain the phase diagram in the parameter space @xmath0 and an accurate estimate of the critical line separating the different phases .
we show several measuments of the magnetization , dimerization , nearest neighbours correlation , and density of energy in the different zones of the phase diagram , as well as a measurement of the string order parameter proposed as the non vanishing phase order parameter characterizing haldane phases .
all these results will be compared in the limit @xmath1 with the behaviour of the @xmath2 bond alternated heisenberg chain ( bahc ) .
the analysis of our data supports the existence of a dimer phase separated by a critical line from a haldane one , which has exactly the same nature as the haldane phase in the @xmath2 bahc . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
in recent years , experimental studies of biological macromolecules have motivated intense research in the field of statistical mechanics of single semiflexible polymer chains . indeed , studies like force - elongation measurements of different biological ( dna , titin , tenascin ) and synthetic ( polyethylene glycol , polyvinyl alcohol ) polymers using atomic force microscopes , optical tweezers and other recently developed tools are abundant@xcite . the measured force - elongation curve is generally fitted to the prediction of the wormlike chain model ( wcm ) of semiflexible chains@xcite , originally proposed by kratky and porod@xcite . from this fit , parameters like the persistence length of the biopolymer are extracted . another kind of experiments has targeted the mechanical properties of eukaryotic cells@xcite . these properties are determined by an assembly of protein fibers called the cytoskeleton . this three dimensional assembly is made of the cytoskeletal polymers ( microtubules , actin filaments , etc . ) . all these polymers are semiflexible polymers at the relevant length scales ( a few microns at most ) . thus , the predictions of the wcm are very relevant for the understanding of the physical behavior of cytoskeletal polymers . the wcm was originally proposed by kratky and porod in 1949@xcite and reformulated using field theoretic methods by saito _ et al_.@xcite in 1967 . in this model , the polymer chain displays resistance to bending deformations . this resistance is modeled using a free energy that penalizes bending the polymer backbone . the free energy depends on parameters ( elastic constants ) that are a consequence of many short - range monomer - monomer interactions . explicitly , the free energy is @xmath3 where @xmath4 is the vectorial field that represents the polymer chain , @xmath5 is the arc of length parameter , @xmath6 is the contour length of the polymer and @xmath7 is the bending modulus . in addition , the local inextensibility constraint @xmath8 must be satisfied . as a consequence of the bending rigidity , a wormlike chain is characterized by a persistence length ( proportional to the bending modulus ) such that , if the length scale is shorter than the persistence length , then the chain behaves like a rod while , if the length scale is larger than the persistence length , then the chain is governed by the configurational entropy that favors the random - walk conformations . the local inextensibility constraint has not allowed researchers to find an exact solution to the wcm . indeed , the constraint @xmath8 is written using a dirac delta distribution in infinite dimensions . depending on how the constraint is written , @xmath8 or @xmath9 , we get an edwards hamiltonian that is non - analytic or non - linear , respectively . consequently , there is no exact solution of this model at present . however , a few properties like the first few moments of the distribution of the end - to - end distance@xcite are known exactly . the aforementioned complexity of the wcm has motivated many approximate treatments of semiflexible polymers . for example , fixman and kovac@xcite developed a modified gaussian model for stiff polymer chains under an external field ( external force ) . in this approach , they computed an approximate distribution for the bond vectors from which they were able to compute the partition function and average end - to - end vector . an alternative approach was proposed by harris and hearst@xcite who developed a distribution for the continuous model from which they were able to compute the two - point correlation function and , consequently , the mean - square end - to - end distance and radius of gyration . another statistical property of interest has been the distribution function of the end - to - end distance or its fourier transform . many approximations for this function have been proposed . for example , different expansions of the distribution in inverse powers of the number of segments have been developed@xcite . similarly , perturbations with respect to the rodlike limit have been derived@xcite . other approaches to the distribution function of the end - to - end distance of semiflexible polymers have led to modified gaussian functions@xcite . finally , many approximations have been proposed for the structure factor@xcite . as stated in the previous paragraph , most of the approximate treatments of the distribution function of the end - to - end distance have been perturbative in nature . indeed , these approximations were perturbation expansions with respect to the flexible or rigid chain limits . a different approach to semiflexible polymers was taken by kholodenko@xcite . in this model , the euclidean version of the dirac propagator is used to predict the conformational properties of semiflexible polymers . in particular , the single chain structure factor has been used to describe experimental data quantitatively@xcite . recently , winkler has proposed another treatment of semiflexible polymers@xcite . in this work , an approximate expression for the distribution function valid for any value of the stiffness of the polymer backbone was developed using the maximum entropy principle . motivated by the experimental studies done on semiflexible polymers and our incomplete understanding of the properties of the wcm , we have developed a new approach that captures many physical properties of the model , like the limits of flexible and rigid polymers , exactly and provides approximate crossover behaviors for all the distribution functions . to accomplish this goal , we have employed a computational technique called the generalized borel transform ( gbt ) which was taken from quantum mechanics and quantum field theories@xcite . this method computes mellin / laplace transforms exactly . we provide a brief summary of this technique in appendix a. this paper is organized as follows . in section ii , we evaluate the characteristic function of the wcm approximately such that some physical constraints are satisfied exactly . furthermore , we evaluate the distribution function ( polymer propagator ) using the gbt and compute the single chain structure factor . in section iii we discuss the results of our calculations which are valid for any value of the semiflexibility of the polymer . section iv contains the conclusions of our work and some speculations about applications to polymer physics of possible extensions of the gbt . the details of the mathematical calculations are presented in the appendices . consider a polymer chain modeled as a sequence of @xmath10 bond vectors @xmath11 connected in a sequential manner . in addition , let us assume that the length of each bond vector is @xmath12 ( = kuhn length ) and that pairs of consecutive bond vectors try to be parallel to each other . this preferential orientation is modeled with a boltzmann weight given by the following expression @xcite @xmath13 where @xmath14 is the strength of the interaction in units of thermal energy @xmath15 . inserting eq . ( [ eq : potential_energy ] ) into the expression for the propagator of the random flight model@xcite , we obtain the following expression for the polymer propagator of semiflexible chains @xmath16 where @xmath17 is the end - to - end vector and @xmath18 is given by the formula @xmath19 the propagator , eq . ( [ eq : definition ] ) , is not normalized . we proceed to express the delta function using its fourier representation@xcite then , eq . ( [ eq : definition ] ) becomes @xmath20},\end{array}\label{eq : propagator_fourier}\ ] ] which can be used to define the characteristic function , @xmath21 , as follows @xmath22 the mathematical expression of the characteristic function is @xmath23 note that since the polymer propagator , eq . ( [ eq : definition ] ) , is not normalized then the characteristic function , eq . ( [ eq : caracprim ] ) , does not approach one when the wave vector goes to zero . instead , it approaches the canonical partition function of the model . furthermore , note that the characteristic function is a fourier transform in a @xmath24-dimensional space . as stated before by yamakawa@xcite , the exact evaluation of the characteristic function ( or the polymer propagator ) for semiflexible chains is not possible at present . therefore , we have developed a new approximation to evaluate this function . this new mathematical approach was developed in such a way that the most relevant physics of the problem is not altered by the approximation . specifically , the proposed approach keeps the thermodynamics ( partition function ) of the model exact . moreover , all the properties of fully flexible chains @xmath25 and infinitely stiff chains @xmath26 are preserved exactly . consequently , this approach captures both asymptotic limits exactly and provides an approximate description of the crossover behavior . in addition , our treatment of the problem uses the exact expression of the mean - square end - to - end distance . consequently , this quantity and the mean - square radius of gyration are exact . another important property of our approach is that it keeps the local inextensibility constraint intact . therefore , our chains have finite extensibility and this model can be used to compute the force - elongation relationship of semiflexible polymers . we describe our approximation hereafter . let us start by computing the following class of integrals @xmath27 which are present in the characteristic function . the wave vector @xmath28 is constant and can be chosen in the direction of the versor @xmath29 . writing all the vectors in spherical coordinates we can express @xmath30 as follows @xmath31 where @xmath32 and @xmath33 are defined as follows @xmath34 the @xmath35-integrals can be done exactly . the result is @xmath36 where @xmath37 is the bessel function of second class@xcite . after we integrate the delta function , the function @xmath30 becomes @xmath38\cos\left(\theta_{j}\right)\right\ } i_{0}\left(\alpha l^{2}\left|\sin\left(\theta_{j}\right)\sin\left(\theta_{j+1}\right)\right|\right).}\label{eq : gj3}\ ] ] we now replace this expression into eq . ( [ eq : caracprim ] ) and obtain the characteristic function @xmath39\cos\left(\theta_{j}\right)\right\ } i_{0}\left(\alpha l^{2}\left|\sin\left(\theta_{j}\right)\sin\left(\theta_{j+1}\right)\right|\right)}.\end{array}\label{eq : characti}\ ] ] the evaluation of @xmath21 is done by iterations . first , we take the term @xmath40 , redefine @xmath41 and @xmath42 , and remove the factor @xmath43 from the definition of @xmath44 in eq . ( [ eq : gj3 ] ) ; then , we can write @xmath45\cos\left(\theta_{1}\right)\right\ } i_{0}\left(\alpha\left|\sin\left(\theta_{1}\right)\sin\left(\theta_{2}\right)\right|\right)},\label{eq : geuno}\ ] ] which is exactly doable@xcite . the result is @xmath46 the next step in the iterative process is the evaluation of @xmath47 given by @xmath48\cos\left(\theta_{2}\right)\right\ } i_{0}\left(\alpha\sin\left(\theta_{2}\right)\sin\left(\theta_{3}\right)\right)g_{1}\left(\theta_{2}\right)}.\label{eq : g2}\ ] ] this integral is not exactly doable . consequently , we proceed to approximate it such that the asymptotic limits of flexible and stiff polymers are captured exactly . thus , the expression that we obtain will give an approximate crossover behavior between the aforementioned limiting regimes . note that in the limit of very stiff chains , @xmath49 , all the segments will be parallel to each other . in other words , when @xmath49 , @xmath50 . then , in this limit we can say that @xmath51^{2}}.\label{eq : g2aprox}\ ] ] in the other limit , @xmath52 , @xmath53 is independent of @xmath54 . therefore , eq . ( [ eq : g2aprox ] ) is also valid in the limit of flexible chains . thus , we conclude that eq . ( [ eq : g2aprox ] ) is a good approximation for @xmath47 since it captures the asymptotic limits exactly and provides an approximate crossover behavior for @xmath47 . the iteration of the aforementioned approximation @xmath55 times leads to the following expression for the characteristic function @xmath56^{n-1}}.\label{eq : chac}\ ] ] note that this expression gives the exact canonical partition function of the model @xmath57 thus , this first part of the approximation preserves both asymptotic behaviors and the thermodynamics of the problem intact . let us now proceed to evaluate the approximate expression of the characteristic function , @xmath58 . the integral in eq . ( [ eq : chac ] ) is not exactly doable thus , we evaluate it using a variational procedure . let us introduce the following anzats @xmath59 where the parameters @xmath60 and @xmath61 are determined from the constraints imposed by the physics of the problem as described below . one of the requirements is that the flexible and rigid limits are captured exactly by the model . this requires that the parameters must behave in the following way @xmath62 using eq . ( [ eq : anzats ] ) , we can approximate the characteristic function as follows @xmath63\right\ } } } \\ { \displaystyle \times\frac{\left[\exp\left\ { \sqrt{\alpha^{2}-k^{2}\nu_{\alpha , n}^{2}}\right\ } -\exp\left\ { -\sqrt{\alpha^{2}-k^{2}\nu_{\alpha , n}^{2}}\left[1-\frac{{\displaystyle 2ikg_{\alpha , n}\cos\left(\theta_{n}\right)}}{{\displaystyle \sqrt{\alpha^{2}-k^{2}\nu_{\alpha , n}^{2}}}}\right]\right\ } \right]^{n-1}}{2^{n}\left[\sqrt{\alpha^{2}-k^{2}\nu_{\alpha , n}^{2}}\right]^{n-1}\left[1+\frac{{\displaystyle ikg_{\alpha , n}\cos\left(\theta_{n}\right)}}{{\displaystyle \sqrt{\alpha^{2}-k^{2}\nu_{\alpha , n}^{2}}}}\right]^{n-1}}}\end{array}\label{eq : characapprox}\ ] ] note that the term @xmath64 goes to zero as @xmath65 in the limit of @xmath49 and , when @xmath52 , it also approaches zero because @xmath66 . thus , neglecting this term does not alter the predictions of the model for the flexible and stiff limits . consequently , we approximate eq . ( [ eq : characapprox ] ) as follows @xmath67^{n-1}}{2\left[\sqrt{\alpha^{2}-k^{2}\nu_{\alpha , n}^{2}}\right]^{n-1}}}\\ { \displaystyle \times\int_{0}^{\pi}d\theta_{n}\sin\left(\theta_{n}\right)\exp\left\ { ik\cos\left(\theta_{n}\right)\left[1+\left(n-1\right)g_{\alpha , n}\right]\right\ } } , \end{array}\label{eq : characapprox1}\ ] ] which is exactly doable . the final expression for the characteristic function is @xmath68^{n-1}}{\left[\sqrt{\alpha^{2}-k^{2}\nu_{\alpha , n}^{2}}\right]^{n-1}k\left[1+\left(n-1\right)g_{\alpha , n}\right]}\sin\left\ { k\left[1+\left(n-1\right)g_{\alpha , n}\right]\right\ } .}\label{eq : chacfinal}\ ] ] we note that the expression of the characteristic function given by eq . ( [ eq : chacfinal ] ) recovers the exact expression of the canonical partition function of the model , eq . ( [ eq : cano ] ) , in the limit of @xmath69 . let us now proceed to determine the values of the parameters @xmath61 and @xmath60 from the physics of the problem . we first look at the force - elongation behavior predicted by this model . this curve is given by the following mathematical expression @xmath70\right\ } } { \partial f}=\frac{1}{k\left(if,\alpha , n\right)}\frac{\partial\left\ { k\left(if,\alpha , n\right)\right\ } } { \partial f},}\label{eq : elongacionsemiflex}\ ] ] where @xmath71 is the applied force and @xmath6 is the average end - to - end distance of the polymer chain in the direction of the force . the physics of the problem imposes the following constraint @xmath72 which represents the finite extensibility of the polymer chain . in other words , the polymer chain can not be stretched more than its total contour length . this constraint , as expressed by eq . ( [ eq : constrain ] ) , results in the following relationship between the parameters @xmath61 and @xmath60 @xmath73 which is in perfect agreement with the required asymptotic behaviors given by eq . ( [ eq : conditionasympt ] ) . equation ( [ eq : secondfit ] ) gives one of the two equations required to determine the parameters @xmath61 and @xmath60 completely . the second equation is obtained from the mean - square end - to - end distance , @xmath74 . we require that our approximation reproduce this statistical quantity exactly . the exact mathematical expression of this average is @xcite , @xmath75},\label{eq : rcuadrado}\ ] ] where @xmath76 is the langevin function @xcite . in order to derive the second relationship between @xmath61 and @xmath60 , we divide the characteristic function , eq . ( [ eq : chacfinal ] ) , by the canonical partition function and expand this ratio in powers of the wave vector @xmath77 to second order . the result is the following @xmath78^{2}+\frac{3\left(n-1\right)\nu_{\alpha , n}^{2}}{\alpha}\mathcal{l}\left(\alpha\right),}\label{eq : firstfit}\ ] ] which completes our approximation . equations ( [ eq : secondfit ] ) , ( [ eq : rcuadrado ] ) and ( [ eq : firstfit ] ) determine @xmath61 and @xmath60 completely . furthermore , the use of the exact expression for @xmath74 assures that our approximation predicts not only @xmath74 exactly , but also @xmath79 since they are related by the equation@xcite @xmath80 the final expression for @xmath61 is @xmath81\left(n^{2}-\left\langle r^{2}\right\rangle _ { \alpha , n}\right)}}{2\left[\left(n-1\right)^{2}+\frac{3\left(n-1\right)}{\alpha}\mathcal{l}\left(\alpha\right)\right]},}\label{eq : nu}\ ] ] and our approximation is complete . replacing the expression given by eq . ( [ eq : chacfinal ] ) into eq . ( [ eq : distri ] ) we obtain the following approximate expression for the polymer propagator @xmath82\right\ } \left[\sinh\left\ { \sqrt{\alpha^{2}-k^{2}\nu_{\alpha , n}^{2}}\right\ } \right]^{n-1}}{2\pi^{2}r\left[1+\left(n-1\right)g_{\alpha , n}\right]\left[\sqrt{\alpha^{2}-k^{2}\nu_{\alpha , n}^{2}}\right]^{n-1}}}.\label{eq : dis}\ ] ] in the limit of @xmath52 , eq . ( [ eq : dis ] ) becomes @xmath83^{n}}{k^{n-1}},}\label{eq : flex}\ ] ] which is the exact expression for the polymer propagator of the random flight model@xcite . similarly , in the limit of @xmath49 , we can perform the following expansion valid for large values of @xmath14 @xmath84^{n-1}}{\left[\sqrt{\alpha^{2}-k^{2}\nu_{\alpha , n}^{2}}\right]^{n-1}}\simeq\frac{\exp\alpha\left(n-1\right)\exp\left[-\left(n-1\right)k^{2}\nu_{\alpha , n}^{2}/\alpha\right]}{\left[2\alpha\right]^{n-1}},}\ ] ] and compute the polymer propagator . the result is @xmath85^{n-1}}\delta\left(r - n\right),}\label{eq : rigido}\ ] ] which is the polymer propagator of an infinitely stiff polymer chain@xcite . this propagator is not normalized . we now combine the approximate expression of the characteristic function , eq . ( [ eq : chacfinal ] ) , with the generalized borel transform to compute the polymer propagator and the single chain structure factor of the model . we start the evaluation of the polymer propagator by rewriting eq . ( [ eq : dis ] ) as follows @xmath86}},}\label{eq : propa}\ ] ] where @xmath87 is defined by the mathematical expression @xmath88\left(\frac{{\displaystyle \sinh\left(\sqrt{\alpha^{2}-w^{2}\nu_{\alpha , n}^{2}}\right)}}{{\displaystyle \sqrt{\alpha^{2}-k^{2}\nu_{\alpha , n}^{2}}}}\right)^{n-1}\right].}}\label{eq : pimf}\ ] ] this integral is evaluated exactly using gbt@xcite . a brief summary of the gbt technique can be found in appendix a. we present the most important steps of the calculation hereafter and leave all the mathematical details for appendix b. we first define an auxiliary function @xmath89 as follows @xmath90,\quad b\geq0}\label{eq : gb}\ ] ] from which the function @xmath87 can be computed as the analytic continuation to the complex plane @xmath91 consequently , @xmath87 can be evaluated from the laplace transform given by eq . ( [ eq : gb ] ) . following the technique we can write @xmath92 the @xmath93 integrations are computed using well - established properties and expansions of the functions @xmath94 and @xmath95^{-\mu}$]@xcite . after some straightforward algebra , we can write the analytical solution of @xmath89 for any even number of segments as follows @xmath96,\left[\beta+\frac{n+1}{2},\beta+\frac{n}{2}\right],-\frac{\left[\left(n-2k-1\right)\nu_{\alpha , n}\right]^{2}}{b^{2}}\right)}}{{\displaystyle \left(n-2k-1\right)^{-n-2\beta+1}2^{n-2}\gamma\left(2\beta+n\right)b}}}},\end{array}\label{eq : gbfinal}\ ] ] where @xmath97,\left[,\right],x\right)$ ] and @xmath98 are the generalized hypergeometric@xcite and gamma@xcite functions , respectively . replacing eq.([eq : gbfinal ] ) into eq.([eq : disg ] ) and computing the analytic continuation to the complex plane through the substitution @xmath99 we obtain @xmath100,\left[\beta+\frac{n+1}{2},\beta+\frac{n}{2}\right],\frac{\left[\left(n-2k-1\right)\nu_{\alpha , n}\right]^{2}}{r^{2}}\right)\right\ } } .\end{array}\label{eq : disim}\ ] ] the imaginary part of the generalized hypergeometric function is calculated using its well - known analytical properties@xcite . this function is an analytic function for values of the modulus of the argument @xmath101 less than one and its continuation to the rest of complex plane generates a cut on the positive real axis starting at @xmath102 . this implies that only values of the argument , @xmath103^{2}}{r^{2}}$ ] , larger or equal to one will contribute to the imaginary part of @xmath104 . consequently , this condition reduces the number of terms in the @xmath105sum such that the last term of eq.([eq : gbfinal ] ) is @xmath106 $ ] . the explicit evaluation of @xmath107 can be found in appendix c. finally , we replace eq.([eq : hyperim ] ) into eq.([eq : disim ] ) to obtain the exact expression for @xmath87 @xmath108}\left(-\right)^{k}\left(\begin{array}{c } n-1\\ k\end{array}\right)\sum_{\beta=0}^{\infty}\frac{\left(\alpha\right)^{2\beta}}{\left[\beta!\right]^{2}}\sum_{l=0}^{\beta}\left(\begin{array}{c } \beta\\ l\end{array}\right)}\\[0.2 in ] { \displaystyle \times\left(\frac{r}{\nu_{\alpha , n}}\right)^{l}\left(\frac{n}{2}-k-\frac{r}{2\nu_{\alpha , n}}-\frac{1}{2}\right)^{2\beta+n-2-l}\frac{{\displaystyle \left(2\beta - l\right)!}}{{\displaystyle \left(2\beta+n-2-l\right)!}}}.\end{array}\label{eq : disfin}\ ] ] note that eq . ( [ eq : disfin ] ) is just a sum of polynomials in @xmath109 . the sum over the index @xmath77 is the one obtained for the random flight model@xcite and imposes the finite extensibility of the polymer chain . the sums over the indexes @xmath33 and @xmath6 are a consequence of the stiffness of the polymer backbone . the expression of @xmath87 can be rewritten as follows @xmath108}\frac{\left(-\right)^{k}}{\left(n-3\right)!}\left(\begin{array}{c } n-1\\ k\end{array}\right)\left(\frac{n}{2}-k-\frac{r}{2\nu_{\alpha , n}}-\frac{1}{2}\right)^{n-2}}\\ { \displaystyle \times\int_{0}^{1}dz\,\left(1-z\right)^{n-3}\ , i_{0}\left(2\alpha\left(\frac{n}{2}-k-\frac{r}{2\nu_{\alpha , n}}-\frac{1}{2}\right)\sqrt{z^{2}+z\frac{\frac{r}{\nu_{\alpha , n}}}{\left(\frac{n}{2}-k-\frac{r}{2\nu_{\alpha , n}}-\frac{1}{2}\right)}}\right),}\end{array}\label{eq : disfin2}\ ] ] which can be used for further approximation if so desired . the expression for @xmath87 given by eq . ( [ eq : disfin ] ) was derived for even number of segments but , its validity for odd number of segments larger than two can be proved by analytic continuation . finally , in order to obtain the polymer propagator for semiflexible chains we have to replace eq . ( [ eq : disfin ] ) into eq . ( [ eq : propa ] ) . observe that , after the replacement , eq . ( [ eq : propa ] ) recovers the known exact solution of the random flight model@xcite when the limit @xmath110 is taken . indeed , the only term different from zero is the one for which @xmath111 . finally , we conclude our calculations of the wcm by computing the single chain structure factor which is defined by the following formula @xcite @xmath112 figures 1 , 2 and 3 show the prediction of the normalized polymer propagator , eq . ( [ eq : propa ] ) , as function of the end - to - end distance for chains with 5 , 10 and 30 kuhn segments , and different values of the semiflexibility parameter @xmath14 . the numerical evaluation of the propagator was done using eq . ( [ eq : disfin ] ) . the sum over the index @xmath33 converges quickly even for large values of the semiflexibility parameter @xmath14 . indeed , even with less than 80 terms in the sum we obtained a relative precision of @xmath113 . the figures clearly show that the location of the peak in the polymer propagator ( multiplied by @xmath114 ) moves toward larger values of @xmath115 when the stiffness of the polymer backbone increases . this behavior is in good qualitative agreement with previous results arising from computer simulation studies@xcite and theoretical approaches based on the maximum entropy principle@xcite . this is the correct result because the stiffer the polymer backbone , the higher the energetic penalty to bend the chain . consequently , those configurations of the macromolecule with small end - to - end distance will be more and more hindered as the stiffness increases while those configurations with large end - to - end distance should be more and more favored . therefore , the peak should shift toward larger values of @xmath115 when the stiffness increases . figure 4 shows the polymer propagator for polymer chains with 5 , 10 and 30 kuhn segments and a fixed value of the semiflexibility parameter @xmath116 . this figure shows that the longer the polymer is , the more it behaves like a flexible chain since the location of peak ( = end - to - end distance divided by the contour length ) moves toward smaller values . in other words , the longer the polymer is , the less relevant the stiffness of the backbone becomes . figures 5 , 6 and 7 show the behavior of @xmath117 , eq . ( [ eq : structuralfactor ] ) , as a function of @xmath77 for different values of @xmath14 and three values of @xmath10 ( 5 , 10 and 30 ) . the figures clearly show a decrease of the single chain structure factor with increasing @xmath77 until it reaches a plateau at infinite @xmath77 . note that our computations predict that the decrease of the single chain structure factor for small values of @xmath77 should be faster in the case of stiff polymers than in the case of flexible ones . this is a consequence of the fact that rigid polymers have a larger radius of gyration than flexible ones for a fixed chain length . in addition , the decrease of the structure factor for large values of @xmath77 is faster for flexible polymers than for stiff ones . indeed , our computations predict that the structure factor goes as @xmath1 for large values of @xmath77 just before the plateau is reached . for values of @xmath10=5 , 10 and 30 , the values of @xmath2 that we got were 1 , 1.08 and 1.3 for @xmath14=10 ( rigid ) and 1.64 , 1.9 and 2 for @xmath14=0.33 ( flexible ) . these results are in good agreement with the fact that the structure factor of polymers with large chain length should scale as @xmath118 for large @xmath77 values where @xmath119 is the fractal dimension of the object ( 2 for a flexible polymer and 1 for a rigid polymer ) . consequently , these results imply that for short polymer chains ( @xmath120 ) , a value of @xmath121 is high enough to make this polymer behave like stiff rod , @xmath122 . on the other hand , as the chain length increases we observe that @xmath121 is not high enough to make the polymer behave as a rigid rod and deviations from the power law @xmath122 are observed . in the case of @xmath123 our calculations predict that a value of @xmath10 equal to five is not high enough to recover the scaling behavior of flexible chains , @xmath124 . but , as the number of segments increases , the exponent approaches the value of two , indicating that the polymer chain behaves more and more like a flexible one . this result was also showed in figure 4 . the figures also show that the approximation we developed in the previous section gives a smooth crossover behavior from the rigid to the flexible limit . in order to make the presentation of our work more balanced and objective , we proceed to compare our results with the predictions of two other models . we start with our prediction for the single chain structure factor , @xmath125 , and compare it with the expression obtained by kholodenko@xcite . it has been shown that kholodenko s result can describe experimental data quantitatively@xcite . thus , a comparison between our expression for the single chain structure factor and kholodenko s will help us gauge the quality of the approximations used in our treatment of the wcm . figures 8 and 9 show the comparison for polymers with @xmath126 and two different values of the semiflexibility parameter ( shown in the plots ) . we have checked that other values of the parameters @xmath14 or @xmath127 and @xmath10 give quantitative agreements of similar quality . in all the cases studied we found that the relationship @xmath128 always gives excellent quantitative agreement between the predictions of both models . thus , our single chain structure factor should agree very well with the experimental data of ballauff and coworkers . still , since both models have different origins , some very small differences can be observed in the case of flexible chains ( figure 8) . we now proceed to compare our expression of the polymer propagator with the one computed by wilhelm and frey@xcite . figure 10 shows this comparison for the case of polymers with @xmath120 . the continuous curves are the results of our calculation with @xmath129 and @xmath130 . the dashed curves were constructed based on the work of wilhelm and frey where we adjusted the bending modulus such that the location of the peak in the propagator matched our results . this gives a better picture of the differences and similarities between both results . figure 10 shows quantitative agreement between both results when the stiffness is low . as the stiffness increases fig . 10 shows that the qualitative behavior of both propagators is still the same . for example , both results predict that the location of the peak moves toward larger values of the end - to - end distance and that the distribution becomes narrower . the main difference between both results is quantitative in nature . our distribution becomes narrower than the one predicted by wilhelm and frey s work by a factor of two approximately which , in turn , generates a higher peak ( the distributions are normalized ) . finally , let us conclude this section by rationalizing the origin of the discrepancy between the propagators and the reason why this does not affect the structure factor significantly . we first note that differences in the propagators are to be expected because both , ours and frey s , calculations are based on different approximations and versions of the model ( continuum or discrete ) . let us now proceed to rationalize the origin of the discrepancy between both predictions . equations ( [ eq : distri ] ) and ( [ eq : structuralfactor ] ) define the polymer propagator and single chain structure factor in terms of the characteristic function . observe that the propagator is a fourier transform of the characteristic function . consequently , the oscillatory nature of the complex exponential generates partial cancellation of the contributions to the integral arising from different parts of the interval of integration . this cancellation magnifies any inaccuracies made in the approximation of the characteristic function . moreover , the consequences of this cancellation become more and more important as the polymer becomes stiffer because the characteristic function itself adopts an oscillatory behavior which , in principle , is out of phase with respect to the complex exponential . therefore , the stiffer the polymer is , the more important the consequences of the approximation become . on the other hand , the evaluation of the structure factor from the characteristic function does not involve any oscillatory function . consequently , any small inaccuracy made in the approximation of the characteristic function will remain small in the expression of the structure factor , as shown previously . the results obtained in this paper show that the generalized borel transform is a very useful computational tool for the statistical mechanics of single semiflexible polymer chains . indeed , the results presented in this paper clearly show that gbt is able to compute polymer propagators for single chain problems exactly . this capability of the technique is a direct consequence of its mathematical simplicity ( the gbt requires elements of basic calculus and some fundamental knowledge of complex variables ) . consequently , it does not add any mathematical complexity to the physics of the starting model . our analysis of the wormlike chain model was based on an approximate expression of the characteristic function . the exact evaluation of this function is not possible at present . therefore , we developed a new approximation that preserves the most relevant physical characteristics of the model intact . specifically , our approach keeps the thermodynamics of the model , the flexible and rigid limits , the mean square end - to - end distance and the finite extensibility of the model intact while providing an approximate expression for the characteristic function for intermediate values of the stiffness of the polymer chain . the polymer propagator was obtained exactly from the approximate characteristic function using gbt . note that the propagator is approximate not because of the gbt , which computes this quantity exactly , but because of the approximate nature of the characteristic function . our expression for the propagator shows a peak that shifts toward larger values of the end - to - end distance as the stiffness of the polymer backbone is increased , in agreement with other theoretical and computational treatments of the model . we also found that , in the low wave vector region , the structure factor decreases faster with increasing wave vector when the stiffness increases . this was rationalized in terms of the behavior of the radius of gyration . similarly , we found that , in the large wave vector region , the structure factor of flexible chains decreases faster than the one of rigid polymers . this was compared with the behavior of very long polymer chains whose behavior for large wave vectors is known exactly . we also compared of the predictions of our calculation with established results for the single chain structure factor and polymer propagator . excellent quantitative agreement was observed between our prediction for the single chain structure factor and the one predicted by kholodenko s model . the polymer propagator was compared with the prediction of wilhelm and frey . very good quantitative agreement was observed between the predictions of both models for low values of the stiffness . for stiff polymers , quantitative deviations were observed and the origin of the deviations was rationalized . the proposed approach to semiflexible polymers can also address semiflexible polymers with other topologies like ring and m - arm star polymers . the procedure should be similar to the one presented in this paper but , the characteristic function will have a different mathematical expression . we conclude this section with a discussion of the characteristic function of the wcm which limits our ability to solve this model exactly . let us rewrite this function . the expression is @xmath131 where we have replaced the wave vector @xmath28 by a group of wave vectors @xmath132 . this expression is the one of the characteristic function when all the @xmath132 are equal to @xmath28 . observe that if we replace @xmath132 by @xmath133 where @xmath134 is the imaginary unit , then @xmath135 which has the form of a laplace transform in @xmath24 dimensions . but , the gbt computes laplace transforms very accurately or even exactly . consequently , a generalization of gbt to many dimensions might lead to an exact or very accurate expression of the characteristic function of the wcm . this expression can be further used to compute the polymer propagator using gbt . thus , we speculate that such extension of the gbt technique might allow us to solve very accurately or even exactly the wormlike chain model . in general , such extension of gbt might allow us to solve other models of polymer chains of the form @xmath136\right)},\ ] ] where @xmath137 $ ] is the hamiltonian of the model . thus , helical wormlike polymers and other models might be mathematically tractable with this generalization of the gbt . we acknowledge the national science foundation , grant # che-0132278 ( career ) , the ohio board of regents action fund , proposal # r566 and the university of akron for financial support . let us briefly present the mathematical aspects of the gbt in connection with the computation of the laplace - mellin transform@xcite . we start with the expression @xmath138 where we have explicitly extracted a factor @xmath139 from the function to be transformed . defining the generalized borel transform ( gbt ) of @xmath140 as @xmath141\left[\frac{1}{\lambda\eta}+1\right]^{-\lambda s}s\left(g , a , n\right)d\left(1/\eta\right),\quad re\left(s\right)<0\label{borel}\ ] ] where @xmath142 is any real positive non - zero value and @xmath143 , we can invert eq . ( [ borel ] ) in the following way @xmath144dwdt,\label{doble}\ ] ] where the explicit expression of @xmath145 is not important for our present purposes ( for more details see ref . @xcite ) . the expression given by eq . ( [ doble ] ) is valid for any non - zero , real and positive value of the parameter @xmath142 . but , the resulting expression for @xmath146 does not depend on @xmath142 explicitly . thus , we can choose the value of this parameter in such a way that it allows us to solve eq . ( [ doble ] ) . the dominant contribution to the double integral is obtained using steepest descent @xcite in the combined variables @xmath147 $ ] . in doing so , one first computes the saddle point @xmath148 and @xmath149 in the limit @xmath150 and then checks the positivity condition @xcite ( the hessian of @xmath151 at this point should be positive ) obtaining @xmath152=t_{o}\left(g , a , n\right)\quad,\quad w_{o}=\ln\left[x_{o}\left(g , a , n\right)\right]=w_{o}\left(g , a , n\right),\label{tw}\ ] ] where @xmath153 is the real and positive solution of the implicit equation coming from the extremes of the function @xmath151 in the asymptotic limit in @xmath142 . therefore , one obtains the following equation @xmath154,\label{gx}\ ] ] where @xmath155}{dx_{o}}.\label{fx}\ ] ] in the range of the parameters where @xmath156 which is fulfilled when @xmath157 , and assuming that there is only one saddle point , we can retain the first order in the expansion of @xmath151 around the saddle point . finally , we obtain the approximate expression for the starting function @xmath146 @xmath158 + 1}}{\sqrt{d\left[x_{o},a , n\right]}}\left[x_{o}\right]^{n+1}h\left[x_{o},a\right]\exp\left[-f\left[x_{o},a , n\right]\right],\label{eq : res}\ ] ] where @xmath159+f\left(x_{o},a , n\right)\left[1+f\left(x_{o},a , n\right)\right].\label{dx}\ ] ] note that the expression given by eq . ( [ eq : res ] ) is valid for functions @xmath160 that fulfill the following general conditions : \1 ) the relation given by eq . ( [ gx ] ) must be biunivocal . \2 ) @xmath161 has to be positive and @xmath162 $ ] has to be negative in @xmath163 . \3 ) @xmath156 . in particular , this condition is fulfilled when @xmath164 these conditions provide the range of values of the parameters where the approximate solution , eq . ( [ eq : res ] ) , is valid . in summary , the gbt provides an approximate solution , eq . ( [ eq : res ] ) , to amplitudes with the mathematical form given by eq . ( [ int ] ) . the calculation consists of solving the implicit equation eq . ( [ gx ] ) for @xmath157 to obtain the saddle point and replace it into eq . ( [ eq : res ] ) . let us now focus on amplitudes with the mathematical form of a laplace transform @xmath165 this kind of amplitudes can be mapped onto expressions of the form given by eq . ( [ int ] ) . in order to use the gbt on eq . ( [ eq : fi ] ) , we use the following relationship between eq . ( [ eq : fi ] ) and eq . ( [ int ] ) @xmath166 which can be inverted to give @xmath167 the finite sum comes from the indefinite integrations . note that all the coefficients vanish whenever the laplace transform , eq . ( [ eq : fi ] ) , fulfills the following asymptotic behavior @xmath168 in addition , the expression , given by eq . ( [ fiapro ] ) , is valid for any value of @xmath169 in particular for @xmath164 consequently , if eq . ( [ eq : asymp ] ) , is fulfilled , then the analytical solution reads @xmath170 where , for @xmath157 , we can use the expression given by eq . ( [ eq : res ] ) . it is important to note that it is the limit @xmath171 that makes the saddle point solution , eq . ( [ eq : res ] ) , an exact solution for eq . ( [ doble ] ) . thus , as long as the @xmath10 indefinite integrals can be done without approximations , as it is in our case , the result for @xmath140 is exact . we start the evaluation of the polymer propagator by rewriting eq . ( [ eq : gb ] ) as follows @xmath172\right\ } _ { c=0}}\\ { \displaystyle = \frac{\partial^{n-1}}{\partial c^{n-1}}\left\ { ga\left(b,\alpha , c\right)\right\ } _ { c=0}},\end{array}\label{eq : gb2}\ ] ] where @xmath173 is @xmath174},\label{eq : ga}\ ] ] and @xmath175 is given by the expression @xmath176 the integral expressed by eq.([eq : ga ] ) satisfies all the requirements of the gbt technique . then , we evaluate it in the following way @xmath177 where @xmath178}.\label{eq : gaproxeneuno}\ ] ] in the asymptotic limit of @xmath179 the gbt provides an analytical solution for eq.([eq : gaproxeneuno ] ) . following the technique , we solve the implicit equation , eq . ( [ gx ] ) , for the saddle point @xmath180 . the asymptotic solution is @xmath181 replacing this expression for @xmath180 in the expression provided by the gbt , eq . ( a7 ) , we obtain @xmath182 furthermore , we replace eq.([eq : gan ] ) into eq.([eq : gaprox ] ) and the resulting expression into eq.([eq : gb2 ] ) , and we exchange the order of the operators in the resulting expression . in other words , we first evaluate the @xmath10 derivatives with respect to @xmath183 and , afterward , we take the limit of @xmath184 to obtain the expression given by eq . ( [ eq : gelimite ] ) . next , we solve the @xmath93 integrations using standard properties and expansions of the functions @xmath94 and @xmath95^{-\mu}$]@xcite , and write @xmath89 as follows @xmath185 where @xmath10 is even and @xmath186^{r - n+1}}{r!}}\\ { \displaystyle \times\sum_{\beta=0}^{\infty}l_{\beta}^{-\frac{n - r-1}{2}-\beta}\left(0\right)\left(-\frac{\alpha^{2}}{n^{2}}\right)^{\beta}\gamma\left(n+1\right)\int db\cdots\int db\frac{1}{b^{2+n - n+r-2\beta}},}\end{array}\label{eq : eme}\ ] ] where @xmath187 are the laguerre polynomials @xcite . note that the only powers on @xmath188 in eq . ( [ eq : eme ] ) that fulfill the asymptotic behavior of the function @xmath89 are those for which the condition @xmath189 is satisfied . therefore , the @xmath93 indefinite integrations are exactly doable . the result is @xmath190 replacing eq . ( [ eq : nint ] ) into eq . ( [ eq : eme ] ) and after the change of variables @xmath191 we can write @xmath192 using the asymptotic properties of the gamma function@xcite @xmath193 eq . ( [ eq : eme2 ] ) finally reads @xmath194 the sum over @xmath195 is exactly doable . the result is @xmath196 where @xmath197,\left[\beta+\frac{n+1}{2},\beta+\frac{n}{2}\right],-\frac{\left[\left(n-2k-1\right)\nu_{\alpha , n}\right]^{2}}{b^{2}}\right)}{\gamma\left(2\beta+n\right)}}\\ { \displaystyle + \frac{2i\left(n-2k-1\right)\nu_{\alpha , n}}{b}\frac{\gamma\left(\beta+\frac{3}{2}\right)\,_{3}f_{2}\left(\left[\beta+\frac{3}{2},1,1\right],\left[\beta+\frac{n}{2}+1,\beta+\frac{n}{2}+\frac{1}{2}\right],-\frac{\left[\left(n-2k-1\right)\nu_{\alpha , n}\right]^{2}}{b^{2}}\right)}{\gamma\left(2\beta+n+1\right)\sqrt{\pi}}.}\end{array}\label{eq : fd}\ ] ] equation ( [ eq : eme3 ] ) clearly shows that the imaginary part affects only the real part of the function @xmath198 . thus , the final expression for @xmath89 is given by eq . ( [ eq : gbfinal ] ) . we evaluate @xmath107 using the following integral representation of the hypergeometric function@xcite @xmath199,\left[\beta+\frac{n+1}{2},\beta+\frac{n}{2}\right],-\frac{\left(n-2k-1\right)^{2}}{b^{2}}\right)=\frac{b^{2\left(\beta+1\right)}}{b\left(1,2\beta+n-1\right)}}\\ { \displaystyle \times\frac{1}{\left(n-2k-1\right)^{2\beta+n-1}}\int_{0}^{n-2k-1}\left[n-2k-1-x\right]^{2\beta+n-2}\left[x^{2}+b^{2}\right]^{-\beta-1}dx},\end{array}\label{eq : otravez}\ ] ] which is valid for values of @xmath10 equal or larger than two . the analytic continuation to the complex plane is done as before through the replacement @xmath99 . then , the imaginary part of the hypergeometric function is @xmath200,\left[\beta+\frac{n+1}{2},\beta+\frac{n}{2}\right],\frac{\left(n-2k-1\right)^{2}}{r^{2}}\right)\right\ } = } \\ { \displaystyle \frac{r^{2\left(\beta+1\right)}\left(-\right)^{\beta+1}}{b\left(1,2\beta+n-1\right)\left(n-2k-1\right)^{2\beta+n-1}}}\\ { \displaystyle \times im\int_{0}^{n-2k-1}\left[n-2k-1-x\right]^{2\beta+n-2}\left[x^{2}-r^{2}\right]^{-\beta-1}dx}.\end{array}\end{array}\label{eq : imaginarypart}\ ] ] the analytical behavior of the function @xmath204 is well known@xcite . it is an analytic function for @xmath205 but , its analytic continuation to the complex plane generates a cut on the real axis in the range @xmath206 which provides its imaginary part . writing @xmath222^{2}}\sum_{l=0}^{\beta}\left(\begin{array}{c } \beta\\ l\end{array}\right)\frac{\left(2\beta - l\right)!}{\left(2\right)^{2\beta+1-l}}}\\ { \displaystyle \frac{r^{l}\left[n-2k - r-1\right]^{2\beta+n-2-l}}{\left(2\beta+n-2-l\right)!}\quad n\geq2}.\end{array}\label{eq : hyperim}\ ] ] 10 m. g. poirier _ _ , _ phys . _ 86 , * 360 * ( 2001 ) ; j. f. leger _ et at . _ , _ phys . lett . _ * 83 * , 1066 ( 1999 ) ; v. parpura and j. m. fernandez , _ biophys . j. _ * 71 * , 2356 ( 1996 ) ; m. d. wang _ et al . _ , _ biophys . j. _ * 72 * , 1335 ( 1997 ) ; a. d. mehta , k. a. pullen and a. spudich , _ febs lett . _ * 430 * , 23 ( 1998 ) ; a. d. mehta , m. rief and j. a. spudich , _ j. biol . chem . _ * 274 * , 14517 ( 1999),t . e. fisher , p. e. marszalek and j. m. fernandez , _ nat . biol . _ * 7 * , 719 ( 2000 ) ; t. e. fisher _ et al . _ , _ trends biochem . * 24 * , 379 ( 1999 ) ; m. carrion - vazquez _ et al . . biophys . biol . _ * 74 * , 63 ( 2000 ) ; t. t. perkins , s. r. quake , d. e. smith and s. chu , _ science _ * 264 * , 822 ( 1994 ) ; s. b. smith , y. cui and c. bustamante , _ science _ * 271 * , 795 ( 1996 ) ; t. t. perkins , d. e. smith and s. chu , _ science _ * 276 * , 2016 ( 1997 ) ; a. f. oberhauser , p. e. marszalek , h. p. erickson and j. m. fernandez , _ nature ( london ) _ * 393 * , 181 ( 1998 ) ; f. oesterhelt , m. rief and h. e. gaub , _ new j. phys . _ * 1 * , 6.1 ( 1999 ) ; h. b. li , w. k. zhang , w. q. xu and x. zhang , _ macromolecules _ * 33 * , 465 ( 2000 ) . j. f. marko and e. d. siggia , _ macromolecules _ * 28 * , 8759 ( 1995 ) ; k. kroy and e. frey , _ phys . rev . lett . _ * 77 * , 306 ( 1996 ) . g. porod , _ monatsh . chem . _ * 80 * , 251 ( 1949 ) ; o. kratky and g. porod , _ rec . chim . _ * 68 * , 1106 ( 1949 ) . l. eichinger _ et al_. , _ biophys . j. _ * 70 * , 1054 ( 1996 ) ; p. janmey _ cell membranes and the cytoskeleton _ , vol . 1a of _ handbook of biological physics _ , chapter 17 , page 805 ( north holland , amsterdam , 1995 ) ; p. a. janmey , _ curr . op . cell . * 2 * , 4 ( 1991 ) . n. saito , k. takahashi and y. yunoki , _ journal of the physical society of japan _ , * 22 * , 219 ( 1967 ) . h. yamakawa , _ helical worm - like chains in polymer solutions _ ( springer , berlin , 1997 ) . m. fixman and j. kovac , _ j. chem . phys . _ * 58 * , 1564(1973 ) . r. a. harris and j. e. hearst , _ j. chem . phys . _ * 44 * , 2595 ( 1966 ) . h. e. daniels , _ proc . edinburgh _ * a63 * , 290 ( 1952 ) ; w. gobush , h. yamakawa , w. h. stockmayer and w. s. magee , _ _ * 57 * , 2839 ( 1972 ) . h. yamakawa , _ pure and appl . chem . _ * 46 * , 135 ( 1976 ) . j. wilhelm and e. frey , _ phys . lett . _ * 77 * , 2581 ( 1996 ) ; p. l. hansen and r. podgornik , _ j. chem . phys . _ * 114 * , 8637 ( 2001 ) . m. g. bawendi and k. f. freed , _ j. chem . phys . _ * 83 * , 2491 ( 1985 ) ; j. b. lagowski , j. noolandi and b. nickel , _ j. chem . phys . _ * 95 * , 1266 ( 1991 ) ; g. a. carri and m. muthukumar , _ j. chem . phys . _ * 109 * , 11117 ( 1998 ) . r. g. winkler , p. reineker and l. harnau , _ j. chem . * 101 * , 8119 ( 1994 ) . c. m. marques and g. h. fredrickson , _ ii france _ * 7 * , 1805 ( 1997 ) . l. harnau , r. g. winkler and p. reineker , _ j. chem . phys . _ * 104 * , 6355 ( 1996 ) ; k. kroy and e. frey , _ phys . rev . e _ * 55 * , 3092 ( 1997 ) . a. l. kholodenko , _ ann . * 202 * , 186 ( 1990 ) ; a. l. kholodenko _ faraday trans . _ * 91 * , 2473 ( 1995 ) ; a. l. kholodenko , _ phys . a _ * 141 * , 351 ( 1989 ) ; a. l. kholodenko , _ phys . lett . a _ * 178 * , 180 ( 1993 ) ; a. kholodenko and t. vilgis _ phys . rev . e _ * 50 * , 1257 ( 1994 ) , a. l. kholodenko and t. a. vilgis _ phys . rev . e _ * 52 * , 3973 ( 1995 ) . a. l. kholodenko , _ macromolecules _ * 26 * , 4179 ( 1993 ) . a. kholodenko , m. ballauff and m. aguero granados , _ physica a _ * 260 * , 267 ( 1998 ) . m. ballauff _ et al . _ , _ macromolecules _ * 30 * , 273 ( 1997 ) . r. g. winkler , _ phys . _ * 118 * , 2919 ( 2003 ) . l.n.epele , h. fanchiotti , c.a . garcia canal and m. marucho , _ phys . lett . _ * b523 * , 102 ( 2001 ) . l.n.epele , h. fanchiotti , c.a . garcia canal and m. marucho , _ phys . lett . _ * b556 * , 87 ( 2003 ) . l.n.epele , h. fanchiotti , c.a . garcia canal and m. marucho , _ nucl . * b583 * , 454 ( 2000 ) . s. m. bhattacharjee and m. muthukumar , _ j. chem . phys . _ * 86 * , 411(1987 ) . k. f. freed in _ advances in chemical physics _ , vol . 22 , pages 1 - 128 , i. prigogine and stuart a. rice editors ( john wiley and sons , new york , 1972 ) . g. arfken , _ mathematical methods for physicists _ ( academic press , inc . , new york , 1985 ) . m. abramowitz and i. stegun , _ handbook of mathematical functions _ ( dover , new york , 1970 ) . i.s . gradshteyn and i.m . ryzhik , _ table of integrals , series , and products _ ( academic press , new york , 2000 ) . h. benoit and p. doty , _ j. phys . chem . _ * 57 * , 958 ( 1953 ) . k. f. freed , _ renormalization group theory of macromolecules _ ( john wiley & sons , new york , 1987 ) . p. g. de gennes , _ biopolymers _ * 6 * , 715 ( 1968 ) . h. bateman , _ higher transcendental functions _ ( mc graw hill , new york , 1953 ) . m. c. wang and e. guth , _ j. chem . _ * 20 * , 1144 ( 1952 ) ; c. hsiung , h. hsiung and a. gordus , _ j. chem . phys . _ * 34 * , 535 ( 1961 ) . w. l. mattice and u. w. suter , _ conformational theory of large macromolecules : the rotational isomeric state model in macromolecular systems _ ( john wiley & sons , new york , 1994 ) . j. wilhelm and e. frey , _ phys . * 77 * , 2581 ( 1996 ) . n. bleistein and r. handelsman , _ asymptotic expansion of integrals _ ( dover , new york , 1986 ) . e. t. copson , _ asymptotic expansions _ ( cambridge university press , 1965 ) . h. jefrey and b. s. jeffrey , _ methods of mathematical physics _ ( cambridge university press , 1966 ) . * normalized polymer propagator @xmath223 versus @xmath224 for @xmath225 . continuous line @xmath226 , dotted line @xmath227 , dashed line @xmath228 , long dashed line @xmath229 and dashed - dotted line @xmath230 . * normalized polymer propagator @xmath223 versus @xmath224 for @xmath231 . continuous line @xmath226 , dotted line @xmath227 , dashed line @xmath228 , long dashed line @xmath229 and dashed - dotted line @xmath230 . * normalized polymer propagator @xmath223 versus @xmath224 for @xmath232 . continuous line @xmath226 , dotted line @xmath227 , dashed line @xmath228 , long dashed line @xmath229 and dashed - dotted line @xmath230 . * normalized polymer propagator @xmath223 versus @xmath224 for @xmath233 . continuous line @xmath234 , dashed line @xmath235 and dashed - dotted line @xmath236 . * single chain structure factor @xmath237 versus wave vector @xmath238 for @xmath120 . continuous line @xmath239 , dotted line @xmath240 , dashed line @xmath241 , dashed - dotted line @xmath242 and circles ( best fits to the power law in the range @xmath243 for @xmath244 and @xmath245 ) . * single chain structure factor @xmath237 versus wave vector @xmath238 for @xmath246 . continuous line @xmath239 , dotted line @xmath240 , dashed line @xmath241 , dashed - dotted line @xmath242 and circles ( best fits to the power law in the range @xmath243 for @xmath244 and @xmath245 ) . * single chain structure factor @xmath237 versus wave vector @xmath238 for @xmath126 . continuous line @xmath239 , dotted line @xmath240 , dashed line @xmath241 , dashed - dotted line @xmath242 and circles ( best fits to the power law in the range @xmath243 for @xmath244 and @xmath245 ) . * single chain structure factor @xmath117 versus wave vector @xmath77 for @xmath126 . ( line ) kholodenko s model with @xmath247 , ( points ) this work with @xmath248 . * single chain structure factor @xmath117 versus wave vector @xmath77 for @xmath126 . ( line ) kholodenko s model with @xmath249 , ( points ) this work with @xmath250 . * normalized polymer propagator @xmath251 versus end - to - end distance @xmath115 in units of kuhn length for @xmath120 . ( continuous lines ) this work , ( dashed lines ) wilhelm and frey s results . normalized polymer propagator @xmath252 versus @xmath253 for @xmath120 . continuous line @xmath254 , dotted line @xmath255 , dashed line @xmath256 , long dashed line @xmath257 and dashed - dotted line @xmath258 . normalized polymer propagator @xmath252 versus @xmath253 for @xmath246 . continuous line @xmath254 , dotted line @xmath255 , dashed line @xmath256 , long dashed line @xmath257 and dashed - dotted line @xmath258 . normalized polymer propagator @xmath252 versus @xmath253 for @xmath126 . continuous line @xmath254 , dotted line @xmath255 , dashed line @xmath256 , long dashed line @xmath257 and dashed - dotted line @xmath258 . single chain structure factor @xmath117 versus wave vector @xmath77 for @xmath120 . continuous line @xmath254 , dotted line @xmath255 , dashed line @xmath256 , dashed - dotted line @xmath258 and circles ( best fits to the power law in the range @xmath263 for @xmath123 and @xmath264 ) . single chain structure factor @xmath117 versus wave vector @xmath77 for @xmath246 . continuous line @xmath254 , dotted line @xmath255 , dashed line @xmath256 , dashed - dotted line @xmath258 and circles ( best fits to the power law in the range @xmath263 for @xmath123 and @xmath264 ) . single chain structure factor @xmath117 versus wave vector @xmath77 for @xmath126 . continuous line @xmath254 , dotted line @xmath255 , dashed line @xmath256 , dashed - dotted line @xmath258 and circles ( best fits to the power law in the range @xmath263 for @xmath123 and @xmath264 ) . | in this paper , we present a new approach to the discrete version of the wormlike chain model ( wcm ) of semiflexible polymers . our solution to the model
is based on a new computational technique called the generalized borel transform ( gbt ) which we use to study the statistical mechanics of semiflexible polymer chains .
specifically , we evaluate the characteristic function of the model approximately .
afterward , we compute the polymer propagator of the model using the gbt and find an expression valid for polymers with any number of segments and values of the semiflexibility parameter .
this expression captures the limits of flexible and infinitely stiff polymers exactly . in between ,
a smooth and approximate crossover behavior is predicted .
another property of our propagator is that it fulfills the condition of finite extensibility of the polymer chain .
we have also calculated the single chain structure factor .
this property is a decreasing function of the wave vector , @xmath0 until a plateau is reached .
our computations clearly show that the structure factor decreases faster with increasing wave vector when the semiflexibility parameter is increased .
furthermore , when the wave vector is large enough , there is a regime where the structure factor follows an approximate power law of the form @xmath1 even for short polymer chains .
@xmath2 is equal to two for flexible polymers and to one for rigid chains .
we also compare our results to the predictions of other models . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
the recovery problem of sparse vectors from a linear underdetermined set of equations has recently attracted attention in various fields of science and technology due to its many applications , for example , in linear regression @xcite , communication @xcite , @xcite , @xcite , multimedia @xcite , @xcite , @xcite , and compressive sampling ( cs ) @xcite , @xcite . in such a sparse representation problem , we have the following underdetermined set of linear equations @xmath12 where @xmath13 is @xmath14 is the dictionary @xmath15 is and @xmath16.[multiblock footnote omitted ] another way of writing is that a large dimensional sparse vector @xmath1 is coded / compressed into a small dimensional vector @xmath17 and the task will be to find the @xmath1 from @xmath17 with the full knowledge of @xmath7 . for this problem , the optimum solution is the sparsest vector satisfying . finding the sparsest vector is however np - hard ; thus , a variety of practical algorithms have been developed . among the most prominent is the convex relaxation approach in which the objective is to find the minimum @xmath4-norm solution to . for the @xmath4-norm minimization , if @xmath1 is @xmath18-sparse , which indicates that the number of non - zero entries of @xmath1 is at most @xmath18 , the minimum @xmath18 that satisfies gives the limit up to which the signal can be compressed for a given dictionary @xmath7 . an interesting question then arises : how does the choice of the dictionary @xmath7 affect the typical compression ratio that can be achieved using the @xmath4-recovery ? recent results in the parallel problem of cs , where @xmath7 acts as a sensing matrix , reveal that the typical conditions for perfect @xmath4-recovery are universal for all random sensing matrices that belong to the rotationally invariant matrix ensembles @xcite . the standard setup , where the entries of the sensing matrix are independent standard gaussian , is an example that belongs to this ensemble . it is also known that the conditions required for perfect recovery do not in general depend on the details of the marginal distribution related to the non - zero elements . on the other hand , we know that correlations in the sensing matrix can degrade the performance of @xmath4-recovery @xcite . this suggests intuitively that using a sample matrix of the rotationally invariant ensembles as @xmath7 is preferred in the recovery problem when we expect to encounter a variety of dense signals @xmath17 . however , the set of matrix ensembles whose @xmath6-recovery performance are known is still limited , and further investigation is needed to assess whether the choice of @xmath7 is indeed so straightforward . the purpose of the present study is to fulfill this demand . specifically , we examine the typical @xmath6-recovery performance of the matrices constructed by concatenating several randomly chosen orthonormal bases . such construction has attracted considerable attention due to ease of implementation and theoretical elegance @xcite , @xcite , @xcite for designing sparsity inducing over - complete dictionaries for natural signals @xcite . for a practical engineering scheme , audio coding ( music source coding ) @xcite uses a dictionary formed by concatenating several modified discrete cosine transforms with different parameters . by using the replica method in conjunction with the development of an integral formula for handling random orthogonal matrices , we show that the dictionary consisting of concatenated orthogonal matrices is also preferred in terms of the performance of @xmath6-recovery . more precisely , the matrices can result in better @xmath6-recovery performance than that of the rotationally invariant matrices when the density of non - zero entries of @xmath1 is not uniform among the orthogonal matrix modules , while the performance is the same between the two types of matrices for the uniform densities . this surprising result further promotes the use of the concatenated orthogonal matrices in practical applications . this paper is organized as follows . in the next section , we explain the problem setting that we investigated . in section 3 , which is the main part of this paper , we discuss the development of a methodology for evaluating the recovery performance of the concatenated orthogonal matrices on the basis of the replica method and an integral formula concerning the random orthogonal matrices . in section 4 , we explain the significance of the methodology through application to two distinctive examples , the validity of which is also justified by extensive numerical experiments . the final section is devoted to a summary . we assume that @xmath0 is a multiple number of @xmath19 ; namely , @xmath20 . suppose a situation in which an @xmath21 dictionary matrix @xmath7 is constructed by concatenating @xmath22 module matrices @xmath23 , which are drawn uniformly and independently from the haar measure on @xmath10 orthogonal matrices , as @xmath24 . \label{lorth}\end{aligned}\ ] ] using this , we compress a sparse vector @xmath25 to @xmath26 following the manner of ( [ eq : sparse_representation_without_noise ] ) . we denote @xmath27 for the concatenation of @xmath22 sub - vectors of @xmath19 dimensions as @xmath28 yielding the expression @xmath29 with full knowledge of @xmath30 @xmath31 and @xmath17 , the @xmath6-recovery is performed by solving the constrained minimization problem @xmath32 where @xmath33 for @xmath34 and @xmath35 generally denotes the minimization of @xmath36 with respect to @xmath37 and @xmath38 @xmath31 . at the minimum condition , @xmath39 constitutes the recovered vector @xmath40 in the manner of ( [ vector_union ] ) . for theoretically evaluating the @xmath6-recovery performance , we assume that the entries of @xmath41 , @xmath42 are distributed independently according to a block - dependent sparse distribution @xmath43 where @xmath44 means the density of the non - zero entries of the @xmath45-th block of the same size @xmath19 in @xmath27 and @xmath46 is a distribution whose second moment about the origin is finite , which is assumed as unity for simplicity . intuitively , as the compression rate @xmath47 decreases , the overall density @xmath48 up to which ( [ l1_recovery ] ) can successfully recover a typical sample of the original vector @xmath27 becomes smaller . however , precise performance may depend on the profile of @xmath49 . the above setting allows us to quantitatively examine how such block dependence of the non - zero density affects the critical relation between @xmath50 and @xmath51 for typically successful @xmath6-recovery of @xmath27 . expressing the solution of ( [ l1_recovery ] ) as @xmath52 where @xmath53 and @xmath54 , constitutes the basis of our analysis . equations ( [ posterior_mean ] ) and ( [ posterior ] ) mean that @xmath40 can be identified with the average of the state variable @xmath1 for the gibbs - boltzmann distribution ( [ posterior ] ) in the vanishing temperature limit @xmath55 . however , as ( [ posterior ] ) depends on @xmath56 and @xmath57 , further averaging with respect to the generation of these external random variables is necessary for evaluating the typical properties of the @xmath6-recovery . evaluation of such `` double averages '' can be carried out systematically using the replica method @xcite . in the replica method , we need to evaluate the average of @xmath58 for @xmath59 with respect to @xmath60 over the uniform distributions of @xmath10 orthogonal matrices . however , this is rather laborious and is easy to yield notational confusions . for reducing such technical obstacles , we introduce a formula convenient for accomplishing this task before going into detailed manipulations . similar formulae have been introduced for handling random eigenbases of symmetric matrices @xcite and random left and right eigenbases of rectangular matrices @xcite . let us assume that @xmath19-dimensional vectors @xmath61 @xmath31 are characterized by their norms as @xmath62 , where @xmath63 denotes the standard euclidean norm of the vector @xmath64 . for these vectors , we define the function @xmath65_{\{{\boldsymbol{o}}_t\ } } \right ) \cr & = & \lim_{m \to \infty } \frac{1}{m}\ln \left ( \frac{\int \left ( \prod_{t=1}^{t}{\cal d}{\boldsymbol{o}}_t \right ) \delta \left ( \sum_{t=1}^{t}{\boldsymbol{o}}_t { \boldsymbol{u}}_t \right ) } { \int \left ( \prod_{t=1}^{t}{\cal d}{\boldsymbol{o}}_t \right ) } \right ) , \label{formula_def}\end{aligned}\ ] ] where @xmath66_x$ ] generally denotes the average of @xmath36 with respect to @xmath37 , and @xmath67 denotes the haar measure of the @xmath10 orthogonal matrices . our claim is that by explicitly using @xmath68 , can be expressed as @xmath69 where @xmath70 generally denotes the extremization of function @xmath36 with respect to @xmath37 . expression ( [ formula_concrete ] ) is derived from the fact that for fixed @xmath71 , @xmath72 moves uniformly on the surface of the @xmath19-dimensional hypersphere of radius @xmath73 when @xmath30 varies according to the uniform distribution of the orthogonal matrices ; therefore , @xmath74 in ( [ formula_def ] ) can be replaced with a spherical measure of @xmath19-dimensional vector @xmath75 . for details , see [ appendix1 ] . function @xmath76 physically represents a characteristic exponent of the probability that @xmath19-dimensional vectors @xmath71 @xmath31 form a closed loop satisfying @xmath77 when they are independently and isotropically sampled under the norm constraints of @xmath78 . for small @xmath22 , the loop condition strongly restricts the region of @xmath79 to which @xmath76 is well defined . in concrete terms , @xmath76 diverges to minus infinity unless an equality @xmath80 and the triangular inequalities @xmath81 are satisfied for @xmath82 and @xmath83 , respectively . in particular , the constraint of ( [ t2 ] ) requires us to deal with the system of @xmath82 in a manner different from that of @xmath84 except for the case of @xmath85 , as discussed in the analysis of section 3.4 . now , we are ready to apply the replica method for analyzing the typical property of the @xmath6-recovery ( [ posterior_mean ] ) . for this , we evaluate the @xmath86-th moment of the partition function using the identity @xmath87 which is valid for only @xmath88 . in the large system limit @xmath89 , the rescaled logarithm of the moment , @xmath90_{\{{\boldsymbol{x}}_t^0\ } , \{{\boldsymbol{o}}_t\}}$ ] , can be accurately evaluated for all @xmath88 by using the saddle point method with respect to macroscopic variables @xmath91 , @xmath92 , and @xmath93 , where @xmath94 and @xmath95 . intrinsic permutation symmetry concerning the replica indices @xmath96 in ( [ power_partition ] ) guarantees that there exists a saddle point of the form @xmath97 , @xmath98 , and @xmath99 , which is often termed the replica symmetric ( rs ) solution . as a simple and plausible candidate , we adopt this solution as the relevant saddle point for describing the typical property of the @xmath6-recovery , the validity of which will be checked in section 3.5 . the detailed computation is carried out as follows . let us consider averaging ( [ power_partition ] ) with respect to @xmath57 and define for each fixed set of @xmath100 : @xmath101_{\{{\boldsymbol{o}}_t\ } } , \label{average_o}\end{aligned}\ ] ] where @xmath102 . when @xmath100 is placed in the configuration of the rs solution , the expression @xmath103^{\rm t } \times \left [ { \boldsymbol{u}}_t^1 \ { \boldsymbol{u}}_t^2 \ \ldots \ { \boldsymbol{u}}_t^n \right ] = \left ( \begin{array}{cccc } mr_t & mr_t & \cdots & mr_t \cr mr_t & mr_t & \cdots & mr_t \cr \vdots & \vdots & \ddots & \vdots \cr mr_t & mr_t & \cdots & m r_t \end{array } \right ) \cr & & = { \boldsymbol{e } } \times \left ( \begin{array}{cccc } m(r_t - r_t+nr_t ) & 0 & \cdots & 0 \cr 0 & m(r_t - r_t ) & \cdots & 0 \cr \vdots & \vdots & \ddots & \vdots \cr 0 & 0 & \cdots & m(r_t - r_t ) \end{array } \right ) \times { \boldsymbol{e}}^{\rm t}\end{aligned}\ ] ] holds for each @xmath45 , where @xmath104 stands for matrix transpose , @xmath105 , and @xmath106 . @xmath107 $ ] denotes an @xmath108 orthogonal matrix composed of the vector @xmath109 and an orthonormal set of @xmath110 vectors @xmath111 that are orthogonal to @xmath112 . this indicates that @xmath113 $ ] may be expressed as @xmath114= \left [ \tilde{{\boldsymbol{u}}}_t^1\ \tilde{{\boldsymbol{u}}}_t^2 \ \ldots \ \tilde{{\boldsymbol{u}}}_t^n \right ] \times { \boldsymbol{e}}^{\rm t } , \label{cordinate_conversion}\end{aligned}\ ] ] by using a set of @xmath86 orthogonal vectors @xmath115 , whose norms are given as @xmath116 and @xmath117 for @xmath118 , along with an @xmath119 orthogonal matrix @xmath120 that does not depend on @xmath45 . this guarantees the equality @xmath121 . furthermore , condition @xmath122 and the orthogonality of @xmath115 among the replica indexes @xmath96 allows us to evaluate the average concerning @xmath57 independently for each index @xmath123 when computing @xmath124 . this , in conjunction with ( [ formula_def ] ) , provides each set @xmath100 of the rs configuration with an expression of ( [ average_o ] ) as @xmath125 the right hand side of ( [ energy ] ) is likely to hold for @xmath126 as well , although ( [ average_o ] ) is defined originally for only @xmath88 . on the other hand , inserting identities @xmath127 , @xmath128 and @xmath129 @xmath130 into ( [ power_partition ] ) and taking an average concerning @xmath56 , in conjunction with integration with respect to dynamical variables @xmath100 , result in the expression @xmath131_{{\boldsymbol{x}}_t^0}\right ) \cr & & = \int d \hat{{\boldsymbol{q } } } \exp \left ( m ( a(\{q_t , q_t , m_t\ } , \{\hat{q}_t^a,\hat{q}_t^{ab},\hat{m}_t^a\ } ) + b(\{\hat{q}_t^a,\hat{q}_t^{ab},\hat{m}_t^a\ } ) \right ) , \label{volume}\end{aligned}\ ] ] for a fixed set of @xmath132 . the conjugate variable @xmath133 is introduced for expressing a delta function as @xmath134 , and similarly for @xmath135 and @xmath136 . notation @xmath137 stands for an integral measure @xmath138 , and the functions on the right hand side are defined as @xmath139 and @xmath140_{x_t^0 } \right ) , \end{aligned}\ ] ] where the average for @xmath141 is taken according to ( [ sparse_dist ] ) . the expression of ( [ volume ] ) indicates that its rescaled logarithm is accurately evaluated using the saddle point method with respect to the conjugate variables in the large system limit @xmath89 . in addition , the replica symmetry guarantees that the relevant saddle point is of the rs form as @xmath142 , @xmath143 , and @xmath144 . as a consequence , the evaluation yields @xmath145_{x_t^0 } \!\right ) \!\right \ } , \label{entropy}\end{aligned}\ ] ] where @xmath146 denotes the gaussian measure . this is also likely to hold for @xmath126 , although ( [ volume ] ) is originally defined for only @xmath88 . the replica method uses the identity @xmath147_{\{{\boldsymbol{x}}_t^0\ } , \{{\boldsymbol{o}}_t\ } } = -\lim_{n\to 0 } ( \partial /\partial n)(\beta m)^{-1 } \ln \left ( \left [ z^n(\beta;\{{\boldsymbol{x}}_t^0\ } , \{{\boldsymbol{o}}_t\}\right . $ ] @xmath148_{\{{\boldsymbol{x}}_t^0\ } , \{{\boldsymbol{o}}_t\ } } \right ) $ ] for evaluating the typical free energy density . the above argument indicates that @xmath149_{\{{\boldsymbol{x}}_t^0\ } , \{{\boldsymbol{o}}_t\ } } \right ) $ ] can be computed by extremizing the sum of ( [ energy ] ) and ( [ entropy ] ) with respect to @xmath132 . furthermore , the obtained expression is likely to hold for @xmath126 , although the calculations are based on ( [ power_partition ] ) that is valid for only @xmath88 . we , therefore , take the limit of @xmath150 utilizing the expressions of ( [ energy ] ) and ( [ entropy ] ) for @xmath151 $ ] as well . in particular , in the limit of @xmath55 , which is relevant in the current problem , the expression of the free energy of the vanishing temperature is expressed as @xmath152_{\{{\boldsymbol{x}}_t^0\ } , \{{\boldsymbol{o}}_t\ } } \cr & & = -\mathop{\rm extr}_{}\left \{\sum_{t=1}^{t}\left ( \frac{\partial f(\{\chi_k\})}{\partial \chi_t } ( q_t-2 m_t+\rho_t ) + \frac{\hat{q}_tq_t}{2}-\frac{\hat{\chi}_t\chi_t}{2}+\hat{m}_tm_t \right . \right .\cr & & \left . . \hspace*{6 cm } -\int dz \left [ \phi \left ( \sqrt{\hat{\chi}_t } z+ { \hat{m}_t } x^0 ; \hat{q}_t \right ) \right ] _ { x^0 } \right ) \right \ } , \label{free_energy}\end{aligned}\ ] ] where @xmath153 and rescaled variables are introduced as @xmath154 , @xmath155 , @xmath156 , and @xmath157 to properly describe the relevant solution in the limit of @xmath55 . extremization is to be performed with respect to @xmath158 . similar to earlier studies , at the extremum characterized by a set of the saddle point equations @xmath159 @xmath160_{x_t^0 } , \label{sp1}\\ & & \chi_t=\int dz \left [ \frac{\partial x^2(\sqrt{\hat{\chi}_t}z+\hat{m}_t x_t^0;\hat{q}_t)}{\partial ( \sqrt{\hat{\chi}_t}z ) } \right ] _ { x_t^0 } , \label{sp2}\\ & & m_t=\int dz \left [ x_t^0x(\sqrt{\hat{\chi}_t}z+\hat{m}_t x_t^0;\hat{q}_t ) \right ] _ { x_t^0 } , \label{sp3}\end{aligned}\ ] ] @xmath161 and @xmath162 @xmath31 physically denote the macroscopic averages of the recovered vector @xmath163 as @xmath164_{\{{\boldsymbol{x}}_k^0\},\{{\boldsymbol{o}}_k\}}$ ] and @xmath165_{\{{\boldsymbol{x}}_k^0\},\{{\boldsymbol{o}}_k\}}$ ] , respectively . here , @xmath166 is provided by the extremum solution of ( [ formula_concrete ] ) for @xmath167 , and @xmath168 for @xmath169 , a lagrange multiplier should be exceptionally introduced in ( [ sph2 ] ) for enforcing @xmath170 . the success of the @xmath6-recovery is characterized by the condition in which @xmath171 is satisfied at the extremum for @xmath172 . therefore , one can evaluate the critical relation between @xmath173 and @xmath174 by examining the thermodynamic stability of the success solution @xmath171 @xmath31 for the saddle point equations ( [ sph1])([sp3 ] ) . we assume @xmath175 for a while , since an exceptional treatment is required for @xmath169 . for obtaining the success solution , it is necessary that @xmath176 and @xmath177 hold . expanding ( [ sp1])([sp3 ] ) under the assumption of @xmath178 yields @xmath179 where @xmath180 , @xmath181 and we used @xmath182_{x_t^0}=\rho_t$ ] , which is derived from the assumption ( [ sparse_dist ] ) . these result in the expression @xmath183 in addition , ( [ formula_concrete ] ) indicates that @xmath184 holds , where @xmath185 is the solution of @xmath22 coupled equations @xmath186 differentiating this yields @xmath187 where we denote @xmath188 and @xmath189 this expression makes it possible to evaluate @xmath190 as @xmath191 where @xmath192 , @xmath193 and @xmath194 . inserting ( [ mse ] ) , ( [ hessian ] ) , and ( [ hessian_inverse ] ) into ( [ sph2 ] ) yields a set of equations to determine @xmath195 for a given set of non - zero densities @xmath196 @xmath197 where we set @xmath198 and used the relation @xmath199 , which is obtained from ( [ sph1 ] ) and ( [ chi_lambda ] ) . equations ( [ sph2 ] ) and ( [ chiasympt ] ) indicate that the critical condition for making @xmath200 stable is expressed as @xmath201 furthermore , the condition @xmath202 must hold by the definition of @xmath203 . for characterizing the critical condition for the success of the @xmath6-recovery , let us suppose that the set of non - zero densities @xmath196 is provided as a function of a single parameter @xmath204 as @xmath205 . for @xmath206 , the critical situation is specified by an appropriate set of @xmath207 variables of @xmath204 , @xmath195 , and @xmath208 . these are provided by @xmath207 conditions of ( [ critical1])([critical3 ] ) . on the other hand , the critical condition for @xmath169 is provided differently from ( [ critical1])([critical3 ] ) because the constraint of @xmath170 for keeping @xmath209 well defined requires @xmath210 for any pair of @xmath211 and @xmath212 . explicitly , the condition is provided by the following four coupled equations : @xmath213 where @xmath214 is a lagrange parameter for enforcing @xmath170 . these determine four variables of @xmath215 at the critical condition . equations ( [ critical1])([critical3 ] ) for @xmath175 and ( [ t2critical1])([t2critical4 ] ) for @xmath169 constitute the main result of this paper . for @xmath169 , an equivalent result can also be obtained in a slightly different manner @xcite . the above calculation can be generalized to arbitrary levels of replica symmetry breaking ( rsb ) . for example , under one - step rsb ( 1rsb ) ansatz , where @xmath86 replicas of each block @xmath45 are classified into @xmath216 groups of an identical size @xmath217 and their overlaps are assumed to be @xmath218 if @xmath123 and @xmath219 belong to the same group and @xmath220 otherwise , ( [ energy ] ) and ( [ entropy ] ) are modified as @xmath221 and @xmath222_{x_t^0 } \!\right \ } , \label{entropy1rsb}\end{aligned}\ ] ] respectively , where @xmath223 . in the 1rsb framework , the rs solution is regarded as a special solution of @xmath224 and @xmath225 ( @xmath95 ) . in the limit of @xmath226 and @xmath55 keeping @xmath227 , the critical condition that a solution of @xmath228 and @xmath229 bifurcates from the rs solution , which corresponds to the de almeida - thouless ( at ) condition @xcite of the current system , is expressed as @xmath230 where @xmath231 and @xmath232_{x_t^0 } ) $ ] . for @xmath206 , @xmath233 holds for the success solution at the critical condition of the @xmath6-recovery . equation ( [ hhath ] ) yields an eigenvalue of unity whose eigenvector is given as @xmath234 , which makes ( [ at ] ) hold . similarly , ( [ at ] ) is also satisfied at the critical condition of the @xmath6-recovery of @xmath169 . these validate our rs evaluation in terms of the local stability analysis , although further justification with other schemes , such as comparison with numerical experiments , is necessary for examining possibilities that the rs solution becomes thermodynamically irrelevant due to discontinuous phase transitions . let us examine the significance of the developed methodology by applying it to two representative examples . we consider the uniform density case of @xmath235 ( @xmath95 ) as the first example , where the uniformity allows us to solve ( [ sph1])([sp3 ] ) setting all variables to be independent of @xmath45 as @xmath236 . in particular , setting @xmath237 simplifies the expressions of ( [ sph1])([sph3 ] ) providing @xmath238 and @xmath239 . this makes it unnecessary to deal with the saddle point problems of @xmath169 in an exceptional manner . as a consequence , the critical condition of the @xmath6-recovery is expressed compactly by using a pair of equations as @xmath240 for both @xmath241 and @xmath169 . by setting @xmath242 , these provide a critical condition identical to that obtained for the rotationally invariance matrix ensembles in earlier studies @xcite . this indicates that for vectors of the uniform non - zero density , the @xmath6-recovery performance of the concatenated orthogonal matrices is identical to that of the standard setup provided by the matrix of independent standard gaussian entries . however , this is not the case when the non - zero density is not uniform . as a distinctive example , we examined the case of localized density , which is characterized by setting @xmath243 and @xmath244 for @xmath245 . table [ table ] and figure [ figure1 ] show critical values of the total non - zero density @xmath48 given the compression rate @xmath242 for the uniform and localized density cases . these show that the concatenated matrices always result in better @xmath6-recovery performance for vectors of the localized densities , and the significance increases as @xmath22 becomes smaller while matrices of rotationally invariant ensembles result in identical performance as long as @xmath51 is unchanged . this indicates that , in addition to their ease of implementation and theoretical elegance , the concatenated orthogonal matrices are preferred for practical use in terms of their high recovery performance for vectors of non - uniform non - zero densities . c|cccccccc @xmath22&2&3&4&5&6&7&8 uniform & 0.1928 & 0.1021 & 0.0668 & 0.0487 & 0.0378 & 0.0308 & 0.0257 localized & 0.2267 & 0.1190 & 0.0780 & 0.0566 & 0.0438 & 0.0354 & 0.0294 ( color online ) critical values @xmath246 of total non - zero density versus compression rate @xmath50 . circles and crosses correspond to the localized and uniform densities , respectively , for @xmath247 . the curve represents the relation between @xmath246 and @xmath50 for the rotationally invariance matrix ensembles . crosses coincide with values of the curve for @xmath242 ( @xmath248 ) . ] to justify our theoretical results , we conducted extensive numerical experiments of the @xmath6-reconstruction . figures [ figure2 ] ( a ) and ( b ) depict the experimental assessment of the critical threshold for @xmath169 and @xmath249 , respectively . the case of an i.i.d . standard gaussian dictionary is also plotted for comparison . given fixed values of @xmath22 and @xmath0 , a trial was started with an empty vector @xmath27 and a concatenated orthogonal dictionary generated from a set of @xmath22 standard i.i.d . @xmath10 gaussian matrices using qr - decomposition . based on the relative densities @xmath250 , one sub - vector @xmath41 was then randomly chosen and assigned a non - zero component drawn from the standard gaussian ensemble . matlab algorithm `` linprog '' from optimization toolbox was used to solve the @xmath6-minimization problem and obtain the reconstruction @xmath40 . the reconstruction was deemed to be a success if @xmath251 and a failure otherwise . given a successful reconstruction , we again randomly chose one sub - vector @xmath41 based on the densities @xmath250 and inserted a non - zero component drawn independently from the standard gaussian ensemble into it . the process was continued until the original vectors @xmath252 had @xmath253 non - zero components and the reconstruction was deemed a failure , that is , @xmath254 . the critical value @xmath255 was recorded and the experiment was started again using a new independent dictionary and an empty vector @xmath27 . for each value of @xmath22 and @xmath0 , we carried out @xmath256 independent trials . the experimental critical density was defined as @xmath257 , where @xmath258 denotes the arithmetic average over the trials . for all system sizes , we also computed the experimental per - block densities @xmath259 and checked that they were close to the desired densities @xmath250 after the @xmath256 trials . for fixed @xmath22 , the experimental data points @xmath260 were fitted with a quadratic function of @xmath261 . extrapolation for @xmath262 provided the experimental estimates of the critical densities , as listed in table [ table2 ] in which the theoretical estimates in table [ table ] are also listed for comparison . ( color online ) experimental assessment of critical densities for @xmath6-reconstruction . experimental data ( see the main text ) were fitted with a quadratic function of @xmath261 and plotted with solid lines . concatenated orthogonal `` o '' and i.i.d . standard gaussian `` g '' basis under uniform and localized densities . filled markers represent the predictions obtained through the replica analysis . extrapolation for @xmath263 provides the estimates for the critical values @xmath246 . ( a ) @xmath264 , where the markers correspond to simulated values @xmath265 , and ( b ) @xmath266 , where the markers correspond to simulated values @xmath267 . , title="fig : " ] ( color online ) experimental assessment of critical densities for @xmath6-reconstruction . experimental data ( see the main text ) were fitted with a quadratic function of @xmath261 and plotted with solid lines . concatenated orthogonal `` o '' and i.i.d . standard gaussian `` g '' basis under uniform and localized densities . filled markers represent the predictions obtained through the replica analysis . extrapolation for @xmath263 provides the estimates for the critical values @xmath246 . ( a ) @xmath264 , where the markers correspond to simulated values @xmath265 , and ( b ) @xmath266 , where the markers correspond to simulated values @xmath267 . , title="fig : " ] c|ccccc @xmath22&2&3&4&5 uniform ( experiment ) & 0.1927 & 0.1019 & 0.0670 & 0.0487 uniform ( theory ) & 0.1928 & 0.1021 & 0.0668 & 0.0487 localized ( experiment ) & 0.2264 & 0.1196 & 0.0779 & 0.0567 localized ( theory ) & 0.2267 & 0.1190 & 0.0780 & 0.0566 comparing the theoretical and experimental results confirms the accuracy of the replica analysis . from figure [ figure2 ] , we observe that for the finite - sized systems , the @xmath22 orthogonal dictionaries seem to always provide higher thresholds @xmath268 than the gaussian one , even for uniform densities . in summary , we investigated the performance of recovering a sparse vector @xmath269 from a linear underdetermined equation @xmath270 when @xmath7 is provided as a concatenation of @xmath271 independent samples of @xmath10 random orthogonal matrices and the @xmath6-recovery scheme is used . performance was measured using a threshold value of the density @xmath51 of non - zero entries in the original vector @xmath27 , below which the @xmath6-recovery is typically successful for given compression rate @xmath272 . for evaluating this , we used the replica method in conjunction with the development of an integral formula for handling the random orthogonal matrices . our analysis indicated that the threshold is identical to that of the standard setup for which matrix entries are sampled independently from identical gaussian distribution when the non - zero entries in @xmath27 are distributed uniformly among @xmath22 blocks of the concatenation . however , it was also shown that the concatenated orthogonal matrices generally provide higher threshold values than the standard setup when the non - zero entries are localized in a certain block . results of extensive numerical experiments exhibited excellent agreement with the theoretical predictions . these mean that , in addition to their ease of implementation and theoretical elegance , the concatenated orthogonal matrices are preferred for practical use in terms of their high recovery performance for vectors of non - uniform non - zero densities . promising future studies include performance evaluation in the case of noisy situations and development of approximate recovery algorithms suitable for the concatenated orthogonal matrices the authors would like to thank erik aurell , mikael skoglund , and lars rasmussen for their useful comments . we also thank csc it center for science ltd . for the allocation of computational resources . this work was partially supported by grants from the japan society for the promotion of science ( kakenhi nos . 22300003 and 22300098 ) ( yk ) and swedish research council under vr grant 621 - 2011 - 1024 ( mv ) . when @xmath30 are sampled independently and uniformly from the haar measure of the @xmath273 orthogonal matrices , @xmath274 @xmath275 are distributed independently and uniformly on the surfaces of the @xmath19-dimensional hyperspheres of radius @xmath73 for a fixed set of @xmath19-dimensional vectors @xmath71 satisfying @xmath276 . this means that the integral of ( [ formula_def ] ) can be expressed as @xmath277 we insert the fourier expressions of @xmath278-function @xmath279 and @xmath280 into the numerator of ( [ spherical_integration ] ) , and carry out the integration with respect to @xmath281 , where @xmath282 . this yields the expression @xmath283 evaluating this by means of the saddle point method with respect to @xmath284 @xmath285 results in @xmath286 similarly , the denominator of ( [ spherical_integration ] ) is evaluated as @xmath287 substituting ( [ spherical_integration ] ) , ( [ numer ] ) , and ( [ denomi ] ) into ( [ formula_def ] ) leads to the expression of ( [ formula_concrete ] ) . 10 url # 1#1urlprefix[2][]#2 miller a 2002 _ subset selection in regression ( second edition ) _ ( chapman and hall / crc ) | we consider the problem of recovering an @xmath0-dimensional sparse vector @xmath1 from its linear transformation @xmath2 of @xmath3 dimension . minimizing
the @xmath4-norm of @xmath1 under the constraint @xmath5 is a standard approach for the recovery problem , and earlier studies report that the critical condition for typically successful @xmath6-recovery is universal over a variety of randomly constructed matrices @xmath7 . for examining the extent of the universality
, we focus on the case in which @xmath7 is provided by concatenating @xmath8 matrices @xmath9 drawn uniformly according to the haar measure on the @xmath10 orthogonal matrices . by using the replica method in conjunction with the development of an integral formula for handling the random orthogonal matrices ,
we show that the concatenated matrices can result in better recovery performance than what the universality predicts when the density of non - zero signals is not uniform among the @xmath11 matrix modules .
the universal condition is reproduced for the special case of uniform non - zero signal densities .
extensive numerical experiments support the theoretical predictions . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
[ p. monsky , 1970 ] a square can not be cut illustrates this . ] into an odd number of triangles of equal areas . the only known proof of this theorem was published by monsky in 1970 @xcite . the proof is based on two key ideas : the sperner s lemma and the coloring of the plane in three colors based on a 2adic valuation . after that , several generalizations of monsky s results appear . the first generalization was conjectured by stein and proved by monsky in 1990 @xcite . it claims that a centrally symmetric polygon can not be cut into an odd number of triangles of equal areas . though it is based on the same idea of 3-coloring , this proof is technically more challenging than the proof in the case of the square and uses a non - trivial homological technique . in 1994 bekker and netsvetaev proved similar statement in higher dimensions @xcite . to state another generalization we need a definition . let us call a finite union of squares of area 1 with integer coordinates of vertices a _ polyomino_. first , stein proved in 1999 @xcite that a polyomino of odd area can not be cut into an odd number of triangles of equal areas , and in 2002 praton @xcite proved the same for an even - area polyomino . in 2000 stein @xcite made a conjectural generalization of theorem 1 , see also @xcite . let @xmath0 be a plane polygon with clockwise oriented boundary . @xmath0 is called _ balanced _ if its edges can be divided into pairs so that in each pair edges are parallel , equal in length and have opposite orientation ( the edges are oriented , their orientation comes from the orientation of the boundary ) . now we are ready to formulate the stein conjecture . [ s. stein , 2000 ] [ stein conjecture ] a balanced polygon can not be cut into an odd number of triangles of equal areas . in this note we will present a proof of a partial case of conjecture [ stein conjecture ] . namely , we will prove the following theorem . [ non - equidissectibility of a balanced lattice polygon ] [ main ] consider a balanced polygon @xmath0 of the integer odd area and assume that the coordinates of all the vertices are integer numbers . then @xmath0 can not be cut into an odd number of triangles of equal areas . for an example of a balanced lattice polygone of area 15 , see fig . [ lattice ] . the proof of nonexistence of equidissection of a balanced lattice polygon consists of several steps . in section @xmath1 we review the coloring of the plane in three colors introduced by monsky . in section @xmath2 we introduce a notion of a _ degree _ of a broken line . it is an integer that depends both on a coloring and a broken line . we prove that if a polygon can be cut into triangles with nonnegative @xmath1adic valuations of araes , then its degree is @xmath3 . in section @xmath4 previous results are applied to the case of a lattice polygon . the proof of the nonexistence of equidissection of a balanced lattice polygon is finished in section @xmath5 . in the appendix we show connections between tropical geometry and @xmath2-colorings of projective plane . . my gratitude goes to sergei tabachnikov for inspiring me to write this article . also to sherman stein for proposing the conjecture and for his constructive criticism of my nascent ideas . i am especially grateful to nikolai mnev , without whose guidance and support this article would not have been possible . the main tool for us will be a special type of coloring of a plane in 3 colors . to begin with , let us recall the notion of a discrete valuation and sketch its basic properties . a function @xmath6 is a 2adic valuation on the field of real numbers if for any two numbers @xmath7 the following properties hold : property 1 : @xmath8 , @xmath9 , property 2 : @xmath10 , property 3 : if @xmath11 then @xmath12 , property 4 : @xmath13 , property 5 : it extends standart 2adic valuation on rationals : @xmath14 the existence of such a function follows from the theorem of the extension of valuations , see @xcite . this function is not unique and its construction is based on the axiom of choice . our goal now is to construct a family of 3colorings of a plane ( we will call these colorings `` tropical '' ) with two properties : * ( p 1 ) * on any line points of only two colors occur . * ( p 2 ) * for any triangle with vertices having all 3 different colors its area has a negative 2adic valuation . let us color points in the plane in three colors @xmath15 according to the following rule : a point @xmath16 with coordinates @xmath17 is colored in color @xmath18 , if @xmath19 , in color @xmath0 , if @xmath20 , in color @xmath21 , if @xmath22 . + this defines a map from the plane to a three - element set @xmath23 on fig . 3 the way of coloring is presented in coordinates @xmath24 . for any area - preserving affine transformation @xmath25 , we can define another coloring @xmath26 by the rule @xmath27 this defines a family of 3colorings , which we will call tropical . [ one ] for the 3coloring @xmath26 properties * p1 * and * p2 * hold . * p2 * @xmath28 * p1*. + if there were three points of different colors on the same line , they would form a triangle of area @xmath3 . because @xmath29 , this is impossible . + * p2*. for a coloring @xmath26 we need to prove that for any triangle @xmath30 , whose image under @xmath31 has vertices of all three different colors , the following holds true : @xmath32 since @xmath31 is area - preserving , it is enough to prove that @xmath33 has area with negative valuation . suppose that the triangle @xmath34 has vertex @xmath35 of color a , @xmath36 of color b and @xmath37 of color c. then , its area is equal to @xmath38 application of the properties of valuation and the definition of the coloring leads to @xmath39 @xmath40 @xmath41 therefore , @xmath42 given a tropical coloring @xmath26 , one can construct a degree map associated with it . it assigns an integer number to any oriented broken line . let @xmath43 be a complete graph with @xmath44 vertices considered as 1dimensional simplicial complex . suppose that we are given a 1dimensional simplicial complex @xmath45 and a map @xmath46 sending vertices of @xmath45 to vertices of @xmath43 . then , this map can be extended to a continuous map from complex @xmath45 to @xmath43 according to the following rules : * vertex @xmath47 is sent to point @xmath48 . * edge @xmath49 is sent to edge @xmath50 by a linear map determined by its ends . this map is , obviously , a continuous simplicial map from one simplicial complex to the other . we will use the same letter for both the original map ( coloring ) and the extended one . for the following , let us fix a tropical coloring @xmath26 . , width=480 ] any closed broken line @xmath51 has a natural structure of a simplicial complex . this complex is homeomorphic to a circle . its vertices are 3colored by @xmath26 . the extension of @xmath26 gives a continuous map from topological circle @xmath52 to topological circle @xmath53 . we denote by @xmath54 its topological degree . see fig . [ class ] . for a polygon @xmath55 we denote its boundary by @xmath56 . [ two ] if a polygon @xmath55 can be cut into triangles , whose areas have non negative 2adic valuations , then for every tropical coloring @xmath26 @xmath57 suppose that polygon @xmath55 has a triangulation @xmath58 with each triangle having non - negative valuation of area . this triangulation carries a natural structure of a 1dimensional simplicial complex , induced from the plane , with vertices colored by @xmath26 . @xmath56 is a subcomplex of @xmath58 homeomorphic to a circle , so there is a class @xmath59 \in h_1(\mathcal{t},\mathbb{z})$ ] corresponding to @xmath56 . the degree @xmath60 is equal to the image of the class @xmath59 $ ] under the map @xmath61 one can orient all triangles in the cut in a coherent way . then the triangles sharing an edge will induce opposite orientations on this edge . adding up the classes of all triangles boundaries we obtain the class of @xmath56 @xmath62=\sum_{\triangle \in \mathcal{t}}[\partial \triangle]\ ] ] and applying @xmath26 we get : @xmath63)=\sum_{\triangle \in \mathcal{t}}\pi^\mathcal{a}([\partial \triangle]).\ ] ] since any @xmath64 has non negative valuation of area , at least two of its vertices are of the same color according to lemma [ one ] . so , @xmath65)=0 $ ] for any triangle in @xmath58 and thus @xmath63)=deg(\partial m,\pi^\mathcal{a})=0.\ ] ] points with integer coordinates in a plane form a two dimensional lattice in @xmath66 , we will call it @xmath67 . we call a polygon @xmath55 or a closed broken line @xmath52 _ lattice _ if all its vertices have integer coordinates . let us denote by @xmath68 a simplicial complex with four vertices labeled by elements of the group @xmath69 and edjes connecting each two of its vertices . we can map @xmath67 to @xmath68 by the map @xmath70 for any lattice broken line @xmath52 we can consider its image under the map @xmath71 using the construction from the previous section . this map induces a map on simplicial homology groups : @xmath72 the image of @xmath73 in the group @xmath74 will be denoted by @xmath75 and called the _ class _ of broken line @xmath76 . [ three ] if a lattice polygon m can be dissected into triangles , whose areas have non - negative 2adic valuations , then the map @xmath71 sends @xmath77 to @xmath3 . let us denote the points of @xmath68 in the following way : @xmath78 the three cycles @xmath79 @xmath80 @xmath81 generate @xmath82 let us suppose that @xmath83 for an integer number its 2adic valuation is always nonnegative and it is equal to zero if and only if the number is odd . from the definition of the 2adic valuation it is clear that for a point @xmath84 @xmath85 @xmath86 @xmath87 @xmath88 this means that the map @xmath89 is correctly defined on @xmath90 and @xmath91 . any area - preserving affine transformation @xmath92 acts on the four vertices of @xmath90 by permutation , a simple check shows that @xmath93 . we will apply lemma [ two ] to the three colorings corresponding to the following area preserving affine transformations : @xmath94 @xmath95 @xmath96 by lemma [ two ] , @xmath97 . @xmath98 @xmath99 @xmath100 so , @xmath101 . similarly , applying the same procedure to affine transformations @xmath102 and @xmath103 , we get @xmath104 and @xmath105 . for two vectors @xmath106 and @xmath107 their _ wedge product _ is defined as the oriented area of the parallelogram formed by these vectors . it can be calculated as the determinant : @xmath108 for any closed broken line @xmath109 we define its generalised area by @xmath110 for a non selfintersecting broken line the notion defined above gives the oriented area of a polygon , which is bounded by the broken line . [ three+ ] for a lattice parallelogram @xmath111 the following is true : * if area of @xmath111 is even , then @xmath112 * if area of @xmath111 is odd , then @xmath113 the parallelogram can be cut into two equal triangles of the integer area . the application of lemma [ two ] gives the first statement . to prove the second statement , we will show that if the parallelogram @xmath111 has odd area , then all its pairs of coordinates of vertices are different modulo @xmath1 . if the vertices of the parallelogram have coordinates @xmath114 , @xmath115 , then its area is equal to @xmath116 @xmath117 from this formula , it is clear that if there are two vertices with both @xmath118 and @xmath119 coordinates being conjugate modulo @xmath1 , then @xmath120 is even . so , vertices of the parallelogram are colored in colors a , b , c , c. depending on the order in which these colors follow each other we obtain one of the cycles @xmath121 @xmath122 @xmath123 . the following lemma generalizes lemma [ three+ ] [ four ] if @xmath0 is a balanced lattice polygon , then the image of its boundary under the map @xmath71 is representing a class @xmath124 in the group @xmath125 , which is lying in a subgroup of index @xmath1 generated by @xmath126 : @xmath127 for some @xmath128 . + furthermore , @xmath129 parallelograms are basic examples of balanced polygons and we have seen that lemma [ four ] holds true for them . now , we are going to show that any balanced polygon is built from parallelograms in some sense . for this we need to describe an action of group @xmath130 on the set of broken lines . for a broken line @xmath109 , let us denote by @xmath131 the side vector of @xmath52 ( here @xmath132 ) . any @xmath133 acts on the set of broken lines according to the rule : @xmath134 with @xmath135 , and for each @xmath136 @xmath137 this action sends balanced broken lines to balanced and lattice to lattice . let @xmath138 denote a transposition @xmath139 . it is well known that a set @xmath140 generates @xmath130 . one can check that @xmath141 and @xmath142 here p is a parallelogram @xmath143 , with @xmath144 these properties guarantee that broken lines @xmath52 and @xmath145 satisfy the conditions of lemma [ four ] simultaneously . since lattice polygon @xmath0 is balanced , number @xmath44 of its vertices is even and sides of @xmath0 can be indexed by numbers @xmath146 and @xmath147 so that @xmath148 and sides with indices @xmath149 and @xmath150 are parallel , equal in length and inherit opposite orientations from the polygons boundary . numbers @xmath149 and @xmath150 are just natural numbers from @xmath151 to @xmath148 , so one can consider permutation @xmath152 in the broken line @xmath153 after any side with an odd number goes the side parallel and equal to it and having the opposite direction . both the area and the class of @xmath153 in @xmath154 is equal to @xmath3 . since @xmath155 can be presented as a product of transpositions @xmath138 , @xmath156 satisfies the conditions of lemma [ four ] . now we are ready to finish the proof of our main result . suppose that for a balanced lattice polygon @xmath0 of integer odd area there exist a cut into an odd number of triangles of equal areas . if @xmath157 and the number of triangles is equal to @xmath158 , then the area of each of them is @xmath159 . since @xmath160 and @xmath158 are odd numbers , @xmath161 according to lemma [ three ] , the class of the broken line @xmath156 in @xmath162 is equal to @xmath3 , and according to lemma [ four ] there exists @xmath128 for which @xmath163 therefore , @xmath164 and @xmath165 . this contradicts the oddness of @xmath160 . in the following section no new results are obtained , so the style will be rather informal . it is more natural to define tropical colorings on @xmath166 the real projective plane . it is well - known that a point of @xmath166 is defined by its homogenious coordinates a triple of real numbers @xmath167 $ ] with not all @xmath168 equal to @xmath3 . for any nonzero @xmath169 triples @xmath167 $ ] and @xmath170 $ ] define the same point . one can define a * momentum map * from the projective plane to a triangle @xmath171 in the plane with vertices ( 1,0),(0,1 ) and ( 0,0 ) : @xmath172 defined by the formula @xmath173)=\frac{2^{-\nu_2(x ) } \big ( 1,0\big ) + 2^{-\nu_2(y ) } \big ( 0,1\big)+ 2^{-\nu_2(z ) } \big ( 0,0\big)}{2^{-\nu_2(x)}+2^{-\nu_2(y)}+2^{-\nu_2(z)}}.\ ] ] one can check that the image of any line in @xmath166 under the momentum map is a union of three segments sharing a common end . for each segment the remaining end is lying on a side of triangle @xmath171 and the whole segment is contained within a line , passing through the vertex of @xmath171 opposite to the side . the image of the line @xmath174 cuts t into three pieces , whose points we color in three colors a , b and c. now we can color each point in @xmath166 in the color of its image under the momentum map . this coloring is the same as coloring @xmath89 constructed in the beginning of the paper if considered on the affine chart of @xmath166 with @xmath175 . [ a. hales , e. straus , 1982 ] [ hs ] let @xmath21 be a set of algebraic curves in @xmath66 having the same newton polygon @xmath111 with @xmath44 integer points inside ( @xmath21 is a @xmath44dimensional linear system of algebraic curves ) . then there exists a coloring of @xmath66 in @xmath176 colors such that no curve in @xmath21 contains all @xmath176 colors and no color is confined within a curve in @xmath21.dimensional linear systems of algebraic curves without based points . ] in @xcite such colorings are constructed algebraically ; here we will give a tropical explanation of this construction . by @xmath177 we mean @xmath178 . let us consider algebraic torus @xmath179 and `` a momentum map '' : @xmath180 where @xmath181 there exists a curve with newton polygon @xmath111 , whose image devides @xmath182 into @xmath183 regions . we assign to them different colors . it can be proved that the image of any other curve with newton polygon @xmath111 intersects at most @xmath44 regions . now , we can color each point in @xmath184 in the color of the region of @xmath66 containing its image . this coloring is the same as that constructed by hales and straus . this illustration shows that colorings constructed in @xcite are natural from the perspective , suggested by tropical geometry . unfortunatelly , it does not lead to an easier way of proving theorem [ hs ] , because of both combinatorial difficulties in analyzing the way in which two tropical curves intersect and algebraic difficulties in extending the colorings from @xmath185 to the whole @xmath186 . | in this paper we show that a lattice balanced polygon of odd area can not be cut into an odd number of triangles of equal areas .
first result of this type was obtained by paul monsky in 1970 .
he proved that a square can not be cut into an odd number of triangles of equal areas . in 2000
sherman stein conjectured that the same holds for any balanced polygon .
we also show connections between the equidissection problem and tropical geometry . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
the universe is filled by light . there are two principal sources of such photons the cosmic microwave background ( cmb ) and the extragalactic background light ( ebl ) . while cmb properties are quite well measured and theoretically understood for more than 20 years ( e.g. @xcite ) , many ebl models with widely different parameters were developed @xcite@xcite ; for instance , the total ebl intensity for these models differs by more than a factor of two @xcite . comparatively recently , however , some of these ebl models started to converge , at least in the 0.510 @xmath2 wavelength region ( e.g. , @xcite and @xcite ) . high - energy @xmath0-rays with primary energy @xmath3/\epsilon ) [ tev]$ ] get absorbed @xcite@xcite ( @xmath4 ) on ebl and cmb photons with energy @xmath5 , reaching maximum cross section at @xmath6/\epsilon ) [ tev]$ ] @xcite . therefore , the improved reliability of ebl models allows for more detailed studies of this fundamental quantum electrodynamics process in the @xmath0-ray energy range of about 500 @xmath1 10 @xmath7 , and of extragalactic @xmath0-ray propagation effects in general . using the cmb for such studies is , in principle , also possible @xcite , but at present is not feasible due to several factors . indeed , the typical mean free path @xmath8 of a @xmath9 @xmath10= 1 @xmath11 @xmath0-ray on the cmb is @xmath1210 @xmath13 ; for such high energies there is still no discovered sources , while using lower energies is not convenient either as in the @xmath14100 @xmath7 energy region the @xmath15 dependence is very strong : @xmath8 falls for about an order of magnitude for every 10 % of @xmath16 decrease . therefore , extremely well knowledge of experimental systematics on primary @xmath0-ray energy would be crucial while using cmb photons at @xmath14100 @xmath7 as a probe of the @xmath4 process . in the present conference contribution we emphasize on the potential importance of the development of electromagnetic cascades on the ebl / cmb to the interpretation of observations in the high energy ( he , @xmath16=100 @xmath17100 @xmath1 ) and the very high energy ( vhe , @xmath16=100 @xmath1100 @xmath7 ) ranges . this short paper does not pretend to claim great originality ; most of its basic ideas were already discussed and most of its main results were already presented in @xcite@xcite . soon after the discovery of the first @xmath7 extragalactic @xmath0-ray source @xcite , the first @xmath0-astronomical constraints on the ebl density were obtained @xcite@xcite , accounting for only the pair - production process as the main cause of transformation of the primary @xmath0-ray spectrum ( for very distant sources , it is necessary to include adiabatic losses ) . this model we call `` the absorption - only model '' . since 1993 , almost every year several papers appear , assuming the absorption - only model , sometimes at a rate of a dozen a year or more . in what follows , any significant deviations from the absorption - only model are called `` anomalies '' , even if they do not fall beyond the conventional physics . several such anomalies were reported @xcite@xcite , with a statistical significance not overwhelming in every single case , but together they indicate that the absorption - only model is incomplete and must be modified in some way ( see @xcite for more details ) . secondary electrons and positrons ( hereafter simply `` electrons '' ) produced in the @xmath4 process radiate cascade @xmath0-rays by means of the inverse compton ( ic ) process ; these @xmath0-rays may contribute to the observable spectrum of an extragalactic source . therefore , besides the properties of background photon fields ( ebl / cmb ) , another very important factor is the strength and structure of the extragalactic magnetic field ( egmf ) . some selected constraints on the egmf strength @xmath18 in voids of the large scale structure ( lss ) , most of them assuming correlation length of the field @xmath19= 1 @xmath20 , are shown in figure [ f1 ] . there are two regions of @xmath18 values that satisfy most of these constraints : @xmath21 @xmath222@xmath23 @xmath24 and @xmath25 @xmath223@xmath26 @xmath24 . the first case corresponds to the absorption - only model regime : for such a strong egmf secondary electrons are , as a rule , strongly deflected and delayed so that cascade photons do not contribute to the point - like image of the source . on the other hand , for the case of the second option such a contribution is still possible , at least in the vhe energy range . the lower bound on the @xmath18 value , @xmath25 @xmath24 , is highly uncertain @xcite-@xcite . very recently a paper @xcite appeared , disfavouring a narrow range of values around @xmath18= @xmath21 @xmath24 for @xmath19= 1 @xmath20 . let us assume for a moment the following simple two - phase model of the egmf . lss voids with @xmath27 fill a fraction of the total volume @xmath28 , while the rest is occupied by the comparatively strong ( @xmath29 @xmath24 ) egmf with @xmath30 @xmath20 . figure [ f2 ] shows several fits to the observed spectral energy distribution ( sed ) of blazar 1es 0229 + 200 ( @xcite for the four lowest - energy bins shown in black and @xcite for other bins ) assuming this model with various values of @xmath31 . cascade component dominates at low energies ( @xmath32300500 @xmath1 ) . details of simulations correspond to the case of figure 15 of @xcite . reasonable fits in the energy range 30 @xmath1 10 @xmath7 may be obtained for the case of @xmath31 between 0.4 and 0.6 . however , below 30 @xmath1 the model intensity exceeds the observed one . this may be explained either by the deflection and/or delay of cascade electrons while travelling the egmf @xcite@xcite or by additional ( with respect to ic ) electron energy losses @xcite . figure [ f3 ] depicts the main spectral signatures of the electromagnetic ( em ) cascade model : 1 ) high - energy cutoff , similar to the absorption - only model , 2 ) an ankle at the intersection of the primary and cascade components ( in fact , this signature is also visible in figure [ f2 ] ) , 3 ) a low - energy cutoff for the case of a non - zero egmf value ( `` magnetic cutoff '' ) , and 4 ) a `` second ankle '' at the low - energy region of the spectrum . we argue that all above - mentioned anomalies may be interpreted within the framework of the em cascade model , namely : + 1 . the apparent excess of observed @xmath0-rays in the highest - energy bins claimed in @xcite@xcite was discussed by @xcite . a prominent ankle ( signature 2 in figure [ f3 ] ) may account for this effect . the unusual spectral hardening towards lower energies @xcite is explained by the existence of a `` magnetic cutoff '' ( signature 3 in figure [ f3 ] ) , as the authors of @xcite themselves note ( however , see @xcite and @xcite itself for other possible explanations ) . the same spectral feature ( a prominent magnetic cutoff ) may account for the result of @xcite , as discussed in @xcite . + 4 . finally , `` halos '' observed around some blazars also may be explained in the context of the em cascade model @xcite . + thus , the em cascade model appears to be the simplest extragalactic @xmath0-ray propagation scenario that coherently explains all these anomalies . other , more exotic models that could account for a part of these anomalies do exist @xcite@xcite ; the main difficulties of these scenarios were discussed by us in @xcite,@xcite . recent observations of some blazars in the 1 @xmath1 10 @xmath7 energy region seem to indicate that the secondary component of cascade @xmath0-rays may constitute a considerable contribution to the observable flux in the @xmath331500 @xmath1 energy range . this work was supported by the rfbr grant 16 - 32 - 00823 . t.d . acknowledges the support of the students and researchers exchange program in sciences ( steps ) , the re - inventing japan project , jsps , and the hospitality of the university of tokyo icrr . peebles , `` principles of physical cosmology '' , princeton university press ( 1993 ) t.m . kneiske et al . , a&a 413 ( 2004 ) 807 f.w . stecker et al . , apj 648 ( 2006 ) 774 a. franceschini et al . , a&a 487 ( 2008 ) 837 j.r . primack et al . , aipcp 1085 ( 2008 ) 71 j.d . finke et al . , apj 712 ( 2010 ) 238 t.m . kneiske & h. dole , a&a 515 ( 2010 ) a19 a. dominguez et al . , mnras 410 ( 2011 ) 2556 r.c . gilmore et al . , mnras 422 ( 2012 ) 3189 y. inoue et al . , apj 768 ( 2013 ) 197 e. dwek & f. krennrich , aph 43 ( 2013 ) 112 a.i . nikishov , sov . jetp 14 ( 1962 ) 393 r.j . gould & g. shreder , phys . 155 ( 1967 ) 1408 j.v . jelley , phys . ( 1966 ) 479 t. dzhatdoev et al . , preprint astro.ph/1609.01013 ( 2016 ) t.a . dzhatdoev , j. phys . ( 2015 ) 012035 t. dzhatdoev , izvestiya rossiiskoi akademii nauk 79 ( 2015 ) 363 ( preprint astro - ph/1501.00259 ) t. dzhatdoev et al . , proc . of the isvhecri-2016 conference , to appear in epj web of conf . ( 2017 ) t. dzhatdoev et al . , proc . of the 2@xmath34 icppa conference , to appear in j. phys . ( 2017 ) m. punch , nature 358 ( 1992 ) 477 f.w . stecker & o.c . de jager , apj 415 ( 1993 ) 71 o.c . de jager et al . , nature 369 ( 1994 ) 294 d. horns & m. meyer , jcap ( 2012 ) 033 d. horns , preprint astro - ph/1602.07499 ( 2016 ) a. neronov et al . , a&a 541 ( 2012 ) a31 a. furniss et al . , mnras 446 ( 2015 ) 2267 w. chen et al . ( 2015 ) 211103 t.c . arlen et al . , apj 796 ( 2014 ) 18 a.e . broderick et al . , apj 752 ( 2012 ) 22 s. archambault et al . , preprint astro - ph/1701.00372 ( 2017 ) ie . vovk et al . , apj lett . 747 ( 2012 ) l14 e. aliu et al . , apj 782 ( 2014 ) 13 a. neronov & ie . vovk , science 328 ( 2010 ) 73 c.d . dermer et al . , apj lett . 733 ( 2011 ) l21 t. dzhatdoev et al . , proc . of the 34@xmath35 russian cosmic ray conference , to appear in izvestiya rossiiskoi akademii nauk ( 2017 ) m.l . ahnen et al . , preprint astro - ph/1612.09472 ( 2016 ) w. essey et al . , phys . 104 ( 2010 ) 141102 a. mirizzi et al . d 76 ( 2007 ) 023001 | we present a very brief overview of some recent @xmath0-ray observations of selected blazars to reveal an indication for a considerable or even dominant contribution of secondary @xmath0-rays from electromagnetic cascades to the observable spectra in the 1500 @xmath1 energy range . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
qcd matter with @xmath1 flavors , @xmath2 colors and zero baryon chemical potential undergoes two finite temperature phase transitions , the chiral transition and the deconfinement . the chiral transition is associated with the spontaneous breaking of the chiral symmetry , @xmath3 , below the critical temperature ( @xmath4 ) for massless quarks . the order parameter for this transition is the chiral condensate . on the other hand the deconfinement transition is associated with the spontaneous symmetry breaking , @xmath5 , above the critical temperature @xmath6 for infinitely heavy quarks . the order parameter for this transition is the polyakov loop expectation value . for finite non - zero quark masses both the chiral symmetry and the @xmath0 symmetry are explicitly broken . nevertheless these two transitions show up as crossover , second or first order transitions depending on the values of the quark masses . the chiral transition depends on the number of quark flavors and the deconfinement transition depends on the number of colors . since the chiral symmetry and the @xmath0 symmetry seem nothing to do with each other one would expect that these two transitions occur independently . however lattice qcd calculations have shown that both these transitions occur simultaneously , @xmath7 [ 1 - 5 ] . furthermore strong correlation between the chiral condensate and the polyakov loop is observed around the phase transition point@xcite . this is clear evidence that there is interaction between these two order parameters . so studying the interaction between the chiral condensate and the polyakov loop is fundamental to understanding the interplay between the chiral transition and deconfinement . there are several studies on the possible causes of the simultaneous occurrence of the chiral transition and the deconfinement transition . mixing between the gauge and matter field operators has been suggested to explain the simultaneous chiral and deconfinement transitions @xcite . some other studies consider that for small quark masses the chiral transition drives the deconfinement transition @xcite . though most of the studies are concerned with why @xmath7 only a few have considered the effect of the interaction on the phase transition itself @xcite . it seems natural to expect that if the interaction between the two order parameters can result in the simultaneous occurrence of the two transitions then the interaction may also have important effect on the phase transition . one of the most interesting cases to study for the possible effects of this interaction is the 2-flavor chiral transition . lattice qcd calculations have not yet been able to settle on the order of this phase transition . lattice calculations by different groups do not agree on the order of this phase transition . some lattice groups find the transition is second order @xcite and other groups find the transition is first order @xcite . conventionally this transition is believed to be second order and in the universality class of @xmath8 heisenberg magnet @xcite . but the effect of interaction between the chiral order parameter ( @xmath9 ) and the polyakov loop ( @xmath10 ) may change the behavior of this transition . so in the present work we investigate the effect of interaction between @xmath9 and @xmath10 on the 2-flavor chiral transition within an effective lagrangian . previously , the effect of interaction between the chiral order parameter and the polyakov loop has been studied in the renormalization group approach @xcite . the main difference between the present work and previous studies is that we consider the explicit breaking of the @xmath0 symmetry . explicit breaking of the @xmath0 symmetry can be introduced in the effective lagrangian by terms such as @xmath11 , @xmath12 . previous studies @xcite have considered interaction term , such as @xmath13 , which respects both the chiral and the @xmath0 symmetry . however , in the chiral limit , the interaction terms need not respect the @xmath0 symmetry . so terms such as @xmath12 should be considered . as we discuss later such a interaction term is always present if the explicit @xmath0 symmetry breaking is large , for example in the chiral limit . for simplicity we consider @xmath14 color qcd . we expect that different @xmath2 will not qualitatively change the physics we are discussing here . for @xmath15 the chiral order parameter @xmath9 is a four component vector field whereas for @xmath14 the polyakov loop @xmath10 is a real scalar field . in this work we basically study the effect of the three terms , @xmath16 , @xmath17 and @xmath18 in the effective lagrangian . our main result is that the strong explicit breaking of the @xmath0 symmetry can make the chiral transition first order . we show that the chiral transition can be first order even at the mean field level . we also carry out numerical monte carlo simulations which confirm the first order phase transition for large enough explicit @xmath0 symmetry breaking . we mention here that the @xmath15 chiral transition can be first order from the interaction term @xmath19 without @xmath0 symmetry breaking @xcite . however lattice qcd results indicate that effect of the @xmath0 symmetric interaction term is small . a possible first order chiral transition can result more likely from the explicit @xmath0 symmetry breaking as we will argue later . conventionally explicit symmetry breaking weakens a phase transition . but our results suggest that for a system with two order parameter fields explicit symmetry breaking can make the transition stronger . it is interesting to note that the chiral order parameter @xmath9 does not couple to gauge fields directly . the gauge fields seem to affect the chiral phase transition indirectly , through the polyakov loop . we mention here that the effect of explicit symmetry breaking discussed here should not be restricted to the @xmath0 symmetry . we expect that the explicit breaking of chiral symmetry may have some effect on the deconfinement transition in the large quark mass region . we mention here that interaction between the chiral order parameter and the diquark fields is considered to study the chiral / color - superconducting transition at low temperature and high density @xcite . this paper is organized as follows . in section - ii we describe the effective lagrangian and discuss the effect of the interaction between @xmath9 and @xmath10 on the chiral transtition . we describe our numerical monte carlo calculations and results in section - iii . the discussions and conclusions are presented in section - iv . we consider the following lagrangian @xcite in 3 dimension for the @xmath9 and the @xmath10 fields , @xmath20 some of the parameters of this reduced 3d lagrangian depend explicitly on temperature . this lagrangian is invariant under @xmath8 rotation of the @xmath9 field . the last two terms of @xmath21 break the @xmath22 symmetry ( @xmath23 ) explicitly . the interactions between the chiral order parameter and the polyakov loop are taken care by 5th and 6th terms in @xmath21 . the signs of the couplings @xmath24 and @xmath25 decides the correlation between the fluctuations of the @xmath9 and the @xmath10 fields . for example when @xmath26 , the thermal average of the correlation between the @xmath9 and @xmath10 fluctuations , @xmath27 , is positive . when @xmath28 , @xmath27 is negative . such `` anti - correlation '' is seen between the fluctuations of the the chiral condensate and the polyakov loop in the results of lattice qcd calculations@xcite . the correlation between the variations of the two order parameters with respect to temperature is of the same sign as the correlation between the fluctuations . note that in the above lagrangian when @xmath29 the @xmath9 field acts like an ordering field for the @xmath10 field . in the chiral symmetric phase the chiral order parameter is small and the polyakov loop expectation value is large . the large expectation value of @xmath10 , in the chiral symmetric phase , can result only from the last term in @xmath21 ( with @xmath30 ) because @xmath31 is small . 0.2 cm as we have mentioned before previous studies @xcite have considered only the interaction term @xmath32 . when the coupling parameters @xmath33 , the chiral transition and the deconfinement transition do not always occur simultaneously . for large value of the coupling @xmath24 the transitions can occur simultaneously and are of first order when coefficients of @xmath34 and @xmath35 are negative in eq.1 @xcite . a large positive @xmath24 would increase the critical temperature for the deconfinement transition as the coefficent of @xmath35 term is negative for high temperatures . lattice results on the other hand show that inclusion of dynamical quarks decrease the deconfinement transition temperature . so in qcd the coupling @xmath24 should be small . for small @xmath24 both the transitions are second order and do not occur simultaneously . 0.2 cm the coupling parameters @xmath25 and @xmath36 represent the strength of the @xmath0 explicit breaking . so they should increase with decreasing quark masses as the @xmath0 breaking becomes severe . this can be seen explicitly in the large quark mass region @xcite . to see the effect of the explicit @xmath0 breaking let us consider @xmath37 and @xmath29 . at the mean field level one can consider the temperature variation of the parameters @xmath38 and @xmath39 . for simplicity we fix @xmath40 , @xmath41 and vary the @xmath38 parameter . to find the @xmath38 dependence of @xmath10 and @xmath9 expectation values one has to solve the following coupled equations , @xmath42 we have checked numerically that for large enough @xmath25 these equations give two solutions which correspond to a degenerate minima of the effective potential @xmath21 at some particular value of @xmath38 . it may seem surprising that the potential @xmath21 has degenerate minima even though there is no cubic term for @xmath9 and @xmath10 in it . however because of the coupling @xmath25 these two fields are mixed . though the mixing anle varies as @xmath38 is varied . with variation of @xmath38 the minimum of @xmath21 , in the @xmath43 plane , moves in directions other than the @xmath31 or @xmath10 axes . to understand how the minimum of @xmath21 behaves one should express @xmath21 in terms of variables which are the linear combinations of @xmath31 and @xmath10 this would invariably lead to cubic terms of the new fields in the effective potential . for some choice of parameters the cubic term may then be important to cause degenerate minima of @xmath21 . even if the explicit symmetry breaking is not strong enough at mean field level fluctuations at higher order can make the transition first order . at one loop the fluctuations of the @xmath9 field will contribute to a non zero 3-point function of the @xmath10 field . this 3-point function can be calculated perturbatively . in the high temperature approximation the 3-point function is given by , @xmath44 assuming zero momentum to all the external @xmath45lines . @xmath46 here is the mass of the @xmath9 field fluctuations . given a suitable choice of parameters in the effective lagrangian the three point function can be significant . the consequence of this is a @xmath47 term in the potential @xmath21 with temperature dependent @xmath48 . this term can cause discontinous change in @xmath10 as temperature varies . when @xmath10 field changes discontinuously the @xmath9 field will also changes discontinuously because of the coupling term @xmath49 . if the coefficient of the @xmath50 term changes from @xmath51 to @xmath52 due to discontinuous change in @xmath10 then @xmath9 will change discontinuously from zero to non - zero . the @xmath53 term can also come from other types of explicit symmetry breaking coupling terms but we think @xmath49 is the simplest term in our model . + the situation with @xmath54 and @xmath55 is same as the one discussed above . when the explicit symmetry breaking parameter @xmath36 is large , @xmath22 symmetry of the @xmath10 field is lost and the @xmath10 field always has non - zero expectation value @xmath56 . in order to study the fluctuations one must expand the potential @xmath21 around @xmath56 , @xmath57 this gives rise to a term like @xmath58 coming from the expansion of @xmath59 around @xmath60 . now the term @xmath58 is similar to the one discussed above . so there can be first order transition when @xmath61 like in the case when @xmath62 . we observed that at the mean field the transition becomes second order for large quartic couplings @xmath63 . now in the following section we discuss the numerical monte carlo simulations of the effective lagrangian ( eq.1 ) and the results . these calculations include higher order as well as non perturbative fluctuations . vs @xmath68,title="fig : " ] [ mf ] -0.3 cm vs @xmath68,title="fig : " ] [ pt ] -0.3 cm -0.3 cm since we take @xmath25 to be positive we see @xmath87 and @xmath94 increase or decrease simultaneously . the choice of the values for the parameters is such that the variation of @xmath87 and @xmath94 are similar in magnitude . for @xmath95 increase in @xmath87 should lead to decrease in @xmath94 . so in this case the hysteresis loop for @xmath87 will look somewhat similar to fig.1 while the hysteresis loop for @xmath94 will be more or less inverted about the y - axis . 0.2 cm for the second set of parameters we choose @xmath96 , @xmath97 , @xmath98 , @xmath99 and @xmath100 . the hysteresis loop of @xmath87 and @xmath94 are observed by again varying the parameter @xmath68 . the choice of @xmath69 and @xmath36 is such that the expectation value of @xmath10 is non - zero and positive . the sign of @xmath24 assures that increase in @xmath87 leads to decrease in @xmath94 and vice - versa as the parameter @xmath68 is varied . the hysteresis curves of the two order parameters are shown in fig.3 and fig.4 . the values of @xmath101 in our calculations are chosen so that we can see first order phase transition clearly and the variation of @xmath87 and @xmath94 are of order o(1 ) across the transition point . note that with suitable choice of @xmath69 and @xmath102 one change the average of the polyakov loop across the transition point . the choice @xmath24 was such that the chiral transition turned out to be second order when the coupling @xmath36 was set to zero . we also did simulations on a 4d lattice . the results in this case are very similar to the 3d simulations . -1.3 cm vs @xmath68,title="fig : " ] [ mf ] -0.3 cm -0.3 cm vs @xmath68,title="fig : " ] [ pt ] -0.3 cm -0.3 cm the results in fig.1 - 4 show strong first order phase transition for the @xmath9 and @xmath10 fields . by fixing the coupling parameters @xmath24 , @xmath25 and @xmath36 we observed that the strength of the transition depends on the values of the quartic couplings @xmath63 and @xmath102 . the transition becomes weaker with increase in any of the quartic couplings @xmath101 . however even for larger quartic couplings a suitable choice of the parameters @xmath24 , @xmath25 and @xmath36 made the transition strong first order . 0.2 cm we also did calculations with small explicit symmetry breaking for the @xmath9 field by considering a linear @xmath9 term in the lagrangian . in this case we found that the hysteresis loops of both @xmath87 and @xmath94 shrinking in size with increase in the coefficient of linear @xmath9 term in the lagrangian . these results suggest that the transition becomes weaker when the chiral symmetry is explicitly broken . using a simple effective lagrangian , which captures the chiral symmetry and @xmath0 symmetry of qcd , we have investigated the effect of the explicit symmetry breaking on the chiral phase transition . since we consider the chiral limit the @xmath22 symmetry of the polyakov loop is explicitly broken . as we incorporate the explicit @xmath22 breaking interaction terms in the effective lagrangian we find the chiral transition becomes first order . we observed that the first order transition becomes weaker when a small explicit symmetry breaking is considered for the chiral order parameter @xmath9 . the results of our calculations show that two flavor chiral transition is of first order for large enough @xmath0 explicit breaking . as we have mentioned before at present some lattice studies suggest that the transition is first order @xcite and some other studies show the transition is second order @xcite . these conflicting results may be because the quark masses studied are not small enough or the lattices used are not big enough . in the previous study , for @xmath15 and @xmath103 @xcite , the second order chiral transition becomes first order when coupled to the polyakov loop . this is because the deconfinement transition is first order for @xmath103 with a cubic term @xmath104 in the effective potential with exact @xmath105 symmetry . however when the quark masses are finite and decrease the deconfinement transition becomes weaker . this can be understood due to explicit breaking of @xmath105 symmetry . already in the large quark mass region the deconfinement transition becomes crossover which implies the the explicit symmetry breaking dominates over the effects of the above @xmath105 symmetric cubic term . for smaller quark masses the explicit @xmath105 symmetry breaking likely grows and expected to be maximal in the chiral limit . so for smaller quark masses , i.e in the chiral limit one should rather consider the effect of the explicit breaking of the @xmath105 symmetry . the effects of explicit symmetry breaking discussed in this work should not be restricted to the explicit breaking of the @xmath0 symmetry . for 2-flavor and 2-color qcd the deconfinement transition in the large quark mass region may have the effects coming from the explicit breaking of the chiral symmetry . it may be possible that this effect change the phase transition behavior of the deconfinement transition in the heavy quark mass region . we have benefited greatly from discussions with t. hatsuda and a. m. srivastava . we would like to thank g. baym , s. datta , m. laine , t. matsuura and m. ohtani for comments . this work is supported by the jsps postdoctoral fellowship for foreign researchers . | we study the effects of the interaction between the chiral condensate and the polyakov loop on the chiral transition within an effective lagrangian .
we find that the effects of the interaction change the order of the phase transition when the explicit breaking of the @xmath0 symmetry of the polyakov loop is large .
our results suggest that the chiral transition in 2-flavor qcd may be first order . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
the goal of the paper is to intiate an approach to the hyperbolic volume conjecture , via asymptotics of solutions of difference equations with a small parameter . the generalized volume conjecture links ( conjecturally ) the ( colored ) jones polynomial of a knot to hyperbolic geometry of its complement . since the colored jones polynomial is a specific solution to a linear @xmath0-difference equation , it follows that the generalized volume conjecture is the wkb limit of a specific solution of a linear difference equation with a small parameter . motivated by this , we study wkb asymptotics of formal and actual solutions of difference equations with a small parameter , under certain regularity asymptions . a knot in 3-space is a smooth embedding of a circle , considered up to isotopy . two of the simplest knots , the trefoil ( @xmath1 ) and the figure eight ( @xmath2 ) are shown here : @xmath3 by the very definition , knots are flexible objects defined up to isotopy , which allows the embedding to move in a smooth and arbitrary way as long as it does not cross itself . in algebraic topology , a common way of studying knots ( and more generally , spaces ) is to associate computable numerical invariants ( such as euler characteristic , or homology ) . invariants are useful in deciding whether two knots are not the same . it is a much harder problem to construct computable invariants that separate knots . the invariant that we will consider in this paper is the jones polynomial of a knot ; @xcite , which is a laurent polynomial with integer coefficients , associated to each knot . the quantum nature of the jones polynomial is apparent both in the original definition of jones ( using temperley - lieb algebras ) and in the reformulation , due to witten , in terms of the expectation value of a quantum field theory ; see @xcite . the combinatorics associated to a planar projection of a knot show that the jones polynomial is a computable invariant . however , it is hard to see from this point of view the relation between the jones polynomial and geometry . in quantum field theory , one often reproduces geometry by moving carefully chosen parameters of the theory to an appropriate limit . in our case , we will introduce a new parameter , a natural number which roughly speaking corresponds to taking a connected @xmath4-parallel of a knot . the resulting invariant is no longer a laurent polynomial , but rather a sequence of laurent polynomials . the _ colored jones function _ of a knot @xmath5 in 3-space is a sequence of laurent polynomials @xmath6.\ ] ] the first term in the above sequence , @xmath7 is the jones polynomial of @xmath5 ; see @xcite . although knots are flexible objects , thurston had the idea that their complements have a unique decomposition in pieces of unique `` crystaline '' shape . the shapes in question are the @xmath8 different geometries in dimension @xmath9 , and the idea in question was termed the `` geometrization conjecture '' . the most common of the @xmath8 geometries is hyperbolic geometry , that is the existence of a complete , finite volume , constant curvature @xmath10 riemannian metric on knot complements . thurston proved that unless the knot is torus or a satellite , then it carries a unique such metric ; see @xcite . the _ hyperbolic volume conjecture _ ( hvc , in short ) connects two very different views of knot : namely quantum field theory and riemannian geometry . the hvc states for every hyperbolic knot @xmath5 @xmath11 where @xmath12 is the _ volume _ of a complete hyperbolic metric in the knot complement @xmath13 . the conjecture was formulated in this form by murakami - murakami @xcite following an earlier version due to kashaev , @xcite . more generally , gukov ( see @xcite ) formulated a generalized hyperbolic volume conjecture that identifies the limit @xmath14 of a hyperbolic knot with known hyperbolic invariants ( such as the volume of cone manifolds obtained by hyperbolic dehn filling ) , for @xmath15-\bq$ ] or @xmath16 . actually , the ghvc is stated for _ complex _ numbers @xmath17 . for simplicity , we will study asympotics for real @xmath18 $ ] . at present , it is not known whether the limit in the hvc exists , let alone that it can be computed . explicit finite multisum formulas for the colored jones function of a knot exist ; see for example @xcite . from these formulas alone , it is difficult to study the above limit . in a sense , the question is to understand the sequence of laurent polynomials that appears in the hvc . if the sequence is in some sense random , then it is hard to expect that the limit exists , or that it can be computed . since the first term of this sequence is the jones polynomial , and since we know little about the possible values of the jones polynomial , one would expect that there is even less to be said about the colored jones function . luckily , the colored jones function behaves in a better way than its first term , namely the jones polynomial . this can be quantified by recent work of ttq le and the first author , who proved that the colored jones function of a knot satisfies a @xmath0-difference equation . in other words , for every knot @xmath5 there exist rational functions @xmath19 ( which of course depend on @xmath5 ) such that for all @xmath20 we have : @xmath21 this opens the possibility of studying the @xmath0-difference equation rather than one of its solutions , namely the colored jones function . although the @xmath0-difference equation is not unique , it was shown by the first author in @xcite that one can choose a unique @xmath0-difference equation , which is a knot invariant . moreover , it was conjectured in @xcite that the characteristic polynomial of this @xmath0-difference equation determines the characters of @xmath22 representations of the knot complement , viewed from the boundary . as was explained by the first author on several occasions , asymptotics of solutions of @xmath0-difference equations would have consequences on the hvc . in this introductory article we review the history of asymptotics of solutions of @xmath0-difference equations . excellent references for differential equations with a parameter are olver s and wasau s books ; @xcite and @xcite . in 1837 , liouville and green independently studied systematically existence of formal ( i.e. , perturbative ) and actual solutions for second order differential equations with a parameter ; see @xcite . second order equations are very important for classical and quantum physics . in 1908 birkhoff had the insight to introduce and study _ arbitrary _ order differential equation with a parameter ( see @xcite ) : @xmath23 where @xmath24 , @xmath25 means @xmath4-th derivative with respect to @xmath26 ( assumed to be restricted to a real interval ) , and @xmath27 is a large complex parameter , and where the coefficient @xmath28 are complex @xmath29 functions with an expansion @xmath30 birkhoff s working assumption was that the eigenvalues @xmath31 of the characteristic equation @xmath32 were distinct but not necessarily nowhere vanishing . in 1926 , three theoretical physicists , wentzel - krammer - brillouin studied the second order differential equation under the assumption that its eigenvalues do not collide , and developed connection formulas linking solutions in the exponential region with those in the oscillatory region . their method is often referred to as the wkb method . as a motivation for our results , let us recall some fundamental results of birkhoff and trjitzinsky from 1930 on difference equations without a parameter ; see @xcite and @xcite . a _ difference equation _ for a discrete function @xmath33 has the form : @xmath34 where @xmath35 are discrete functions so that @xmath36 for all @xmath20 . we will assume the existence of asymptotic expansions of @xmath37 around @xmath38 for all @xmath39 : @xmath40 where @xmath41 . this certainly holds for @xmath42 if @xmath43 are rational functions of @xmath4 , as is often the case in combinatorial problems . due to the nowhere vanishing of @xmath44 , it follows that the set of solutions of is a vector space of dimension @xmath45 . there are two main problems of difference equations : * existence of _ formal series solutions _ @xmath46 to . * existence of a basis @xmath47 of solutions so that @xmath48 is asymptotic to @xmath49 for large @xmath4 . in @xcite , birkhoff solved the existence of formal solutions in complete generality ( that is , without any assumptions on the eigenvalues of the characteristic equation ) . in a sequel paper @xcite , birkhoff - trjitzinsky solved the second problem in complete generality . among other things , the formal solutions of birkhoff lead to the development of differential galois theory , see @xcite . decades later , the results of birkhoff and trjitzinsky on difference equations have found applications to enumerative combinatorics and numerical analysis ; see for example wimp and zeilberger in @xcite and references therein . it is not surprising that difference equations are used in numerical analysis , since difference equations are numerical schemes of approximating differential equations . in enumerative combinatorics and complexity theory , difference equations appear in recursive computation . for example , the number @xmath50 of involutions of @xmath51 ( that is , permutations which are a product of 1 and 2-cycles ) is given by @xmath52 with @xmath53 , @xmath54 . using the results of birkhoff - trjitzinsky and the fact that @xmath50 is monotone , it follows that @xmath55 for nonzero constants @xmath56 and some @xmath57 . actually , the @xmath58 can be computed recursively from the difference equation ; see @xcite . by some historical accident , asymptotics of solutions of difference equations with a parameter was not discussed a century ago . the first paper that discusses second order difference equation with a parameter appears to be due to deift - mclaughlin ( see @xcite ) which was generalized by costin - costin to arbitrary order difference equations , @xcite . the purpose in this paper is to show that for regular @xmath0-difference equations , a regular solution has a well - defined and computable exponential growth rate in terms of a relative entropy of the characteristic polynomial of the @xmath0-difference equation ; see theorem [ thm.1 ] below . this subject is classical and has been reinvented over the past hundred years by several groups , often unaware of each others results . in a sense , the problem of formal solutions of @xmath0-difference equations is a problem in differential galois theory ; @xcite , and a problem in numerical analysis ; see for example @xcite . our results are hardly new and are contained or can be obtained by minor modifications from results of costin - costin or from work of birkhoff and collaborators , @xcite . since the presentation in the above papers varies by time and taste , we have decided to give a self - contained account of the theory with complete proofs . hopefully , this will benefit the researchers in quantum topology and in analysis . in this paper , we will describe asymptotics of solutions of @xmath0-difference equations . a _ @xmath0-difference equation _ for a sequence @xmath59 of smooth functions of @xmath0 has the form : @xmath60 where @xmath61 are smooth functions and @xmath62 . before we proceed further , let us remark that @xmath0 is a _ variable _ in and not a complex number of absolute value less ( or more ) than @xmath63 . in the usual analytic theory of @xmath0-difference equations , @xmath0 is a complex number inside or outside the unit disk . moreover , in the ghvc , we need to compute the @xmath4th term @xmath64 in the above @xmath0-difference equation , and then evaluate it at @xmath65 , for @xmath17 fixed . in other words , in the ghvc , @xmath66 is a complex number that varies with @xmath4 in such a way that it stays in the unit circle and approaches @xmath63 as @xmath38 . with this in mind , @xmath67-difference equations ( defined below ) are obtained from @xmath0-difference equations by the substitution @xmath68 where @xmath67 is a small nonnegative real number , that plays the role of planck s constant . the _ characteristic polynomial _ of the @xmath0-difference equation is @xmath69 we will say that is _ regular _ if @xmath70 for all @xmath71 , where @xmath72 is the _ discriminant _ of @xmath73 , which is a polynomial in the coefficients of @xmath73 . let @xmath74 denote the roots of the characteristic polynomial , which we call the _ eigenvalues _ of . it turns out that is regular iff the eigenvalues @xmath74 never collide and never vanish , for every @xmath71 . moreover , it follows by the implicit function theorem that the roots are smooth functions of @xmath71 . since we are interested in asymptotics of solutions of @xmath0-difference equations which , as we shall see , are governed by the magnitude of the eigenvalues , we need to partition the circle according to the magnitudes of the eigenvalues . let @xmath75 denote a partition of @xmath76 into a finite union of closed arcs ( with nonoverlapping interiors ) , such that the magnitude of the eigenvalues does not change in each arc . in other words , for each @xmath77 , there is a permutation @xmath78 of the set @xmath79 such that latexmath:[\[|\l_{\s_p(1)}(v)| \geq |\l_{\s_p(2)}(v)| \geq \dots \geq the following definition introduces a locally fundamental set of solutions of @xmath0-difference equations . fix a partition of @xmath81 as above . a set @xmath82 is a _ locally fundamental set of solutions _ of iff for every solution @xmath83 for every @xmath77 and for every @xmath84 there exist smooth functions @xmath85 such that @xmath86 assume that is regular . then , there exists a locally fundamental set of solutions @xmath82 such that * for every @xmath84 and @xmath87 such that @xmath88 we have @xmath89 * for some smooth functions @xmath90 with uniform ( with respect to @xmath91 $ ] ) _ asymptotic expansion _ @xmath92 where @xmath93 for all @xmath94 * and leading term @xmath95 where we have chosen a branch for the logarithm of @xmath96 . for every @xmath39 the smooth functions @xmath97 for positive @xmath94 are uniquely determined from the coefficients @xmath98 of by a hierarchy of first - order differential equations along with specified initial conditions . on the other hand , the smooth functions @xmath90 are not uniquely determined , since they are obtained by a smooth interpolation . thus , the locally fundamental set of solutions is not uniquely determined from the @xmath0-difference equation , although its asymptotic behavior is . it follows from theorem [ thm.1 ] that each locally fundamental solution @xmath99 of the @xmath0-difference equation satisfies the ghvc in the sense that for every @xmath18 $ ] we have : @xmath100 fix a solution @xmath83 of . theorem [ thm.1 ] expresses @xmath83 as a linear combination of @xmath101 s in each arc @xmath102 . for every @xmath103 , let @xmath104 later ( in section [ sub.assregularq ] ) we will define the notion of a regular solution to a @xmath0-difference equation . as a prototypical example , consider an @xmath0-difference equation that satisfies @xmath105 for all @xmath106 and all @xmath71 . then , any solution that satisfies @xmath107 ( or more generally , @xmath108 has a nonvanishing derivative at @xmath109 ) is regular . it is possible that @xmath110 . in other words , the restriction of a fixed solution @xmath83 to different intervals @xmath102 may be a linear combination of different @xmath111s . this is an important phenomenon , referred by the name of _ stokes phenomenon _ ; see @xcite . our next definition captures the growth rate of regular solutions to regular @xmath0-difference equations . fix a collection @xmath112 of subsets of @xmath113 . the @xmath114-_entropy _ @xmath115 \to \br\ ] ] of the @xmath0-difference equation is defined by @xmath116 where @xmath117 \to \br$ ] is defined by @xmath118 the _ entropy _ of is the set of functions @xmath119\to\br \quad | \quad s \subset \{1,\dots , d\ } \}.\ ] ] notice that the entropy of a @xmath0-difference equation is not a real number , but rather a finite collection of functions . the main result is the following if @xmath120 is an @xmath114-regular solution of the regular @xmath0-difference equation , then for every @xmath18 $ ] we have : @xmath121 finally , let us define the @xmath122-entropy of a knot . in @xcite the first author showed that to every knot @xmath5 one can associate a canonical @xmath0-difference equation of degree @xmath45 , and a specific solution of it , namely the colored jones function of @xmath5 . the @xmath0-difference equation itself is an invariant of a knot , which ( by definition ) is determined by the colored jones function of the knot . thus , any invariant of the @xmath0-difference equation is also an invariant of a knot , which is determined by the colored jones function of the knot . the @xmath122-entropy of a knot is the entropy of its associated @xmath0-difference equation . we denote the @xmath122-entropy of a knot @xmath5 by @xmath123\to\br \quad | \quad s \subset \{1,\dots , d\ } \}.\ ] ] the paper was written in the spring of 2004 . since then , a number of papers that discuss the asymptotics of the colored jones function have appeared ; see @xcite . the results of this paper were announced in the jami 2003 meeting in johns hopkins . the first author wishes to thank j. morava for the invitation , and p. deligne who suggested the asymptotic behavior of solutions of @xmath0-difference equations . the first author wishes to thank d. boyd for sharing and explaining his unpublished work and also a. riese , t. morley , and d. zeilberger . in this section , we will translate asymptotics of solutions of @xmath0-difference equation in terms of asymptotics of solutions of _ @xmath67-difference equations_. the latter are defined as follows . fix a positive number @xmath124 , a compact interval @xmath81 of @xmath125 and a natural number @xmath45 . we will consider functions @xmath126 with domain @xmath127 , \quad k \e \in i \}.\ ] ] consider the @xmath67-difference equation for a function @xmath128 @xmath129 where @xmath130)$ ] . we will assume that for all @xmath131 , @xmath132 has a uniformly ( with respect to @xmath26 ) asymptotic expansion @xmath133 where @xmath134 . as we mentioned before , @xmath67-difference equations are obtained from @xmath0-difference equations by the substitution @xmath68 where @xmath67 is a small nonnegative real number , that plays the role of planck s constant . the _ characteristic polynomial _ of is @xmath135 we will say that if @xmath136 for all @xmath137 . let @xmath138 denote the roots of the characteristic polynomial , which we call the _ eigenvalues _ of . it turns out that is regular iff the eigenvalues @xmath138 never collide and never vanish , for every @xmath137 . moreover , it follows by the implicit function theorem that the roots are smooth functions of @xmath137 . since we are interested in asymptotics of solutions of @xmath67-difference equations which , as we shall see , are governed by the magnitude of the eigenvalues , we need to partition the interval @xmath81 according to the magnitudes of the eigenvalues . let @xmath139 denote a partition of @xmath81 into a finite union of closed intervals ( with nonoverlapping interiors ) , such that the magnitude of the eigenvalues does not change in each interval . in other words , for each @xmath77 , there is a permutation @xmath78 of the set @xmath79 such that latexmath:[\[|\l_{\s_p(1)}(x)| \geq |\l_{\s_p(2)}(x)| \geq \dots \geq the following definition introduces a locally fundamental set of solutions of @xmath67-difference equations . fix a partition of @xmath81 as above . a set @xmath82 is a _ locally fundamental set of solutions _ of iff for every solution @xmath141 , for every @xmath77 and for every @xmath84 there exist smooth functions @xmath142 $ ] such that @xmath143 here , the notation @xmath85 does not indicate the @xmath144th power of @xmath145 . the next theorem summarizes the results of costin - costin . ( @xcite ) assume that is regular . then , there exists a positive @xmath146 and a locally fundamental set of solutions @xmath82 of such that * for every @xmath84 and @xmath147 we have @xmath148 * for some smooth functions @xmath149)$ ] with uniform ( with respect to @xmath137 ) _ asymptotic expansion _ @xmath150 where @xmath93 for all @xmath94 * and leading term @xmath151 fix a solution @xmath83 of . theorem [ thm.2 ] expresses @xmath83 as a linear combination of the @xmath111 s in each interval @xmath102 . for every @xmath103 , let @xmath152 later ( in section [ sub.assregulare ] ) we will define the notion of a regular solution to an @xmath67-difference equation . as a prototypical example , consider an @xmath67-difference equation that satisfies @xmath153 for all @xmath106 and all @xmath154 $ ] . then , any solution that satisfies @xmath155 ( or more generally , @xmath108 has a nonvanishing derivative at @xmath156 ) is regular . our next definition captures the growth rate of regular solutions to regular @xmath67-difference equations . fix a collection @xmath112 of subsets of @xmath113 . the @xmath114-_entropy _ @xmath157 of the @xmath67-difference equation is defined by @xmath158 where @xmath159 is defined by @xmath160 if @xmath83 is an @xmath114-regular solution to a regular @xmath67-difference equation , and @xmath137 , we have : @xmath161 the next remarks concern the uniqueness of a set of locally fundamental solutions to . for every @xmath84 the smooth functions @xmath162 for positive @xmath94 are uniquely determined by and the initial condition @xmath163 . indeed , applying taylor series ( with respect to @xmath67 ) in and collecting terms , we get for example : @xmath164 where @xmath165 and @xmath166 . similarly , for @xmath167 we have : @xmath168 where @xmath169 is a function of derivatives of @xmath170 and @xmath171 for @xmath172 . notice that the denominator vanishes nowhere since the roots do not collide and do not vanish for every @xmath137 . if the coefficients @xmath132 of the regular @xmath67-difference equation are analytic functions , then the functions @xmath162 of theorem [ thm.2 ] are also analytic , for every @xmath173 and @xmath94 . this follows by induction from the differential hierarchy which these functions satisfy , and from the fact that the eigenvalues are analytic functions . even though @xmath162 is analytic for every @xmath173 and @xmath94 , the series @xmath174 is in general divergent , and the functions @xmath175 of theorem [ thm.2 ] are not analytic . even though the functions @xmath162 are uniquely determined by the @xmath67-difference equation , the smooth functions @xmath176 ( and thus the locally fundamental set of solutions @xmath101 ) are not uniquely determined by the @xmath67-difference equation . the problem is that smooth interpolation is not unique . recently developed ideas of exponentially small corrections might construct a unique set of locally fundamental solutions when the coefficients of are analytic functions . we will elaborate on this in a separate occasion . the translation of @xmath0-difference equations to @xmath67-difference equations is as follows . if @xmath120 satisfies the @xmath0-difference equation @xmath177 then set @xmath178 and consider the @xmath67-difference equation for a function @xmath179 ( with domain @xmath180 for some @xmath181 and @xmath182 $ ] ) : @xmath183 the following lemma , although elementary , is the key to translating @xmath0-difference equations to @xmath67-difference equations . for every @xmath184 $ ] we have : @xmath185 consequently , for every @xmath186 $ ] , we have : @xmath187 thus , theorem [ thm.2 ] implies theorem [ thm.1 ] . observe that @xmath188 , thus @xmath189 satisfies the equation @xmath190 and so does @xmath191 . since solutions with matching initial conditions are unique , follows . equation follows from equation by the substitution @xmath192 : @xmath193 in this section we will review some linear algebra . it is obvious that the complex roots of a monic polynomial uniquely determine it . it is also known @xcite that the eigenvalues of a companion matrix uniquely determine it , in case they are distinct . consider a _ companion _ @xmath45 by @xmath45 matrix @xmath194 the characteristic polynomial of @xmath195 is @xmath196 with roots @xmath197 . let @xmath198 be the vandermonde matrix , and @xmath199 be the diagonal matrix with diagonal entries @xmath200 . if a companion matrix has distinct eigenvalues , then with the above notation we have : @xmath201 observe that @xmath202 is an eigenvector of @xmath195 with eigenvalue @xmath203 . thus , @xmath204 and @xmath205 . the result follows . now , consider a companion matrix @xmath206 whose entries in the bottom row are smooth functions in a variable @xmath207 , with roots @xmath208 which we assume are distinct for all @xmath207 . the next lemma is a key estimate for the norm of long products of slowly varying matrices . in the language of physics , @xmath206 is the _ transfer matrix _ and @xmath209 is the _ transition matrix_. assume that the eigenvalues @xmath208 of @xmath206 are distinct for all @xmath207 and @xmath210 then for @xmath211 , @xmath212 , we have @xmath213 by lemma [ lem.0 ] , we have : @xmath214 if @xmath211 , it follows that @xmath215 which implies that @xmath216 now , we have @xmath217 if @xmath218 $ ] , using the fact that @xmath219 , we obtain : @xmath220 if @xmath221 and @xmath222 are vandermonde matrices , such that @xmath223 is nonsingular , then @xmath224 in this section we will prove that has a unique set of formal solutions . let us define those first . a _ formal series _ @xmath225 is one of the form @xmath226 where @xmath227 are smooth functions for all @xmath94 . note that @xmath228 lies in the ring @xmath229 $ ] of formal power series with coefficients smooth functions on @xmath81 . note further that if @xmath225 is a formal series , so is @xmath230 for every @xmath231 , where the latter may be defined using the taylor series of @xmath232 : @xmath233 it follows that if @xmath225 is a formal series , then the ratio @xmath234 lies in the ring @xmath229 $ ] . using the language of _ difference galois theory _ ( see @xcite ) this implies that @xmath229 $ ] is a finite difference ring , under the map @xmath235 . we say that a formal series @xmath236 of is a _ formal series solution _ to iff @xmath237.\ ] ] it is easy to see that if @xmath236 is a formal solution to , then the leading term @xmath238 satisfies the equation @xmath239 where @xmath240 is an eigenvalue of . assume that is regular . then , has @xmath45 unique formal series solutions @xmath241 with leading terms corresponding to the eigenvalues of . first we need to show that is indeed an equation in the power series ring @xmath229 $ ] , i.e. , that the terms involving negative powers of @xmath67 cancel . suppose that @xmath236 is given by . it follows from the calculation preceding lemma [ lem.galois ] that for every @xmath242 , we have : @xmath243 where @xmath244 denotes the coefficient of @xmath245 in a power series @xmath246 , and where @xmath247 is a polynomial in the derivatives of @xmath248 for @xmath172 . expand the terms of equation into power series in @xmath67 using the above equation and , and collect terms of powers of @xmath67 . it follows that is equivalent to a hierarchy of first order differential equations : @xmath249 where for positive @xmath94 , @xmath250 is a polynomial in the derivatives of @xmath248 and @xmath170 for @xmath172 . now fix an @xmath251 , and choose @xmath252 such that @xmath253 , where @xmath254 are the eigenvalues of . since is regular , it follows that the roots @xmath138 of the characteristic polynomial @xmath255 never collide , and never vanish for @xmath137 . thus , @xmath256 for all @xmath137 . thus , after we choose @xmath252 , it follows that we can find functions @xmath162 for @xmath257 that satisfy the above hierarchy . moreover , for every @xmath173 , the sequence of functions @xmath162 is uniquely determined by the above hierarchy and the initial conditions @xmath163 . in this section we present an alternative , and slightly more general form , of formal series . in case of regular @xmath67-difference equations this alternative form will not be needed . however , when eigenvalues collide or vanish , one must use this alternative form of formal series . thus , in the present paper we will not use this alternative form of formal series , and the reader may skip this section . an _ alternative formal series _ @xmath225 is one of the form @xmath258 where @xmath259 are smooth functions for all @xmath94 , and @xmath260 for all @xmath137 . in the remainder of this subsection , we will refer to alternative formal series simply by formal series . note that if @xmath225 is a formal series , so is @xmath230 for any @xmath231 , where the latter may be defined using the taylor series of @xmath261 and @xmath262 around @xmath26 . it follows that @xmath263 moreover , if @xmath264 is a formal series , then the ratio @xmath234 lies in the ring @xmath229 $ ] . this follows from and the following computation , valid for every @xmath265 : @xmath266 in analogy with lemma [ lem.galois ] , this implies that @xmath229 $ ] is a finite difference ring , under the map @xmath235 . we say that a formal series @xmath236 of is a _ formal series solution _ to iff @xmath267.\ ] ] it is easy to see that if @xmath236 is a formal solution to , then the leading term @xmath179 satisfies the equation @xmath268 where @xmath240 is an eigenvalue of . in analogy with proposition [ prop.formal ] , we have the following : assume that is regular . then , has @xmath45 unique formal series solutions @xmath241 with leading terms corresponding to the eigenvalues of . in this section we prove theorem [ thm.2 ] . the strategy is to * prove that there exists a solution @xmath269 with the stated properties where @xmath270 is an eigenvalue with maximum magnitude . * use this solution @xmath269 to reduce theorem [ thm.2 ] to the case of a @xmath67-difference equation of degree one less than the original one . * prove that the constructed set of solutions is a locally fundamental set . without loss of generality , we will assume that the eigenvalues of satisfy the inequality : @xmath271 for all @xmath137 . otherwise , we can partition @xmath81 into subintervals where this is true . consider first a formal solution @xmath272 of given in proposition [ prop.formal ] , which satisfies and . consider the smooth functions @xmath273 of . the proof of the following lemma ( due to borel in case @xmath274 are constant functions , for all @xmath94 ) can be found in ( * ? ? ? * lemma 2.5 ) : there exists a smooth function @xmath275)$ ] such that we have ( uniformly in @xmath137 ) : @xmath276 now , consider the unique solution @xmath269 of with initial conditions @xmath277 and for small enough @xmath67 , where without loss of generality , we assume that @xmath278 $ ] . of course , for large @xmath279 it may not be true that @xmath280 . the next proposition estimates the error , uniformly with respect to @xmath279 : there exists an @xmath281 and constants @xmath282 such that for all @xmath283 and all @xmath242 , we have ( uniformly in @xmath279 ) : @xmath284 let us make a change of variables : @xmath285 where @xmath286 we will show that for a fixed @xmath287 , and for every @xmath94 there exists a constant @xmath282 such that for all @xmath288 we have : @xmath289 since @xmath269 satisfies , it follows that @xmath290 satisfies @xmath291 where @xmath292 it is easy to see that * @xmath293)$ ] , has uniform ( with respect to @xmath26 ) @xmath67-asymptotic expansion as in , * @xmath294 , * and since @xmath236 is a formal series solution to and @xmath295 is given by lemma [ lem.borel ] , if follows that for every @xmath94 we have : @xmath296 the characteristic polynomial of has roots @xmath297 for @xmath39 . if is regular , so is . notice that @xmath270 vanishes nowhere since is regular . we now show . let us write the difference equation in matrix form @xmath298 where @xmath299 and @xmath300 . iterating , we obtain that @xmath301 for all @xmath302 , where @xmath303 , a column vector with all entries equal to @xmath63 . equation gives : @xmath304 where @xmath305 uniformly in @xmath279 and @xmath67 . feeding in the above equation , we obtain : @xmath306 now , let us look at the roots @xmath307 of @xmath308 since @xmath309 , it follows that @xmath310 since @xmath311 lies in @xmath81 , a compact set , lemma [ lem.1 ] and equation imply that @xmath312 for all @xmath313 , where @xmath314 . this completes the proof of . equations and differ in the presence of @xmath287 . it is easy to see that if @xmath120 is a function such that for a fixed @xmath287 and any @xmath242 we have : @xmath315 then @xmath316 this observation shows that implies and concludes the proof of the proposition . @xmath317 there exists a smooth function @xmath318)$ ] such that * for all @xmath283 , we have : @xmath319 * @xmath320 has an asymptotic expansion ( uniform with respect to @xmath26 ) : @xmath321 where @xmath274 are as in lemma [ lem.borel ] . as a result , we have an asymptotic expansion ( uniform with respect to @xmath279 ) : @xmath322 consider the change of variables @xmath290 as in . due to our choice of initial conditions it follows that for every fixed @xmath323 , the function @xmath324 is smooth . using this and the smoothness of the coefficients of , it follows that for every fixed @xmath279 , the function @xmath324 is smooth . so far , the function @xmath290 is defined on @xmath325 : @xmath326 which is a union of line segments in a rectangle @xmath327 $ ] , and satisfies . the complement of these line segments in @xmath328 $ ] consists of an infinite union of open triangles , together with the horizontal segment @xmath329 . we can smoothly interpolate @xmath290 inside these open triangles so that it is defined on @xmath330 $ ] and @xmath331 $ ] and all @xmath242 . let us extend @xmath290 to @xmath327 $ ] by defining @xmath332 for all @xmath137 . we claim that @xmath290 is smooth on @xmath327 $ ] . we need only to check this at the points @xmath333 for @xmath137 . this follows easily from . for example , to check continuity at @xmath333 , consider a sequence @xmath334 such that @xmath335 . then , for @xmath336 implies that @xmath337 , thus @xmath290 is continuous at @xmath333 . using for @xmath338 it follows that @xmath339 for all @xmath340 , and we find that @xmath290 has an @xmath67-asymptotic expansion ( uniform with respect to @xmath26 ) : @xmath341 restricting further @xmath342 if needed , we may assume that @xmath343 for all @xmath344 $ ] ; in other words @xmath345 makes sense for all @xmath344 $ ] . now , we can finish the proof of proposition [ prop.step1 ] . let us define @xmath346 then , implies ( a ) . since @xmath347 is asymptotic to @xmath63 ( uniformly on @xmath26 ) , it follows that @xmath348 is asymptotic to @xmath349 . using the asymptotic of @xmath295 given by lemma [ lem.borel ] , ( b ) follows . we will now prove theorem [ thm.2 ] by induction on the degree @xmath45 of the @xmath67-difference equation . for @xmath350 , it follows from proposition [ prop.step1 ] . the inductive step is the next proposition . assume that theorem [ thm.2 ] holds for regular @xmath67-difference equations of degree less than @xmath45 . then it holds for regular @xmath67-difference equations of degree @xmath45 . _ proof . _ consider a @xmath67-difference equation of degree @xmath45 . we will use the solution @xmath269 of it constructed in proposition [ prop.step1 ] to reduce it to an equivalent equation of degree @xmath351 , and an inhomogeneous @xmath67-difference equation of degree @xmath63 . consider the dependent change of variables @xmath352 this is well - defined since @xmath269 is nowhere zero . then , @xmath179 satisfies iff @xmath290 satisfies @xmath353 where @xmath354 the characteristic polynomials of and are related by @xmath355 as in the proof of proposition [ prop.step1 ] , it is easy to see that is a regular @xmath67-difference equation . moreover , it is easy to see that theorem [ thm.2 ] holds for iff it holds for . indeed , check that the change of variables given by preserves the asymptotics of the solutions of and . thus , it suffices to work with . in that case , @xmath356 is a solution of , since @xmath83 is a solution of . it follows that @xmath357 ( compare this with ) . let us define @xmath358 then , we get that @xmath359 is a solution of the @xmath67-difference equation @xmath360 where @xmath361 the characteristic equations of and are related by @xmath362 since @xmath363 ( by ) and @xmath364 , the same arguments of proposition [ prop.step1 ] imply that is regular , assuming that is regular . by the induction hypothesis , it follows that satisfies theorem [ thm.2 ] . for the remainder of this section , fix a solution @xmath359 of which satisfies the properties of theorem [ thm.2 ] . in other words , @xmath359 satisfies and @xmath365 where @xmath366 is a smooth function with uniform ( with respect to @xmath26 ) asymptotics : @xmath367 there exists a formal solution @xmath368 of such that @xmath369 we need to solve the formal power series equation @xmath370 for @xmath371 in terms of @xmath366 . using the taylor expansion @xmath372 it is easy to see that the above equation equals to @xmath373 from which follows that @xmath374 . dividing the equation by @xmath375 we get an equation in formal power series with nonnegative powers of @xmath67 . moreover , the coefficient of @xmath245 in that power series ( for @xmath257 ) equals to @xmath376 where @xmath169 is a function of @xmath377 and @xmath378 for @xmath379 . since @xmath380 is an eigenvalue of , it is never equal to @xmath63 . this and induction prove the lemma . \(a ) there exists a solution to the equation @xmath381 for @xmath382 in terms of @xmath359 with appropriate initial condition . ( b ) for all @xmath383 we have : @xmath384 where @xmath385 fix @xmath386 and let @xmath218 $ ] . consider a natural number @xmath279 such that @xmath313 and @xmath387 . this is equivalent to @xmath388 where @xmath389 and @xmath390 are natural numbers that depend on @xmath67 and @xmath81 , although we do not indicate this in our notation . then equation implies that @xmath391 summing up , we obtain that @xmath392 choose initial conditions so that @xmath393 . this completes part ( a ) . part ( b ) follows by a telescoping calculation . to finish the proof of proposition [ prop.step2 ] , it suffices to show that the solution @xmath359 of lemma [ lem.stepb ] is asymptotic to the formal solution @xmath394 of lemma [ lem.stepa ] . since @xmath384 and @xmath395 it follows by the definition of @xmath290 given in lemma [ lem.stepb ] and by a telescopic sum , that : @xmath396 this concludes the proof of proposition [ prop.step2 ] . let us summarize what we have obtained so far . consider a partition @xmath397 of @xmath398 $ ] given by @xmath399 $ ] for @xmath400 , and consider a permutation @xmath78 of @xmath401 such that latexmath:[\[|\l_{\s_p(1)}(x)| \geq |\l_{\s_p(2)}(x)| \geq \dots \geq constructed solutions smooth functions @xmath90 for @xmath84 with asymptotic expansion given by and . let us define @xmath403 where @xmath90 are smooth functions with asymptotic expansions as in equations and . moreover , we have shown that for every interval @xmath102 ( as in the discussion prior to theorem [ thm.2 ] ) , @xmath404 is a set of solutions of when @xmath405 . fix a solution @xmath406 of and an interval @xmath102 . the following lemma certainly implies that @xmath407 is a locally fundamental set of solutions of . this concludes the proof of theorem [ thm.2 ] . in addition , the next lemma motivates the definition of a regular solution , given in the following section . \(a ) fix @xmath83 and @xmath102 as above . then , there exist smooth functions @xmath85 such that @xmath408 for all @xmath405 . ( b ) moreover , for every @xmath144 and @xmath84 we have @xmath409 for some smooth functions @xmath410 , with the understanding that @xmath411 . here $ ] is the _ largest integer smaller than _ @xmath26 , and @xmath413 + 1.\ ] ] without loss of generality , let us assume that @xmath414 for @xmath39 . equation is a linear equation in @xmath85 , with solutions @xmath415 where @xmath416 $ ] , @xmath417 and @xmath418 where the @xmath83 s are in the @xmath173th column of @xmath419 . we will show that for small enough @xmath67 , @xmath420 is nonsingular . using equations , and , it follows that @xmath421 uniformly in @xmath26 , where @xmath96 are the eigenvalues of . since is regular , its eigenvalues never collide . similarly , using equation , we have : @xmath422 where @xmath423 where the @xmath83 s are in the @xmath173th column of @xmath424 . thus , @xmath425 the idea now is to move the recursion relation backwards @xmath45 times . using the solution @xmath406 for @xmath426 will allow us to compute the smooth functions @xmath85 . in detail , consider the matrix @xmath427 and move the recursion relation backwards once . using equation and the fact that the the @xmath428th column of @xmath429 for @xmath430 is an eigenvector of @xmath431 ( up to @xmath432 ) with eigenvalue @xmath433 , it follows that @xmath434 iterating @xmath351 more times , it follows that @xmath435 since @xmath436 , it follows ( as in the computation of @xmath437 above ) that : @xmath438 this , together with equation proves . in this section we discuss regular solutions of @xmath0 and @xmath67-difference equations and their asymptotics . according to lemma [ lem.certainly ] , a solution @xmath83 to determines a collection @xmath112 of subsets of @xmath113 , where @xmath439 fix a collection @xmath112 of subsets of @xmath113 . we say that a solution @xmath83 of a regular equation is @xmath114-_regular _ iff for every @xmath77 we have : * @xmath440 if @xmath441 . * @xmath410 are _ not flat _ at @xmath109 for all @xmath442 . that is , some derivative of @xmath410 at @xmath443 does not vanish . * for every @xmath77 there exists an element @xmath444 such that @xmath445 in other words , in the interior of the interval @xmath102 , and among the eigenvalues @xmath446 for @xmath447 , there is a unique eigenvalue of strictly maximum magnitude . we will say that a solution to is _ regular _ if it is @xmath114-regular for some @xmath114 . ( of theorem [ thm.ass ] ) let @xmath83 be an @xmath114-regular solution to . let us assume that @xmath448 , and @xmath449 of the closed interval @xmath278 $ ] . fix an @xmath137 . then , we have : @xmath450 for @xmath451 , where @xmath452 , and @xmath453 . then , we have : @xmath454 recall from theorem [ thm.2 ] that @xmath455 thus , @xmath456 combined with @xmath457 for @xmath458 , and @xmath459 , it follows that @xmath460 for all @xmath137 and @xmath461 therefore , @xmath462 and @xmath463 thus , equation implies that @xmath464 the result follows . in the general case , we partition the interval @xmath102 as in the discussion prior to theorem [ thm.2 ] and repeat the above proof using equation . the result follows . first , we need to define what is a regular solution to a @xmath0-difference equation . consider a solution @xmath83 of a @xmath0-difference equation and a partition of @xmath76 as in section [ sub.qdiff ] . then , at each interval @xmath102 , we can write the solution as a linear combination of fundamental solutions , as in equation . let @xmath465 be the indexing set of the fundamental solutions that we use in each interval @xmath102 ; see . suppose that the partition of @xmath76 is given by @xmath466 $ ] for @xmath467 , with @xmath468 . then , with @xmath469 it turns out that for every @xmath144 and @xmath84 we have @xmath470 for some smooth functions @xmath410 , with the understanding that @xmath411 . fix a collection @xmath112 of subsets of @xmath113 . we say that a solution @xmath83 of a regular equation is @xmath114-_regular _ iff for every @xmath77 we have : * @xmath440 if @xmath441 . * @xmath410 are _ not flat _ at @xmath109 for all @xmath442 . that is , some derivative of @xmath410 at @xmath443 does not vanish . * for every @xmath77 there exists an element @xmath444 such that @xmath471 in other words , in the interior of the interval @xmath102 , and among the eigenvalues @xmath472 for @xmath447 , there is a unique eigenvalue of strictly maximum magnitude . we will say that a solution to is _ regular _ if it is @xmath114-regular for some @xmath114 . ( of theorem [ thm.assq ] ) it follows from equation of lemma [ lem.translate ] and theorem [ thm.ass ] . the coefficients of the @xmath0-difference equations are rational functions of @xmath0 and @xmath473 . in order to simplify the typesetting , we will give the @xmath0-difference equation @xmath474 in operator form @xmath475 where @xmath476 and where the operators @xmath477 , @xmath478 and @xmath0 , act on a discrete function @xmath479 $ ] by @xmath480 note that @xmath0 commutes with @xmath478 and @xmath477 , and that @xmath481 . in @xcite , the first author showed that the colored jones function @xmath488 of a knot @xmath5 satisfies an essentially unique smallest degree @xmath0-difference equation @xmath489 where the coefficients @xmath490 of @xmath491 are rational functions of @xmath207 and @xmath492 with rational coefficients . the @xmath195-polynomial of @xmath5 parametrizes the moduli space of characters of @xmath495 representations of @xmath496 , restricted to the boundary torus @xmath497 . the @xmath195-polynomial of a knot is an important ingredient to the geometrization of the knot complement and its dehn fillings . * it has integer coefficients and even powers of @xmath223 , that is @xmath499 $ ] . * it is reciprocal , that is , @xmath500 . * it is tempered , that is the edge polynomials of its newton polygon are cyclotomic . * it specializes to @xmath501 for some integers @xmath502 . * @xmath503 is always a factor of @xmath494 , that corresponds to @xmath504 representations . if the colored jones function of a knot was an @xmath114-regular solution to a regular @xmath0-difference equation , and if the aj conjecture were true , then it follows that for every @xmath18 $ ] we have : @xmath505 where @xmath506 is the _ @xmath195-entropy _ of a knot , defined as follows . for a knot @xmath5 in @xmath498 , let @xmath507 for @xmath508 denote the roots of the equation @xmath509 for @xmath510 $ ] , where @xmath45 is the @xmath511-degree of @xmath494 . fix a partition @xmath512 of @xmath513 $ ] by closed intervals with nonoverlapping interiors and a permutation @xmath78 of the set @xmath401 such that latexmath:[\[|l_{\s_p(1)}(t)| \geq |l_{\s_p(2)}(t)| \geq \dots \geq it is natural to ask how the @xmath195-entropy of a hyperbolic knot ( evaluated at @xmath16 ) compares to the hyperbolic volume . the answer to this question is essentially contained in work of d. boyd , @xcite , which we quote without proof here . we urge the reader to look in @xcite for beautiful and suggestive calculations . more generally , among the roots @xmath507 there is a distinguished one , @xmath522 , corresponding to the discrete faithful representation when @xmath523 . let @xmath524 denote the eigenvalue corresponding to the @xmath504 representations . boyd informs us that for @xmath525-bridge knots @xmath5 ( in particular , for the @xmath1 and @xmath2 knots ) , it is true that @xmath526 it follows by ( s4 ) that the eigenvalues collide at @xmath527 . moreover , boyd informs us that for @xmath525-bridge knots there exists a @xmath528 such that @xmath529 for all @xmath428 and all @xmath530 . in this section we discuss in detail @xmath0-difference equation of the the colored jones function of the two simplest knots , namely the trefoil @xmath1 and the figure eight @xmath2 . the former is not hyperbolic , and the latter is . in @xcite , the first author computed that the colored jones function @xmath531 ( resp . @xmath532 ) satisfies the second ( resp . third ) order @xmath0-difference equation @xmath533 where the noncommutative @xmath195-polynomials @xmath534 and @xmath535 are given by : for @xmath2 knot , we have 3 eigenvalues @xmath540 , @xmath541 and @xmath542 . assuming appropriate choices for the branches of the eigenvalues , the plot of @xmath543 and @xmath544 for @xmath545 $ ] is given by : since the hvc is true for the @xmath2 knot , it suggests that the colored jones function lies in a strictly smaller subspace of the vector space of solutions to the @xmath0-difference equation @xmath489 . using work of murakami @xcite , one can figure out exactly the selection principle ; that is which locally fundamental solutions contribute to the colored jones function . * the eigenvalues collide at @xmath527 ( since @xmath548 ) , at @xmath549 ( since @xmath550 ) and by symmetry at @xmath551 and @xmath552 . * there is resonance on the interval @xmath553 $ ] where all three eigenvalues have equal magnitude . * there is vanishing of the coefficients at @xmath554 ( since the denominator @xmath555 of the coefficients is singular at @xmath554 ) . | in this paper we develop an asymptotic analysis for formal and actual solutions of @xmath0-difference equations , under a regularity assumption , namely the non - collision and non - vanishing of the eigenvalues .
in particular , evaluations of regular solutions of regular @xmath0-difference equations have an exponential growth rate which can be computed from the @xmath0-difference equation .
the motivation for the paper comes from a problem in quantum topology , the hyperbolic volume conjecture , which states that a sequence on laurent polynomials ( the so - called colored jones function of a knot ) , appropriately evaluated , becomes a sequence of complex numbers that grows exponentially .
moreover , the exponential growth rate is proportional to the volume of the knot complement .
the connection of the hyperbolic volume conjecture with the paper comes from the fact that the colored jones function of a knot is a solution of a @xmath0-difference equation , as was proven by ttq .
le and the author . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
dwarf galaxies are at the focus of a major cosmological problem affecting the @xmath3cdm scenario : the number of dark matter subhalos around milky way - type galaxies predicted by @xmath3cdm simulations is much larger than the number of observed satellite dwarf galaxies @xcite . most proposals to overcome this problem stem from the idea that the smallest halos would have formed very few stars or failed to form stars at all , and that gas would have been removed in an early epoch . in this way , the lowest mass sub - halos would be either completely dark , and thus undetectable , or extremely faint . two main mechanisms are usually invoked as responsible of the smallest sub - halos failing to have an extended star formation history ( sfh ) : heating from cosmic ultraviolet ( uv ) background radiation arising from the earliest star formation in the universe @xcite and internal sn feedback following the first star formation episodes in the host dwarf galaxy . the cosmic uv background raises the entropy of the intergalactic medium around the epoch of reionization , preventing baryons from falling into the smallest sub - halos and it can also heat and evaporate the interstellar medium of larger sub - halos which have managed some star formation . the former would never form stars while the latter would presently show only a very old stellar population . recent high resolution simulations of dwarf galaxy formation show that the cosmic uv radiation field can also still suppress star formation , even when it can not evaporate the gas from the halos , by simply preventing gas from becoming dense enough to form molecular clouds @xcite , verifying a previous proposal by @xcite . it has also been proposed that ram pressure stripping in the diffuse corona of the host massive galaxy could very rapidly remove the ism already heated by the cosmic uv even over a large range of dwarf galaxy masses @xcite . however , such a mechanism would become dominant later , during the main accretion phase of typical milky way - sized halo , at @xmath4 , in principle allowing star formation to extend for at least a couple of gyr beyond the epoch of reionization . although consensus exists on the important role played by the two former mechanisms , less clear is the mass range of the affected sub - halos ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the fact that most or all of the recently discovered ultra - faint dwarfs ( ufds ) appear to host only a small population of very old stars , points to them as possible fossils of this process , but some of the classical dsph galaxies may also be affected . besides heating the gas , uv photons produce the global cosmic reionization . the redshift at which the universe was fully reionized was @xmath5 , as obtained from the presence of the gunn - peterson trough in quasars , although there is increasing evidence that this process was inhomogeneous @xcite . according to models , the minimum circular velocity for a dwarf halo to accrete and cool gas in order to produce star formation is in the range of @xmath6 to @xmath7 kms@xmath8 , which corresponds to a total mass of @xmath9 m@xmath2 . however , while most dwarf galaxies in the local group show circular velocities below this range and dynamical masses smaller than @xmath10 m@xmath2 ( see e.g. , * ? ? ? * ) , many of them have cmds that have been interpreted as indicating the presence of star formation activity extended well beyond the reionization epoch , even in old dsph galaxies . two main mechanisms have been proposed to overcome this problem . the first one is that dwarf halos could have been much larger in the past and have lost a significant amount of mass due to tidal harassment @xcite . this scenario is further supported by detailed simulations of the tidal interaction between satellites and the host which includes also the baryonic component and ram pressure stripping @xcite . however , counter - arguments exists pointing to dwarf halos being resilient to tidal harassment @xcite . the second one is that a self - shielding mechanism would be at work , protecting the gas in the central denser regions of the dwarf galaxy , where gas can be optically thick to the impinging radiation field . the first mechanism is robust , since it is a natural consequence of hierarchical accretion as dwarf satellites move on highly eccentric orbits , suffering strong tidal shocks from the host potential . the second mechanism is more subtle , since models neglect a related , competitive effect , namely that the local uv radiation from the primary galaxy or nearby proto - clusters could have been much higher than the mean cosmic ionizing flux at @xmath11 @xcite . models by show that milky way satellites with total luminosities @xmath12 are very unlikely to be true fossils of the reionization epoch , and that they probably are the result of hierarchical build - up from smaller halos extending beyond the reionization epoch . more specifically , conclude that the simulated properties of true fossils , i.e. , those which have not undergone any merging events after reionization , agree with those of a subset of the ultra - faint dwarf satellites of andromeda and the milky way . also , they found that most classical dsph satellites are unlikely true fossils , although they have properties in common with them : diffuse , old stellar populations and no gas . we note that , while it would seem natural to associate ufds to reionization fossils , alternative explanations for their origin have recently appeared in the literature in which at least a fraction of them could be remnants of the oldest , most heavily stripped population of galaxy satellites accreting at @xmath13 onto the milky way halo @xcite . used the fraction of star formation produced before reionization as a test to distinguish true fossil galaxies from non - fossils , defining the former as those having produced at least 70% of their stars by @xmath14 . this criterion was defined after the analysis of the theoretical simulations and has the advantage of allowing a thorough comparison of models and observations . in turn , @xcite have obtained the sfhs of 38 local group dwarf galaxies with stellar masses in the range @xmath15 , finding that only five of them are consistent with forming the bulk of their stars before reionization and that only two out of the 13 predicted true fossils by show a star formation quenched by reionization . however , it should be noted that the results of @xcite are affected by limited time resolution at early ages while the predictions by are free of the these effects . these effects are expected to significantly modify the sfhs obtained from observational data , as can be seen in the simulations shown by e.g. , @xcite or @xcite among others . as a consequence of this , either observational effects should be simulated in theoretical models or they should be taken into consideration when comparing observational results with the theoretical models . the second is the objective of this paper . in this paper , we discuss how the temporal resolution limitations of a sfh derived from a cmd might be accounted for . in particular , the goal of this paper is to obtain an estimate of the maximum fraction of mass , _ consistent with the observations _ , which has been converted into stars by @xmath14 and of the time by which 70% of the baryonic mass has been converted into stars in each galaxy . these values can be directly compared with the predictions of any theoretical model of star formation in dwarf galaxies and we do this comparison , for illustrative purposes , with some of the most recent ones , currently available in the literature . we have used high resolution sfhs obtained for a set of isolated local group dwarfs by the lcid collaboration ( * ? ? ? * and references therein ) . our approach opens a new window to the possibility of testing the effects that cosmic ultraviolet ( uv ) radiation and internal sn feedback have on the early sfh of dwarf galaxies ( whether or not combined with self - shielding or other effects ) . this is an exploratory paper in which we provide details of the method and apply it to a sample of very well studied dwarf galaxies . environmental effects , such as tides and ram pressure , affect the mappings between present - day and pre - infall mass satellites and between baryon content and star formation . therefore , we specifically concentrate on a sample of relatively isolated faint dwarfs . the chosen sample is diverse enough to contain both dirrs , transition dwarfs and dsphs . in future works , the method will be applied to an extended sample including ultra faint dwarfs . the structure of the paper is as follows . in section [ secdat ] observational data are described . in section [ secdeconvol ] , the proposed method is explained and applied to a sample of four local group dwarf galaxies . sfhs from section [ secdeconvol ] are compared in section [ secmod ] with a representative set of state - of - the - art theoretical models of galaxy formation . the results are discussed in section [ secsum ] , together with the main conclusions . as with the previous lcid papers , cosmological parameters of @xmath16 kms@xmath8mpc@xmath8 , @xmath17 , and a flat universe with @xmath18 are assumed @xcite . for this work , regarding the observational material , we have used the sfhs of the galaxies cetus @xcite , tucana @xcite , lgs-3 @xcite , and phoenix @xcite , obtained by the lcid collaboration . data were obtained with the acs and wfpc2 onboard hst . the sfhs used in this paper were derived using the iac method , based on the suite of codes iac - star / iac - pop / minniac . for a more detailed analysis , we have divided the three galaxies with larger field coverages , cetus , tucana , and lgs-3 , into two regions : an inner one , within one scale length from the center , and an outer one located beyond two scale lengths . the properties of the galaxies are summarized in table [ t1 ] . the total mass in stellar objects ( @xmath19 ) , @xmath20 luminosity ( @xmath21 ) , velocity dispersion ( @xmath22 ) , and metallicity ( @xmath23 $ ] ) are given . @xmath21 , @xmath22 , and @xmath23 $ ] are from @xcite . the @xmath19 values are calculated scaling @xmath21 with the mass - luminosity relation obtained from the sfh solution of each galaxy . the @xmath21 values range from @xmath24 to @xmath25 , bracketing the limiting value obtained by for galaxies likely to be true fossils . robust sfhs are derived from the cmds of resolved stellar populations , but they are still affected by several sources of uncertainty that limit time resolution . in short , these sources are of three kinds : ( 1 ) those affecting data , ( 2 ) those linked to physical properties , and ( 3 ) those inherent to the methodology used to derive the sfh . limitations of the first kind are related to photon counting statistics , defective flat - field corrections , or psf sampling , among others . sources of the second kind are mainly distance to the object ( contributing to blending and crowding ) , background and foreground contamination , and differential reddening . the term _ observational effects _ has often been used by our team to refer to these two kinds combined and we will adopt it here from now on . they result in limiting the photometry depth and completeness , which in turn vary with stellar colors and magnitudes . they are modeled ( e.g. , with artificial star tests ) and accounted for when synthetic populations are computed . effects of the third kind refer to the robustness of the method and include the accuracy of the stellar evolution libraries from which synthetic populations are simulated as well as the way in which the best solution is reached . other items closely related to the physics of the problem like the age - metallicity degeneracy in the cmd are also involved . all these effects combined result in time resolution limitations and age inaccuracies in the final sfh solutions . to derive the sfh from the cmd of a resolved stellar population one makes a reliable simulation of observational effects ( 1 and 2 above ) in the cmd of one or several synthetic stellar populations . these cmds are in turn compared with the observational cmd to obtain the sfh . and @xcite made an analysis of the effects of this process on the time resolution . to a good approximation it can be assumed that all together , the effects on the sfh are similar to a temporal shift plus a convolution with a gaussian function , @xmath26 . * it should be noticed that average ages of older populations tend to be biased to younger values . this is mainly the effect of the time limit imposed by the method , that does not allow ages older than the age of the universe in the solution . * the function @xmath26 can be obtained according to the following recipe ( see * ? ? ? first , a single burst stellar population with no age dispersion ( or a very small one ) is computed with the age at which @xmath26 is searched . second , the observational effects obtained for the galaxy in the artificial star tests are simulated . third , the sfh of this synthetic population is derived . the result can be taken as @xmath26 . the @xmath26 function can be used to partially remove the observational effects from the sfh derived for a galaxy . to do so , one can proceed parametrically . we do this by : ( i ) computing a large number of model sfhs of given shape , peak , and duration ; ( ii ) convolving them with @xmath26 , and ( iii ) selecting those producing results compatible with the observationally derived sfh . the result of this inverse procedure is not the real , free of errors sfh . the problem we are facing is in fact a bias - variance tradeoff one ( see e.g. , * ? ? ? our objective is removing bias , but the intrinsic variance of the problem remains . nevertheless it provides a better sfh for comparing to the theoretical models of galaxy formation . this is especially true for the oldest ages , where the limitations on temporal resolution are greatest . specifically , we proceed as follows for each observed galaxy field . first , we obtain @xmath26 functions for a range of input ages from 5.0 to 13.0 @xmath27 and we select the one whose resulting peak age is closest to the galaxy sfh peak age . input ages of the selected @xmath26 functions range from 11.0 ( inner lgs-3 field ) , to 13.0 gyr ( outer cetus and tucana fields ) . second , we have computed a large number of sfhs shaped according to gaussian functions of amplitude @xmath28 , mean age @xmath29 , and standard deviation @xmath22 . we will refer to these trial model sfhs simply as _ trial models_. values of @xmath29 have been sampled from 6.0 to 13.5 gyr with step of 0.1 gyr . in turn , @xmath22 has been sampled from 0.1 to 6 gyr with steps of 0.1 gyr and with the additional criterion that the resulting functions are truncated for age larger than 13.5 gyr . trial models include a simple but realistic simulation of the metallicity distribution of the real galaxy for the model age . to this , a gaussian metallicity distribution is used . the mean metallicity is the metallicity observationally obtained for the galaxy at age @xmath29 and the sigma is the metallicity dispersion obtained for the galaxy and the same age . each trial model shape has been computed a total of 201 times , each one with a different value of @xmath28 . a total of about @xmath30 trial models have been computed for each galaxy field . in the third step , each trial model has been convolved with the selected @xmath26 function . we will refer by _ convolved models _ to the results of these convolutions . the fourth step has been to select all the convolved modles that are compatible with the sfh obtained for the galaxy field within the error intervals of the latter , including its integral . to apply this criterion , only the age interval @xmath31 gyr has been considered . the final step has been to average all the compatible convolved modles and , in turn , all the trial models originating them . figure [ f0 ] shows a summary of the models , including the ones selected for the case of the outer field of the tucana galaxy . general results are given in figure [ f1 ] , which shows that , except for the case of phoenix , the average of the _ good _ trial models peaks at older ages and is narrower than the convolved models or the observational sfhs . following this , the averages of the good trial models will be used to make direct comparisons to theoretical models of galaxy formation . the gaussian functions used above are clearly a simplified representation of the sfh . it is not expected to reproduce extended sfhs or those in which recursive star formation bursts are going on for an extended period of time . indeed , figure [ f1 ] shows that intermediate ages for phoenix are not well reproduced . nonetheless it can still be a good approximation for old stellar populations which is the purpose of this exercise . to check for the dependence of the solutions with the trial model shape , we have repeated the same experiments using in turn triangular and step functions . in all the cases results are similar and do not significantly change the conclusions of our work . since we are interested here in the fraction of stars formed before a given time , it is better to use the cumulative sfh . we define it as @xmath32 , where @xmath33 is the age assumed for the onset of the star formation and @xmath34 is the star formation rate figure [ f2 ] shows the observed and the good trial model cumulative sfhs for all the galaxy fields . the latter is the information needed to make a direct comparison with theoretical models of galaxy formation . in figure [ f3 ] the average of the good trial models of the galaxies are compared with a few , representative theoretical models of galaxy formation , namely models 2 , 7 , and 22 by @xcite , model _ doc _ by @xcite , and two models by @xcite representative of old and old+intermediate age dwarf galaxies . the latter are those given by the authors by the green curves in their figure 5 . @xcite have presented high - resolution hydrodynamical simulations of the formation and evolution of isolated dwarf galaxies including the most relevant physical effects , namely , metal - dependent cooling , star formation , feedback from type ii and ia sne , uv background radiation , and internal self - shielding . models 2 and 7 include uv background radiation and internal self - shielding together with sne feedback . model 22 includes only sne feedback , i.e. , it is representative of the case that uv background radiation has no effect on the sfh . @xcite have carried out fully cosmological , very high resolution , @xmath3cdm simulations of a set of field dwarf galaxies . model _ doc _ corresponds to a virial mass of @xmath35 . finally , @xcite have used the cosmological hydrodynamical simulation of the local group carried on as part of the clues project . in all the cases , the reader is referred to the source papers for details . some relevant properties of the model galaxies are summarized in table [ t1 ] . the total mass in stellar objects ( @xmath19 ) , @xmath20 luminosity ( @xmath21 ) , velocity dispersion ( @xmath22 ) , and metallicity ( @xmath23 $ ] ) are given . @xmath19 is provided by @xcite and @xcite , respectively . regarding @xmath21 , the values given by @xcite are listed for their models . for the @xcite models , the @xmath21 are obtained with iac - star using the model sfhs provided by these authors as input . no data are given for the @xcite models , since these authors do not provide them . figure [ f3 ] shows that the average of the cumulative good trial models of cetus , tucana , and lgs-3 lay between models sm2 and sm7 ( both including reionization ) , on the one side , and models sm22 ( without reionization ) and bl-1 ( average of their oldest models ) , on the other side . furthermore , the outer parts of cetus and tucana show a reasonable correspondence with sm7 , indicating that reionization could have played a role in quenching the star formation in those regions . however , according to this figure , the inner regions of both galaxies plus lgs-3 ( inner and outer ) and phoenix seem difficult to reconcile with the @xcite models with reionization . a good correspondence exists between the inner region of lgs-3 and models bl-1 and sm22 , while no good match is found for phoenix with any of the models considered here . finally , models bl2 , doc , and bashful are , by far , too young to reproduce any of the observed galaxies and are likely to be more suitable for much younger galaxies , like ic-1613 ( see * ? ? ? have defined true fossil galaxies as those having formed all or most of their stars before the reionization era , at @xmath14 . they used the cumulative fraction of star formation produced before reionization as a test to distinguish true fossil galaxies from non - fossils , defining the former as those having produced at least 70% of their stars at @xmath14 , which , for the cosmological parameters adopted here ( see above ) , corresponds to approximately @xmath36 gyr ago . table [ t2 ] gives , for each galaxy field and model , the age at which @xmath37 and the value of @xmath38 for redshift @xmath14 . columns 2 and 3 list the values corresponding to the observed sfhs . columns 4 and 5 give those corresponding to the average of the good trial models . errors have been obtained from the error bands shown in figure [ f2 ] . two main conclusions can be obtained from table [ t2 ] . first , the age for which the cumulative sfh , @xmath38 reaches 70% is increased by observational effects by @xmath39 on average , while the value of @xmath38 at @xmath14 is increased by a factor of @xmath40 , on average , although it can be larger than two in some cases . this shows that working with the average of good trial models is useful if the earliest evolution of dwarf galaxies is sought . second , taken at face value , none of the four galaxies analyzed here fulfill the criterion by of having @xmath41 at @xmath14 , to be considered true fossils from the pre - reionization era , although the outer regions of cetus and tucana might marginally qualify , within a 2@xmath22 error interval . in addition , also concluded that galaxies with @xmath42 are unlikely true fossils , while those with @xmath43 remain reasonable candidates to be fossils of the first galaxy generation . however , three out of the four galaxies considered here have @xmath44 ( tucana , lgs-3 , and phoenix ) and are not true - fossils . in summary , we have presented a method to address the limitations of the temporal resolution of the early sfhs of galaxies derived from deep cmd modeling . we have applied the method to the analysis of the duration of the early star formation activity in a sample of local group dwarf galaxies with the purpose of testing whether or not they are true fossils of the pre - reionization era . for a study of this kind to be accurate , the affects of limited time resolution need to be accounted for . for testing for fossils , using sfhs directly derived from deep cmd modeling may produce biased results due to limited time resolution . we have shown elsewhere @xcite that the sfh derived for a synthetic , single age and metallicity stellar population can be simulated by a gaussian whose sigma depends on age ( @xmath26 ) . the method presented here consists in computing a large number of stellar populations with gaussian sfhs and metallicity distribution similar to the on obtained for the galaxy for the same age interval ( that we call _ trial models _ ; about @xmath30 per galaxy in this case ) of varying mean , sigma , and amplitude , and selecting all those that , convolved with @xmath26 , produce results compatible with the optimal sfh directly derived from observations . the average of the good trial models is an improved approach to the real sfh of the galaxy at the very earliest times . in general , the ages of the averages of the good trial models are about @xmath45 gyr older than the optimal sfh solutions for the case of predominantly old galaxies . we have applied our method to four local group dwarfs : cetus , tucana , lgs-3 , and phoenix and we have compared our results with predictions made by a number of recent cosmological hydrodynamical simulations for the formation and evolution of dwarf galaxies . a relatively sharp exhaustion of the star formation is necessary at an early epoch , close to @xmath14 , to account for the sfh obtained for the outer region of tucana and cetus . the inner part of both galaxies , as well as lgs-3 and , more clearly , phoenix , are compatible with models predicting more extended star formation activity for dwarf galaxies . however , none of the galaxies and fields studied here , except perhaps the outer regions of cetus and tucana , fulfill the criterion to be considered as fossil remnants of the pre - reionization era . as a cautionary note , at present , it can not be excluded that cetus and tucana were affected by environmental effects to some extent . their distances to the massive spirals of the lg , while placing them outside the virial radius of either today , is still small enough to admit scenarios in which these galaxies are on nearly radial orbits and they may have already had a close pericenter passage with their host @xcite . if they are bound satellites , their relatively large velocity dispersion , @xmath46 km / s ( see table 1 ) , would make them comparable to the brightest dsph satellites such as fornax , and implies their halos could have been much more massive before tides began to prune them substantially . if this were the case , it would be likely that their halo mass before infall was high enough to place them clearly above the threshold mass for reionization to play a role , strengthening further the conclusion that they can not be reionization fossils . the authors thanks the anonymous referee for her / his comments and suggestions , that help to improve the paper clarity . this work has been funded by the economy and competitiveness ministry of the kingdom of spain ( grants aya2010 - 16717 and aya2013 - 42781-p ) and by the instituto de astrofsica de canarias ( grant p3/94 ) . aparicio , a. , & gallart , c. 2004 , , 128 , 1465 aparicio , a. , gallart , c. , & bertelli , g. 1997 , , 114 , 680 aparicio , a. , & hidalgo , s. l. 2009 , , 138 , 558 becker , r. h. , fan , x. , white , r. l. , et al . 2001 , , 122 , 2850 becker , g. d. , bolton , j. s. , madau , p. , et al . 2015 , , 447 , 3402 bentez - llambay , a. , navarro , j. f. , abadi , m. g. , et al . 2015 , , 450 , 4207 bovill & ricotti 2011a , , 741 , 17 bovill & ricotti 2011b , , 741 , 18 bullock , j. s. , kravtsov , a. v. , & weinberg , d. h. 2000 , , 539 , 517 gnedin , n. y. 2000 , , 542 , 535 gnedin , n. y. , & kravtsov , a. v. 2006 , , 645 , 1054 grebel , e. k. , & gallagher , j. s. iii 2004 , , 610 , l89 hastie , t. , tibshirani , r. , friedman , j. 2009 _ the elements of statistical learning _ , springer - verlag , new york . hidalgo , s. l. , aparicio , a. , martnez - delgado , d. , & gallart , c. 2009 , , 705 , 704 hidalgo , s. l. , aparicio , a. , skillman , e. , et al . 2011 , , 730 , 14 hidalgo , s. l. , monelli , m. , aparicio , a. , et al . 2013 , , 778 , 103 iliev , i. t. , moore , b. , gottlber , s. et al . 2011 , , 413 , 2093 kauffmann , g. , white , s. d. m. , & guiderdoni , b. 1993 , , 264 , 201 kazantzidis , s. , mayer , l. , mastropietro , c. , et al . 2004 , , 608 , 663 kazantzidis , s. , okas , e. l. , callegari , s. , et al . 2011 , , 726 , 98 klypin , a. , kravtsov , a. v. , valenzuela , o. , & prada , f. 1999 , , 522 , 82 komatsu , e. , dunkley , j. , nolta , m. r. , et al . 2009 , , 180 , 330 kravtsov , a. v. , gnedin , o. y. , & klypin , a. a. 2004 , , 609 , 482 loeb , a. , & barkana , r. 2001 , , 39 , 19 mac low , m .- m . , & ferrara , a. 1999 , , 513 , 142 mayer , l. , kazantzidis , s. , mastropietro , & c. , wadsley , 2006 , , 445 , 738 mayer , l. , 2010 , advances in astronomy , art . 278434 mcconnachie , a. w. 2012 , , 144 , 4 monelli , m. , hidalgo , s. l. , stetson , p. b. , et al . 2010a , , 720 , 1225 monelli , m. , gallart , c. , hidalgo , s. l. , et al . 2010b , , 722 , 1864 moore , b. , ghigna , s. , governato , f. , et al . 1999 , , 524 , l19 pearrubia , j. , navarro , j. f. , & mcconnachie , a. w. 2008 , , 673 , 226 ricotti , m. , & gnedin , n. y. 2005 , , 629 , 259 sawala , t. , scannapieco , c. , maio , u. , & white , s. 2010 , , 402 , 1599 schaye , j. 2001 , , 562 , l95 shen , s. , madau , p. , conroy , c. , et al . 2014 , , 792 , 99 skillman , e. d. , hidalgo , s. l. , weisz , d. r. , et al . 2014 , , 786 , 44 sobacchi , e. , & mesinger , a. 2015 , , 453 , 1843 spitler , l. r. , romanowsky , a. j. , diemand , j. , et al . 2012 , , 423 , 2177 susa , h. , & umemura , m. 2004 , , 610 , l5 tomozeiu , m. , mayer , l. , & quinn , t. 2015 , arxiv:1506.02140 weisz , d. r. , dolphin , a. e. , skillman , e. d. , et al . 2014a , , 789 , 147 weisz , d. r. , dolphin , a. e. , skillman , e. d. , et al . 2014b , , 789 , 148 ) the used models . the light grey region shows the sfh of the galaxy including the 1@xmath22 error interval estimate . the dark grey region shows the average of convolved - sfhs compatible with empirical results . the wideness shows the 1-@xmath22 dispersion . thick , black line shows the average of the good trial models . thin , black lines show the 1-@xmath22 dispersion while thin dashed lines are the envelope of all the good trial models . , width=529 ] error interval estimate . dark grey regions show the average of convolved models compatible with empirical results . the wideness of these regions show the 1-@xmath22 dispersion . thick , black lines show the average of the good trial models . thin , black lines show the 1-@xmath22 dispersions . , width=529 ] ) compared with the same for a number of theoretical models of galaxy formation . numbers in the figure correspond to 1 : tucana - outer , 2 : cetus - outer , 3 : tucana - inner , 4 : lgs-3-outer , 5 : cetus - inner , 6 : lgs-3-outer , and 7 : phoenix - inner . regarding to models , sm2 , sm7 and sm22 are respectively models 2 , 7 and 22 by @xcite . model `` doc '' is by @xcite . models `` bl1 '' and `` bl2 '' are models 1 and 2 by @xcite . , width=529 ] ccccc cetus & 7.17 & 2.58 & @xmath47 & @xmath48 + tucana & 1.66 & 0.54 & @xmath49 & @xmath50 + lgs-3 & 2.08 & 0.94 & @xmath51 & @xmath52 + phoenix & 1.29 & 0.78 & & @xmath53 + s2 & 0.96 & 0.43 & 7.3 & @xmath54 + s7 & 10.02 & 4.50 & 9.1 & @xmath55 + s22 & 2.58 & 1.34 & 7.1 & @xmath56 + shen13-doc & 34.0 & 34.04 & & + shen13-bashful & 115.0 & 135.52 & & + bl-1 & & & & + bl-2 & & & & + cccccccc cetus inner & @xmath57 & @xmath58 & @xmath59 & @xmath60 + cetus outer & @xmath61 & @xmath62 & @xmath63 & @xmath64 + tucana inner & @xmath65 & @xmath66 & @xmath67 & @xmath68 + tucana outer & @xmath69 & @xmath70 & @xmath71 & @xmath72 + lgs-3 inner & @xmath73 & @xmath74 & @xmath75 & @xmath76 + lgs-3 outer & @xmath77 & @xmath78 & @xmath79 & @xmath80 + phoenix inner & @xmath81 & @xmath82 & @xmath83 & @xmath84 + sm2 & & & @xmath36 & @xmath85 + sm7 & & & @xmath86 & @xmath87 + sm22 & & & @xmath88 & @xmath89 + shen13-doc & & & @xmath90 & @xmath91 + shen13-bashful & & & @xmath92 & @xmath93 + bl - old & & & @xmath94 & @xmath95 + bl - young & & & @xmath96 & @xmath97 + | the analysis of the early star formation history ( sfh ) of nearby galaxies , obtained from their resolved stellar populations is relevant as a test for cosmological models .
however , the early time resolution of observationally derived sfhs is limited by several factors .
thus , direct comparison of observationally derived sfhs with those derived from theoretical models of galaxy formation is potentially biased .
here we investigate and quantify this effect .
for this purpose , we analyze the duration of the early star formation activity in a sample of four local group dwarf galaxies and test whether they are consistent with being true fossils of the pre - reionization era ; i.e. , if the quenching of their star formation occurred before cosmic reionization by uv photons was completed .
two classical dsph ( cetus and tucana ) and two dtrans ( lgs-3 and phoenix ) isolated galaxies with total stellar masses between @xmath0 to @xmath1 m@xmath2 have been studied .
accounting for time resolution effects , the sfhs peak as much as 1.25 gyr earlier than the optimal solutions .
thus , this effect is important for a proper comparison of model and observed sfhs .
it is also shown that none of the analyzed galaxies can be considered a true - fossil of the pre - reionization era , although it is possible that the _
outer regions _ of cetus and tucana are consistent with quenching by reionization . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
it has long been appreciated that high energy @xmath2-rays from sources at cosmological distances will be absorbed via electron - positron pair production @xcite on the diffuse background of long wavelength photons produced over the history of the universe . now that several extragalactic tev sources have been discovered , it is beginning to become possible to use this process to probe the extragalactic background light ( ebl ) . this technique will become increasingly powerful as many more sources will presumably be discovered at greater distances with the new generation of @xmath2-ray telescopes ( glast , celeste , stacee , magic , hess , veritas , and milagro ) . broadly speaking , there are three approaches to studying the ebl/@xmath2-absorption connection , represented by the three speakers on this topic here at the veritas workshop : * limits on the ebl , and on models for its production , from @xmath2-absorption data @xcite ; * semi - empirical estimates of the ebl @xcite ; and * prediction of the ebl and @xmath2-absorption from physical theories of galaxy formation and evolution in a cosmological framework . the advantage of the last of these approaches , which we will follow in this talk , is that it permits one to deduce from @xmath2-ray absorption data a great deal about galaxy formation and evolution , including the effects of the stellar initial mass distribution and of dust extinction and reradiation . it is also arguably the best way to estimate the extent of @xmath2-ray absorption at various energies , as shown by the correct prediction from a simplified model of this sort ( * ? ? ? * hereafter mp96 ) that there would be rather little absorption of tev @xmath2-rays from the nearest extragalactic sources , mrk 421 and mrk 501 , at redshifts of only @xmath3 . the calculations reported here ( and in more detail in @xcite and @xcite ) are based on state - of - the - art semi - analytic models ( sams ) of galaxy formation , which we summarize briefly below . but it will be useful to start by summarizing our earlier calculations ( mp96 ) . although there is a long history of spectral synthesis models leading to predictions for the ebl ( e.g. , @xcite ) , such models typically attempted to account only for the star formation history of the galaxies existing today i.e. , they are pure luminosity evolution models . moreover in spectral synthesis , the star formation history of each galaxy is _ not _ determined from its cosmological history , taking into account the fact that gas is not available to form stars until it has cooled within a dense collapsed structure . but the evidence is certainly increasing that there was a great deal of galaxy formation and merging in the past , plausibly in agreement with the predictions of hierarchical models of galaxy formation of the cdm type . the motivation of the approach used by mp96 was to obtain theoretical predictions for the ebl and the resulting absorption of @xmath2-rays in the context of hierarchical theories of structure formation , specifically within the cdm family of cosmological models . it is relatively straightforward to calculate the evolution of structure in the dark matter component within the cdm paradigm . in mp96 , the number density of dark matter halos as a function of mass and redshift and its dependence on cosmology was modeled using press - schechter theory , which agrees fairly well with the predictions of n - body simulations . however , obtaining the corresponding radiation field as a function of time and wavelength involves complicated astrophysics with many unknown parameters . this problem was addressed empirically , using the simple assumption that each dark matter halo hosts one galaxy , with the galaxy luminosity assumed to be a monotonic function of the halo mass . to model the spectrum of each galaxy , the star formation rate ( sfr ) was assumed to be @xmath4 , with ellipticals ( assumed to be 28% of the galaxies ) having @xmath5 gyr , and spirals ( the remaining 72% ) having @xmath6 gyr . the actual sfr for each galaxy was determined so that a desired local luminosity function ( llf ) was reproduced at redshift @xmath7 . as there is considerable variation in the observed b - band luminosity function derived from different redshift surveys , mp96 considered three representative choices . stellar emission was modeled in a simplified way , with each stellar population of a given mass and age treated as a black body with the appropriate temperature . three different power - law initial mass functions ( imfs ) @xmath8 describing the differential distribution of the initial stellar masses were considered : the standard salpeter imf with @xmath9 , and also steeper imfs with @xmath10 and -2.0 . the evolution of the gas content and metallicity within each galaxy was treated in the instantaneous recycling approximation . dust absorption was treated in a standard way @xcite , assuming that the mass of dust increases with the galaxy metallicity and gas fraction . the extinction curve was similar to a galactic extinction curve , but was scaled according to the metallicity ( galaxies with lower metallicities have steeper extinction curves in the uv , as indicated by observations of the lmc and smc ) . energy was conserved , so that any energy absorbed by dust was reradiated . the dust emission spectrum was modeled @xcite with three components : pah molecules ( @xmath11 to 30 @xmath12 m ) , warm dust from active star forming regions ( 30 to 70 @xmath12 m ) , and cold `` cirrus '' dust ( 70 to 1000 @xmath12 m ) . two cosmological models were considered , a standard ( cluster - normalized , @xmath13 ) @xmath14 cold dark matter ( scdm ) model , and a cobe - normalized cold + hot dark matter ( chdm ) model with the then - favored hot dark matter fraction @xmath15 . since both were @xmath14 models , the hubble parameter was chosen to be @xmath16 ( @xmath17 km s@xmath18 mpc@xmath18 ) in order to obtain a universe with an age of 13 gyr . galaxy formation occurs fairly early in the scdm model ( because of the large amount of power on small scales ) , and considerably more recently in the chdm model . the main conclusion from this study was that cosmology is the dominant factor influencing the ebl in the range 1 - 10 @xmath12 m , which is the range most relevant for absorption of @xmath0 tev @xmath2-rays from nearby extragalactic sources . in this wavelength range , the most extreme differences between the three different llfs and three different imfs considered were less than the difference between the scdm and chdm cosmological models , representing early and late galaxy formation respectively . it is not difficult to understand why this happens : since the optical luminosity at @xmath7 was fixed for each assumed llf , the main factor determining the ebl in the near infrared was the star formation history . because galaxies were assumed to trace halos in a simple way , the star formation history was almost entirely determined by the cosmology . as mp96 explained , the scdm model predicted a larger ebl flux because ( 1 ) the stars have put out more light since they have been shining longer than in chdm , ( 2 ) there was more redshifting of their light from the optical to the near infrared , and ( 3 ) the scdm galaxies are older at a given redshift and hence are composed of more evolved stars , producing more flux in the red and near - infrared . as expected , when the optical depth for @xmath2-rays due to @xmath19 production was calculated , more absorption was predicted for scdm than for chdm . but for sources as near as mrk 421 and 501 , the predicted absorption only steepens the spectrum a little in the 300 gev - 10 tev range for which results have thus far been published , with curvature noticeable mainly above about 3 tev . these predictions appear to be consistent with the observations @xcite , unlike those from earlier @xcite and later @xcite calculations based on a semi - empirical approach . the predictions of our new , more complete treatment are qualitatively consistent with this earlier simplified approach . our new approach is based on semi - analytic models ( sams ) of galaxy formation , which allow one to model the astrophysical processes involved in galaxy formation in a simplified but physical way within the framework of the hierarchical structure formation paradigm . the semi - analytic models used here are described in detail in @xcite , @xcite , and @xcite . these models are in reasonably good agreement with a broad range of local galaxy observations , including the relation between luminosity and circular velocity ( the tully - fisher relation ) , the b - band luminosity function , cold gas contents , metallicities , and colors @xcite . our basic approach is similar in spirit to the models originally presented by @xcite and @xcite , and subsequently developed by these groups in numerous other papers ( reviewed in @xcite and @xcite ) . significant improvements included in @xcite are that we assumed a lower stellar mass - to - light ratio ( in better agreement with observed values ) , included the effects of dust extinction , and developed an improved `` disk - halo '' model for supernovae feedback . with these new ingredients , we were able to overcome some of the difficulties of previous models , which did not simultaneously reproduce the tully - fisher relation and b - band luminosity function , and produced bright galaxies that were too blue . instead of assuming a one - to - one relationship between galaxies and dark matter halos , as in mp96 , we now determine the galaxy population residing in halos of a given mass by constructing the `` merging history '' of each halo using an extension of the press - schechter technique . using the method described in @xcite , we create monte - carlo realizations of the masses of progenitor halos and the redshifts at which they merge to form a larger halo . these `` merger trees '' ( each branch in the tree represents a halo merging event ) reflect the collapse and merging of dark matter halos within a specific cosmology and have been shown to agree fairly well with merger trees extracted from n - body simulations @xcite . each halo at the top level of the hierarchy is assumed to be filled with hot gas , which cools radiatively and collapses to form a gaseous disk . the cooling rate is calculated from the density , metallicity , and temperature of the gas . cold gas is turned into stars using a simple recipe , depending on the mass of cold gas present and the dynamical time of the disk . supernovae inject energy into the cold gas and may expell it from the disk and/or halo if this energy is larger than the escape velocity of the system . chemical evolution is traced assuming a constant yield of metals per unit mass of new stars formed . the spectral energy distribution ( sed ) of each galaxy is then obtained by assuming an imf and using stellar population models ( e.g. @xcite ; in the present work we use the updated gissel98 models with solar metallicity ) . when halos merge , the galaxies contained in each progenitor halo retain their seperate identities until they either fall to the center of the halo due to dynamical friction and merge with the central galaxy , or until they experience a binding merger with another satellite galaxy orbiting within the same halo . all galaxies are assumed to start out as disks , and major ( nearly equal mass ) mergers result in the formation of a spheriod . new gas accretion and star formation may later form a new disk , resulting in a variety of bulge - to - disk ratios at late times . this may be used to divide galaxies into rough morphological types , and seems to reproduce observational trends such as the morphology - density relation and color - morphology trend @xcite . the recipes for star formation , feedback , and chemical evolution contain free parameters , which we set by requiring an average fiducial `` milky way '' galaxy to have an i - band magnitude , cold gas mass , and metallicity as dictated by observations of nearby galaxies . the star formation and feedback processes are some of the most uncertain elements of these models , and indeed of any attempt to model galaxy formation . we have investigated several different combinations of recipes for star formation and supernova feedback ( sf / fb ) , discussed in detail in @xcite , @xcite and @xcite . the star formation history for several different scenarios in shown in figure [ fig : sfrz ] . note that the three models shown in figure [ fig : sfrz ] are for the same scdm cosmology ; the only difference is in the mechanism used to convert cold gas into stars . this illustrates that , unlike in the previous approach of mp96 , the star formation history of the universe is quite sensitive to the assumed astrophysics and not only the cosmology . here we will discuss results for a single choice of sf / fb recipe , which corresponds to the fiducial `` santa cruz '' model discussed in @xcite , and is similar to the models of kauffmann et al . ( e.g. @xcite ) . we shall elaborate on the effects of changing the sf / fb recipes on the ebl and gamma ray absorption in @xcite and @xcite . in @xcite , we included dust extinction using the same approach as mp96 . as shown in @xcite , this led to better results for the b - band luminosity function , and improved galaxy colors . using a similar approach to modelling dust extinction , @xcite confirmed these results and demonstrated that in addition the inclusion of dust greatly improves the agreement of the galaxy - galaxy correlation function with observations . the inclusion of dust extinction and the re - radiation of absorbed light at longer wavelengths is of course a crucial ingredient in modeling the ebl . the current dust model , discussed in @xcite , is an improved version of the one used in mp96 , and is very similar to the approach used by guiderdoni et al . @xcite . as in mp96 , all of the absorbed starlight is re - radiated by the dust , assuming a three - component blackbody emission spectrum . however , in mp96 the shape of the dust emission spectrum was chosen to match that of the galaxy , which is inconsistent with the global data for iras galaxies . in the current models , the temperatures and relative contributions of the three components are determined by requiring the colors at 12 , 25 , and 60 @xmath12 m to match the observed colors of iras galaxies @xcite . details will be given in @xcite . the stellar initial mass function determines the wavelength distribution of starlight produced by a stellar population of a given age , and as such it is an important ingredient in the calculation of the ebl . in mp96 we considered only the salpeter imf and steeper power - law imfs , but here we will discuss results both for the salpeter imf and the scalo imf ( see figure [ fig : imfs ] ) . these are two of the most commonly used imfs , although recent studies ( e.g. , @xcite ) indicate that a better representation of the observed imf may be a salpeter - like slope at @xmath20 , with a flattening at @xmath21 . the most important difference between the salpeter and scalo imfs for our present purposes is that if both are normalized to the same total mass of stars , the fraction of high - mass stars is higher with the salpeter imf . since only high - mass stars emit significant amounts of ultraviolet light , this results in much more ultraviolet in the spectrum of a typical galaxy with salpeter imf as compared with scalo imf , as shown in figure [ fig : galspectra ] , both without and with inclusion of the effects of dust . note that the much greater amount of absorbed ultraviolet light with the salpeter imf results in much more reradiated infrared light at long wavelengths . also note that changing the imf has a relatively small effect on the predictions in the 1 - 10 @xmath12 m range . [ cols="<,<,<,<,<,<,<,<,<,^,^",options="header " , ] [ tab : cosmo ] previous sam calculations of properties of local galaxies ( e.g. @xcite ) assumed a scalo imf . however , in @xcite and @xcite , it was shown that a more `` top - heavy '' imf ( more high - mass stars ) such as the salpeter imf is favored by observations of uv - bright galaxies at very high redshift ( lyman - break galaxies ) . in @xcite , we investigate the effects of using different imfs ( scalo or salpeter ) on the observable properties of galaxies at @xmath7 . in particular , we calculate the luminosity function at 2000 , b , r , and k , the corresponding 2000 - b , b - v , b - i , and b - k colors predicted by our models , and the tully - fisher relation in various bands . we find that because the mass - to - light ratio in the longer wavebands ( i to k ) is significantly higher for the salpeter imf , the luminosity of a galaxy with a given velocity dispersion is smaller . this makes it very difficult to obtain a bright enough tully - fisher zero - point in cosmologies with @xmath14 , in which the baryon fraction is low ( we assume a fixed value of @xmath22 , as suggested by observations @xcite ; the baryon fraction @xmath23 is therefore higher in low-@xmath24 cosmologies ) . if the mass - to - light ratios predicted by the current generation of stellar population models are accurate and the salpeter imf is really representative of typical galaxies , this may suggest that the baryon fraction in bright galaxies must be @xmath25 , similar to the value in groups and clusters . in @xcite , we calculate the predicted ebl for several cosmologies and also investigate the effects of the assumed imf and star formation recipe , and the variations among stellar population models compiled by different groups . here we can only present a subset of preliminary results . the cosmological models considered are summarized in table 1 . the scdm model is shown for comparison with other work , but is ruled out by many independent observational considerations . the three remaining models represent currently favored variants of the cdm family of models . the shape and normalization of the luminosity function that we obtain from the sams depends on the cosmological model , as shown in figure [ fig : lfk ] . the k - band luminosity function shown in the figure is relatively insensitive to dust and the star formation history . to put the models on an equal footing , following a similar logic to the b - band local luminosity function normalization of mp96 , we renormalize each model to give the same integrated luminosity in the k - band . the common normalization is obtained by integrating the observed local luminosity function from ref . the resulting correction factors range from 0.42 for scdm to 1.7 for lcdm . note that this renormalization also sidesteps the known inaccuracy of the press - schechter model used to estimate the number density of dark matter halos . the factor of 0.42 for scdm is consistent with the rule - of - thumb factor of 0.5 determined from comparison with n - body simulations @xcite . figure [ fig : lcdmsalscalo ] shows the ebl for the model for the scalo and salpeter imfs . as expected from our previous discussion , the predicted ebl is much higher for the salpeter imf at both short and long wavelengths , but the predictions are very similar in the 1 - 10 @xmath12 m band that is most relevant for @xmath0 tev @xmath2-ray attenuation from relatively nearby sources , in agreement with mp96 . note that both ebl curves are consistent with the lower limits from source counts at ultraviolet , optical , and near - infrared wavelengths ( filled symbols ) , but neither curve is high enough to agree with the new dirbe ebl detections at 140 and 240 @xmath12 m @xcite . the remaining results that we present are all for the salpeter imf . figure [ fig : ebl_all ] shows the predicted ebl for the four cosmological models discussed above . note that the three models in which galaxy and star formation are relatively early predict rather similar ebl , while chdm predicts generally lower ebl . figure [ fig : evol ] shows the reason for this in more detail . only about 10% of the ebl in the 1 - 10 @xmath12 m band comes from @xmath26 for the chdm model , compared to 20 - 40% for the model , in which galaxies form considerably earlier . ( an alternative way of looking at the evolution of the ebl is given in figure 8 of @xcite . ) keep in mind , however , that even in a model with an `` early formation '' cosmology like but which assumes less efficient star formation at high redshift ( for example , the model with the star formation history indicated by the light solid line in figure [ fig : sfrz ] ) , the ebl will show a steeper evolution than that shown in figure [ fig : evol ] for lcdm . so we expect some degeneracy between cosmology and astrophysics . however , to the extent that the background cosmology may soon be determined by other methods , measurements of the ebl will provide useful constraints on the star formation history of the universe . all of the models fall short of the dirbe detection at 140 @xmath12 m by at least a factor of @xmath27 . this may be due to a number of effects that we have not yet included in our modeling . much of the far - infrared and sub - mm light is probably produced by heavily extinguished ultra - luminous starburst galaxies . the phenomenological work of guiderdoni et al . @xcite suggests that the contribution from this population may increase with redshift . our independent work suggests a physical reason for this : the galaxy interactions that probably trigger these starburst events are more frequent at higher redshift because of the higher density of the universe , and galaxies are more gas rich so the starburst events may be more dramatic @xcite . observations of nearby starburst galaxies ( cf . @xcite ) suggest that the galactic / smc model that we have used here does not provide a good description of dust extinction in actively star forming galaxies . additional contributions to the far - ir flux that we have not included may come from agn and from energy that is injected into the gas by supernovae and later radiated at long wavelength . we will include the contribution of starburst galaxies and address these other effects in improved models that we are developing in collaboration with bruno guiderdoni and julien devriendt . however , we do not expect that this improved treatment will have much effect on the absorption of @xmath2-rays with energies @xmath28 tev , which we now discuss briefly . here we have space to present only the results for one case , lcdm with salpeter imf . figure [ fig : tau ] shows the optical depth of the universe @xmath29 as a function of @xmath2-ray energy for varying source redshifts , @xmath30 . the corresponding attenuation factors , @xmath31 , are shown in figure [ fig : attenuation ] . note that for sources as near as mrk 421 and 501 ( @xmath3 ) , attenuation is predicted to be rather small between 1 - 10 tev , with little curvature in the spectrum . gamma - ray energies @xmath32 tev for local sources , or sources at @xmath33 for lower energies , will likely be needed in order to see significant attenuation . in @xcite we will present results for several cosmological models , and discuss dependence on imf and star formation prescriptions . bullock et al . @xcite discusses the difference in predicted @xmath2-attenuation between salpeter and scalo imf for this same lcdm model . the early universe is much more transparent to 10 - 100 gev @xmath2-rays with the scalo imf , since the fraction of high - mass stars is lower , and the ultraviolet flux density is correspondingly reduced ( cf . * semi - analytic models ( sams ) of galaxy formation provide a convenient and powerful theoretical framework to determine how input assumptions e.g. , cosmology , star formation history , and imf affect the predicted extragalactic background light ( ebl ) and the resulting @xmath0tev @xmath2-ray attenuation . * the 1 - 10 @xmath12 m ebl and the resulting attenuation of few - tev @xmath2-rays reflect mainly the history of star formation in the universe , with less attenuation for models such as chdm in which galaxies form relatively late . * the ebl at @xmath34 @xmath12 m and @xmath35 @xmath12 m is significantly affected by the imf and the modeling of the absorption and reradiation by dust . * gamma - ray energies @xmath32 tev and/or sources at @xmath33 will probably be needed to provide clear evidence of attenuation due to @xmath36 . * therefore , both space- and ground - based @xmath2-ray telescopes will be required to probe the spectra of agns at various redshifts , in order to determine both * * the unabsorbed spectra , which will help determine how these @xmath2-rays are produced , and * * the intergalactic absorption , which as we have shown is affected by cosmology , star formation history , imf , and dust . jrp and jsb were supported by nsf and nasa grants at ucsc , and rss was university fellowship from the hebrew university . jrp thanks avishai dekel for hospitality at hebrew university . donn macminn contributed significantly to an early stage of this work , especially the dust modeling . his life was tragically cut short by a hit - and - run driver as he was bicycling near chicago on august 30 , 1997 . | we present here the extragalactic background light ( ebl ) predicted by semi - analytic models of galaxy formation , and show how measurements of the absorption of gamma rays of @xmath0 tev energies via pair production on the ebl can probe cosmology and the formation of galaxies . semi - analytic models permit a physical treatment of the key processes of galaxy formation including gravitational collapse and merging of dark matter halos , gas cooling and dissipation , star formation , supernova feedback and metal production and have been shown to reproduce key observations at low and high redshift . using this approach
, we investigate the consequences of variations in input assumptions such as the stellar initial mass function and the underlying cosmology .
we conclude that observational studies of the absorption of @xmath1 tev gamma rays will help to constrain the star formation history of the universe , and the nature and extent of the extinction of starlight due to dust and reradiation of the absorbed energy at infrared wavelengths .
cosmology : observations , diffuse radiation infrared : galaxies galaxies : evolution gamma rays : theory |
You are an expert at summarizing long articles. Proceed to summarize the following text:
in the papers @xcite @xcite we started the systematic study of the special class of effective theories of strong interactions , which we call _ localizable_. roughly speaking , these are theories with the @xmath0-matrix which can be obtained in a _ perturbative way _ in the frame of an _ effective field theory _ that contains auxiliary resonance fields along with the fields of true asymptotic states ( those stable with respect to the strong interaction ) . our goal is to construct an efficient method for calculating the amplitudes of physical processes . this means that we need to develop the systematic scheme of perturbative calculations in the framework of infinite component effective field theory . there are few obstacles that usually prevents the effective theory concept to become a useful computational tool . the main one is the presence of an infinite number of coupling constants , which requires introducing an infinite number of renormalization prescriptions . it is clear that we hardly get some predictive power for our theory unless find some _ regularity _ in the system of those prescriptions . further , if one admits an unlimited number of the resonance fields in a theory ( as we do ) , then even the amplitude of a given loop order ( say , tree level ) acquires contributions from the infinite number of graphs . the problem of convergence of the latter _ functional _ series is the problem of correct definition of the loop expansion . our approach suggests a solution . first , we systematize the set of renormalization prescriptions . in @xcite we demonstrated that only the combinations of coupling constants that we call as _ resultant _ parameters require fixing to obtain renormalized @xmath0-matrix at any given loop order . next , we show that even these renormalization prescriptions can not be taken arbitrary it is the convergence requirements giving a hand . here is the basic idea . any requirement of convergence of functional series always can be thought as a _ restriction for the parameters _ appearing in those series . in other words , _ the couplings , and so the renormalization prescriptions , are unavoidably restricted for each step of perturbation expansion to make sense_. it is this circumstance that eventually gives rise to the system of bootstrap equations for physical parameters ( see @xcite ) and allows to make numerical predictions . briefly speaking , we develop dyson s perturbation scheme for the infinite component effective theories taking seriously the problems of mathematical correctness and self - consistency . in this talk we try to make a short overview of our strategy and explain the main postulates . the first ( and , probably , the main ) step toward a classification of the renormalization prescriptions is transition to the _ minimal parametrization _ discussed in @xcite . this allows one to rewrite every given graph in terms of the minimal propagators and minimal effective vertices . the numerator of the minimal propagator is just a covariant spin sum considered as a function of four independent components of momentum . at tree level the minimal vertices are just the hamiltonian on - shell vertices with the wave functions crossed out . to be precise , one shall consider an @xmath0-matrix element of the formal infinite sum of all the hamiltonian items constructed from a given set of , say , @xmath1 ( normally ordered ) field operators . the members of this sum differ from each other by the hamiltonian coupling constants , by number of differentiation operators and/or , possibly , by their matrix structure . the matrix element under consideration should be calculated on the mass shell , presented in a lorentz - covariant form and considered as a function of @xmath2 independent components of the particle momenta . the wave functions should be crossed out . the resulting structure we call as the @xmath1-leg _ minimal effective vertex of the 0-the level_. every minimal effective vertex presents a finite sum of scalar formfactors dotted by the corresponding tensor structures , each formfactor being a formal power series , or even just a polynomial in relevant scalar kinematical variables . l - th level _ effective vertex only differs from that described above by the presence of bubbles ( @xmath3 loops in total ) attached to the same point as the external legs . the numerical coefficients ( eventually supplied with the index @xmath3 ) in the formal power series that describe the corresponding formfactors are called as the _ @xmath3-th level minimal parameters_. the above - mentioned resultant parameters are certain sums of the minimal ones . the resultant parameters are all independent , as far as one considers as independent all the hamiltonian couplings . besides , as shown in @xcite , every _ amplitude graph _ depends only on minimal parameters . in turn , the full sum of @xmath4-th loop order graphs describing certain scattering process can be expressed solely in terms of the resultant parameters with level index @xmath5 . at least a few words should be said here about the renormalization prescriptions . in @xcite it is shown that , if the renormalization point is taken on shell , the resultant parameters are the only quantities that require formulating renormalization prescriptions . actually , our use of the minimal parametrization implies that we rely upon the scheme of _ renormalized perturbation theory_. in this scheme one starts from the physical action written in terms of _ physical _ parameters and adjusts the counter terms in such a way that the numerical values of those parameters remain unchanged after renormalization . it is this fact that allows us to obtain the bootstrap restrictions for the physical ( in principle _ measurable _ ) parameters later on . one of the most important requirements which we make use of when constructing the meaningful items of dyson s perturbation series is that of polynomial boundedness . namely , the full sum of @xmath0-matrix graphs with a given set of external lines and fixed number @xmath4 of loops must be polynomially bounded in every pair energy at fixed values of the other kinematical variables . there are two basic reasons for imposing this limitation . first , from the general postulates of quantum field theory ( see , e.g. , @xcite ) it follows that the full ( non - perturbative ) amplitude must be a polynomially bounded function of its variables . second , from experiment it follows that this is quite a reasonable requirement . since we never fit data with non - perturbative expressions for the amplitude , it is natural to impose the polynomial boundedness requirement on a sum of terms up to any fixed loop order and , hence , on the sum of terms of each order . similar argumentation also works with respect to the bounding polynomial degrees . to avoid unnecessary mutual contractions between different terms of the loop series , it makes sense to attract the following _ asymptotic uniformity _ requirement . _ the degrees of the bounding polynomials which specify the asymptotics of the amplitude of a given loop order must be equal to those specifying the asymptotics of the full ( non - perturbative ) amplitude of the process under consideration . _ surely , this latter degree may depend on the type of the process as well as on the values of the variables kept fixed . the condition of asymptotic uniformity ( or , simply , uniformity ) is concerned with the asymptotic behavior of the items corresponding to different loop orders . it tells us not too much about the rules needed to convert the disordered sum of graphs with the same number of loops ( and , of course , describing the same process ) into the well - defined ( _ summable _ ) functional series . to solve the latter problem we rely upon another general principle which we call as _ summability requirement_. it is formulated as follows . _ in every sufficiently small domain of the complex space of kinematical variables there must exist an appropriate order of summation of the formal sum of contributions coming from the graphs with a given number of loops , such that the reorganized series converges . altogether , these series must define a unique analytic function with only those singularities which are presented in contributions of individual graphs . _ at first glance , the summability ( analyticity ) requirement may seem somewhat artificial . this is not true . there are certain mathematical and field - theoretical reasons for taking it as the guiding principle that provides a possibility to manage infinite formal sums of graphs in a way allowing to avoid inconsistencies . it is , actually , _ both _ the summability and uniformity principles that allow us to use the cauchy formula and obtain well defined expression for the amplitude of a given loop order @xcite . we would like to stress that the requirements of uniformity and summability are nothing but independent subsidiary conditions fixing the type of perturbation scheme which we only work with . surely , there is no guarantee that on this way one can construct the most general expressions for the s - matrix elements in the case of effective theory . nevertheless , there is a hope to construct at least a meaningful ones presented by the dyson s type perturbation series only containing the well - defined items . the explicit calculations are better illustrated by the concrete examples @xcite , here we just briefly sketch the strategy . one needs to consider any process and classify all the resultant parameters contributing to it at a given loop order . then , using the stated above principles of uniformity and summability , one applies the well known cauchy formula to the amplitude of the given order at various regions in the space of kinematical variables . several things happen during this process . first , the summability principle helps to identify the parameters of the amplitude singularities as a combination of resultant parameters which are already fixed by renormalization conditions . with this in hands , and with the degree of bounding polynomial given by uniformity principle , the cauchy formula allows one to write down a _ well defined expression for the amplitude in a given domain ( layer ) of the space of kinematical variables . _ next , one obtains _ different expressions for the same amplitude in different layers _ , different couplings and masses contributing to each of them . the layers intersect , so one should equate the expressions in the common domains of validity to ensure self - consistency ( usually , it appears to be equivalent to requirement of crossing symmetry of the given loop order amplitude ) . the latter step gives an infinite number of numerical relations for the resultant parameters and thus , as explained above , for the renormalization prescriptions or , the same , for the physical parameters of a theory . we call these relations as the _ bootstrap equations _ of a given loop order . it should be noted , that , although the bootstrap for each loop order amplitude gives restrictions for _ physical _ parameters ( the latter are independent of the loop order ) , the structure of bootstrap equations themselves varies from order to order . the search for the _ solution _ of all these equations is still beyond our scope , what we can only do so far is to test them with experimental data . the results are presented in two other talks , see @xcite . we are grateful to v. cheianov , h. nielsen , s. paston , j. schechter , a. vasiliev and m. vyazovski for stimulating discussions . 100 a. vereshagin , _ `` explicit equations for renormalization prescriptions in the case of pion - nucleon scattering '' _ , this conference ; k. semenov - tian - shansky , _ `` renormalization in effective theories : prescriptions for kaon - nucleon resonance parameters '' _ , this conference . a. vereshagin and v. vereshagin , phys . d * 59 * ( 1999 ) 016002 . a. vereshagin , v. vereshagin , and k. semenov - tian - shansky , zap . nauchn . sem . pomi * 291 * , part 17 , ( 2002 ) 78 ( in russian ) . hep - th/0303242 ( the english version will appear in `` j. math . ( ny ) * 125 * , iss . 2 ( 2005 ) 114 ) . a. vereshagin and v. vereshagin , phys . rev . d * 69 * ( 2004 ) 025002 . s. weinberg , physica a * 96 * ( 1979 ) 327 . s. weinberg , _ the quantum theory of fields _ , vols . 1 - 3 ( cambridge university press , cambridge , england , 2000 ) . n. n. bogoliubov , a. a. logunov and i. t. todorov . _ foundations of the axiomatic method in quantum field theory _ ( nauka , moskow , 1969 ; in russian ) . j. f. chew . _ the analytic s - matrix _ ( w. a. benjamin , inc . new york , 1966 ) . | we summarize our latest developments in perturbative treating the effective theories of strong interactions .
we discuss the principles of constructing the mathematically correct expressions for the s - matrix elements at a given loop order and briefly review the renormalization procedure .
this talk shall provide the philosophical basement as well as serve as an introduction for the material presented at this conference by a. vereshagin and k. semenov - tian - shansky @xcite . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
the phase transition from partonic degrees of freedom ( quarks and gluons ) to interacting hadrons is a central topic of modern high - energy physics . in order to understand the dynamics and relevant scales of this transition laboratory experiments under controlled conditions are presently performed with ultra - relativistic nucleus - nucleus collisions . hadronic spectra and relative hadron abundancies from these experiments reflect important aspects of the dynamics in the hot and dense zone formed in the early phase of the reaction . furthermore , as has been proposed early by rafelski and mller @xcite the strangeness degree of freedom might play an important role in distinguishing hadronic and partonic dynamics . in fact , estimates based on the bjorken formula @xcite for the energy density achieved in central au+au collisions suggest that the critical energy density for the formation of a quark - gluon plasma ( qgp ) is by far exceeded during a few fm / c in the initial phase of the collision at relativistic heavy ion collider ( rhic ) energies @xcite , but sufficient energy densities ( @xmath4 0.7 - 1 gev/@xmath8 @xcite ) might already be achieved at alternating gradient synchrotron ( ags ) energies of @xmath4 10 @xmath0gev @xcite . more recently , lattice qcd calculations at finite temperature and quark chemical potential @xmath9 @xcite show a rapid increase of the thermodynamic pressure @xmath10 with temperature above the critical temperature @xmath11 for a phase transition to the qgp . the crucial question is , however , at what bombarding energies the conditions for the phase transition ( or cross - over ) might be fulfilled . presently , transverse mass ( or momentum ) spectra of hadrons are in the center of interest . it is experimentally observed that the transverse mass spectra of kaons at ags and sps energies show a substantial _ flattening _ or _ hardening _ in central au+au collisions relative to @xmath1 interactions ( cf . @xcite ) . in order to quantify this effect , the spectra are often parametrised as : @xmath12 where @xmath13 is the transverse mass and @xmath14 is the inverse slope parameter . this hardening of the spectra is commonly attributed to strong collective flow , which is absent in the @xmath1 or @xmath15 data . the authors of refs . @xcite have proposed to interpret the approximately constant @xmath16 slopes above @xmath17 a@xmath3gev the step as an indication for a phase transition following an early suggestion by van hove @xcite . this interpretation is also based on a rather sharp maximum in the @xmath2 ratio at @xmath4 20 to 30 a@xmath3gev in central au+au ( pb+pb ) collisions ( the horn @xcite ) . however , it is presently not clear , if the statistical model assumptions invoked in refs . @xcite hold to be reliable . we will demonstrate in this contribution that the pressure needed to generate a large collective flow to explain the hard slopes of the @xmath16 spectra as well as the horn in the @xmath2 ratio is not produced in the present models by the interactions of hadrons in the expansion phase of the hadronic fireball . in our studies we use two independent transport models that employ hadronic and string degrees of freedom , i.e. , urqmd ( v. 1.3 ) @xcite and hsd @xcite . they take into account the formation and multiple rescattering of hadrons and thus dynamically describe the generation of pressure in the hadronic expansion phase . this involves also interactions of leading pre - hadrons that contain a valence quark ( antiquark ) from a hard collision ( cf . @xcite ) . the urqmd transport approach @xcite includes all baryonic resonances up to masses of 2 gev as well as mesonic resonances up to 1.9 gev as tabulated by the particle data group @xcite . for hadronic continuum excitations a string model is used with hadron formation times in the order of 1 - 2 fm / c depending on the momentum and energy of the created hadron . in the hsd approach nucleons , @xmath18 s , n@xmath19(1440 ) , n@xmath19(1535 ) , @xmath20 , @xmath21 and @xmath22 hyperons , @xmath23 s , @xmath24 s and @xmath25 s as well as their antiparticles are included on the baryonic side whereas the @xmath26 and @xmath27 octet states are included in the mesonic sector . high energy inelastic hadron - hadron collisions in hsd are described by the fritiof string model @xcite whereas low energy hadron - hadron collisions are modeled based on experimental cross sections . both transport approaches reproduce the nucleon - nucleon , meson - nucleon and meson - meson cross section data in a wide kinematic range . we point out , that no explicit parton - parton scattering processes ( beyond the interactions of leading quarks / diquarks ) are included in the studies below contrary to the multi - phase transport model ( ampt ) @xcite , which is currently employed from upper sps to rhic energies . in order to explore the main physics from central @xmath29 reactions it is instructive to have a look at the various particle multiplicities relative to scaled @xmath1 collisions as a function of bombarding energy . for this aim we show in fig . [ multppaa ] the total multiplicities of @xmath30 and @xmath31 ( i.e. , the @xmath32 yields ) from central au+au ( at ags ) or pb+pb ( at sps ) collisions ( from urqmd and hsd ) in comparison to the scaled total multiplicities from @xmath1 collisions versus the kinetic energy per particle @xmath33 . the general trend from both transport approaches is quite similar : we observe a slight absorption of pions at lower bombarding energy and a relative enhancement of pion production by rescattering in heavy - ion collisions above @xmath410 a@xmath3gev . kaons and antikaons from @xmath34 collisions are always enhanced in central reactions relative to scaled @xmath1 multiplicities , which is a consequence of strong final state interactions . thus , the kink in the pion ratio as well as the @xmath16 enhancement might result from conventional hadronic final state interactions . fig . [ fig_yield ] shows the excitation function of @xmath35 and @xmath36 yields ( midrapidity ( l.h.s . ) and rapidity integrated ( r.h.s ) ) from central au+au ( pb+pb ) collisions in comparison to the experimental data . note that all data from the na49 collaboration at 30 a@xmath3gev have to be considered as preliminary. as can be seen from fig . [ fig_yield ] the differences between the independent transport models are less than 20% . the maximum deviations between the models and the experimental data are less than @xmath17% . in addition , a systematic analysis of the results from both models and experimental data for central nucleus - nucleus collisions from 2 to 160 @xmath0gev in ref . @xcite has shown that also the longitudinal rapidity distributions of protons , pions , kaons , antikaons and hyperons are quite similar in both models and in reasonable agreement with available data . the exception are the pion rapidity spectra at the highest ags energy and lower sps energies , which are overestimated by both models @xcite . for a more detailed comparison of hsd and urqmd calculations with experimental data at rhic energies we refer the reader to refs . @xcite . in fig . [ fig_rat ] we present the excitation function of the particle ratios @xmath37 and @xmath38 from central au+au ( pb+pb ) collisions in comparison to experimental data . the deviations between the transport models and the data are most pronounced for the midrapidity ratios ( left column ) since the ratios are very sensitive to actual rapidity spectra . the @xmath2 ratio in urqmd shows a maximum at @xmath4 8 a@xmath3gev and then drops to a constant ratio of 0.11 at top sps and rhic energies . in the case of hsd a continuously rising ratio with bombarding energy is found for the midrapidity ratios which partly is due to a dip in the pion pseudo - rapidity distribution at rhic energies ( cf . fig . 1 in ref . the 4@xmath39 ratio in hsd is roughly constant from top sps to rhic energies , however , larger than the ratio from urqmd due to the lower amount of pion production in hsd essentially due to an energy - density cut of 1 gev/@xmath8 , which does not allow to form hadrons above this critical energy density @xcite and a slightly higher @xmath40 yield ( cf . [ fig_yield ] ) . nevertheless , the experimental maximum in the @xmath2 ratio is missed , which we address dominantly to the excess of pions in the transport codes rather than to missing strangeness production . qualitatively , the same arguments - due to strangeness conservation - also hold for the @xmath41 ratio , where the pronounced experimental maxima are underestimated due to the excess of pions in the transport models at top ags energies ( for hsd ) and above @xmath4 5 a@xmath3gev ( for urqmd ) . since the @xmath31 yields are well reproduced by both approaches ( cf . [ fig_yield ] ) the deviations in the @xmath42 ratios at sps and rhic energies in urqmd can be traced back to the excess of pions ( see discussion above ) . we stress that the maximum in the @xmath41 ratio is essentially due to a change from baryon to meson dominated dynamics with increasing bombarding energy . similar arguments hold for the experimentally observed maxima in the ratio @xmath43 ( cf . however , the horn in the @xmath2 ratio at @xmath430 a@xmath3gev is not described by neither of our transport models . we now focus on transverse mass spectra of pions and kaons / antikaons from central au+au ( pb+pb ) collisions from 2 @xmath0gev to 21.3 @xmath0tev and compare to recent data ( cf . @xcite ) . without explicit representation we mention that the agreement between the transport calculations and the data for @xmath1 and for central c+c and si+si is quite satisfactory @xcite ; no obvious traces of new physics are visible . the situation , however , changes for central au+au ( or pb+pb ) collisions . whereas at the lowest energy of 4 @xmath0gev the agreement between the transport approaches and the data is still acceptable , severe deviations are visible in the @xmath16 spectra at sps energies of 30 and 160 @xmath0gev @xcite . we note that the @xmath44 spectra are reasonably described at all energies while the inverse slope @xmath14 of the @xmath16 transverse mass spectra in eq . ( [ slope ] ) is underestimated severely by about the same amount in both transport approaches ( within statistics ) . the increase of the inverse @xmath16 slopes in heavy - ion collisions with respect to @xmath1 collisions , which is generated by rescatterings of produced hadrons in the transport models , is small because the elastic meson - baryon scattering is strongly forward peaked and therefore gives little additional transverse momentum at midrapidity . the question remains whether the underestimation of the @xmath16 slopes in the transverse mass spectra @xcite might be due to conventional hadronic medium effects . in fact , the @xmath45 slopes of kaons and antikaons at sis energies ( 1.5 to 2 @xmath0gev ) were found to differ significantly @xcite . as argued in @xcite the different slopes could be traced back to repulsive kaon - nucleon potentials , which lead to a hardening of the @xmath40 spectra , and attractive antikaon - nucleon potentials , which lead to a softening of the @xmath31 spectra . however , the effect of such potentials was calculated within hsd and found to be of minor importance at ags and sps energies @xcite since the meson densities are comparable to or even larger than the baryon densities at ags energies and above . additional self energy contributions stem from @xmath16 interactions with mesons ; however , @xmath46-wave kaon - pion interactions are weak due to chiral symmetry arguments and @xmath47-wave interactions such as @xmath48 transitions are suppressed substantially by the approximately thermal pion spectrum @xcite . furthermore , we have pursued the idea of refs . @xcite that the @xmath16 spectra could be hardened by string - string interactions , which increase the effective string tension @xmath49 and thus the probability to produce mesons at high @xmath45 @xcite . in order to estimate the largest possible effect of string - string interactions we have assumed that for two overlapping strings the string tension @xmath49 is increased by a factor of two , for three overlapping strings by a factor of three etc . here the overlap of strings is defined geometrically assuming a transverse string radius @xmath50 , which according to the studies in ref . @xcite should be @xmath51 0.25 fm . based on these assumptions ( and @xmath50=0.25 fm ) , we find only a small increase of the inverse slope parameters at ags energies , where the string densities are low . at 160 @xmath0gev the model gives a hardening of the spectra by about 15% , which , however , is still significantly less than the effect observed in the data . our findings are summarized in fig . [ fig_t ] , where the dependence of the inverse slope parameter @xmath14 ( see eq . ( [ slope ] ) ) on @xmath52 is shown and compared to the experimental data @xcite for central au+au ( pb+pb ) collisions ( l.h.s . ) and @xmath1 reactions ( r.h.s . ) . the upper and lower solid lines ( with open circles ) on the l.h.s . in fig . [ fig_t ] correspond to results from hsd calculations , where the upper and lower limits are due to fitting the slope @xmath14 itself , an uncertainty in the repulsive @xmath16-pion potential or the possible effect of string overlaps . the slope parameters from @xmath1 collisions ( r.h.s . in fig . [ fig_t ] ) are seen to increase smoothly with energy both in the experiment ( full squares ) and in the hsd calculations ( full lines with open circles ) . the urqmd results for @xmath1 collisions are shown as open triangles connected by the solid line and systematically lower than the slopes from hsd at all energies . we mention that the rqmd model @xcite gives higher inverse slope parameters for kaons at ags and sps energies than hsd and urqmd , which essentially might be traced back to the implementation of effective resonances with masses above 2 gev as well as color ropes which decay isotropically in their rest frame @xcite . a more detailed discussion of this issue is presented in ref.@xcite . the na49 collaboration @xcite has recently observed a vanishing elliptic flow of protons in pb+pb collisions at 40 a@xmath3gev at midrapidity for all centralities ( fig . [ fig_v2pr40 ] ) . this observation of the apparent collapse of the collective flow @xmath7 is remarkable because the proton elliptic flow @xmath7 at top ags ( 11 a@xmath3gev ) @xcite and top sps energies ( 160 a@xmath3gev ) @xcite is non - zero for mid - central collisions and large for peripheral collisions . this experimental observation of a minimum of the collective flow excitation function has been predicted as a signature for a first order phase transition @xcite . this still leaves us with the question of the origin of the rapid increase of the @xmath16 slopes with invariant energy for central au+au collisions at ags energies and the constant slope at sps energies ( the step ) , which is missed in both transport approaches . we recall that higher transverse particle momenta either arise from repulsive self energies in mean - field dynamics or from collisions , which reduce longitudinal momenta in favor of transverse momenta @xcite . as shown above in fig . [ fig_t ] conventional hadron self - energy effects and hadronic binary collisions are insufficient to describe the dramatic increase of the @xmath16 slopes as a function of @xmath52 . this indicates additional mechanisms for the generation of the pressure that is observed experimentally . here we propose that additional pre - hadronic / partonic degrees of freedom might be responsible for this effect already at @xmath4 5 @xmath0gev . our arguments are based on a comparison of the thermodynamic parameters @xmath14 and @xmath54 extracted from the transport models in the central overlap regime of au+au collisions @xcite with the experimental systematics on chemical freeze - out configurations @xcite in the @xmath55 plane . the solid line in fig . [ fig_qcd ] characterizes the universal chemical freeze - out line from cleymans et al . @xcite whereas the full dots with errorbars denote the experimental chemical freeze - out parameters - determined from the fits to the experimental yields - taken from ref . @xcite . the various symbols ( in vertical sequence ) stand for temperatures @xmath14 and chemical potentials @xmath54 extracted from urqmd transport calculations in central au+au ( pb+pb ) collisions at 21.3 a@xmath3tev , 160 , 40 and 11 a@xmath3gev @xcite as a function of the reaction time ( from top to bottom ) . the open symbols denote nonequilibrium configurations and correspond to @xmath14 parameters extracted from the transverse momentum distributions , whereas the full symbols denote configurations in approximate pressure equilibrium in longitudinal and transverse direction . during the nonequilibrium phase ( open symbols ) the transport calculations show much higher temperatures ( or energy densities ) than the experimental chemical freeze - out configurations at all bombarding energies ( @xmath56 11 a@xmath3gev ) . these numbers are also higher than the tri - critical endpoints extracted from lattice qcd calculations by karsch et al . @xcite ( large open circle ) and fodor and katz @xcite ( star ) . though the qcd lattice calculations differ substantially in the value of @xmath54 for the critical endpoint , the critical temperature @xmath11 is in the range of 160 mev in both calculations , while the energy density is in the order of 1 gev/@xmath8 or even below . nevertheless , this diagram shows that at rhic energies one encounters more likely a cross - over between the different phases when stepping down in temperature during the expansion phase of the hot fireball. this situation changes at lower sps or ags ( as well as new gsi sis-300 ) energies , where for sufficiently large chemical potentials @xmath54 the cross over should change to a first order transition @xcite , i.e. , beyond the tri - critical point in the ( @xmath55 ) plane . nevertheless , fig . [ fig_qcd ] demonstrates that the transport calculations show temperatures ( energy densities ) well above the phase boundary ( horizontal line with errorbars ) in the very early phase of the collisions , where hadronic interactions practically yield no pressure , but pre - hadronic degrees of freedom should do . this argument is in line with the studies on elliptic flow at rhic energies , that is underestimated by 30% at midrapidity in the hsd approach for all centralities @xcite . only strong early stage pre - hadronic interactions might cure this problem . in fig . [ fig_qcd ] we also show the baryon chemical potential @xmath54 for different rapidity intervals at rhic energies as obtained from a statistical model analysis by the brahms collaboration based on measured the antihadron to hadron yield ratios @xcite . for midrapidity , @xmath57 , whereas for forward rapidities @xmath54 increases up to @xmath58 mev at @xmath59 . thus , the forward rapidity measurement allows to probe large @xmath54 at the same bombarding energies . hence , at rhic only a rather limited chemical potential range is accessible experimentally . to reach the probable first order phase transition region , the international facility at gsi seems to be the right place to go . it is of great interest , of course , to investigate whether the above - mentioned observations could be due to a phase transition of strongly interacting matter . the natural effective theory for exploring the effects from phase transitions on the production and phase - space distribution of hadrons is hydrodynamics : the equation of state enters directly by closing the system of continuity equations for energy , momentum and charge conservation . typically , first - order phase transitions are modelled by matching the pressure of the low - density massive ( symmetry broken ) phase to that of the high - density massless ( symmetric ) phase along a ` phase boundary ' in the @xmath53 plane ( cf . [ fig_qcd ] ) . on the phase transition line , the system is in a mixed state where both phases coexist and where their relative fractions are determined from gibbs s conditions of phase equilibrium . this construction assumes that the phase transition is a quasi - static , reversible process ( entropy is conserved ) near equilibrium . entropy is produced only in the initial compression stage which ends with the formation of a locally equilibrated fireball of hot and dense matter which subsequently expands and cools . the crossover from suppressed to increased pion production in central nuclear collisions relative to scaled @xmath1 collisions reflects the excess entropy produced at higher energies @xcite , as also seen in fig . 1 . somewhat surprisingly though , the excitation function of entropy production turns out to be rather smooth , without exhibiting ` discontinuities ' from crossing the phase boundary at some energy . aside from some dynamical effects , the main reason for this behavior is that in _ baryon - rich _ matter the specific entropy is a smooth function of temperature without a pronounced ` jump ' @xcite . since the entropy produced right at impact ( on a time scale of order @xmath60 in the cm frame ) increases smoothly with energy , all hadron abundance ratios will behave correspondingly and the sharp ` horn ' in @xmath61 seen in the data can not be reproduced . this holds for typical hydro models with a first - order phase transition @xcite as well as for hadronic transport models ( see discussion above ) . if early - stage entropy poduction can not account for the sharp peak of @xmath62 ratio then perhaps the phase transition _ back _ to the broken phase ( which occurs later on after some cooling ) can ? this might be possible indeed if one abandons the equilibrium phase transition based on the macroscopic gibbs construction and , in turn , introduces a dynamical microscopic treatment of phase transitions into hydrodynamics . it is well - known that first - order phase transitions lead to inhomogeneities such as high - density ` nuggets ' , surrounded by low - density ` voids ' . analogous effects are frequently discussed within the context of the qcd transition in the early universe , where inhomogeneities of the entropy ( or baryon to photon ratio ) might affect bbn . the usual mixed - phase construction applies on scales much larger than the size and separation of inhomogeneities , and on such scales the matter and entropy distributions appear smooth . on small scales however , for example in heavy - ion collisions , inhomogeneous density distributions have significant effects on observables which are non - linear functions of the density : take a large homogeneous system , split it in half , and move all baryons into one half , then let each half equilibrate . suppose you can not measure the hadron multiplicities in each half separately , just the total . the obvious measurement , namely of baryon number , does nt reflect the presence of the high - density nugget because the total baryon number is the same as for the homogeneous distribution . however , the total yield of @xmath40 over the total yield of @xmath63 will be larger than for the homogeneous system ! this is because the ratio is enhanced in the high - density nugget by a much bigger factor than it is suppressed in the low - density half of the system . ( other hadrons like ( multi-)strange baryons of course compensate the strangeness and are also more abundant than for the homogeneous system . ) the effect diminishes rapidly when the entropy per ( net ) baryon becomes large , that is , in the meson - dominated high - energy regime . to investigate the formation of inhomogeneities during the phase transition we solve for the coupled evolution of an order parameter field such as the chiral condensate @xmath64 and the thermalized matter fields @xcite : @xmath65 here , @xmath66 is the energy - momentum tensor of the fluid , @xmath67 that of the classical modes of the chiral condensate , and @xmath68 is the effective potential obtained by integrating out the thermalized degrees of freedom . we focus first on energy - density inhomogeneities and present solutions of these coupled equations for vanishing baryon density @xcite . as initial condition we chose a homogeneous energy density above the critical energy density for the transition to the broken phase . however , the condensate @xmath64 exhibits ` primordial ' gaussian fluctuations on length scales @xmath69 fm on top of a smoothly varying mean field . these fluctuations are then propagated through a first - order chiral phase transition and leave a rather inhomogeneous ( energy- ) density distribution in the wake of the transition , as seen in fig . [ edens_chhyd ] . evidently , the scale for such fluctuations is not tiny and so it would not be appropriate to assume a homogeneous density distribution . on the other hand , they are too small to be resolved in rapidity space because the scale factor is large at times long after the initial impact . to resolve individual hot / dense spots would require a resolution better than one unit of rapidity , which is roughly equal to the thermal width of the local particle momentum distributions . however , additional hints for the existence of large density inhomogeneities created in the course of the transition to the broken phase remain to be explored . ( inhomogeneities from fluctuations of particle production in the primary nucleon - nucleon collisions should be largely washed out until decoupling by hydrodynamic transport of matter due to pressure gradients , see e.g. @xcite . ) clearly , the yields of other hadron species depend non - linearly on the density as well , and their behavior has to be tested for consistency . moreover , coordinate - space fluctuations of the energy - momentum tensor of matter produced by a phase transition are uncorrelated to the reaction plane and therefore should act to reduce out - of - plane collective flow ( @xmath70 ) as compared to equilibrium hydrodynamics , cf . the discussion in @xcite . finally , hanbury - brown twiss correlations could provide valuable coordinate - space information on the regions from which particles are emitted . in this regard , note the stunning result of ceres @xcite according to which pions decouple when their mean - free path is @xmath69 fm . this is inconceivable in standard equilibrium hydrodynamics without density perturbations because there particles decouple only when their mean - free path exceeds the scale of spatial homogeneity , which is about an order of magnitude larger @xcite . the ceres analysis indicates that density ( and perhaps velocity ) gradients in coordinate space are 1/1 fm rather than 1/10 fm . summarizing this contribution , we point out that baryon stopping @xcite and hadron production in central au+au ( or pb+pb ) collisions is quite well described in the independent transport approaches hsd and urqmd . also the longitudinal rapidity distributions of protons , pions , kaons , antikaons and hyperons are similar in both models and in reasonable agreement with available data . the exception are the pion rapidity spectra at the highest ags energy and lower sps energies , which are overestimated by both models @xcite . as a consequence the hsd and urqmd transport approaches underestimate the experimental maximum of the @xmath2 ratio ( horn ) at @xmath4 20 to 30 a@xmath3gev . however , we point out that the maxima in the @xmath2 and ( @xmath71 ratios partly reflect a change from baryon to meson dominated dynamics with increasing bombarding energy . we have found that the inverse slope parameters @xmath14 for @xmath16 mesons from the hsd and urqmd transport models are practically independent of system size from @xmath1 up to central pb+pb collisions and show only a slight increase with collision energy , but no step in the @xmath16 transverse momentum slopes . the rapid increase of the inverse slope parameters of kaons for collisions of heavy nuclei ( au+au ) found experimentally in the ags energy range , however , is not reproduced by neither model ( see fig . [ fig_t ] ) . since the pion transverse mass spectra which are hardly effected by collective flow are described sufficiently well at all bombarding energies @xcite , the failure has to be attributed to a lack of pressure . this additional pressure should be generated in the early phase of the collision , where the transverse energy densities in the transport approaches are higher than the critical energy densities for a phase transition ( or cross - over ) to the qgp . the interesting finding of our analysis is , that pre - hadronic degrees of freedom might already play a substantial role in central au+au collisions at ags energies above @xmath4 5 @xmath0gev . the more astonishing is the experimentally observed collapse of the collective flow @xmath7 at 40 a@xmath3gev - such a behavior has been predicted as signature for a first order phase transition long ago . j. rafelski and b. mller , _ phys . _ * 48 * ( 1982 ) 1066 . bjorken , _ phys . rev . _ d * 27 * 1983 140 . phys . _ a * 715 * ( 2003 ) 1 . f. karsch _ _ , _ nucl . _ b * 502 * ( 2001 ) 321 . h. stcker and w. greiner , _ phys . rep . _ * 137 * ( 1986 ) 277 . w. cassing , e.l . bratkovskaya , and s. juchem , _ nucl . phys . _ a * 674 * ( 2000 ) 249 . z. fodor and s. d. katz , jhep * 0203 * , 014 ( 2002 ) ; z. fodor , s. d. katz , and k. k. szabo , phys . b * 568 * , 73 ( 2003 ) . v. friese _ et al . _ , na49 collaboration , _ j. phys . _ g * 30 * ( 2004 ) 119 . m. i. gorenstein , m. gadzicki , and k. bugaev , _ phys . _ b * 567 * ( 2003 ) 175 . m. gadzicki and m. i. gorenstein , _ acta phys . _ b * 30 * ( 1999 ) 2705 . l. van hove , _ phys . _ b * 118 * ( 1982 ) 138 . s.a . bass _ et al . _ , _ prog . * 42 * ( 1998 ) 255 . m. bleicher _ et al . _ , _ j. phys . _ g * 25 * ( 1999 ) 1859 . j. geiss , w. cassing , and c. greiner , _ nucl . phys . _ a * 644 * ( 1998 ) 107 . w. cassing and e. l. bratkovskaya , _ phys . _ * 308 * ( 1999 ) 65 . h. weber , e.l . bratkovskaya , w. cassing , and h. stcker , _ phys . c * 67 * ( 2003 ) 014904 . k. hagiwara _ et al . _ , ( review of particle properties ) , _ phys . rev . d * 66 * ( 2002 ) 010001 . b. andersson _ _ , _ z. phys . c _ * 57 * ( 1993 ) 485 . z. w. lin _ et al . _ , _ nucl . phys . _ a * 698 * ( 2002 ) 375 . et al . _ , e866 and e917 collaboration , _ phys . _ b * 476 * ( 2000 ) 1 ; _ ibid . _ * 490 * ( 2000 ) 53 . j. l. klay _ et al . _ , e895 collaboration , _ phys . _ c * 68 * ( 2003 ) 054905 . s. ahmad _ et al . _ , e891 collaboration , _ phys . _ b * 382 * ( 1996 ) 35 ; c. pinkenburg _ et al . _ , e866 collaboration , _ nucl . phys . _ a * 698 * ( 2002 ) 495c . s. v. afanasiev _ _ , na49 collaboration , _ phys . _ c * 66 * ( 2002 ) 054902 . a. mischke _ et al . _ , na49 collaboration , _ j. phys . _ g. * 28 * ( 2002 ) 1761 ; _ nucl . phys . _ a * 715 * ( 2993 ) 453 . f. antinori _ et al . _ , wa97 collaboration , _ nucl . phys . _ a * 661 * ( 1999 ) 130c . d. ouerdane _ et al . _ , brahms collaboration , _ nucl . phys . _ a * 715 * ( 2003 ) 478 ; j. h. lee _ _ , _ j. phys . _ g * 30 * ( 2004 ) s85 . s. s. adler _ et al . _ , phenix collaboration , preprint nucl - ex/0307010 ; preprint nucl - ex/0307022 . _ , star collaboration , preprint nucl - ex/0206008 ; o. barannikova _ et al . _ , _ nucl . phys . _ a * 715 * ( 2003 ) 458 ; k. filimonov _ _ , preprint hep - ex/0306056 . e. l. bratkovskaya _ et al . _ , preprint nucl - th/0402026 . e. l. bratkovskaya , w. cassing and h. stcker , _ phys . rev . c * 67 * ( 2003 ) 054905 . s. soff _ _ , _ phys . _ b * 551 * ( 2003 ) 115 . k. redlich , j. cleymans , h. oeschler , and a. tounsi , _ acta phys . b * 33 * ( 2002 ) 1609 . e. l. bratkovskaya , s. soff , h. stcker , m. van leeuwen , and w. cassing , _ phys . * 92 * ( 2004 ) 032302 . a. frster _ _ , kaos collaboration , _ j. phys . _ g * 28 * ( 2002 ) 2011 . b. v. martemyanov _ _ , nucl - th/0212064 . h. sorge , _ phys . rev . _ c * 52 * ( 1995 ) 3291 . _ , _ phys . _ b * 471 * ( 1999 ) 89 . _ , _ phys . b * 447 * ( 1999 ) 31 . i. kraus _ et al . _ , na49 collaboration , _ j. phys . _ g * 30 * ( 2004 ) 5583 . _ , na44 collaboration , preprint nucl - ex/0202019 . m. kliemant , b. lungwitz , and m. gadzicki , preprint hep - ex/0308002 . h. van hecke _ _ , _ phys . * 81 * ( 1998 ) 5764 . et al . _ , e895 collaboration , _ phys . rev . _ * 84 * ( 2000 ) 5488 . c. alt _ et al . _ , na49 collaboration , preprint nucl - ex/0303001 , _ phys . _ c , in press . j. hofmann , h. stcker , u. heinz , w. scheid , and w. greiner , _ phys . * 36 * ( 1976 ) 88 . w. cassing and u. mosel , _ prog . _ * 25 * , 235 ( 1990 ) . j. cleymans and k. redlich , _ phys . _ c * 60 * ( 1999 ) 054908 . l. v. bravina _ _ , _ phys . _ c * 60 * ( 1999 ) 024904 . phys . _ a * 698 * ( 2002 ) 383 . f. karsch , talk given in _ quark matter 2004 _ , oakland , january 11 - 17 , 2004 ; http://qm2004.lbl.gov i. g. bearden _ et al . _ , brahms collaboration , _ phys . * 90 * ( 2003 ) 102301 . e. v. shuryak , _ nucl . _ a * 661 * ( 1999 ) 119c . m. reiter _ et al . _ , _ nucl . phys . _ a * 643 * ( 1998 ) 99 . k. paech , h. stcker and a. dumitru , _ phys . _ c * 68 * ( 2003 ) 044907 ; k. paech , preprint nucl - th/0308049 . m. bleicher _ et al . _ , _ nucl . phys . _ a * 638 * ( 1998 ) 391 . d. adamova _ et al . _ , ceres collaboration , _ phys . _ * 90 * ( 2003 ) 022301 . s. soff , s. a. bass and a. dumitru , _ phys . * 86 * ( 2001 ) 3981 . h. weber , e.l . bratkovskaya and h. stcker , _ phys . _ b * 545 * ( 2002 ) 285 . | we investigate hadron production and transverse hadron spectra in nucleus - nucleus collisions from 2 @xmath0gev to 21.3 @xmath0tev within two independent transport approaches ( urqmd and hsd ) based on quark , diquark , string and hadronic degrees of freedom .
the enhancement of pion production in central au+au ( pb+pb ) collisions relative to scaled @xmath1 collisions ( the kink ) is described well by both approaches without involving a phase transition .
however , the maximum in the @xmath2 ratio at 20 to 30 a@xmath3gev ( the horn ) is missed by @xmath4 40% .
also , at energies above @xmath4 5 a@xmath3gev , the measured @xmath5 @xmath6-spectra have a larger inverse slope than expected from the models .
thus the pressure generated by hadronic interactions in the transport models at high energies is too low .
this finding suggests that the additional pressure - as expected from lattice qcd at finite quark chemical potential and temperature - might be generated by strong interactions in the early pre - hadronic / partonic phase of central heavy - ion collisions .
finally , we discuss the emergence of density perturbations in a first - order phase transition and why they might affect relative hadron multiplicities , collective flow , and hadron mean - free paths at decoupling .
a minimum in the collective flow @xmath7 excitation function was discovered experimentally at 40 a@xmath3gev - such a behavior has been predicted long ago as signature for a first order phase transition . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
observational manifestations of magnetic fields in intermediate- and high - mass stars with radiative envelopes differ considerably from the magnetism of solar - type and low - mass stars . as directly observed for the sun and inferred for many late - type stars , vigorous envelope convection and differential rotation give rise to ubiquitous intermittent magnetic fields , which evolve on relatively short time - scales and generally exhibit complex surface topologies . although details of the dynamo operation in late - type stars , in particular the relative importance of the convective and tachocline dynamo mechanisms is a matter of debate @xcite and probably depends on the position in the h - r diagram , it is understood that essentially every cool star is magnetic . chromospheric and x - ray emission and surface temperature inhomogeneities , which are responsible for characteristic photometric variability , provide an indirect evidence of the surface magnetic fields in cool stars . in contrast , stars hotter than about mid - f spectral type and more massive than @xmath01.5@xmath1 are believed to lack a sizable convective zone near the surface and therefore are incapable of generating observable magnetic fields through a dynamo mechanism . nevertheless , about 10% of o , b , and a stars exhibit very strong ( up to 30 kg ) , globally organized ( axisymmetric and mostly dipolar - like ) magnetic fields that appear to show no intrinsic temporal variability whatsoever . this phenomenon is usually attributed to the so - called fossil stellar magnetism a hitherto unknown process ( possibly related to initial conditions of stellar formation or early stellar mergers ) by which a fraction of early - type stars become magnetic early in their evolutionary history . by far the most numerous among the early - type magnetic stars are the a and b magnetic chemically peculiar ( ap / bp ) stars . these stars were the first objects outside our solar system in which the presence of magnetic field was discovered @xcite . ap / bp stars are distinguished by slow rotation @xcite and are easy to recognize spectroscopically by the abnormal line strengths of heavy elements in their absorption spectra . these spectral peculiarities are related to distinctly non - solar surface chemical composition of these stars and non - uniform horizontal ( e.g. * ? ? ? * ; * ? ? ? * ) and vertical distributions of chemical elements ( e.g. * ? ? ? * ; * ? ? ? these chemical structures are presumably formed by the magnetically - controlled atomic diffusion @xcite operating in stable atmospheres of these stars . the chemical spot distributions and magnetic field topologies of ap stars remain constant ( frozen in the atmosphere ) . yet , all these stars show a pronounced and strictly periodic ( with periods from 0.5 d to many decades ) spectroscopic , magnetic and photometric variability due to rotational modulation of the aspect angle at which stellar surface is seen by a distant observer . a subset of cool magnetic ap stars rapidly oscillating ap ( roap ) stars also varies on much shorter time scales ( @xmath010 min ) due to the presence of @xmath2-mode oscillations aligned with the magnetic field @xcite . a large field strength and lack of intrinsic variability facilitates detailed studies of the field topologies of individual magnetic ap stars and statistical analyses of large stellar samples . in this review i outline common methodologies applied to detecting and modeling surface magnetic fields in early - type stars and summarize main observational results . closely related contributions to this volume include an overview of massive - star magnetism ( wade , grunhut ) , a discussion of the stability and interior structure of fossil magnetic fields ( braithwaite ) , and an assessment of the chemical peculiarities and magnetism of pre - main sequence a and b stars ( folsom ) . with a few exceptions , investigations of the magnetism of cool stars have to rely on high - resolution spectropolarimetry and to engage in a non - trivial interpretation of the complex polarization signatures inside spectral lines in order to characterize the field topologies @xcite . in contrast , a key advantage of the magnetic field studies of early - type stars with stable global fields is availability of a wide selection of magnetic observables that are simple to derive and interpret , but are still suitable for a coarse analysis of the surface magnetic field structure . the simplest approach to detecting the presence of the field in early - type stars is to perform spectroscopic observation with a zeeman analyzer equipped with a quarter - wave retarder plate and a beamsplitter . the resulting pair of left- and right - hand circularly polarized spectra will exhibit a shift proportional to the land factors of individual spectral lines and to _ the mean longitudinal magnetic field _ the line - of - sight field component averaged over the stellar disk . various versions of this longitudinal field diagnostic technique have been applied by @xcite , @xcite , and @xcite to medium - resolution spectra . @xcite and @xcite have extended it to , respectively , photopolarimetric and low - resolution spectropolarimetric measurements of polarization in the wings of hydrogen lines . the mean longitudinal magnetic field represents a particular example of an integral measurement derived from a moment of stokes @xmath3 profile ( the first moment in this case ) . @xcite have generalized the moment technique to other stokes @xmath4 and @xmath3 profile moments . in practice , only _ the mean quadratic field _ ( the second moment of stokes @xmath4 ) and _ crossover _ ( the second moment of stokes @xmath3 ) , in addition to longitudinal field , were systematically studied by mathys and collaborators using medium - resolution observations of ap stars ( e.g. * ? ? ? observations are offset vertically . clear circular and linear polarization signatures are evident in many individual spectral lines . detailed analysis of these observations has been published by @xcite . ] the three aforementioned magnetic observables can be related to the disk - averaged properties of stellar magnetic field under a number of simplifying and restrictive assumptions ( weak lines , weak field , no chemical spots ) . at the same time , an entirely assumption - free method to diagnose magnetic fields in early - type stars is to measure a separation of the resolved zeeman - split components in the intensity profiles of magnetically sensitive spectral lines . the resulting _ mean field modulus _ measurements have been obtained for many ap stars showing strong magnetic fields in combination with a particularly slow rotation @xcite . a different approach can be applied to diagnose the transverse magnetic field components which give rise to linear polarization . cooler ap stars exhibit measurable _ net linear polarization _ due to differential saturation of the @xmath5 and @xmath6 components of the strong spectral lines . the resulting net @xmath7 and @xmath8 signals can be detected with a broad - band photopolarimetric technique and related to disk - averaged characteristics of the transverse magnetic field @xcite . with the advent of high - resolution spectropolarimeters at the 24 m - class telescopes it became possible to directly record and interpret the circular and linear polarization signatures in individual spectral lines . a multi - line lsd ( least - squares deconvolution ) technique @xcite is often used in conjunction with such observations to obtain very high signal - to - noise ( s / n ) ratio mean intensity and polarization profiles . lsd analysis greatly facilitates detection of weak magnetic fields ( e.g. * ? ? ? * ) and allows to derive the mean longitudinal field , net linear polarization and other profile moments for direct comparison with historic studies . the first high - resolution full stokes vector investigations were carried out for ap stars with now decommissioned musicos spectropolarimeter @xcite . more recent analyses took advantage of espadons , narval @xcite and harpspol @xcite instruments . an example of exceptionally high quality ( @xmath9 , @xmath10 , 16 rotation phases , coverage of 38006910 wavelength region ; see @xcite for details ) harpspol stokes spectra of the roap star hd24712 is illustrated in fig . these observations represent the highest quality full stokes vector spectra available for any star other than the sun while covering a much wider wavelength domain than typical for solar polarization observations . fitting the phase curves of one or several magnetic observables constitutes the basic method of constraining the stellar magnetic field parameters . a limited information content of integral observables and their relatively simple sinusoidal variation in most ap stars justifies describing the stellar magnetic field topology with a small number of free parameters . by far the most common approximation is a simple rigidly rotating dipolar field geometry @xcite , characterized by an inclination angle of the stellar rotational axis @xmath11 , magnetic obliquity @xmath12 , and a polar field strength @xmath13 . observations of the phase variation of the mean longitudinal magnetic field alone allows one to constrain @xmath13 and @xmath12 , provided that @xmath11 is known and not too close to 90@xmath14 . in the latter case longitudinal field measurements constrain only the product @xmath15 . occasional deviations of the longitudinal field curves from the sinusoidal shape expected for a dipolar field and the requirement to fit simultaneous measurements of the longitudinal and mean surface fields led to the development of more complex field geometry models , described with additional free parameters . different low - order multipolar field parameterizations have been considered in the literature . this included a dipolar field offset along its axis ( e.g. * ? ? ? * ) , an arbitrary offset dipole @xcite , an axisymmetric combination of the aligned dipole , quadrupole , and octupole components @xcite , a general non - axisymmetric quadrupolar field @xcite , and a potential field geometry formed by a superposition of an arbitrary number of point - like magnetic sources @xcite . the choice between these different multipolar parameterizations is typically subjective . stellar observations themselves frequently do not allow one to make a clear - cut distinction between multipolar models established in the framework of different parameterizations , even when several magnetic observables are available for a given star . indeed , it was demonstrated that the same set of observed magnetic curves can be successfully interpreted with very different actual surface magnetic field distributions , depending on which multipolar parameterization is used @xcite . nevertheless , systematic applications of multipolar modeling to a large number of stars allowed to reach interesting conclusions . using centered dipole fits to the longitudinal field curves , @xcite established the existence of a lower field limit of @xmath16 g for ap stars . this threshold of global fossil field strength is likely to be of fundamental importance for understanding the magnetism of intermediate - mass stars ( see lignires , this volume ) . among other notable findings one can mention the work by @xcite , who demonstrated that magnetic field axis tends to be more aligned with the stellar rotation axis for ap stars with long ( @xmath17 d ) rotation periods . @xcite confirmed this result using a different multipolar field parameterization . they also found a certain dependence of the relative orientation of the dipolar and quadrupolar components on the stellar rotation rate . both studies fitted the observed curves of the mean field modulus , longitudinal field , crossover , and quadratic field . despite the overall statistical agreement , in many individual cases the surface field maps resulting from the application of landstreet s and bagnulo s parameterizations appear very different for the same stars . furthermore , some of the observables are poorly reproduced by either multipolar model , which can be ascribed to the presence of more complex field structures , an unaccounted influence of chemical abundance spots or to shortcomings of the basic assumptions of the moment technique or to a combination of all these effects . some applications of the multipolar fitting procedure have incorporated detailed polarized radiative transfer ( prt ) modeling of the zeeman - split stokes @xmath4 profiles into solving for the surface field geometry @xcite . this approach enables an independent validation of the magnetic field topology and makes possible to deduce a schematic horizontal distribution of chemical spots in addition to studying the field geometry . however , a feedback of chemical spots on the magnetic observables is not taken into account by these studies . the most sophisticated and well - constrained non - axisymmetric multipolar models were developed by @xcite for the ap stars @xmath12 crb and 53 cam using all integral magnetic observables available from the stokes @xmath18 spectra together with the broad - band linear polarization measurements . however , even such detailed models do not guarantee a satisfactory description of the same stokes parameter spectra from which the magnetic observables are obtained . as found by @xcite , the multipolar models derived from magnetic observables provide a rough qualitative reproduction of the phase variation of the stokes @xmath3 profiles but sometimes fail entirely in matching the stokes @xmath19 signatures observed in individual metal lines . this problem points to a significant limitation of the multipolar models : a successful fit of the phase curves of all magnetic observables is often non - unique and is generally insufficient to guarantee an adequate description of the high - resolution polarization spectra . on the other hand , discrepancies between the model predictions and observations in partially successful multipolar fits can not be easily quantified in terms of deviations from the best - fit magnetic field geometry model . modeling of high - resolution observations of polarization signatures in individual spectral lines or in mean line profiles represents the ultimate method of extracting information about stellar magnetic field topologies . the wide - spread usage of the lsd processing of high - resolution polarization spectra stimulated development of various stokes @xmath3 profile fitting methodologies @xcite . these studies usually deal with weak - field early - type stars without prominent chemical spots ( e.g. magnetic massive or herbig ae / be stars , but not typical ap stars ) . the observed lsd profiles are approximated with a dipolar field topology , using a simplified analytical treatment of the polarized line formation . eventual variations caused by chemical spots or other surface features are not considered . so far , this modeling approach has been applied to a few stars , but it has a potential of providing constraints on magnetic field and other stellar parameters ( inclination , @xmath20 ) beyond what can be obtained from the longitudinal field curves @xcite . a more rigorous approach to the problem of finding the stellar surface magnetic field geometry from spectropolarimetric observations is to perform a full magnetic inversion known as magnetic ( zeeman ) doppler imaging ( mdi ) . in the mdi methodology developed by @xcite and @xcite the time - series observations in two or four stokes parameters are interpreted with detailed prt calculations , taking surface chemical inhomogeneities into account . simultaneous reconstruction of the magnetic field topology and chemical spot distributions is carried out by solving a regularized inverse problem . regularization limits the range of possible solutions and is needed to stabilize the iterative optimization process and to exclude small - scale surface structures not justified by the data . different versions of regularization have been applied for magnetic mapping of early - type stars . @xcite needed only the local tikhonov regularization ( imposing a correlation between neighboring surface pixels ) to achieve a reliable reconstruction of an arbitrary magnetic field map from full stokes vector spectra . however , a more restrictive multipolar regularization @xcite or a spherical harmonic field expansion @xcite is required to reconstruct a low - order multipolar field in the case when only stokes @xmath4 and @xmath3 observations are available . magnetic imaging of ap stars was recently coupled to a calculation of the atmospheric models that take into account horizontal variations of the atmospheric structure due to chemical spots @xcite . however , while the self - consistency between spots and atmospheric models is critical for magnetic mapping of cool stars @xcite and may be needed for mapping he inhomogeneities in he - rich stars , it is generally unnecessary for treating metal spots in ap stars . it should be emphasized that , in contrast to temperature or chemical spot imaging from intensity spectra , the mdi with polarization data is not limited to rapid rotators . polarization is strongly modulated by the stellar rotation even for magnetic stars with negligible @xmath20 . numerical experiments and studies of real stars demonstrated that this modulation is sufficient for recovering the field structure at least at the largest spatial scales ( e.g. * ? ? ? * ; * ? ? ? several studies @xcite applied mdi to high - quality time - series circular polarization spectra of several early - type magnetic stars . these stokes @xmath3 analyses did not find any major deviations from dipolar field topologies . at the same time , they found numerous examples of chemical spot maps showing diverse and complex distributions of chemical elements , often not correlating in any meaningful way to the underlying simple magnetic field geometry . these results are difficult to explain in the framework of atomic diffusion theory because the latter expects a very similar behavior for different elements and a definite correlation between the spots and magnetic field @xcite . a couple of other studies have attempted to examine the surface magnetic field structure in b - type stars with fields deviating significantly from dipolar geometry . a study of the he - peculiar star hd37776 @xcite has simultaneously interpreted a longitudinal field curve and moderate - resolution stokes @xmath3 spectra . this analysis inferred a decisively non - axisymmetric , complex and strong ( up to 30 kg locally ) magnetic field , but ruled out a record @xmath0100 kg quadrupolar field proposed for this star by previous longitudinal field curve fits . an mdi study of the early b - type star @xmath21 sco @xcite revealed the presence of weak complex magnetic field configuration , which exhibits no appreciable temporal variation @xcite . these two studies have proven that stable complex fields can exist in early - type stars and tend to be found in the most massive objects . despite these impressive mdi results , it should be kept in mind that the stokes @xmath18 inversions are intrinsically non - unique and their outcome is highly sensitive to additional constraints adopted to stabilize inversions . details of the magnetic field maps of hd37776 and @xmath21 sco are likely to change if different regularizations or different forms of spherical harmonic expansion are adopted for magnetic imaging . numerical tests of mdi inversions @xcite have concluded that reconstruction of stellar magnetic field topologies from the full stokes vector data should be considerably more reliable and resistant to cross - talk and non - uniqueness problems in comparison to the stokes @xmath18 imaging . in particular , a four stokes parameter inversion is able to recover the field structure without imposing any _ a priori _ constraints on the global field geometry . the first stokes @xmath22 mdi studies exploiting this possibility were carried out for the ap stars 53 cam @xcite and @xmath23 cvn @xcite using the musicos spectra collected by @xcite . both studies succeeded in reproducing the phase variation of the circular and linear polarization signatures in metal lines with the magnetic maps containing small - scale deviations from the dominant dipolar - like field component . interestingly , it was the inclusion of stokes @xmath19 profiles in the magnetic inversions that allowed to ascertain the presence of complex fields . the deviations from dipolar field configurations occur on much smaller spatial scales than can be described by a quadrupolar field . thus , the widely adopted dipole+quadrupole expansion may not be particularly useful for interpreting the stokes @xmath22 spectra of ap stars . cvn reconstructed by silvester et al . ( 2013 , submitted ) using mdi . these magnetic inversions were based on a set of high - resolution four stokes parameter spectra described by @xcite . the star is shown at five different rotation phases and an inclination angle @xmath24 . the spherical maps show a ) surface distribution of the magnetic field strength , b ) distribution of the radial magnetic field component , and c ) vector plot of magnetic field . the field complexity is evident , especially in the field strength map . ] cvn with a dipolar field model ( red thin line ) and a non - axisymmetric dipole plus quadrupole magnetic configuration ( blue thick line ) . it is evident that none of the models fits the @xmath19 observations at all rotational phases , thus requiring a more structured field geometry for this star ( see fig . [ fig2 ] ) . however , the presence of this small - scale field can not be recognized from the stokes @xmath18 data alone . ] the limited resolution , s / n ratio , and wavelength coverage of the musicos spectra allowed us to model the stokes @xmath22 profiles of only 23 saturated metal lines . a new generation of mdi studies is currently underway , taking advantage of the higher - quality stokes profile data available from espadons , narval , and harpspol spectropolarimeters . in particular , silvester et al . ( submitted ) have reassessed the magnetic field topology of @xmath23 cvn using new observations and extending the prt mdi modeling to a large number of weak and strong fe and cr lines . the resulting magnetic maps ( fig . [ fig2 ] ) show some dependence of the mapping results on the spectral line choice but generally demonstrate a very good agreement with the magnetic topology found by @xcite from observations obtained about 10 years earlier . thus , the small - scale magnetic features discovered in ap stars by mdi studies do not exhibit any temporal evolution . the new four stokes parameter observations of @xmath23 cvn also demonstrate very clearly the necessity of going beyond a low - order multipolar field model and the role of stokes @xmath19 spectra in recognizing this field complexity . as illustrated by fig . [ fig3 ] , an attempt to reproduce the observations of @xmath23 cvn with either a pure dipole or dipole+quadrupole geometries fails for stokes @xmath19 while providing a reasonable fit to stokes @xmath18 . despite an improved sensitivity to complex fields , not all four stokes parameter magnetic inversions point to local deviations from dipolar field topologies . the ongoing harpspol study of the cool ap star hd24712 does not reveal any significant non - dipolar field component ( @xcite and in this volume ) . the preliminary conclusion of this work is that the previous mdi analysis of this star carried out by @xcite using the stokes @xmath4 and @xmath3 data and assuming a dipolar field did not miss any significant aspects of the field topology . hd24712 is much cooler , hence less massive and/or older than 53 cam and @xmath23 cvn , raising an intriguing possibility of the mass and/or age dependence of the degree of magnetic field complexity in early - type stars . the presence of very complex non - dipolar fields only in relatively massive b - type magnetic stars ( hd37776 , @xmath21 sco ) agrees with this trend . modeling of magnetic fields in early - type stars with radiative envelopes has traditionally assumed low - order multipolar field configurations and focused on interpretation of the phase curves of the mean longitudinal magnetic field and other integral magnetic observables derived from moderate - quality circular polarization spectra . the studies based on this methodology have reached several important statistical conclusions about the nature of magnetic fields in ap and related stars . this includes the discovery of a lower threshold of the surface magnetic field strength , demonstration of an alignment of the magnetic and rotational axes in stars with long rotation periods , and confirmation of long - term stability stability of fossil magnetic fields . as observational material improves and high - resolution spectra in several stokes parameters become more widely available , the focus of magnetic field modeling studies gradually shifts to direct interpretation of the polarization signatures in spectral line profiles . the most powerful version of this methodology magnetic doppler imaging inversions based on detailed calculation of polarized spectra has been applied to a handful of ap stars observed in all four stokes parameters . these studies revealed significant local deviations from a dominant dipolar field topology , suggesting that the magnetic field structure of early - type stars with fossil fields is more complex than thought before and that the degree of field complexity increases with stellar mass . with the exception of a couple of massive stars with distinctly non - dipolar fields , these small - scale field structures could be recognized and fully characterized only using spectropolarimetric observations in all four stokes parameters . | stars with radiative envelopes , specifically the upper main sequence chemically peculiar ( ap ) stars , were among the first objects outside our solar system for which surface magnetic fields have been detected
. currently magnetic ap stars remains the only class of stars for which high - resolution measurements of both linear and circular polarization in individual spectral lines are feasible .
consequently , these stars provide unique opportunities to study the physics of polarized radiative transfer in stellar atmospheres , to analyze in detail stellar magnetic field topologies and their relation to starspots , and to test different methodologies of stellar magnetic field mapping . here
i present an overview of different approaches to modeling the surface fields in magnetic a- and b - type stars .
in particular , i summarize the ongoing efforts to interpret high - resolution full stokes vector spectra of these stars using magnetic doppler imaging . these studies reveal an unexpected complexity of the magnetic field geometries in some ap stars . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
cataclysmic variables ( cvs ) are close binary systems transferring matter from a low - mass dwarf secondary to a white dwarf . the transferred matter forms an accretion disk . thermal instability od the disk caused by partial ionization of hydrogen results outbursts in dwarf novae ( dne ) , a subclass of cvs . tidal instability of the disk caused by the 3:1 resonance with the orbiting secondary is considered to develop an eccentric ( or flexing ) disk in su uma - type dwarf novae , a subclass of dne . this eccentric disk is responsible for superhumps , which have periods a few percent longer than the orbital period [ see e.g. @xcite ; @xcite ; @xcite ] . the enhanced mass accretion by tidal instability causes long - lasting superoutbursts [ thermal tidal instability ( tti ) model : @xcite ; @xcite ] . although there had been intensive discussions whether superoutbursts are a result of tidal instability or an enhanced mass - transfer from the secondary [ e.g. @xcite , @xcite , @xcite ] , recent detailed analyses of high - precision kepler observations have favored the tti model as the only viable model for ordinary su uma - type dwarf novae ( @xcite ; @xcite ; @xcite ) . [ for general information of cvs , dne , su uma - type dwarf novae and superhumps , see e.g. @xcite ] . this paper is one of series of papers @xcite , @xcite , @xcite , @xcite , @xcite and @xcite . these papers originally intended to clarify the period variations of superhumps in su uma - type dwarf novae and first succeeded in identifying superhump stages ( stages a , b and c ) in @xcite . among them , stage a superhumps have recently been identified to be reflect the precession of the eccentric disk at the radius of the 3:1 resonance , and have been one of the most promising tools in determining the mass - ratios in su uma - type dwarf novae and in following the terminal evolution of cvs @xcite . continuing this project , we report observations of superhumps and associated phenomena in su uma - type dwarf novae whose superoutbursts were observed in 20142015 . in this paper , we report basic observational materials and discussions in relation to individual objects . general discussion related to wz sge - type dwarf novae , which are a subclass of su uma - type dwarf novae with infrequent , large amplitude superoutbursts , will be given as in a planned summary paper by @xcite . starting from @xcite , we have been intending these series of papers to be also a source of compiled information , including historical , of individual dwarf novae since there have been no compiled publication since @xcite . the material and methods of analysis are given in section [ sec : obs ] , observations and analysis of individual objects are given in section [ sec : individual ] , including some discussions particular to the objects , the general discussion is given in section [ sec : discuss ] and the summary is given in section [ sec : summary ] . the data were obtained under campaigns led by the vsnet collaboration @xcite . for some objects , we used the public data from the aavso international databasehttp://www.aavso.org / data - download@xmath2 . ] . the majority of the data were acquired by time - resolved ccd photometry by using 30cm - class telescopes located world - wide , whose observational details will be presented in future papers dealing with analysis and discussion on individual objects of interest . the list of outbursts and observers is summarized in table [ tab : outobs ] . the data analysis was performed just in the same way described in @xcite and @xcite and we mainly used r softwarehttp://cran.r - project.org/@xmath2 . ] for data analysis . in de - trending the data , we used both lower ( 15th order ) polynomial fitting and locally - weighted polynomial regression ( lowess : @xcite ) . the times of superhumps maxima were determined by the template fitting method as described in @xcite . the times of all observations are expressed in barycentric julian days ( bjd ) . the abbreviations used in this paper are the same as in @xcite : @xmath3 means the orbital period and @xmath4 for the fractional superhump excess . following @xcite , the alternative fractional superhump excess in the frequency unit @xmath5 has been introduced because this fractional superhump excess can be directly compared to the precession rate . we therefore used @xmath6 in referring the precession rate . we used phase dispersion minimization ( pdm ; @xcite ) for period analysis and 1@xmath7 errors for the pdm analysis was estimated by the methods of @xcite and @xcite . we present evidence for an su uma - type dwarf nova by presenting period analysis and averaged superhump profile if the paper provides the first solid presentation of individual objects as such . the resultant @xmath8 , @xmath9 and other parameters are listed in table [ tab : perlist ] in same format as in @xcite . the definitions of parameters @xmath10 and @xmath9 are the same as in @xcite . and @xmath11 ) for the stages b and c given in the table sometimes overlap because of occasional observational ambiguity in determining the stages . ] comparisons of @xmath0 diagrams between different superoutbursts are also presented whenever available , since this comparison was one of the main motivations in of these series papers ( cf . @xcite ) . combined @xmath0 diagrams also help identifying superhump stages particularly when observations are insufficient . in drawing combined @xmath0 diagrams , we usually used @xmath12 for the start of the superoutburst , which usually refers to the first positive detection of the outburst . this epoch usually has an accuracy of @xmath11 d for well - observed objects , and if the outburst was not sufficiently observed , we mentioned in the figure caption how to estimate @xmath13 in such an outburst . we also present @xmath0 diagrams and light curves especially for wz sge - type dwarf novae , which are not expected to undergo outbursts in the near future . in all figures , the binned magnitudes and @xmath0 values are accompanied by 1@xmath7 error bars , which are omitted when the error is smaller than the plot mark . we used the same terminology of superhumps summarized in @xcite . we especially call attention to the term `` late superhumps '' . although this term has been used to refer to various phenomena , we only used the concept of `` traditional '' late superhumps when an @xmath10.5 phase shift is detected [ @xcite ; see also table 1 in @xcite for various types of superhumps ] , since we suspect that many of the past claims of detections of `` late superhumps '' were likely stage c superhumps [ cf . @xcite ; note that the kepler observation of v585 lyr also demonstrated this persistent stage c superhumps without a phase shift @xcite , and most recently it is confirmed in another kepler cv by @xcite ] . early superhumps are double - wave humps seen during the early stages of wz sge - type dwarf novae , and have period close to the orbital periods ( @xcite ; @xcite ; @xcite ) . we are going to discuss this phenomenon in the planned paper @xcite . we used the period of early superhumps as approximate orbital period @xcite . as in @xcite , we have used coordinate - based optical transient ( ot ) designations for some objects , such as apparent dwarf nova candidates reported in the transient objects confirmation page of the central bureau for astronomical telegramshttp://www.cbat.eps.harvard.edu / unconf / tocp.html@xmath2 . ] and listed the original identifiers in table [ tab : outobs ] . for objects detected in the catalina real - time transient survey ( crts ; @xcite)http://nesssi.cacr.caltech.edu / catalina/@xmath2 . for the information of the individual catalina cvs , see @xmath14http://nesssi.cacr.caltech.edu / catalina / allcv.html@xmath2 . ] transients , we preferably used the names provided in @xcite . if these names are not yet available , we used the international astronomical union ( iau)-format names provided by the crts team in the public data releasehttp://nesssi.cacr.caltech.edu / datarelease/@xmath2 . ] since asas - sn detectors have relatively poor angular resolutions ( 7.5 arcsec / pixel ) , we provided coordinates from our own astrometry and astrometric catalogs for asas - sn cvs . we used sdss , the initial gaia source list ( igsl , @xcite ) and guide star catalog 2.3.2 . the coordinates used in this paper are j2000.0 . questions have been raised to our surveys how to select the periods among the aliases and what is the uncertainty . such question are natural if one only sees pdm diagrams . we should note that pdm ( and most of other period finding algorithms ) assumes `` uncorrelated '' ( in time ) observations and defines the statistics . the actual data have more information , such as the superhump timing data . even if the pdm result shows strong aliases , we can resolve the alias problem if we have sufficiently long continuous observations ( continuous data produce no aliases ) . our period selection is mostly based on this principle , when alias selection is inconclusive by the pdm analysis only . the second approach is to examine the trends of @xmath0 values against the trial periods . a period longer than the actual one produces systematically decreasing @xmath0 values within each night , and our measurements of the superhump maxima have typical errors of 0.001 d , which is usually sufficient to select one - day aliases when multiple superhumps were detected each night . if the case is not , we describe the uncertainty of the selection . we show an example how our method works by using the data of fi cet ( subsection [ obj : ficet ] ) . a pdm analysis over a wider range of periods ( figure [ fig : ficetshpdm2 ] ) gives an impression that there are many period candidates and a period of 0.057 d and 0.060 d are equally acceptable . if we rely on statistics assuming temporarily `` uncorrelated '' observations , this would give equal significance to these periods . an example for fi cet is shown in figure [ fig : ficetoccomp ] . scatter in the @xmath0 values and systematic trends within each night are apparent for trial period 0.06033 d , which is rejected . although we do not show similar figures for other objects due to the limitation of space , we made similar analysis for objects when ambiguities in period selection remained . readers may be interested in the result of least absolute shrinkage and selection operator ( lasso ) ( @xcite ; @xcite ) analysis , which has been proven to be very effective in detecting rapidly varying periods in unevenly sampled data ( e.g. @xcite ; @xcite ; @xcite ; @xcite ) . the result is shown in figure [ fig : ficetlasso ] , and the impression is so different from the pdm result ( figure [ fig : ficetshpdm2 ] ) . however , we did nt widely used this method in selecting the aliases since our model used in lasso analysis also assumed temporarily uncorrelated observations , and the suppression of the aliases is simply a result of highly non - linear characteristics of compressed sensing . this figure ( if compared to classical figures ) would give misleading impression that there is no possibility for aliases and we have not used this type of figure in this paper . ( 88mm,70mm)fig1.eps ( 88mm,70mm)fig2.eps ( 88mm,70mm)fig3.eps ccccl subsection & object & year & observers or references & i d + [ obj : kxaql ] & kx aql & 2014 & rpc & + [ obj : nncam ] & nn cam & 2014 & dpv & + [ obj : v342cam ] & v342 cam & 2014 & oku , kis , ioh & + [ obj : oycar ] & oy car & 2014 & spe & + & & 2015 & spe & + [ obj : ficet ] & fi cet & 2014 & hac & + [ obj : zcha ] & z cha & 2014 & hac & + [ obj : yzcnc ] & yz cnc & 2014b & aavso & + [ obj : v337cyg ] & v337 cyg & 2014 & imi & + [ obj : v503cyg ] & v503 cyg & 2014 & kis , dpv & + & & 2014b & rpc & + [ obj : bcdor ] & bc dor & 2015 & hac , coo & + [ obj : v660her ] & v660 her & 2014 & ioh & + [ obj : cthya ] & ct hya & 2015 & spe , kis & + [ obj : lyhya ] & ly hya & 2014 & hac & + [ obj : mmhya ] & mm hya & 2014 & mdy , ioh & + [ obj : rzlmi ] & rz lmi & 2014 & mic & + [ obj : aylyr ] & ay lyr & 2014 & ous , aka & + [ obj : v453nor ] & v453 nor & 2014 & hac & + [ obj : dtoct ] & dt oct & 2014 & hac & + [ obj : uvper ] & uv per & 2014 & ioh , dpv , imi , mdy , kis , & + & & & nov , rpc & + [ obj : hypsc ] & hy psc & 2014 & oku , ioh , imi , kis & + + + + [ cols="^,^,^,^,<",options="header " , ] ( 160mm,110mm)fig151.eps in @xcite , we showed from the kepler data of v1504 cyg and v344 lyr that superhumps start to develop following precursor outbursts . this finding was examined in more detail using the kepler data of the same objects in @xcite , @xcite . the same phenomenon was also confirmed in the kepler data of the background dwarf nova of kic 4378554 and v516 lyr @xcite . these findings have strengthened the universal application of the tti model to various su uma - type dwarf novae . in this paper , we have observed similar evolution of superhumps following the precursor outburst in z cha . although only one superhump was recorded , superhumps likely started to grow just before the final rise to the main superoutburst also in cy uma . an @xmath0 analysis of su uma also suggested that superhumps likely started to appear around the maximum of the precursor outburst . all of these findings are in good agreement with what the tti model predicts ( cf . @xcite ) . in recent years , an increasing number of am cvn - type objects [ for recent reviews of am cvn - type objects , see e.g. @xcite ; @xcite ] have been recorded in outbursts and superhumps were detected during outbursts : cr boo ( observation in atypical state in @xcite , more regular superoutburst in @xcite ) ; v803 cen ( observation in atypical state in @xcite ) ; kl dra @xcite ; v406 hya @xcite ; sdss [email protected] ( @xcite ; @xcite ) ; yz lmi ( partial coverage in @xcite ) ; ptf1 [email protected] @xcite ; cp eri ( @xcite ; @xcite ) ; sdss [email protected] @xcite ; css j045019.7@xmath16093113 @xcite . in @xcite , we also studied superhumps and outburst patterns in an am cvn star cr boo and found that the basic @xmath0 variation of superhumps in am cvn - type superoutburst is the same as in hydrogen - rich systems . also detected stage a superhumps in cr boo , first time in am cvn - type objects with typical supercycles . in this paper , we added new examples of am cvn - type systems ptf1 j071912 , sdss j090221 ( see also @xcite ) and sdss j173047 . during the present survey , two further objects asassn-14ei and asassn-14mv have been identified as am cvn - type objects showing superoutbursts and multiple rebrightenings (; for spectroscopic identifications , see @xcite , see also vsnet - alert 18160 ) . asassn-14cn was also found to be an eclipsing am cvn - type object in outburst ( vsnet - alert 17879 ; superhumps are yet to be detected ) . asassn-14fv was also found to be an outbursting am cvn - type object ( @xcite for spectroscopic identification ; superhumps are yet to be detected ) . these data suggest that about 8% of objects showing dwarf nova - type outbursts are am cvn - type objects . this fraction is much larger than the hitherto known statistics ( about 1% in rkcat edition 7.21 @xcite ) . am cvn - type objects may be more populous than have been considered . there have also been an increasing number of ei psc - type objects ( cvs containing but depleted hydrogen with orbital period below the period minimum ) : v485 cen ( @xcite ; @xcite ) ; ei psc ( @xcite ; @xcite ; @xcite ) ; crts j102842.9@xmath16081927 ( @xcite ; @xcite ; @xcite ) ; crts j112253.3@xmath16111037 @xcite ; sbs 1108@xmath15574 ( @xcite ; @xcite ; @xcite ) ; css j174033.5@xmath15414756 ( ) . candidate systems include crts j233313.0@xmath16155744 @xcite . we have added another object of this class , v418 ser , in this paper . in all well - observed cases , the development of superhumps in these systems follow the same pattern as in hydrogen - rich systems . although the number of these objects is smaller than am cvn - type objects , these object may be more abundant than have been considered . in this paper , we studied long - term variation of supercycles in mm hya ( subsection [ obj : mmhya ] ) and cy uma ( subsection [ obj : cyuma ] ) . in mm hya , variations of the supercycle in the range of 330 d and 386 d were recorded . in cy uma , a sudden decrease of the supercycle from 362(3 ) d to 290(1 ) d was observed in 2003 . in both systems , the supercycle tended to be constant for several to ten years , and there was a tendency of a sudden switch to a different period . similar systematic variations of cycle lengths were studied in ss cyg - type dwarf novae [ cf . ar and , uu aql , ru peg : @xcite ] . @xcite suggested the solar - type activity as a possible cause , but the phenomenon still remained a puzzle . in recent years , @xcite recorded variations of supercycles from 43.6 to 59.2 d in er uma . the variations of the supercycle in er uma was an order similar to those in the two systems studied in this paper . although there was a possibility that a disk tilt , which is supposed to produce negative superhumps ( @xcite ; @xcite ) , could affect the outburst properties , @xcite could not find a correlation between the appearance of negative superhumps and the supercycle length . since a disk tilt is less likely to occur in systems with lower mass - transfer rate like mm hya and cy uma supposing a theoretical interpretation by @xcite , we consider it less likely that a disk tilt can explain the variations of the supercycle commonly seen in a variety of dwarf novae . the mechanism needs to be sought further . in addition to results of observations superhumps of the objects studied in this paper , the major findings we obtained can be summarized as follows . * the contribution of various surveys in detecting su uma - type dwarf novae have dramatically changed in the last ten years . the advantage of a wide - field survey on nightly basis ( such as asas - sn survey ) has become obvious . * thanks to the increase of samples of su uma - type dwarf novae , we could clarify the distribution of orbital periods in su uma - type dwarf novae , which approximate the cv population with short orbital periods . there is a sharp cut - off at a period of 0.053 d , which is considered to be the period minimum . there is a high concentration of objects just above the period minimum , which can be interpreted as the `` period spike '' . the distribution monotonically decreases towards the longer period , and there is little indication of the period gap . * we observed a precursor outburst in z cha and found that superhumps developed just following the precursor outburst . @xmath0 analyses in cy uma and yz cnc also support this finding . the agreement of the time of growing superhumps with the rising branch following the precursor outburst has been confirmed in all su uma - type systems so far studied . this finding provides a strong support to the thermal - tidal disk instability model which predicts that a superoutburst develops as a result of the development of superhumps . * we detected possible negative superhumps in z cha . during the phase of negative superhumps , the outburst cycle apparently lengthened . this finding seems to support a suggestion that a disk tilt suppresses normal outbursts . * we studied secular variations of the quiescent brightness and the amplitude of orbital humps throughout one supercycle in z cha . the quiescent brightness decreased as the system approached the next superoutburst . there was no enhanced orbital humps just before the superoutburst . * we studied long - term trends in supercycles in mm hya and cy uma and found systematic variations of supercycles of @xmath120% . this degree and characteristics os variations are similar to those recorded in other ss cyg - type dwarf novae and the su uma - type system er uma . a disk tilt is unlikely a common source of this variation . * the wz sge - type object asassn-15bp showed a phase jump of superhumps during the plateau phase . * a sizable number of am cvn - type objects ( ptf1 j071912 , sdss j090221 , sdss j173047 ) were studied in this paper and four more am cvn - type objects ( asassn-14cn , asassn-14ei , asassn-14fv and asassn-14mv ) were found in this period . this number suggests that about 8% of objects showing dwarf nova - type outbursts are am cvn - type objects . * we have added another ei psc - type object v418 ser in this paper . these object may be more abundant than have been considered . * css j174033 , an ei psc - type object , showed a similar type of superoutbursts in 2012 and 2014 , comprising of a dip and the second plateau phase . this finding suggests that the same type of superoutbursts tend to be reproduced in ei psc - type objects as in hydrogen - rich systems . * ot j213806 , a wz sge - type object , exhibited a remarkably different peak brightness and @xmath0 diagrams between the 2010 and 2014 superoutbursts . the fainter superoutburst in 2014 was shorter and the difference was most striking in the later part of the plateau phase . * master j085854 showed two post - superoutburst rebrightenings . this object had a rather exceptionally short superhump period among the objects showing multiple rebrightenings . * four deeply eclipsing su uma - type dwarf novae were identified ( asassn-13cx , asassn-14ag , asassn-15bu , nsv 4618 ) . asassn-14id also showed shallow eclipses . this work was supported by the grant - in - aid `` initiative for high - dimensional data - driven science through deepening of sparse modeling '' ( 25120007 ) from the ministry of education , culture , sports , science and technology ( mext ) of japan . the authors are grateful to observers of vsnet collaboration and vsolj observers who supplied vital data . we acknowledge with thanks the variable star observations from the aavso international database contributed by observers worldwide and used in this research . we are also grateful to the vsolj database . this work is deeply indebted to outburst detections and announcement by a number of variable star observers worldwide , including participants of cvnet and baa vss alert . the ccd operation of the bronberg observatory is partly sponsored by the center for backyard astrophysics . we are grateful to the catalina real - time transient survey team for making their real - time detection of transient objects available to the public . r. modic acknowledge the bradford robotic telescope for the detection of the 2014 outburst of qz vir . the work by a. sklyanov is partially performed according to the russian government program of competitive growth of kazan federal university . | continuing the project described by @xcite , we collected times of superhump maxima for 102 su uma - type dwarf novae observed mainly during the 20142015 season and characterized these objects .
our project has greatly improved the statistics of the distribution of orbital periods , which is a good approximation of the distribution of cataclysmic variables at the terminal evolutionary stage , and confirmed the presence of a period minimum at a period of 0.053 d and a period spike just above this period .
the number density monotonically decreased toward the longer period and there was no strong indication of a period gap .
we detected possible negative superhumps in z cha .
it is possible that normal outbursts are also suppressed by the presence of a disk tilt in this system .
there was no indication of enhanced orbital humps just preceding the superoutburst , and this result favors the thermal - tidal disk instability as the origin of superoutbursts .
we detected superhumpsin three am cvn - type dwarf novae .
our observations and recent other detections suggest that 8% of objects showing dwarf nova - type outbursts are am cvn - type objects .
am cvn - type objects and ei psc - type object may be more abundant than previously recognized . ot j213806 , a wz sge - type object , exhibited a remarkably different feature between the 2010 and 2014 superoutbursts .
although the 2014 superoutburst was much fainter the plateau phase was shorter than the 2010 one , the course of the rebrightening phase was similar .
this object indicates that the @xmath0 diagrams of superhumps can be indeed variable at least in wz sge - type objects .
four deeply eclipsing su uma - type dwarf novae ( asassn-13cx , asassn-14ag , asassn-15bu , nsv 4618 ) were identified .
we studied long - term trends in supercycles in mm hya and cy uma and found systematic variations of supercycles of @xmath120% . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
hadrons ; su(3 ) multiplets ; mass quantization . although the standard model is in excellent agreement with the rich experimental data , it leaves many questions unanswered . the observed mass spectrum of the elementary particles is one of the fundamental questions that has so far defied any reasonable explanation in the standard model@xcite@xmath0 @xcite . the distribution of the elementary particle masses is quite bizzare and is spread from a fraction of ev s for neutrinos to hundreds of gev s for the top quark . apart from few patterns based on su(3 ) symmetry that led to the gell - mann okubo@xcite and coleman - glashow@xcite formulae interelating the masses of pseudoscalar mesons and octet baryons respectively , the standard model has not revealed any general order in the elementary mass spectrum . on the other hand the associated physical mass is the best known and most fundamental characteristic property of the elementary particles , the recognition of empirical systematics and of any general regularities in the mass spectrum of these particles , irrespective of their nature or scheme of classication is of tremendous importance in understanding the intricacies of mass spectrum of elementary particles . the lowest mass states , as a general rule in physical systems , are considered to be the building blocks of more complex systems and hence in some sense the most fundamental . the most stable and least massive particles i.e. electron , muon and pion to which other particles decay after a transient existence , are the natural candidiates to search for a systematic regularity within the mass spectrum of elementary particles@xcite . empirical and theoritical investigations based on experimental data reveal the electron , muon and the pion to serve as basic units for exploring the discrete nature of the mass distribution of elementary particles@xcite@xmath1@xcite . to search for an order , we perform a specific systematic analysis of the mass spectrum of elementary particles and reveal that mass differences among particles when arranged in the ascending order of mass have a general tendency to be close integral / half integral multiple of mass difference between a neutral pion and a muon i.e. 29.318 mev . the mass differences between unstable leptons and between baryons were shown to be quantized as integral multiples of this basic unit of 29.318@xcite . in the present study , we evaluate the applicability of this result to the su(3 ) hadron multiplets and to neutral hadrons . we reveal that mass unit of about 29.318 mev is a determining factor for the distribution of mass of elementary particles by showing that 29.318 mev integral multiplicity of mass differences to be valid for these hadrons . this reinforces our earlier result that elementary particles do not occur randomly and are linked through the mass difference between first two massive elementary particles i.e. a muon and a neutral pion . the database for the present study is the latest version of the particle data group listings@xcite . here we investigate relationship the pion - muon mass difference has with the mass structure of the 1 ) hadrons which are classified into multiplets on the basis of su(3 ) symmetry and 2 ) neutral hadrons . the masses of the baryon octet members with spin j and parity p such that @xmath2=@xmath3@xmath1 are : @xmath4=938.27203 mev , @xmath5=939.56536 mev , @xmath6=1115.683 mev , @xmath7=1189.37 mev , @xmath8=1197.449 , @xmath9=1314.83 mev and @xmath10=1321.31 mev . the successive mass differences are tabulated in column 1 of table 1 with numerical value in mevs given in column 2 . the small mass differences between the different members of an isospin charge multiplet are known to arise from the electromagnetic interaction@xcite . however , the masses of the members of different isospin multiplets differ considerably . column 4 shows the integral multiples of 29.318 mev that are close to the observed mass difference between successive members of the octet . the integers being shown in column 3 . the deviations of the observed value from the closest integral multiple of 29.318 mev are given in column 5 . it is observed that the mass difference between @xmath11 and @xmath12 i.e. 176.118 mev differs from the nearest predicted value of 175.908 mev by only 0.21 mev . same is true of the mass difference i.e. 117.381 mev between the particles @xmath13 and @xmath14 which differs from the predicted value of 117.272 mev by only 0.109 mev . however , observed mass interval of @xmath15 and @xmath11 differs from the predicted value by about 14.264 mev . interestingly , this large value turns out to be half integral multiple of the mass difference between a @xmath16 and a @xmath17 . as can be clearly seen from the row 3 of table 1 , the observed mass difference between @xmath15 and @xmath11 i.e. 73.69 differs from the half integral ( @xmath18 ) multiple of pion and muon mass difference by only 0.39 mev . the maximum mass splitting within the baryon octet i.e. mass difference of 383.037 mev , between the heaviest member @xmath19 and the lightest baryon @xmath20 is close integral multiple of 29.318 mev , differing from the predicted value by only 1.904 mev . it may be pointed out that 29.318 mev multiplicity also holds for the mass intervals among any of the octet members@xcite . clearly the 29.318 mev multiplicity holds with great precision for the baryon octet members . the analysis for the baryon decuplet members with @xmath2=@xmath21@xmath1 is detailed in table 2 . it may be pointed out that while all the members of the baryon octet are non - resonant states , for the baryon decuplet all the members execpt for the @xmath22 are resonances . since the particle data group reports an average mass for the four charged states of the @xmath23 baryons and individual masses for the different charge states of @xmath24 and @xmath25 baryons , we cosider the average masses of each isospin multiplet which are as follows : @xmath26=1232 mev , @xmath27=1384.56 mev , @xmath28=1533.4 mev , and @xmath29=1672.45 mev . the first three mass differences in the table 2 are those between the successive members of the decuplet . the equal spacing rule for the su(3 ) decuplet predicts masses of successive isospin multiplets to be equidistant@xcite i.e. @xmath27 - @xmath26 = @xmath28 - @xmath27 = @xmath29 - @xmath28 . however , this rule is not strictly obeyed in the decuplet , since the mass separations are not exactly same as evident from table 2 . further , although the mass spacing among the successive decuplet members deviate from the closest integral ( 5 ) multiples of 29.318 mev by 5.97 , 2.25 and 7.54 mev respectively , it is important to note that the average mass spacing among successive members i.e. 146.816 mev is very close to 146.59 mev , a value obtained on integral ( 5 ) multiplication of 29.318 mev . the difference between the observed and predicted values being 0.226 mev only . this may be compared with 140 mev pionic mass interval@xcite among the decuplet members which deviates from the average mass spacing by about 6.816 mev@xcite . the @xmath22 - @xmath23 is the difference between the mass of lightest resonance member @xmath23 and the heaviest non - resonant member @xmath22 of the decuplet . this observed mass interval of 440.45 mev is very close to 439.77 mev , obtained on integral ( 15 ) multiplication of the mass difference between a neutral pion and a muon . the difference between the observed and expected value being only 0.68 mev . the mesons with spin zero and odd parity i.e. @xmath2 = @xmath30 are organized into a multiplet contaning nine states to form pseudoscalar meson nonet . these mesons have the lowest rest energy . in column 1 of table 3 , the k@xmath31 - @xmath32 , @xmath33 - k@xmath31 and @xmath34 - @xmath33 are the mass difference between the successive members of the pseudoscalar meson nonet with numerical values given in column 2 . the deviation of 0.182 mev between the observed and predicted @xmath34 - @xmath33 mass difference may be compared with that of 0.730 mev obtained from the integral multiple ( 3 ) of the 137 mev@xcite . both the observed mass intervals of @xmath33 - @xmath16 and @xmath33 - @xmath32 deviate from the 14 multiples of 29.318 mev by about 2 mev . however the difference 410.2364 mev between the mass of @xmath33 and average pion mass of 137.27339 mev differs by only 0.215 mev from predicted value 410.452 obtained by the multiplication of 29.318 mev by integer 14 . again this departure of the observed mass difference between @xmath33 and average pion mass may be comapared with the difference of 0.736 mev bewteen the observed and that expected when mass intervals are taken as integral multiples of average pion mass i.e. 137 mev@xcite . the observed mass interval between the @xmath34 and average pion mass is 820.506 mev . this value deviates from the predicted value of 820.904 mev obtained from the integral multiple ( 28 ) of 29.318 mev by only 0.3973 mev only . whereas the predicted value 822 mev on the basis of integral pion mass differs from the observed value by 1.4933 mev@xcite . the difference @xmath34 - @xmath16 is the mass difference between the lightest and heaviest member of the pseudoscalar meson nonet . as can be seen from the table 3 , the observed mass spacings are close integral multiples of the mass difference between a neutral pion and a muon . the masses for the pseudoscalar mesons taken from the particle data group listings are : @xmath35=134.9766 mev , @xmath36=139.57018 mev , @xmath37=493.677 mev , @xmath38= 497.648 mev , @xmath39=547.51 mev , @xmath40=957.78 mev . the nine vector mesons with spin one and odd parity i.e. @xmath2 = @xmath41 form a multiplet called vector meson nonet . the analysis for the mass differences among the vector meson nonet members is detailed in table 4 . the mass difference between successive members ( column 1 ) of the vector meson nonet is given in the column 2 . as is evident from the table 4 , the observed mass differences k@xmath42 - @xmath43 and @xmath44 - @xmath45 are close integral multiples of 29.318 mev . although the mass difference between isospin triplet @xmath43 meson and isospin singlet @xmath45 meson 7.15 mev is nonelectromagnetic in origin but is of the order of electromagnetic mass splitting . the deviation of 2.226 mev of the observed @xmath44 - @xmath45 mass difference from the integral multiple of 29.318 mev is in better agreement than that of 8.19 mev obtained when mass interval is predicted as half integral multiple of a mass unit of about 70 mev i.e about half the pion mass@xcite . since the particle data group lists average mass for the isospin triplet @xmath43 states and separate masses for two charge states of k@xmath42 meson , we consider the average mass of vector mesons for the analysis which are : @xmath46=893.83 mev , @xmath47=775.5 mev , @xmath48=782.65 mev and @xmath49=1019.460 mev . the applicabilty of 29.318 mev multiplicity of elementary particle mass intervals extends beyond the mass differences between the successive members of a particular su(3 ) multiplet and applies to mass intervals among the members of multiplets with different spin and parity characteristics . the analysis for the mass intervals between the octet and decuplet baryons is detailed in table 5 . the lightest member of decuplet i.e @xmath23(1232 ) resonance is the first excited state of the proton and the observed mass interval between the two is accounted by the hyperfine splitting due to colour magnetic interaction among the quarks@xcite . however , from table 5 it is seen that the observed mass interval of 293.728 mev between @xmath23(1232 ) and proton is very close to 293.180 mev , a value obtained on integral multiplication of 29.318 by 10 . the difference between the two values being only 0.548 mev . the relation of @xmath23(1232 ) and proton is expected as @xmath23 baryons are considered to be the excited states of nucleon but there is no relation between @xmath23(1232 ) and @xmath11 . however , from our analysis it follows that the observed mass interval between the two 116.317 mev differs from 117.272 mev , the closest integral ( 4 ) multiple of 29.318 mev by 0.955 mev only . thus @xmath23(1232 ) can be obtained by taking four excitations of 29.318 mev from the @xmath11 . similarly the observed mass inteval of 556.767 bewteen two unrelated baryons @xmath22 and @xmath11 deviates from 557.042 mev , a value obtained as 19 multiples of pion - muon mass difference by only 0.275 mev . further , the observed mass difference 351.14 mev between the heaviest member of baryon octet @xmath19 and that of baryon decuplet i.e. @xmath22 the only non - resonant member , differs from 351.816 mev , a value obtained on integral ( 12 ) multiplication of 29.318 mev , by only 0.676 mev . the detailed anlysis of the mass differences among the pseudoscalar and vector mesons is given in table 6 . as seen from the table 6 the 29.318 mev mass interval multiplicity is valid for mesons also . the neutral hadrons are placed in the column 1 of the table 7 in the ascending order of their associated physical mass . since no bound system of top quarks has been detected , the neutral hadrons listed in table 7 span the quark structure from the light u , d , s quarks to the heaviest charm and bottom quarks . the masses of the heavier mesons are : @xmath50 = 3096.916 mev , @xmath51 = 3686.093 mev , @xmath52 = 9460.30 mev . the differences amongst mesons @xmath16 , @xmath33 , @xmath34 and @xmath45 reported in tables 3 and 6 are reproduced here for completeness . the @xmath53 is the only baryon in the list . in column 2 we give the mass differences between the successive particles . the integral ( column 3 ) multiples of 29.318 mev , the mass difference between a neutral pion and a muon , closest to the observed mass differences are tabulated in column 4 and in column 5 we give the deviation of the observed value from the predicted value . as evident from the table the agreement of the predicted values as integral multiples of pion muon mass difference with the observed values is extraordinary . the observed mass difference between @xmath53 and @xmath34 is 234.862 mev . this value is very close to 234.544 mev obtained on multiplication of 29.318 mev by integer 8 . the difference between the two values being only 0.318 mev . similarly 1904.274 mev , the observed difference between the mass of j/@xmath54 and that of @xmath53 deviates from 29 multiples of 29.318 mev i.e. 1905.67 mev by only 1.396 mev . the j/@xmath54 is the lowest - lying member of the charm - anticharm meson series and @xmath55 is its first excited state . from table 7 it can be seen that the mass difference between @xmath55 and j/@xmath54 is close integral multiple of the mass difference between a neutral pion and muon . similarly @xmath56 is the first member of the bottom - antibottom meson series . from table 13 , it is seen that the observed mass difference between the @xmath56 and j/@xmath54 is 6363.384 which is very close to 217 multiples of 29.318 mev i.e. 6362.006 . the difference between the two values being only 1.378 mev . the fact that the mass differences of the lightest hadron @xmath16 with the heaviest mesons j/@xmath54 , @xmath55 and @xmath56 are close integral multiples of the mass difference between a neutral pion and muon indicates that the 29.318 mev mass systematics is valid over the wider range of the mass spectrum . in tables 8 to 13 we detail the analysis for the mass intervals of each neutral hadron with respect to the remaining hadrons . the mass intervals for each neutral hadron already listed in table 7 have been excluded in the analysis detailed in 8 - 13 . from tables 1 to 13 it is clear that the hadron mass intervals fall into two classes 1 ) those integral multiples of the mass difference between a neutral pion and a muon and 2 ) those in which the difference between the observed and predicted values are large but turn out to be exact half integral multiples of 29.318 mev . the inability of the standard model to explain the observed spectra of mass of elementary particle has led to the development of models and theories for particle masses based on direct analysis of the elementary particle data to understand different intricacies of the mass spectrum . the recognition of the pionic mass intervals among the particles@xcite combined with the fact that ratios of life times of elementary particles scale as powers of @xmath57 , the fine structue constant , indicated the fundamental role of @xmath57 in the generation of mass of elementary particles and that its domain is not restricted to leptons only , but also extends to include hadrons@xcite . the statistical analysis of masses of elementary particles when grouped according to their quark composition and @xmath58 quantum numbers reveals that the masses are quantized as integral multiples of a mass unit of about 35 mev . this 35 mev mass quantization combined with the analysis of the distribution of particle lifetimes as a function of mass reveals a shell structure of hadrons@xcite . a standing wave model assuming the particles to be held together in a cubic nuclear lattice has been proposed to explain empirical fact that meson and baryon masses are integral multiples of the neutral pion@xcite . from the preceeding discussion , it is clear that the understanding of any pattern in a physical system is a driving force for the development of a fundamental theory . the development proceeds from the experimental measurements of the phenomena , first to a recognition of regularities amongst the measurements , then to the physical insight which gives some understanding of these regularities and finally to a fundamental theory , which allows the totality of the phenomena to be understood from a few general principles@xcite . the failure to understand the regularities within a reasonable data lays the foundation for the neccessity of the underlying theory . the above mentioned studies are among many other attempts dedicated to pin down the fundamental relation of least massive and most stable particles , pion and muon , to the elementary particle mass spectrum pointed by their presence in the decay products of almost all the heavier elemenatry particle mass states@xcite@xmath1@xcite . the mass intervals among elementary particles have been shown to be integral multiples of the pion mass@xcite and some sequances of particles are also repoted with muonic mass intervals@xcite . the baryon and meoson masses are reported to be integral / half integral multiples of the pion mass@xcite@xmath1@xcite . the mass unit of 35 mev equal to one - third of muon mass and one - fourth of the pion mass serves as basic unit for the quantization of the elementary particle masses@xcite . on the other hand we look at the elementary particle mass data from a different prespective and reveal the fundamental nautre of mass difference between a neutral pion and a muon to the elementary particle mass spectrum . the electron and muon mass are the first two levels in the mass spectrum . indeed the muon is treated to be an excited state of the electron@xcite@xmath0 @xcite@xmath0 @xcite . however , the mass of electron being much smaller than the muon mass , the mass interval between the two is obscured by the large mass of muon and is thus equal to muon mass itself . hence we regard 29.318 mev , the mass interval between neutral pion ( a hadron ) and muon ( a lepton ) to be the first excitation within elementary particle mass spectrum . we investigate whether the mass excitations among the elementary particles are related to the first excitation . we have satisfactorily shown that mass differences among the elementary particles in general when arranged in the ascending order of mass have a striking tendency to be integral / half integral multiples of 29.318 mev@xcite . same is shown to be true for the mass differences between any of the baryons and mass interval between the unstable charged leptons . from the present study , it follows that the `` fine structure '' of mass i.e. mass differences betweend different isospin members of a given hadron multiplet are close integral multiples of the mass difference between a neutral pion and a muon . this is also true for the mass separation of the lightest and heaviest member . the mass intervals between the members of different su(3 ) multiplets , i.e. between octet and decuplet baryons and between the pseudoscalar and vector mesons are also integral multiples of the mass unit of 29.318 mev . the 29.318 mev discretness of the mass intervals is applicable to elementary particles irrespective of their classification into leptons , mesons and baryons on the basis of their structure , associated quantum numbers and type of interaction . it covers a wide range of mass spectrum , a fact borne out by the precise agreement of the observed and predicted mass intervals for the heavy charm - anticharm and bottom - antibottom mesons . although the present findings are not in line with the current theories , yet the large scale agreement between the observed mass differences and those calculated as integral multiples of the mass difference between a neutral pion and a muon clearly rules out it to be a mere coincidence . the important intriguing result of the present study is that the mass difference between the lightest composite particle pion and the second lightest lepton muon interlinks the masses of both leptons and hadrons . clearly the occurence of elementary particle states is not random but seems to follow a definite order such that the mass excitations from one particle to other are always in units of 29.318 mev i.e. the mass difference between a neutral pion and a muon . the mass differences among the baryon octet members have been found to be close integral multiples of the mass difference between a neutral pion and a muon . this also holds true for the average mass separation among the successive members of the baryon decuplet , for the mass intervals among the pseudoscalar meson nonet and among the vector meson nonet members . we also pointed out mass intervals among neutral hadrons to be integral multiples of mass difference between a neutral pion and a muon . our study reveals that inter particle mass excitations are quantized as integral multiples of 29.318 mev thereby indicating the fundamental nature of this mass quantum for the elementary particle mass spectrum . 00 v. christianto and f. smarandache , progress in physics , 4 , 112 ( 2007 ) h. fritzsch , arxiv:0802.0099 , hep - ph/0201198 s. okubo , prog . 27 , 949 ( 1962 ) s. coleman and s. l. glashow , phys . review lett . 6 , ( 1961 ) m. h. mac gregor , il nuovo cimento . a103 , 983 ( 1990 ) g. n. shah and t. a. mir , hep - ph/0702140 , mod . a23 , 53 ( 2008 ) y. nambu , prog . phys . 7 , 595 ( 1952 ) r. m. sternheimer , phys . review lett . 10 , 7 ( 1963 ) , 13 , 1 ( 1964 ) , 13 , 10 ( 1964 ) r. m. sternheimer , phys . b2 , 138 ( 1965 ) , 143 , 4 ( 1966 ) r. m. sternheimer , phys . review , 143 , 1262 ( 1966 ) , 5 , 170 ( 1968 ) f. r. tangherlini , prog . th . phys . 58 , 2002 ( 1977 ) b. g. sidharth , physics/0309037 , physics/0306010 m. h. mac gregor , il nuovo cimento . a58 , 159 ( 1980 ) m. h. mac gregor , int a20 , 719 ( 2005 ) p. palazzi , int . j. mod a22 , 546 ( 2007 ) , ( url : http://www.particlez.org ) e. l. koschmieder , physics/0602037 , physics/0002179 r. frosch , nuovo cimento . a104 , 913 ( 1991 ) a. o. barut , phys . b73 , 310 ( 1978 ) , phys . 42 , 1251 ( 1979 ) w. a. rodrigues and j. vaz , hep - th/9607231 d. akers , hep - ph/0310239 , hep - ph/0311031 , hep - ph/0303139 , hep - ph/0303025 w.- m. yao et al , ( particle data group ) , j. phys . g. 33 , 1 , ( 2006 ) ( url : http://pdg.lbl.gov ) d. h. perkins , introduction to high energy physics , pp . 126 - 129 , 4th edn . ( cambridge university press , 2000 ) y. oh , h. kim and s. h. lee , hep - ph/0310117 l. j. hall , hep - ph/9303217 | the study on the linkage of elementary particle mass differences with pion - muon mass difference is explored further . in the present study
we show this linkage to be equally true for the mass differences amongst the members of su(3 ) hadron multiplets and the hadrons belonging to multiplets having different spin and parity characteristics .
this reinforces our contention that inter particle mass excitations are quantized as integral multiples of basic mass unit of 29.318 mev and thereby indicates the fundamental nature of pion - muon mass difference in the elementary particle mass distribution . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
one major task in the theory of dynamics is to establish some kind of dynamic irreducibility of a system . of principal interest are the systems that display some kind of persistent irreducibility . the two main examples of this concept are robust transitivity and stable ergodicity . blenders were introduced by c. bonatti and l. daz in @xcite , to produce a large class of examples of non - hyperbolic robustly transitive diffeomorphisms . there , they showed that these objects appeared in a neighborhood of the time - one map of any transitive anosov flow . these systems are partially hyperbolic . to establish robust transitivity , they showed that the strong invariant manifolds entered a small ball ( the _ blender _ ) , where things got mixed . this blending was then distributed all over the manifold by means of the strong invariant manifolds . this phenomenon is robust , whence they get robust transitivity . in @xcite , c. bonatti and l. daz showed that blenders appear near co - index one heterodimensional cycles . this provides a local source of robust transitivity . in ( * ? ? ? * theorem c ) , the authors showed that , surprisingly , blenders also provide a local source of stable ergodicity . this arouses some interest in the appearance of blenders in the conservative setting . indeed , the presence of blenders near pairs of periodic points of co - index one in the conservative setting , allows the authors to prove a special case of a longstanding conjecture by c. pugh and m. shub , namely that stable ergodicity is @xmath1-dense among partially hyperbolic diffeomorphisms with 2-dimensional center bundle @xcite . the aim of this paper is to obtain conservative diffeomorphisms admitting blenders near conservative diffeomorphisms with a pair of hyperbolic periodic points with co - index one . this result is crucial in the proof of the pugh - shub conjecture @xcite . let us remark that , in fact , what we need for the proof of the pugh - shub conjecture is a very special case of theorem [ main.theorem ] . we think that in view of the new importance of blenders for the conservative setting , it is interesting to state the result in its full generality . the main result of this paper is the following . [ main.theorem ] let @xmath2 be a @xmath0 diffeomorphism preserving a smooth measure @xmath3 such that @xmath2 has two hyperbolic periodic points @xmath4 of index @xmath5 and @xmath6 of index @xmath7 . then there are @xmath0 diffeomorphisms arbitrarily @xmath1-close to @xmath2 which preserve @xmath3 and admit a @xmath8-blender associated to the analytic continuation of @xmath4 . the proof of theorem [ main.theorem ] closely follows the scheme in @xcite . all preliminary concepts are in section [ section.preliminaries ] . let @xmath2 be a @xmath0 diffeomorphism preserving a smooth measure @xmath3 such that @xmath2 has two hyperbolic periodic points @xmath4 of index @xmath5 and @xmath6 of index @xmath7 . a first step is to prove : [ proposition.heterodimensional.cycle ] let @xmath2 be a @xmath0 conservative diffeomorphism such that @xmath2 has two hyperbolic periodic points @xmath4 of index @xmath5 and @xmath6 of index @xmath7 , then @xmath1-close to @xmath2 there is a @xmath0 conservative diffeomorphism such that the analytic continuations of @xmath4 and @xmath6 form a co - index one heterodimensional cycle . the proof of this involves a combination of recurrence results by c. bonatti and s. crovisier , the connecting lemma , and a recent result by a. vila concerning approximation of conservative @xmath1 diffeomorphisms by smooth conservative diffeomorphisms . we prove this proposition in section [ section.real.central.eigenvalues ] . the goal of of the rest of the paper is to reduce this heterodimensional cycle to a standard form , in which perturbations are easily made . note that in a co - index one heterodimensional cycle associated to the points @xmath4 and @xmath6 with periods @xmath9 and @xmath10 , both @xmath11 and @xmath12 have @xmath13 contracting eigenvalues and @xmath7 expanding eigenvalues . there is a remaining ( central ) eigenvalue which is expanding for @xmath11 and contracting for @xmath14 . this center eigenvalue , however , could be complex or have multiplicity bigger than one . this could complicate getting a simplified model of the cycle . next step is to show , like in @xcite , the following : [ teo.real.central.eigenvalues ] let @xmath2 be a @xmath0 conservative diffeomorphism having a co - index one cycle associated to the periodic points @xmath4 and @xmath6 . then @xmath2 can be @xmath1-approximated by @xmath0 conservative diffeomorphisms having co - index one cycles with real central eigenvalues associated to the periodic points @xmath15 and @xmath16 which are homoclinically related to the analytic continuation of @xmath4 and @xmath6 . the fact that @xmath15 and @xmath16 are homoclinically related to the analytic continuation of @xmath4 and @xmath6 is important due to the fact that if @xmath17 is a @xmath8-blender associated to @xmath4 and @xmath15 is homoclinically related to @xmath4 then @xmath17 is also a @xmath8-blender associated to @xmath15 . see remark [ remark.blender.homoclinic ] . the proof of theorem [ teo.real.central.eigenvalues ] is in section [ section.real.central.eigenvalues ] . the proof of theorem [ main.theorem ] shall now follow after proving theorems [ teo.strong.homoclinic.intersection ] and [ teo.blenders ] below . [ teo.strong.homoclinic.intersection ] let @xmath2 be a @xmath0 conservative diffeomorphism having a co - index one cycle with real central eigenvalues . then @xmath2 can be @xmath1-approximated by @xmath0 conservative diffeomorphisms having strong homoclinic intersections associated to to a hyperbolic periodic point with expanding real central eigenvalue . theorem [ teo.strong.homoclinic.intersection ] is the more delicate part . next theorem is now standard after c. bonatti , l. daz and m. viana s work @xcite : [ teo.blenders ] let @xmath2 be a @xmath0 diffeomorphism preserving @xmath3 with a strong homoclinic intersection associated to hyperbolic periodic point @xmath4 with expanding real center eigenvalue . then @xmath2 can be @xmath1-approximated by a @xmath0 diffeomorphism preserving @xmath3 and having a @xmath8-blender associated to @xmath4 . in the creation of conservative blenders , we closely follow the scheme in @xcite . it will be proved that after a @xmath1-perturbation , we obtain a @xmath0 conservative diffeomorphism such that the co - index one cycle with real central eigenvalue has local coordinates where the dynamics of the cycle is affine and partially hyperbolic , with one dimensional central direction . this is called a _ simple cycle _ , see definition [ definition.simple.cycle ] . let us note that , unlike in @xcite , we can not use sternberg s theorem to linearize , due to the obvious resonance . we use instead the pasting lemmas by a. arbieto and c. matheus @xcite . the construction of simple cycles @xmath1-close co - index one cycles with real center eigenvalues is proved in section [ section.simple.cycle ] . we shall afterwards produce a continuous family of perturbations @xmath18 of @xmath2 shifting the unstable manifold of @xmath6 in @xmath19 so that it does not intersect @xmath20 , thus breaking the cycle . these perturbations preserve the bundles @xmath21 , @xmath22 and @xmath23 . in this form , they induce maps of the interval on @xmath22 . by carefully choosing a small parameter @xmath24 , and possibly touching " the center expansion / contraction , we may obtain that there is an @xmath25 plane that is periodic by two different itineraries of @xmath26 . now , the dynamics @xmath26 on this periodic plane is hyperbolic , using a markovian property , we get two periodic points that are homoclinically related within this plane . using the @xmath27-lemma we obtain a periodic point with a strong homoclinic intersection . this is done in section [ section.strong.homoclinic.intersection ] . in this way we can apply the well - known techniques of bonatti , daz and viana @xcite and of bonatti , daz which give blenders near points with strong homoclinic intersections @xcite . note that these perturbations can be trivially made so that the resulting diffeomorphism be @xmath0 and conservative . from now on , we shall consider a smooth measure @xmath3 on a smooth manifold @xmath28 , and a @xmath0 diffeomorphism @xmath29 preserving @xmath3 . we shall say that @xmath2 is conservative if @xmath2 preserves @xmath3 . given a hyperbolic periodic point @xmath4 of @xmath2 , the index of @xmath4 is number of expanding eigenvalues of @xmath11 , counted with multiplicity , where @xmath9 is the period of @xmath4 . [ definition.heterodimensional.cycle ] a diffeomorphism @xmath2 has a heterodimensional cycle associated to two hyperbolic periodic points @xmath4 and @xmath6 of @xmath2 if their indices are different , and the stable manifold @xmath20 of @xmath4 meets the unstable manifold @xmath30 of @xmath6 , and the unstable manifold @xmath31 of @xmath4 meets the stable manifold @xmath32 of @xmath6 . when the indices of @xmath4 and @xmath6 differ in one , we say @xmath4 and @xmath6 are a co - index one heterodimensional cycle , or co - index one cycle . we shall say that @xmath2 is partially hyperbolic on an @xmath2-invariant set @xmath33 , or @xmath33 is partially hyperbolic for @xmath2 if there is a @xmath34-invariant splitting @xmath35 such that for all @xmath36 and all unit vectors @xmath37 , @xmath38 we have : @xmath39 for some suitable riemannian metric on @xmath28 . we require that both @xmath40 and @xmath41 be non - trivial . our goal is to produce a @xmath1-perturbation admitting a blender . here is the definition of blender we shall be using : [ definition.blender ] let @xmath4 be a partially hyperbolic periodic point for @xmath2 such that @xmath34 is expanding on @xmath22 and @xmath42 . a small open set @xmath17 , near @xmath4 but not necessarily containing @xmath4 , is a @xmath8-blender associated to @xmath4 if : 1 . every @xmath5-strip well placed in @xmath17 transversely intersects @xmath20 . 2 . this property is @xmath1-robust . moreover , the open set associated to the periodic point contains a uniformly sized ball . a @xmath5-strip is any @xmath5-disk containing a @xmath7-disk @xmath43 , so that @xmath43 is centered at a point in @xmath17 , the radius of @xmath43 is much bigger than the radius of @xmath17 , and @xmath43 is almost tangent to @xmath41 , i.e. the vectors tangent to @xmath43 are @xmath1-close to @xmath41 . a @xmath5-strip is well placed in @xmath17 if it is almost tangent to @xmath44 . see fgure [ figure.blender ] . naturally , it makes sense to talk about robustness of these properties and concepts , since there is an analytic continuation of the periodic point @xmath4 and of the bundles @xmath40 , @xmath22 and @xmath41 . we can define @xmath45-blenders in a similar way . for @xmath45-blenders we will consider a partially hyperbolic point such that @xmath22 is one - dimensional and @xmath34 is contracting on @xmath22 . @xmath8-blender associated to @xmath4,width=226 ] @xmath8-blender associated to @xmath4,width=226 ] this is the definition of blender we shall be using in this work , and , in particular what we obtain in theorem [ main.theorem ] . we warn the reader that there are other definitions of blenders . in @xcite , chapter 6.2 . there is a complete presentation on the different ways of defining these objects . our definition corresponds to definition 6.11 of @xcite ( the operational viewpoint ) . in some works of c. bonatti and l. daz , see for instance @xcite and @xcite , what we call @xmath17 is known as the _ characteristic region of the blender _ ; and what they call _ blender _ is in fact a hyperbolic set which is the maximal invariant set of a small neighborhood of @xmath6 ( definition 6.9 . of @xcite ) . however , let us remark that , under the hypothesis of theorem [ main.theorem ] , we obtain a @xmath1-perturbation preserving @xmath3 and admitting blenders also in the sense of definition 6.9 of @xcite , and of @xcite . the existence of blenders in the sense of @xcite implies the existence of blenders in the sense of definition [ definition.blender ] . [ definition.blender.homoclinic ] let @xmath15 be a partially hyperbolic periodic point for @xmath2 such that @xmath34 is expanding on @xmath22 , with @xmath42 . a small open set @xmath46 is called @xmath8-blender associated to @xmath15 if @xmath47 , where @xmath4 is a partially hyperbolic periodic point homoclinically related to @xmath4 and @xmath17 is a @xmath8-blender near @xmath4 . we shall also denote @xmath48 the @xmath8-blender associated to @xmath15 . [ remark.blender.homoclinic ] let us note that if @xmath48 is a @xmath8-blender associated to a hyperbolic periodic point @xmath15 and @xmath17 is a @xmath8-blender near @xmath4 , where @xmath4 is homoclinically related to @xmath15 , then it follows from the @xmath27-lemma that @xmath49 transversely intersects every @xmath5-strip well placed in @xmath17 , and that this property is robust . the @xmath27-lemma also implies that a @xmath8-blender associated to @xmath15 is also a @xmath8-blender associated to @xmath50 if @xmath15 and @xmath50 are homoclinically related . if @xmath2 is a partially hyperbolic diffeomorphism in @xmath28 , then we can define @xmath8-blender associated to a periodic point @xmath15 directly as in definition [ definition.blender ] , without requiring that @xmath48 be near @xmath15 . indeed , this requirement is only needed to guarantee the existence of the @xmath5-strips well placed in @xmath48 . in the case that @xmath2 is partially hyperbolic , the @xmath41 and @xmath22 bundles are globally defined , and so definition [ definition.blender ] makes sense even without asking that @xmath48 be near @xmath15 . let us consider two hyperbolic periodic points @xmath4 of index @xmath5 and @xmath6 of index @xmath7 , with periods @xmath9 and @xmath10 respectively . let us denote by @xmath51 , @xmath52 the eigenvalues of @xmath11 and @xmath12 respectively , with @xmath53 , ordered in such a way that : @xmath54 s are contracting and the @xmath55 s are expanding . moreover , @xmath56 is expanding for @xmath4 and contracting for @xmath6 . note that @xmath56 could be equal to @xmath57 , and could be a complex eigenvalue . a co - index one heterodimensional cycle associated to two hyperbolic periodic points @xmath4 and @xmath6 as described above has real central eigenvalues if @xmath58 and @xmath59 . note that when a cycle has real central eigenvalues , then @xmath60 and @xmath61 have multiplicity one . in the remainder of this subsection , we shall work with heterodimensional cycles with real central eigenvalues . in this case , both orbits of @xmath4 and @xmath6 admit a partially hyperbolic splitting @xmath62 , with @xmath42 , @xmath63 and @xmath64 . we shall denote @xmath65 the strong stable manifold of @xmath4 , this means , the invariant manifold tangent to @xmath66 that has dimension @xmath13 . analogously we define @xmath67 , the strong unstable manifold of @xmath4 . note that the unstable manifold of @xmath4 could be of dimension @xmath5 , in which case it would contain @xmath67 . a partially hyperbolic periodic point @xmath4 has a strong homoclinic intersection if @xmath68 the first goal is to obtain a @xmath1-perturbation of @xmath2 admitting a co - index one cycle with some local coordinates such that the dynamics in a neighborhood of the points are affine : [ definition.simple.cycle ] a co - index one cycle associated to the periodic points @xmath4 of index @xmath5 and @xmath6 of index @xmath7 is called a simple cycle if it has real central eigenvalues and : 1 . @xmath4 and @xmath6 admit neighborhoods @xmath19 and @xmath69 on which the expressions of @xmath70 and @xmath71 are linear and partially hyperbolic , with central dimension 1 . we denote the coordinates of a point @xmath72 or @xmath73 according to wether it belongs to @xmath19 or @xmath69 . 2 . there is a quasi - transverse heteroclinic point @xmath74 such that @xmath75 for some @xmath76 , and a neighborhood @xmath77 of @xmath78 satisfying @xmath79 for which @xmath80 is an affine map preserving the partially hyperbolic splitting , which is contracting in the @xmath13-direction , expanding in the @xmath7-direction , and an isometry in the central direction . there is a point @xmath81 , with @xmath82 such that @xmath83 with @xmath84 for some @xmath85 , and a neighborhood @xmath86 of @xmath87 , such that @xmath88 and @xmath89 is an affine map preserving the partially hyperbolic splitting , and is contracting in the @xmath13-direction , expanding in the @xmath7-direction , and an isometry in the central direction . 4 . there is a segment @xmath90_p^c$ ] contained in @xmath91 , such that the form of @xmath92 in @xmath69 is @xmath93^c_q$ ] . we call the affine maps @xmath94 and @xmath95 the transitions of the simple heterodimensional cycle . in this subsection we state the preliminary results that shall be used in this work . the following , theorem 1.3 of @xcite , allows us to approximate conservative diffeomorphisms by transitive diffeomorphisms preserving @xmath3 : [ teo.bonatti.crovisier ] there exists a residual set of the set of diffeomorphisms preserving @xmath3 such that all diffeomorphisms in this set is transitive . moreover , @xmath28 is the unique homoclinic class . we shall use this theorem in combination with the connecting lemma below : [ connecting.lemma ] let @xmath96 be hyperbolic periodic points of a @xmath0 transitive diffeomorphism @xmath2 preserving a smooth measure @xmath3 . then , there exists a @xmath1-perturbation @xmath97 preserving @xmath3 such that @xmath98 . the recent remarkable result by a. vila allows us to approximate @xmath1 conservative diffeomorphisms by @xmath99 conservative diffeomorphisms : [ a. vila @xcite][teo.avila ] @xmath99 diffeomorphisms are dense in the set of @xmath1 diffeomorphisms preserving @xmath3 . this following conservative version of franks lemma is proposition 7.4 of @xcite : [ franks.lemma ] let @xmath2 be a @xmath0 diffeomorphism preserving a smooth measure @xmath3 , @xmath100 be a finite set . assume that @xmath46 is a conservative @xmath101-perturbation of @xmath34 along @xmath100 . then for every neighborhood @xmath102 of @xmath100 there is a @xmath1-perturbation @xmath103 preserving @xmath3 , coinciding with @xmath2 on @xmath100 and out of @xmath102 , such that @xmath104 is equal to @xmath46 on @xmath100 . the fundamental tools in order to adapt the construction of c. bonatti and l. daz @xcite to the conservative case are the pasting lemmas of a. arbieto and c. matheus @xcite . we shall need two such lemmas , one for vector fields and the other for diffeomorphisms . the following is theorem 3.1 . of @xcite , and states that we can paste " two sufficiently @xmath1-close @xmath0 vector fields , so that one gets the value of the first one on one set and the value of the second one on a disjoint set : [ pasting.lemma.para.campos ] given @xmath105 and @xmath106 there exists @xmath107 such that if @xmath108 are two @xmath0 vector fields preserving a smooth measure @xmath3 that are @xmath109 @xmath1-close on a neighborhood @xmath110 of a compact set @xmath111 , then there exist an @xmath3-preserving vector field @xmath112 @xmath101 @xmath1-close to @xmath113 and two neighborhoods @xmath102 and @xmath114 of @xmath111 such that @xmath115 satisfying @xmath116 and @xmath117 . we shall also need the pasting lemma for diffeomorphisms , which states that we can produce a @xmath0 diffeomorphism by pasting " a conservative diffeomorphism @xmath2 with its derivative on a neighborhood of a point : [ pasting.lemma.para.difeos]if @xmath2 is a @xmath118 diffeomorphism preserving a smooth measure @xmath3 and @xmath119 is a point in @xmath28 , then for any @xmath120 there exists a @xmath0 diffeomorphism @xmath121 preserving @xmath3 , @xmath101-@xmath1-close to @xmath2 and two neighborhoods @xmath102 and @xmath110 of @xmath119 such that @xmath122 and @xmath123 and @xmath124 ( in local charts ) . let @xmath2 be a @xmath0 diffeomorphism preserving a smooth measure @xmath3 , with two hyperbolic periodic points @xmath4 of index @xmath5 and @xmath6 of index @xmath7 . our first step is to show that @xmath2 can be @xmath1-approximated by a @xmath0 diffeomorphism preserving @xmath3 such that the analytic continuations of @xmath4 and @xmath6 form a co - index one heterodimensional cycle . namely : [ lemma.heterodimensional.cycle ] let @xmath2 be a @xmath0 diffeomorphism preserving a smooth measure @xmath3 such that @xmath2 has two hyperbolic periodic points @xmath4 and @xmath6 , then @xmath2 is @xmath1-approximated by @xmath0 diffeomorphisms preserving @xmath3 such that the continuations of @xmath4 and @xmath6 either form a heterodimensional cycle in case they have different indices , or else they are homoclinically related . let us consider the case where @xmath4 has unstable index @xmath7 and @xmath6 has unstable @xmath125 . the case @xmath126 is simpler , and follows analogously . we shall apply the connecting lemma ( theorem [ connecting.lemma ] ) to produce the heterodimensional cycle . however , this lemma has transitivity as a hypothesis . so , let us consider a conservative diffeomorphism @xmath127 @xmath1-close to @xmath2 so that @xmath127 is transitive . this @xmath127 exists due to theorem [ teo.bonatti.crovisier ] by c. bonatti and s. crovisier . note that @xmath127 is a priori only @xmath1 and has two hyperbolic periodic points @xmath128 and @xmath129 which are the analytic continuations of @xmath4 and @xmath6 . now , due to its transitivity , the connecting lemma applies , and we can find a @xmath1-close conservative diffeomorphism @xmath130 such that @xmath131 intersects @xmath132 , where @xmath133 and @xmath134 are the analytic continuations of @xmath4 and @xmath6 . we can even ask that this intersection be transverse ( it will be @xmath135-dimensional ) . note that since it is transverse , this intersection persists under @xmath1-perturbations . we will also ask , by using theorem [ teo.bonatti.crovisier ] again , that @xmath130 be transitive . then , we apply the connecting lemma again , and obtain a new @xmath1 conservative diffeomorphism @xmath136 @xmath1-close to @xmath130 so that @xmath137 and @xmath138 , we can even ask that this last intersection be quasi - transverse , i.e. @xmath139 does not contain a non - trivial vector . but @xmath136 could be not @xmath0 a priori . theorem [ teo.avila ] of a. vila yields a @xmath99 conservative diffeomorphism @xmath140 @xmath1-close to @xmath136 . since the stable and unstable manifolds of @xmath141 and @xmath142 vary continuously , we obtain that @xmath143 , and @xmath144 is close to @xmath145 near the point @xmath146 . there is a @xmath99 conservative diffeomorphism @xmath147 , @xmath1-close to @xmath140 , so that @xmath148 intersects @xmath149 . this last perturbation can be made in fact @xmath99 . this gives the desired heterodimensional cycle . theorem [ teo.real.central.eigenvalues ] follows from lemma [ lemma.real.central.eigenvalues ] below and lemma [ lemma.heterodimensional.cycle ] . the idea of lemma [ lemma.real.central.eigenvalues ] is as in lemma 4.2 . of @xcite . see also its generalization to dimension @xmath150 in lemmas 1.9 and 4.16 of @xcite . the only difference is that we shall apply the pasting lemmas and the conservative franks lemma instead of the corresponding results . [ lemma.real.central.eigenvalues ] let @xmath2 be a conservative @xmath0 diffeomorphism with a hyperbolic periodic point @xmath4 . then @xmath2 is @xmath1-approximated by a @xmath0 conservative diffeomorphism with a hyperbolic periodic point @xmath15 having the same index as @xmath4 such that the inequalities in equation ( [ equation.real.central.eigenvalues ] ) are strict . let us assume @xmath4 is a hyperbolic fixed point for @xmath2 and let us denote the eigenvalues of @xmath152 as in equation ( [ equation.real.central.eigenvalues ] ) . we may assume , by using the conservative franks lemma ( proposition [ franks.lemma ] ) that the multiplicity of all the eigenvalues , complex or real , is one ; and that any pair of complex eigenvalues with the same modulus are conjugated , and have rational argument . we shall also assume that @xmath153 , and that @xmath60 and @xmath154 are complex conjugated expanding eigenvalues . we shall prove that there is @xmath15 such that @xmath155 , whence @xmath15 will have index @xmath5 , just like @xmath4 . in fact , this is all we need , since the same argument applies to show there is @xmath16 of index @xmath7 with real central ( contracting ) eigenvalue of multiplicity one . lemma [ lemma.real.central.eigenvalues ] follows from an inductive argument , since the fact that the eigenvalue is the central one is not used in the argument . using the connecting lemma ( theorem [ connecting.lemma ] ) and genericity arguments we obtain a transverse homoclinic intersection @xmath156 . using the pasting lemma for diffeomorphisms ( theorem [ pasting.lemma.para.difeos ] ) we can linearize @xmath2 in a small neighborhood @xmath102 of @xmath4 , so that @xmath2 remains the same outside a neighborhood @xmath157 . by considering sufficiently large iterates , we may assume that @xmath158 , and @xmath159 are such that the tangent spaces to @xmath20 at @xmath146 and to @xmath31 at @xmath119 are close enough to @xmath66 and @xmath160 . the tangent space to @xmath20 at @xmath119 and to @xmath31 at @xmath146 are @xmath66 and @xmath160 due to the linearization . let @xmath167 and @xmath168 . applying the conservative franks lemma we obtain a perturbation such that @xmath169 a new perturbation allows us to fix " the eigenspace associated to @xmath60 and @xmath154 , and obtain normal bases on which the derivative @xmath170 has the form : @xmath171 where @xmath172 is a @xmath173 matrix acting on a subspace @xmath174 , where @xmath175 is the 2-dimensional eigenspace associated to the eigenvalues @xmath60 and @xmath154 . for simplicity we shall assume that @xmath4 and @xmath6 are fixed points , and that the co - index cycle is a simple cycle . we shall call @xmath239 and @xmath240 . we shall assume that @xmath241 and @xmath242 , since no greater complication appears in the cases @xmath243 and @xmath244 . since the cycle is simple , there are coordinates @xmath245 and @xmath246 in suitable neighborhoods of @xmath247 and @xmath248 on which the expression of @xmath2 is : @xmath249 where @xmath250 , @xmath251 are contractions , and @xmath252 , @xmath253 are expansions . we recall that there are points @xmath78 in the quasi - transverse intersection of @xmath254 , and @xmath255 in the transverse intersection of @xmath190 such that on suitable neighborhoods @xmath77 and @xmath86 the transitions @xmath256 and @xmath257 have the form : @xmath258 and @xmath259 where @xmath260 , @xmath261 are contractions and @xmath262 , @xmath263 are expansions . we shall produce a continuous family of perturbations @xmath18 of @xmath2 shifting the unstable manifold of @xmath6 in @xmath19 so that it does not intersect @xmath65 , see figure [ figure.perturbation ] . these perturbations preserve the bundles @xmath21 , @xmath22 and @xmath23 . in this form , they induce maps of the interval on @xmath22 . by eventually changing the original @xmath264 and @xmath241 ( so that @xmath26 continues to be conservative ) , and carefully choosing a small parameter @xmath24 , we may obtain that the @xmath25 plane containing the point @xmath255 is periodic by two different ( large ) itineraries . now , the dynamics @xmath26 on this periodic plane is hyperbolic , then using a markovian property , we get two periodic points that are homoclinically related within this plane . the @xmath27-lemma gives now a periodic point with a strong homoclinic intersection . proceeding as in section [ section.simple.cycle ] we take a divergence - free vector field @xmath113 supported in a small neighborhood of @xmath265 , so that the composition @xmath26 of @xmath2 with the time-@xmath266 map of @xmath113 form a @xmath1-family of @xmath0 conservative diffeomorphisms admitting transitions @xmath227 of the form ( [ equation.transition2 ] ) and @xmath267 of the form : @xmath268 where @xmath230 is as in formula ( [ equation.transition1 ] ) since @xmath269 and @xmath270 , the formulas ( [ equation.linear.expresion.f ] ) hold for all small @xmath24 . note that if the composition @xmath271 makes sense for some point and takes a small neighborhood of @xmath255 into @xmath19 then , due to the formulas above , its center coordinate takes the form : @xmath272\qquad\mbox{where}\quad\delta y = y^--y^+\ ] ] conversely , we have the following : [ lemma.periodic.points.m.n ] if @xmath273 are such that @xmath274 with sufficiently large @xmath275 , then there is a point @xmath276 in the @xmath25 plane through @xmath255 that is @xmath26-periodic , with period @xmath277 . its center eigenvalue is @xmath278 . let @xmath24 be a small parameter , and let @xmath284 be sufficiently large . suppose that @xmath285 . then it is easy to see that the @xmath286 image of the @xmath287-disc @xmath288^s\times\{y^+\}\times[-1,1]^u$ ] contains a cylinder of the form @xmath289^u,\ ] ] for some sufficiently small @xmath290 , where @xmath291 denotes the @xmath13-disc of radius @xmath109 centered at @xmath119 . analogously , we obtain that the @xmath286 pre - image of the @xmath287-disc @xmath288^s\times\{y^+\}\times[-1,1]^u$ ] contains a cylinder of the form @xmath292^s\times \{y^+\}\times b^u_\delta(z),\ ] ] for some @xmath293 and some suitable small @xmath109 , which can be taken equal to the previous one . @xmath294 denotes the @xmath7-disc of radius @xmath109 centered at @xmath295 . this implies the existence of a periodic point @xmath283 of period @xmath277 . the fact that the transitions are isometries on the bentral direction implies that @xmath296 . if @xmath279 are such that @xmath280 , then the previous argument gives us a periodic point @xmath297 which , by construction , is different from @xmath283 . due to linearity , the unstable manifolds of @xmath283 , @xmath297 are , respectively , the @xmath7-discs @xmath298^u$ ] and @xmath299^u$ ] . also , @xmath300^s\times(y^+,z_{m , n})$ ] and @xmath301^s\times(y^+,z_{m',n'})$ ] . the @xmath7-discs transversely intersect the @xmath7-discs in the @xmath287-plane . therefore , @xmath283 and @xmath297 are homoclinically related in the @xmath287-plane , that is @xmath302 the @xmath27-lemma implies the existence of a point @xmath303 in the intersection of the strong stable and strong unstable manifolds of @xmath283 . the expression of @xmath26 in @xmath19 implies that this intersection is quasi - transverse . this is proved in lemma 3.11 of @xcite . the proof of theorem [ teo.strong.homoclinic.intersection ] now ends by taking @xmath312 so that @xmath313 . with the techniques used in section [ section.simple.cycle ] we can produce a @xmath1 perturbation so that to obtain a @xmath0 conservative diffeomorphism where the linear expressions ( [ equation.linear.expresion.f ] ) are such that the center coordinate expansions are , respectively @xmath314 and @xmath315 . theorem [ teo.blenders ] has been proved in @xcite and in theorem 2.1 of @xcite , see also section 4.1 of @xcite . note that the perturbations can be trivially made so that the resulting diffeomorphism be @xmath0 and conservative . rrrrrr a. arbieto , c. matheus , a pasting lemma and some applications for conservative systems , _ erg . th . & dyn . * 27 * , 5 ( 2007 ) 1399 - 1417 . arnaud , cration de connexions en topologie @xmath316 , _ erg . th . & dyn . systems _ , * 21 * , 2 ( 2001),339 - 381 . a. vila , on the regularization of conservative maps , _ preprint _ ( 2008 ) arxiv:0810.1533 c. bonatti , s. crovisier , rcurrence et gnricit , _ inv . math . _ , * 158 * , 1 ( 2004 ) , 33 - 104 . c. bonatti , l. daz , persistent nonhyperbolic transitive diffeomorphisms . ( 2 ) * 143 * ( 1996 ) , no . 2 , 357 - 396 . c. bonatti , l. daz , robust heterodimensional cycles and @xmath1-generic dynamics , _ journal of the inst . of math . jussieu _ * 7 * ( 3 ) ( 2008 ) 469 - 525 . c. bonatti , l. daz , e. pujals , a @xmath1-generic dichotomy for diffeomorphisms : weak forms of hyperbolicity or infinitely many sinks or sources , _ ann . math . _ * 158 * ( 2003 ) 355 - 418 . c. bonatti , l. daz , e. pujals , j. rocha , robustly transitive sets and heterodimensional cycles , _ astrisque _ * 286 * ( 2003 ) 187 - 222 . c. bonatti , l. daz , m. viana , discontinuity of the hausdorff dimension of hyperbolic sets , _ c. r. acad . paris sr . _ * 320 * ( 1995 ) , no . 6 , 713 - 718 . c. bonatti , l. daz , m. viana , _ dynamics beyond uniform hyperbolicity . a global geometric and probabilistic perspective . _ encyclopaedia of mathematical sciences , 102 . mathematical physics , iii . springer - verlag , berlin , 2005 . l. daz , e. pujals , r. ures , partial hyperbolicity and robust transitivity , _ acta math _ , * 183 * ( 1999 ) 1 - 43 . f. rodriguez hertz , m. rodriguez hertz , a. tahzibi , r. ures , a criterion for ergodicity of non - uniformly hyperbolic diffeomorphisms , _ era - ms _ * 14 * , ( 2007 ) 74 - 81 . f. rodriguez hertz , m. rodriguez hertz , a. tahzibi , r. ures , new criteria for ergodicity and non - uniform hyperbolicity , _ preprint _ we may assume now that @xmath2 is a @xmath0 conservative diffeomorphism with a co - index cycle having real central eigenvalues . our main result will be established if we prove theorems [ teo.strong.homoclinic.intersection ] and [ teo.blenders ] . to do this , we shall perturb that in order to obtain a simplified model of the cycle , that is , a simple cycle . the creation of simple cycles follows the same arguments as in ( * ? ? ? * proposition 3.5 ) and ( * ? ? ? * lemma 3.2 ) . the only difference is that we shall use the pasting lemmas to linearize in the conservative setting . the goal of this section is to produce a simple cycle . let us suppose that @xmath4 and @xmath6 are hyperbolic fixed points of indices @xmath5 and @xmath7 respectively . we may also assume that @xmath32 and @xmath31 have a non - trivial transverse intersection , and @xmath20 and @xmath30 have a point of quasi transverse intersection . on @xmath178 we have a partially hyperbolic splitting @xmath62 , where @xmath64 , @xmath42 and @xmath63 with @xmath179 . using the pasting lemma for diffeomorphisms , we obtain two neighborhoods @xmath19 and @xmath69 on which we can linearize @xmath2 . we call this new @xmath0 conservative diffeomorphism @xmath121 . @xmath121 equals @xmath152 on @xmath19 and @xmath180 on @xmath69 , the strong stable and unstable manifolds are , respectively , the @xmath13- and @xmath7-planes parallel to @xmath66 and @xmath160 , or @xmath181 and @xmath182 . the center lines parallel to @xmath183 and @xmath184 are also invariant under @xmath121 . we can choose @xmath121 so that there is a point of transverse intersection @xmath113 of @xmath31 and @xmath32 , and a point of quasi - transverse intersection @xmath185 of @xmath20 and @xmath30 . there is a sufficiently large iterate @xmath186 so that @xmath187 and @xmath188 . take @xmath189 . now , generically @xmath190 is transverse to the strong unstable in @xmath19 and to the strong stable foliation in @xmath69 . so , take a curve @xmath191 and iterates @xmath186 so large that @xmath192 is the graphic of a map @xmath193 , and @xmath194 is the graphic of a map @xmath195 . @xmath196 and @xmath197 are small segments contained , respectively in @xmath198 and @xmath199 . note that @xmath192 approaches @xmath198 exponentially faster than it approaches @xmath4 ; analogously @xmath200 approaches @xmath199 exponentially faster than it approaches @xmath6 . hence we can choose @xmath186 so large that @xmath201 and @xmath202 and are @xmath1-close to zero . let us define @xmath0 vector fields @xmath203 and @xmath204 in suitable neighborhoods of @xmath205 and @xmath206 . we define @xmath203 as a vector field that is constant the hyperplanes parallel to @xmath207 , such that @xmath208 . that is , @xmath203 assigns to each point its center coordinate . since @xmath203 is constant along the hyperplanes parallel to @xmath207 it is divergence free and it is very close to the null vector field . then we can apply the pasting lemma for flows ( theorem [ pasting.lemma.para.campos ] ) and paste @xmath100 with the null vector field obtaining a @xmath0-vector field @xmath209 that is @xmath1-close to the null vector field . by composing our diffeomorphism with the time - one map of @xmath210 we have a perturbation of @xmath121 ( which we continue to call @xmath121 ) such that @xmath211 is contained in @xmath198 . analogously , we obtain a @xmath1-perturbation , so that @xmath212 . in this way , we have obtained so far the points @xmath78 , @xmath213 , @xmath87 and @xmath214 . let us apply the pasting lemma for diffeomorphisms again , so that we obtain a new perturbation for which the transitions @xmath215 and @xmath216 are affine maps , where @xmath102 and @xmath114 are small neighborhoods of @xmath78 and @xmath87 respectively . we loose no generality in assuming that the images of the hyperplanes @xmath21 , @xmath23 , @xmath217 , @xmath218 and @xmath219 are in general position . by taking @xmath76 sufficiently large , one obtains that the image of the center - unstable foliation becomes very close to the @xmath219 in @xmath19 . a small perturbation using the pasting lemma for vector fields like in the previous paragraph gives us an invariant center - unstable foliation . indeed , there exists a matrix @xmath220 with @xmath221 , close to the identity , taking the image of the center - unstable foliation in @xmath86 into the center - unstable foliation of @xmath222 . but now , there exists a vector field @xmath223 such that the time - one map of @xmath223 is @xmath220 . we use the pasting lemma for vector fields to paste @xmath223 in a neighborhood of @xmath214 in @xmath224 with the identity outside of @xmath224 . composing @xmath121 with the time one map of this vector field , we get a @xmath0 diffeomorphisms @xmath1-close to @xmath121 such that @xmath225 leaves the center - unstable foliation invariant . we replace @xmath224 by @xmath226 , and @xmath227 by this new affine transition . let us continue to call @xmath121 this new @xmath0 conservative diffeomorphism , and @xmath227 the new transition . in order to get the invariance of the strong stable foliation , note that by the previous construction , the center unstable foliation is preserved by backward iterations . the backward iterations of the strong stable foliation approach the strong stable foliation in @xmath19 . if necessary , we may replace @xmath87 by a large backward iterate , and @xmath114 by a corresponding iterate in @xmath19 . there is a matrix @xmath46 with @xmath228 close to the identity , that preserves the center - unstable foliation and is such that @xmath229 preserves the strong stable and center - stable foliations . proceeding as in the previous paragraph , we obtain a @xmath1-close diffeomorphism preserving these foliations . we can repeat now the same argument inside the center - unstable foliation in order to get the invariance of the strong unstable and the center foliations . in this way we obtain an affine partially hyperbolic transition @xmath227 preserving @xmath21 , @xmath23 and @xmath22 . analogously we obtain @xmath230 . we only need to show that we can perturb in order to obtain that the transitions @xmath230 and @xmath227 are isometries on the center foliations . now we can replace @xmath230 by @xmath231 with large @xmath232 on a suitable small neighborhood of @xmath233 . there are infinitely many @xmath232 such that the center eigenvalues of @xmath234 are in a bounded away from zero finite interval . considering @xmath235 and @xmath236 sufficiently large , and changing @xmath78 by a point @xmath113 with coordinates of the same form , we obtain a @xmath1-perturbation in a small neighborhood of the segment of orbit @xmath113 , @xmath237 , where @xmath238 , such that the action of @xmath234 in the central direction is an isometry . the perturbation is produced using the pasting lemma for vector fields as in the previous paragraphs . in analogous way we obtain a transition @xmath227 acting as an isometry in the central direction . let @xmath2 be a @xmath0 conservative diffeomorphism having a co - index one cycle with real central eigenvalues associated to the points @xmath4 and @xmath6 . then @xmath2 can be @xmath1-approximated by @xmath0 conservative diffeomorphisms having simple cycles associated to @xmath4 and @xmath6 . | in this work we prove that each @xmath0 conservative diffeomorphism with a pair of hyperbolic periodic points of co - index one can be @xmath1-approximated by @xmath0 conservative diffeomorphisms having a blender . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
johansen s ( 1988 , 1991 ) likelihood ratio ( lr ) test for cointegration rank is a very popular econometric technique . however , it is rarely applied to systems of more than three or four variables . on the other hand , there exist many applications involving much larger systems . for example , davis ( 2003 ) discusses a possibility of applying the test to the data on seven aggregated and individual commodity prices to test lewbel s ( 1996 ) generalization of the hicks - leontief composite commodity theorem . in a recent study of exchange rate predictability , engel , mark , and west ( 2015 ) contemplate a possibility of determining the cointegration rank of a system of seventeen oecd exchange rates . banerjee , marcellino , and osbat ( 2004 ) emphasize the importance of testing for no cross - sectional cointegration in panel cointegration analysis ( see breitung and pesaran ( 2008 ) and choi ( 2015 ) ) , and the cross - sectional dimension of modern macroeconomic panels can easily be as large as forty . the main reason why the lr test is rarely used in the analysis of relatively large systems is its poor finite sample performance . even for small systems , the test based on the asymptotic critical values does not perform well ( see johansen ( 2002 ) ) . for large systems , the size distortions become overwhelming , leading to severe over - rejection of the null in favour of too much cointegration as shown in many simulation studies , including ho and sorensen ( 1996 ) and gonzalo and pitarakis ( 1995 , 1999 ) . in this paper , we study the asymptotic behavior of the sample canonical correlations that the lr statistic is based on , when the number of observations and the system s dimensionality go to infinity simultaneously and proportionally . we show that the empirical distribution of the squared sample canonical correlations almost surely converges to the so - called _ wachter distribution _ which also arises , albeit with different parameters , as the limit of the empirical distribution of the squared sample canonical correlations between two independent high - dimensional white noises ( see wachter ( 1980 ) ) . our analytical findings explain the observed over - rejection of the null hypothesis by the lr test , shed new light on the workings and limitations of the bartlett - type correction approach to the problem ( see johansen ( 2002 ) ) , and lead us to propose a very simple graphical device , similar to the scree plot , for a preliminary analysis of the validity of cointegration hypotheses in large vector autoregressions . the basic framework for our analysis is standard . consider a @xmath0-dimensional var in the error correction form @xmath1 where @xmath2 and @xmath3 are vectors of deterministic terms and zero - mean gaussian errors with unconstrained covariance matrix , respectively . the lr statistic for the test of the null hypothesis of no more than @xmath4 cointegrating relationships between the @xmath0 elements of @xmath5 against the alternative of more than @xmath4 such relationships is given by @xmath6 where @xmath7 is the sample size , and @xmath8 are the squared sample canonical correlation coefficients between residuals in the regressions of @xmath9 and @xmath10 on the lagged differences @xmath11 @xmath12 and the deterministic terms . in the absence of the lagged differences and deterministic terms , the @xmath13 s are the eigenvalues of @xmath14 where @xmath15 and @xmath16 are the sample covariance matrices of @xmath9 and @xmath17 respectively , while @xmath18 is the cross sample covariance matrix . more substantively , @xmath19 is the largest possible squared sample correlation coefficient between arbitrary linear combinations of the entries of @xmath9 and the entries of @xmath17 @xmath20 is the largest squared correlation among linear combinations restricted to be orthogonal to those yielding @xmath19 , and so on ( see muirhead ( 1982 ) , ch . johansen ( 1991 ) shows that the asymptotic distribution of @xmath21 under the asymptotic regime where @xmath22 while @xmath0 remains fixed , can be expressed in terms of the eigenvalues of a matrix whose entries are explicit functions of a @xmath23-dimensional brownian motion . unfortunately , for relatively large @xmath0 , this asymptotics does not produce good finite sample approximations , as evidenced by the over - rejection phenomenon mentioned above . therefore , in this paper , we consider a _ simultaneous _ asymptotic regime @xmath24 where both @xmath0 and @xmath7 diverge to infinity so that @xmath25 , \label{asymptotic regime}\ ] ] while @xmath0 remains no larger than @xmath7 . our monte carlo analysis shows that the corresponding asymptotic approximations are relatively accurate even for such small sample sizes as @xmath26 and @xmath27 . the basic specification for the data generating process ( [ model ] ) that we consider has @xmath28 . in the next section , we discuss extensions to more general vars with low - rank @xmath29 matrices and additional common factor terms . we also explain there that our main results hold independently from whether a deterministic vector @xmath2 with fixed or slowly - growing dimension is present or absent from the var . our study focuses on the behavior of the empirical distribution function ( d.f . ) of the squared sample canonical correlations,@xmath30 where @xmath31 denotes the indicator function . we find that , under the null of @xmath4 cointegrating relationships , as @xmath24 while @xmath32 almost surely ( a.s . ) , @xmath33 where @xmath34 denotes the weak convergence of d.f.s ( see billingsley ( 1995 ) , p.191 ) , and @xmath35 denotes the _ wachter _ d.f . with parameters @xmath36 and @xmath37 . the _ wachter distribution _ was derived by wachter ( 1980 ) as the limit of the empirical distribution of the eigenvalues of the multivariate beta matrix of growing dimension and degrees of freedom . it has a simple density , which is introduced in the next section , and , for @xmath38 and/or @xmath39 point masses at zero and/or one , respectively . weak convergence ( [ convergence ] ) and the fact that the squared sample canonical correlations are no larger than unity imply the a.s . convergence of averages @xmath40 for any @xmath41 which is bounded and continuous on @xmath42 $ ] . by definition , the likelihood ratio statistic scaled by @xmath43 has this form ( with omitted first @xmath4 summands ) , where @xmath44 is continuous but unbounded function . therefore , ( [ convergence ] ) can guarantee an a.s . asymptotic lower bound for the scaled lr statistic . for the lr statistic scaled by @xmath45 we have , almost surely,@xmath46 in contrast , we show that , under the ( standard ) asymptotic regime where@xmath22 while @xmath0 is held fixed@xmath47 @xmath48 concentrates around @xmath49 for relatively large @xmath0 . ) , our weak convergence results only guarantee that @xmath50 is a lower bound , but we conjecture that it is also the limit of the scaled lr statistic as first @xmath22 and then @xmath51 . this conjecture is supported by monte carlo evidence . ] a direct calculation reveals that @xmath49 is smaller than the lower bound ( [ bound ] ) , for all @xmath52 , with the gap growing as @xmath53 increases . that is , the standard asymptotic distribution of the lr statistic is centered at a too low level , especially for relatively large @xmath0 . this explains the tendency of the asymptotic lr test to over - reject the null . the reason for the poor centering delivered by the standard asymptotic approximation is that it classifies terms @xmath54 in the asymptotic expansion of the likelihood ratio statistic as @xmath55 when @xmath0 is relatively large , such terms can substantially contribute to the finite sample distribution of the statistic , but will be ignored as asymptotically negligible . in contrast , the _ simultaneous _ _ asymptotics _ classifies all terms @xmath54 as @xmath56 they are not ignored asymptotically , which improves the centering of the simultaneous asymptotic approximation relative to the standard one . it is possible to use bound ( [ bound ] ) , with @xmath53 replaced by @xmath57 , to construct a bartlett - type correction factor for the standard lr test . as we show below , for @xmath58 the value of such a theoretical correction factor is very close to the simulation - based factor described in johansen , hansen and fachin ( 2005 ) . however , for larger @xmath57 , the values diverge , which may be caused by the fact that johansen , hansen and fachin s ( 2005 ) simulations do not consider combinations of @xmath0 and @xmath7 with @xmath59 and the functional form that they use to fit the simulated correction factors does not work well uniformly in @xmath57 . the weak convergence result ( [ convergence ] ) can be put to a more direct use by comparing the quantiles of the empirical distribution of the squared sample canonical correlations with the quantiles of the limiting wachter distribution . under the null , the former quantiles plotted against the latter ones should form a 45@xmath60 line , asymptotically . deviations of such a wachter quantile - quantile plot from the line indicate violations of the null . creating wachter plots requires practically no additional computations beyond those needed to compute the lr statistic , and we propose to use this simple graphical device for a preliminary analysis of cointegration in large vars . to the best of our knowledge , our study is the first to derive the limit of the empirical d.f . of the squared sample canonical correlations between random walk @xmath10 and its innovations @xmath9 . wachter ( 1980 ) shows that @xmath35 is the weak limit of the empirical d.f . of the squared sample canonical correlations between @xmath61- and @xmath62-dimensional independent gaussian white noises with the size of the sample @xmath63 when @xmath64 so that @xmath65 and @xmath66 . yang and pan ( 2012 ) show that wachter s ( 1980 ) result holds without the gaussianity assumption for i.i.d . data with finite second moments . our proofs do not rely on those previous results . the values of parameters @xmath36 and @xmath37 in ( [ convergence ] ) imply that the limiting d.f . for the case of @xmath7 observations of @xmath0-dimensional random walk and its innovations , that we consider in this paper , is the same as the limiting d.f . for the case of @xmath67 observations of two independent white noises - one @xmath0-dimensional and the other @xmath68-dimensional . it is tempting to think that there exists a deep connection between the two cases , even though we were unable to uncover it so far . our paper opens up a new direction for the asymptotic analysis of panel var cointegration tests based on the sample canonical correlations . one such test is developed in larsson and lyhagen ( 2007 ) . it generalizes larsson , lyhagen , and lothgren ( 2001 ) and groen and kleibergen ( 2003 ) by allowing for cross - unit cointegration , which is important from the empirical perspective . larsson and lyhagen ( 2007 ) are reluctant to recommend their test for large vars and suggest that for the analysis of relatively large panels it may be better to rely on tighter parameterized models , such as that of bai and ng ( 2004 ) . in the recent review of the panel cointegration literature , choi ( 2015 ) expresses a related concern that , with the large number of cross - sectional units , larsson and lyhagen s test may not work well even with the bartlett s correction . we speculate that the larsson - lyhagen test , as well as johansen s lr test , based on the _ simultaneous _ asymptotics would work well in panels with comparable cross - sectional and temporal dimensions . the results of this paper can be used to describe only the appropriate centering of the corresponding test statistics . the next step would be to derive the _ simultaneous _ asymptotic distribution of scaled deviations of such statistics from the centering values . we conjecture that the _ simultaneous _ asymptotic distribution of @xmath21 is gaussian , as is often the case for averages of regular functions of eigenvalues of large random matrices ( see bai and silverstein ( 2010 ) and paul and aue ( 2014 ) ) . we are currently undertaking work to validate this conjecture . the rest of this paper is structured as follows . in section 2 , we prove the convergence of @xmath69 to the _ wachter _ d.f . and use this result to derive the asymptotic lower bound for @xmath70 section 3 derives the sequential limit of the empirical d.f . of the squared sample canonical correlations as , first @xmath22 and then @xmath51 . it then uses differences between the obtained sequential asymptotic limit and the simultaneous limit derived in section 2 to explain the over - rejection phenomenon , and to design a theoretical bartlett - type correction factor for the lr statistic in high - dimensional vars . section 4 contains a monte carlo study that confirms good finite sample properties of the wachter asymptotic approximation . it also illustrates the proposed wachter quantile - quantile plot technique using a relatively high - dimensional macroeconomic panel . section 5 concludes and points out directions for future research . all proofs are given in the appendix . consider the following basic version of ( [ model])@xmath71 with @xmath72-dimensional vector of deterministic regressors @xmath2 . let @xmath73 and @xmath74 be the vectors of residuals from the ols regressions of @xmath9 on @xmath75 and @xmath10 on @xmath75 respectively . define@xmath76 and let @xmath8 be the eigenvalues of @xmath77 the main goal of this section is to establish the a.s . weak convergence of the empirical d.f . of the @xmath13 s to the wachter d.f . , under the null of @xmath4 cointegrating relationships , when @xmath24 . the wachter distribution with d.f . @xmath35 and parameters @xmath78 has density@xmath79 on @xmath80 \subseteq\left [ 0,1\right ] $ ] with@xmath81 and atoms of size @xmath82 at zero , and @xmath83 at unity . we shall assume that model ( [ econometricians model ] ) may be misspecified in the sense that the true data generating process is described by the following generalization of ( [ model])@xmath84 where @xmath85 @xmath86 are still i.i.d . @xmath87 with arbitrary @xmath88 @xmath89 but @xmath90 is not necessarily unity , and @xmath91 is a @xmath92-dimensional vector of deterministic or stochastic variables that does not necessarily coincide with @xmath2 . for example , some of the components of @xmath91 may be common factors not observed and not modelled by the econometrician . further , we do not put any restrictions on the roots of the characteristic polynomial associated with ( [ general model ] ) . in particular , explosive behavior and seasonal unit roots are allowed . finally , no constraints on @xmath93 and the initial values @xmath94 , apart from the asymptotic requirements on @xmath92 and @xmath90 as spelled out in the following theorem , are imposed . [ main]suppose that the data are generated by ( [ general model ] ) , and let @xmath95 @xmath96 be the eigenvalues of @xmath14 where @xmath97 are as defined in ( [ sss ] ) . further , let @xmath69 be the empirical d.f . of the @xmath13 s , and let @xmath98 $ ] . if@xmath99 as @xmath24 while @xmath0 remains no larger than @xmath7 , then , almost surely,@xmath100 condition ( [ wistles ] ) requires the number @xmath72 of deterministic regressors in the econometrician s model ( [ econometricians model ] ) , the dimensionality @xmath92 of @xmath91 , the number @xmath4 of the cointegrating relationships under the null , the order @xmath90 of the data generating var , and the dimensionality of the union of the column spaces of the matrix coefficients on further lags in ( [ general model ] ) to be either fixed or growing less than proportionally to the dimensionality @xmath0 or , equivalently , to the sample size @xmath7 . this condition rules out situations where some or all lags which are omitted from the econometrician s model ( [ econometricians model ] ) have full rank coefficients @xmath29 . the simplest special situation where ( [ wistles ] ) is clearly satisfied corresponds to the pure random walk data @xmath101 . the reason why the limit of the empirical d.f . @xmath102 does not change when the data generating process ( [ general model ] ) changes so that ( [ wistles ] ) remains true is that the corresponding changes in the matrix @xmath103 have rank that is less than proportional to @xmath0 ( and to @xmath7 ) . by the so - called rank inequality ( theorem a43 in bai and silverstein ( 2010 ) ) , the lvy distance between the empirical d.f . of eigenvalues corresponding to versions of @xmath103 that differ by a matrix of rank @xmath104 is no larger than @xmath105 which converges to zero as @xmath106 since the lvy distance metrizes the weak convergence ( see billingsley ( 1995 ) , problem 14.5 ) , the limiting d.f . is not affected . for further details , see the proof of theorem [ main ] in the appendix . in standard cases where @xmath2 is represented by @xmath107 it is customary to impose restrictions on @xmath108 so that there is no quadratic trend in @xmath5 ( see johansen ( 1995 ) , ch . then , the lr test of the null of @xmath4 cointegrating relationships is based on the eigenvalues of @xmath109 defined similarly to @xmath103 by replacing @xmath10 with @xmath110 and regressing @xmath9 and @xmath110 on constant only to obtain @xmath73 and @xmath111 the empirical distribution function of so modified eigenvalues still converges to @xmath112 because the difference between matrices @xmath113 and @xmath103 has small rank . figure [ illustrationmain ] shows quantile plots of the wachter distribution with parameters @xmath114 and @xmath115 for different values of @xmath53 . for @xmath116 the dimensionality of the data constitutes 20% of the sample size . the corresponding wachter limit of @xmath117is supported on @xmath118 $ ] . in particular , we expect @xmath19 be larger than @xmath119 for large @xmath120 even in the absence of any cointegrating relationships . for @xmath121 the upper boundary of support of the wachter limit is unity . this accords with gonzalo and pitarakis ( 1995 , lemma 2.3.1 ) finding that as @xmath122 @xmath123 for @xmath124 the wachter limit has mass @xmath125 at unity . illustrationmain.eps wachter ( 1980 ) derives @xmath35 as the weak limit of the empirical d.f . of eigenvalues of the @xmath0-dimensional beta matrix @xmath126 with @xmath127 degrees of freedom as @xmath128 so that @xmath129 and @xmath130 the eigenvalues of multivariate beta matrices are related to many important concepts in multivariate statistics , including canonical correlations , multiple discriminant ratios , and manova . in particular , the squared sample canonical correlations between @xmath61- and @xmath62-dimensional independent gaussian samples of size @xmath131 are jointly distributed as the eigenvalues of @xmath132 where @xmath133 and @xmath134 . therefore , their empirical d.f . weakly converges to @xmath135 with @xmath136 and @xmath137 as mentioned above . since the squared canonical correlations in theorem [ main ] are between random walk and its innovations rather than independent white noises , the convergence to the wachter distribution came to us as a pleasant surprise . in the context of multiple discriminant analysis , wachter ( 1976b ) proposes to use a quantile - quantile ( qq ) plot , where the multiple discriminant ratios are plotted against quantiles of @xmath138 , as a simple graphical method that helps one recognize hopeless from promising analyses at an early stage . a plot that clearly deviates from the 45@xmath60 line suggests that the data are at odds with the null hypothesis of the homogeneous population , and a further analysis of the heterogeneity is useful . nowadays , such qq plots are called _ wachter plots _ ( see johnstone ( 2001 ) ) . theorem [ main ] implies that the wachter plot can be used as a simple preliminary assessment of cointegration hypotheses in large vars . as an illustration , figure [ wachterplot ] shows a wachter plot of the simulated sample squared canonical correlations corresponding to a @xmath139-dimensional var(1 ) model ( [ econometricians model ] ) with @xmath140 so that there are three white noise and seventeen random walk components of @xmath5 . no deterministic terms are included . we set @xmath141 so that @xmath142 . the graph clearly shows three canonical correlations that destroy the 45@xmath60 line fit , so that the null hypothesis of no cointegration is compromised . wachterplot.eps theorem [ main ] does not provide any explanation to the fact that exactly three canonical correlations deviate from the 45@xmath60 line in figure [ wachterplot ] . to interpret deviations of the wachter plots from the 45@xmath60 line , it is desirable to investigate behavior of @xmath143 under various alternatives . so far , we were able to obtain a clear result only for the extreme alternative , where @xmath5 is a vector of independent white noises . under such an alternative , @xmath144 we plan to publish a full proof of this and some related results elsewhere . interestingly , for @xmath121 the wachter limits ( [ 5again ] ) and ( [ white noise ] ) corresponding to random walk and white noise nulls , respectively , coincide . hence , as @xmath53 approaches @xmath145 not only the largest sample canonical correlation converges to one and the lr test breaks down , but also the wachter plot looses the ability to differentiate between opposite cointegration hypotheses . for smaller values of @xmath146 however , the wachter limits ( [ 5again ] ) and ( [ white noise ] ) become well separated . we provide monte carlo analysis of the behavior of @xmath147 under some alternative hypotheses in section 4 below . the almost sure weak convergence of @xmath69 established in theorem [ main ] implies the almost sure convergence of bounded continuous functionals of @xmath148 an example of such a functional is the scaled pillai - bartlett statistic for the null of no more than @xmath4 cointegrating relationships ( see gonzalo and pitarakis ( 1995))@xmath149 which is asymptotically equivalent to the lr statistic under the standard asymptotic regime where @xmath0 is fixed and @xmath22 . since , by definition , @xmath150 , $ ] we have@xmath151 where @xmath41 is the bounded continuous function@xmath152{cc}0 & \text{for } \lambda<0\\ \lambda & \text{for } \lambda\in\left [ 0,1\right ] \\ 1 & \text{for } \lambda>1 . \end{array } \right . .\ ] ] as long as @xmath153 as @xmath154 the second term on the right hand side of ( [ pbrepresentation ] ) converges to zero . therefore , theorem [ main ] implies that @xmath155 almost surely converges to @xmath156 a direct calculation based on ( [ densityw ] ) , which we report in the supplementary appendix , yields the following corollary . [ pb]under the assumptions of theorem [ main ] , as @xmath157 almost surely,@xmath158 a similar analysis of the lr statistic is less straightforward because@xmath159 and @xmath160 is unbounded on @xmath161 .$ ] in fact , for @xmath162 @xmath21 is ill - defined because a non - negligible proportion of the squared sample canonical correlations exactly equal unity . however for @xmath163 we can obtain the almost sure asymptotic lower bound on @xmath164 note that for such @xmath146 the upper bound of the support of @xmath165 equals @xmath166 . let @xmath167{cc}0 & \text{for } \lambda<0\\ \log(1-\lambda ) & \text{for } \lambda\in\left [ 0,b_{+}\right ] \\ \log(1-b_{+ } ) & \text{for } \lambda > b_{+}. \end{array } \right . \label{truncated log}\ ] ] clearly , @xmath168 is a bounded continuous function and@xmath169 hence , we have the following a.s . lower bound on @xmath170 ( the corresponding calculations are reported in the supplementary appendix ) . [ lrsim]under the assumptions of theorem [ main ] , for @xmath163 as @xmath154 almost surely,@xmath171 [ conjectureremark]we conjecture that the lower bound reported in the corollary is , in fact , the a.s . limit of @xmath164 to prove this conjecture , one needs to show that @xmath172 is almost surely bounded away from unity so that the unboundedness of @xmath173 is not consequential . we leave this as an important topic for future research . corollaries [ pb ] and [ lrsim ] suggest appropriate centering points for pb and lr statistics scaled by @xmath174 for relatively large and comparable @xmath0 and @xmath175 as we show in the next section , the standard asymptotic distribution of the scaled pb and lr statistics are likely to concentrate around very different points when @xmath0 becomes large . as will be seen below , this difference sheds new light on the over - rejection phenomenon discussed above and on the workings and limitations of the bartlett correction for the lr statistic . to study the concentration of the standard asymptotic distributions of the scaled pb and lr statistics as @xmath0 grows , we will consider the _ sequential _ asymptotic regime , where first @xmath176 and then @xmath51 . to obtain useful results under the sequential asymptotics , we shall study eigenvalues of the scaled matrix@xmath177 note that under the simultaneous asymptotic regime @xmath154 the asymptotic behavior of the scaled and unscaled eigenvalues is the same up to the factor @xmath178 . however , as first @xmath22 while @xmath0 remains fixed , the unscaled eigenvalues converge to zero , while scaled ones do not . we shall denote the empirical d.f . of eigenvalues of the scaled matrix as @xmath179 . without loss of generality , we focus on the case of simple data generating process @xmath180 and on the situation , where the econometrician does not include any deterministic regressors in his or her model , that is @xmath181 . there is no loss of generality in such simplifications because , as follows from lemma [ misspec ] and the rank inequality used in the proof of lemma [ rank ] in the appendix , the lvy distance between the versions of @xmath179 that correspond to the simplified and the general cases is bounded from above by a fixed multiple of @xmath182 . we shall assume that the latter expression goes to zero as @xmath183 therefore , whatever the sequential asymptotic limit of @xmath184 is under the above simplification , it must also be the sequential asymptotic limit under the general case . for simplicity , in the rest of this section , we shall assume that @xmath185 and will consider statistics @xmath186 rather than more general @xmath70 under the above simplifications , johansen s ( 1988 , 1991 ) results imply that , as @xmath22 while @xmath0 is held fixed , the eigenvalues of the scaled matrix ( [ rawmatrix ] ) jointly converge in distribution to the eigenvalues of@xmath187 where @xmath188 is a @xmath0-dimensional brownian motion . we denote the eigenvalues of ( [ johansenlimit ] ) as @xmath189 and their empirical d.f . as @xmath190 it is not unreasonable to expect that , as @xmath51 , @xmath191 becomes close to the limit of the empirical distribution of eigenvalues of ( [ rawmatrix ] ) under a simultaneous , rather than sequential , asymptotic regime @xmath192 where @xmath193 is close to zero . we shall denote such a limit as @xmath194 this expectation turns out to be correct in the sense that the following theorem holds . [ levy ] let @xmath195 be the weak limit as @xmath196 of @xmath197 then , as @xmath198 @xmath199 weakly converges to @xmath200 in probability . importantly , the weak limit @xmath195 is not the wachter d.f . instead , the following proposition holds . [ mpproposition ] @xmath195 corresponds to a distribution supported on @xmath201 $ ] with @xmath202 and having density@xmath203 a reader familiar with large random matrix theory ( see bai and silverstein ( 2010 ) ) might recognize that @xmath195 is the cumulative distribution function of the continuous part of a special case of the marchenko - pastur distribution ( marchenko and pastur ( 1967 ) ) . the general marchenko - pastur distribution has density@xmath204 over @xmath201 $ ] with @xmath205 and a point mass @xmath206 at zero . density ( [ mp density ] ) is two times @xmath207 with @xmath208 and @xmath209 the multiplication by two is needed because the mass @xmath210 at zero is not a part of the distribution @xmath211 . recall that , as @xmath22 while @xmath0 remains fixed , the lr statistic converges in distribution to @xmath0 times the trace of matrix ( [ johansenlimit]):@xmath212 on the other hand , according to theorem [ levy ] , for any @xmath213 and all sufficiently large @xmath214@xmath215 a direct calculation , which we report in the supplementary appendix , shows that @xmath216 hence , we have the following corollary . [ table reduction]as first @xmath176 and then @xmath217 the lower probability bound on @xmath218 is unity in the following sense . as @xmath22 while @xmath0 is held fixed , @xmath218 converges in distribution to @xmath219 further , for any @xmath220 and all sufficiently large @xmath214 the probability that @xmath221 is no smaller than @xmath222 is no smaller than @xmath223 the reason why we only claim the lower bound on @xmath224 is that theorem [ levy ] is silent about the behavior of the individual eigenvalues @xmath189 the largest of which may , in principle , quickly diverge to infinity . we suspect that @xmath49 is not just the lower bound , but also the probability limit of @xmath225 , so that the sequential probability limit of @xmath218 is unity . verification of this conjecture requires more work , similar to that discussed in remark [ conjectureremark ] . corollary [ table reduction ] is consistent with the numerical finding of johansen , hansen and fachin ( 2005 , table 2 ) that , as @xmath7 becomes large while @xmath0 is being fixed , the sample mean of the lr statistic is well approximated by a polynomial @xmath226 ( see also johansen ( 1988 ) and gonzalo and pitarakis ( 1995 ) ) . the value of @xmath227 depends on how many deterministic regressors are included in the var . our theoretical result captures only the ` highest order ' sequential asymptotic behavior of the lr statistic , which remains ( bounded below by ) @xmath228 independent on the number of the deterministic regressors . another piece of numerical support for @xmath228 being not only the lower bound but also the first order sequential asymptotic approximation to the lr statistic is provided by the tables of the asymptotic critical values for johansen s lr test ( see , for example , mackinnon , haug and michelis ( 1999 ) ) . the critical values in such tables become uncomfortably large for @xmath229 . of course , the reason for such an unpleasant growth is that those critical values are of order @xmath228 . the transformation@xmath230 makes the lr statistic ` well - behaved ' under the sequential asymptotics . the division by @xmath0 reduces the ` second order behavior ' to @xmath231 while subtracting @xmath68 eliminates the remaining explosive ` highest order term ' . we report the corresponding transformed 95% critical values alongside the original ones in table [ cv ] . the transformed critical values resemble 97 - 99 percentiles of @xmath232 . since the lr test is one - sided , the resemblance is coincidental . however , we do expect the sequential asymptotic distribution of the transformed lr statistic ( as well as its simultaneous asymptotic distribution ) to be normal ( possibly with non - zero mean and non - unit variance ) . our expectation is based on the fact that @xmath233 behaves as the eigenvalue average ( see ( [ tlimit ] ) ) , which is a special case of the so - called linear spectral statistic . the asymptotic normality of linear spectral statistics for relatively simple classes of high - dimensional random matrices is a well established result in the large random matrix theory ( see bai and silverstein ( 2010 ) ) . extending it to the linear spectral statistics of matrices of form ( [ johansenlimit ] ) is left as an important direction for future research . [ c]l|l|l@xmath0 & unadjusted cv & cv/@xmath234 + & & + & & + & & + & & + & & + & & + & & + & & + & & + & & + & & + & & + in this subsection , let us assume that the following conjecture holds . [ conjecture]the simultaneous and sequential asymptotic lower bounds for the scaled lr statistics derived in corollaries [ lrsim ] and [ table reduction ] represent the corresponding simultaneous and sequential asymptotic limits . specifically , for @xmath235,@xmath236@xmath237 figure [ overrejection ] plots the right hand side of ( [ simlim ] ) against the value of @xmath238 as demonstrated by the monte carlo analysis of the next section , in finite samples with comparable values of @xmath0 and @xmath7 , simultaneous asymptotics provides a better approximation to the finite sample behavior of the lr statistic than the sequential asymptotics . therefore , ` typical ' finite sample values of the lr statistic are concentrated around the solid line in figure [ overrejection ] , and above the dashed line , which represents the points of concentration of the ` standard ' asymptotic critical values for the lr test . in other words , the standard asymptotic distribution of the lr statistic is centered at a too low level . this leads to the over - rejection of the null of no cointegration . overrejection1.eps gonzalo and pitarakis ( 1995 ) propose an interesting approach to address the problem . using monte carlo , they find that , in contrast to the lr test , the pillai - bartlett test based on the pb statistic under - rejects the null . therefore , they propose to test cointegration hypotheses using the average of the lr and pb statistics . according to corollary [ pb ] , under the simultaneous asymptotics @xmath239 almost surely . this convergence holds independent on whether conjecture [ conjecture ] is true or not . the fact that @xmath240 is smaller than one , explains the under - rejection of the test based on the pb statistic . more interestingly , the average of the simultaneous asymptotic limits of the lr and pb statistics ( divided by @xmath228 ) turns out to be numerically close to one , and hence to the point of the concentration of the standard critical values ( divided by @xmath228 ) , at least for @xmath241 figure [ pbaver ] shows such an average . this explains the much better performance of the ( lr+pb)/2 test relative to the lr test in gonzalo and pitarakis ( 1995 ) monte carlo experiments . pbaverage.eps a more systematic and popular approach to addressing the over - rejection problem is based on the bartlett - type correction of the lr statistic . it was explored in much detail in various important studies , including johansen ( 2002 ) . the idea is to scale the lr statistic so that its finite sample distribution better fits the asymptotic distribution of the unscaled statistic . specifically , let @xmath242 be the mean of the asymptotic distribution under the fixed-@xmath0 , large-@xmath7 asymptotic regime . then , if the finite sample mean , @xmath243 , satisfies@xmath244 the scaled statistic is defined as @xmath245 by construction , the match between the scaled mean and the original asymptotic mean is improved by an order of magnitude . although , as shown by jensen and wood ( 1997 ) in the context of unit root testing , the match between higher moments does not improve by an order of magnitude , it may become substantially better ( see nielsen ( 1997 ) ) . a theoretical analysis of the adjustment factor @xmath246 can be rather involved . in general , @xmath247 will depend not only on @xmath0 , but also on all the parameters of the var . however , for gaussian var(1 ) without deterministic terms , under the null of no cointegration , @xmath247 depends only on @xmath0 . for @xmath248 the exact expression for @xmath247 was derived in larsson ( 1998 ) . given the difficulty of the theoretical analysis of @xmath249 johansen ( 2002 ) proposes to numerically evaluate the bartlett correction factor @xmath250 by simulation . johansen , hansen and fachin ( 2005 ) simulate @xmath251 for various values of @xmath252 and @xmath253 and fit a function of the form@xmath254 \right\}\ ] ] to the obtained results . for relatively large values of @xmath255 the term @xmath256 $ ] in the above expression is small . when it is ignored , the fitted function becomes particularly simple:@xmath257 our simultaneous and sequential asymptotic results shed light on the workings of @xmath258 given that conjecture [ conjecture ] holds,@xmath259 therefore , for non - negligible @xmath260 we expect @xmath251 to be well approximated by @xmath261 where @xmath262 is the finite sample analog of @xmath263 figure [ bartlettnew ] superimposes the graphs of @xmath264 and @xmath265 as functions of @xmath266 for @xmath267 there is a strikingly good match between the two curves , with the maximum distance between them @xmath268 . for @xmath269 the quality of the match quickly deteriorates . this can be explained by the fact that all @xmath120-pairs used in johansen , hansen and fachin s ( 2005 ) simulations are such that @xmath270 . bartlettnew.eps further , the good match between @xmath265 and @xmath264 observed for @xmath270 would be impossible had johansen , hansen and fachin s ( 2005 ) specified the bartlett correction factor as a linear function of @xmath57 . note that the standard theoretical choice for the bartlett correction factor , @xmath246 from ( [ bartlett theory ] ) , can be viewed as a linear function of @xmath57 with a slope possibly varying with @xmath0 . this is obvious when @xmath271 is represented as @xmath272 with @xmath273 . figure [ bartlettnew ] shows that such theoretical correction factors can not work well uniformly with respect to @xmath57 . uniformly good correction factors must include terms @xmath54 with @xmath274 . under the fixed-@xmath214 large-@xmath7 asymptotics , such terms are of lower order than @xmath275 but under the simultaneous asymptotics , they are of order @xmath276 . although the bartlett - type correction approach may deliver good results for high - dimensional systems with carefully chosen correction factor , we believe that tests based on the simultaneous asymptotics of the appropriately scaled and centered lr statistic would be preferable for relatively large @xmath0 . in this section , we describe results of small - scale monte carlo experiments that assess the finite sample quality of the wachter asymptotic approximation . in addition , we illustrate the wachter qq plot technique using a macroeconomic dataset of relatively high dimensions . first , we generate pure random walk data with zero starting values for @xmath277 and @xmath278 throughout this section , the analysis is based on 1000 monte carlo replications . the generated random walk data are ten - dimensional so that there are ten corresponding squared sample canonical correlations , @xmath13 . figure [ mcboxplot ] shows the tukey boxplots summarizing the mc distribution of each of the @xmath95 @xmath279 ( sorted in the ascending order throughout this section)@xmath280 the boxplots are superimposed with the quantile function of the wachter limit with @xmath142 for the left panel and @xmath281 for the right panel . precisely , for @xmath282 we show the value the @xmath283 quantile of the wachter limit . for @xmath284 these are the 5-th,15-th, ... ,95-th quantiles of @xmath285 even for such small values of @xmath0 and @xmath255 the theoretical quantiles track the location of the mc distribution of the empirical quantiles very well . the smallest sample canonical correlation is an exception . its distribution lies mostly below the corresponding theoretical quantile . the dispersion of the mc distributions around the theoretical quantile is quite large for the chosen small values of @xmath0 and @xmath175 to see how such a dispersion changes when @xmath0 and @xmath7 increase while @xmath57 remains fixed , we generated pure random walk data with @xmath286 and @xmath287 for @xmath288 , and with @xmath289 and @xmath290 for @xmath291 instead of reporting the tukey boxplots , we plot only the 5-th and 95-th percentiles of the mc distributions of the @xmath95 @xmath292 against @xmath293 quantiles of the corresponding wachter limit . the plots are shown on figure [ mcqqplot ] . mcboxplot.eps mcqqplot.eps we see that the [ 5%,95% ] ranges of the mc distributions of @xmath294 are still considerably large for @xmath295 these ranges become much smaller for @xmath296 interestingly , the distribution of @xmath19 remains below the wachter limit even for @xmath296 this does not contradict our theoretical results because a weak limit of the empirical distribution of @xmath13 s is not affected by an arbitrary change in a finite ( or slowly growing ) number of them . in fact , we find it somewhat surprising that only the distribution of @xmath19 is not well - alligned with the derived theoretical limit . our proofs are based on several low rank alterations of the matrix @xmath103 , and there is nothing in them that guarantee that only one eigenvalue of @xmath103 behaves in a special way . in future work , it would be interesting to investigate the behavior of @xmath19 and other extreme eigenvalues of @xmath103 theoretically . next , we explore the effect of the deterministic regressor on the quality of the wachter approximation . we generate data with and without constant in the data generating process ( [ general model ] ) . that is , we consider two cases : @xmath297 and @xmath298 the coefficient @xmath299 on @xmath91 is a @xmath300 vector independent across different mc replications . we also consider two models ( [ econometricians model ] ) contemplated by the econometrician : one with @xmath301 , and the other with @xmath302 if @xmath303 the econometrican s model is misspecified . figure [ mcmisspec ] shows the wachter plots similar to those reported in figure [ mcqqplot ] . the dimensions of the data are @xmath304 and @xmath305 if the data generating process ( dgp ) contains constant ( @xmath297 ) , but the econometrician does not include it in his or her model , then the largest @xmath306 @xmath307 start to significantly deviate from the 45@xmath60 line on the wachter plot ( lower right panel ) . if the econometrician s model is over - specified ( lower left panel ) , there are no dramatic deviations from the line . mcqqdeterministic.eps our next monte carlo experiment simulates data that are not random walk . instead , the data are stationary var(1 ) with zero mean , zero initial value , and @xmath308 we consider three cases of @xmath309 @xmath310 and @xmath311 figure [ mcalternatives ] shows the wachter plots with solid lines representing 5th and 95-th percentiles of the mc distributions of @xmath312 plotted against the @xmath313 quantiles of the corresponding wachter limit . the dashed line correspond to the null case where the data are pure random walk ( shown for comparison ) . mcqqalternatives.eps the lower panel of the figure corresponds to the most persistent alternative with @xmath314 samples with @xmath304 seem to be too small to generate substantial differences in the behavior of wachter plots under the null and under such persistent alternatives . the less persistent alternative with @xmath315 is easily discriminated against by the wachter plot for @xmath288 ( left panel ) . the discrimination power of the plot for @xmath316 ( central panel ) is weaker . for @xmath317 there is still some discrimination power left , but the location of the wachter plot under alternative switches the side relative to the 45@xmath60 line . the plots easily discriminate against white noise ( @xmath318 ) alternative for @xmath142 and @xmath116 but not for @xmath319 in accordance to the result that we announced above , and plan to publish elsewhere , the wachter limit for @xmath281 approximates equally well the empirical distribution of the squared sample canonical correlations based on random walk and on white noise data . results reported in figure [ mcalternatives ] indicate that for relatively small @xmath0 and @xmath260 wachter plots can be effective in discriminating against alternatives to the null of no cointegration , where the cointegrating linear combinations of the data are not very persistent . further , tests of no cointegration hypothesis that may be developed using simultaneous asymptotics would probably need to be two - sided . it is because the location of the wachter plot under the alternative may switch sides relative to the 45@xmath60 depending on the persistence of the data under the alternative . finally , cases with @xmath53 close to 1/2 must be analyzed with much care . for such cases , the behavior of the sample canonical correlations become similar under extremely different random walk and white noise data generating processes . furthermore , the largest sample canonical correlations are close to unity , which can result in an unstable behavior of the lr statistic . our final mc experiment studies the finite sample behavior of the scaled lr statistic @xmath320 we simulate pure random walk data with @xmath26 and @xmath321 and @xmath7 varying so that @xmath57 equals 1/10,2/10, ... ,5/10 . corollary [ lrsim ] shows that the simultaneous asymptotic lower bound on @xmath218 has form@xmath322 figure [ mclr ] shows the tukey boxplots of the mc distributions of @xmath218 corresponding to @xmath323 with @xmath26 ( left panel ) , and @xmath321 ( right panel ) . the boxplots are superimposed with the plot of the line representing the above displayed formula for the lower bound ( with @xmath53 replaced by @xmath57 ) . for the case @xmath324 we also show ( horizontal dashed line ) the 95% asymptotic critical value ( scaled by @xmath325 ) of the standard johansen trace test taken from mackinnon et al ( 1999 , table ii ) . for @xmath326 critical values for the standard test are not available , and we show the dashed horizontal line at unit height instead . this is the sequential asymptotic lower bound on @xmath327 as established in corollary [ table reduction ] . mclr.eps the reported results support our conjecture that the simultaneous asymptotic lower bound ( [ simultaneous lower bound ] ) is , in fact , the simultaneous asymptotic limit of @xmath218 for @xmath328 interestingly , the bound is located near the center of the mc distribution of the scaled lr statistic even for the case @xmath319 the left panel of figure [ mclr ] illustrates the over - rejection phenomenon . the horizontal dashed line that corresponds to the 95% critical value of the standard test is just above the interquartile range of the mc distribution of @xmath224 for @xmath329 is below this range for @xmath330 and is below all 1000 mc replications of the scaled lr statistic for @xmath331 . although the lower bound ( [ simultaneous lower bound ] ) seems to provide a very good centering point for the scaled lr statistic , the mc distribution of this statistic is quite dispersed around such a center for @xmath332 as discussed above , we suspect that the scaled statistic centered by ( [ simultaneous lower bound ] ) and appropriately rescaled has gaussian simultaneous asymptotic distribution . optimistically , the tukey plots on figure [ mclr ] , that correspond to @xmath235 , look reasonably symmetric although some skewness is present for the left panel where @xmath332 our first example uses @xmath333 quarterly observations ( 1973q2 - 1998q4 , with the initial observation 1973q1 ) on bilateral us dollar log nominal exchange rates for @xmath334 oecd countries : australia , austria , belgium , canada , denmark , finland , france , germany , japan , italy , korea , netherlands , norway , spain , sweden , switzerland , and the united kingdom . the data are as in engel , mark , and west ( 2015 ) , and were downloaded from charles engel s website at http : // www.ssc.wisc.edu / cengel /. that data are available for a longer time period up to 2008q1 , but we have chosen to use only the early sample that does not include the euro period . engel , mark , and west ( 2015 ) point out that log nominal exchange rates are well modelled by random walk , but may be cointegrated , which can be utilized to improve individual exchange rate forecasts relative to the random walk forecast benchmark . they propose to estimate the common stochastic trends in the exchange rates by extracting a few factors from the panel . in principle , the number of factors to extract can be determined using johansen s test for cointegrating rank , but engel , mark , and west ( 2015 ) do not exploit this possibility , referring to ho and sorensen ( 1996 ) that indicates poor performance of the test for large @xmath335 figure [ erfigure ] shows the wachter plot for the log nominal exchange rate data . the squared sample canonical correlations are computed as the eigenvalues of @xmath14 where @xmath97 are defined as in ( [ sss ] ) with @xmath73 and @xmath74 being the demeaned changes and the lagged levels of the log exchange rates , respectively . the dashed lines correspond to the 5-th and 95-th percentiles of the mc distribution of the squared canonical correlation coefficients under the null of no cointegration . precisely , we generated data from model ( [ econometricians model ] ) with @xmath336 @xmath333 , @xmath337 @xmath338 and @xmath108 being i.i.d . @xmath300 vectors across the mc repetitions . log exchange rates for 1973q1 was used as the initial value of the generated series . erfigure.eps the figure shows a mild evidence for cointegration in the data with the largest five @xmath13 s being close to the corresponding 95-th percentiles of the mc distributions . if we interpret this as the existence of five cointegrating relationships in the data , we would be lead to conclude that there are twelve stochastic trends . recall , however , that the ability of the wachter plot to differentiate against highly persistent cointegration alternatives with @xmath339 is very low , so there well may be many more cointegrating relationships in the data . whatever such relationships are , the deviations from the corresponding long - run equilibrium are probably highly persistent as no dramatic deviations from the 45@xmath60 line are present in the wachter plot . very different wachter plots ( shown in figure [ ipcpi ] ) correspond to the log industrial production ( ip ) index data and the log consumer price index ( cpi ) data for the same countries plus the us . these data are still the same as in engel , mark , and west ( 2015 ) . we used the long sample 1973q2:2008q1 @xmath340 because the ip and cpi data are not affected by the introduction of the euro to the same degree as the exchange rate data . for the cpi data , we included both intercept and trend in model ( [ econometricians model ] ) for the first differences because the level data seem to be quadratically trending . the plots clearly indicate that the ip and cpi data are either stationary or cointegrated with potentially many cointegrating relationships , short run deviations from which are not very persistent . in this paper , we consider the simultaneous , large-@xmath214 large-@xmath7 , asymptotic behavior of the squared sample canonical correlations between @xmath0-dimensional random walk and its innovations . we find that the empirical distribution of these squared sample canonical correlations almost surely weakly converges to the so - called _ wachter distribution _ with parameters that depend only on the limit of @xmath57 as @xmath106 in contrast , under the sequential asymptotics , when first @xmath22 and then @xmath198 we establish the convergence in probability to the so - called marchenko - pastur distribution . the differences between the limiting distributions allow us to explain from a theoretical point of view the tendency of the lr test for cointegration to severely over - reject the null when the dimensionality of the data is relatively large . furthermore , we derive a simple analytic formula for the bartlett - type correction factor in systems with relatively large @xmath57 ratio . we propose a quick graphical method , the wachter plot , for a preliminary analysis of cointegration in large - dimensional systems . the monte carlo analysis shows that the quantiles of the wachter distribution constitute very good centering points for the finite sample distributions of the corresponding squared sample canonical correlations . the quality of the centering is excellent even for such small @xmath0 and @xmath7 as @xmath26 and @xmath341 however , for such small values of @xmath0 and @xmath255 the empirical distribution of the squared sample canonical correlation can considerably fluctuate around the wachter limit . as @xmath0 increases to 100 , the fluctuations become numerically very small . our analysis leaves many open questions . first , it is very important to study the fluctuations of the empirical distribution around the wachter limit . we conjecture that linear combinations of reasonably smooth functions of the squared sample canonical correlations , including the @xmath342 used by the lr statistic , will be asymptotically gaussian after appropriate centering and scaling . the centering can be derived from the results obtained in this paper . a proof of the asymptotic gaussianity would require different methods from those used here . we are currently investigating this research direction . further , it would be important to remove the gaussianity assumption on the data . we believe that the existence of the finite fourth moments is a sufficient condition for the validity of the wachter limit . next , it would be interesting to study the simultaneous asymptotic behavior of a few of the largest sample canonical correlations . this may lead to a modification of johansen s maximum eigenvalue test . another interesting direction of research is to study situations where the number of cointegrating relationships under the null is growing proportionally with @xmath0 and @xmath175 the simultaneous asymptotics of the empirical distribution of the squared sample canonical correlations under various alternatives , as well under the null in var(k ) with @xmath343 , also deserves further study . still another , totally different , research direction is to investigate the quality of bootstrap when @xmath0 is large . our own very preliminary analysis indicates that the currently available non - parametric bootstrap procedures ( see , for example , cavaliere , rahbek , and taylor ( 2012 ) ) do not work well for @xmath57 as large as , say , 1/3 . however , further analysis is needed before we can claim any specific results . we hope that this paper opens up an interesting and broad area for future research . let @xmath344 and @xmath345 be distribution functions that may depend on @xmath0 and @xmath7 and are possibly random . we shall call them asymptotically equivalent if the a.s . weak convergence @xmath346 to some non - random d.f . @xmath347 implies similar a.s . weak convergence for @xmath348 and vice versa . let @xmath349 and @xmath350 with @xmath351 be , possibly random , matrices that may depend on @xmath0 and @xmath7 such that @xmath349 and @xmath350 are a.s . positive definite for @xmath352 below , we shall often refer to the following auxiliary lemma . [ rank]if , almost surely , @xmath353 as @xmath24 for @xmath354 then @xmath344 and @xmath355 are asymptoically equivalent , where @xmath344 and @xmath356 are the empirical d.f . of eigenvalues of @xmath357 and @xmath358 respectively . let @xmath359 the a.s . convergence @xmath353 implies the a.s . convergence @xmath360 on the other hand , by the rank inequality ( theorem a43 in bai and silverstein ( 2010 ) ) , @xmath361 where @xmath362 is the lvy distance between @xmath344 and @xmath363 recall that the lvy distance metrizes the weak convergence . therefore , the almost sure convergence @xmath364 yields the asymptotic equivalence of @xmath344 and @xmath365 now , let @xmath366 and @xmath367 and let@xmath368 since @xmath73 and @xmath369 which enter the definition ( [ sss ] ) of @xmath97 , are the residuals in the regressions of @xmath9 on @xmath2 and @xmath10 on @xmath2 , respectively , we have@xmath370 by assumption , @xmath371 as @xmath154 so that by lemma [ rank ] , @xmath102 is asymptotically equivalent to the empirical d.f . of eigenvalues of @xmath372 therefore , we may and will replace @xmath73 and @xmath74 in the definitions ( [ sss ] ) of @xmath97 by @xmath9 and @xmath17 respectively , without loss of generality . furthermore , scaling @xmath97 by @xmath7 does not change the product @xmath373 and thus , in the rest of the proof , we shall work with@xmath374 next , we show that , still without loss of generality , we may replace the data generated process ( [ general model ] ) by pure random walk with zero initial value . indeed , let @xmath375,$ ] where @xmath376 are arbitrary and @xmath5 with @xmath377 are generated by ( [ general model ] ) . further , let @xmath378 be zero vectors , @xmath379 for @xmath380 and @xmath381.$ ] a proof of this lemma is given in the supplementary appendix . it is based on the representation of @xmath5 as a function of the initial values , @xmath383 and @xmath384 ( see theorem 2.1 in johansen ( 1995 ) ) , and requires only elementary algebraic manipulations . lemmas [ misspec ] and [ rank ] together with the assumption ( [ wistles ] ) imply that replacing @xmath385 and @xmath10 in ( [ sssnew ] ) by @xmath386 and @xmath387 respectively , does not change the weak limit of @xmath388 hence , in the rest of the proof of theorem [ main ] , without loss of generality , we shall assume that the data are generated by@xmath389 assuming that @xmath13 s are the eigenvalues of @xmath103 with @xmath97 satisfying ( [ sssnew ] ) and ( [ rw dgp ] ) , we can interpret them as the squared sample canonical correlations between lagged values of a random walk @xmath10 and its current innovations @xmath3 . since the sample canonical correlations are invariant with respect to the multiplication of the data by any invertible matrix , we assume without loss of generality that the variance of @xmath3 equals @xmath390 further , we assume that @xmath7 is even . the case of odd @xmath7 can be analyzed similarly , and we omit it to save space . let @xmath391 $ ] and let @xmath392 be the upper - triangular matrix with ones above the main diagonal and zeros on the diagonal . then @xmath393 , $ ] so that @xmath394 we shall show that the empirical d.f . of the @xmath13 s , @xmath395 is asymptotically equivalent to the empirical d.f . @xmath396 of eigenvalues of @xmath397 where @xmath398 @xmath399 is a diagonal matrix,@xmath400 and @xmath401 is a block - diagonal matrix,@xmath402 here @xmath403 is the 2-dimensional identity matrix , and @xmath404 are defined as follows . let @xmath405 then for @xmath406@xmath407{cc}\cos j\theta & -\sin j\theta\\ \sin j\theta & \cos j\theta \end{array } \right ) , \ ] ] whereas @xmath408 @xmath409 let @xmath411 be the circulant matrix ( see golub and van loan ( 1996 , p.201 ) ) with the first column @xmath412 direct calculations show that @xmath413 and @xmath414 where @xmath415 is the @xmath416-th column of @xmath417 and @xmath418 is the vector of ones . using these identities , it is straightforward to verify that@xmath419 now , let us define @xmath420 using identities ( [ sssu ] ) for @xmath97 and lemma [ rank ] , we conclude that @xmath102 is asymptotically equivalent to @xmath421 , where @xmath422 is the empirical d.f . of the eigenvalues of @xmath423 further , ( [ identity1 ] ) and ( [ identity2 ] ) yield@xmath424 applying lemma [ rank ] one more time , we obtain the asymptotic equivalence of @xmath425 and @xmath426 where @xmath427 is the empirical d.f . of the eigenvalues of @xmath428 with@xmath429 as is well known ( see , for example , golub and van loan ( 1996 ) , chapter 4.7.7 ) , @xmath430 circulant matrices can be expressed in terms of the discrete fourier transform matrices@xmath431 with @xmath405 precisely,@xmath432 where @xmath433 and the star superscript denotes transposition and complex conjugation . for the @xmath434-th diagonal elements of @xmath435 and @xmath436 we have@xmath437 note that @xmath438 for @xmath439 if @xmath7 is even , as we assumed above , then there are @xmath440 pairs @xmath441 , and there is one pair @xmath442 that correspond to@xmath443 define a permutation matrix @xmath444 so that the equal diagonal elements of @xmath445 are grouped in adjacent pairs . precisely , let @xmath446 , where@xmath447{l}1\text { if } t=2s-1\text { for } s=1, ... ,t/2\\ 1\text { if } t=2\left ( t - s+2\right ) \operatorname{mod}t\text { for } s = t/2 + 1, ... ,t\\ 0\text { otherwise}\end{array } \right.\ ] ] and let @xmath448 be the unitary matrix@xmath449{cc}i_{2 } & 0\\ 0 & i_{t/2}\otimes z \end{array } \right ) \text { with } z=\frac{1}{\sqrt{2}}\left ( \begin{array } [ c]{cc}1 & 1\\ \mathrm{i } & -\mathrm{i}\end{array } \right ) , \ ] ] where @xmath450 denotes the kronecker product . further , let @xmath451 . as is easy to check , @xmath452 is an orthogonal matrix . furthermore,@xmath453 where @xmath399 and @xmath401 are as defined in ( [ d1 ] ) and ( [ d2 ] ) . combining this with ( [ c2d2 ] ) and using lemma [ rank ] once again , we obtain the asymptotic equivalence of @xmath454 and @xmath455 where @xmath456 is the empirical d.f . of the eigenvalues of @xmath457 with@xmath458 because of the rotational invariance of the gaussian distribution , the distributions of @xmath459 and @xmath383 are the same . hence , @xmath460 is asymptotically equivalent to @xmath461 and thus , @xmath462 is asymptotically equivalent to @xmath143.@xmath463 our proof of the almost sure weak convergence of @xmath462 to the wachter distribution consists of showing that the stieltjes transform of @xmath410,@xmath464 almost surely converges pointwise in @xmath465 to the stieltjes transform @xmath466 of the wachter distribution . to establish such a convergence , we show that , if @xmath62 is a limit of @xmath467 along any _ subsequence _ of @xmath154 then it must satisfy a system of equations with unique solution given by @xmath466 . the almost sure convergence of @xmath410 ( and thus , also of @xmath143 ) to the wachter distribution follows then from the continuity theorem for the stieltjes transforms ( see , for example , corollary 1 in geronimo and hill ( 2003 ) ) . we shall write @xmath468 for the stieltjes transform @xmath467 to simplify notation . let @xmath469 then by definition ( [ mhat def ] ) , @xmath468 must satisfy the following equations@xmath470 , \label{equation2}\\ \hat{m } & = \frac{1}{p}\operatorname*{tr}\left [ d\tilde{m}^{-1}\right ] .\label{equation4}\ ] ] let us study the above traces in detail . define@xmath471 , \text { } j=1, ... ,t/2.\ ] ] we now show that the traces in ( [ equation2 ] ) and ( [ equation4 ] ) can be expressed as functions of the terms having form @xmath472 where @xmath473 is independent from @xmath474 then , we argue that @xmath475 i_{2}\ ] ] a.s . converge to zero , and use this fact to derive equations that the limit of @xmath476 if it exists , must satisfy . first , consider ( [ equation2 ] ) . note that@xmath477 = \frac{1}{p}{\displaystyle\sum_{j=1}^{t/2 } } \operatorname*{tr}\left [ \varepsilon_{(j)}^{\prime}m^{-1}\varepsilon _ { ( j)}\right ] .\label{germ of eq2}\ ] ] let us introduce new notation:@xmath478@xmath479@xmath480 in addition , let@xmath481 a straightforward algebra that involves multiple use of the sherman - morrison - woodbury formula ( see golub and van loan ( 1996 ) , p.50 ) @xmath482 and the identity@xmath483 establishes the following equality@xmath484\omega_{j}[v_{j},u_{j}^{\prime}]^{\prime},\label{stem of eq2}\ ] ] where@xmath485{cc}\frac{1}{1-z}i_{2}+v_{j } & \frac{1}{1-z}r_{j}\delta_{2j}^{\prime}+u_{j}^{\prime}\\ \frac{1}{1-z}r_{j}\delta_{2j}+u_{j } & \frac{z}{1-z}r_{j}i_{2}-s_{j}+w_{j}\end{array } \right ) ^{-1}.\ ] ] a derivation of ( [ stem of eq2 ] ) can be found in the supplementary appendix . let us define @xmath486 , \text { } \hat{u}=\frac{1}{t}\operatorname*{tr}\left [ d^{-1}c^{\prime}m^{-1}\right ] , \\ \hat{v } & = \frac{1}{t}\operatorname*{tr}\left [ m^{-1}\right ] , \text { and } \\ \hat{w } & = \frac{1}{t}\operatorname*{tr}\left [ d^{-1}c^{\prime}m^{-1}cd^{-1}\right ] .\end{aligned}\ ] ] we have the following lemma , where @xmath487 denotes the spectral norm . its proof is given in the supplementary appendix . [ the second equation]there exists @xmath491 such that , for any @xmath492 with zero real part , @xmath493 , and the imaginary part satisfying @xmath494 we have@xmath495@xmath496 and @xmath497 , as @xmath106 consider a @xmath498 matrix @xmath499 that is obtained from @xmath500 by replacing @xmath501 and @xmath502 in ( [ stem of eq2 ] ) with @xmath503 and @xmath504 respectively . we have@xmath505\hat{\omega}_{j}[\hat { v}i_{2},\hat{u}i_{2}]^{\prime},\ ] ] where@xmath506{cc}\frac{1}{1-z}i_{2}+\hat{v}i_{2 } & \frac{1}{1-z}r_{j}\delta_{2j}^{\prime}+\hat{u}i_{2}\\ \frac{1}{1-z}r_{j}\delta_{2j}+\hat{u}i_{2 } & \frac{z}{1-z}r_{j}i_{2}+(\hat { w}-\hat{s})i_{2}\end{array } \right ) ^{-1}.\ ] ] a simple algebra and the identity @xmath507 yield@xmath508{cc}\frac{z}{1-z}r_{j}i_{2}+(\hat{w}-\hat{s})i_{2 } & -\frac{1}{1-z}r_{j}\delta_{2j}^{\prime}-\hat{u}i_{2}\\ -\frac{1}{1-z}r_{j}\delta_{2j}-\hat{u}i_{2 } & \frac{1}{1-z}i_{2}+\hat{v}i_{2}\end{array } \right ) , \label{omegatilde}\ ] ] and@xmath509 by definition , @xmath510 in the proof of lemma [ rigour1 ] , we show that the norms @xmath511 @xmath512 and @xmath513 almost surely remain bounded as @xmath106 hence , @xmath514 @xmath515 @xmath516 and @xmath517 are also almost surely bounded . further , by definition,@xmath518 where @xmath519 is an orthogonal matrix , so that @xmath520 and @xmath521 are clearly bounded uniformly in @xmath522 therefore , the norm of matrix @xmath523 almost surely remains bounded as @xmath24 , uniformly in @xmath522 regarding @xmath524 which appear in the denominator on the right hand side of ( [ omega ] ) , the supplementary appendix establishes the following result . the above results imply that , for @xmath492 with @xmath493 and @xmath494 @xmath526 almost surely remains bounded as @xmath24 , uniformly in @xmath522 therefore , by lemma [ rigour1],@xmath527 where @xmath528 as @xmath154 uniformly in @xmath522 a straightforward algebra reveals that@xmath529 using this in equations ( [ approximation1 ] ) and ( [ germ of eq2 ] ) , we obtain@xmath530 where , in the latter expression , the term corresponding to @xmath531 is included in the @xmath532 term to take into account the special definition of @xmath533 as follows from lemma [ small delta lemma ] and the boundedness of @xmath534 and @xmath535 the derivative @xmath536 almost surely remains bounded by absolute value as @xmath154 uniformly in @xmath537 .$ ] therefore@xmath538 the statement of proposition [ the second equation ] now follows by noting that the latter integral is one quarter of the integral over @xmath539 .\square$ ] a similar analysis of equation ( [ equation4 ] ) gives us another proposition , describing @xmath468 as function of @xmath540 and @xmath541 where@xmath542 , \text { } \tilde{u}=\frac{1}{t}\operatorname*{tr}\left [ a^{-1}c\tilde{m}^{-1}\right ] , \\ \tilde{v } & = \frac{1}{t}\operatorname*{tr}\left [ \tilde{m}^{-1}\right ] , \text { and } \\ \tilde{w } & = \frac{1}{t}\operatorname*{tr}\left [ a^{-1}c\tilde{m}^{-1}c^{\prime}a^{-1}\right ] .\end{aligned}\ ] ] we omit the proof because it is very similar to that of proposition [ the second equation ] . although we now have two asymptotic equations for @xmath476 ( [ asyeq1 ] ) and ( [ asyeq2 ] ) , they contain many unknowns : @xmath545 and the corresponding variables with tildes . the following result establishes simple relationships between the unknowns with hats and tildes . the identity @xmath547 is established by the following sequence of equalities@xmath548 the relationship @xmath549 is obtained as follows@xmath550 the identity @xmath551 is obtained similarly to @xmath549 by interchanging the roles of @xmath552 and @xmath553.@xmath463 the identities ( [ connection identities ] ) imply the following equality@xmath554 we denote the reciprocal of the common value of the right and left hand sides of this equality as @xmath555 a direct calculation shows that@xmath556 and the asymptotic relationships ( [ asyeq1 ] ) and ( [ asyeq2 ] ) can be written in the following form@xmath557{l}\hat{m}=\frac{1}{2\pi c}\int_{0}^{2\pi}\hat{h}\left ( z,\varphi\right ) \left ( \left ( z\tilde{v}-4\sin^{2}\varphi\right ) \hat{v}-\hat{u}^{2}\right ) \mathrm{d}\varphi+o(1)\\ \hat{m}=\frac{1}{2\pi c}\int_{0}^{2\pi}\hat{h}\left ( z,\varphi\right ) \left ( \left ( z\hat{v}-1\right ) \tilde{v}-\hat{u}^{2}\right ) \mathrm{d}\varphi+o(1 ) \end{array } \right . .\label{system of two}\ ] ] this can be viewed as an asymptotic system of two equations with four unknowns : @xmath558 and @xmath559 we shall now complete the system by establishing the other two asymptotic relationships connecting these unknowns . multiplying both sides of the identity@xmath560 by @xmath561 taking trace , dividing by @xmath0 , and rearranging terms , we obtain@xmath562 .\label{equation1}\ ] ] next , we analyze ( [ equation1 ] ) similarly to the above analysis of ( [ equation2 ] ) . that is , first , we note that@xmath563 = \frac { 1}{p}{\displaystyle\sum_{j=1}^{t/2 } } \operatorname*{tr}\left [ \delta_{2j}^{\prime}\varepsilon_{(j)}^{\prime}d^{-1}c^{\prime}m^{-1}\varepsilon_{(j)}\right ] .\label{germ of eq1}\ ] ] then elementary algebra , based on the sherman - morrison - woodbury formula ( [ identity ] ) , yields@xmath564 \omega_{j}\left [ v_{j},u_{j}^{\prime } \right ] ^{\prime}\right ) \label{stem of eq1}\\ & + r_{j}\left ( r_{j}i_{2}+s_{j}\right ) ^{-1}\left ( u_{j}-\left [ u_{j},w_{j}\right ] \omega_{j}\left [ v_{j},u_{j}^{\prime}\right ] ^{\prime } \right ) .\nonumber\end{aligned}\ ] ] multiplying both sides of ( [ stem of eq1 ] ) by @xmath565 and replacing @xmath566 and @xmath502 by @xmath567 and @xmath504 respectively , yields an asymptotic approximation to @xmath568 which can be used in ( [ germ of eq1 ] ) and ( [ equation1 ] ) to produce the following result . its proof , as well as the proof of ( [ stem of eq1 ] ) , are given in the supplementary appendix . one might think that the remaining asymptotic relationship can be obtained by using the identity @xmath570 which parallels ( [ another identity ] ) . unfortunately , following this idea delivers a relationship equivalent to ( [ asyeq3 ] ) . therefore , instead of using ( [ parallel identity ] ) , we consider the identity@xmath571 = \frac{1}{p}\operatorname*{tr}\left [ dd^{-1}c^{\prime}m^{-1}\right ] , \label{last identity}\ ] ] which yields@xmath572 = \frac{1}{p}{\displaystyle\sum_{j=1}^{t/2 } } \operatorname*{tr}\left [ \delta_{1j}\varepsilon_{(j)}^{\prime}d^{-1}c^{\prime}m^{-1}\varepsilon_{(j)}\right ] .\label{germ of eq4}\ ] ] then , we proceed as in the above analysis of ( [ germ of eq1 ] ) and ( [ germ of eq2 ] ) to obtain the remaining asymptotic relationship . the proof of the following proposition is given in the supplementary appendix . summing up the results in propositions [ the second equation ] , [ the fourth equation ] , [ the first equation ] , and [ the last equation ] , the unknowns @xmath574 and @xmath575 must satisfy the following system of asymptotic equations@xmath557{l}\hat{m}=\frac{1}{2\pi c}\int_{0}^{2\pi}\hat{h}\left ( z,\varphi\right ) \left ( \left ( z\tilde{v}-4\sin^{2}\varphi\right ) \hat{v}-\hat{u}^{2}\right ) \mathrm{d}\varphi+o(1)\\ \hat{m}=\frac{1}{2\pi c}\int_{0}^{2\pi}\hat{h}\left ( z,\varphi\right ) \left ( \left ( z\hat{v}-1\right ) \tilde{v}-\hat{u}^{2}\right ) \mathrm{d}\varphi+o(1)\\ 1+z\hat{m}=\frac{1}{2\pi c}\int_{0}^{2\pi}\hat{h}\left ( z,\varphi\right ) \left ( 2\hat{u}\sin^{2}\varphi+z\tilde{v}\hat{v}-\hat{u}^{2}\right ) \mathrm{d}\varphi+o(1)\\ 0=\frac{1}{2\pi c}\int_{0}^{2\pi}\hat{h}\left ( z,\varphi\right ) \left ( 4\hat{v}\sin^{2}\varphi+2\hat{u}\right ) \mathrm{d}\varphi+o(1 ) \end{array } \right . .\label{asymptotic system}\ ] ] recall that the unknowns @xmath574 and @xmath575 in the asymptotic relationships ( [ asymptotic system ] ) depend on @xmath576 the definition ( [ mhat def ] ) of @xmath468 implies that @xmath577 is bounded by @xmath578 further , as shown in the proof of proposition [ the second equation ] , @xmath575 and @xmath579 are a.s . bounded by absolute value , and it can be similarly shown that @xmath580 is a.s . bounded by absolute value . therefore , there exist a subsequence of @xmath120 along which @xmath574 and @xmath575 a.s . converge to some limits @xmath581 and @xmath582 these limits must satisfy a non - asymptotic system of equations@xmath557{l}m=\frac{1}{2\pi c}\int_{0}^{2\pi}h\left ( z,\varphi\right ) \left ( \left ( zy-4\sin^{2}\varphi\right ) v - u^{2}\right ) \mathrm{d}\varphi\\ m=\frac{1}{2\pi c}\int_{0}^{2\pi}h\left ( z,\varphi\right ) \left ( \left ( zv-1\right ) y - u^{2}\right ) \mathrm{d}\varphi\\ 1+zm=\frac{1}{2\pi c}\int_{0}^{2\pi}h\left ( z,\varphi\right ) \left ( 2u\sin^{2}\varphi+zvy - u^{2}\right ) \mathrm{d}\varphi\\ 0=\frac{1}{2\pi c}\int_{0}^{2\pi}h\left ( z,\varphi\right ) \left ( 2v\sin ^{2}\varphi+u\right ) \mathrm{d}\varphi \end{array } \right . , \label{the system}\ ] ] where@xmath583 ^{-1}.\ ] ] let us consider , until further notice , only such @xmath492 that have zero real part , @xmath493 , and the imaginary part satisfying @xmath494 for some @xmath491 . let us solve system ( [ the system ] ) for @xmath584 adding two times the last equation to the first one , and subtracting the second equation we obtain@xmath585 note that @xmath586 otherwise , from the second equation of ( [ the system ] ) , we have @xmath587 which can not be true because @xmath468 is the stieltjes transform of the empirical distribution of the squared canonical correlations , all of which lie between zero and one . indeed , clearly , for any @xmath588 and @xmath492 with @xmath589@xmath590 therefore , @xmath591 and @xmath468 can not converge to @xmath592 . since @xmath593 ( [ zero ] ) yields @xmath594 with @xmath595 and @xmath596 ( if one of them equals zero , the other equals zero too , and @xmath592 by the second equation of ( [ the system ] ) , which is impossible ) . since @xmath597 the last equation implies that @xmath598 as well . further , subtracting from the third equation the sum of @xmath492 times the second and @xmath599 times the last equation , and using ( [ reduction1a ] ) , we obtain@xmath600 this equation , together with the second equation of ( [ the system ] ) yield@xmath601 next , for the integrand in the last equation of ( [ the system ] ) , we have@xmath602 this assumes that @xmath603 if not , then @xmath604 ^{-1}\ ] ] would not depend on @xmath605 and the last equation of ( [ the system ] ) would imply that @xmath606 the latter equation and the equality @xmath607 would yield @xmath608 which when combined with the second equation of ( [ the system ] ) would give us @xmath609 which can not be true because @xmath610 being a limit of @xmath468 , must satisfy @xmath611 for @xmath612 finally , elementary calculations given in the supplementary appendix show that@xmath615 where @xmath616 $ ] . using ( [ fact ] ) , ( [ germ of red2 ] ) , and the definition of @xmath617 , we obtain the following relationship@xmath618 that holds as long as@xmath619 .\ ] ] the latter inclusion holds because otherwise @xmath617 is not a bounded function of @xmath620 which would contradict lemma [ small delta lemma ] . equation ( [ possibility1 ] ) can not hold because otherwise , ( [ reduction4 ] ) would imply that @xmath624 which is impossible as argued above . equation ( [ possibility2 ] ) taken together with ( [ reduction3 ] ) implies that@xmath625 which was ruled out above . this leaves us with ( [ possibility3 ] ) , so that , using ( [ reduction4 ] ) , we get@xmath626 for @xmath627 with @xmath589 the imaginary part of the right hand side of ( [ second possibility ] ) is negative when ` @xmath628 ' is used in front of the square root . here we choose the branch of the square root , with the cut along the positive real semi - axis , which has positive imaginary part . since @xmath629 can not be negative , we conclude that@xmath630 but the right hand side of the above equality is the value of the limit of the stieltjes transforms of the eigenvalues of the multivariate beta matrix @xmath631 as @xmath106 this can be verified directly by using the formula for such a limit , given for example in theorem 1.6 of bai , hu , pan and zhou ( 2015 ) . as follows from wachter ( 1980 ) , the weak limit of the empirical distribution of the eigenvalues of the multivariate beta matrix @xmath632 as @xmath24 equals @xmath633 . equation ( [ the limit ] ) shows that , for @xmath492 with @xmath493 and @xmath494 any converging subsequence of @xmath468 converges to the same limit . hence , @xmath468 a.s . converges for all @xmath492 with @xmath493 and @xmath634 note that @xmath468 is a sequence of bounded analytic functions in the domain @xmath635 where @xmath636 is an arbitrary positive number . therefore , by vitaly s convergence theorem ( see titchmarsh ( 1939 ) , p.168 ) @xmath468 a.s . converges to @xmath610 described by ( [ the limit ] ) , for any @xmath637 the almost sure convergence of @xmath462 ( and thus , also of @xmath69 ) to the wachter distribution follows from the continuity theorem for the stieltjes transforms ( see , for example , corollary 1 in geronimo and hill ( 2003 ) ) . first , let us show that the weak limit @xmath195 of @xmath638 as @xmath196 exists and equals the continuous part of the marchenko - pastur distribution with density ( [ mp density ] ) . by definition and theorem [ main ] , @xmath639 is the ( scaled ) wachter d.f . @xmath640 therefore , by ( [ densityw ] ) and ( [ support boundaries ] ) , the density , @xmath641 , and the boundaries of the support , @xmath642 , $ ] of the distribution @xmath643 equal @xmath644@xmath645 as @xmath646 @xmath647 where @xmath648 as in ( [ mp boundaries ] ) , and @xmath641 converges to the density given by ( [ mp density ] ) . this implies the weak convergence of @xmath638 to @xmath195 with @xmath211 supported on @xmath201 $ ] and having density ( [ mp density ] ) . to establish the theorem , it remains to show that , as @xmath51 , @xmath649 weakly converges to @xmath650 in probability . recall that the weak convergence is metrized by the lvy distance @xmath651 . we need to show that for any @xmath652 there exists @xmath653 such that ( s.t . ) for all @xmath654@xmath655 let @xmath656 be so small that@xmath657 for any @xmath214 let @xmath658 be the smallest even integer satisfying @xmath659 that is,@xmath660 for any @xmath661 by the triangle inequality , we have@xmath662 where @xmath663 and @xmath664 denote the empirical distributions of eigenvalues of@xmath665 with @xmath666 and @xmath667 respectively . by theorem [ main ] , @xmath668 a.s . converges to zero as @xmath183 therefore , for all sufficiently large @xmath0 , we have@xmath669 further , as shown by johansen ( 1988 , 1991 ) , for any @xmath214 as @xmath670 the eigenvalues of ( [ rawmatrixagain ] ) with @xmath671 jointly converge in distribution to those of@xmath672 therefore , for any @xmath0 and all sufficiently large @xmath673 we have@xmath674 let us denote the sum of @xmath675 @xmath676 and @xmath677 as @xmath678 by ( [ triangle ] ) , we have@xmath679 inequalities ( [ levy1 ] ) , ( [ levy2 ] ) , and ( [ levy4 ] ) show that for any @xmath652 there exists @xmath680 such that ( s.t . ) for any positive @xmath681 there is a @xmath682 s.t . for any @xmath683 there is a @xmath684 s.t . for any @xmath685@xmath686 the subscripts in @xmath687 @xmath682 and @xmath684 signify dependence on the value of the corresponding parameter . inequalities ( [ levy124 ] ) and ( [ two terms ] ) would establish ( [ need to show ] ) as long as we are able to show that for any @xmath652 there exists @xmath688 s.t . for any positive @xmath689 there is a @xmath690 s.t . for any @xmath691 _ and any _ @xmath692 there exists @xmath693 s.t.@xmath694 let us denote @xmath695 where @xmath383 is a @xmath696 matrix with i.i.d . @xmath697 entries , as defined in section 2 . we shall assume that , as @xmath120 change , @xmath698 represents @xmath696 sections of a fixed infinite array of i.i.d . standard normal random variables . consider@xmath699 so defined matrix @xmath700 is identical to the real symmetric matrix @xmath701 the above definition is formulated in terms of @xmath698 to clarify that @xmath700 depends on @xmath7 not only via the term @xmath702 but also through @xmath703 and @xmath704 note that @xmath705 and @xmath664 are the empirical distributions of eigenvalues of @xmath706 and @xmath707 respectively . the following lemma is established in the supplementary appendix . [ alpha12 ] for any @xmath708 there exists @xmath709 s.t . for any positive @xmath710 , there is a @xmath690 s.t . for any @xmath711 and any @xmath692 there exists @xmath693 s.t . with probability larger than @xmath712 @xmath713 can be represented as the sum of two real symmetric matrices @xmath714 and @xmath104,@xmath715 where @xmath716 @xmath717 , and @xmath718 is an absolute constant . finally , let @xmath719 be the empirical distribution of eigenvalues of @xmath720 then , by theorem a45 ( norm inequality ) of bai and silverstein ( 2010),@xmath721 whereas by their theorem a43 ( rank inequality),@xmath722 therefore , by lemma [ alpha12 ] and the triangle inequality , for any @xmath708 there exists @xmath709 s.t . for any positive @xmath723 , there is a @xmath690 s.t . for any @xmath691 and any @xmath692 there exists @xmath724 s.t.@xmath725 for @xmath726 this inequality implies ( [ levy3 ] ) with @xmath727 combining ( [ levy3 ] ) with ( [ levy124 ] ) yields ( [ need to show ] ) , which completes the proof . gonzalo , j. and j - y pitarakis ( 1999 ) dimensionality effect in cointegration analysis , cointegration , causality , and forecasting . a festschrift in honour of clive wj granger , oxford university press , oxford , pp . 212 - 229 ho , m.s . , and b.e . sorensen ( 1996 ) finding cointegration rank in high dimensional systems using the johansen test : an illustration using data based mote carlo simulations , _ review of economics and statistics _ 78 , 726 - 732 . wachter , k. w. ( 1976b ) . probability plotting of multiple discriminant ratios , " _ proceedings of the social statistics section of the american statistical association , part ii _ 830 - 833 . prindle , weber and schmidt , boston . | johansen s ( 1988 , 1991 ) likelihood ratio test for cointegration rank of a gaussian var depends only on the squared sample canonical correlations between current changes and past levels of a simple transformation of the data .
we study the asymptotic behavior of the empirical distribution of those squared canonical correlations when the number of observations and the dimensionality of the var diverge to infinity simultaneously and proportionally .
we find that the distribution almost surely weakly converges to the so - called _
wachter distribution_. this finding provides a theoretical explanation for the observed tendency of johansen s test to find spurious cointegration .
it also sheds light on the workings and limitations of the bartlett correction approach to the over - rejection problem .
we propose a simple graphical device , similar to the scree plot , for a preliminary assessment of cointegration in high - dimensional vars . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
recent years witnessed large developments in the kinetic theory methods applied to mathematical physics and more recently also to mathematical biology . among important branches of the kinetic theory are optimal transportation problems and related to them wasserstein metrics , and monge - kantorovich metrics @xcite . partial differential equations in metric spaces are being applied to transportation problems @xcite , gradient flows @xcite and structured population models @xcite . output of mathematical modeling can often be described as radon measures . comparing the results of the models requires then a definition of distance in the space of measures . the desired properties of such metrics depend on the structure of the considered problem . in most of the cases , the topology of total variation is too strong for applications , and weaker metrics had to be introduced , see @xcite for details . well known @xmath2-wasserstein metric , on the other hand , is only applicable to processes with mass conservation . to cope with growth in the process , various modifications have been proposed , including flat metric and centralized wasserstein metric . in the present paper , we additionally introduce a normalized wasserstein distance . for comparison of different metrics , their interpretation and examples see table [ table ] . even though , all those distances can be computed using linear programming ( lp ) , its computational complexity becomes often larger than the complexity of solving the original problem . for example , the equations for which the stability of numerical algorithm is proven in @xmath3 require an efficient algorithm for the flat metric to find the residual error . algorithms proposed in this paper are designed to compute efficiently wasserstein - type distance between two radon measures in the form of @xmath4 , where @xmath5 is the dirac delta . the algorithms can be also applied for the case of an arbitrary pair of measures , by approximating those measures by the sum of dirac deltas . in the process of finding numerical solutions to partial differential equations , the distance between a discrete and absolutely continuous measures is needed to evaluate the quality of the initial condition approximation , while the distance between two discrete measure is needed to find the residual error . the paper is organized as follows . in section [ metrics ] we introduce and compare four wasserstein - type metrics . additionally , we introduce a two - argument function , based on the wasserstein distance , which is not a metric , however , provides a good tool to estimate the flat metric from above . section [ algorithms ] is devoted to numerical algorithms proposed to calculate distances between two measures in respect to the considered metrics . we justify the algorithms and provide respective pseudocodes . the novelty of this paper is the algorithm for flat metric , which has been recently proven to be adequate for the escalator boxcar train ( ebt ) method for solving transport equations @xcite . we propose an efficient algorithm which computation cost is @xmath6 and optimize it further to @xmath1 . to judge the efficiency of the algorithms , we compare the times needed to compute the flat distance between two sums of a large number of dirac deltas randomly distributed over [ -1,1 ] . the framework of wasserstein metric in the spaces of probability measures has proven to be very useful for the analysis of equations given in a conservative form , such as , for example , transport equation @xcite , fokker - planck @xcite or nonlinear diffusion equation @xcite . it was originally defined using the notion of optimal transportation , but we focus on its dual representation . @xmath2-wasserstein distance between two measures @xmath7 and @xmath8 is given by @xmath9 observe that the integral @xmath10 is invariant with respect to adding a constant to function @xmath11 . consequently , for any @xmath12 holds @xmath13 furthermore , the following two properties hold : * @xmath2-wasserstein distance is scale - invariant @xmath14 * @xmath2-wasserstein distance is translation - invariant @xmath15 if @xmath16 , then @xmath17 by the dual representation definition . it makes this metric useless for processes not preserving the conservation of mass . in this section we introduce a simple normalization that leads to a definition of translation - invariant metric suitable for non - local models we define normalized @xmath2-wasserstein distance between two measures @xmath7 and @xmath8 as @xmath18 where @xmath19 denotes the total variation of the measure @xmath7 . the distance defined by is a metric . let @xmath7 , @xmath8 and @xmath20 be radon measures . then , it holds * @xmath21 if and only if @xmath22 . + indeed , either @xmath23 or @xmath24 imply that @xmath22 . * @xmath25 , * since @xmath26 to show the triangle inequality , we consider four possibilities @xmath27 this metric can be easily computed numerically using the algorithm for @xmath2-wasserstein distance . however , it lacks the scaling property , which holds for the wasserstein distance , and which is useful for applications . nonetheless , the following weaker property holds . let @xmath28 and @xmath29 be two sequences of radon measures and @xmath30 , @xmath31 then @xmath32 in this section we present a different modification of the @xmath2-wasserstein distance , which is scale - invariant . we define centralized @xmath2-wasserstein distance between two measures @xmath7 and @xmath8 as @xmath33\right\}.\ ] ] the metric was introduced in @xcite for analysis of the structured population models in the spaces of non - negative radon measures , @xmath34 , on @xmath35 with integrable first moment , i.e. , @xmath36 this metric satisfies the scaling property , but on the contrary to wasserstein metric , it is not invariant with respect to translations . it is therefore only useful for modeling specific phenomena , for example structured population dynamics where mass generation can only occur in one specific point of space . another solution of the problem of comparing two measures of different masses is the flat metric which is defined as follows flat distance between two measures @xmath7 and @xmath8 is given by @xmath37 the flat metric , known also as a bounded lipschitz distance @xcite , corresponds to the dual norm of @xmath38 , since the test functions used in the above definition are dense in @xmath39 . this metric satisfies the scaling property and it is translation invariant . it has proven to be useful in analysis of structured population models and in particularly , lipschitz dependence of solutions on the model parameters and initial data @xcite . the flat metric has been recently used for the proof of convergence and stability of ebt , which is a numerical algorithm based on particle method , for the transport equation with growth terms @xcite . consequently , an implementation of the ebt method requires an algorithm to compute the flat distance between two measures . computing of flat metric exactly is more costly than computing of @xmath2-wasserstein metric ( @xmath1 vs @xmath40 , see section [ sectionflatalg ] ) . moreover , often in applications it is sufficient to calculate upper bound of the distance , for example when computing residual error . we propose the following function , which requires only linear computing time . @xmath41 let @xmath7 , @xmath8 be radon measures on @xmath0 . then , the following estimate holds @xmath42 assume that @xmath43 . then , using the estimate of the flat metric by @xmath2-wasserstein distance for the measures with equal masses provides @xmath44 the last equality results from the representation of radon distance and positivity of measure @xmath8 . the upper bound function is more useful if it can be estimated from above by @xmath45 for some constant @xmath46 . in a general case , however , such constant does not exist , since by taking @xmath47 , @xmath48 and passing @xmath49 we obtain @xmath50 nevertheless , the desired estimate can be shown on a bounded set . let @xmath7 , @xmath8 be non - negative radon measures on a compact set @xmath51 , then the following estimate holds @xmath52 let @xmath53 and @xmath54 be such point that @xmath55 for any @xmath56 . then , assuming @xmath43 , we obtain @xmath57 in the following table we compare the introduced metrics and provide examples and interpretations in terms of optimal transport . @xmath58 & scale - invariance & translation - invariance & intuition of @xmath59 & compute complexity + wasserstein & @xmath60 & yes & yes & the cost of optimal transportation of distribution @xmath7 to a state given by @xmath8 , assuming that moving mass @xmath61 by @xmath62 requires @xmath63 energy . & @xmath40 + wasserstein normalized & @xmath64 @xmath65 & & & minimum of the cost of annihilating @xmath7 and generating @xmath8 ; and of the cost of generating the difference in mass between @xmath7 and @xmath8 and transporting @xmath66 to @xmath67 , assuming that generating / annihilating mass at any position requires the cost equal to the mass . & @xmath40 + wasserstein centralized & @xmath68 & yes & no & the cost of generating the difference in mass in point @xmath69 in space added to the cost of transporting @xmath70 to @xmath8 . & @xmath40 + flat & @xmath71 & yes & yes & the cost of optimal transporting and/or generating and/or annihilating mass to form @xmath8 from @xmath7 . & @xmath1 + radon & @xmath72 & yes & yes & the cost of generating and/or annihilating mass to form @xmath8 from @xmath7 & @xmath40 + in the remainder of this paper , we present algorithms for computing and approximating of the introduced metrics . we start with introducing the algorithm for computation of @xmath2-wasserstein distance . the algorithm is well - known and straightforward . nevertheless , we present it here , since the specific approach used in this section will be also applied later in the proof of correctness of more involved algorithms for other distances . define @xmath77}{n}}}&=&\sum_{i=1}^{n}\delta_{i / n}\mu\left[\frac{i-1}{n},\frac{i}{n}\right).\\\end{aligned}\ ] ] then , we estimate @xmath78}{n } } ) \leq \sum_{i=1}^{n}\frac{1}{n } \mu\left[\frac{i-1}{n},\frac{i}{n}\right ) \leq \frac{1}{n } \mu[0,1],\ ] ] and the right - hand side tends to @xmath69 with @xmath79 . consequently , @xmath2-wasserstein distance between an arbitrary pair of finite radon measures can be estimated by the distances between their discrete approximations . from this point on , we assume that @xmath80 define @xmath85 then , obviously @xmath86 . denote @xmath87 , and observe that @xmath88 @xmath89}w_{1}^{1}(x)=a_{2}x+a_{1}x+a_{1}\cdot sgn(a_{1})d_{1}=\\ & = & \left(a_{1}+a_{2}\right)x+|a_{1}|d_{1}.\end{aligned}\ ] ] it can be shown by induction that @xmath90 notice that the value @xmath91 is not used in this formula . it is , however , involved indirectly , as @xmath92 . we assume @xmath108 define @xmath109 as already proven @xmath110 from lp representation of the metric @xmath111 it can be deduced that @xmath112}\left(\underline{w}_{1}^{m+1}(x)+\overline{w}_{1}^{m+1}(x)-a_{m+1}x\right),\ ] ] so the distance is given by the formula @xmath113 each iteration of each loop takes a constant time . the total number of iterations in all three loops is equal to @xmath131 . computational complexity of this algorithm is therefore @xmath102 , while the memory complexity is @xmath103 . the algorithm for flat distance requires storing the shape of functions analogous to @xmath132 as they get more complicated when @xmath61 increases . in section [ flat_alg ] we provide a recursive formula for the sequence of these functions . the pseudocode in section [ flat_pseudo ] implements the algorithm using an abstract data structure to store previously defined functions . the computational complexity depends on the choice of this structure . in further sections we provide two solutions that require @xmath6 and @xmath1 operations . define @xmath138 obviously , it holds by definition @xmath139}f^{n}(x).\ ] ] observe that @xmath140\cap[-1,1]}f^{1}(x ) = a_{2}x+\min(|a_{1}|,a_{1}x+|a_{1}|d_{1}),\\\\\\ f^{m}(x ) & = & a_{m}x+\sup_{f_{m-1}\in[x - d_{m-1},x+d_{m-1}]\cap[-1,1]}f^{m-1}(x).\end{aligned}\ ] ] computing of @xmath141 based on @xmath142 is more complex than in previous cases , as @xmath142 is not necessarily monotonic . @xmath146\cap[-1,1]}f^{n}(x).\ ] ] choose @xmath147 $ ] . then , there exist @xmath148,\ , y'\in b(y , d)\cap[-1,1]$ ] such that @xmath149 because @xmath143 is concave , it holds @xmath150 the last inequality follows from @xmath151 . it is now proven that @xmath152 is convex , as it is a sum of a linear function and a convex function @xmath153 . for each @xmath61 the function @xmath143 is piecewise linear on @xmath61 intervals and it holds for some point @xmath154 @xmath155}}\\ f^{m-1}(x_m ) & \mbox{on \ensuremath{[x_m - d_{m-1},x_m+d_{m-1}]}}\\ f^{m-1}(x - d_{m-1 } ) & \mbox{on \ensuremath{[x_m+d_{m-1},1 ] } } \end{cases}\ ] ] @xmath156 is a linear function , so it can be described by its values in @xmath157 . assume that @xmath143 can be described by at most @xmath158 points and is linear between those points . as @xmath143 is concave , there exists a point @xmath159 $ ] such that @xmath160 for every @xmath62 . the maximum of @xmath143 on an interval whose both ends are smaller than @xmath161 is taken on its right end . similarly , if both ends of the intervals are larger than @xmath161 , the maximum is taken on its left end . finally , if the interval contains @xmath161 , the maximum is in @xmath161 . these considerations prove the formula for @xmath162 . it follows that @xmath163 is piecewise linear and it can be described by as many points as @xmath164}$ ] plus @xmath2 . the simplest implementation of @xmath186 uses an array of pairs @xmath187 sorted by @xmath169 in ascending order and by @xmath188 in the reverse order in the same time . this is possible based on lemma [ concave ] . the first block of instructions can be performed in @xmath189 by simply shifting all elements such that @xmath190 to the right , and modifying @xmath169 by iterating over all elements of @xmath186 . the next block ( computing of @xmath191 ) can be computed with the same complexity , as @xmath192 is simply the next element after @xmath169 in the ordered array . finally , every instruction in the last block can be performed in @xmath189 by iterating over all its elements . in this implementation @xmath186 is represented by global variables @xmath193 and a bst of values @xmath194 where @xmath188 is the key . an entry @xmath187 in this structure is represented as @xmath195 so obtaining an element of @xmath186 may take linear time . the advantages of this structure can be easily seen when analyzing the first block of the code . the division of @xmath186 by the value of @xmath188 ( at first @xmath69 ) can be achieved in @xmath196 . shifting all elements of those subsets can then be done in a constant time by modifying first elements of those sets . adding the extra node also requires @xmath196 operations . removing nodes with the first coordinate @xmath197 is obviously done in @xmath40 in total . identifying nodes with the first coordinate @xmath198 might seem problematic . it is , however , known that for the least @xmath188 the value of @xmath169 is equal to @xmath199 . relevant nodes can be , therefore , removed from the back in @xmath40 . adding @xmath91 to the second coordinate of each node is done by adding it to global variable @xmath193 . performance of the algorithm depends on the choice of @xmath186 data structure . theoretic bounds for computational complexity are , however , not sufficient to argue about performance of these two options . the first reason is that the each operation in @xmath185 algorithm is much faster than in @xmath1 in terms of number of instructions . secondly , hardware architectures provide solutions in which iterating over large tables is accelerated . finally , the algorithm does reach its theoretical bound only if many points concentrate on a small interval . a gap of size @xmath201 between two points completely cleans @xmath186 data structure . all that is shown in numerical tests . to measure performance we have used a single core of amd athlon ii x4 605e processor clocked at 2.3ghz with 8 gb of memory . the results are presented in figs.[fig1]-[fig2 ] . dirac deltas randomly distributed over @xmath202 $ ] . the plot shows how the time of computation depends on @xmath203 . for each input size 100 independent tests were executed to demonstrate how sensitive the algorithms are to input data distribution . results of @xmath204 algorithm are depicted as red dots , and results of @xmath6 algorithm as blue dots.,width=547 ] dirac deltas with a distributed over a large domain , i.e. distance between each two masses is larger than @xmath201 . in this case both algorithms are in fact linear , as the funcdescription structure has at most two elements . the plot demonstrates the overhead of using bst structures . results of @xmath204 algorithm are depicted as red dots , and results of @xmath6 algorithm as blue dots.,width=547 ] * acknowledgments * + jj was supported by the international ph.d . projects program of foundation for polish science operated within the innovative economy operational program 2009 - 2015 funded by european regional development fund ( ph.d . program : mathematical methods in natural sciences ) . am - c was supported by the erc starting grant `` biostruct '' no . 210680 and the emmy noether program of the german research council ( dfg ) . l. ambrosio and g. crippa . _ existence , uniqueness , stability and differentiability properties of the flow associated to weakly differentiable vector fields _ in transport equations and multi - d hyperbolic conservation laws , lect . notes unione mat . springer 5 : 357 , 2008 . , _ an introduction to the nonlinear boltzmann - vlasov equation _ , in kinetic theories and the boltzmann equation , springer , berlin , lecture notes in math . 1048 : 60110 , 1981 . , , 5 : 873900 , 2012 . . variational particle schemes for the porous medium equation and for the system of isentropic euler equations _ math . 44 : 133166 , 2010 . , springer - verlag , berlin , 2009 | in this paper numerical methods of computing distances between two radon measures on @xmath0 are discussed .
efficient algorithms for wasserstein - type metrics are provided . in particular , we propose a novel algorithm to compute the flat metric ( bounded lipschitz distance ) with a computational cost @xmath1 . the flat distance has recently proven to be adequate for the escalator boxcar train ( ebt ) method for solving transport equations with growth terms . therefore ,
finding efficient numerical algorithms to compute the flat distance between two measures is important for finding the residual error and validating empirical convergence of different methods .
* keywords : * metric spaces , flat metric , wasserstein distance , radon measures , optimal transport , linear programming , minimum - cost flow |
You are an expert at summarizing long articles. Proceed to summarize the following text:
it is well known that the symmetry - adapted solution of the nonlinear hartree - fock ( hf ) equations of an electronic system is sometimes unstable . an unstable solution corresponds to a saddle point of the energy as a function of the orbital parameters , and breaking of space and/or spin symmetries of the wave function then necessarily leads to one or several lower - energy hf solutions . the stability conditions of the hf equations were first formulated by thouless @xcite , and the different instabilities were first categorized by iek and paldus @xcite . for closed - shell systems , one may encounter `` singlet instabilities '' when only spatial symmetry is broken , and `` triplet ( or nonsinglet ) instabilities '' when spin symmetry is also broken . there is thus a symmetry dilemma @xcite in choosing between the symmetry - adapted wave function of higher hf energy and a symmetry - broken wave function of lower hf energy , in particular as a reference for a post - hartree - fock calculation . a clear example is provided by closed - shell hydrogen rings h@xmath0 with equal bond lengths @xcite ( see , also , ref . ) . the symmetry - adapted hf solution exhibits singlet instabilities for sufficiently large numbers of hydrogen atoms , and one can obtain symmetry - broken hf solutions with orbitals localizing on either the atoms or the bonds . however , both mller - plesset perturbation theory and linearized coupled cluster doubles theory ( also called cepa0 or dmbpt@xmath1 ) give a lower total energy when starting from the symmetry - adapted solution than when starting from the symmetry - broken solutions , which casts doubts on the physical significance of the symmetry - broken solutions . of course , the symmetry dilemma would be removed with a full configuration - interaction calculation which must give one unique solution , independent of the orbitals used . quantum monte carlo ( qmc ) approaches are alternatives to the traditional quantum chemistry methods @xcite . the two most commonly used variants are variational monte carlo ( vmc ) which simply evaluates the energy of a flexible trial wave function by stochastic sampling , and fixed - node diffusion monte carlo ( dmc ) which improves upon vmc by projecting the trial wave function onto the ground state subject to the condition that the nodes of the projected wave function are the same as those of the trial wave function . for large systems , the most common form of the trial wave function is a jastrow factor multiplied by a fixed hf determinant ( though for small systems one can do much better by replacing the hf determinant by a linear combination of optimized slater determinants ) . if a system exhibits hf instabilities , then qmc also faces the symmetry dilemma in choosing between different hf wave functions . indeed , different hf wave functions necessarily lead to different energies not only in vmc , but also in dmc since the nodes of these hf wave functions are generally different . this symmetry dilemma in dmc is only due to the fixed - node approximation , since without this approximation dmc would give one unique solution , independent of the trial wave function . of course , for systems that are not very large , symmetry breaking could probably be avoided in the first place by optimizing the orbitals within vmc @xcite , instead of using fixed hf orbitals . in this work , we study the impact of the hf symmetry dilemma for qmc in hydrogen rings h@xmath0 . in sec . ii , we review the hf symmetry - breaking problem in these systems , and discuss the effect of using a slater basis versus a gaussian basis . in sec . iii , we explain the qmc methodology and report our vmc and dmc results . our conclusions are summarized in sec . in previous studies @xcite , the electronic structure of periodic rings of @xmath2 evenly spaced hydrogen atoms ( with a fixed distance of @xmath3 ) has been investigated . the number of hydrogen atoms is restricted to @xmath2 in order to obtain a possible closed - shell single - determinant solution with @xmath4 occupied orbitals . the symmetry - adapted hf wave function has a metallic character ( in the limit of an infinite ring ) and can be expressed with either delocalized canonical orbitals or localized wannier orbitals ( which do not have an exponential decay ) . the canonical orbitals , except for the lowest one , are doubly degenerate , and in a minimal basis the orbital coefficients are fixed by the cyclic symmetry . besides the symmetry - adapted ( sa ) solution , two different symmetry - broken hf solutions of lower energy can be obtained beyond critical ring sizes , when using unit cells of 2 hydrogen atoms . one solution corresponds to orbitals localized on hydrogen atoms and is referred to as the symmetry - broken atom - centered ( sb - ac ) solution , while the other corresponds to orbitals localized on bonds and is referred to as the symmetry - broken bond - centered ( sb - bc ) solution . the sb - bc solution has the lowest hf energy . the three solutions can be schematically described as @xmath5h@xmath5h@xmath5 , @xmath5h@xmath6h@xmath7 , and @xmath5h h@xmath5 . in each case , the symmetry breaking is accompanied by an opening of an energy gap between occupied and virtual orbitals , and orbitals decay much more rapidly than for the symmetry - adapted solution @xcite , in agreement with the theoretical result of kohn @xcite . in order to distinguish the three different wave functions , one may look at the one - particle density matrix @xmath8 @xmath9 containing the coefficients @xmath10 of the occupied molecular orbitals @xmath11 , @xmath12 expanded in a minimal set of atom - centered basis functions , i.e. one single basis function @xmath13 per hydrogen atom . as depicted in figure [ densmat ] , for the sa solution , we see equal elements on the diagonal and the sub - diagonals of the density matrix . for the sb - ac solution , an alternation of element values on the diagonal of the density matrix is obtained , but equal elements on the first sub - diagonal , and for the sb - bc solution we have equality of the diagonal elements and alternation on the first sub - diagonal . in ref . , a minimal gaussian basis set ( five @xmath14 gaussian functions contracted to one single basis function for each hydrogen atom ) was used . however , gaussian basis functions are not appropriate for all - electron qmc calculations . they give large statistical fluctuations due to their incorrect vanishing gradient at the nuclear positions . it is thus much preferable to use slater basis functions which correctly have a non - zero gradient on the nuclei and an exponential decay at large distance . in this work , we use a minimal slater basis set ( one @xmath15 slater function on each hydrogen atom ) with an exponent of 1.17 , which is smaller than the optimal exponent of 1.24 for an isolated h@xmath16 molecule . spin - restricted hf ( and mp2 ) calculations were performed with an experimental code for ring systems , employed already for the previous studies @xcite . the necessary integrals over slater functions have been calculated with the program smiles @xcite . in order to obtain the symmetry - broken hf solutions , we start from a set of localized wannier orbitals describing either an ionic situation or an explicit bond in the two - atom unit cell , and use an iterative configuration interaction procedure using singly excited determinants @xcite instead of diagonalizing a fock operator . [ cols="<,^,^,^,^ " , ] [ tab : hfmp2vmcdmc ] figure [ dmc ] shows the corresponding dmc results . the energy ordering is the same as in vmc and mp2 , the sa wave function giving the lowest dmc total energy , and thus the smallest fixed - node error . as shown in table [ tab : hfmp2vmcdmc ] , the energy splittings between the different solutions are much smaller in dmc . this indicates that dmc is less sensitive to symmetry breaking than other correlation methods . it is an interesting feature for cases where symmetry breaking can not be avoided . of course , symmetry breaking would probably be avoided if the orbitals were reoptimized in the presence of the jastrow factor within vmc , but this is computationally expensive for large systems . when hf trial wave functions are used in qmc calculations , in case of hf instabilities qmc faces the hf symmetry dilemma in choosing between the symmetry - adapted solution of higher hf energy and symmetry - broken solutions of lower hf energies . in this work , we have examined the hf symmetry dilemma in hydrogen rings h@xmath0 which present hf singlet instabilities for sufficiently large ring sizes . we have shown that using a slater basis set , instead of a gaussian basis set , delays the onset of hf symmetry breaking until larger rings and slightly reduces the energy splittings between the symmetry - adapted and symmetry - broken wave functions . when using these different hf wave functions in vmc and dmc , we have found that the energy ordering is reversed ; the symmetry - adapted wave function always giving the lowest energy . this confirms previous post - hartree - fock studies in showing that these symmetry - broken solutions are bad starting wave functions for correlated calculations . the fact that the symmetry - adapted wave function gives the lowest dmc energy indicates that this wave function has more accurate nodes than the symmetry - broken wave functions . the present experience thus suggests that spatial symmetry is an important criterion for selecting good trial wave functions . for systems that are not very large , the symmetry - breaking problem could probably be avoided altogether by optimizing the orbitals within the quantum monte carlo calculation , rather than using fixed hf orbitals . this work has been financed mainly through the deisa network , project stop - qalm . all qmc calculations have been performed on the ibm bluegene machines in jlich and munich ( germany ) . c.j.u . was supported in part by the nsf ( grant nos . dmr-0908653 and che-1004603 ) . the authors thank the staff of idris ( orsay , france ) for technical assistance to install , test and run the qmc program champ on these machines . we also acknowledge using the slater integral code smiles ( madrid , spain ) for obtaining the hf starting wave functions . discussions with p. gori - giorgi ( amsterdam , netherlands ) and j .- p . malrieu ( toulouse , france ) were very helpful for the project . | when using hartree - fock ( hf ) trial wave functions in quantum monte carlo calculations , one faces , in case of hf instabilities , the hf symmetry dilemma in choosing between the symmetry - adapted solution of higher hf energy and symmetry - broken solutions of lower hf energies . in this work , we have examined the hf symmetry dilemma in hydrogen rings which present singlet instabilities for sufficiently large rings .
we have found that the symmetry - adapted hf wave function gives a lower energy both in variational monte carlo and in fixed - node diffusion monte carlo .
this indicates that the symmetry - adapted wave function has more accurate nodes than the symmetry - broken wave functions , and thus suggests that spatial symmetry is an important criterion for selecting good trial wave functions . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
[ sec : intro ] the classical area of _ combinatorial designs _ ( see , e.g. , @xcite ) asks about extremal subset systems of a finite set that have specified intersection patterns . in this paper , we are interested in extremal families of curves in a closed , orientable surface with specified intersection patterns . more specifically , benson farb and chris leininger brought the following topological variant to our attention : [ farbq ] let @xmath1 be a closed oriented surface of genus @xmath0 , and let @xmath2 be a collection of pairwise non - isotopic essential simple closed curves @xmath3 such that @xmath4 intersects @xmath5 at most once for any @xmath6 . how large can @xmath2 be ? in fact , this question goes back further , to work of juvan , malni , and mohar @xcite , which we learned about after circulating an earlier version @xcite of the present paper . juvan , et al . @xmath7-system _ of curves on a genus @xmath0 closed , oriented surface to be a collection of non - isotopic , essential , simple closed curves @xmath8 such that each of the @xmath4 intersects any @xmath5 , @xmath9 at most @xmath7 times . we define @xmath10 to be the maximum size of any @xmath7-system on a closed surface @xmath1 of genus @xmath0 . the most important related results from @xcite are : * for all @xmath11 and @xmath12 , @xmath10 is finite . * @xmath13 . * @xmath14 asymptotics for @xmath7 and @xmath0 growing are also given in @xcite . we study @xmath15 for @xmath7 fixed . we are particularly interested in the farb - leininger case @xmath16 , for which we can prove : [ theo : generalbounds ] for @xmath17 , @xmath18 and for all @xmath19 , @xmath20 both the upper and lower bounds are , to the best of our knowledge , the best known . juvan , et al . do nt optimize the bound in ( * ? ? ? * theorem 3.3 ) for @xmath16 , but examining their arguments , we get an upper bound on the order of @xmath21 . the upper bound of theorem [ theo : generalbounds ] is based on the @xmath22-homology of curves ( proposition [ prop : mod2classupper ] ) . thus same upper bound statement and proof applies if we replace `` @xmath23-system '' with @xmath7-system in which all pairs of curves are either disjoint or intersect an odd number of times . thus the upper bound in theorem [ theo : generalbounds ] has some built - in slack , and it is plausible that , for large enough @xmath7 , this upper bound is nearly tight . reducing the gap will require a proof method that can see the difference between `` one intersection '' and `` an odd number of intersections '' . [ [ surfaces - with - boundary - components ] ] surfaces with boundary components + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + for a surface with boundary components , a @xmath7-system is defined similarly to the closed case , except we add the condition that none of the curves is allowed to to be homotopic to a boundary curve . define @xmath24 to be the maximum size of a @xmath7-system in a surface of genus @xmath0 with @xmath25 boundary components . holding @xmath0 fixed ( along with @xmath7 ) , we can determine @xmath26 very precisely in terms of @xmath27 : [ theo : boundaryasymptotics ] fix some @xmath28 . then , for all @xmath29 , @xmath30 for @xmath31 and @xmath32 @xmath33 relaxing the definition of a @xmath7-system to allowed arcs as well as curves on a surface with boundary seems to make the problem as hard as determining @xmath27 . [ [ low - genus ] ] low genus + + + + + + + + + on the torus , it is straightforward to show that a maximal @xmath23-system has @xmath34 curves , i.e. @xmath35 . in genus two , we present an elegant geometric argument , based on the hyperelliptic involution , which shows that : [ theo : genus2 ] n(1,2 ) = 12 . moreover , there are exactly two mapping class orbits of maximal @xmath23-systems on a genus-@xmath36 surface . the proof gives exact structural information about the maximal @xmath23-systems on a genus-@xmath36 surface : they are lifts of @xmath37-vertex triangulations of the sphere drawn in the quotient of the surface by the hyperelliptic involution . [ [ nearly - all - curves - intersecting ] ] nearly all curves intersecting + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + given a @xmath7-system @xmath38 on a closed surface @xmath1 of genus @xmath0 , we define its _ odd intersection graph _ @xmath39 to be the graph that has one vertex for each of the @xmath4 and an edge between vertices @xmath40 and @xmath41 if @xmath4 intersects @xmath5 an odd number of times . with assumptions on @xmath42 , we can get much tighter upper bounds . [ theo : alloddupper ] let @xmath38 be a @xmath7-system with odd intersection graph @xmath39 . then for @xmath43 and @xmath44 : * if @xmath39 is the complete graph @xmath45 , then @xmath46 , and the bound is sharp for all @xmath47 . * if @xmath39 has average degree @xmath48 , then @xmath49 . the first statement is proved using a @xmath22-homology argument and the second statement follows via an application of turn s theorem @xcite to @xmath42 s complement , an idea we learned from van vu s paper @xcite . [ [ nearly - all - curves - non - intersecting ] ] nearly all curves non - intersecting + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + given a @xmath7-system @xmath38 we define its _ intersection graph _ to be the graph that has one vertex for each @xmath4 and an edge between vertices @xmath40 and @xmath41 if @xmath4 intersects @xmath5 . [ theo : lowdegree ] let @xmath38 be a @xmath7-system on a closed surface of genus @xmath0 with intersection graph @xmath42 . for @xmath50 and @xmath44 . if @xmath42 has average degree @xmath51 , then @xmath52 . in the case where @xmath53 , this is just the well - known fact that a maximal system of disjoint nonisotopic simple closed curves has at most @xmath54 curves jm , ir , and lt received support for this work from rivin s nsf cdi - i grant dmr 0835586 . lt s final preparation was supported by the european research council under the european union s seventh framework programme ( fp7/2007 - 2013 ) / erc grant agreement no 247029-sdmodels . our initial work on this problem took place at ileana streinu s 2010 barbados workshop at the bellairs research institute . [ sec : gen2 ] in this section , we give a geometric argument to prove theorem [ theo : genus2 ] . the argument generalizes to give a sub - quadratic lower bound in higher genus . the key tool we need is the _ hyperelliptic involution_. here are the facts we need , which can all be found in @xcite . [ prop : hyper - exists ] every riemann surface @xmath1 of genus @xmath36 admits a `` hyperelliptic involution '' with @xmath37 fixed points . the fixed points of the hyperelliptic involution are called _ weirstrass points _ , and we denote them @xmath55 . [ prop : hyper - quotient ] let @xmath1 be a closed riemann surface of genus @xmath36 . the quotient @xmath56 of @xmath1 by the hyperelliptic involution is a sphere with six double points corresponding to the weierstrass points ; i.e. , @xmath56 is an orbifold of signature @xmath57 . simple closed geodesics in the riemann surface @xmath1 have nice representatives in @xmath56 . [ prop : hyper - geodesics ] let @xmath1 and @xmath56 be as in proposition [ prop : hyper - quotient ] . then any non - separating simple closed geodesic @xmath58 in @xmath1 goes through exactly two of the wierstrass points and projects to a geodesic segment going between exactly two of the doubled points in @xmath56 . by proposition [ prop : nosep ] proven below , there is no separating curve in a maximal @xmath23-system , and by proposition [ prop : hyper - geodesics ] , the @xmath23-systems in @xmath1 that have only non - separating curves correspond to isomorphism classes of simple planar graphs on @xmath37 vertices . since planar triangulations on @xmath37 vertices have @xmath59 edges , it follows that @xmath60 . there are two graph isomorphism classes of planar triangulations on @xmath37 vertices , namely , the octahedron and doubly stellated tetrahedron . since these graphs are @xmath34-connected , there are only two triangulations up to homeomorphism of the sphere ( permuting weierstrass points ) . birman and hilden @xcite have shown that the mapping class group of the genus @xmath36 surface modulo the hyperelliptic involution is isomorphic to the mapping class group of the @xmath37-times punctured sphere . consequently , there are two mapping class group orbits of maximal @xmath23-systems . @xmath61 the lower bound of theorem [ theo : genus2 ] generalizes to surfaces @xmath1 of @xmath62 : there are @xmath63 weierstrass points corresponding to double points in the quotient by the hyperelliptic involution , implying that any planar graph on @xmath63 vertices corresponds to some @xmath23-system . thus , we obtain the lower bound for @xmath17 from theorem [ theo : generalbounds ] . [ sec : structure ] we will need the following result on the structure of maximal @xmath23-systems : [ prop : nosep ] let @xmath1 be a closed surface of genus @xmath43 . then , any maximal @xmath23-system in @xmath1 contains no separating curves . from now on in this section , we assume that @xmath1 satisfies the hypotheses of proposition [ prop : nosep ] . we also use the standard notions of _ minimal position _ and _ geometric intersection number _ for curves , which may be found in ( * ? ? ? * section 1.2.3 ) . we denote the geometric intersection number of ( the isotopy classes of ) @xmath64 and @xmath65 by @xmath66 . a @xmath7-system is in minimal position when the curves in it are pairwise in minimal position . every @xmath7-system has such a representative ( * ? ? ? * corollary 1.9 ) . the auxiliary lemma [ lemma : annulipoly ] , which describes the complementary regions of a maximal @xmath7-system is also of independent interest . let @xmath2 be a @xmath7-system and define the _ complementary regions of @xmath2 _ to be the components of the surface @xmath67 obtained by cutting @xmath1 along the curves in @xmath2 . [ lemma : annulipoly ] let @xmath2 be a maximal @xmath7-system such that the curves in @xmath2 are in minimal position . then , the complementary regions are polygons and annuli . furthermore , each annulus has a boundary component consisting of a single curve in @xmath2 . intuitively , one would expect that if proposition [ prop : nosep ] were to fail , an inductive argument would establish an @xmath68 upper bound on the size of any @xmath23-system , contradicting the fact that there are @xmath23-systems with @xmath69 curves . however , we are unaware of a rigorous proof along these lines , so instead we use lemma [ lemma : annulipoly ] . the difficult part of the proof of lemma [ lemma : annulipoly ] will be to rule out pairs of pants as complementary regions . this next lemma is straightforward and gives the starting point . [ lemma : nobiggerthanpants ] suppose @xmath2 is a maximal @xmath7-system in minimal position . then , the complementary regions are all polygons , annuli , or pairs of pants . furthermore , each annulus has a boundary component consisting of a single curve in @xmath2 , and every boundary component of the pairs of pants are curves in @xmath2 . if a complementary region has nonzero genus , then it contains a simple closed curve that is not isotopic to a boundary curve which we could add to @xmath2 to obtain a larger @xmath7-system . a similar fact is true if a complementary region is a sphere with at least @xmath70 boundary components . this proves that the complementary regions are all polygons , annuli , or pairs of pants . now consider a complementary region that is an annulus . if neither boundary component is a single curve in @xmath2 , then the core curve of this annulus is , by our assumption of minimal position , not isotopic to any curve in @xmath2 and does not intersect any curves in @xmath2 . this contradicts the maximality of @xmath2 . similar reasoning shows that all boundary curves of a pair of pants must be elements of @xmath2 . continuing from lemma [ lemma : nobiggerthanpants ] , it suffices to rule out pants as a complementary region . comparing the ( classical ) upper bound , theorem [ theo : all - disjoint ] , of the size of a @xmath71-system and the lower bound on the size of a @xmath23-system from theorem [ theo : generalbounds ] , we see that not all the complementary regions can be pairs of pants . thus , if @xmath2 has some pair of pants as a complementary region , then connectedness of @xmath1 and lemma [ lemma : nobiggerthanpants ] implies that some curve @xmath58 in @xmath2 is disjoint from the rest of @xmath2 and bounds a pair of pants @xmath72 on one side and an annulus @xmath73 on the other . the other boundary component of @xmath73 consists of arcs of curves in @xmath2 , and there are at least two such arcs . let @xmath74 be two curves with consecutive arcs in @xmath75 . @xmath76 at 50 140 @xmath77 at 58 21 @xmath78 at 110 170 @xmath79 at 130 58 @xmath58 at 37 25 @xmath80 at 76 50 @xmath81 at 150 89 @xmath82 at 48 104 @xmath83 at 140 145 and arcs @xmath84.,scaledwidth=50.0% ] we now show that @xmath58 may be replaced with two new curves to obtain a larger @xmath23-system ; the construction is depicted in figure [ figure : newcurves ] . let @xmath85 be the other boundary components ( aside from @xmath58 ) in the pair of pants @xmath72 , and let @xmath82 be a simple arc connecting @xmath76 to @xmath80 and lying in @xmath86 . the regular neighborhood of @xmath87 is a pair of pants where two of the boundary curves are isotopic to @xmath76 and @xmath80 ; let @xmath77 be the other boundary component . similarly , let @xmath83 be a simple arc which connects @xmath78 to @xmath81 , lies in @xmath86 , and is disjoint from @xmath82 ; let @xmath79 be defined similarly to @xmath77 . to show @xmath88 is a @xmath7-system of larger size , we must show pairwise intersections are less than or equal to @xmath7 and that @xmath77 and @xmath79 are not isotopic to other curves in @xmath2 or each other . it is possible to do the latter by establishing the following geometric intersection numbers since the numbers are an invariant of isotopy classes . * for all curves @xmath89 , we have @xmath90 and @xmath91 * @xmath92 * @xmath93 before proving ( 1)-(3 ) , let us see that they establish that @xmath94 is a @xmath7-system . statements ( 1 ) and ( 2 ) show @xmath94 is a @xmath7-system . ( note that ( 1 ) also says @xmath95 and similarly for @xmath96 . ) statement ( 3 ) implies @xmath77 and @xmath79 are not isotopic to any curves in @xmath2 since they are all disjoint from @xmath58 , and statement ( 1 ) implies @xmath97 which establishes that @xmath77 and @xmath79 are not isotopic . [ [ proof - of-1 ] ] proof of ( 1 ) + + + + + + + + + + + + we only show that @xmath90 since the proof applies also to @xmath78 and @xmath79 . if @xmath98 , this is immediate since @xmath77 and @xmath76 are disjoint . now suppose @xmath99 . homotope @xmath77 to a simple closed curve @xmath100 which is the union of an arc @xmath101 from @xmath76 and an arc @xmath102 which lies in @xmath86 , is disjoint from all curves in @xmath103 , and cuts @xmath72 into two annuli . ( see figure [ figure : nopantsfig2 ] . ) clearly , the arc @xmath101 and curve @xmath104 cross @xmath105 times , so by the bigon criterion ( ( * ? ? ? * proposition 1.7 ) ) , it suffices to show that it is impossible to bound a disc with an arc from @xmath104 and an arc from @xmath100 . since @xmath2 is in minimal position , @xmath76 and @xmath104 have no bigons , and so any bigon made from @xmath104 and @xmath100 must use the arc @xmath102 . however , on each side of @xmath102 , we may connect @xmath102 to either @xmath80 or @xmath106 with an arc disjoint from @xmath104 ( since @xmath104 is disjoint from @xmath72 ) ; this would be impossible if an arc containing @xmath102 made a bigon with an arc from @xmath104 . this established the first equality ; the second follows by a similar argument . indicated in gray.,scaledwidth=50.0% ] [ [ proof - of-2 ] ] proof of ( 2 ) + + + + + + + + + + + + let @xmath100 be as above . by further homotoping , we can ensure that @xmath102 is disjoint from @xmath79 and so @xmath107 [ [ proof - of-3 ] ] proof of ( 3 ) + + + + + + + + + + + + again , we use the bigon criterion . by construction , @xmath58 and @xmath77 each cut each other into two arcs , @xmath108 and @xmath109 respectively , where w.l.o.g . @xmath110 lies in @xmath72 . the curves @xmath111 and @xmath112 are each homotopic to some boundary component in @xmath72 and hence can not be trivial . if @xmath113 bounded a disc , then we could homotope @xmath77 to lie in @xmath72 which is impossible since @xmath78 and @xmath77 have nonzero geometric intersection number . @xmath61 suppose , for a contradiction , that @xmath2 is a maximal @xmath23-system with a separating curve @xmath58 ; w.l.o.g . , we may assume that @xmath2 is in minimal position . lemma [ lemma : annulipoly ] then applies , so @xmath58 is incident on two complementary regions @xmath73 and @xmath114 that are both annuli , and each of @xmath73 and @xmath114 has a boundary component consisting of arcs from at least two different curves in @xmath2 . we will show that by removing @xmath58 , two new curves may be added to get a larger @xmath23-system . because @xmath2 is a @xmath23-system , we may slightly strengthen the conclusion of lemma [ lemma : annulipoly ] : the boundary components of @xmath73 and @xmath114 that are not @xmath58 span arcs from at least three different curves in @xmath2 . let @xmath74 be curves with consecutive arcs in @xmath75 , and let @xmath80 be a curve contributing a boundary arc to @xmath114 . let @xmath115 be the surface obtained by cutting along @xmath58 . there is an arc @xmath101 starting and ending at @xmath116 such that @xmath101 `` follows '' @xmath76 . more precisely , there is an arc @xmath101 such that @xmath101 , @xmath76 and a part of @xmath116 cobound an annulus and @xmath101 has the same intersection numbers with the other curves in @xmath103 . similarly , there is such an arc @xmath117 for @xmath78 , and let @xmath118 be two disjoint such arcs for @xmath80 . we can homotope the arcs @xmath119 so that : @xmath76 at 50 100 @xmath78 at 110 130 @xmath58 at 102 26 @xmath58 at 331 26 @xmath80 at 284 110 @xmath101 at 48 48 @xmath117 at 140 95 @xmath120 at 285 62 @xmath121 at 280 30 * @xmath101 and @xmath117 intersect exactly once transversely and * the endpoints of @xmath101 match up with @xmath120 and @xmath117 with @xmath121 when gluing @xmath115 back together . ( this is depicted in figure [ figure : arcsfornosep ] . ) let @xmath77 and @xmath79 be the resulting arcs from gluing @xmath101 with @xmath120 and @xmath117 with @xmath121 respectively . since @xmath77 and @xmath79 intersect exactly once transversely , they are distinct non - trivial simple closed curves . furthermore both @xmath77 and @xmath79 intersect some curve exactly once on either side of @xmath58 . since no other curve in @xmath2 does this , @xmath77 and @xmath79 are isotopically distinct from all other curves in @xmath2 . thus @xmath122 is a larger @xmath23-system than @xmath2 . @xmath61 [ sec : upper - hom ] now we turn to the general case of a surface @xmath1 of genus @xmath50 . the point of this section is to prove the following proposition , from which the upper bound in theorem [ theo : generalbounds ] follows readily . [ prop : mod2classupper ] let @xmath1 be a closed oriented surface of genus @xmath50 . then any @xmath23-system of curves on @xmath1 has at most @xmath123 curves in any nontrivial @xmath22-homology class . the main lemma we need is : [ lemma : disj - all - sep ] let @xmath43 and let @xmath1 be a genus @xmath0 surface with @xmath36 boundary components . let @xmath2 be a @xmath71-system of separating curves in @xmath1 such that each curve separates the boundary components . then , @xmath124 . we argue by induction . suppose @xmath31 . in this case , the only separating simple curves which are not boundary parallel cut @xmath1 into a torus with a single boundary component and a three - holed sphere . such a curve does not separate the boundary components of @xmath1 . suppose @xmath125 . cutting @xmath1 along a curve @xmath58 in @xmath2 yields two surfaces @xmath126 with genus @xmath127 adding up to @xmath0 and each with @xmath36 boundary components . we note that @xmath103 deposits a @xmath71-system in each of @xmath115 and @xmath128 with the same properties as @xmath2 . thus @xmath129 . it suffices to show that one can find at most @xmath123 mutually disjoint curves in the same @xmath22-homology class . let @xmath1 be a closed genus @xmath50 surface , and @xmath2 be a @xmath71-system of curves all in the same @xmath22-homology class . we reduce the proposition to lemma [ lemma : disj - all - sep ] . cut along some @xmath130 to obtain a surface @xmath115 of genus @xmath131 with two boundary components corresponding to @xmath58 ; fill in each boundary component with a disk to get a closed surface @xmath128 . the curves in @xmath132 in @xmath128 are all null - homologous in @xmath133 . this means that their homology classes in @xmath134 are all either non - primitive or trivial . the former implies the the classes are not primitive in @xmath135 , which is disallowed by the hypothesis that all the curves in @xmath2 are simple ( see ( * ? ? ? * proposition 6.2 ) ) . thus , any curve in @xmath103 separates @xmath128 . furthermore , any @xmath136 must separate the boundary components of @xmath115 . if it did not , then it would bound a subsurface in @xmath115 with a single boundary component , and thus also in @xmath1 . this however would imply that @xmath81 is null - homologous as an element of @xmath137 , a contradiction . consequently , we are in the situation of lemma [ lemma : disj - all - sep ] and are done . @xmath61 by proposition [ prop : nosep ] , a maximal @xmath23-system contains no separating curve and thus no null - homologous curve . the curves must then all lie in the @xmath138 nontrivial @xmath22-homology classes , and , by proposition [ prop : mod2classupper ] , each of these contains at most @xmath123 curves in a @xmath23-system . [ sec : extreme ] in this section we give much sharper upper bounds when the intersection graph @xmath42 of a @xmath23-system @xmath2 is either very sparse or very dense . the key cases are when @xmath42 or its complement are complete . we recall the following classical and widely known fact , which may be found in ( * ? ? ? * section 8.3.1 ) . [ theo : all - disjoint ] let @xmath139 and let @xmath1 be a closed surface of genus @xmath0 , and let @xmath2 be a @xmath71-system . then @xmath140 . to bound the size of @xmath23-systems with all pairs of curves intersecting , we continue along the lines of section [ sec : upper - hom ] . let @xmath141 and @xmath142 be vectors in @xmath143 and let @xmath144 denote the standard symplectic pairing . [ prop : linalg ] let @xmath145 be non - zero vectors in @xmath143 with the property that , for all @xmath6 , @xmath146 . then @xmath46 . suppose there is a linear dependence @xmath147 among the @xmath148 , with not all the @xmath149 zero . suppose further that the dependence is non - trivial and that @xmath150 and @xmath151 . pairing both sides of with @xmath152 tells us the number of non - zero @xmath149 is odd ; similarly pairing both sides of with @xmath153 tells us the number of non - zero @xmath149 is even . the resulting contradiction implies that , in fact , for any non - trivial linear dependence among the @xmath148 , the @xmath149 are all one . thus , the @xmath148 are either independent , implying @xmath154 , or there is a unique linear dependence with full support among them , implying @xmath155 . as a corollary , we obtain : [ prop : all - intersecting ] let @xmath44 , and let @xmath2 be a @xmath7-system in a closed genus @xmath0 surface with any pair of curves intersecting an odd number of times . then @xmath156 . if the minimal geometric intersection number between essential , non - isotopic , simple closed curves is odd , then their algebraic intersection number mod @xmath36 is @xmath23 and , furthermore , their @xmath22-homology classes must be distinct . apply proposition [ prop : linalg ] . we prove theorem [ theo : alloddupper ] , since the proof of theorem [ theo : lowdegree ] is nearly identical . the upper bound when the graph @xmath39 is @xmath45 is proposition [ prop : all - intersecting ] . if @xmath39 has average degree @xmath48 , then its complement has average degree @xmath51 , and by turn s theorem @xcite , must contain an independent set of size @xmath157 . applying proposition [ prop : all - intersecting ] to the curves in @xmath2 represented by the corresponding clique in @xmath39 , we see that @xmath158 . @xmath61 the bound of proposition [ prop : linalg ] is tight , as the following example shows . define @xmath159 , @xmath160 , and @xmath161 . for @xmath50 , inductively define @xmath162 for @xmath163 $ ] and then @xmath164 , @xmath165 , and @xmath166 . ( the semi - colons mean concatenating vectors . ) .,scaledwidth=70.0% ] [ prop : linalg - tight ] let @xmath43 , and let @xmath167 be defined as above . for all @xmath9 , we have @xmath168 . for @xmath169 , this is an easy computation , and the @xmath50 cases follow by induction . figure [ fig : curves - g2 ] shows curves with @xmath22-homology classes given by the @xmath170 . the vectors @xmath167 are canonical . [ prop : linalg - canonical ] let @xmath43 and suppose that @xmath171 have the property that , for @xmath6 , @xmath172 . then there is a symplectic automorphism @xmath73 of @xmath143 such that @xmath173 . in the proof , we need the basic fact : [ lemma : linalg - indep ] let @xmath174 and @xmath175 be linearly independent sets of vectors such that @xmath176 for all @xmath40 and @xmath41 . then there is a symplectic automorphism @xmath73 of @xmath143 such that @xmath173 . let @xmath73 be the automorphism from lemma [ lemma : linalg - indep ] applied to the @xmath177 and @xmath178 from the statement of the theorem . this is allowed , because the proof of proposition [ prop : linalg ] says that for any collections meeting the hypothesis of the @xmath167 and @xmath179 , we have @xmath180 with the first @xmath181 vectors independent . thus , we see that @xmath182 we can explicitly describe a @xmath23-system with @xmath183 curves all intersecting pairwise . [ theo : canonical - curves ] for any genus @xmath184 closed surface , there is a @xmath23-system of @xmath183 curves which all pairwise intersect . a genus @xmath0 surface is homeomorphic to a regular @xmath185-gon with opposite sides identified . we obtain @xmath181 simple closed curves from simple arcs connecting opposite sides of the @xmath185-gon . after identifying sides , all vertices become identified , so we can add a diagonal to the configuration as well . later , we will use the following slightly stronger statement which is clear from the construction . [ cor : canonical - curves - same - point ] the @xmath23-system from theorem [ theo : canonical - curves ] can be chosen so that all curves pairwise intersect at the same point on the surface . in both genus @xmath23 and @xmath36 , it can be shown that , up to the mapping class group , there is only one maximal configuration of curves all of which pairwise intersect exactly once . in genus @xmath36 , the configuration corresponds to a star graph in the quotient under hyperelliptic involution . it would be interesting to know if this generalizes . in higher genus surfaces , is there only one mapping class orbit of @xmath23-systems such that each pair of curves intersects ? we now prove the upper bound of theorem [ theo : boundaryasymptotics ] . first , we improve proposition [ prop : all - intersecting ] [ lemma : all - intersecting - boundary ] the statement of proposition [ prop : all - intersecting ] holds for a genus @xmath0 surface with any number of boundary components . suppose @xmath2 is a @xmath7-system in a genus @xmath0 surface with @xmath25 boundary components with any pair of curves intersecting an odd number of times . glue in discs into all the boundary components to obtain a closed surface @xmath1 . since each curve intersects any other curve transversely an odd number of times , no pair of curves of @xmath2 can be pairwise isotopic in @xmath1 . consequently , @xmath2 is a @xmath7-system satisfying the hypothesis of proposition [ prop : all - intersecting ] and hence @xmath186 . we argue by induction and show that @xmath187 let @xmath2 be a maximal @xmath23-system in @xmath188 , the surface of genus @xmath0 and @xmath25 boundary components . glue in a disc @xmath51 to one of the boundary components to obtain @xmath189 . since intersection numbers did not increase , curves in @xmath2 still pairwise intersect at most once , but some curves may have become isotopic as a result of gluing in the disc . we will show that a @xmath23-system @xmath190 on @xmath189 may be obtained by removing at most @xmath183 curves from @xmath2 . then , we have @xmath191 if two curves become isotopic after gluing in @xmath51 , then since @xmath2 is a @xmath23-system , they are disjoint and hence bound an annulus which necessarily contains @xmath51 . in particular , any curve can become isotopic to at most @xmath23 other curve , so we must understand how many pairs of isotopic curves can occur . if @xmath192 and @xmath193 are two such pairs , then both pairs bound an annulus containing @xmath51 and the annuli must intersect . consequently , each curve in an isotopic pair intersects every curve in the other pairs . if we construct a set @xmath94 by taking one curve from each pair , then @xmath94 is as in lemma [ lemma : all - intersecting - boundary ] and so @xmath194 . thus @xmath195 is the @xmath23-system on @xmath189 as desired . in the case of the torus , we have @xmath196 . indeed , if two curves on a one - holed torus become isotopic after gluing in the disc @xmath51 , then , as before , they bound an annulus containing @xmath51 ; however , since the surface is a torus , they also bound an annulus , not containing @xmath51 , on the other side , and so they must have already been isotopic in the one - holed torus . @xmath61 in the next section , we will see that obtaining our lower bounds essentially amounts to reversing the argument in the previous paragraph . [ sec : lower ] in this section we prove the lower bounds of theorem [ theo : generalbounds ] and theorem [ theo : boundaryasymptotics ] . we will first prove theorem [ theo : boundaryasymptotics ] . the lower bound of theorem [ theo : generalbounds ] then follows easily by attaching handles . the base case is corollary [ cor : canonical - curves - same - point ] . suppose we have constructed @xmath2 for @xmath188 . let @xmath51 be a small disc which contains @xmath199 and intersects only the curves from @xmath94 . via a homeomorphism , we can identify @xmath51 with the standard unit disc in @xmath200 and the arcs from @xmath94 as straight - line diagonals all intersecting at the center @xmath201 , none of which is vertical . remove a small disc @xmath202 directly above @xmath199 , and construct @xmath183 new simple closed curves as follows . for each arc @xmath101 in @xmath51 , place a new arc @xmath203 in @xmath51 parallel to @xmath101 but above @xmath202 . see figure [ figure : lowerbound ] . obtain a simple closed curve @xmath77 by continuing @xmath203 outside of @xmath51 along the curve @xmath76 in @xmath94 which contains @xmath101 . this can be done over all @xmath204 so that : * for all @xmath205 , the corresponding new curves @xmath206 deposit straight lines in @xmath51 and otherwise outside of @xmath51 , the curves @xmath207 are pairwise disjoint . hence , pairwise intersections among @xmath207 are at most @xmath23 . * for all @xmath204 and @xmath208 , the new curve @xmath77 has intersection @xmath209 let @xmath190 be @xmath2 with the newly constructed curves added . each new curve @xmath77 bounds with its `` parent '' @xmath76 an annulus containing the ( removed ) disc @xmath202 . since @xmath28 , the complement of the annulus has genus at least @xmath23 , and thus is not an annulus ; consequently @xmath76 and @xmath77 are non isotopic . furthermore , @xmath77 is not isotopic to any other curve in @xmath190 since if it were , then , after replacing @xmath202 , we would see that @xmath76 were isotopic to some other curve in @xmath2 , a contradiction . note that @xmath190 has exactly @xmath183 more curves and @xmath210 still has the desired properties . notice that this argument fails in @xmath31 only because if @xmath211 , then the complement of the annulus containing @xmath51 would in fact be another annulus . however , once @xmath32 , the complement would contain a boundary component , and the argument proceeds mutatis mutandis , but with the smaller lower bound of @xmath212 . @xmath61 the cases @xmath213 were established in section [ sec : gen2 ] so assume @xmath214 . let @xmath215 if @xmath0 is even and @xmath216 if @xmath0 is odd and let @xmath217 . by theorem [ theo : boundaryasymptotics ] , there is a @xmath23-system @xmath2 of size @xmath218 on a surface of genus @xmath219 and @xmath220 boundary components . gluing @xmath25 handles to the @xmath220 boundary components does not cause curves to become isotopic , the system consisting of @xmath2 and the @xmath25 curves going around the handles is a @xmath23-system of at least @xmath221 curves on a genus @xmath0 surface . @xmath61 joan s. birman and hugh m. hilden . on the mapping class groups of closed surfaces as covering spaces . in _ advances in the theory of riemann surfaces ( proc . conf . , stony brook , n.y . , 1969 ) _ , pages 81115 . ann . of math . studies , no . princeton univ . press , princeton , n.j . , 1971 . andrew haas and perry susskind . the geometry of the hyperelliptic involution in genus two . _ , 1050 ( 1):0 159165 , 1989 . issn 0002 - 9939 . doi : 10.2307/2046751 . url http://dx.doi.org/10.2307/2046751 . m. juvan , a. malni , and b. mohar . systems of curves on surfaces . _ j. combin . theory ser . b _ , 680 ( 1):0 722 , 1996 . issn 0095 - 8956 . doi : 10.1006/jctb.1996.0053 . url http://dx.doi.org/10.1006/jctb.1996.0053 . van h. vu . extremal set systems with weakly restricted intersections . _ combinatorica _ , 190 ( 4):0 567587 , 1999 . issn 0209 - 9683 . doi : 10.1007/s004939970008 . url http://dx.doi.org/10.1007/s004939970008 . | we give an exponential upper and a quadratic lower bound on the number of pairwise non - isotopic simple closed curves can be placed on a closed surface of genus @xmath0 such that any two of the curves intersects at most once . although the gap is large , both bounds are the best known for large genus . in genus one and two , we solve the problem exactly .
our methods generalize to variants in which the allowed number of pairwise intersections is odd , even , or bounded , and to surfaces with boundary components .
= 1 |
You are an expert at summarizing long articles. Proceed to summarize the following text:
binary models like ising - type simulation have a long history . they have been applied by schelling to describe the ghetto formation in the inner cities of the usa , i.e. , to study phase separation between black and white @xcite . in the sociophysics context , recently , many social phenomena such as election , propagation of information , predicting features of traffic , migration , opinion dynamics and formation in a social group have been successful modelled based on ising spin systems using models and tools of statistical physics . with this respect , particularly successful models have been developed by sznajd @xcite , deffuant et al.@xcite and hegselmann and krause @xcite . among those three models , the one developed by sznajd is the most appropriate for simulation in networks and lattices , since it consider just the interactions between the nearest neighbors . indeed , the sznajd model has been successfully applied to model sociophysical and economic systems @xcite . on the other hand , several modifications of the sznajd model have been studied using different rules or topologies starting from different initial opinion densities @xcite . all these models are static ( i.e. not dynamic ) and they allow for consensus ( one final opinion ) , polarization ( two final opinion ) , and fragmentation ( more than two final opinions ) , depending on how tolerant people are to different opinions . more recently the striking sociophysical model has been suggested by aydiner @xcite in order to explain the time evolution of resistance probability of a closed community in a one - dimensional sznajd like model based on ising spin system . it has been shown that resistance probability in this model decay as a stretched exponential with time . in that model spins does not move on the lattice sites during the simulation , so this model was so - called static . however , in a realistic case , spins i.e. , people move in the community i.e. , in the space . social or opinion formation formed depend upon dynamics of the system . because , there must be a direct connection between opinion dynamics and formation in a social system since the social formation is determined by the dynamics . meyer - ortmanns @xcite studied recent work in which the condition for ghetto formation in a population with natives and immigrants by using kawasaki - exchange dynamics in a two dimensional ising model . she showed that ghetto formation can be avoided with a temperature increasing with time . similarly , schulze have also generalized meyer - ortmanns work to up to seven different ethnic groups to explain ghetto formation in a multi - cultural societies in a potts - like model @xcite . in this study , we have developed a dynamic version of the aydiner @xcite model by combining the aydiner and meyer - ortmanns @xcite models based on one - dimensional ising model . in one - dimensional static model @xcite , each site carriers a spin which is either spin up ( + 1 ) or spin down ( -1 ) randomly . spin up ( + 1 ) represent the host people and spin down ( -1 ) represent the soldier . the host people always against occupation , and , on the other hand , soldier always willing to continue occupation , who always have the opinion opposite of that of the host people . furthermore , the community member i.e. , spins does nt also move on the lattice during the process . in this model , initially , it was assumed that there was a over all consensus among member of the community against occupation even if some exceptions exist . one expects that host people obey to this consensus at least initially . in this sense , community behaves as polarized at zero social temperature @xcite against occupation just like ising ferromagnet at zero temperature . it was conjectured that host people are influenced by soldiers even though they against occupation owing to they are exposed to intensive biased information or propagation . soldiers affect the host people and force to change their opinion about occupation . effected people may change their own opinions depending on resistance probability of the nearest neighbors about occupation . moreover , effected host people affect neighbors . such a mechanism depolarize the polarization ( resistance probability ) of all host people . hence social polarization destroy . however , soldiers , unlike host people , have not been influenced by the host people . their opinion about justifying the occupation does not change during the occupation process , since they may be stubborn , stable or professional etc . , who behaves like persistent spins in ising spin system . it is means that the probability of the against occupation of a soldier is always zero . if we summarize , we can say that none spins does flip fully in the system . spin up always remains spin up , and spin down always remains spin down . in this respect , the probability of against occupation of host people can be interpreted as a survival probability of opinion of host people about occupation under above considerations . in this sense , the survival probability @xmath0 of opinion of host people indicate equal to @xmath1 at least initially and , on the other hand , the probability of against occupation of soldier equal to zero , which means that soldier behaves as a trap point lattice which depolarize the survival probability of opinion of host people . of course , one may suggest that there are many different number of opinions in society , however , it is possible to find that a society being formed two - state opinion in a real case . therefore this model is a good example for two - state opinion model as well galam contrarian model @xcite even though it seems that it is very simple . furthermore , in real social systems , people move on the space , i.e. , lattice . therefore , in this study , we assumed that people i.e. , spins randomly move on the lattice through the kawasaki - exchange dynamics contrary to previous model . the survival probability @xmath2 for a people at site @xmath3 at the next time @xmath4 is determined with the survival probability of nearest - neighbors with previous time @xmath5 as @xmath6.\label{eq1}\ ] ] we note that the survival probability for all site are calculated as synchronously . randomly motion of the spins i.e. , people on the lattice through the kawasaki - exchange dynamics . firstly , a spin pair is chosen randomly and then it is decided whether spin pair exchange with each other or not . in this approach , the nearest - neighbor spins are exchanged under heat - bath dynamics , i.e. , with probability @xmath7 , where @xmath8 is the energy change under the spin exchange , @xmath9 is the boltzmann constant , and @xmath10 is the temperature i.e. , social temperature or tolerance . hence , to obtain probability @xmath11 we need to calculate @xmath12 and @xmath13 which correspond to energy of the spin pair at first position and after exchange with position of spins , respectively . energy @xmath12 and @xmath13 can be calculated in terms of the survival probability instead of spin value as @xmath14 @xmath15 where @xmath16\ ] ] and @xmath17\ ] ] energy difference is written as @xmath18 from eq . ( [ eq2a ] ) and ( [ eq2b ] ) . in addition , the total survival probability of opinion of host people at the any time @xmath5 can be obtained over each person for any @xmath19 configuration as @xmath20 where @xmath21 is the initial number of host people . on the other hand , the averaged survival probability at the any time @xmath5 can be obtained from eq . ( [ eq3 ] ) over the independent configuration as @xmath22 where @xmath23 is the number of different configurations . we have adopted the monte carlo simulation technique to the one - dimensional sociophysical model using the lattice size @xmath24 with periodic boundary condition , and independent configuration @xmath25 for the averaged results . the simple algorithm for the simulation is as follows : i ) at the @xmath26 , eq . ( [ eq4 ] ) is initially calculated , ii ) for @xmath27 a spin pair is randomly chosen , and then it is decided whether the spin pair exchange or not with the probability @xmath28 , this step is repeated @xmath29 times , iii ) after ii - steps are completed , eq . ( [ eq4 ] ) is recalculated again , and to continue this procedure goes to step ii . the simulation results are as follow : we have firstly plotted simulation data versus time in fig . [ fig1 ] in a several manner . it is explicitly seen from figs . [ fig1](a)-(c ) that there are no power , exponential and logarithmic law dependence in our simulation data , respectively . however , as seen fig . [ fig1](d ) , data well fit to the stretched exponential function as @xmath30 where @xmath31 is the relaxation constant , and @xmath32 is the decay exponent of the survival probability . this result indicate that the time evaluation of survival probability of the opinion of the host people in a closed community has stretched exponential character i.e. , kohlraush - william - watts ( kww ) decay law @xcite . it should be tested whether fig . [ fig1](d ) satisfies to stretched exponential or not @xcite . because , as noted by stauffer , the fig . [ fig1](d ) would work as stretched exponential , if pre - factor of eq . ( [ eq5 ] ) is equal to 1 . however , if pre - factor is less than @xmath1 , it may give the impression of stretched exponential form , even for @xmath33 . therefore , it can be plotted @xmath34 versus suitable powers of @xmath5 , like @xmath35 , @xmath36 , etc . , and find out the best straight line among the powers of @xmath5 for long times . hence , @xmath34 was plotted versus powers of @xmath5 for @xmath37 then the best straight fitting line for long times was obtained for @xmath38 for @xmath39 , @xmath40 for @xmath41 , and @xmath42 for @xmath43 as seen in fig . [ fig3](a)-(c ) respectively . these results confirm to this method used to find out stretched exponential exponents in fig . [ fig1](d ) , and also all figures in fig . [ fig2 ] as mentioned . also , this test indicates that prefactor in eq . ( [ eq5 ] ) does not effect results presented in this paper . it is concluded that results for high temperatures also consistent with static model @xcite . but , unlike the static model , time crossover has been observed in dynamic model at low temperatures . in order to investigate the transition we have plotted survival probability versus time for different social temperature @xmath10 in fig . it is clearly seen that the time crossover occurs depend on social temperature . when social temperature decreases , the crossover become more clear . such a behavior was not observed in a static model . we can bridge the short time regime and the long time regime by a scaling function @xmath44 @xmath45 where @xmath46 indicates the time crossover . for our simulation data , the scaling relation ( 6 ) can be written for very long and very short time intervals as @xmath47{c}e^{-\left ( t/\tau\right ) ^{\beta_{1}}}\\ e^{-\left ( t/\tau\right ) ^{\beta_{2}}}\end{array } \begin{array } [ c]{l}if \hspace{0.3 cm } t<<t_{c}\\ if \hspace{0.3 cm } t>>t_{c}.\end{array}\ ] ] on the other hand , in order see how the decay exponent @xmath32 depend on soldier density @xmath48 , and social temperature @xmath10 , we have plotted @xmath32 versus soldier density @xmath48 in fig . [ fig4](a ) for @xmath49 and @xmath50 in account to taken different social temperatures , and social temperature @xmath10 in fig . [ fig4](b ) for a fixed value of density @xmath48 , respectively . as seen from fig . [ fig4](a ) that @xmath51 and @xmath52 are linearly depend on soldier density both of two regimes at low social temperature . on the other hand , the decay exponent has two different character for @xmath49 and @xmath50 depend on social temperature @xmath10 in fig . [ fig4](b ) , the decay exponent @xmath51 decreases with increasing temperature @xmath10 for @xmath49 , whereas @xmath52 increases with increasing temperature @xmath10 for @xmath50 at low temperatures . however , for relatively high temperatures we roughly say that @xmath51 approach to @xmath52 for both two regimes obey to eq . ( [ eq7 ] ) . finally , to understand the social temperature and soldier density dependence of the time crossover @xmath53 , we have plotted @xmath53 versus social temperature @xmath10 in fig . [ fig5](a ) for a fixed soldier density @xmath48 , and versus soldier density in fig . [ fig5](b ) for fixed social temperature @xmath10 , respectively . it seems from fig . [ fig5](a ) that the crossover transition @xmath53 quite rapidly decrease with increasing @xmath10 , on the other hand , it seems from fig . [ fig5](b ) that it slowly decrease with increasing soldier density @xmath48 . we note that as seen inserted figure in fig . [ fig5](b ) the crossover transition @xmath53 depends on soldier density with power law for fixed social temperature . we suggest that the stretched exponential behavior of decay must be originated from model system . the persistent spins i.e. , the soldiers does nt flip during simulation , therefore they behave as a trap in the system . hence they play a role diminishing the survival probability of the neighbor spins in the system . consequently , decay characteristic of the system can be explain due to the trapping states . another say , this characteristic behavior does nt depend on either diffusion dynamics of spins or interaction rules between spins . another unexpected behavior is the time crossover in @xmath32 contrast to previous model @xcite . we supposed that this amazing result originated from opinion dynamics depend on social temperature . model allows to the opinion formation with time . indeed , there is a direct connection between opinion dynamics and formation in a social system since the social formation is determined by the dynamics as depend on the social temperature . for example , in a real spin system , decreasing temperature phase separation may occur in the system . in the sociophysical sense , it means that people who have different opinion are separated each other with decreasing social tolerance , and therefore the ghetto formation or polarization may occur in the system . it is expected that interactions between soldier and host people is maximum when soldiers are randomly distributed in the community . as social temperature , i.e. , tolerance is decreased , however , phase separation occur with time , so this leads to decreasing of the interactions . in our opinion , the ghetto formation in the system does nt leads crossover transition in time because of the ghetto formation is randomly distributed relatively . on the other hand , the time average of survival probability over different configuration effect of ghetto formation may probably destroy . so we do nt hope that ghetto formation is not responsible crossover transition . however , polarization must be occurred at low temperature leads to meaningful phase separation in the system . such a polarization may leads to crossover transition in time . stretched exponential behavior indicates mathematically that decay for the relatively short times is fast , but for relatively long times it is slower . one can observe that this mathematical behavior corresponds to occupation processes in the real world . in generally , a military occupation is realized after a hot war . the community does not react to occupation since it occurs as a result of defeat . people are affected easily by propaganda or other similar ways . therefore , it is not surprised that resistance probability decrease rapidly at relatively short times . on the other hand , spontaneous reaction may begin against occupation in the community after the shock . hence , community begins by regaining consciousness and more organized resistance may display difficulties for occupants . for long times , the resistance probability decreases more slowly . this means that resistance against occupation extends to long times in practice . at this point , the number of soldiers is also important , because the density of soldiers determines the speed of decaying . the different regimes have been observed in the decay of the survival probability . these regimes clearly appear particularly at low temperatures . in the case of the social temperature is very low , @xmath51 is bigger than @xmath52 which indicates the decay of the survival probability for relatively short time is slower than for relatively long time . this can be interpreted that the resistance of host people against occupation may be broken spontaneously if soldier can wait enough time . of course , the mechanism considered in this work can be regarded as simple , but , it would be useful to understand the time evolution of the resistance probability of the community against to occupation in the one - dimensional model under some considerations . we remember that simple social rules lead to complicated social results , hence we believe that the obtained results and model can be applied the real social phenomena in the societies to understand the basis of them . authors are grateful dietrich stauffer for the suggestions in the preparation of this paper . 000 g. deffuant , d. neau , f. amblard , and g. weisbuch , adv . compl . syst . * 3 * , 87 ( 2000 ) ; g. deffuant , f. amblard , g. weisbuch , and t. faure , artificial societies and social simulation * 5 * ( 4 ) , 1 ( 2002 ) ( jass.soc.surrey.ac.uk ) . | the time dependence of the survival probability of an opinion in a closed community has been investigated in accordance with social temperature by using the kawasaki - exchange dynamics based on previous study in ref . [ 1 ] .
it is shown that the survival probability of opinion decays with stretched exponential law consistent with previous static model .
however , the crossover regime in the decay of the survival probability has been observed in this dynamic model unlike previous model .
the decay characteristics of both two regimes obey to stretched exponential .
* keywords : * ising model ; politics ; random walk ; sociophysics ; sznajd model . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
this is a research on the interface between topology and graph theory with applications to motion planning algorithms in robotics . we consider moving objects as zero - size points travelling without collisions along fixed tracks forming a graph , say on a factory floor or road map . we prefer to call these objects ` robots ' , although the reader may use a more neutral and abstract word like ` token ' . for practical reasons we study discrete analogues of configuration spaces of graphs , where robots can not be very close to each other , roughly one edge apart . this discrete approach reduces the motion planning of real ( not zero - size ) vehicles to combinatorial questions about ideal robots moving on a subdivided graph . first we recall basic notions . a _ graph _ @xmath0 is a 1-dimensional finite cw complex , whose 1-cells are supposed to be open . the 0-cells and open 1-cells are called _ vertices _ and _ edges _ , respectively . if the endpoints of an edge @xmath1 are the same then @xmath1 is called a _ loop_. a _ multiple _ edge is a collection of edges with the same distinct endpoints . the topological _ closure _ @xmath2 of an edge @xmath1 is the edge @xmath1 itself with its endpoints . the _ degree _ @xmath3 of a vertex @xmath4 is the number of edges attached to @xmath4 , i.e. a loop contributes 2 to the degree of its vertex . vertices of degrees 1 and 2 are _ hanging _ and _ trivial _ , respectively . vertices of degree at least 3 are _ essential_. a _ path _ ( a _ cycle _ , respectively ) of length @xmath5 in @xmath0 is a subgraph consisting of @xmath5 edges and homeomorphic to a segment ( a circle , respectively ) . a _ tree _ is a connected graph without cycles . the direct product @xmath6 ( @xmath7 times ) has the product structure of a ` cubical complex ' such that each product @xmath8 is isometric to a euclidean cube @xmath9^k$ ] , where @xmath10 is the topological closure of a cell of @xmath0 . the dimension @xmath5 is the number of the cells @xmath11 that are edges of @xmath0 . the _ diagonal _ of the product @xmath12 is @xmath13 [ def : topconfigurationspaces ] let @xmath0 be a graph , @xmath7 be a positive integer . the _ ordered topological _ configuration space @xmath14 of @xmath7 distinct robots in @xmath0 is @xmath15 . the _ unordered topological _ configuration space @xmath16 of @xmath7 indistinguishable robots in @xmath0 is the quotient of @xmath14 by the action of the permutation group @xmath17 of @xmath7 robots . the ordered topological space @xmath18,2)$ ] is the unit square without its diagonal @xmath19 ^ 2 { { \ ; | \;}}x\neq y\}$ ] , which is homotopy equivalent to a disjoint union of 2 points . topological spaces @xmath20 are _ homotopy _ equivalent if there are continuous maps @xmath21 , @xmath22 such that @xmath23 , @xmath24 can be connected with @xmath25 , @xmath26 , respectively , through continuous families of maps . in particular , @xmath27 is _ contractible _ if @xmath27 is homotopy equivalent to a point . a space @xmath27 can be homotopy equivalent to its subspace @xmath28 through a _ deformation retraction _ that is a continuous family of maps @xmath29 , @xmath30 $ ] , such that @xmath31 , i.e. all @xmath32 are fixed on @xmath28 , @xmath33 and @xmath34 . the unordered topological space @xmath35,2)\approx \{(x , y)\in[0,1]^2 { { \ ; | \;}}x < y\}$ ] is contractible to a single point . more generally , @xmath18,n)$ ] has @xmath36 contractible connected components , while @xmath35,n)$ ] deformation retracts to the standard configuration @xmath37 , @xmath38 , in @xmath9 $ ] . if a connected graph @xmath0 has a vertex of degree at least 3 then the configuration spaces @xmath14 , @xmath16 are path - connected . we swap robots @xmath39 near such a vertex as shown in figure [ fig : permuterobotstriod ] . without collisions on the triod @xmath40 ] [ def : graphbraidgroups ] given a connected graph @xmath0 having a vertex of degree at least @xmath41 , the _ graph braid _ groups @xmath42 and @xmath43 are the fundamental groups @xmath44 and @xmath45 , respectively , where arbitrary base points are fixed . for the triod @xmath40 in figure [ fig : permuterobotstriod ] , both configuration spaces @xmath46 , @xmath47 are homotopy equivalent to a circle , see example [ exa : topconf2pointt ] , i.e. @xmath48 , @xmath49 , although @xmath50 can be considered as an index 2 subgroup @xmath51 of @xmath48 . [ def : discconfigurationspaces ] the _ ordered discrete _ space @xmath52 consists of all the products @xmath53 such that each @xmath11 is a cell of @xmath0 and @xmath54 for @xmath55 . the _ unordered discrete _ space @xmath56 is the quotient of @xmath52 by the action of @xmath17 . the _ support _ @xmath57 of a subset @xmath58 is the minimum union of closed cells containing @xmath59 . for instance , the support of a vertex or open edge coincides with its topological closure in @xmath0 , while the support of a point interior to an open edge @xmath1 is @xmath2 , i.e. the edge @xmath1 with its endpoints . a configuration @xmath60 is _ safe _ if @xmath61 whenever @xmath55 . then @xmath52 consists of all safe configurations : @xmath62 . a path in a graph @xmath0 is _ essential _ if it connects distinct essential vertices of @xmath0 . a cycle in @xmath0 is _ essential _ if it contains a vertex of degree more than 2 . since only connected graphs are considered , a non - essential cycle coincides with the whole graph . subdivision theorem [ the : subdivision ] provides sufficient conditions such that the configuration spaces @xmath63 deformation retract to their discrete analogues @xmath64 , respectively . then @xmath65 . [ the : subdivision ] ( * ? ? ? * theorem 2.1 ) let @xmath0 be a connected graph , @xmath66 . the discrete spaces @xmath64 are deformation retracts of the topological configuration spaces @xmath67 , respectively , if both conditions ( [ the : subdivision]a ) and ( [ the : subdivision]b ) hold : ( [ the : subdivision]a ) every essential path in @xmath0 has at least @xmath68 edges ; ( [ the : subdivision]b ) every essential cycle in @xmath0 has at least @xmath68 edges . the conditions above imply that @xmath0 has at least @xmath7 vertices , so @xmath69 . a strengthened version of subdivision theorem [ the : subdivision ] for @xmath70 only requires that @xmath0 has no loops and multiple edges ( * ? ? ? * theorem 2.4 ) . hence the topological configuration spaces of 2 points on the kuratowski graphs @xmath71 deformation retract to their smaller discrete analogues , which are easy to visualise , see figure [ fig : kuratowskigraphs ] . in @xmath72 , if the 1st robot is moving along an edge @xmath73 , then the 2nd robot can be only in the triangular cycle @xmath74 , which gives in total 10 triangular tubes @xmath75 forming the oriented surface of genus 6 . similarly , computing the euler characteristic , we may conclude that @xmath76 is the oriented surface of genus 4 . these are the only graphs without loops whose discrete configuration spaces @xmath77 are closed manifolds , see ( * ? ? ? * corollary 5.8 ) . and @xmath78 there are two different approaches to computing graph braid groups suggested by abrams @xcite and farley , sabalka ( * ? ? ? * theorem 5.3 ) . in the former approach a graph braid group splits as a graph of simpler groups , which gives a nice global structure of the group and proves that , for instance , the graph braid groups are torsion free ( * ? ? ? * corollary 3.7 on p. 25 ) . the latter approach based on the discrete morse theory by forman @xcite writes down presentations of graph braid groups retracting a big discrete configuration space to a smaller subcomplex . we propose another local approach based on classical seifert van kampen theorem [ the : seifertvankampen ] . presentations are computed step by step starting from simple graphs and adding edges one by one , which allows us to update growing networks in real - time . resulting algorithm [ alg : graphbraidgroups ] expresses generators of graph braid groups in terms actual motions of robots , i.e. as a list of positions at discrete time moments . we also design motion planning algorithm [ alg : motionplanningunordered ] connecting any configurations of @xmath7 robots . its complexity is linear in the number of edges and quadratic in the number of robots . [ alg : graphbraidgroups ] there is an algorithm writing down a presentation of the graph braid group @xmath43 and representing generators by actual paths between configurations of robots , see step - by - step instructions in subsection [ subs : motionplanningunordered ] . according to ( * ? ? ? * theorem 5.6 ) , the braid groups of planar graphs having only disjoint cycles have presentations where each relator is a commutator , not necessarily a commutator of generators . demonstrating the power of algorithm [ alg : graphbraidgroups ] , we extend this result to a wider class of light planar graphs . a planar connected graph @xmath0 is called _ light _ if any cycle @xmath79 has an open edge @xmath80 such that all cycles from @xmath81 do not meet @xmath82 . any loop or multiple edge provides an edge @xmath80 satisfying the above condition . figure [ fig : triangulargraph ] shows a non - light planar graph with 4 choices of a ( dashed ) edge @xmath80 and corresponding ( fat ) cycles from @xmath81 . removing the closure @xmath83 from @xmath0 is equivalent to removing the endpoints of @xmath80 and all open edges attached to them . ] [ cor:2pointgroups ] the braid group @xmath84 of any light planar graph @xmath0 has a presentation where each relator is a commutator of motions along disjoint cycles . a stronger version of corollary [ cor:2pointgroups ] with a geometric description of generators and relators is given in proposition [ pro:2pointgroupsunordered ] in the case of unordered robots . * outline . * in section [ sect : discretespaces ] we consider basic examples and recall related results . section [ sect : fundamentalgroupsunordered ] introduces the engine of propositions [ pro : addhangingedgeunordered ] , [ pro : stretchhangingedgeunordered ] , [ pro : createcyclesunordered ] updating presentations of graph braid groups by adding edges one by one . section [ sect : computinggroupsunordered ] lists step - by - step instructions to compute a presentation of an arbitrary graph braid group . as an application , we geometrically describe presentations of 2-point braid groups of light planar graphs . further open problems are stated in subsection [ subs : openproblems ] . * acknowledgements . * the author thanks michael farber for useful discussions and lucas sabalka sending an early version of his manuscript @xcite . in this section we discuss discrete configuration spaces in more details and construct them recursively in lemmas [ lem : recursiveconstructionunordered ] and [ lem : recursiveconstructionordered ] . further we assume that @xmath66 . in this subsection we describe configuration spaces of 2 points on the triod @xmath40 comprised of 3 hanging edges @xmath85 attached to the vertex @xmath4 , see figure [ fig : triod2pointproducts ] . and @xmath86 , @xmath87 , @xmath88 ] [ exa : topconf2pointt ] the ordered topological space @xmath46 is the union of three 3-page books @xmath89 , @xmath90 , @xmath91 shown in the right pictures of figure [ fig : triod2pointproducts ] without the diagonal @xmath92 . then @xmath46 consists of the 6 symmetric rectangles @xmath93 ( @xmath55 ) and 6 triangles from the squares @xmath94 , @xmath95 , after removing their diagonals , see the left picture of figure [ fig : triod2pointspaces ] and ( * ? ? ? * example 6.26 ) . and its discrete analogue @xmath96 ] [ exa : discconf2pointt ] the ordered topological space @xmath46 deformation retracts to the polygonal circle in the right picture of figure [ fig : triod2pointspaces ] , which is the ordered discrete space @xmath96 having 12 vertices @xmath97 ( @xmath55 ) and @xmath98 , @xmath99 , @xmath95 , symmetric under the permutation of factors . the unordered spaces @xmath100 are quotients of the corresponding ordered spaces by the rotation through @xmath101 and are homeomorphic to the same spaces @xmath102 , respectively . hence the graph braid groups @xmath48 , @xmath49 can be computed using the simpler discrete spaces @xmath103 , which is reflected in subdivision theorem [ the : subdivision ] . in this subsection we explain recursive constructions of discrete configuration spaces that will be used in section [ sect : fundamentalgroupsunordered ] to compute their fundamental groups . [ exa : recursive2pointt ] we show how to construct the unordered space @xmath104 adding the closed edge @xmath105 to the subgraph @xmath106 $ ] . if both robots @xmath39 are not in the open edge @xmath107 , then @xmath108 , where @xmath109 $ ] , i.e. either @xmath110 , @xmath111 $ ] or @xmath112,2)$ ] . the robot @xmath113 can not be close to @xmath114 by definition [ def : discconfigurationspaces ] , e.g. if @xmath115 then @xmath116 , i.e. @xmath117 or @xmath118 . then @xmath119\times v_1)\cup { { \mathrm{\bf ud}}}([0,1],2)\cup(\{v_2,v_3\}\times\bar e_1),\ ] ] where the segments @xmath120 and @xmath121 are glued at the endpoints @xmath122 and @xmath123 , respectively . up to a homeomorphism , we get 2 arcs attached at theirs endpoints to a solid triangle without one side , see the left picture of figure [ fig : attachcylinder ] . ] the argument of example [ exa : recursive2pointt ] motivates the following notion . the _ neighbourhood _ @xmath124 of an open edge @xmath125 consists of @xmath2 and all open edges attached to the endpoints of @xmath1 . for instance , the complement to the neighbourhood @xmath126 in the triod @xmath40 consists of the hanging vertices @xmath127 , see the left picture of figure [ fig : triod2pointproducts ] . [ exa : recursive2pointg ] extending the recursive idea of example [ exa : recursive2pointt ] , we construct the unordered 2-point space @xmath128 of any connected graph @xmath0 . fix an open edge @xmath129 with vertices @xmath130 and consider the case when one of the robots , say @xmath114 , stays in @xmath1 , then @xmath131 , because @xmath113 can not be in the same edge @xmath1 and also in the edges adjacent to @xmath1 . if both robots @xmath39 are not in @xmath1 then @xmath132 is in the smaller unordered space @xmath133 . then @xmath128 is a union of smaller subspaces : @xmath134 where the cylinder @xmath135 is glued to @xmath133 along the subgraphs @xmath136 and @xmath137 . the reduction above extends to a general recursive construction in lemma [ lem : recursiveconstructionunordered ] . lemmas [ lem : recursiveconstructionunordered ] and [ lem : recursiveconstructionordered ] are discrete analogues of ghrist s construction of the ordered topological space @xmath14 ( * ? ? ? * lemma 2.1 ) . [ lem : recursiveconstructionunordered ] let a graph @xmath0 have an open edge @xmath1 with vertices @xmath130 . then the unordered discrete space @xmath56 is homeomorphic to ( see figure [ fig : attachcylinder ] ) @xmath138 the cylinder @xmath139 is glued to @xmath140 along @xmath141 in the space @xmath56 of all safe configurations @xmath142 consider the smaller subspace @xmath140 , where @xmath143 for each @xmath38 . the complement @xmath144 consists of configurations with(say ) @xmath145 . here the index @xmath7 is not important since the robots are not ordered . by definition [ def : discconfigurationspaces ] , the other robots @xmath146 , i.e. the complement is @xmath147 the bases of the last cylinder are subspaces of the smaller configuration space : @xmath148 the cylinder @xmath149 represents motions when the @xmath7-th robot moves along @xmath1 , while the other robots remain in @xmath150 . further in sections [ sect : fundamentalgroupsunordered ] and [ sect : computinggroupsunordered ] the simpler unordered case is considered . we believe that our approach literally extends to the ordered case using similar lemma [ lem : recursiveconstructionordered ] with @xmath7 cylinders indexed by @xmath38 since the robots are ordered . [ lem : recursiveconstructionordered ] let a graph @xmath0 have an open edge @xmath1 with vertices @xmath130 . then the ordered discrete space @xmath52 is homeomorphic to ( see figure [ fig : attachcylinder ] ) @xmath151 @xmath152 is glued to @xmath153 @xmath154 @xmath155 in this subsection we recall general results on homotopy types of configuration spaces . recall that a topological space @xmath27 is _ aspherical _ or a @xmath156 space if it has a contractible universal cover , in particular @xmath157 for @xmath158 . a covering @xmath159 is _ universal _ if the cover @xmath28 is simply connected . then the covering @xmath160 has the _ universal _ property that , for any covering @xmath161 , there is another covering @xmath162 whose composition with @xmath161 gives the original covering @xmath159 . [ pro : asphericity ] _ ( asphericity of configuration spaces , ghrist ( * ? ? ? * corollary 2.4 , theorem 3.1 ) for topological spaces and abrams @xcite for discrete spaces ) _ every component of @xmath163 is aspherical . ghrist ( * ? ? ? * corollary 2.4 , theorem 3.1 ) proves the above result for the ordered topological space @xmath14 , which implies the same conclusion for @xmath16 , because the universal cover of a component of @xmath16 is a universal cover of some component of @xmath14 as mentioned by abrams ( * ? ? ? * the proof of corollary 3.6 ) . proposition [ pro : dimension ] implies that the homotopy type of discrete spaces depends on the graph @xmath0 , but not on the number @xmath7 of robots . it was proved by ghrist ( * ? ? ? * theorems 2.6 and 3.3 ) for the ordered topological space @xmath14 , which easily extends to the unordered case . the circle @xmath164 is excluded below , because its unordered space @xmath165 is contractible , while @xmath166 deformation retracts to a disjoint union of @xmath167 configurations indexed by permutations of @xmath7 robots up to cyclic shifts . [ pro : dimension ] _ ( homotopy type of topological configuration spaces ) _ if a connected graph @xmath0 is not homeomorphic to @xmath164 and has exactly @xmath168 essential vertices , then @xmath14 and @xmath16 deformation retract to @xmath168-dimensional complexes . for instance , the configuration spaces of 2 robots in the triod @xmath40 having a single essential vertex deformation retract to a 1-dimensonal circle , see examples [ exa : topconf2pointt ] , [ exa : discconf2pointt ] . in this section we compute graph braid groups showing how their presentations change by seifert van kampen theorem [ the : seifertvankampen ] after adding new edges to a graph . let @xmath20 be open path - connected subsets of @xmath169 such that @xmath170 is also path - connected . if @xmath20 are not open in @xmath169 , they usually can be replaced by their open neighbourhoods that deformation retract to @xmath20 , respectively . assume that @xmath171 have a common base point . if @xmath172 is a finite vector of elements then a group presentation has the form @xmath173 , where the relator @xmath174 ( a vector of words in the alphabet @xmath172 ) denotes the vector relation @xmath175 . we give the practical reformulation of the seifert van kampen theorem ( * ? ? ? * theorem 3.6 on p. 71 ) . [ the : seifertvankampen ] _ ( seifert van kampen theorem ( * ? ? ? * theorem 3.6 on p. 71 ) ) _ + if presentations @xmath176 , @xmath177 are given and @xmath178 is generated by ( a vector of ) words @xmath172 , then the group @xmath179 has the presentation @xmath180 , where @xmath181 are obtained from the words @xmath172 by rewriting them in the alphabets @xmath182 , @xmath183 , respectively . as an example , consider the 2-dimensional torus @xmath169 , where @xmath27 is the complement to a closed disk @xmath184 , while @xmath28 is a open neighbourhood of @xmath184 , i.e. @xmath185 is an annulus . then @xmath27 is homotopically equivalent to a wedge of 2 circles , i.e. @xmath186 is free , @xmath187 is trivial and @xmath188 , hence @xmath189 as @xmath190 represents the boundary of @xmath184 . we will write down presentations of the fundamental groups @xmath191 step by step adding edges to the graph and watching the changes in the presentations . the base of our recursive computation is the contractible space @xmath192,n)$ ] of @xmath7 robots in a segment whose fundamental group is trivial . in proposition [ pro : addhangingedgeunordered ] we glue a hanging edge to a vertex of degree at least 2 , e.g. to an internal vertex of @xmath9 $ ] , which may create an essential vertex . in proposition [ pro : stretchhangingedgeunordered ] we add a hanging edge to a hanging vertex of degree 1 , which does not create an essential vertex . in example [ exa : createcyclesunordered ] and proposition [ pro : createcyclesunordered ] we attach an edge creating cycles . algorithm [ alg : graphbraidgroups ] computing graph braid groups is essentialy based on propositions [ pro : addhangingedgeunordered ] , [ pro : stretchhangingedgeunordered ] , [ pro : createcyclesunordered ] showing how a presentation is gradually becoming more complicated . we start with the degenerate case when a tree @xmath59 is obtained by adding a hanging edge @xmath1 to some internal vertex @xmath4 of @xmath9 $ ] . assume that @xmath9 $ ] is subdivided into at least @xmath193 subedges , otherwise the discrete configuration space @xmath194 since @xmath7 robots occupy at least @xmath7 distinct vertices . choose a hanging ( open ) edge @xmath195 attached to a hanging vertex @xmath196 and vertex @xmath4 of degree at least 3 . if the vertex @xmath4 has degree @xmath3 then @xmath197 consists of @xmath198 disjoint subtrees , some of them could be points . hence @xmath199 splits into @xmath198 subspaces @xmath200 , where @xmath201 may vary from @xmath202 to @xmath198 . fix base points : @xmath203 to a non - hanging vertex @xmath4 ] we also fix a base point @xmath204 , which can be chosen as @xmath205 for simplicity . in @xmath199 find a path @xmath206 from @xmath207 to @xmath208 , a path @xmath209 from @xmath210 to @xmath211 , @xmath212 , see figure [ fig : addhangingedge ] and motion planning algorithm [ alg : motionplanningunordered ] in subsection [ subs : motionplanningunordered ] . the base configurations @xmath213 are connected by the motion @xmath214 when @xmath193 robots stay fixed at @xmath215 and 1 robot moves along @xmath2 , see figure [ fig : addhangingedge ] . adding @xmath216 at the start and end of the motion @xmath214 , respectively , we get the @xmath198 paths @xmath217 going from @xmath207 to @xmath210 in @xmath218 , @xmath212 . for a loop @xmath219 representing a motion of @xmath193 robots , the loop @xmath220 denotes the motion when @xmath193 robots follow @xmath221 and one robot remains fixed at @xmath196 . [ pro : addhangingedgeunordered ] _ ( adding a hanging edge @xmath1 to a non - hanging vertex @xmath4 ) _ + in the notations above and for presentations @xmath222 and @xmath223 the group @xmath224 is generated by @xmath225 , @xmath226 , @xmath227 , @xmath228 by the recursive construction from lemma [ lem : recursiveconstructionunordered ] one has @xmath229 since @xmath230 splits into the vertex @xmath196 and the remaining subgraph @xmath231 , then the space @xmath232 consists of the 2 connected components @xmath233 , where all robots are in @xmath231 , and @xmath234 , where one robot is at @xmath196 . the non - connected cylinder @xmath235 splits into @xmath198 cylinders @xmath236 connecting @xmath233 and @xmath234 since the complement @xmath197 is obtained from @xmath59 by removing @xmath130 and all open edges attached to the vertex @xmath4 of degree @xmath3 . add the cylinders @xmath236 to the subspace @xmath233 , which does not affect the group @xmath237 , because the cylinders deformation retract to their bases @xmath238 . to apply seifert van kampen theorem [ the : seifertvankampen ] correctly , add all the paths @xmath217 to the resulting union , which gives the @xmath239 new generators @xmath240 , @xmath241 . consider the space @xmath234 as a subspace of @xmath218 . formally a loop @xmath242 becomes the loop @xmath243 from @xmath244 , where one robot remains fixed at @xmath196 . the same argument applies to the relator @xmath245 . no other relations appear as the intersection of @xmath246 and @xmath247 contracts to @xmath207 . now take the union with the remaining subspace @xmath234 , which adds the generators and relations of @xmath248 . the resulting intersection deformation retracts to the wedge of the @xmath198 bases @xmath249 , so each generator @xmath250 gives a relation between the words representing the loops @xmath251 in the spaces @xmath233 and @xmath234 . in the latter space the loop can be conjugated by @xmath217 , which replaces @xmath210 by the base point @xmath252 , we may set @xmath253 . notice that the loops @xmath254 live in @xmath233 with the base point @xmath207 and can be expressed in terms of the generators @xmath255 . so the last equality in the presentation is a valid relation between new generators . in this subsection we show how the presentation of a braid group changes after stretching a hanging edge of a tree . first we consider the degenerate case of stretching a hanging edge @xmath1 of the triod @xmath40 in the top left picture of figure [ fig : stretchhangingedge ] . [ exa : stretchhangingedgeunordered ] let @xmath59 be the tree obtained by adding a hanging edge @xmath256 to the hanging vertex @xmath196 of the triod @xmath40 in the top left picture of figure [ fig : stretchhangingedge ] , i.e. @xmath257 , where @xmath258 is the only hanging vertex of @xmath256 in the tree @xmath59 . the complement @xmath259 consists of 2 hanging edges distinct from @xmath1 and meeting at the centre @xmath4 of the triod @xmath40 . we compute the braid group @xmath260 using @xmath48 from example [ exa : discconf2pointt ] . by lemma [ lem : recursiveconstructionunordered ] the unordered space @xmath261 has the form @xmath262 where the 2 components of @xmath263 are connected by the band @xmath264 . first we apply seifert van kampen theorem [ the : seifertvankampen ] to the union @xmath265 , which keeps the fundamental group unchanged , i.e. isomorphic to @xmath48 , because the union deformation retracts to @xmath104 . then we apply the same trick taking the union with @xmath266 , which leads to @xmath267 for the same reasons . ] proposition [ pro : stretchhangingedgeunordered ] below extends example [ exa : stretchhangingedgeunordered ] to a general tree @xmath59 . choose an ( open ) edge @xmath268 with a hanging vertex @xmath258 and vertex @xmath196 of degree 2 . fix a base point : @xmath269 let @xmath270 be the motion from @xmath271 to @xmath272 in @xmath218 , when @xmath193 robots stay fixed at @xmath207 , while 1 robot moves along @xmath273 , see the right picture of figure [ fig : stretchhangingedge ] . then , for a loop @xmath274 , both loops @xmath275 and @xmath276 pass through the base point @xmath277 . [ pro : stretchhangingedgeunordered ] _ ( stretching a hanging edge ) _ + in the notations above and for presentations @xmath278 and @xmath279 @xmath280 @xmath281 by the recursive construction from lemma [ lem : recursiveconstructionunordered ] one has @xmath282 where the cylinder @xmath283 is glued to @xmath284 along the bases @xmath285 and @xmath286 . since @xmath256 is hanging then @xmath287 has 2 components : the hanging vertex @xmath258 and remaining tree @xmath257 , hence @xmath288 . since the edge @xmath1 is hanging in @xmath289 before stretching then the complement @xmath287 and cylinder @xmath290 are connected . adding the cylinder to @xmath291 does not change the presentation of the fundamental group , because the cylinder deformation retracts to its base in @xmath291 . then add @xmath292 meeting the previous union along @xmath285 . by seifert van kampen theorem [ the : seifertvankampen ] to get a presentation of @xmath224 with the base point @xmath271 , we add the generators @xmath293 and relations @xmath294 coming from the group @xmath295 . add the new relations @xmath296 saying that the generators of the group @xmath297 after adding the stationary @xmath7-th robot become homotopic through the subspace @xmath290 . in this subsection we extend our computations to graphs containing cycles . first we show how the braid group changes if an edge is added at 2 vertices of a triod . [ exa : createcyclesunordered ] let @xmath0 be the graph obtained from the triod @xmath40 in the top left picture of figure [ fig : createcycles ] by adding the edge @xmath80 at the vertices @xmath298 . by lemma [ lem : recursiveconstructionunordered ] one has @xmath299 geometrically the band @xmath300 is glued to the hexagon @xmath104 as shown in the bottom left picture of figure [ fig : createcycles ] . to compute the graph braid group @xmath84 we first add to the band @xmath300 the motions @xmath301 connecting the base configuration @xmath302 to @xmath303 , @xmath304 , respectively . this adds a generator to the trivial fundamental group of the contractible band @xmath300 . second we add the union @xmath305 to @xmath104 , which gives @xmath128 . the intersection of the spaces attached above has the form @xmath306 and is contractible , i.e. @xmath84 is the free product of @xmath307 and @xmath308 . creating cycles ] proposition [ pro : createcyclesunordered ] extends example [ exa : createcyclesunordered ] to a general graph excluding the case @xmath309 . choose an ( open ) edge @xmath310 with vertices @xmath298 such that @xmath311 is connected . let @xmath312 consist of @xmath5 connected components . then @xmath313 splits into @xmath5 subspaces @xmath314 , where @xmath315 . fix base points @xmath316 and @xmath317 . denote by @xmath318 the motion such that one robot goes along the path @xmath319 from @xmath320 to @xmath321 , while the other robots remain fixed at @xmath317 , see the right picture of figure [ fig : createcycles ] in the case @xmath322 when we may skip the index @xmath201 . take paths @xmath323 going from @xmath207 to @xmath324 , respectively , in @xmath325 , see algorithm [ alg : motionplanningunordered ] . then @xmath326 is a loop with the base point @xmath207 in the space @xmath56 . [ pro : createcyclesunordered ] _ ( adding an edge @xmath80 creating cycles ) _ given presentations @xmath327 the group @xmath328 is generated by @xmath172 , @xmath326 subject to @xmath175 and @xmath329 the @xmath5 subspaces @xmath314 can be disconnected , but they are in a 1 - 1 correspondence with the connected components of @xmath312 . each of the cylinders @xmath330 meets the subspace @xmath325 at the bases @xmath331 and @xmath332 first we add to each cylinder @xmath330 the union of the paths @xmath333 connecting the bases to @xmath334 , see figure [ fig : createcycles ] . the fundamental group of @xmath335 is isomorphic to the free product of @xmath336 and @xmath337 generated by the loop @xmath326 . second we add to @xmath325 each union @xmath335 . the intersection of the spaces attached above has the form @xmath338 and is homotopically a wedge of 2 copies of the base @xmath314 . by siefert van kampen theorem [ the : seifertvankampen ] we express the loops @xmath339 and @xmath340 generating the fundamental group of the intersection in terms of the loops from @xmath341 in the latter space these loops are conjugated by @xmath326 as required , i.e. homotopic through the cylinder @xmath330 . if the vector of generators @xmath174 is empty , i.e. the groups @xmath342 are trivial , then no new relations are added in proposition [ pro : createcyclesunordered ] . at the end of subsection [ subs : motionplanningunordered ] we give step - by - step instructions of algorithm [ alg : graphbraidgroups ] computing presentations of graph braid groups . the computing algorithm is based on the technical propositions from section [ sect : fundamentalgroupsunordered ] and auxiliary algorithms from subsection [ subs : motionplanningunordered ] below . as a theoretical application , in proposition [ pro:2pointgroupsunordered ] we extend the result about 2-point braid groups of graphs with only disjoint cycles ( * ? ? ? * theorem 5.6 ) to a wider class of graphs including all light planar graphs . proposition [ pro : addhangingedgeunordered ] requires a collision free motion connecting two configurations of @xmath7 robots . take a connected graph @xmath0 and number its vertices . we will work with discrete configuration spaces assuming that at every discrete moment all robots are at vertices of a graph @xmath0 and in one step any robot can move to an adjacent vertex if it is not occupied . the output contains positions of all robots at every moment . to describe planning algorithm [ alg : motionplanningunordered ] we introduce auxiliary definitions and searching algorithms [ alg : extremerobots ] , [ alg : neighbourrobot ] . the @xmath343-th robot is called _ extreme _ in a given configuration @xmath344 if the remaining robots are in one connected component of @xmath345 . one configuration may have several extreme robots , e.g. on a segment there are always 2 extreme robots , while on a circle every robot is extreme . for each robot @xmath348 we visit all vertices of @xmath345 remembering the robots we have seen . if not all robots were seen then the robot @xmath348 is not extreme and we check a robot from a smaller connected component of @xmath345 , which has fewer edges than @xmath0 . hence we will inevitably find an extreme robot , which requires in total not more than @xmath346 steps for each @xmath38 . a robot @xmath349 is a _ neighbour _ of a robot @xmath348 if a shortest path from @xmath349 to @xmath348 has the minimal number of edges among all shortest paths from @xmath349 to robots @xmath350 for @xmath351 . for @xmath7 robots on a segment each of the 2 extreme robots has a unique neighbour , while on a circle each robot has 2 neighbours . a shortest path to a neighbour does not contain other robots , i.e. the corresponding motion is collision free . we travel on @xmath0 in a ` spiral way ' starting from @xmath348 , i.e. first we visit all vertices adjacent to @xmath348 and check if there is another robot @xmath349 at one of them , which can be a neighbour of @xmath348 . if not then repeat the same procedure recursively for all these adjacent vertices . in total we pass through not more than @xmath346 edges of @xmath0 . for simplicity we assume that all robots are at vertices of degree 2 , otherwise we may subdivide edges of the graph @xmath0 and move a robot to an adjacent vertex of degree 2 . this increases the number @xmath346 of edges by not more than @xmath354 . _ assume that the found extreme robot , say @xmath356 , is from the final configuration , otherwise swap the roles of initial and final positions . using algorithm [ alg : extremerobots ] of complexity @xmath352 , find a shortest path from @xmath356 to its neighbour , say @xmath357 , from the initial configuration . then safely move @xmath357 towards @xmath356 along the shortest path avoiding collisions and keeping fixed all other robots from the initial configuration . _ step 3 . _ remove from the graph @xmath0 the robot @xmath356 at a vertex of degree 2 and all open edges attached to @xmath356 reducing the problem to a smaller graph with @xmath193 robots . the new graph remains connected since the robot @xmath356 was extreme . return to _ step 1 _ applying the recursion @xmath193 times , which gives @xmath353 operations in total . in algorithm [ alg : motionplanningunordered ] the quadratic complexity in the number of robots seems to be asymptotically optimal , because avoiding collisions between @xmath7 robots should involve some analysis of their pairwise positions . start from @xmath7 robots on a segment subdivided into @xmath193 subsegments , when the configuration space @xmath192,n)$ ] is a single point and @xmath358,n)$ ] is trivial . construct the graph @xmath0 adding edges one by one and updating presentations of resulting graph braid groups by propositions [ pro : addhangingedgeunordered ] , [ pro : stretchhangingedgeunordered ] and [ pro : createcyclesunordered ] . when we need a motion connecting 2 configurations , we apply motion planning algorithm [ alg : motionplanningunordered ] . every generator is represented as a list of vertices where robots are located at every discrete moment . the first part of lemma [ lem:2pointtreegroupsunordered ] without computing the rank was obtained by the global approach of abrams ( * ? ? ? * corollary ) . the second part was claimed by farber ( * ? ? ? * theorems 9 , 10 ) . both parts follow from our local step - by - step computations . induction on the number of edges of @xmath59 . the base @xmath360 $ ] is trivial . in the inductive step notice that trees are contractible , hence their fundamental groups are trivial and for @xmath70 the vectors @xmath361 ( with indices @xmath201 ) are empty in propositions [ pro : addhangingedgeunordered ] and [ pro : stretchhangingedgeunordered ] . the vectors @xmath174 are also empty , because they can only come from 2-point braid groups of smaller trees . so the braid group @xmath260 is free . the only generators of @xmath260 are @xmath240 , @xmath362 , coming from proposition [ pro : addhangingedgeunordered ] , which gives @xmath363 generators in total after attaching all edges to each vertex @xmath4 of degree @xmath3 . the kuratowski graphs @xmath71 in figure [ fig : kuratowskigraphs ] do not satisfy lemma [ lem : chooseedge ] , because the complement to the neighbourhood of any edge @xmath73 ( @xmath364 , respectively ) is the triangular ( rectangular , respectively ) cycle intersecting any cycle @xmath365 . [ lem : chooseedge ] any light planar graph can be constructed from a tree by adding edges as follows : an open edge @xmath80 added to the new graph @xmath0 creates a cycle @xmath82 not meeting any cycle from @xmath312 having all its cycles in one connected component . recall that a planar connected graph @xmath0 is light if any cycle @xmath79 has an edge @xmath80 such that all cycles from @xmath81 ( or , equivalently , @xmath312 ) do not meet @xmath82 . for a given light planar graph @xmath0 , take any cycle @xmath82 and corresponding edge @xmath80 . the smaller graph @xmath311 is light planar , because it has fewer cycles satisfying the same condition . we may also assume that all cycles of the subgraph @xmath312 are in one connected component , otherwise it splits as in the left picture of figure [ fig : chooseedge ] . indeed , the open edge @xmath80 can not split @xmath0 since @xmath80 belongs to the cycle @xmath79 . then we may choose another cycle from a component of @xmath312 with a smaller number of edges etc . remove edges one by one until the light planar graph becomes a tree . the original graph can be reconstructed by reversing the procedure above . the construction from lemma [ lem : chooseedge ] is also applicable to some non - light planar graphs . the right picture of figure [ fig : chooseedge ] shows 3 stages of such a construction , where the closed edge @xmath83 is dashed and the corresponding subgraph @xmath312 has fat edges . the biggest graph fails to be light planar because of the cycle bounding the grey triangle . for the same graph and dashed edge @xmath80 , one can choose another cycle @xmath82 that does not meet the only ( triangular ) cycle from @xmath312 . lemma [ lem : chooseedge ] implies that corollary [ cor:2pointgroups ] for unordered robots is a particular case of more technical proposition [ pro:2pointgroupsunordered ] , which holds for all graphs constructed as described above . [ pro:2pointgroupsunordered ] for any graph @xmath0 constructed from a tree as in lemma [ lem : chooseedge ] , let @xmath168 be the first betti number of @xmath0 . the braid group @xmath84 has a presentation with @xmath366 generators subject to commutator relations , where the sum is over all vertices @xmath367 of degree at least 3 . a geometric description follows . @xmath368 at each vertex @xmath367 fix an edge @xmath369 . for any unordered pair of other edges @xmath370 at the same vertex @xmath4 , @xmath212 , one generator of @xmath84 swaps 2 robots in the triod @xmath371 using the collision free motion shown in figure [ fig : permuterobotstriod ] . @xmath368 denote by @xmath372 disjoint open edges of @xmath0 such that @xmath373 is a tree . the remaining @xmath168 generators of @xmath84 correspond to cycles @xmath374 passing through the selected edges @xmath372 , respectively , when one robot stays at a base point and the other robot moves along a cycle @xmath375 without collisions . by subdivision theorem [ the : subdivision ] to compute the 2-point braid group @xmath84 , we may assume that @xmath0 has no loops and multiple edges removing extra trivial vertices of degree 2 . induction on the first betti number @xmath168 . base @xmath376 is lemma [ lem:2pointtreegroupsunordered ] , where every generator @xmath240 coming from proposition [ pro : addhangingedgeunordered ] is represented by a loop swapping 2 robots near a vertex of degree at least 3 as shown in figure [ fig : permuterobotstriod ] . in the induction step , for an edge @xmath310 from lemma [ lem : chooseedge ] , we show how a presentation of @xmath84 differs from a presentation of @xmath377 satisfying the conditions by the induction hypothesis . since all cycles of @xmath312 are in one connected component then @xmath322 in proposition [ pro : createcyclesunordered ] and we skip the index @xmath201 . so we add 1 new generator @xmath378 that conjugates the loops @xmath379 and @xmath380 . geometrically , @xmath378 represents a motion when the 1st robot stays away from the 2nd robot that completes a cycle @xmath381 containing @xmath80 . it remains to show that the loops @xmath382 and @xmath383 are homotopic , i.e. the new relator is a commutator . take the cycle @xmath365 from the construction of lemma [ lem : chooseedge ] . since @xmath82 does not meet all cycles from @xmath312 , then we may move the 2nd robot along @xmath384 from @xmath385 to @xmath386 without collisions with the 1st robot moving along the cycles @xmath182 generating @xmath387 . this gives a free homotopy from @xmath382 to @xmath388 . during the motion @xmath389 the 1st robot is fixed at the base point @xmath210 in @xmath312 , the 2nd moves along @xmath384 avoiding all cycles of @xmath312 . in proposition [ pro : createcyclesunordered ] we may choose the path @xmath390 from @xmath207 to @xmath391 in @xmath392 so that @xmath393 . then the loops @xmath379 and @xmath380 are homotopic with the fixed base point @xmath394 . our experience shows that presentations of planar graph braid groups may naturally contain relators that are not commutators if there are no enough disjoint cycles . so we state the problem opposite to ( * ? ? ? * conjecture 5.7 ) saying that all 2-point braid groups of planar graphs have presentations where all relators are commutators . | we design an algorithm writing down presentations of graph braid groups .
generators are represented in terms of actual motions of robots moving without collisions on a given graph .
a key ingredient is a new motion planning algorithm whose complexity is linear in the number of edges and quadratic in the number of robots .
the computing algorithm implies that 2-point braid groups of all light planar graphs have presentations where all relators are commutators . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
18.5pt in a recent article @xcite blte and nienhuis performed numerical investigations of what they termed the fully - packed loop ( fpl ) model . this is a statistical model where the ensemble is the set of all combinations of closed paths on the honeycomb lattice that visit every vertex and do not intersect . the boltzmann weight of such a filling set of paths is just the exponential of the number of paths , i.e. the energy of a configuration is the number of closed loops used to cover the lattice . an example of a fully - packed configuration of loops on this lattice is shown in figure [ example ] . the partition function for this model may be represented as @xmath1 where the sum is over all @xmath2 , the coverings of the vertices of the hexagonal lattice by closed nonintersecting paths , @xmath3 is the number of paths in the covering @xmath2 , and @xmath4 is a generalized activity . this model was originally studied for its interest as the low - temperature limit of the @xmath5 vector lattice models @xcite . in this limit , the dimensionality of vectors @xmath4 is just the activity @xmath4 in equation ( [ zfpl ] ) . the partition function ( [ zfpl ] ) is apparently the generating function for the numbers of ways to cover the hexagonal lattice by any number of closed paths . its calculation in the thermodynamic limit is an interesting combinatorial problem . more recently , batchelor , suzuki and yung @xcite pointed out that previous authors @xcite had exploited an identification of the fpl model with the integrable lattice model associated to the quantum group @xmath0 @xcite . this integrable model is a vertex model on the square lattice where each link of the lattice can be in one of three states , and the vertex weights are given by the r - matrix for @xmath0 as in figure [ weights ] . the r - matrix depends on a deformation parameter @xmath6 , as well as a spectral parameter @xmath7 typical of integrable theories . denoting the @xmath0 partition function as @xmath8 , the precise idenfication is @xmath9 where @xmath10 is the volume of the lattice ( the number of hexagonal faces ) . since the model is integrable , much exact information can be derived . in particular , the model s bethe equations have been constructed and solved . ( 400,100 ) ( 50,25)(0,1)50 ( 25,50)(1,0)50 ( 50,25)a ( 50,75)b ( 25,50)i ( 75,50)j ( 100,50 ) @xmath11 \\ \end{array } $ ] one of the more important results that have been derived in this way is the existence of a phase transition in the model ( [ zfpl ] ) at @xmath12 @xcite . at larger @xmath4 , larger numbers of loops are favored and at smaller @xmath4 configurations with fewer loops are favored . it has been conjectured that this transition is between a large-@xmath4 phase where the average loop length is finite and a small-@xmath4 phase where this average is infinite . in section [ flsection ] a simple relation between the free energy of the fpl model and the ensemble average length of loops is derived . from the known solution to the bethe equations of the @xmath0 integrable lattice model , the free energy is identified and used to graph the exact value of the average loop length as a function of @xmath4 . identifying @xmath13 , it becomes apparent that @xmath14 corresponds to the integrable model for @xmath15 real and @xmath16 corresponds to @xmath15 purely imaginary . the former phase is known @xcite to be massive , in the sense that there is a gap in the spectrum of eigenvalues of the transfer matrix between the leading eigenvalue and the next - leading eigenvalue . by standard arguments @xcite , this implies a finite correlation length . the gap tends to zero as @xmath15 goes to zero , showing that @xmath12 is a critical point of the model ( [ zfpl ] ) . in section [ xisection ] this correlation length is studied by considering the spectrum of eigenvalues of this transfer matrix . the spectrum may be deduced directly from the model s bethe equations . in the case that the transfer matrix is symmetric and therefore has real eigenvalues , the correlation length is related to the maximum eigenvalue @xmath17 and next - leading eigenvalue @xmath18 by @xmath19 18.5pt the r - matrix of the @xmath21 quantum group is an @xmath22 matrix that may be interpreted as a matrix of boltzmann weights of a vertex model on the square lattice as shown in figure [ weights ] . since this matrix satisfies the yang - baxter equation , the associated transfer matrix commutes with itself evaluated at differing values of the spectral parameter and the model is exactly solvable by a recursive set of @xmath23 nested bethe anstze @xcite . the formula for eigenvalues of the transfer matrix for the range of parameters , @xmath24 and @xmath15 real and positive , is @xmath25 where the product is over roots , @xmath26 of a set of bethe equations and we have neglected terms that do not contribute in the thermodynamic limit . in this limit the @xmath26 are distributed in the interval @xmath27 $ ] with the density , @xmath28}.\ ] ] for eigenvalues near the maximum eigenvalue , the changes in the distribution are parameterized by the locations of holes @xmath29 , @xmath30 , @xmath31 , according to @xmath32}{\sinh[(d+1)m\gamma ] } \sum_{h=1}^{n_q } e^{-2im\theta^q_h}. \label{density}\ ] ] the numbers of holes are constrained to satisfy the relation , @xmath33 the formulas ( [ eigenvalue ] ) , ( [ density ] ) may be combined in the thermodynamic limit to yield a formula for eigenvalues of the transfer matrix of the integrable model , @xmath34 \,d\lambda . \label{formula}\ ] ] 18.5pt the formulas of the preceding section in the case @xmath35 yield directly the free energy density of the model ( [ zfpl ] ) for the @xmath14 phase as the logarithm of the maximum eigenvalue of the transfer matrix , rescaled by the factor of equation ( [ identification ] ) . this free energy was actually derived in 1970 by baxter @xcite as the solution to a weighted three - coloring problem on the honeycomb lattice . the free energy density of the fpl model in the @xmath14 phase is @xmath36 where @xmath37 , and @xmath38 . this function has an essential singularity at @xmath39 . the free energy for @xmath40 and with periodic boundary conditions is given in integral form in @xcite . it is interesting to note that for both phases , the free energy density gives the ensemble average length of loops . since a configuration @xmath2 on a lattice of @xmath10 faces has @xmath41 occupied links , the total length of loops is always @xmath41 . the average loop length of configuration @xmath2 is therefore @xmath42 . if we define the ensemble average loop length @xmath43 by @xmath44 then from inspection of equation ( [ zfpl ] ) it is clear that @xmath45 = \frac{2n}{n } z_{fpl}(n ) . \label{leqn}\ ] ] the general solution to this equation can be written up to quadrature by direct integration : @xmath46 where the lower limit of integration is an undetermined constant . in terms of the free energy density @xmath47 , this becomes @xmath48 the integral in equation ( [ lintegral ] ) can be evaluated by steepest descent . the result is @xmath49 } { n \frac{df_{fpl}}{dn}(n ' ) } \right]^n_c.\ ] ] the constant of integration may now be determined from the known value of @xmath43 at @xmath50 . as will be shown in section [ perturbsection ] , in this limit @xmath51 , @xmath52 , and @xmath53 . these imply that @xmath54 , so in the thermodynamic limit @xmath55 in this calculation we have neglected corrections of order @xmath56 to @xmath43 . a graph of the ensemble average loop length versus @xmath4 in the large-@xmath4 phase is shown in figure [ lengthgraph ] . this verifies the conjecture of @xcite that loop length diverges at the critical point . 18.5pt to obtain the correlation length , we must compute the expression ( [ formula ] ) for the minimal hole distribution . when @xmath35 , there are two choices for the @xmath57 . either @xmath58 and @xmath59 , or @xmath60 and @xmath61 . in each case , the eigenvalue gap is minimized for holes at @xmath62 where the sum in equation ( [ density ] ) after integration in ( [ formula ] ) is oscillatory . the transfer matrix of the model is symmetric at the point @xmath63 , and its eigenvalues are then real . after setting @xmath7 to this value , the correlation length of the model is given by equation ( [ xidef ] ) . considering the case @xmath58 and @xmath59 , we denote the next - leading eigenvalue for this hole distribution as @xmath64 . the equation ( [ formula ] ) together with the formula for densities ( [ density ] ) gives @xmath65 where @xmath66 are the integrals over roots , @xmath67 \,d\lambda . \label{integral}\ ] ] the integral in equation ( [ integral ] ) can easily be performed by contour integration . after introducing the variables @xmath68 and @xmath69 , the result for @xmath70 is @xmath71 \,d\lambda = \left\ { \begin{array}{ll } \frac{\pi}{m } [ 1-(z^2 q^{-1})^m ] , & m>0 \\ -2\pi\log z , & m=0 \\ \frac{\pi}{m } [ 1-(z^2 q)^m ] , & m<0 \end{array } \right . .\ ] ] substituting this result into equation ( [ formula2 ] ) gives @xmath72 after expanding the demoninator of the summand in a power series in @xmath73 , this may be resummed to the form , @xmath74.\ ] ] this form is now convergent at the symmetric point , @xmath75 or equivalently @xmath76 . we may therefore evaluate it there to obtain the correlation length according to equation ( [ xidef ] ) , @xmath77 this is the desired result , the correlation length of the fpl model where @xmath37 and @xmath78 , or equivalently @xmath79 . the other possible choice of holes , @xmath60 and @xmath61 may be computed in the same way to give @xmath80 this quantity is greater than ( [ xi ] ) for all @xmath78 , so it is not the inverse correlation length . for large @xmath81 , the inequality may be seen by considering the limiting forms of expressions ( [ xi ] ) and ( [ other ] ) . rigorously , the multiplicands in ( [ other ] ) may be seen to be greater than those in ( [ xi ] ) term by term in @xmath82 . 18.5pt the fpl model has a natural large-@xmath4 expansion which allows simple perturbative verifications of results . when @xmath4 is large , the dominant configurations are those with large numbers of loops . the perturbative procedure is to approximate the sum over states by including the configurations with the highest numbers of loops . on a hexagonal lattice with number of faces @xmath10 a multiple of three , there are three configurations with the maximum possible number of loops . in these states , one out of every three faces has a small loop around it and these small loops lie on a triangular lattice . a sample is shown in figure [ nodefects ] . these three configurations differ by translations and each has @xmath83 loops . ( 140,153 ) ( 20,0)(140,153 ) ( 60,0)(-20,0)(3,5)10(-10,17)(1,0)20(20,0)(-3,5)10 ( 120,0)(-20,0)(3,5)10(-10,17)(1,0)20(20,0)(-3,5)10 ( 30,51)(-10,17)(1,0)20(20,0)(-3,5)10(20,0)(-3,-5)10(10,-17)(-1,0)20 ( 90,51)(20,0)(-3,5)10(20,0)(-3,-5)10(10,-17)(-1,0)20(-20,0)(3,-5)10(-20,0)(3,5)10(-10,17)(1,0)20 ( 150,51)(10,-17)(-1,0)20(-20,0)(3,-5)10(-20,0)(3,5)10(-10,17)(1,0)20 ( 60,102)(20,0)(-3,5)10(20,0)(-3,-5)10(10,-17)(-1,0)20(-20,0)(3,-5)10(-20,0)(3,5)10(-10,17)(1,0)20 ( 120,102)(20,0)(-3,5)10(20,0)(-3,-5)10(10,-17)(-1,0)20(-20,0)(3,-5)10(-20,0)(3,5)10(-10,17)(1,0)20 ( 30,153)(20,0)(-3,-5)10(10,-17)(-1,0)20 ( 90,153)(20,0)(-3,-5)10(10,-17)(-1,0)20(-20,0)(3,-5)10 ( 150,153)(10,-17)(-1,0)20(-20,0)(3,-5)10 the smallest change in the number of loops that can be made is to introduce a defect somewhere in one of the maximal configurations , as shown in figure [ onedefect ] . there are @xmath84 different such defects that can be introduced and each reduces the number of loops by 2 . ( 140,153 ) ( 20,0)(140,153 ) ( 60,0)(-20,0)(3,5)10(-10,17)(1,0)20(20,0)(-3,5)10 ( 120,0)(-20,0)(3,5)10(-10,17)(1,0)20(20,0)(-3,5)10 ( 30,51)(-10,17)(1,0)20(20,0)(-3,5)10(20,0)(-3,-5)10(10,-17)(-1,0)20 ( 90,51)(20,0)(-3,5)10(20,0)(-3,-5)10(10,-17)(-1,0)20(-20,0)(3,-5)10(-20,0)(3,5)10 ( 150,51)(10,-17)(-1,0)20(-20,0)(3,-5)10(-20,0)(3,5)10(-10,17)(1,0)20 ( 60,102)(20,0)(-3,5)10(10,-17)(-1,0)20(-20,0)(3,-5)10(-20,0)(3,5)10(-10,17)(1,0)20 ( 120,102)(-20,0)(3,5)10(-10,17)(1,0)20(20,0)(-3,5)10(20,0)(-3,-5)10(10,-17)(-1,0)20 ( 30,153)(20,0)(-3,-5)10(10,-17)(-1,0)20 ( 90,153)(20,0)(-3,-5)10(10,-17)(-1,0)20(-20,0)(3,-5)10 ( 150,153)(10,-17)(-1,0)20(-20,0)(3,-5)10 ( 90,85)(20,0)(-3,-5)10(-20,0)(3,-5)10(-10,17)(1,0)20 introducing defects in this way , we can reach all possible configurations . to see that this is so , we can represent a configuration by labelling the links on the lattice that do not contain part of a path . one of every three links is unoccupied , and every vertex touches one unoccupied link . these unoccupied links form a dimer configuration for the vertices of the lattice . if we draw rhombuses around every dimer and interpret the resulting picture as the projection of the edges of a stack of cubes , we see that a fpl configuration is equivalent to a stack of cubes . such an identification is shown in figure [ cubes ] . ( 300,150 ) ( 0,34 ) ( 0,0)(20,0)(-3,-5)10 ( 30,-17)(20,0)(-3,-5)10 ( 60,-34)(20,0)(-3,5)10 ( 0,34)(20,0)(-3,-5)10(10,-17)(-1,0)20 ( 30,17)(20,0)(-3,5)10(20,0)(-3,-5)10(10,-17)(-1,0)20 ( 60,0)(20,0)(-3,5)10(10,-17)(-1,0)20 ( 90,-17)(20,0)(-3,5)10 ( 0,68)(20,0)(-3,5)10(20,0)(-3,-5)10(10,-17)(-1,0)20 ( 30,51)(20,0)(-3,5)10(10,-17)(-1,0)20 ( 60,34)(20,0)(-3,5)10(20,0)(-3,-5)10 ( 90,17)(20,0)(-3,5)10(10,-17)(-1,0)20 ( 30,85)(20,0)(-3,5)10(20,0)(-3,-5)10 ( 60,68)(20,0)(-3,5)10(10,-17)(-1,0)20 ( 90,51)(20,0)(-3,-5)10 ( 60,102)(20,0)(-3,-5)10 ( 90,85)(20,0)(-3,-5)10(10,-17)(-1,0)20 ( 120,68)(10,-17)(-1,0)20 ( 150,34 ) ( 0,0)(0,0)(0,1)34(0,0)(5,-3)30 ( 30,-17)(0,0)(0,1)34(0,0)(5,-3)30 ( 60,-34)(0,0)(0,1)34 ( 0,34)(0,0)(0,1)34(0,0)(5,-3)30 ( 30,17)(0,0)(0,1)34(0,0)(5,-3)30 ( 60,0 ) ( 90,-17)(0,0)(0,1)34(0,0)(-5,-3)30 ( 0,68)(0,0)(5,-3)30 ( 30,51 ) ( 60,34)(0,0)(0,1)34(0,0)(5,-3)30(0,0)(-5,-3)30 ( 90,17)(0,0)(-5,-3)30 ( 120,0)(0,0)(0,1)34(0,0)(-5,-3)30 ( 30,85)(0,0)(5,-3)30(0,0)(-5,-3)30 ( 60,68)(0,0)(-5,-3)30 ( 90,51)(0,0)(0,1)34(0,0)(5,-3)30(0,0)(-5,-3)30 ( 120,34)(0,0)(0,1)34(0,0)(-5,-3)30 ( 60,102)(0,0)(5,-3)30(0,0)(-5,-3)30 ( 90,85)(0,0)(5,-3)30(0,0)(-5,-3)30 ( 120,68 ) in this new representation , the action of inserting a defect is just the action of adding or removing a cube . this identification is exhibited in figure [ cubedefect ] . the result then follows that since every stack of cubes can be made by adding or removing cubes , every fpl configuration can be made from one of the maximal ones by inserting some combination of defects . ( 200,200 ) ( 0,0)(0,0)(0,1)34(0,0)(5,-3)30 ( 30,-17)(0,0)(0,1)34 ( 0,34)(0,0)(5,-3)30 ( 30,17 ) ( 60,0)(0,0)(0,1)34(0,0)(-5,-3)30 ( 30,51)(0,0)(5,-3)30(0,0)(-5,-3)30 ( 60,34)(0,0)(-5,-3)30 ( 120,0 ) ( 0,0)(0,0)(0,1)34(0,0)(5,-3)30 ( 30,-17 ) ( 0,34 ) ( 30,17)(0,0)(0,1)34(0,0)(5,-3)30(0,0)(-5,-3)30 ( 60,0)(0,0)(0,1)34(0,0)(-5,-3)30 ( 30,51)(0,0)(5,-3)30(0,0)(-5,-3)30 ( 60,34 ) ( 0,120 ) ( 0,0)(20,0)(-3,-5)10(-10,17)(1,0)20 ( 60,0)(-20,0)(3,-5)10(-10,17)(1,0)20 ( 0,34)(20,0)(-3,5)10(10,-17)(-1,0)20 ( 60,34)(10,-17)(-1,0)20(-20,0)(3,5)10 ( 30,17)(20,0)(-3,5)10(10,-17)(-1,0)20(-20,0)(3,5)10 ( 120,120 ) ( 0,0)(20,0)(-3,-5)10(-10,17)(1,0)20 ( 60,0)(-20,0)(3,-5)10(-10,17)(1,0)20 ( 0,34)(20,0)(-3,5)10(10,-17)(-1,0)20 ( 60,34)(10,-17)(-1,0)20(-20,0)(3,5)10 ( 30,17)(20,0)(-3,-5)10(-20,0)(3,-5)10(-10,17)(1,0)20 to obtain an approximation for the free energy , consider first the maximal state shown in figure [ nodefects ] . for a lattice of @xmath10 faces , this configuration has @xmath85 loops . there are 3 such configurations corresponding to the three - fold translational degeneracy of the state . to lowest order then @xmath86 $ ] . this result was used in section [ flsection ] to determine the asymptotics of the average loop length . allowing defects , there are @xmath87 locations for a defect and each defect reduces the number of loops by two . defects may be applied in any number and in any combination , giving the usual sum over disconnected diagrams . we can write this as the exponential of the connected diagram ( one defect ) and we will be correct except for the effects of excluded volumes which begin with two - defect connected diagrams and are therefore higher order . to the next order , @xmath88 \exp[o(n^{-4})]$ ] . perturbatively calculating the fpl free energy , we see that @xmath89 in conformity with baxter s result shown in equation ( [ ffpl ] ) . this point of view incidentally leads to a simple expression for the entropy density of the fpl configurations at @xmath90 . at this point , all configurations are weighted equally and @xmath91 is just the number of configurations , or the exponential of the entropy . then calculating the partition function is just the problem of counting the number of coverings of the honeycomb lattice by paths , which is the number of different possible stacks of cubes , which is the old combinatorial problem of counting plane partitions . elser @xcite has calculated the asymptotics of plane partitions for large arrays of numbers . the result applied to this case is entirely dependent on the shape of the boundary , even in the thermodynamic limit . this is to be expected when @xmath90 , because this is in the small-@xmath4 phase where the model is critical . for a lattice of @xmath10 faces and free boundary conditions , the maximum entropy is obtained for a hexagon - shaped boundary and in that case the partition function is asymptotically @xmath92.\ ] ] 18.5pt the ground state of the @xmath93 integrable lattice model is @xmath94-fold degenerate . this implies the existence of a notion of interfacial tension @xmath95 away from the critical point between regions of differing antiferromagnetic polarization . by considering finite - size corrections , de vega @xcite has derived transcendental equations for this interfacial tension and computed the asymptotic behavior of @xmath96 in the limits @xmath97 and @xmath98 . scaling arguments originally due to widom @xcite predict that the scaling relation , @xmath99 should hold near the critical point , @xmath100 or equivalently @xmath101 . it would be interesting to test this relation in this case , but we know of no explicit expression for the interfacial tension . away from the critical point however , a comparison can be made . the asymptotic behavior of the interfacial tension for @xmath98 was extracted by devega , and the result is @xmath102 in the case of the fpl model , @xmath35 , @xmath103 , and @xmath104 this result may be compared with a perturbative calculation . consider the sum over fpl states at large-@xmath4 with the constraint that boundary conditions are fixed to cause frustration in the bulk , as in figure [ interface ] . the configuration in that figure has the maximum number of loops possible and is the analog of the configuration shown in figure [ nodefects ] . denoting the sum over defects in this configuration by @xmath105 , the interfacial tension is defined to be the change in free energy per unit length of the interface : @xmath106 where @xmath107 is the vertical size of the lattice . for a lattice of n faces , the maximum number of loops possible in the presence of the constraint is @xmath108 instead of @xmath85 . the maximal state in the presence of the constraint is now @xmath109-fold degenerate , because there are @xmath110 locations near the interface where defects may be freely introduced without changing the number of loops . to lowest order therefore , @xmath111 reading off the exponents , we have from equation ( [ sdef ] ) the result that @xmath112 equation ( [ s ] ) is apparently consistent with the large-@xmath15 asymptotics derived in @xcite . equation ( [ s ] ) together with equation ( [ xi ] ) show that @xmath113 in the fpl model . more generally , from the correlation length calculation it is clear that for large @xmath15 the leading behavior of the correlation length for any value of @xmath23 will always be @xmath15 , and the leading behavior of @xmath96 is always @xmath114 . h. w. j. blte , b. nienhuis , _ phys . lett _ * 72 * , 1372 ( 1994 ) . b. nienhuis , _ phys . lett . _ * 49 * , 1062 ( 1982 ) . e. domany , d. mukamel , b. nienhuis , a. schwimmer , _ nucl . phys . _ * b190 * , 279 ( 1981 ) . m. t. batchelor , j. suzuki , c. m. yung , preprint cond - mat 9408083 , august 1994 . s. o. warnaar , b. nienhuis , _ j. phys . * 26 * , 2301 ( 1993 ) . n. yu . reshetikhin , _ j. phys . * 24 * , 2387 ( 1991 ) . o. babelon , h. j. de vega , c. m. viallet , _ nucl . phys . _ * b200 * 266 ( 1982 ) ; _ nucl . phys . _ * b220 * 283 ( 1983 ) . r. j. baxter , _ j. math . phys . _ * 11 * , 784 ( 1970 ) . h. j. de vega , _ j. phys . a : math . gen . _ * 20 * , 6023 ( 1987 ) . r. j. baxter , _ exactly solved models in statistical mechanics _ , new york , academic , 1982 . b. widom , _ j. chem . phys . _ * 43 * , 3892 ( 1965 ) . v. elser , ph.d . thesis , university of california , berkeley , 1984 . | the fully - packed loop model of closed paths covering the honeycomb lattice is studied through its identification with the @xmath0 integrable lattice model .
some known results from the bethe ansatz solution of this model are reviewed .
the free energy , correlation length , and the ensemble average loop length are given explicitly for the many - loop phase .
the results are compared with the known result for the model s surface tension .
a perturbative formalism is introduced and used to verify results .
8.5 in 6.5 in -.35 in arch - ive/9411132 + .2 in * correlation length and average loop length of the fully - packed loop model * .5 in anton kast + .5 in |
You are an expert at summarizing long articles. Proceed to summarize the following text:
in the continuum euclidean path - integral , chiral anomaly comes from jacobian of the fermion measure @xcite . the point is that @xmath0 of the transformed fermion measure gives nonzero contribution because of the infinite degrees of freedom . on the other hand , lattice field theory is a framework to simulate the continuum theory with a finite number of lattice sites . chiral anomaly can not be reproduced on the finite lattice in the same way as the continuum theory . we should definitely distinguish finite and infinite lattice formulations and concentrate on reproducing chiral anomaly based on the finite lattice for practical numerical studies . the species doubling problem of the lattice fermion @xcite is closely related with chiral anomaly @xcite . according to the nielsen - ninomiya theorem , a single weyl fermion can not exist on the lattice @xcite . when formulating lattice dirac fermion , one has almost no choice but to break chiral symmetry explicitly using wilson terms @xcite . otherwise , one has to give up one of the other assumptions of the theorem as seen in the literature . lscher s implementation of lattice chiral symmetry based on the ginsparg - wilson relation @xcite was a breakthrough @xcite . the most significant feature to be stressed is that chiral anomaly can be devised even with a finite lattice . in the lscher s formulation , new chiral transformation is introduced and chiral anomaly is obtained from jacobian of the fermion measure in a different way from the continuum theory . the lattice index theorem holds for arbitrary lattice spacing @xcite and chiral anomaly agrees with the continuum result in the continuum limit @xcite . the lesson to be learned is that one can modify the axial current in order to reproduce the correct chiral anomaly on the finite lattice . in the electroweak theory , left- and right - handed fermions couple to gauge fields in different ways . this type of formulation is called chiral gauge theory . for consistent construction of chiral gauge theories , gauge symmetry needs to be maintained at the quantum level ( see ref . @xcite , for example ) . introduction of wilson terms has trouble because the mixing of left- and right - handed fermions complicates discussion of gauge anomaly cancelation @xcite . slac fermion partially simplifies the problem because it does not use wilson terms @xcite . however , it has not been successful due to breakdown of locality and lorentz invariance associated with axial currents in the continuum limit @xcite . there have been discussions that deny these defects and the non - conservation of the axial current @xcite . the defects originate in the definition of axial currents with a derivative different from the slac derivative . careful and consistent treatment is necessary when discussing species doubler and chiral anomaly with slac fermion . in this paper , we give a method to save the slac derivative . on a finite lattice , a derivative is defined as a discrete fourier transform of momentum in a similar way to the conventional slac derivative . to remove doubler modes , antiperiodic boundary conditions are chosen for the derivative in real space . the doubler modes at the momentum boundary are lifted and the dispersion relation of the continuum theory is reproduced discretely . although long - range hopping interactions appear , the pathology of slac fermion is avoided and the correct continuum limit is guaranteed . however , long - range interactions are not useful in practical numerical calculations because sparser dirac operator is better for numerical efficiency . by using the lanczos factor technique demonstrated in ref . @xcite , we effectively truncate long - range hopping interactions and improve the fermion propagator . since the proposed lattice derivative does not use wilson terms , left- and right - handed fermions are completely independent . as a result , the dirac operator constructed with the derivative has exact chiral symmetry . this means that the dirac fermion does not reproduce chiral anomaly because the fermion measure of path - integral is trivially invariant under the chiral transformation on the finite lattice . to simulate the correct chiral anomaly , we modify chiral transformation using the neuberger s solution to the ginsparg - wilson relation @xcite . as a result , we obtain an anomalous ward identity for the modified chiral transformation . the zero mode of the axial current divergence is generally non - zero and gives the index theorem . the absence of chiral anomaly under the usual chiral transformation implies the existence of a single anomaly - free weyl fermion . chiral gauge theories can be constructed using the single weyl fermion as a building block . this work is a tricky extension of slac fermion . in the conventional slac fermion , periodic boundary conditions and an infinite lattice are assumed , where the doubler remains as a singularity at the momentum boundary . on the other hand , our formulation is based on antiperiodic boundary conditions and a finite lattice and therefore free from species doubler . in addition , the axial current is defined with the modified chiral transformation and evaluated nonperturbatively . the nielsen - ninomiya theorem does not apply to our implementation of chiral anomaly because the modified chiral transformation is used to define a symmetry . this paper is organized as follows . in sec . [ lattice_derivative ] , the lattice derivative is defined based on the finite lattice formulation . locality of the derivative is discussed . long - range hopping interactions are truncated using the lanczos factor . in sec . [ chiral_anomaly ] , a modified chiral transformation is introduced to reproduce the correct chiral anomaly . a way of constructing gauge - invariant chiral gauge theories is given in a nonperturbative way . [ sumamry_and_discussions ] is devoted to summary and discussions . we define lattice derivative as a discrete fourier transform of momentum @xmath1 where @xmath2 represents lattice sites and takes integer values between @xmath3 and @xmath4 . the lattice size @xmath5 is a finite even number and @xmath6 corresponds to momentum . @xmath7 antiperiodic boundary conditions have been chosen in real space , @xmath8 . in eq . ( [ nabla ] ) , the summation can be carried out easily . @xmath9,\ ] ] where @xmath10 . the lattice derivative ( [ nabla ] ) is doubler - free and reproduces discretely the dispersion relation of the continuum theory for arbitrary lattice spacing . in the large @xmath5 limit , eq . ( [ nabla ] ) becomes an integral @xmath11 where @xmath12 is a lattice spacing and @xmath13 is a space coordinate . in the continuum limit @xmath14 , we obtain the first order derivative of the continuum theory . @xmath15 the lattice derivative ( [ nabla ] ) is local in the continuum limit . figure [ fig1 ] plots the lattice derivative @xmath16 with @xmath10 in a range @xmath17 for two lattice sizes @xmath18 and @xmath19 . the absolute value of @xmath20 is large around @xmath21 and @xmath22 and small around @xmath23 . the points @xmath21 and @xmath22 are equivalent because of antiperiodicity . therefore , @xmath20 around @xmath24 does not mean severe nonlocality . the points @xmath23 give the most long - range hopping . as expected from eq . ( [ nabla3 ] ) , locality of the derivative @xmath20 is quite good when @xmath25 . on the other hand , decay of the derivative @xmath20 is slow when the lattice size is small . we are interested in constructing a theory with better locality for a practical purpose . in order for eq . ( [ nabla ] ) to be a useful lattice derivative , long - range hopping interactions need to be truncated . however , truncation of hopping interactions may cause errors . we need to find a systematic way to reproduce spectra effectively with only short - range hopping interactions . for simplicity , let us consider a classical action for a free massless fermion in one - dimensional space . @xmath26 fourier transforms of the one - component fermions are @xmath27 where antiperiodicity is assumed @xmath28 then we have @xmath29 where @xmath30 with the inverse transform @xmath6 of eq . ( [ nabla ] ) @xmath31 to obtain this , antiperiodicity of @xmath32 , @xmath33 , and @xmath34 has been used . antiperiodicity in real space gives rise to periodicity in momentum space , @xmath35 , @xmath36 , and @xmath37 . some explanations would be necessary for the derivation of eq . ( [ pl0 ] ) . consider the matrix contained in eq . ( [ pll1 ] ) @xmath38 which has periodicity , @xmath39 . in fig . [ fig2 ] , the matrix @xmath40 is shown schematically ( see the upper diagram ) . some examples for the indices @xmath41 are given . the matrix elements on the dotted lines are not contained in eq . ( [ pll1 ] ) . in the upper diagram , the vertices @xmath42 and @xmath43 correspond to the points with @xmath24 in fig . [ fig2 ] . using the periodicity of the matrix , the triangles 1 and 2 can be moved to form the parallelogram ( see the lower diagram ) . as a result , ( [ pll1 ] ) can be evaluated by calculating the summation for each @xmath44 , which draws a line segment parallel to the oblique sides of the parallelogram . the last term of eq . ( [ pl0 ] ) is a contribution of the term with @xmath45 , which corresponds to the left oblique side of the parallelogram . in eq . ( [ pl0 ] ) , we truncate long - range terms with a parameter @xmath46 , which represents the largest distance of fermion hopping . @xmath47 . \label{pl1}\ ] ] the truncation can be implemented to eq . ( [ pll1 ] ) . @xmath48 the indices @xmath49 and @xmath2 run from @xmath3 to @xmath4 . the prime symbol means that the summation is taken over @xmath50 , @xmath51 , and @xmath52 . fermion hopping has been restricted to a finite range . figure [ fig3 ] plots @xmath6 of eq . ( [ pl1 ] ) as a function of @xmath53 for @xmath54 , @xmath55 , and @xmath56 with a lattice size @xmath18 . @xmath57 gives the exact result with no truncation , which satisfies the dispersion relation of the continuum theory . if one does not mind inclusion of long - range hopping , doubler - free formulation of a single weyl fermion is possible maintaining the correct dispersion relation . when @xmath58 is small , there is also no doubler modes because of antiperiodic boundary conditions . although some modes around the momentum boundary deviate from the correct dispersion , those are not so harmful because there is no genuine degeneracy with low - lying modes . however , @xmath6 oscillates around the exact result . the oscillation tends to be large as @xmath58 goes to small . the small oscillation around the correct dispersion comes from the truncated terms having large @xmath2 s in eq . ( [ pl1 ] ) . as shown in ref . @xcite , such oscillation can be removed by introducing the lanczos factor , which is used in fourier analysis to cancel the gibbs phenomenon . @xcite we modify eq . ( [ pl1 ] ) as follows : @xmath59 . \label{pl2}\ ] ] where @xmath60 is the lanczos factor . as a result , ( [ pll2 ] ) becomes @xmath61 the final form of the action is given as @xmath62 figure [ fig4 ] plots @xmath6 of eq . ( [ pl2 ] ) improved with the lanczos factor as a function of @xmath53 for @xmath54 , @xmath55 , and @xmath56 with @xmath18 , which are compared with the exact result ( @xmath57 ) shown in fig . [ fig3 ] . as before , there is no doubler for every @xmath58 . in addition to this , the oscillation has been removed with the lanczos factor . as @xmath58 goes to large , the deviation around the momentum boundary tends to be small . in this way , we can construct a doubler - free ultralocal derivative . if the lanczos factor is introduced , ultralocal formulation of a single weyl fermion is possible maintaining almost correct dispersion relation . on the finite four - dimensional euclidean lattice , consider an effective action @xmath63 $ ] @xmath64}=\int { \cal d}\psi { \cal d}\bar{\psi } e^{-s[u ] } , \label{ea}\ ] ] where @xmath65 is a classical action for a massless dirac fermion coupled to gauge . @xmath66 is a dirac operator and the euclidean dirac matrices satisfy @xmath67 and @xmath68 . chirality is defined with @xmath69 . the fermion variables @xmath70 and @xmath33 are grassmann valued . the indices @xmath49 and @xmath2 are four - component numbers to indicate lattice sites and each component runs from @xmath3 to @xmath4 as before . the lattice covariant derivative @xmath71 is diagonal with respect to the space - time indices @xmath49 and @xmath2 except for the @xmath72-th ones . the dirac operator @xmath73 is antihermitian and satisfies @xmath74 . the classical action is invariant under the usual chiral transformation . the derivative @xmath32 can be replaced with the truncated one in the same way as eq . ( [ action1 ] ) , if ultralocal construction is preferred . the gauge variable @xmath75 is a product of all link variables that compose a line segment between the two sites @xmath49 and @xmath2 parallel to the @xmath72-th direction . when connecting the two sites with link variables along the @xmath72-th direction , there are two ways because of periodicity of the action . the most natural choice is the shorter path . one of the two ways is chosen depending on the distance between two sites . when @xmath76 , @xmath77 which corresponds to the hexagon sandwiched between the triangles 1 and 2 in fig . @xmath78 is a unit vector in the @xmath72-th direction . when @xmath79 , @xmath80 which corresponds to the triangles 1 and 2 in fig . [ fig2 ] and therefore intersects the boundary . the link variables @xmath81 are elements of a gauge group and satisfy @xmath82 and @xmath83 . in the continuum limit , the lattice covariant derivative becomes @xmath84 where @xmath85 and @xmath86 , and @xmath87 with a parameterization @xmath88 . ( [ lcd ] ) reduces to the covariant derivative of the continuum theory in the continuum limit . with our dirac operator , the usual chiral transformation does not reproduce chiral anomaly because the classical action and the fermion measure is invariant . to simulate the correct chiral anomaly on the finite lattice , we introduce the modified chiral transformation with @xmath89 @xmath90 where @xmath91 the operator @xmath92 is the neuberger s solution @xcite to the ginsparg - wilson relation @xmath93 and has nothing to do with the dirac operator ( [ lcd ] ) . the modified chiral transformation is a symmetry of the effective action @xmath94 for arbitrary lattice spacing independent of whether the transformation is local or global . when the transformation is global @xmath95 , it is also a symmetry of the classical action @xmath96 in the continuum limit @xmath14 because the variation of the lagrangian density induced by the transformation is proportional to lattice spacing . @xmath97 although the variation is classically zero in the continuum limit , the vacuum expectation value of the variation gives the index theorem for arbitrary lattice spacing . the axial current divergence is defined as a variation of the classical action under the local chiral transformation with @xmath98 . @xmath99.\end{aligned}\ ] ] the axial current is obtained by inverting the derivative @xmath100,\end{aligned}\ ] ] which is gauge invariant . in the free theory , the currents are local in the continuum limit because the leibniz rule holds in the limit . the fermion measure transforms as @xmath101 chiral anomaly @xmath102 is a gauge - invariant quantity . as shown in ref . @xcite , the index theorem holds for arbitrary lattice spacing . @xmath103 ( see also ref . @xcite . ) since the effective action @xmath94 is invariant under transformation of the integration variables , the modified axial current @xmath104 does not conserve . @xmath105 the ward identity for the modified chiral transformation holds also for the zero mode , which gives the index theorem and corresponds to the variation under the global transformation ( [ var ] ) . ) is necessary for the existence of the non - vanishing zero mode of the axial current divergence . ] in the continuum limit , the chiral anomaly agrees with the continuum result @xcite @xmath106 where @xmath107 and @xmath108 $ ] . the effect of chiral anomaly can be implemented to physical quantities via the modified anomalous axial current . construction of anomaly - free chiral gauge theory is easy if our lattice derivative is used . in our formulation , weyl fermions are defined with the ordinary @xmath109 ( not @xmath98 ) . as a result , the fermion measure of the path integral do not depend on gauge variables . the weyl fermions are free from gauge anomaly beforehand . consider a classical action for a dirac fermion @xmath110 where only the right - handed fermion is coupled to gauge . the left - handed fermion is redundant and does not contribute to the effective action . once the absence of gauge anomaly is confirmed , the left - handed fermion can be integrated out . the definition of the dirac operator @xmath111 is same as eq . ( [ lcd ] ) except that gauge is defined as a product of link variables @xmath112 where @xmath113 and @xmath114 . @xmath111 is no longer antihermitian . under local gauge transformation , the classical action ( [ action2 ] ) is invariant . on the finite lattice , infinitesimal local gauge transformation @xmath115 does not change the fermion measure @xmath116 { \cal d}\psi { \cal d}\bar{\psi } = { \cal d}\psi { \cal d}\bar{\psi}\ ] ] and hence the effective action . a single weyl fermion can exist on the lattice without violating gauge symmetry . we have constructed a doubler - free covariant derivative and an anomalous ward identity for the modified chiral symmetry on the lattice . the index theorem holds for arbitrary lattice spacing and the dependence of chiral anomaly on gauge fields agrees with the continuum result in the continuum limit . the zero mode of the anomalous ward identity gives the index theorem . in this formulation , a single weyl fermion can exist on the lattice maintaining gauge symmetry . chiral gauge theories can be constructed nonperturbatively using the single anomaly - free weyl fermion as a building block . the proposed lattice derivative is a simple matrix that gives the correct dispersion . we have shown that introduction of the lanczos factor enables us to construct lattice derivatives with good locality . in this case , almost the correct dispersion is reproduced except for deviation at high momentum . it depends on fermion mass how large the truncation parameter @xmath58 should be . the correct chiral anomaly has been obtained without violating locality and lorentz invariance in the continuum limit in spite of the existence of long - range hopping interactions . this is a result of a nonperturbative formulation based on the modified chiral transformation ( or equivalently the modified axial current ) . to reproduce chiral anomaly , wilson terms have to be introduced somewhere . we have introduced them only in the modified chiral transformation to maintain complete independence of left- and right - handed fermions in the classical action . when calculating physical quantities dependent on chiral anomaly such as the @xmath117 meson mass , @xmath109 needs to be replaced with @xmath98 in the concerned vertex operators to include the effect of chiral anomaly . it depends on the construction of the operator @xmath92 how precisely the effect of chiral anomaly is implemented with finite lattice spacing . in lattice qcd , the modification of the axial current does not affect physical quantities independent of chiral anomaly because the axial current does not couple to gauge directly . this research was supported in part by riken . 0 k. fujikawa , phys . lett . * 42 * , 1195 ( 1979 ) ; phys . d * 21 * , 2848 ( 1980 ) ; phys . rev . d * 22 * , 1499 ( 1980 ) . j. kogut and l. susskind , phys . d * 11 * , 395 ( 1975 ) ; s. d. drell , m. weinstein , and s. yankielowicz , phys . rev . d * 14 * , 487 ( 1976 ) . k. g. wilson , in _ new phenomena in subnuclear physics _ , erice , 1975 , edited by a. zichichi ( plenum , new york , 1977 ) . d. b. kaplan , phys . b * 288 * , 342 ( 1992 ) ; m. f. golterman , k. jansen and d. b. kaplan , phys . b * 301 * , 219 ( 1993 ) . y. shamir , nucl . b * 406 * , 90 ( 1993 ) ; v. furman and y. shamir , nucl . b * 439 * , 54 ( 1995 ) . h. neuberger , phys . b * 417 * , 141 ( 1998 ) ; _ ibid _ * 427 * , 353 ( 1998 ) . h. b. nielsen and m. ninomiya , phys . b * 105 * , 219 ( 1981 ) ; nucl . phys . b 185 , 20 ( 1981)[e : b 195 , 541 ( 1982 ) ] ; b 193 , 173 ( 1981 ) . l. h. karsten and j. smit , nucl . b * 144 * , 536 ( 1978 ) ; phys . b * 85 * , 100 ( 1979 ) . l. h. karsten and j. smit , nucl . b * 183 * , 103 ( 1981 ) . e. seiler and i. o. stamatescu , phys . d * 25 * , 2177 ( 1982 ) [ erratum - ibid . d * 26 * , 534 ( 1982 ) ] . p. h. ginsparg and k. g. wilson , phys . d * 25 * , 2649 ( 1982 ) . m. lscher , phys . b * 428 * , 342 ( 1998 ) . p. hasenfratz , v. laliena and f. niedermayer , phys . b * 427 * , 125 ( 1998 ) m. lscher , nucl . phys . b * 549 * , 295 ( 1999 ) ; * 568 * , 162 ( 2000 ) . y. kikukawa and a. yamada , phys . b * 448 * , 265 ( 1999 ) . k. fujikawa , nucl . b * 546 * , 480 ( 1999 ) . h. suzuki , prog . * 102 * , 141 ( 1999 ) . d. h. adams , annals phys . * 296 * , 131 ( 2002 ) . r. a. bertlmann , _ anomalies in quantum field theory _ ( oxford university press , oxford , 1996 ) . m. lscher , arxiv : hep - th/0102028 . k. melnikov and m. weinstein , phys . rev . d * 62 * , 094504 ( 2000 ) j. m. rabin , phys . rev . d * 24 * , 3218 ( 1981 ) . m. ninomiya and c. i. tan , phys . lett . * 53 * , 1611 ( 1984 ) . t. sugihara , phys . d * 68 * , 034502 ( 2003 ) . g. b. arfken , _ mathematical methods for physicists _ ( academic press , new york , 1970 ) . t. w. chiu , phys . d * 58 * , 074511 ( 1998 ) k. fujikawa , phys . d * 60 * , 074505 ( 1999 ) e. witten , phys . b * 117 * , 324 ( 1982 ) . | we combine a pair of independent weyl fermions to compose a dirac fermion on the four - dimensional euclidean lattice .
the obtained dirac operator is antihermitian and does not reproduce anomaly under the usual chiral transformation . to simulate the correct chiral anomaly ,
we modify the chiral transformation .
we also show that chiral gauge theories can be constructed nonperturbatively with exact gauge invariance .
the formulation is based on a doubler - free lattice derivative , which is a simple matrix defined as a discrete fourier transform of momentum with antiperiodic boundary conditions .
long - range fermion hopping interactions are truncated using the lanczos factor . #
10= 0=0 1= 1=1 0>1 # 1 / |
You are an expert at summarizing long articles. Proceed to summarize the following text:
tidal interaction between the lmc , the smc , and the galaxy have long been considered to play vital roles not only in dynamical and chemical evolution of the magellanic clouds ( mcs ) but also in the formation of the magellanic stream ( ms ) and bridge ( mb ) around the galaxy ( e.g. , westerland 1999 ; murai & fujimoto 1981 ; bekki & chiba 2005 , b05 ) . although previous theoretical and numerical studies on the lmc - smc - galaxy tidal interaction discussed extensively the origin of dynamical properties of the mb ( e.g. , gardiner & noguchi 1995 , g96 ) , they have not yet investigated so extensively the long - term formation histories of field stars and star clusters ( scs ) in the mcs . therefore , long - standing and remarkable problems related to the interplay between the lmc - smc - galaxy interaction and the formation histories of stars and scs remain unsolved ( see bekki et al . 2004a , b for the first attempts to challenge these problems ) . one of intriguing and unexplained observations on scs in the lmc is that an intermediate - age sc ( ngc 1718 ) with the estimated age of @xmath2 gyr has a distinctively low metallicity of [ fe / h]@xmath3 among intermediate - age scs ( geisler et al . 2003 , g03 ; grocholski et al . 2006 , g06 ) . santos & piatti ( 2004 , s04 ) investigated integrated spectrophotometric properties of old and young scs and found that several young scs with ages less than 200 myr have metallicities smaller than @xmath4 . three examples of these low - metallicity objects including rolleston et al . ( 1999 , r99 ) are listed in the table 1 . given the fact that the stellar metallicity of the present lmc is about @xmath5 in [ fe / h ] ( e.g. , van den bergh 2000 , v00 ; cole et al . 2005 ) , the above examples of low - metallicity , young scs are intriguing objects . no theoretical attempts however have been made to understand the origin of these intriguing objects in the lmc . the purpose of this letter is to show , for the first time , that the observed distinctively low metallicities in intermediate - age and young scs in the lmc can be possible evidences for accretion and infall of low - metallicity gas onto the lmc from the smc . based on dynamical simulations of the lmc - smc - galaxy interaction for the last 2.5 gyr , we investigate whether gas stripped from the smc as a result of the tidal interaction can pass through the central region of the lmc and consequently can play a role in the star formation history of the lmc . based on the results of the simulations , we discuss how the sporadic accretion / infall of metal - poor gas onto the lmc from the smc ( referred to as `` the magellanic squall '' ) can control recent star formation activities of the lmc . .examples of distinctively metal - poor stars and scs for the lmc and the inter - cloud region close to the lmc . [ cols="^,^,^,^ " , ] recent observations on stellar kinematics of old stars in the smc have suggested that the smc is _ not _ a dwarf irregular with a strongly rotating stellar disk but a dwarf spheroidal / elliptical with little rotation ( harris & zaritsky 2006 ) . the smc however has been modeled as a low - luminosity disk system in previous simulations ( g96 ) . considering the above observations , we model the smc s stellar component either as a dwarf elliptical ( de ) with a spherical shape and no rotation or as a dwarf irregular ( di ) with a disky shape and rotation in the present study . the smc s stellar ( gaseous ) component with the size of @xmath6 ( @xmath7 ) and the mass of @xmath8 ( @xmath9 ) is embedded by a massive dark matter halo with the total mass of @xmath10 set to be roughly equal to @xmath11 and the `` universal '' density distribution ( navarro , frenk & white 1996 ) . the projected density profile of the stellar component has an exponential profile with the scale length of @xmath12 for the de and the di models . @xmath6 is fixed at 1.88 kpc so that almost no stellar streams can be formed along the ms and the mb . many dwarfs are observed to have extended hi gas disks ( e.g. , ngc 6822 ; de blok & walater 2003 ) . the smc is therefore assumed to have an outer gas disk with an uniform radial distribution , @xmath13 ( @xmath14 ) , and @xmath15 ( @xmath16 ) being key parameters that determine the dynamical evolution of the gas . the rotating gas disk is represented by _ collisionless particles _ in the present simulations , firstly because we intend to understand purely tidal effects of the lmc - smc - galaxy interaction on the smc s evolution and secondly because we compare the present results with previous ones by g96 and connors et al . ( 2006 ) for which the `` gas '' was represented by collisionless particles . although we investigate models with different @xmath17 and @xmath18 , we show the results of the models with @xmath19 and 3 and @xmath20 and 4 for which the magellanic stream with a gas mass of @xmath21 can be reproduced reasonably well . the baryonic mass fraction ( @xmath22 ) thus changes according to the adopted @xmath17 . owing to the adopted @xmath20 and 4 , a very little amount of stars in the smc can be transferred into the lmc for the last 2.5 gyr . the initial spin of the smc s gas disk in a model is specified by two angles , @xmath23 and @xmath24 , where @xmath23 is the angle between the @xmath25-axis and the vector of the angular momentum of the disk and @xmath24 is the azimuthal angle measured from @xmath26-axis to the projection of the angular momentum vector of the disk onto the @xmath27 plane . although these @xmath23 and @xmath24 are also considered to be free parameters , models with limited ranges of these parameters can reproduce the ms and the mb ( e.g. , connors et al . the gas disk is assumed to have a _ negative _ metallicity gradient as the stellar components has ( e.g. , piatti et al . the gradient represented by @xmath28}_{\rm g}(r)$ ] ( dex kpc@xmath29 ) is given as ; @xmath30}_{\rm g}(r)= \alpha \times r + \beta,\ ] ] where @xmath31 ( in units of kpc ) is the distance from the center of the smc , @xmath32 , and @xmath33 . these values of @xmath34 and @xmath35 are chosen such that ( i ) the metallicity of the _ central _ region of the smc can be consistent with the observed one ( @xmath28 } \sim -0.6 $ ] ; v00 ) and ( ii ) the slope is well within the observed range of @xmath34 for very late - type , gas - rich galaxies ( zaritsky et al . if we adopt a stellar gradient ( i.e. , smaller @xmath34 ) in a model , gas particles stripped from the smc show a smaller mean metallicity . we investigate ( i ) the time ( @xmath36 ) when gas particles stripped from the smc pass through the lmc s central 7.5 kpc ( corresponding to the disk size with the scale length of 1.5 kpc , v00 ) and ( ii ) the metallicities ( [ fe / h ] ) of the particles for models with different morphological types ( de or di ) , @xmath37 , @xmath17 , @xmath18 , @xmath23 , and @xmath24 in the smc . such stripped smc s particles are referred to as `` accreted particles '' in the present study just for convenience . we also examine the mean metallicity and the mass fraction of the `` accreted particles '' ( @xmath28}_{\rm acc}$ ] and @xmath38 , respectively ) in each of the six models for which values of model parameters are shown in the table 2 . the present simulations with no gas dynamics , no star formation , and a point - mass particle for the lmc can not precisely predict how much fraction of the `` accreted particles '' can be really accreted onto the lmc s gas disk and consequently used for star formation . we however believe that the present models enables us to grasp essential ingredients of gas transfer between the mcs for the last few gyrs . we mainly show the results for the `` standard model '' ( i.e. , model 1 ) which shows typical behaviors of gas stripping in the smc . in the followings , the time @xmath39 is measured with respect to the present - time ( @xmath40 ) : for example , @xmath41 gyr means 1.5 gyr ago in the present study . fig.1 shows , for the standard model ( model 1 ) , the time evolution of the total gas mass ( stripped from the smc ) which reaches and is just located within the central 7.5 kpc of the lmc at each time step , @xmath42 . it is noted that @xmath42 is not an accumulated gas mass but is changeable with time as gas particles can pass through the lmc in the current collisionless simulation . it is clear that the @xmath42 evolution shows a number of peaks with the first peak about @xmath43 gyr ( @xmath44 ) , just after the first pericenter passage of the smc with respect to the galaxy in the 2.5 gyr evolution . the highest peak is seen at @xmath45 myr ( @xmath46 ) , when the lmc and the smc interact the most strongly . since the gas mass ( @xmath42 ) at its peak is not negligibly small compared with the present - day hi mass of the lmc ( @xmath47 ; v00 ) , accretion and infall of the gas onto the lmc s gas disk can increase local gas densities and consequently can possibly trigger star formation in the lmc . 2 demonstrates the epoch of the `` magellanic squall '' , when the stripped gas particles of the smc are falling onto the disk of the lmc . 3 shows the initial locations of the smc s gas particles ( with respect to the smc s center ) with @xmath48 myr @xmath49 and @xmath50 myr , where @xmath36 denotes the time when a particle passed through the central region of the lmc last time . the particles with @xmath51 myr are initially located in the outer part of the smc so that they can be stripped from the smc and consequently pass through the lmc earlier . owing to the small pericenter distance of the lmc - smc orbital evolution at @xmath45 myr , the smc is strongly disturbed to lose gas particles not only from its outer part but from its inner one . as a result of this , gas initially located throughout the gas disk of the smc can pass through the central region of the lmc at @xmath45 myr and thus show @xmath48 myr @xmath49 . the abovementioned differences in the initial spatial distributions between gas particles with @xmath48 myr @xmath49 and @xmath50 myr can cause the differences in metallicity distributions of the gas between the two populations , because the smc s gas disk is assumed to have a negative metallicity gradient . 4 shows that the gas particles with @xmath50 myr have a larger fraction of metal - poor gas with @xmath52[fe / h]@xmath53 and a mean metallicity of [ fe / h]@xmath54 . the particles with @xmath55 gyr has a mean metallicity of [ fe / h]@xmath56 , because they are initially located in the outermost part of the smc s gas disk . 4 also shows that the gas particles with @xmath48 myr @xmath49 have a peak around [ fe / h]@xmath57 with a mean metallicity of [ fe / h]@xmath58 . the particles with @xmath59 myr has a mean metallicity of [ fe / h]@xmath54 . these results clearly suggest that the lmc can replenish gas supplies through accretion and infall of _ metal - poor gas from the smc _ onto the lmc s disk . it should be stressed here that the metallicities of accreted gas from the smc at @xmath59 myr can be appreciably higher than the above , if we consider chemical evolution of the smc due to star formation for the last 2.5 gyr . 5 shows that relative velocities ( @xmath60 ) of the smc s gas particles within the central 7.5 kpc of the lmc with respect to the lmc velocity range from 40 to 150 km s@xmath29 at @xmath45 myr . this result indicates that if the particles can infall onto the lmc s disk , they can give strong dynamical impact on the hi gas of the lmc and possibly cause shock energy dissipation owing to @xmath60 much higher than the sound velocities of cold gas . previous numerical simulations showed that cloud - cloud collisions with moderately high relative velocities ( @xmath61 km s@xmath29 ) can trigger the formation of scs ( bekki et al . therefore the above result implies that some fraction of the particles passing through the lmc s central region can be responsible for the formation of new scs in the lmc . the parameter dependences of @xmath28}_{\rm acc}$ ] and @xmath38 are briefly summarized as follows . firstly @xmath28}_{\rm acc}$ ] and @xmath38 do not depend so strongly on baryonic fractions , gas mass fractions , and orbital configurations ( see the table 2 ) : @xmath28}_{\rm acc}$ ] ( @xmath38 ) ranges from @xmath62 ( 0.34 ) to @xmath63 ( 0.47 ) for a fixed size ratio of @xmath18 ( @xmath64 ) . secondly , @xmath28}_{\rm acc}$ ] and @xmath38 are _ both larger _ in the model with smaller @xmath18 ( model 6 ) for which a smaller amount of gas particles can be tidally stripped from the smc . the reason for the larger @xmath38 is that a significantly larger fraction of particles once stripped from the smc can pass through the lmc in model 6 . thirdly , the morphological type of the smc in the present study is not important for @xmath28}_{\rm acc}$ ] and @xmath38 . given the fact that only 0.2% of gas can be converted into strongly bound scs ( rather than into field stars ) in the evolution of the mcs ( b05 ) , these results imply that the maximum possible mass of scs formed from smc s gas in the lmc is roughly @xmath65 in the present models . owing to the very short time scale ( @xmath66 yr ) of sc formation from gas clouds during the tidal interaction ( bekki et al . 2004b ) , the stripped smc s gas clouds are highly likely to be accreted onto the lmc within the dynamical time scale of the lmc ( @xmath67 yr ) and then converted into scs within @xmath66 yr after the accretion . ngc 1718 with an estimated age of @xmath2 gyr has a low metallicity ( [ fe / h]@xmath68 ) about 0.3 dex smaller than those of other scs with similar ages in the lmc ( e.g. , g03 ; g06 ) . if the interstellar medium ( ism ) of the lmc about @xmath69 gyr ago was very inhomogeneous in terms of chemical abundances , some fraction of stars could be born from quite low - metallicity gas clouds with [ fe / h]@xmath68 . the distinctively low metallicity therefore could be due to the abundance inhomogeneity of the ism in the lmc about @xmath69 gyr ago . however , intermediate - age scs _ other than _ ngc 1718 have very similar metallicities of [ fe / h]@xmath70 and a small metallicity dispersion of only 0.09 dex in the lmc ( g06 ) . the observed low - metallicity of ngc 1718 thus seems to be unlikely to be due to the abundance inhomogeneity of the ism . we suggest that the origin of the ngc 1718 can be closely associated with the magellanic squall about @xmath69 gyr ago . since gaseous abundance patterns ( e.g. , [ mg / fe ] ) of the smc about a few gyr ago might well be very different from those of the lmc , ngc 1718 could have abundance patterns quite different from those of other gcs . it should be here stressed that the simulated peak of the squall ( @xmath71 gyr ) is not very consistent with the observed age of ngc 1718 ( @xmath72 gyr , g03 ) . s04 recently have reported that eight young scs with ages less than 200 myr have metallicities smaller than [ fe / h]@xmath73 that is a typical stellar metallicity of the lmc ( e.g. , v00 ) . although there could be some observational uncertainties in age and metallicity determination based solely on integrated spectrophotometric properties of scs ( s04 ) , their results imply that these scs could have been formed from metal - poor gas in the lmc quite recently . the present numerical results imply that ngc 1711 , ngc 1831 , ngc 1866 , and ngc 1984 , all of which are observed to have possible metallicities smaller than [ fe / h]@xmath74 , can be formed as a result of the magellanic squall . since the chemical abundances of the outer gas disk of the smc can be significantly different from those of the present lmc s gas disk , the detailed abundances ( e.g. , [ c / fe ] , [ n / fe ] , and [ mg / fe ] ) of the above four clusters can be significantly different from those of other young scs with `` normal '' metallicities with [ fe / h]@xmath75 in the lmc . the observed young , metal - poor stars ( [ fe / h]@xmath76 ) in the inter - cloud region close to the lmc ( r99 ) will be equally explained by the gas - transfer between the mcs ( see also bekki & chiba 2007 ) . the present study has first pointed out that the magellanic squall can also play a role in the relatively recent star formation history of the lmc . sporadic infall of metal - poor gas like the magellanic squall might well be also important for recent star formation histories in pairs of interacting galaxies . previous hydrodynamical simulations showed that high - velocity collisions of hi gas onto a galactic disk can create hi holes and shells ( e.g. , tenorio - tagle et al . the magellanic squall , which inevitably can cause high - velocity impact of the gas clouds stripped from the smc on the lmc , can thus be responsible for _ some _ of the observed hi holes in the lmc ( e.g. , staveley - smith et al . we plan to investigate how collisions between low - metallicity gas clouds from the smc and those initially in the lmc trigger the formation of stars and scs in the lmc s disk based on more sophisticated , high - resolution hydrodynamical simulations with pc - scale star formation processes . our future studies thus will enable us to understand more deeply how the magellanic squall influences pc - scale star formation processes in the lmc . we are grateful to the referee , daisuke kawata , for his valuable comments , which contribute to improve the present paper . k.b . acknowledges the large australian research council ( arc ) . numerical computations reported here were carried out on grape system kindly made available by the astronomical data analysis center ( adac ) of the national astronomical observatory of japan . | we first show that a large amount of metal - poor gas is stripped from the small magellanic cloud ( smc ) and fallen into the large magellanic cloud ( lmc ) during the tidal interaction between the smc , the lmc , and the galaxy over the last 2 gyrs .
we propose that this metal - poor gas can closely be associated with the origin of lmc s young and intermediate - age stars and star clusters with distinctively low - metallicities with [ fe / h ] @xmath0 .
we numerically investigate whether gas initially in the outer part of the smc s gas disk can be stripped during the lmc - smc - galaxy interaction and consequently can pass through the central region ( @xmath1 kpc ) of the lmc .
we find that about 0.7 % and 18 % of the smc s gas can pass through the central region of the lmc about 1.3 gyr ago and 0.2 gyr ago , respectively .
the possible mean metallicity of the replenished gas from the smc to lmc is about [ fe / h ] = -0.9 to -1.0 for the two interacting phases .
these results imply that the lmc can temporarily replenish gas supplies through the sporadic accretion and infall of metal - poor gas from the smc .
these furthermore imply that if these gas from the smc can collide with gas in the lmc to form new stars in the lmc , the metallicities of the stars can be significantly lower than those of stars formed from gas initially within the lmc .
[ firstpage ] magellanic clouds galaxies : structure galaxies : kinematics and dynamics galaxies : halos galaxies : star clusters |
You are an expert at summarizing long articles. Proceed to summarize the following text:
in public health studies , clustered or longitudinal binary responses may be collected on a group of individuals where only a subset of these individuals are susceptible to having a positive response . for example , questionnaires may ask teenagers who are dating to answer a series of questions about dating violence . as in the next generation health study , a larger proportion of all zero responses are observed than would occur by chance ; presumably many individuals who are not dating filled in all zeros on the questionnaire ( also known as `` structural zeros '' ) . while there may be alternative reasons for structural zeros , for example , participants giving socially desirable responses , we believe this accounts for only a small fraction of zero inflation . interest is in making inference about the correlated binary responses for those who are susceptible ( i.e. , inference about dating violence among individuals who were dating ) . there is an extensive literature on zero - inflated poisson and binomial models [ @xcite ; @xcite ] that provide early references , along with more recent work on zero - inflated ordinal data [ @xcite ] and zero - inflated sum score data with randomized responses [ @xcite ] . @xcite reviewed various statistical models incorporating zero inflation in both discrete and continuous outcomes for cross - sectional data . @xcite discussed cross - sectional binary regression with zero inflation , and proved the model identifiability when at least one covariate is continuous . @xcite first considered longitudinal or clustered data with zero - inflated binomial or poisson outcomes . they incorporated a random effect structure to model the within - subject correlation and proposed an em algorithm to estimate the parameters . @xcite extended the work of @xcite by proposing a generalized estimation equation ( gee ) approach to model several zero - inflated distributions in a longitudinal setting . @xcite presented a hurdle model with random effects for repeated measures of zero - inflated count data . there has been no work , however , on zero - inflated clustered binary data . a component of the next generation health study examines the prevalence and correlates of dating violence among 2787 tenth - grade students , following them over seven years . dating violence is common among adolescents , may impact adolescent expectations regarding adult intimate relationships [ @xcite ] , and has been found to be associated with increased risk of depression and engagement in high - risk behaviors [ @xcite and @xcite ] . thus , dating violence among adolescents merits interest from both developmental and public health perspectives [ @xcite ] . investigators involved in the next study are primarily interested in identifying the risk factors associated with dating violence . @xcite found a relationship between high - risk behaviors ( i.e. , depressive symptoms , alcohol use , smoking and drug use ) , gender and the prevalence of dating violence victimization . a total of 10 questions were asked about dating violence . five of the questions were on dating violence victimization : did your partner ( 1 ) insult you in front of others , ( 2 ) swear at you , ( 3 ) threaten you , ( 4 ) push or shove you , or ( 5 ) throw anything that could hurt you ; the other five were similar questions on perpetration : did you ( 1 ) insult your partner in front of others , ( 2 ) swear at your partner , ( 3 ) threaten your partner , ( 4 ) push or shove your partner , or ( 5 ) throw anything that could hurt your partner ? as illustrated in figure [ fig1 ] , the distribution of the number of `` yes '' responses is clumped at zero . when we fit the frequencies with a zero - inflated binomial distribution , the zero - inflation probability is estimated to be about 58% . the binomial distribution yields a poor fit to the frequencies for two reasons . first , the prevalence of `` yes '' responses is unequal across different questions ; second , the responses from the same subject are correlated . but this only serves as an intuitive visualization of zero inflation . one can argue that the clump of zeros might be due to the high correlation of the binary responses within the same subject ; and , therefore , we also fit the generalized linear mixed model ( glmm ) and plot the fitted frequencies in figure [ fig1 ] . glmm attempts to fit the spike at 0 , and hence tends to overestimate the within - subject correlation . in this paper , we hope to explore whether zero inflation exists while allowing for the cluster effects . we propose maximum - likelihood ( ml ) and gee approaches to simultaneously account for the zero inflation and clustering in the multiple binary responses . the major difference between our work and the previous work is that @xcite , @xcite , and @xcite all considered the zero inflation at the observation level , while in our paper the zero inflation is at the subject level ( meaning that with a structural zero , all the binary responses from a subject are zero ) . for our dating violence example , subjects have all zero responses because they are not susceptible to the condition ( e.g. , in a relationship ) . the proposed methods are evaluated and compared in simulation studies . we then reexamine the relationships between high - risk behaviors and dating violence among teenagers using the proposed analysis strategy accounting for zero inflation . in section [ sec2 ] we present both maximum - likelihood and gee approaches for parameter estimation . section [ sec3 ] discusses the identifiability of the proposed model and proposes a likelihood ratio test for zero inflation . simulation study results are presented in section [ sec4 ] . the next dating violence data is analyzed in section [ sec5 ] , and a discussion follows in section [ sec6 ] . let @xmath0 be the multivariate binary outcome for subject @xmath1 ( @xmath2 ) , and @xmath3 be the corresponding matrix of covariates . let @xmath4 be the latent class , so that @xmath5 always takes the value of @xmath6 ( structural zero ) if @xmath7 , and @xmath5 follows a multivariate binary distribution with density @xmath8 if @xmath9 , where @xmath10 is a vector of parameters . we suppress the subscript @xmath1 when there is no confusion . let @xmath11 be the prevalence of the latent class 1 . in our example , @xmath9 indicates that subject @xmath1 is susceptible to the possibility of dating violence ( i.e. , potential of answering the dating violence questions in a positive fashion ) , while @xmath7 indicates that the subject is not susceptible . if both @xmath12 and @xmath13 are observed , the individual contribution to the full data likelihood is @xmath14 the observed likelihood of @xmath12 is then given by @xmath15 here we assume that the zero - inflation probability @xmath16 is the same across all the subjects in the sample . this could easily be extended to allow @xmath16 to depend on covariates , for example , with a logistic regression model . we use a generalized linear mixed effects model ( glmm ) to describe the multivariate distribution , @xmath17 : @xmath18 where @xmath19 , @xmath20 is the vector of random effects following the multivariate normal distribution @xmath21 , @xmath22 is the design matrix of the random effects , and @xmath23 is the known link function . the parameter vector @xmath10 consists of the parameter of interest @xmath24 and the nuisance parameters in the variance component @xmath25 . assume @xmath26 s are mutually independent given @xmath27 and @xmath20 , and let @xmath28 be the probability density function of @xmath20 . then the likelihood for subject @xmath1 becomes @xmath29 the integral with respect to the random effects can be approximated by gaussian hermite quadrature as @xmath30 where @xmath31 is the @xmath32th quadrature grid point and @xmath33 is the associated weight [ @xcite ] . the parameter estimation for @xmath16 and @xmath10 can be found by maximizing the log - likelihood for all @xmath34 subjects , @xmath35 . the variance estimation is calculated from the inverse of the observed information : @xmath36 and can be implemented by the ` optim ` function in ` r ` [ @xcite ] . likelihood - based inference makes full distributional assumptions on @xmath37 . when these assumptions are correct , the estimator gains efficiency ; otherwise , classical inference has poor statistical properties . we explore the estimating equations approach [ @xcite ] that only specifies a structure for the conditional mean @xmath38 . suppose @xmath39 where @xmath23 is the known link function and @xmath40 is the regression coefficients of interest . unconditional on @xmath4 , the `` marginal '' mean of @xmath5 is given by @xmath41 the estimating equations can then be written as @xmath42 where @xmath43 and @xmath44 is the working covariance matrix for @xmath5 [ @xcite ] . we can decompose @xmath44 as @xmath45 with @xmath46 being the diagonal matrix of the variance of @xmath26 [ which is @xmath47 and @xmath48 being the working correlation matrix specified by some nuisance parameter @xmath49 . if the mean model ( [ eq0-mean ] ) is correct , the estimating equations ( [ eq1-gee1 ] ) are always consistent regardless of the working correlation , and choosing an approximately correct working correlation generally leads to improved efficiency . in the context of zero - inflated regression , we propose two ways to specify the working correlation : marginal and conditional specification . the marginal correlation directly makes assumptions on @xmath48 , which is similar to the standard gee : the _ marginal independent correlation _ assumes @xmath50 , the @xmath51-dimensional identity matrix ; the _ marginal exchangeable correlation _ assumes that @xmath52 , where @xmath53 is the @xmath54 square matrix of ones . we refer to these two different approaches as gee - mi and gee - me , respectively . the conditional correlation exploits the zero - inflated structure and utilizes the conditional covariance , @xmath55 , to derive the unconditional covariance @xmath56 . a similar idea was first used by @xcite to derive their gee estimator for observation - level zero inflation . by the law of total covariance , for @xmath57 , @xmath58 where @xmath59 is the @xmath60 element of @xmath61 , and @xmath62 is the @xmath63th element of @xmath64 . the variance of @xmath26 is given by @xmath65 . conditional independence correlation _ assumes that @xmath66 , so the working correlation is @xmath67 with the @xmath60 element as @xmath68 the _ conditional exchangeable correlation _ assumes that @xmath69 that is , a correlation of @xmath70 between any @xmath26 and @xmath71 _ given _ @xmath72 . therefore , the @xmath60 element of the working correlation @xmath73 is @xmath74 we refer to these conditional gee approaches as gee - ci and gee - ce , respectively . similar to the ordinary gee , an unstructured working correlation can be assumed that allows for distinct correlations for each pair of outcomes . with the unstructured gee , the marginal and conditional specification of working correlation are equivalent , that is , @xmath75 we refer to this approach as gee - un . with each of the five forms of working correlation matrices , we could solve ( [ eq1-gee1 ] ) using the newton raphson method to obtain the corresponding parameter estimates @xmath76 . with the exchangeable or unstructured correlation structure , we iteratively update @xmath70 from its moment estimator and @xmath40 from equation ( [ eq1-gee1 ] ) [ @xcite ] . according to the standard theory of gee , the variance of the estimated @xmath76 has the usual sandwich form @xmath77 , where @xmath78 we note that the regression parameters in the glmm and gee are not directly comparable as they have different interpretations . the former is interpreted as the `` subject - specific effect '' conditional on a subject @xmath1 , while the latter is the `` population - averaged effect '' or `` marginal effect '' [ @xcite ] . thus , glmm and gee are not compatible for nonidentity link functions . in other words , if the glmm is true , the marginal expectation by integrating out the random effects @xmath20 may not preserve the linear additive form of the covariates . however , for binary regression with a probit link and random intercept , glmm and gee are compatible . we adopt a probit random effects model for both the simulations and example analysis . let @xmath79 and @xmath80 be the c.d.f . and p.d.f . of the standard normal distribution . consider the generalized linear mixed effects model with a probit link and a random intercept only , @xmath81 by integrating out @xmath82 , the marginal probability of @xmath26 is computed as follows : @xmath83 while glmm estimates @xmath84 , gee estimates @xmath85 . the latter is a probit regression model as well , with the regression coefficients , @xmath86 . this allows us to compare the performance of glmm and gee by comparing the marginal effects of the covariates , which is our interest in the dating violence analysis of the next study . in general , zero - inflated models are mixtures of two parametric parts , a point mass at zero ( equivalently , a binary distribution with @xmath87 ) and a parametric distribution for the nonstructural zero part . typically , zero - inflated models are identified by observing a larger number of zeros than would be consistent with the parametric model . for example , with poisson or binomial outcomes , one can observe excessive proportion of zeros with a histogram . for a single binary outcome , zero inflation can not be distinguished from rare events , unless covariate dependence is introduced . when there is a continuous covariate @xmath88 , zero inflation is identified because of the linear effect of @xmath88 on the binary response through a known link function . @xcite proved a weaker sufficient condition for identifiability when covariates are all categorical : to identify a two component mixture of logistic regressions with a binary response , the covariate vector needs to take at least 7 distinct values . @xcite also used the same argument to prove the identifiability of zero - inflated ordinal regression . single binary outcome can be seen as a special case of our proposed model with @xmath89 and @xmath90 . as more information is available with @xmath91 , our model is also identified under follmann and lambert s condition . @xcite proved the model identifiability for the zero - inflated binary regression with at least one continuous covariate . using a similar technique , we can prove our model identifiability . for gee with a probit link , consider @xmath92 and @xmath93 to be two parameter vectors that yield the same conditional mean @xmath94 , that is , @xmath95 equivalently , @xmath96 . suppose the @xmath97th component of @xmath98 ( i.e. , @xmath99 ) is continuous , then we can take the partial derivative with respect to @xmath99 , which yields @xmath100 taking the partial derivative on both sides of ( [ eq2-iden ] ) with respect to @xmath99 , and with some algebra , it follows that @xmath101 , and hence @xmath102 . from ( [ eq1-iden ] ) , we further get @xmath103 . this proves the identifiability gee - ci and gee - mi . in gee - un , the association parameters are indeed obtained from a moment estimator of the correlation between @xmath26 and @xmath71 . since the mean model is identified , the variance and correlation are also identified . for exchangeable working correlation , the association parameter is the `` average '' correlation between all @xmath26 and @xmath71 pairs with @xmath57 , which is identifiable as well . we now prove the identifiability of the random effects model ( [ eq0-glmm ] ) with a probit link . with normally distributed random effects , the mean of @xmath26 could be marginalized as @xmath104 , where @xmath25 is the variance covariance matrix of random effects @xmath20 . we further assume that a continuous covariate is contained in @xmath98 but not in @xmath105 . then the same argument of ( [ eq2-iden ] ) still applies by denoting @xmath106 as @xmath40 , which proves the identifiability of the regression coefficients up to a scale . now it suffices to prove the identifiability of @xmath25 . denote @xmath107 as the correlation coefficient of @xmath26 and @xmath71 ( @xmath57 ) given @xmath72 . note that from section [ sec2.2 ] , we have @xmath108 since @xmath40 is identifiable , @xmath109 is also identifiable . therefore , if two parameter vectors @xmath110 and @xmath111 lead to the same @xmath112 and @xmath113 , @xmath107 must be the same . furthermore , the regular glmm is identifiable , suggesting that the correlation structure @xmath107 conditional on @xmath72 is uniquely defined by @xmath25 . hence , we prove @xmath114 , and , consequently , the identifiability of the ml estimator is established . we also note that when @xmath90 and no covariates are available , the repeated binary counts could be collapsed into a binomial distribution . the problem then reduces to the zero - inflated binomial model , which is clearly identifiable . in the presence of the random effects , collapsing the binary counts leads to an over - dispersed binomial distribution . @xcite discussed the zero - inflated beta - binomial model , where the over - dispersion is controlled by a beta distributed random intercept . our model assumes that the over - dispersion comes from a normal distributed random intercept . another way to view the proposed model is a mixture of random effect distributions . recall that @xmath26 follows a bernoulli distribution with probability @xmath115 , given by @xmath116 instead of introducing the latent class @xmath4 , we assume that @xmath20 is a mixture of normal distribution and a point mass at @xmath117 : @xmath118 when @xmath119 , the probability @xmath115 is always 0 for @xmath120 , so @xmath5 is the structural zero . it is easy to show that the likelihood is exactly the same as the proposed model . in practice , one may wish to test for the existence of zero inflation , which can be performed under the parametric model framework . the likelihood ratio statistic is given by @xmath121 where @xmath122 is the maximized log - likelihood for the zero - inflated model , and @xmath123 is the maximized log - likelihood for the ordinary glmm . as the null hypothesis ( @xmath87 ) is on the boundary of the parameter space , the asymptotic null distribution of @xmath124 is a mixture of @xmath125 and point mass at 0 , with equal mixture probabilities [ @xcite ] . theoretically , we could also construct a score test statistic similar to the test proposed by @xcite for zero inflation in a poisson distribution . however , for our problem , the likelihood function involves intractable integrals , making the score and information matrix both difficult to evaluate . so in our application , we apply the likelihood ratio test . motivated by the next study , the data generation for the simulation studies mimics the real example . to evaluate the statistical properties of the above methods , simulation studies of the true model and a misspecified model were run with two different levels of within - cluster correlation . a sample size of @xmath126 with a cluster size of @xmath127 questions is considered . the simulations were repeated 5000 times to compare the performance of the naive estimator ( glmm , where the zero - inflation is ignored ) , the maximum - likelihood ( ml ) estimator and the five gee estimators ( gee - mi , gee - ci , gee - me , gee - ce and gee - un ) . we calculated the average ( mean ) and standard deviation ( sd ) of the estimated parameters , average of the estimated standard errors ( se ) and 95% ci coverage rates ( cover ) based on the wald intervals to evaluate the robustness and efficiency of the gee and the maximum - likelihood approaches . twenty gaussian hermite quadrature points were used for computing the glmm and ml estimators . we also tried 10 and 40 quadrature points as well as the adaptive quadrature with 250 simulated data sets . in our simulations , the results are very similar for differing number of quadrature points . our experience for generalized linear mixed models with the logit link function is that gaussian quadrature works very well , and in most situations agq is not needed . in terms of numerical efficiency , we found that the computation time for agq is about 1020 times longer than the fixed quadrature . @@lcd2.3d2.3ccc@ & * parameter * & & & & * se * & * cover * + ml & @xmath128 & 0.700 & 0.700 & 0.014 & 0.014 & 0.949 + & @xmath129 & 0.500 & 0.499 & 0.040 & 0.039 & 0.953 + & @xmath130 & 0.000 & 0.000 & 0.039 & 0.040 & 0.959 + & @xmath131 & 0.894 & 0.895 & 0.027 & 0.027 & 0.949 + & @xmath132 & -0.447 & -0.448 & 0.051 & 0.051 & 0.951 + & @xmath133 & -0.358 & -0.358 & 0.050 & 0.051 & 0.956 + & @xmath134 & 0.179 & 0.179 & 0.051 & 0.050 & 0.949 + & @xmath135 & 0.358 & 0.358 & 0.050 & 0.051 & 0.948 + glmm & @xmath129 & 0.500 & 1.352 & 0.047 & 0.046 & 0.000 + & @xmath130 & 0.000 & -0.444 & 0.029 & 0.030 & 0.000 + & @xmath131 & 0.894 & 0.570 & 0.028 & 0.025 & 0.000 + & @xmath132 & -0.447 & -0.301 & 0.034 & 0.034 & 0.013 + & @xmath133 & -0.358 & -0.239 & 0.033 & 0.034 & 0.060 + & @xmath134 & 0.179 & 0.114 & 0.032 & 0.032 & 0.489 + & @xmath135 & 0.358 & 0.224 & 0.031 & 0.032 & 0.015 + the estimated parameters for ml and glmm methods were marginalized , as we described in section [ sec2.3 ] . in the following sections , we evaluate the performance of the maximum likelihood and gee under a correctly specified and a misspecified model . additional simulation results are reported in the supplementary material [ @xcite ] , including ( a ) the performance of the proposed model with a smaller sample size ( @xmath136 ) ; ( b ) sensitivity of assuming a constant zero - inflation probability when the probability is affected by covariates ; ( c ) performance of zero - inflated beta - binomial model . we generated a continuous subject - level covariate @xmath137 from a standard normal distribution and categorical covariate @xmath138 to denote the questions for each subject . the zero - inflation indicator @xmath4 was generated from @xmath139 with @xmath140 . the outcome @xmath26 was generated from a probit random effects model : @xmath141 where @xmath142 is the indicator function and @xmath82 is the random intercept following a normal distribution @xmath143 . we fixed the regression parameters @xmath144 . the variance component @xmath145 was taken to be @xmath146 and @xmath147 , respectively , to reflect weak ( pearson correlation of about 0.1 ) and strong ( pearson correlation of about 0.45 ) within - cluster correlations . the simulation results are shown in tables [ tab1 ] and [ tab2 ] , where the true regression parameters are the marginal covariate effects given by @xmath148 . @@lcd2.3d2.3ccc@ & * parameter * & & & & * se * & * cover * + gee - mi & @xmath128 & 0.700 & 0.704 & 0.041 & 0.040 & 0.951 + & @xmath130 & 0.000 & -0.002 & 0.096 & 0.096 & 0.951 + & @xmath131 & 0.894 & 0.897 & 0.061 & 0.061 & 0.947 + & @xmath132 & -0.447 & -0.448 & 0.060 & 0.060 & 0.950 + & @xmath133 & -0.358 & -0.358 & 0.056 & 0.057 & 0.952 + & @xmath134 & 0.179 & 0.179 & 0.055 & 0.054 & 0.952 + & @xmath135 & 0.358 & 0.358 & 0.059 & 0.059 & 0.946 + gee - ci & @xmath128 & 0.700 & 0.701 & 0.029 & 0.029 & 0.948 + & @xmath130 & 0.000 & 0.001 & 0.069 & 0.070 & 0.955 + & @xmath131 & 0.894 & 0.897 & 0.044 & 0.043 & 0.953 + & @xmath132 & -0.447 & -0.449 & 0.055 & 0.056 & 0.953 + & @xmath133 & -0.358 & -0.359 & 0.053 & 0.054 & 0.958 + & @xmath134 & 0.179 & 0.179 & 0.052 & 0.052 & 0.951 + & @xmath135 & 0.358 & 0.360 & 0.056 & 0.056 & 0.946 + gee - me & @xmath128 & 0.700 & 0.701 & 0.032 & 0.032 & 0.953 + & @xmath130 & 0.000 & 0.001 & 0.077 & 0.078 & 0.953 + & @xmath131 & 0.894 & 0.898 & 0.049 & 0.049 & 0.949 + & @xmath132 & -0.447 & -0.449 & 0.058 & 0.058 & 0.955 + & @xmath133 & -0.358 & -0.359 & 0.055 & 0.056 & 0.956 + & @xmath134 & 0.179 & 0.180 & 0.054 & 0.054 & 0.952 + & @xmath135 & 0.358 & 0.359 & 0.057 & 0.058 & 0.950 + gee - ce & @xmath128 & 0.700 & 0.701 & 0.029 & 0.029 & 0.950 + & @xmath130 & 0.000 & 0.001 & 0.069 & 0.070 & 0.955 + & @xmath131 & 0.894 & 0.897 & 0.044 & 0.043 & 0.953 + & @xmath132 & -0.447 & -0.449 & 0.055 & 0.056 & 0.954 + & @xmath133 & -0.358 & -0.359 & 0.052 & 0.054 & 0.958 + & @xmath134 & 0.179 & 0.179 & 0.052 & 0.052 & 0.951 + & @xmath135 & 0.358 & 0.360 & 0.056 & 0.056 & 0.946 + gee - un & @xmath128 & 0.700 & 0.702 & 0.036 & 0.036 & 0.952 + & @xmath130 & 0.000 & 0.000 & 0.085 & 0.085 & 0.952 + & @xmath131 & 0.894 & 0.897 & 0.053 & 0.053 & 0.948 + & @xmath132 & -0.447 & -0.448 & 0.058 & 0.059 & 0.954 + & @xmath133 & -0.358 & -0.358 & 0.055 & 0.056 & 0.954 + & @xmath134 & 0.179 & 0.179 & 0.055 & 0.054 & 0.952 + & @xmath135 & 0.358 & 0.359 & 0.058 & 0.058 & 0.948 + @@lcd2.3d2.3ccc@ & * parameter * & & & & * se * & * cover * + ml & @xmath128 & 0.700 & 0.701 & 0.024 & 0.024 & 0.951 + & @xmath129 & 1.500 & 1.503 & 0.088 & 0.089 & 0.955 + & @xmath130 & 0.000 & -0.001 & 0.053 & 0.053 & 0.950 + & @xmath131 & 0.555 & 0.555 & 0.033 & 0.033 & 0.949 + & @xmath132 & -0.277 & -0.278 & 0.037 & 0.038 & 0.952 + & @xmath133 & -0.222 & -0.222 & 0.037 & 0.037 & 0.949 + & @xmath134 & 0.111 & 0.111 & 0.036 & 0.036 & 0.954 + & @xmath135 & 0.222 & 0.222 & 0.037 & 0.037 & 0.951 + glmm & @xmath129 & 1.500 & 2.248 & 0.071 & 0.073 & 0.000 + & @xmath130 & 0.000 & -0.419 & 0.032 & 0.029 & 0.000 + & @xmath131 & 0.555 & 0.384 & 0.029 & 0.026 & 0.000 + & @xmath132 & -0.277 & -0.204 & 0.027 & 0.027 & 0.230 + & @xmath133 & -0.222 & -0.163 & 0.027 & 0.027 & 0.387 + & @xmath134 & 0.111 & 0.079 & 0.025 & 0.026 & 0.763 + & @xmath135 & 0.222 & 0.155 & 0.025 & 0.026 & 0.271 + gee - mi & @xmath128 & 0.700 & 0.712 & 0.083 & 0.088 & 0.949 + & @xmath130 & 0.000 & -0.002 & 0.161 & 0.170 & 0.966 + & @xmath131 & 0.555 & 0.560 & 0.072 & 0.075 & 0.962 + & @xmath132 & -0.277 & -0.279 & 0.050 & 0.050 & 0.949 + & @xmath133 & -0.222 & -0.223 & 0.046 & 0.046 & 0.950 + & @xmath134 & 0.111 & 0.111 & 0.040 & 0.040 & 0.954 + & @xmath135 & 0.222 & 0.223 & 0.047 & 0.049 & 0.953 + gee - ci & @xmath128 & 0.700 & 0.709 & 0.064 & 0.064 & 0.954 + & @xmath130 & 0.000 & -0.006 & 0.121 & 0.123 & 0.969 + & @xmath131 & 0.555 & 0.556 & 0.056 & 0.057 & 0.958 + & @xmath132 & -0.277 & -0.278 & 0.045 & 0.045 & 0.951 + & @xmath133 & -0.222 & -0.222 & 0.042 & 0.042 & 0.946 + & @xmath134 & 0.111 & 0.111 & 0.038 & 0.039 & 0.952 + & @xmath135 & 0.222 & 0.223 & 0.044 & 0.045 & 0.951 + gee - me & @xmath128 & 0.700 & 0.706 & 0.058 & 0.057 & 0.947 + & @xmath130 & 0.000 & -0.001 & 0.115 & 0.114 & 0.954 + & @xmath131 & 0.555 & 0.557 & 0.053 & 0.053 & 0.952 + & @xmath132 & -0.277 & -0.279 & 0.045 & 0.044 & 0.948 + & @xmath133 & -0.222 & -0.223 & 0.042 & 0.042 & 0.947 + & @xmath134 & 0.111 & 0.111 & 0.039 & 0.039 & 0.955 + & @xmath135 & 0.222 & 0.223 & 0.043 & 0.044 & 0.953 + @@lcd2.3d2.3ccc@ & * parameter * & & & & * se * & * cover * + gee - ce & @xmath128 & 0.700 & 0.707 & 0.056 & 0.056 & 0.951 + & @xmath130 & 0.000 & -0.003 & 0.111 & 0.111 & 0.956 + & @xmath131 & 0.555 & 0.556 & 0.052 & 0.051 & 0.951 + & @xmath132 & -0.277 & -0.278 & 0.043 & 0.043 & 0.952 + & @xmath133 & -0.222 & -0.222 & 0.041 & 0.041 & 0.950 + & @xmath134 & 0.111 & 0.111 & 0.038 & 0.038 & 0.955 + & @xmath135 & 0.222 & 0.223 & 0.043 & 0.043 & 0.954 + gee - un & @xmath128 & 0.700 & 0.709 & 0.068 & 0.068 & 0.950 + & @xmath130 & 0.000 & -0.003 & 0.132 & 0.134 & 0.959 + & @xmath131 & 0.555 & 0.558 & 0.060 & 0.060 & 0.952 + & @xmath132 & -0.277 & -0.278 & 0.046 & 0.046 & 0.948 + & @xmath133 & -0.222 & -0.223 & 0.044 & 0.043 & 0.948 + & @xmath134 & 0.111 & 0.111 & 0.039 & 0.039 & 0.951 + & @xmath135 & 0.222 & 0.223 & 0.045 & 0.046 & 0.952 + both the ml and the five gee methods perform reasonably well , in terms of small bias and good ci coverage rate . glmm is seriously biased with poor ci coverage . it can be seen that the ml method is the most efficient , as it makes use of the full distributional assumption of the observed data . on the contrary , gee only relies on the first moments of the outcome . in estimating @xmath16 , the zero - inflated probability , the ses of the gee approaches are more than twice as large as the se of the ml method . the ses for other parameters are also significantly smaller for the ml method . of the five gee methods , we found that gee - ce is the most efficient with the smallest se , while gee - mi is the least efficient . by exploiting the correlation structure induced by the zero - inflation process , the conditional independence and exchangeable working correlation both gain a substantial amount of efficiency , compared to their marginal counterparts . this result is consistent with the simulation results in @xcite . the ses for gee - ce and gee - ci are quite close , implying that adding working dependence to the outcome given @xmath9 would not help much as long as the dependence due to zero inflation is accounted for . we did observe a bigger improvement of gee - ce versus gee - ci for the strong correlation case . but the improvement of gee - ci versus gee - mi is even larger . therefore , we recommend that it is more important to make use of the zero - inflation structure in the gee estimators . although gee - un has the most flexible form of working correlation , it is not as efficient as gee - ci or gee - ce , probably due to estimating a larger number of nuisance parameters . we found that gees may occasionally not have a solution or have a boundary solution ( @xmath149 ) in about 12% of the simulations with @xmath150 . our experience is that nonconvergence or boundary solutions occur more often when the covariate effects are weaker , the within - cluster correlation is stronger , the true zero - inflation probability is closer to 1 , or the model is more severely misspecified . we consider a misspecified model where only the first three questions are correlated and the last two are independent . the data generation for @xmath26 with @xmath151 was the same as in section 3.1 , but @xmath26 for @xmath152 was generated as follows : @xmath153 the random intercept @xmath82 was not added to the last two questions , but a factor of @xmath154 was divided to the coefficients to keep the marginalized regression coefficients the same . in this case , the ml estimator is from a misspecified model since the random intercept model imposes correlation among all the questions . for gee , only the working correlation is misspecified , while the first moment of @xmath26 is still correct . the simulation results are presented in tables [ tab3 ] and [ tab4 ] . @@lcd2.3d2.3ccc@ & * parameter * & & & & * se * & * cover * + ml & @xmath128 & 0.700 & 0.706 & 0.013 & 0.013 & 0.925 + & @xmath129 & 0.500 & 0.273 & 0.047 & 0.047 & 0.000 + & @xmath130 & 0.000 & -0.008 & 0.038 & 0.039 & 0.953 + & @xmath131 & 0.894 & 0.899 & 0.024 & 0.024 & 0.948 + & @xmath132 & -0.447 & -0.446 & 0.051 & 0.053 & 0.963 + & @xmath133 & -0.358 & -0.357 & 0.049 & 0.053 & 0.964 + & @xmath134 & 0.179 & 0.177 & 0.053 & 0.052 & 0.949 + & @xmath135 & 0.358 & 0.355 & 0.053 & 0.053 & 0.949 + glmm & @xmath129 & 0.500 & 1.176 & 0.042 & 0.040 & 0.000 + & @xmath130 & 0.000 & -0.441 & 0.029 & 0.030 & 0.000 + & @xmath131 & 0.894 & 0.570 & 0.028 & 0.024 & 0.000 + & @xmath132 & -0.447 & -0.303 & 0.035 & 0.036 & 0.019 + & @xmath133 & -0.358 & -0.241 & 0.034 & 0.035 & 0.080 + & @xmath134 & 0.179 & 0.114 & 0.034 & 0.034 & 0.519 + & @xmath135 & 0.358 & 0.223 & 0.033 & 0.034 & 0.021 + gee - mi & @xmath128 & 0.700 & 0.704 & 0.041 & 0.040 & 0.951 + & @xmath130 & 0.000 & -0.002 & 0.095 & 0.095 & 0.951 + & @xmath131 & 0.894 & 0.896 & 0.059 & 0.060 & 0.949 + & @xmath132 & -0.447 & -0.447 & 0.059 & 0.060 & 0.953 + & @xmath133 & -0.358 & -0.357 & 0.056 & 0.057 & 0.953 + & @xmath134 & 0.179 & 0.178 & 0.057 & 0.057 & 0.950 + & @xmath135 & 0.358 & 0.358 & 0.061 & 0.062 & 0.948 + @@lcd2.3d2.3ccc@ & * parameter * & & & & * se * & * cover * + gee - ci & @xmath128 & 0.700 & 0.701 & 0.030 & 0.030 & 0.950 + & @xmath130 & 0.000 & 0.000 & 0.070 & 0.071 & 0.955 + & @xmath131 & 0.894 & 0.897 & 0.044 & 0.044 & 0.951 + & @xmath132 & -0.447 & -0.449 & 0.055 & 0.056 & 0.956 + & @xmath133 & -0.358 & -0.358 & 0.053 & 0.054 & 0.957 + & @xmath134 & 0.179 & 0.179 & 0.055 & 0.055 & 0.949 + & @xmath135 & 0.358 & 0.360 & 0.059 & 0.059 & 0.951 + gee - me & @xmath128 & 0.700 & 0.702 & 0.034 & 0.033 & 0.951 + & @xmath130 & 0.000 & 0.000 & 0.079 & 0.080 & 0.952 + & @xmath131 & 0.894 & 0.897 & 0.049 & 0.050 & 0.949 + & @xmath132 & -0.447 & -0.448 & 0.058 & 0.058 & 0.955 + & @xmath133 & -0.358 & -0.358 & 0.055 & 0.056 & 0.954 + & @xmath134 & 0.179 & 0.179 & 0.057 & 0.057 & 0.952 + & @xmath135 & 0.358 & 0.359 & 0.061 & 0.061 & 0.951 + gee - ce & @xmath128 & 0.700 & 0.701 & 0.030 & 0.030 & 0.951 + & @xmath130 & 0.000 & 0.000 & 0.070 & 0.071 & 0.955 + & @xmath131 & 0.894 & 0.897 & 0.044 & 0.044 & 0.950 + & @xmath132 & -0.447 & -0.448 & 0.055 & 0.056 & 0.955 + & @xmath133 & -0.358 & -0.358 & 0.053 & 0.054 & 0.956 + & @xmath134 & 0.179 & 0.179 & 0.055 & 0.055 & 0.950 + & @xmath135 & 0.358 & 0.359 & 0.059 & 0.059 & 0.952 + gee - un & @xmath128 & 0.700 & 0.703 & 0.037 & 0.036 & 0.952 + & @xmath130 & 0.000 & -0.001 & 0.086 & 0.087 & 0.951 + & @xmath131 & 0.894 & 0.897 & 0.053 & 0.054 & 0.948 + & @xmath132 & -0.447 & -0.448 & 0.058 & 0.059 & 0.953 + & @xmath133 & -0.358 & -0.358 & 0.055 & 0.056 & 0.954 + & @xmath134 & 0.179 & 0.179 & 0.057 & 0.057 & 0.950 + & @xmath135 & 0.358 & 0.359 & 0.061 & 0.062 & 0.950 + @@lcd2.3d2.3ccc@ & * parameter * & & & & * se * & * cover * + ml & @xmath128 & 0.700 & 0.730 & 0.015 & 0.015 & 0.472 + & @xmath129 & 1.500 & 0.624 & 0.037 & 0.038 & 0.000 + & @xmath130 & 0.000 & -0.046 & 0.039 & 0.039 & 0.776 + & @xmath131 & 0.555 & 0.561 & 0.024 & 0.024 & 0.941 + & @xmath132 & -0.277 & -0.273 & 0.036 & 0.046 & 0.986 + & @xmath133 & -0.222 & -0.218 & 0.036 & 0.045 & 0.987 + & @xmath134 & 0.111 & 0.103 & 0.047 & 0.044 & 0.933 + & @xmath135 & 0.222 & 0.208 & 0.046 & 0.045 & 0.934 + glmm & @xmath129 & 1.500 & 1.201 & 0.038 & 0.039 & 0.000 + & @xmath130 & 0.000 & -0.416 & 0.029 & 0.030 & 0.000 + & @xmath131 & 0.555 & 0.378 & 0.024 & 0.022 & 0.000 + & @xmath132 & -0.277 & -0.206 & 0.027 & 0.034 & 0.425 + & @xmath133 & -0.222 & -0.164 & 0.027 & 0.034 & 0.613 + & @xmath134 & 0.111 & 0.079 & 0.034 & 0.033 & 0.816 + & @xmath135 & 0.222 & 0.153 & 0.033 & 0.032 & 0.432 + gee - mi & @xmath128 & 0.700 & 0.712 & 0.080 & 0.082 & 0.948 + & @xmath130 & 0.000 & -0.005 & 0.153 & 0.159 & 0.963 + & @xmath131 & 0.555 & 0.558 & 0.066 & 0.067 & 0.954 + & @xmath132 & -0.277 & -0.278 & 0.048 & 0.049 & 0.946 + & @xmath133 & -0.222 & -0.222 & 0.045 & 0.045 & 0.949 + & @xmath134 & 0.111 & 0.111 & 0.054 & 0.054 & 0.951 + & @xmath135 & 0.222 & 0.223 & 0.058 & 0.06 & 0.954 + gee - ci & @xmath128 & 0.700 & 0.707 & 0.066 & 0.066 & 0.944 + & @xmath130 & 0.000 & -0.001 & 0.129 & 0.130 & 0.966 + & @xmath131 & 0.555 & 0.558 & 0.058 & 0.058 & 0.960 + & @xmath132 & -0.277 & -0.279 & 0.046 & 0.046 & 0.948 + & @xmath133 & -0.222 & -0.223 & 0.043 & 0.043 & 0.950 + & @xmath134 & 0.111 & 0.112 & 0.053 & 0.052 & 0.950 + & @xmath135 & 0.222 & 0.224 & 0.057 & 0.058 & 0.951 + gee - me & @xmath128 & 0.700 & 0.709 & 0.069 & 0.068 & 0.948 + & @xmath130 & 0.000 & -0.004 & 0.132 & 0.133 & 0.951 + & @xmath131 & 0.555 & 0.557 & 0.060 & 0.060 & 0.952 + & @xmath132 & -0.277 & -0.278 & 0.046 & 0.046 & 0.946 + & @xmath133 & -0.222 & -0.222 & 0.043 & 0.043 & 0.946 + & @xmath134 & 0.111 & 0.112 & 0.054 & 0.054 & 0.950 + & @xmath135 & 0.222 & 0.224 & 0.058 & 0.059 & 0.953 + @@lcd2.3d2.3ccc@ & * parameter * & & & & * se * & * cover * + gee - ce & @xmath128 & 0.700 & 0.709 & 0.067 & 0.066 & 0.946 + & @xmath130 & 0.000 & -0.004 & 0.129 & 0.130 & 0.959 + & @xmath131 & 0.555 & 0.556 & 0.058 & 0.058 & 0.954 + & @xmath132 & -0.277 & -0.279 & 0.045 & 0.046 & 0.948 + & @xmath133 & -0.222 & -0.223 & 0.043 & 0.043 & 0.948 + & @xmath134 & 0.111 & 0.111 & 0.053 & 0.052 & 0.948 + & @xmath135 & 0.222 & 0.224 & 0.057 & 0.058 & 0.951 + gee - un & @xmath128 & 0.700 & 0.711 & 0.073 & 0.073 & 0.951 + & @xmath130 & 0.000 & -0.005 & 0.140 & 0.142 & 0.954 + & @xmath131 & 0.555 & 0.557 & 0.061 & 0.061 & 0.952 + & @xmath132 & -0.277 & -0.278 & 0.047 & 0.047 & 0.947 + & @xmath133 & -0.222 & -0.222 & 0.044 & 0.044 & 0.948 + & @xmath134 & 0.111 & 0.111 & 0.053 & 0.053 & 0.950 + & @xmath135 & 0.222 & 0.223 & 0.058 & 0.059 & 0.953 + with @xmath155 , the ml approach is almost unbiased for estimating @xmath16 as well as the regression coefficients . when @xmath129 increases to 1.5 , the ml estimator becomes slightly biased with poor ci coverage , especially for @xmath16 and @xmath130 . the estimation of other parameters appears to be robust to the model misspecification , except that the ses for @xmath132 and @xmath133 overestimate the true variability . on the other hand , the five gee methods all perform quite well , in terms of little bias and close - to - nominal coverage rates . similar to the previous simulation study , we observed that about 1% of the gee simulations did not converge for @xmath150 . although the maximum - likelihood approach is biased , its standard error is much smaller than the gee approaches . for example , the ml estimator for @xmath16 in table [ tab4 ] has a se only a quarter as large as that of the gee - ci and gee - ce estimators . as a result of the variance - bias trade - off , the mean squared error for the ml estimator is still smaller than gee . if the interest is in estimation , one can still argue that the ml performs better ; but if the interest is in hypothesis testing , gee methods are preferred , as they are more robust and preserve the correct type i error rate . from the above two sets of simulation studies , we would generally recommend the ml estimator in practice because of its high efficiency . the correlation structure of the outcome is critical in identifying the zero - inflation process . therefore , a full parametric assumption for the correlation can lead to good efficiency in the estimation . however , if this parametric assumption does not hold , the ml estimator could have poor ci coverage rates . in order to perform hypothesis testing , we would prefer the gee approaches , which only rely on the correct mean model and are not sensitive to the working correlation assumption . among the five gee approaches , the gee - ci and gee - ce are the most favorable , because they are more efficient by exploiting the dependence structure induced by the zero - inflation process . in practice , the gee - ci and gee - ce estimators may be computed in conjunction with the ml estimator as a sensitivity analysis . in this section we fit our proposed ml and gee models to the dating violence example , together with the naive glmm model . a total of 2787 students were enrolled in the study , among which 664 left all the dating violence questions blank . these 664 subjects are either not in a relationship , and thus skipped these questions , or they did not respond at all to the whole survey . in the remaining 2123 subjects , 39 were excluded because they only answered part of the dating violence questions , and 61 were excluded because they have missing data in the covariates . the final analysis sample was @xmath156 . the clustered outcomes of interest ( @xmath26 ) are the ten questions of dating violence , including five victimization and five perpetration questions . we can see from figure [ fig1 ] that the frequency histogram of `` yes '' responses shows a huge spike at 0 . it seems likely that some students who answered all the questions with `` no '' were not in a relationship , that is , a zero inflation of the outcome . define the latent variable @xmath9 if the subject is in a relationship and 0 otherwise . we included gender ( @xmath157 ) , depressive symptoms ( @xmath158 ) , family relationship ( @xmath159 ) and family influence ( @xmath160 ) as the predictors of @xmath26 given @xmath9 . the ds score comes from the questionnaire of depressive symptoms and is on the continuous scale ranging from 1 to 5 , with the larger score indicating worse depressive symptoms . the fr ( ranging from 0 to 10 ) measures the participant s satisfaction with the relationship in his / her family , with 10 being a very good relationship . the fi score ( ranging from 1 to 7 ) is the family influence on the participants not verbally or physically abusing their romantic partner , with a higher score being greater influence . we adjust for question number as a factor and question type ( victimization vs. perpetration ) , in order to account for different prevalence of yes responses . the interactions between question type and other covariates ( @xmath161 , @xmath162 , @xmath163 and @xmath164 ) are also included . the summary statistics of these variables are described in table [ tab5 ] . @@ld2.6@ * variable * & + gender : female & 56.6% + ds score & 2.0 ( 1.0 ) + family relationship & 7.4 ( 2.3 ) + family influence & 5.7 ( 1.8 ) + question 1v insult you & 18.5% + question 1p insult your boyfriend / girlfriend & 16.8% + question 2v swear at you & 31.3% + question 2p swear at your boyfriend / girlfriend & 26.1% + question 3v threaten you & 7.2% + question 3p threaten your boyfriend / girlfriend & 5.6% + question 4v push you & 13.5% + question 4p push your boyfriend / girlfriend & 9.9% + question 5v throw object at you & 4.5% + question 5p throw object at your boyfriend / girlfriend & 3.8% + denote @xmath27 to be the design matrix including all the covariates and interaction terms mentioned above . we fit the probit random effects model @xmath165 where @xmath166 is the random intercept . this model has the same marginal mean as the marginal probit regression model : @xmath167 as @xmath168 for @xmath169 . for comparative purposes , we report the marginal regression coefficients @xmath40 for all the analyses . = @@ld2.10d2.10d2.10d2.10d2.10d2.10d2.10@ * parameter * & & & & & & & + ( intercept ) & -0.192 ( 0.176 ) & -0.746 ( 0.177 ) & -0.055 ( 0.254 ) & -0.159 ( 0.225 ) & -0.497 ( 0.222 ) & -0.341 ( 0.221 ) & -0.201 ( 0.232 ) + question 2 & 0.528 ( 0.037 ) & 0.374 ( 0.024 ) & 0.578 ( 0.071 ) & 0.508 ( 0.057 ) & 0.475 ( 0.058 ) & 0.488 ( 0.057 ) & 0.529 ( 0.061 ) + question 3 & -0.754 ( 0.043 ) & -0.577 ( 0.031 ) & -0.814 ( 0.069 ) & -0.750 ( 0.061 ) & -0.716 ( 0.063 ) & -0.729 ( 0.061 ) & -0.768 ( 0.063 ) + question 4 & -0.332 ( 0.035 ) & -0.250 ( 0.026 ) & -0.360 ( 0.053 ) & -0.335 ( 0.048 ) & -0.315 ( 0.047 ) & -0.324 ( 0.047 ) & -0.339 ( 0.049 ) + question 5 & -1.003 ( 0.050 ) & -0.773 ( 0.036 ) & -1.069 ( 0.083 ) & -0.996 ( 0.073 ) & -0.949 ( 0.077 ) & -0.969 ( 0.074 ) & -1.013 ( 0.075 ) + question type & 0.222 ( 0.122 ) & 0.158 ( 0.089 ) & 0.264 ( 0.173 ) & 0.234 ( 0.150 ) & 0.215 ( 0.146 ) & 0.223 ( 0.145 ) & 0.215 ( 0.155 ) + ds score & 0.192 ( 0.036 ) & 0.178 ( 0.035 ) & 0.250 ( 0.043 ) & 0.220 ( 0.038 ) & 0.226 ( 0.037 ) & 0.219 ( 0.037 ) & 0.230 ( 0.039 ) + gender & 0.132 ( 0.067 ) & 0.153 ( 0.054 ) & 0.171 ( 0.076 ) & 0.118 ( 0.068 ) & 0.186 ( 0.067 ) & 0.160 ( 0.066 ) & 0.164 ( 0.070 ) + family relationship & -0.037 ( 0.014 ) & -0.035 ( 0.012 ) & -0.042 ( 0.019 ) & -0.039 ( 0.017 ) & -0.041 ( 0.015 ) & -0.041 ( 0.016 ) & -0.041 ( 0.017 ) + family influence & -0.114 ( 0.017 ) & -0.087 ( 0.015 ) & -0.137 ( 0.021 ) & -0.135 ( 0.018 ) & -0.095 ( 0.017 ) & -0.112 ( 0.017 ) & -0.123 ( 0.018 ) + question type@xmath170ds score & 0.037 ( 0.025 ) & 0.026 ( 0.018 ) & 0.034 ( 0.032 ) & 0.037 ( 0.029 ) & 0.028 ( 0.026 ) & 0.034 ( 0.028 ) & 0.035 ( 0.029 ) + question type@xmath170gender & -0.409 ( 0.051 ) & -0.300 ( 0.037 ) & -0.446 ( 0.066 ) & -0.392 ( 0.057 ) & -0.373 ( 0.056 ) & -0.379 ( 0.056 ) & -0.399 ( 0.059 ) + question type@xmath170family & -0.003 ( 0.010 ) & -0.002 ( 0.008 ) & -0.009 ( 0.014 ) & -0.007 ( 0.012 ) & -0.006 ( 0.012 ) & -0.006 ( 0.012 ) & -0.005 ( 0.013 ) + relationship + question type@xmath170family & 0.025 ( 0.012 ) & 0.019 ( 0.009 ) & 0.031 ( 0.015 ) & 0.025 ( 0.013 ) & 0.026 ( 0.013 ) & 0.024 ( 0.013 ) & 0.028 ( 0.013 ) + influence + @xmath171 & 0.571 ( 0.031 ) & 1 & 0.523 ( 0.062 ) & 0.626 ( 0.075 ) & 0.690 ( 0.104 ) & 0.663 ( 0.089 ) & 0.587 ( 0.070 ) + @xmath129 & 1.033 ( 0.065 ) & 1.627 ( 0.055 ) & & & & & + the results of the ml , glmm and gee estimations are listed in table [ tab6 ] . from the ml estimation , we can see that all four subject - level covariates are significant : the probability of dating violence perpetration was higher for females , those who are more depressed , those who have a worse relationship in their family , and those who are less influenced by their guardians . the interaction terms between question type and gender , and question type and family influence were both significant , suggesting ( a ) boys are more likely to be the victims of dating violence , and ( b ) the family influence has a slightly higher impact on dating violence perpetration than victimization . the finding regarding greater male dating violence victimization in this age is in line with previous studies [ @xcite ; @xcite ] . the impacts of depression score and family relationship are similar regardless of question type . the directions of association are expected and consistent with some of the findings in @xcite . the zero - inflation probability is estimated to be 0.571 , that is , we expect about 43% of the sample ( about 860 subjects ) to be structural zeros . we suspect that a majority of them were not in a relationship , but answered all the dating violence questions with `` no . '' however , there may be alternative reasons for the zero inflation , for example , some kids may give socially desirable answers in the survey and hence underreport dating violence . however , we believe that this only accounts for a small fraction of the structural zeros . the likelihood ratio test statistic for zero inflation is @xmath172 ( @[email protected] ) . the parameter estimations by gee are generally close to ml , but the standard errors are larger . the naive glmm method estimated smaller covariate effects , which could be biased due to ignoring the zero - inflated nature of the data . as pointed out by a referee , the zero - inflation problem could be avoided by including a filter question of asking whether the subject had a relationship or not . the filter question was not included because the study investigators felt that it was an unreliable question to ask . relationships between teenagers today can not easily be characterized , and the investigators felt that explicitly asking this question may limit important responses about violence [ @xcite ] . there are other cases where the susceptible population can not be ascertained accurately . for example , in drug abuse studies , specifically asking whether individuals are drug abusers might not be a question that would result in a reliable response . but we may be interested in whether the abusers seek particular types of treatment . additionally , this approach may be relevant to questions regarding immigration status , where there may be legal implications ( perceived or real ) in answering , and in questions on mental health status , where the respondent may have a reduced ability to accurately report their status . in public health research , excessive zero responses may occur if the population which is susceptible to respond is not carefully screened or is unknown . the resulting zero inflation may have an effect on the results obtained by conventional methods of analysis . in the next generation health study , investigators were interested in identifying predictors of dating violence in teenagers . examining these regression relationships are of interest for those individuals who are in a relationship ( i.e. , the susceptible condition ) . many more individuals completed this study component than investigators thought would be in a relationship at this age . this led to what appears to be zero - inflated clustered binary data . we developed both ml and gee approaches for analyzing such data . through simulations and analysis of the real data example , we found that the ml approach is substantially more efficient than the gee approaches . however , under moderate model misspecification , the ml approach may result in biased inference . it is recommended that , as a sensitivity analysis , both ml and gee approaches be applied in applications . in the gee approach , we treat the regression parameters and nuisance working correlation as orthogonal , that is , a gee1 approach with the parameters in the working correlation estimated by a moment estimator . potential efficiency gain could be achieved using an improved version of gee1 [ @xcite ] or gee2 [ prentice and zhao ( @xcite ) ; liang , zeger and qaqish ( @xcite ) ] , by establishing another set of estimating equations on the second moment . however , gee2 requires a correct variance structure with working third and fourth moment model , which is hard to verify with the presence of zero inflation , and results in bias under second , third and fourth moment misspecification . the previous work of @xcite adopted prentice and zhao s gee2 that maintains the parameter orthogonality in their second moment estimating equations , and they argued that only making a first moment assumption may lead to parameter nonidentifiability . this is true in their case , where the zero - inflation probability is at the observation level , so @xmath174 and @xmath62 might be confounded in @xmath175 . however , in the case of subject - level zero inflation , we have shown the model identifiability of the gee1 estimators . in principle , zero - inflated models can not be identified nonparametrically ; parametric assumptions for the nonzero part play a fundamental role in model estimation . for example , zero inflation in poisson and binomial data can be determined by the lack of fit in the zero cells of these respective distributions . in this paper , we assume that the nonzero distribution is given by a generalized linear mixed model with normal random effects . the ml approach exploits the correlation structure in order to distinguish structural zeros and random zeros . intrinsically , gee only uses the mean structure of the binary data in order to estimate regression parameters and , unlike ml , does not use the entire distribution for estimation . in our application , there were very few missing data . however , in many studies with sensitive psychological or behavioral questions , there may be informative missingness . an advantage of the ml approach is that it can more easily be extended to account for informative missing responses [ see @xcite , e.g. ] . the proposed methodology implicitly assumes that the subjects answer the questions truthfully . if this assumption does not hold , the parameter estimation is likely to be biased . we could formulate a likelihood approach if we had good prior information about the distribution of false negative occurrence across the questions . this would require a verification subsample corresponding to the survey questions ( maybe obtained through interviews of parents and friends ) on a fraction of teenagers . the zero - inflation probability in our model is assumed to be constant , but it is straightforward to include covariates in both ml and gee approaches . in addition , our focus is on a cross - sectional inference , that is , analyzing the dating violence data at one point in time ( 11th grade ) . understanding behavior change from adolescence over time is interesting but also challenging . in the longitudinal setting , the zero - inflation probability is time - dependent that can probably be modeled by a latent markov process . we will leave it for future exploration . we greatly appreciate and enjoyed reading the very insightful comments and suggestions by the associate editor and two anonymous referees . their comments not only remarkably improved the quality of the paper , but also motivated our deeper thoughts of possible extensions of the methodology in the future . | the next generation health study investigates the dating violence of adolescents using a survey questionnaire .
each student is asked to affirm or deny multiple instances of violence in his / her dating relationship .
there is , however , evidence suggesting that students not in a relationship responded to the survey , resulting in excessive zeros in the responses .
this paper proposes likelihood - based and estimating equation approaches to analyze the zero - inflated clustered binary response data .
we adopt a mixed model method to account for the cluster effect , and the model parameters are estimated using a maximum - likelihood ( ml ) approach that requires a gaussian hermite quadrature ( ghq ) approximation for implementation .
since an incorrect assumption on the random effects distribution may bias the results , we construct generalized estimating equations ( gee ) that do not require the correct specification of within - cluster correlation . in a series of simulation studies ,
we examine the performance of ml and gee methods in terms of their bias , efficiency and robustness .
we illustrate the importance of properly accounting for this zero inflation by reanalyzing the next data where this issue has previously been ignored . , , |
You are an expert at summarizing long articles. Proceed to summarize the following text:
the grnwald formula approximation and the @xmath5 approximation of the caputo derivative have been regularly used for numerical solution of fractional differential equations @xcite . the grnwald formula approximation has weights @xmath6 and accuracy @xmath7 . the @xmath5 approximation has order @xmath4 and weights @xmath8 . the weights of the @xmath5 approximation are linear combinations of terms which have power @xmath9 . in the present paper we construct approximations of the caputo derivative whose weights consist of terms which have power @xmath10 and @xmath11 . the accuracy of the numerical solution of order @xmath4 is influenced by the coefficient of the term @xmath12 in the expansion of the approximation . in table 1 , table 3 and table 9 we compute the error and the order of the numerical solutions of the fractional relaxation equation which use approximations , and of the caputo derivative . in all experiments the accuracy of the numerical solutions using approximation is higher than the accuracy of the numerical solutions using approximations and . the caputo derivative of order @xmath13 , where @xmath14 is defined as @xmath15 when the first derivative of the function @xmath16 is a bounded function the caputo derivative at the initial point @xmath17 is equal to zero . the exponential function has caputo derivative @xmath18 and @xmath19 where @xmath20 is the mittag - leffler function ( @xmath21 ) @xmath22 the analytical solutions of ordinary and partial differential equations are often expressed with the mittag - leffler function . the ordinary fractional differential equation @xmath23 has exact solution @xmath24 the finite - difference schemes for numerical solution of ordinary and partial fractional differential equations use an approximation for the fractional derivative . let @xmath25 and @xmath26 be the value of the function @xmath16 at the point @xmath27 . the @xmath5 approximation of the caputo derivative is constructed by dividing the interval @xmath28 $ ] to subintervals of equal length @xmath29 and approximating the first derivative on each subinterval using a second - order central difference approximation . @xmath30 where @xmath31 , @xmath32 and @xmath33 approximation has accuracy @xmath34 when @xmath35 $ ] ( @xcite ) . the numbers @xmath36 have properties @xmath37 where @xmath38 and @xmath39 . .error and order of numerical solution @xmath40 $ ] of equation i with @xmath41 , equation ii with @xmath42 and equation iii with @xmath43 . [ cols="<,^,^,^,^,^,^ " , ] from , approximation has asymptotic expansion @xmath45 in order to construct approximations for the caputo derivative we express expansion formula in the following form @xmath46 in the previous section we showed that @xmath47-s_n[{\alpha}]+{\dfrac{n^{1-{\alpha}}}{{\alpha}(1-{\alpha})}}.\ ] ] in this section we obtain the formula for the coefficient @xmath48 and an approximation for the caputo derivative of order @xmath1 . by changing the order of summation in @xmath49 we have that @xmath50 . then @xmath51 when @xmath52 we obtain @xmath53 from the identity @xmath54 we obtain @xmath55 @xmath56 @xmath57}-n{\left [ } { \dfrac{n^{1-{\alpha}}}{1-{\alpha } } } \sum_{m=1}^{\infty}\binom{1-{\alpha}}{2m}{\dfrac{b_{2m}}{n^{2m}}}{\right ] } \\ & + { \dfrac{1}{2}}{\left [ } { \dfrac{n^{2-{\alpha}}}{2-{\alpha } } } \sum_{m=1}^{\infty}\binom{2-{\alpha}}{2m}{\dfrac{b_{2m}}{n^{2m}}}{\right ] } .\end{aligned}\ ] ] from @xmath58+{\dfrac{n^{-{\alpha}}}{{\alpha}}}+{\dfrac{1}{2n^{1+{\alpha}}}},\ ] ] @xmath59-{\dfrac{n^{2-{\alpha}}}{2-{\alpha}}}+{\dfrac{1}{2n^{{\alpha}-1}}}.\ ] ] then @xmath60+{\dfrac{n^{-{\alpha}}}{{\alpha}}}+{\dfrac{1}{2n^{1+{\alpha}}}}{\right ] } -n{\left [ } s_n[{\alpha}]-{\dfrac{n^{1-{\alpha}}}{1-{\alpha}}}+{\dfrac{1}{2n^{\alpha}}}{\right ] } \\ & + { \dfrac{1}{2}}{\left [ } s_n[{\alpha}-1]-{\dfrac{n^{2-{\alpha}}}{2-{\alpha}}}+{\dfrac{1}{2n^{{\alpha}-1}}}{\right ] } , \end{aligned}\ ] ] @xmath61-n s_n[{\alpha}]+{\dfrac{1}{2}}s_n[{\alpha}-1]+{\dfrac{n^{2-{\alpha}}}{{\alpha}({\alpha}-1)({\alpha}-2)}}.\end{aligned}\ ] ] approximation has expansion of order @xmath1 @xmath62 by approximating @xmath63 in using the approximations @xmath64 @xmath65 we obtain the approximation for the caputo derivative of order @xmath1 @xmath66 where @xmath67 @xmath68 @xmath69 @xmath70 when @xmath71 , approximation has weights @xmath72 when @xmath73 : @xmath74 @xmath75 @xmath76 @xmath77 in the previous sections computed the numerical solutions of the fractional relaxation equation of order @xmath78 and @xmath79 using the second - order approximation for the value @xmath80 . in order to obtain a third - order approximation for @xmath80 we need to compute the values of @xmath81 and @xmath82 . by applying fractional differentiation of order @xmath9 to equation we obtain @xmath83 then @xmath84 . in @xcite we showed that @xmath85 by differentiating we obtain @xmath86 @xmath87 when the solution of equation has a bounded second derivative , the value of @xmath82 is computed with the formula @xmath88 now we compute the values of the derivatives @xmath89 of the exact solutions of equation i , equation ii and equation iii . @xmath90 @xmath91 then @xmath92 . @xmath93 then @xmath94 the exact solution of equation i satisfies @xmath95 and @xmath96 . @xmath97 @xmath98 then @xmath99 . @xmath100 @xmath101 the exact solution of equation ii satisfies @xmath102 . @xmath103 @xmath104 @xmath105 then @xmath106 . @xmath107 @xmath108 the exact solution of equation iii satisfies @xmath109 . let @xmath110 from taylor s formula @xmath111 is an approximation for @xmath80 with accuracy @xmath112 . the accuracy of numerical solution @xmath113 $ ] with initial values @xmath114 is @xmath115 . in table 10 we compute the maximum error and the order of numerical solution @xmath113 $ ] of equation i , equation ii and equation iii . in figure 3 we compare numerical solutions @xmath116,ns[40]$ ] and @xmath113 $ ] of equation iii and @xmath117 . $ ] -red , and @xmath118$]-blue , and @xmath116$]-green for @xmath119 and @xmath120.,scaledwidth=55.0% ] in the present paper we obtained approximations of the caputo derivative of order @xmath121 and @xmath1 whose weights consist of terms which have power @xmath10 and @xmath11 . in all experiments the accuracy of the numerical solutions which use approximation for the caputo derivative is higher than the accuracy of the numerical solutions using approximations and . a question for future work is to construct an approximation of order @xmath4 whose weights consist of terms which have power @xmath10 , where the accuracy of the numerical solutions of equation i , equation ii and equation iii is higher than the accuracy of the numerical solutions using the @xmath5 approximation . 99 m. abramowitz , i. a. stegun : _ handbook of mathematical functions with formulas , graphs , and mathematical tables , _ dover , new york , 1964 . d. baleanu , k. diethelm , e. scalas and j. j. trujillo : _ fractional calculus : models and numerical methods , _ world scientific , ( 2012 ) . k. diethelm : _ the analysis of fractional differential equations : an application - oriented exposition using differential operators of caputo type , _ springer ; 2010 . y. dimitrov : _ numerical approximations for fractional differential equations , _ journal of fractional calculus and applications , * 5(3s ) * ( 2014 ) , no . 22 , 145 . y. dimitrov : _ a second order approximation for the caputo fractional derivative , _ * arxiv:1502.00719 * ( 2015 ) . y. dimitrov : _ higher - order numerical solutions of the fractional relaxation - oscillation equation using fractional integration , _ * arxiv:1603.08733 * ( 2016 ) . h. ding , c. li : _ high - order numerical algorithms for riesz derivatives via constructing new generating functions , _ * arxiv:1505.03335 * , ( 2015 ) . h. ding , c. li : _ high - order algorithms for riesz derivative and their applications ( iii ) _ , fract . , * 19(1 ) * , ( 2016 ) , 1955 . g. h. gao and z. z. sun , h. w. zhang : _ a new fractional numerical differ - entiation formula to approximate the caputo fractional derivative and its applications , _ j. comput . * 259 * ( 2014 ) , 3350 . r. gorenflo , f. mainardi : _ fractional calculus : integral and differential equations of fractional order _ , cism lecture notes ( 2008 ) . m. glsu , y. ztrk and a. anapali : _ numerical approach for solving fractional relaxation - oscillation equation , _ applied mathematical modelling , * 37(8 ) * ( 2013 ) , 59275937 . m. ishteva : _ properties and applications of the caputo fractional operator , _ master thesis , university of karlsurhe ( 2005 ) . b. jin , r. lazarov and z. zhou : _ an analysis of the l1 scheme for the subdiffusion equation with nonsmooth data , _ i m a j. numer . , * 36 ( 1 ) * ( 2016 ) , 197221 . c. li , r. wu , h. ding : _ high - order approximation to caputo derivatives and caputo - type advection - diffusion equations , _ communications in applied and industrial mathematics , * 6(2 ) * ( 2014 ) , e-536 . c. li , f. zeng : _ numerical methods for fractional calculus , _ chapman and hall / crc ( 2015 ) . y. lin , c. xu : _ finite difference / spectral approximations for the time - fractional diffusion equation , _ journal of computational physics , * 225(2 ) * ( 2007 ) , 15331552 . c. lubich : _ discretized fractional calculus _ , siam j. math . , * 17 * ( 1986 ) , 704719 . y. ma : _ two implicit finite difference methods for time fractional diffusion equation with source term , _ journal of applied mathematics and bioinformatics , * 4(2 ) * ( 2014 ) , 125145 . f , mainardi : _ fractional relaxation - oscillation and fractional diffusion - wave phenomena , _ chaos , solitons @xmath122 fractals , * 7(9 ) * ( 1996 ) , 14611477 . f. mainardi , r. gorenflo : _ time - fractional derivatives in relaxation processes : a tutorial survey _ , fractional calculus and applied analysis , * 10(3 ) * ( 2007 ) , 269308 . d. a. murio , implicit finite difference approximation for time fractional diffusion equations , computers & mathematics with applications , * 56(4 ) * ( 2008 ) , 11381145 . z. odibat : _ approximations of fractional integrals and caputo fractional derivatives , _ applied mathematics and computation , * 178(2 ) * ( 2006 ) , 527533 . i. podlubny : _ fractional differential equations , _ academic press , san diego ( 1999 ) . h. m. srivastava , j. choi : _ series associated with the zeta and related functions _ , kluwer academic publishers ( 2001 ) . c. tadjeran , m. m. meerschaert and h. p. scheffer : _ a second - order accurate numerical approximation for the fractional diffusion equation , _ journal of computational physics , * 213 * ( 2006 ) , 205213 . tian , h. zhou and w. deng : _ a class of second order difference approximation for solving space fractional diffusion equations , _ mathematics of computation , * 84(294 ) * ( 2012 ) , 17031727 . r. wu , h. ding and c. li : _ determination of coefficients of high - order schemes for riemann - liouville derivative , _ the scientific world journal ( 2014 ) , article i d 402373 . y. yan , k. pal and n. j. ford : _ higher order numerical methods for solving fractional differential equations , _ bit numerical mathematics , * 54(2 ) * ( 2014 ) , 555584 . | in this paper we construct approximations for the caputo derivative of order @xmath0 and @xmath1 .
the approximations have weights @xmath2 and @xmath3 , and the higher accuracy is achieved by modifying the initial and last weights using the expansion formulas for the left and right endpoints .
the approximations are applied for computing the numerical solution of ordinary fractional differential equations .
the properties of the weights of the approximations of order @xmath4 are similar to the properties of the @xmath5 approximation . in all experiments presented in the paper the accuracy of the numerical solutions using the approximation of order @xmath4 which has weights
@xmath3 is higher than the accuracy of the numerical solutions using the @xmath5 approximation for the caputo derivative . *
2010 math subject classification : * 26a33 , 34a08 , 34e05 , 41a25 + * key words and phrases : * fractional derivative , approximation , fourier transform , fractional differential equation . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
asteroseismology of solar - type stars has been one of the major successes of the nasa _ kepler _ mission ( gilliland et al . the nominal mission provided data of exquisite quality for unprecedented numbers of low - mass main - sequence stars and cool subgiants . asteroseismic detections were made in more than 600 field stars ( chaplin et al . 2011a ; 2014 ) , including a sample of _ kepler _ planet hosts ( huber et al . these data have enabled a range of detailed asteroseismic studies ( see chaplin & miglio 2013 and references therein ) , many of which are ongoing . the nominal mission ended in 2013 may with the loss of a second of the spacecraft s four onboard reaction wheels . this meant the spacecraft could no longer maintain three - axis stabilized pointing . however , thanks to the ingenuity of the mission teams , _ kepler _ data collection has continued as a new ecliptic - plane mission , k2 ( howell et al . targeting stars in the ecliptic minimizes the now unconstrained roll about the spacecraft boresight , thereby helping to compensate for the loss of full three - axis stability . the degraded photometric performance presents particular challenges for the detection of oscillations in solar - type stars . the oscillations are stochastically excited and intrinsically damped by near - surface convection . while this mechanism gives rise to a rich spectrum of potentially observable overtones , having periods of the order of minutes , it also limits the modes to tiny amplitudes , typically several parts - per - million in brightness . the opportunity to continue asteroseismic studies of solar - type stars with k2 would provide fresh data on stars in the solar neighborhood for application to both stellar and galactic chemical evolution studies . the new fields have also led to the possibility of detecting oscillations of solar - type stars in open clusters and eclipsing binaries . this would provide independent data to test the accuracy of asteroseismic estimates of fundamental stellar properties . other specific targets of interest would potentially benefit from the provision of asteroseismic data , known exoplanet host stars being obvious examples . in this paper we report the detection of oscillations in several subgiants using k2 short - cadence ( sc ) data collected during campaign1 ( c1 ) . we describe the target selection and data analysis , and also discuss the implications of our results for future k2 campaigns . our selected target list started with the hipparcos catalog ( van leeuwen 2007 ) . use of these data allows us to make robust predictions for many bright , potential k2 targets in the ecliptic . effective temperatures were estimated from the @xmath2 color data in the catalog , using the calibration of casagrande et al . ( 2010 ) , and luminosities , @xmath3 , were estimated from the parallaxes . these calculations used reddening estimates from drimmel et al . ( 2003 ) ( negligible for many of our targets ) . we adopted @xmath4 ( torres 2010 ) , and consistent bolometric corrections from the flower ( 1996 ) polynomials presented in torres ( 2010 ) , which use the estimated @xmath5 as input . we also applied a cut on parallax , selecting only those stars having fractional parallax uncertainties of 15% or better . stellar radii were then estimated from @xmath3 and @xmath5 , and approximate masses were estimated from a simple power law in @xmath3 ( which was sufficient for selecting targets ) . the estimated hipparcos - based stellar properties were used as input to well - tested procedures ( chaplin et al . 2011b ) that enabled us to predict seismic parameters and relevant detectability metrics . we narrowed down the sample to 23 well - characterized bright ( @xmath6 to 9 ) solar - type stars to be proposed for k2 observations . all targets were predicted to show solar - like oscillations on timescales of the order of minutes , necessitating sc observations . we also collected ground - based spectroscopic data on our selected c1 targets to help us check the hipparcos - based predictions , and to better understand the final yield of asteroseismic detections . observations were made using the tres spectrograph ( frsz 2008 ) on the 1.5-m tillinghast telescope at the f. l. whipple observatory . atmospheric parameters were derived using the stellar parameter classification pipeline ( spc ; see buchhave et al . spc was used to match observed spectra taken at a resolution of 44000 to sets of synthetic model spectra to derive estimates of @xmath7 , @xmath8 , metallicity , and @xmath9 . in what follows we assume that relative metal abundances [ m / h ] returned by spc are equivalent to relative iron abundances , [ fe / h ] . table [ tab : tab1 ] contains the derived spectroscopic parameters . there are four rapidly rotating stars in the sample , and some caution is advised regarding their estimated parameters . overall , we found good agreement between the spectroscopic parameters and the hipparcos - based values . table [ tab : tab1 ] also includes the hipparcos - based estimates of the luminosities . to understand the limits on k2 performance in c1 , we deliberately sampled the region of the hr diagram across which detections had been made in the nominal mission , as shown in the top panel of fig . [ fig : fig1 ] . the symbols denote stars that provided firm asteroseismic detections ( black ) , marginal detections ( gray ) , no detections ( open ) or no detections with a high measured @xmath9 ( red asterisks ) . details are given below ( notably in section [ sec : det ] ) . llcccccc 201162999 & 56884 & @xmath10 & @xmath11 & @xmath12 & @xmath13 & @xmath14 & @xmath15 + & & & & & & & + 201164031 & 56907 & @xmath16 & @xmath17 & @xmath18 & @xmath19 & @xmath20 & @xmath21 + & & & & & & & + 201182789 & 57275 & @xmath22 & @xmath23 & @xmath24 & @xmath25 & @xmath26 & @xmath27 + & & & & & & & + 201215315 & 57456 & @xmath28 & @xmath29 & @xmath30 & @xmath31 & @xmath32 & @xmath33 + & & & & & & & + 201343968 & 55379 & @xmath34 & @xmath35 & @xmath36 & @xmath37 & @xmath38 & @xmath39 + & & & & & & & + 201353392 & 55288 & @xmath40 & @xmath41 & @xmath42 & @xmath43 & @xmath44 & @xmath45 + & & & & & & & + 201367296 & 58093 & @xmath46 & @xmath47 & @xmath48 & @xmath49 & @xmath50 & @xmath51 + & & & & & & & + 201367904 & 58191 & @xmath52 & @xmath53 & @xmath24 & @xmath54 & @xmath55 & @xmath56 + & & & & & & & + 201421619 & 55438 & @xmath57 & @xmath58 & @xmath59 & @xmath60 & @xmath61 & @xmath62 + & & & & & & & + 201436411 & 56282 & @xmath63 & @xmath23 & @xmath64 & @xmath65 & @xmath66 & @xmath67 + & & & & & & & + 201592408 & 56755 & @xmath68 & @xmath69 & @xmath70 & @xmath71 & @xmath72 & @xmath73 + & & & & & & & + 201601162 & 54675 & @xmath74 & @xmath75 & @xmath36 & @xmath76 & @xmath77 & @xmath78 + & & & & & & & + 201602813 & 55022 & @xmath79 & @xmath80 & @xmath81 & @xmath82 & @xmath83 & @xmath84 + & & & & & & & + 201614568 & 54857 & @xmath85 & @xmath41 & @xmath86 & @xmath87 & @xmath88 & @xmath89 + & & & & & & & + 201620616 & 58643 & @xmath90 & @xmath91 & @xmath92 & @xmath71 & @xmath93 & @xmath94 + & & & & & & & + 201626704 & 54541 & @xmath95 & @xmath96 & @xmath97 & @xmath98 & @xmath99 & @xmath100 + & & & & & & & + 201698809 & 55638 & @xmath101 & @xmath102 & @xmath97 & @xmath103 & @xmath104 & @xmath105 + & & & & & & & + 201729267 & 55574 & @xmath106 & @xmath107 & @xmath108 & @xmath109 & @xmath110 & @xmath111 + & & & & & & & + 201733406 & 55467 & @xmath112 & @xmath107 & @xmath113 & @xmath114 & @xmath115 & @xmath116 + & & & & & & & + 201756263 & 57034 & @xmath117 & @xmath80 & @xmath118 & @xmath119 & @xmath120 & @xmath121 + & & & & & & & + 201820830 & 55778 & @xmath122 & @xmath123 & @xmath124 & @xmath125 & @xmath126 & @xmath127 + & & & & & & & + 201853942 & 57136 & @xmath128 & @xmath129 & @xmath86 & @xmath130 & @xmath131 & @xmath132 + & & & & & & & + 201860743 & 57676 & @xmath133 & @xmath134 & @xmath118 & @xmath65 & @xmath135 & @xmath136 + & & & & & & & + [ tab : tab1 ] each target was observed by k2 for just over 82days in c1which lasted from 2014 may 30 to 2014 august 21with data collected in sc mode ( gilliland et al . 2010b ) . we used the k2p@xmath137 pipeline ( lund et al . 2015 ) to prepare sc lightcurves for asteroseismic analysis . in brief , the pipeline took the sc target pixel data as input . masks for all targets in a given frame were defined manually , and flux and position data were then extracted for our chosen targets of interest . corrections were then applied to the lightcurves to mitigate the impact of changes of the target positions on the ccd . finally , additional corrections were made using the filtering described by handberg & lund ( 2014 ) . the lightcurves were distributed to several teams , who each attempted to detect signatures of solar - like oscillations in the power spectra of the data . a complementary range of well - tested analysis codes was used , which had been applied extensively to data from the nominal _ kepler _ mission ( e.g. christensen - dalsgaard et al . 2008 ; huber et al . 2009 ; mosser & appourchaux 2009 ; roxburgh 2009 ; hekker et al . 2010 ; kallinger et al . 2010 ; mathur et al . 2010 ; gilliland et al . 2011 ; benomar et al . 2012 ; campante 2012 ; see also comparison of methods in verner et al . 2011 ) . in cases where oscillations were detected , each team was asked to return estimates of the two most commonly used global or average asteroseismic parameters : @xmath138 , the average frequency spacing between consecutive overtones of the same angular degree ; and @xmath139 , the frequency at which the oscillations present their strongest observed amplitudes . we checked the asteroseismic detection yield using the spectroscopic data . the bottom panel of fig . [ fig : fig1 ] provides a visual summary of these checks . the horizontal axis shows the predicted @xmath139 for each target , made using the spectroscopic @xmath7 and @xmath140 as input . estimates were calculated using the widely - used scaling relation ( brown et al . 1991 , kjeldsen & bedding 1995 ) : @xmath141 with the solar value @xmath142 ( see chaplin et al . 2014 ) providing the absolute calibration . these predicted @xmath139 are given in table [ tab : tab1 ] . the vertical axis on the bottom panel of fig . [ fig : fig1 ] relates to our ability to detect solar - like oscillations . in the frequency domain , where the analysis of the k2 data is conducted , peaks due to the oscillations are superimposed on a slowly varying , broad - band background . our ability to make a detection depends on the prominence of the oscillation peaks above that underlying background . for solar - like oscillators , the background in the frequency range occupied by the most prominent modes has contributions from granulation , shot noise , and other instrumental noise . the symbols with error bars show measured background levels , @xmath143 , in the k2 spectra at the predicted @xmath144 . solar - like oscillators with lower frequencies of maximum variability show larger amplitude oscillations : the lower @xmath144 , the larger is the maximum amplitude ( i.e. , brightness variation , in ppm ) and hence the easier it is to make a detection for a given @xmath145 . using the results from over 600 stars from the nominal mission we can predict oscillation amplitudes as a function of @xmath139 . with the predicted amplitudes in hand , we may estimate threshold background levels that would permit a significant detection of the oscillations . we calculated these thresholds using the detection recipe in chaplin et al . ( 2011b ) . the lines in the bottom panel of fig . [ fig : fig1 ] show the threshold levels , @xmath146 , below which the observed backgrounds @xmath145 must lie to have a @xmath147 ( black dashed ) and @xmath148 ( black dotted ) chance of making a detection . the gray curves show uncertainties on the plotted thresholds . the results returned by the mode - detection teams indicated that we had four good asteroseismic detections . these stars are plotted in fig . [ fig : fig1 ] using filled black symbols ; numbers are the final three digits of the associated k2 ecliptic plane catalog ( epic ) numbers ( huber et al . , in preparation ) . two other targets showed marginal detections , and they are shown in gray . it is important to stress that these stars all lie in the part of the bottom panel where we would expect to make detections , i.e. , where @xmath149 . even though there are other targets that show lower or similar background levels in their k2 spectra , they lie at higher predicted @xmath139 where the intrinsic oscillation amplitudes , and hence the chances of making a detection , are lower . the observed @xmath139 were in good agreement with the spectroscopic predictions , at the level of precision of the data . note that the spectroscopic predictions of @xmath139 have fractional uncertainties of @xmath150 , which are significantly larger than the typical uncertainties given by the asteroseismic measurements ( which are in contrast at the few - percent level ) . [ fig : fig2 ] shows the k2 power spectra of the four stars with firm detections . each spectrum has been smoothed with a @xmath151 boxcar filter . sets of vertical gray solid and dashed lines are separated by the estimated average @xmath138 , and mark the spacing on which we would expect to see modes . the power envelope of the oscillation spectrum of epic201820830 is somewhat flatter in frequency than the more classic gaussian - like envelopes shown by the other three stars . this is a characteristic of hot f - type stars ( e.g. , arentoft et al . the oscillations are strongly damped and this tends to wash out the visual appearance of the oscillation spectrum . the insets show the power spectrum of the power spectrum ( psps ) of each star , computed from the region around @xmath139 . the significant peak in each psps lies at @xmath152 , and is the detected signature of the near - regular spacing of oscillation peaks in the frequency spectrum . these detection signatures persisted for each star when the respective c1 lightcurves were divided into two equal halves in the time domain and analyzed separately . there are two stars in the the bottom panel of fig . [ fig : fig1 ] that did not yield firm or even marginal detections , but might be expected to do so . one of these stars ( epic201756263 ) is plotted in red . there are also three other stars shown in red on the diagram . these are all rapidly rotating solar - type stars with @xmath153 in the range @xmath154 to @xmath155 . they are therefore presumably young and may be very active , which is known to lead to significant suppression of the oscillation amplitudes ( e.g. , garca et al . 2010 ; chaplin et al . 2011c ; campante et al . 2014 ) making it much harder to detect oscillations . the detection recipe does not yet make allowance for suppression of the oscillation amplitudes by activity . the other star that might be expected to show a detection ( epic201602813 , plotted with an open symbol ) is noteworthy because it is by far the most metal - poor star in the sample . this may lead to attenuation of the observed amplitudes , relative to the basic predictions ( e.g. , houdek et al . 1999 ; samadi et al . other explanations are that the @xmath139 scaling underpredicts the @xmath139 value for this star ; or that the spectroscopic parameters are incorrect . we think that this latter explanation is unlikely . the target - selection ( hipparcos - based ) prediction for @xmath139 was in good agreement with the spectroscopic estimate . the target is also the primary component of a single - lined spectroscopic binary ( carney et al . 2001 ) , which will be spatially unresolved in the k2 pixel data . however , flux contamination from the secondary ( a suspected white dwarf ) is likely to be low . it is of course also possible that residual artifacts and other problems relating to the lightcurve extraction , preparation and filtering may have prevented a good detection being made . there are clearly persistent artifact peaks in some of the frequency spectra . moreover , the high - frequency noise levels for all stars are not close to being shot - noise limited , unlike the nominal _ kepler _ data . analysis of the k2 sc spectra therefore demands some degree of careful , manual scrutiny to be sure that a claimed detection is not the result of a chance combination of noise peaks . not including the active target epic201756263 , of the seven cases for which we would hope to detect oscillations , there are firm detections in four and marginal detections in two this is a good return given the challenges posed by the k2 photometry . we consolidated the @xmath138 and @xmath139 estimates returned by the mode detection teams to give final asteroseismic parameters ready for asteroseismic modeling . the final parameters were those returned by an updated version of the octave pipeline ( chaplin et al . , in preparation ; see also hekker et al . its estimates had the smallest average deviation from the median estimates in a global comparison made over all pipelines and all stars . uncertainties on each final parameter were given by adding ( in quadrature ) : the formal parameter uncertainty given by the chosen pipeline ; the standard deviation of the parameter estimates given by all other pipelines ; and a small contribution to account for uncertainties in the solar reference values ( which are employed in the grid - modeling ; see chaplin et al . 2014 for further discussion ) . the final parameters are listed in table [ tab : tab2 ] . a second set of teams were then asked to independently apply grid - based modeling to estimate fundamental properties of the four stars with firm detections . these teams , like their mode - detection counterparts , used codes that have been applied extensively to nominal - mission _ kepler _ data ( e.g. , see : stello et al . 2009 ; basu et al . 2010 ; kallinger et al . 2010 ; quirion et al . 2010 ; gai et al . 2011 ; bazot et al . 2012 ; creevey et al . 2013 ; hekker et al . 2013 ; lundkvist et al . 2014 ; miglio et al . 2013 ; serenelli et al . 2013 ; hekker & ball 2014 ; rodrigues et al . 2014 ; silva aguirre et al . further details may also be found in chaplin et al . ( 2014 ) and pinsonneault et al . ( 2014 ) . we tested the impact on the estimated stellar properties of using different sets of inputs , i.e. , a first set with \{@xmath138 , @xmath139 , @xmath5 , [ fe / h ] } , a second set with \{@xmath138 , @xmath5 , [ fe / h ] } , and further sets with the parallax - based luminosities also included . we found very good agreement between the properties given by the different sets , and by the different pipelines . results from the first two sets were consistent with the independent luminosity estimates . using the luminosities as an additional input constraint did not have a significant impact on the results . the parallax uncertainties which range from @xmath156 to 12%are too large to add anything useful to the seismic constraints . the fact that the second set of inputs provided results that were consistent with the estimated @xmath139 lends further confidence to the claimed detections and suggests that , at least for these data , potential bias in the @xmath139 estimates arising from spurious noise peaks is not a significant cause for concern . table [ tab : tab2 ] gives final values for the estimated properties , using the first set of inputs . the properties were calculated using the bespp pipeline ( serenelli et al . 2013 ) , which uses individual model frequencies to calculate model predictions of @xmath138 for comparison with the observations . the uncertainties include a contribution from the scatter between pipelines ( following the procedure outlined above for the input seismic parameters ; see also the in - depth discussions in chaplin et al . 2014 ) . these grid - modeling results demonstrate that k2 has returned solid results that allow asteroseismic modeling to be performed on the targets . two cases here epic201367296 and epic201367904 will be amenable to more in - depth modelling studies since it will be possible to extract precise and robust individual frequencies of several overtones of each star . cccccccc 201367296 & 58093 & @xmath157 & @xmath158&@xmath159 & @xmath160 & @xmath161 & @xmath162 + 201367904 & 58191 & @xmath163 & @xmath164&@xmath165 & @xmath166 & @xmath167 & @xmath168 + 201820830 & 55778 & @xmath169 & @xmath170&@xmath171 & @xmath172 & @xmath173 & @xmath174 + 201860743 & 57676 & @xmath175 & @xmath176&@xmath159 & @xmath177 & @xmath178 & @xmath179 [ tab : tab2 ] we analysed k2 short - cadence ( sc ) data for 23 solar - type stars observed in c1 . of the seven targets where we would hope to detect oscillations , there are firm asteroseismic detections in four cases , and marginal detections in a further two . this represents a good return , in spite of the challenges posed by the k2 photometry . in sum , we have a very good understanding of the asteroseismic yield . the results put us in a good position to hone target selections for future campaigns . current performance levels mean we can detect oscillations in sub - giants , but not in main - sequence stars . changes to the operation of the fine - guidance sensors are expected to give significant improvements in the high - frequency performance of k2 from c3 onwards . the high - frequency noise is currently a crucial limitation to making asteroseismic detections , in particular in main - sequence stars . with reference to the bottom panel of fig . [ fig : fig1 ] , we note that a reduction of excess high frequency noise by a factor of two - and - a - half in amplitude ( just over six in power ) would lead to consistent detections of oscillations in main sequence stars with @xmath139 as high as @xmath1 , as well as converting marginal detection cases to ones for which detailed modeling could be performed . the prospects are therefore very encouraging . solar - type stars in the pleiades and hyades open clusters have already been observed by k2 in sc during c4 . more stars will be observed in sc during c5 in the open clusters m44 and m67 . there is also now clear potential to build up a statistical sample of solar - type field stars in the ecliptic that have good asteroseismic data , and to target specific stars of interest for asteroseismic study such as bright eclipsing binaries and known exoplanet host stars . funding for this discovery mission is provided by nasa s science mission directorate . the authors wish to thank the entire _ kepler _ team , without whom these results would not be possible . we also thank all funding councils and agencies that have supported the activities of kasc working group1 . | we present the first detections by the nasa k2 mission of oscillations in solar - type stars , using short - cadence data collected during k2 campaign1 ( c1 ) .
we understand the asteroseismic detection thresholds for c1-like levels of photometric performance , and we can detect oscillations in subgiants having dominant oscillation frequencies around @xmath0 .
changes to the operation of the fine - guidance sensors are expected to give significant improvements in the high - frequency performance from c3 onwards .
a reduction in the excess high - frequency noise by a factor of two - and - a - half in amplitude would bring main - sequence stars with dominant oscillation frequencies as high as @xmath1 into play as potential asteroseismic targets for k2 . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
this paper derives an exact deterministic nonlinear observer to compute the continuous state of an inertial navigation system based on partial discrete measurements . the main analysis tool is nonlinear contraction theory @xcite . recent work on nonlinear observer design for mechanical systems based on nonlinear contraction theory can be found in @xcite . specifically , we consider the classical strap - down problem in inertial navigation @xcite , where angular position ( euler angles ) @xmath0 and inertial position * r * are computed from the body turn rate @xmath1 and inertial acceleration @xmath2 , measured continuously in intrinsic ( body - fixed ) coordinates , @xmath3 with @xmath4\ ] ] and @xmath5\ ] ] as made precise in @xcite such a system lies at the boundary between convergence and divergence , much like a triple integrator . in this paper , the continuous measurements of @xmath1 and @xmath2 are augmented by _ discrete _ measurements of @xmath6 and @xmath7 , leading to a globally exponentially convergent nonlinear observer design . such combinations of measurements are typical in inertial navigation , whether for vehicles or robots ( see e.g. @xcite for a recent discussion ) . the human vestibular system also features a similar structure , with otolithic organs measuring linear acceleration and semi - circular canals estimating angular velocity through heavily damped angular acceleration signals , an information then combined with visual data at much slower update rate . section [ sec : general ] introduces the basic observer designs . we build simple observers to compute @xmath8 based on partial discrete measurements @xmath9 and @xmath10 . in section [ sec : extension ] we discuss extensions , such as the use of nonlinear measurements , and the effects of system disturbance and measurement disturbance @xcite . we also study the case where the inertial navigation system is expressed in quaternion form @xcite . section [ sec : simulation ] presents simulation results on a 3-dimensional system . brief concluding remarks are offered in section [ sec : conclusion ] . a very brief review of basic results in contraction theory is presented in the appendix . in this section , we construct a discrete observer for system ( [ eq : system ] ) , which consists of a hierarchy of three sub - systems , mirroring the hierarchical nature of systems physics ( [ eq : system ] ) . the observer is based on the partial - measurements of the state * x * and * r * at a series of instants @xmath11 . + + * first * , based on the discrete measurement @xmath12 , compute * x * with the observer @xmath13 where the first equation describes a continuous update between measurements , and the second equation a discrete measurement incorporation . virtual displacements , which are systematically used in mathematical physics and optimization theory , also represent basic tools in contraction theory ( see appendix ) . computing virtual displacements in ( [ eq : observer_ep1 ] ) leads to @xmath14 based on @xcite , define @xmath15 with @xmath16 . this implies that @xmath17 + from ( [ eq : delta_x_hat ] ) , we have @xmath18 from hybrid contraction condition ( [ eq : hybrid_condition ] ) in the appendix , if @xmath19 where @xmath20 , then both @xmath21 and @xmath22 tend to zero exponentially . so @xmath23 tends to * x * exponentially . + + * second * , based on the discrete measurement of @xmath24 , compute * v * with the observer @xmath25 from ( [ eq : observer_ep2 ] ) and the first step , we get @xmath26 since @xmath27 tends exponentially to a constant , we have @xmath28 using ( [ eq : dp_delta_v ] ) , this implies that @xmath29 , which by continuity implies that the constant which @xmath27 tends to must be zero . we thus have , exponentially , @xmath30 since by design @xmath31 is a particular solution of ( [ eq : observer_ep2 ] ) , this implies that @xmath32 tends to @xmath33 exponentially . + + * third * , based on the discrete measurement @xmath34 , use the observer @xmath35 since we know @xmath27 tends to zero exponentially , we have @xmath36 if @xmath37 , i.e. @xmath38 where @xmath39 is the largest eigenvalue of @xmath40 . so @xmath41 tends to * r * exponentially . + * extension 1 * : when we compute * v * and * r * , we only use the discrete - time measurement @xmath34 without @xmath42 . this allows @xmath42 and @xmath34 to be measured at different instants , with the same computation . + + * extension 2 * : the metric can also be written @xmath43 since * a * is orthogonal . so we can simply use @xmath44 . + + * extension 3 * : assume that in ( [ eq : observer_ep1 ] ) we replace the discrete update law by the more general @xmath45 where @xmath46 and @xmath47 commute . then @xmath48 the hybrid contraction condition ( [ eq : condition_ep1 ] ) becomes @xmath49 where @xmath50 is the largest eigenvalue of @xmath51 . note that because the generalized jacobians are zero at each step of the hierarchy , the hybrid contraction conditions simply define the _ metrics _ in which the discrete measurement incorporation steps should be contracting . as we shall see later , the flexibility offered within this constraint will allow us to trade - off model error vs measurement error , similarly in spirit to a standard kalman filter . discussions about full discrete measurements , use of a linear observer , nonlinear measurement , disturbance effects , and quaternion representation are offered in this section . an observer based on full measurement is described in section [ sec : full ] . effects of system disturbance and measurement disturbance are discussed in section [ sec : error ] . section [ sec : nonlinear ] we develop a more general discrete observer applicable to nonlinear measurements . use of quaternions is studied in section [ sec : quaternion ] . assume that _ all _ states * x * , * v * , and * r * are actually measured , at a series of discrete instants @xmath52 . then steps 1 and 3 are unchanged , but we can replace step 2 ( the estimation of @xmath33 ) by the observer @xmath53 since we know @xmath22 tends to zero exponentially , we have @xmath54 with @xmath55 , we have @xmath56 where @xmath57 is the largest eigenvalue of @xmath58 . so @xmath32 tends to * v * exponentially . note that in some cases one only needs to estimate orientation @xmath6 and velocity @xmath33 , and that the discrete measurement of @xmath33 may be obtained from optical flow , which can be computationally `` expensive '' and thus infrequent . effects of bounded inputs and measurement disturbances can be quantified and obeserver gains chosen accordingly . consider input disturbance @xmath59 and measurement disturbance @xmath60 , with @xmath61 and @xmath62 , leading to the modified system @xmath63 using the basic robustness result in @xcite , we can quantify the corresponding quadratic bounds @xmath64 on the estimation error @xmath65 where @xmath66 is the largest eigenvalue of the symmetric part of @xmath67 . define the objective function ( @xmath68 ) @xmath69 then , @xmath70 , where @xmath71 and @xmath72 . we know @xmath73 should also satisfy @xmath74 define @xmath75 as an upper bound of @xmath73 . therefore , @xmath76 where @xmath77 . finally , we obtain the minimum of @xmath78 @xmath79 where @xmath71 and @xmath72 . when different measurements are available , the above formulas can also be used to select _ a priori _ the most informative measurement . this can be the case for instance for selecting the direction of gaze of the eyes in hopping robot @xcite . this can also be the case when the measurements are `` expensive '' , for instance computationally . + * extension * : the discussions above will still work when the bounds of input disturbance and measurement disturbance are time - varying . if @xmath80 and @xmath81 when @xmath82 . similar to the above , we have @xmath83 where @xmath71 and @xmath84 . for the system @xmath85 , consider the observer @xmath86 where @xmath87 we have @xmath88 defining @xmath89 , we have @xmath90 . using equation ( [ eq : nonlinear_delta ] ) yields @xmath91 where @xmath92 and @xmath93 . the sufficient contraction condition on hybrid systems can be written @xmath94 where @xmath95 and @xmath66 is the largest eigenvalue of the symmetric matrix @xmath96 . if condition ( [ eq : condition_gn ] ) is satisfied by an appropriate choice of @xmath97 , then @xmath98 will tend to * x * exponentially . a a simple illustration , consider using distance measurements instead of direct cartesian position measurements . in the 3-dimensional space , measure the distances from one point @xmath99 to four time - varying reference points @xmath100^t$ ] , @xmath101^t$ ] , @xmath102^t$ ] , and @xmath103^t$ ] , @xmath104 the discrete - update part of observer ( [ eq : nonlinear_observer ] ) can be built up as below , @xmath105 = \left [ \begin{array}{l } \hat{x}_{1,i}^- \\ \hat{x}_{2,i}^- \\ \hat{x}_{3,i}^- \end{array } \right ] -\frac{1}{2 } { \bf k}_i \left [ \begin{array}{l } ( \hat{y}_{1,i}^-)^2-(\hat{y}_{2,i}^-)^2-(y_{1,i}^2-y_{2,i}^2 ) \\ ( \hat{y}_{2,i}^-)^2-(\hat{y}_{3,i}^-)^2-(y_{2,i}^2-y_{3,i}^2 ) \\ ( \hat{y}_{3,i}^-)^2-(\hat{y}_{4,i}^-)^2-(y_{3,i}^2-y_{4,i}^2 ) \end{array } \right]\ ] ] where @xmath106 is a 3 by 3 time - varying gain matrix . using equation ( [ eq : nonlinear_y ] ) yields @xmath107 @xmath108\ ] ] where subscript @xmath109 refers to the value at time @xmath110 . + assume @xmath111 is non - singular . then we can choose @xmath112 with equation ( [ eq : nonlinear_dvarying ] ) , we have @xmath113 by choosing @xmath114 , we can make @xmath115 satisfy the following contraction contidion that makes @xmath21 tends to zero exponentially.@xmath116 where @xmath117 and @xmath66 is the largest eigenvalue of the symmetric matrix @xmath96 . therefore , @xmath22 will tend to zero , and @xmath98 will tend to * x * exponentially . + * remark * when @xmath111 is singular , one has @xmath118 + equation ( [ eq : nonlinear_singular1 ] ) is equivalent to @xmath119 \cdot \left| \begin{array}{ccc } { \bf i } & { \bf j } & { \bf k } \\ ( c_{1i}-b_{1i } ) & ( c_{2i}-b_{2i } ) & ( c_{3i}-b_{3i } ) \\ ( d_{1i}-c_{1i } ) & ( d_{2i}-c_{2i } ) & ( d_{3i}-c_{3i } ) \end{array } \right|=0\ ] ] + which we can write @xmath120 this means that points a , b , c , and d are in the same plane , and therefore that the geometry does not contain enough information to infer position . + to compute velocity , one can rewrite observer ( [ eq : observer_ep2 ] ) as @xmath121 - { \bf k}_i \left [ \begin{array}{c } ( \hat{y}_{1,i}^+)^2-(\hat{y}_{2,i}^+)^2-(y_{1,i}^2-y_{2,i}^2 ) \\ ( \hat{y}_{2,i}^+)^2-(\hat{y}_{3,i}^+)^2-(y_{2,i}^2-y_{3,i}^2 ) \\ ( \hat{y}_{3,i}^+)^2-(\hat{y}_{4,i}^+)^2-(y_{3,i}^2-y_{4,i}^2 ) \end{array } \right ] \ } \end{array } \right.\ ] ] where @xmath122 and @xmath123^{-1 } ~~\textrm{and}~~ { \bf k}_{i+1}= \left [ \begin{array}{ccc } ( b_{1i+1}-a_{1i+1 } ) & ( b_{2i+1}-a_{2i+1 } ) & ( b_{3i+1}-a_{3i+1 } ) \\ ( c_{1i+1}-b_{1i+1 } ) & ( c_{2i+1}-b_{2i+1 } ) & ( c_{3i+1}-b_{3i+1 } ) \\ ( d_{1i+1}-c_{1i+1 } ) & ( d_{2i+1}-c_{2i+1 } ) & ( d_{3i+1}-c_{3i+1 } ) \end{array } \right]^{-1}\ ] ] we then have @xmath124 which is the same as equation ( [ eq : dp_delta_v ] ) . similarly to the second step of section [ sec : general ] , this shows that @xmath125 tends to * v * exponentially . + note that the geometry problem of going from distances to positions is solved by a dynamic system , the observer , rather than explicitly at each instant . in general , one may also use linear measurements at some instants and nonlinear ones at others . note that if a measurement is delayed , the algorithms work similarly but the actual information is available after the delay ( i.e. the measurement is incorporated at some past time and the forward simulation runs instantly to the current time ) . consider now , extending section [ sec : error ] , the effect of model and measurement errors . for the system @xmath126 , with the following nonlinear observer , @xmath127 where@xmath128 and @xmath129 + + so we have @xmath130 we know the quadratic bounds @xmath64 on the estimation error @xmath131 where @xmath132 , @xmath133 , and @xmath134 is the largest eigenvalue of the symmetric part of @xmath67 . + + we can choose the most relevant discrete update function @xmath135 which will best contribute to improving the estimate @xmath98 ( i.e. , to minimize @xmath136 ) . angular position can be expressed in quaternion form , avoiding representation singularities @xcite . quaternions express a rotation of angle @xmath137 about the unit vector @xmath138 as @xmath139 . with @xmath140 the quaternion vector , this leads to @xmath141 where @xmath142\ ] ] and @xmath143\ ] ] in this representation , the fact that the dynamics of * q * is indifferent is obvious , since @xmath144 is skew - symmetric . the observers can be derived as earlier , simply by replacing ( [ eq : observer_ep1 ] ) by @xmath145 based on the discrete measurements @xmath146 . computing virtual displacements @xmath147 and because the dynamics of * q * is indifferent , we only need @xmath148 where @xmath50 is the largest eigenvalue of @xmath149 . under condition ( [ eq : condition_qp1 ] ) , @xmath150 tends to zero exponentially , and @xmath151 tends to * q * exponentially . the other two steps are unchanged , with @xmath152 being replaced by @xmath153 . all the above variations and extensions can of course be combined . in this section , we will do a 3-dimentional simulation about system ( [ eq : system ] ) based on the discrete measurement @xmath42 and the nonlinear distance measurements @xmath154 , @xmath155 , @xmath156 , and @xmath157 , as in section [ sec : nonlinear ] . consider system ( [ eq : system ] ) in the 3-dimensional case . where @xmath158~~\textrm{and}~~ { \bf \gamma}=~ \left [ \begin{array}{c } \cos(2 t ) \\ \frac{1 + 2\sin t}{3 } \end{array } \right]\ ] ] four time - varying reference points are chosen as below ( all move on circular trajectories ) , @xmath159 @xmath160 observer ( [ eq : observer_ep1 ] ) with @xmath161 is used to compute * x*. observer ( [ eq : observer_ep2_distance ] ) with gain ( [ eq : nonlinear_gain ] ) is used to compute * v*. using observer ( [ eq : nonlinear_observer],[eq : nonlinear_od1 ] ) and gain ( [ eq : nonlinear_k_i ] ) , we choose @xmath162 to satisfy condition ( [ eq : condition_gn ] ) , thus we can compute * r*. figure [ fig : example_ep ] shows @xmath163 tends to @xmath164 exponentially . observers similar to those developed in this paper can in principle be applied to other continuous nonlinear systems besides inertial navigation systems , although much simplification was afforded by exploiting the hierarchical structure of the system physics . an animation of the basic observer as applied to head stabilization @xcite in a simulated robot hopper @xcite can also be found in _ http://web.mit.edu / nsl / www / hopping_robot.mpg_. * acknowledgement * this paper benefited from stimulating discussions with dr . agostino martinelli . 0 aghannan , n. , rouchon , p. , an intrinsic observer for a class of lagrangian systems , _ i.e.e.e . trans . control _ , * 48(6 ) * ( 2003 ) . berthoz , a. , the brain s sense of movement , _ harvard university press , cambridge , massachusetts _ ( 2000 ) . egeland , o. , kristiansen , e. , and nguyen , t.d . , observers for euler - bernoulli beam with hydraulic drive , _ i.e.e.e . conf . ( 2001 ) . goldstein , h. , classical mechanics , _ addison - wesley _ ( 1980 ) . grassia , f.s . , practical parameterization of rotations using the exponential map , _ the journal of graphics tools _ , vol . 3.3 ( 1998 ) . hestenes , d. , new foundations for classical mechanics , _ kulwer academic publishers , dordrecht , the netherlands _ ( 1986 ) . jouffroy , j. , and j. opderbecke , underwater vehicle trajectory estimation using contracting pde - based observers , _ american control conference , boston , massachusetts _ ( 2004 ) . kristiansen , d. , and egeland , o. , time and spatial discretization methods that preserve passivity properties for systems described by partial differantial equations , _ in proceedings of the 2000 american control conference , chiago , illinois _ ( 2000 ) lohmiller , w. , and slotine , j.j.e . , on contraction analysis for nonlinear systems , _ automatica _ , * 34(6 ) * ( 1998 ) . lohmiller , w. , contraction analysis of nonlinear systems , _ ph.d . thesis , department of mechanical engineering , mit _ ( 1999 ) . lohmiller , w. , and slotine j.j.e . , nonlinear process control using contraction theory , _ a.i.ch.e . journal _ ( 2000 ) . lohmiller , w. , and slotine j.j.e . , control system design for mechanical systems using contraction theory , _ i.e.e.e . trans . control _ , * 45(5 ) * ( 2000 ) . slotine , j.j.e . , and li , w. , applied nonlinear control , _ prentice - hall _ ( 1991 ) . slotine , j.j.e . , and lohmiller , w. , modularity , evolution , and the binding problem : a view from stability theory , _ neural networks _ , * 14(2 ) * ( 2001 ) . raibert , m.h . , legged robots that balance , _ the m.i.t . press , cambridge , massachusetts _ ( 1986 ) . ryu , j. , rossetter e.j . , and gerdes , j.c . , vehicle sideslip and roll parameter estimation using gps . _ the proceedings of avec 2002 , hiroshima , japan _ ( 2002 ) . varshalovich , d.a . , moskalev , a.n . , and khersonskii , v.k . , description of rotation in terms of the euler angles , _ singapore : world scientific _ ( 1988 ) . the main theorem of contraction analysis @xcite can be stated as is uniformly negative definite , then all system trajectories then converge exponentially to a single trajectory , with convergence rate @xmath169 , where @xmath170 is the largest eigenvalue of the symmetric part of @xmath171 . the system is said to be contracting . [ th : theoremf ] it can be shown conversely that the existence of a uniformly positive definite metric with respect to which the system is contracting is also a necessary condition for global exponential convergence of trajectories . in the linear time - invariant case , a system is globally contracting if and only if it is strictly stable , with @xmath171 simply being a normal jordan form of the system and @xmath172 the coordinate transformation to that form . furthermore , since where is @xmath174 the symmetric part of @xmath171 , all transformations @xmath175 corresponding to the same @xmath176 lead to the same eigenvalues for @xmath177 , and therefore to the same contraction rate @xmath169 . consider now a hybrid case @xcite , consisting of a continuous system @xmath178 which is switched to a discrete system @xmath179 every @xmath180 for one discrete step . letting , in the _ same coordinate system @xmath172 _ , @xmath66 be the largest eigenvalue of the symmetric matrix @xmath96 , and @xmath181 be the largest eigenvalue of @xmath182 ( the corresponding discrete - time quantity , where @xmath183 , see @xcite ) , a sufficient condition for the overall system to be contracting is @xmath184 contraction theory proofs and this paper make extensive use of _ virtual displacements _ , which are differential displacements at fixed time borrowed from mathematical physics and optimization theory . formally , if we view the position of the system at time @xmath185 as a smooth function of the initial condition @xmath186 and of time , @xmath187 , then @xmath188 . | we derive an exact deterministic nonlinear observer to compute the continuous state of an inertial navigation system based on partial discrete measurements , the so - called strapdown problem .
nonlinear contraction is used as the main analysis tool , and the hierarchical structure of the system physics is sytematically exploited .
the paper also discusses the use of nonlinear measurements , such as distances to time - varying reference points . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
transiting extrasolar planets ( teps ) provide unique opportunities to study the physical properties of planetary mass objects outside of the solar system . by combining time - series photometric observations taken during transit with radial velocity ( rv ) measurements of the star , it is possible to precisely measure the mass and radius of the planet , if the stellar mass and radius can be determined by other means . the bulk density of the planet may then be compared with the predictions of theoretical planetary structure models ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) to infer the structure of the planet . discoveries of planets that fall outside the predicted mass - radius range ( e.g. inflated hot jupiters such as tres-4b ; * ? ? ? * ) lead in turn to refinements of these models . teps also provide unique opportunities to study planetary atmospheres , including their composition ( e.g. * ? ? ? * ) , and their thermal profiles ( e.g. * ? ? ? it is also possible to measure the projection of the angle between the orbital axis and the stellar spin axis for these planets ( e.g. * ? ? ? * ) , which may be used to test theories of planetary migration @xcite . to date more than 40 tep discoveries have been published , with the majority coming from dedicated photometric surveys . these planets span a range covering more than two orders of magnitude in mass from a super - earth tep ( corot-7b ; * ? ? ? * ) and super - neptunes ( gj 436b and hat - p-11b ; * ? ? ? * ; * ? ? ? * ) to brown dwarf size objects ( corot-3b , xo-3b ; * ? ? ? * ; * ? ? ? focusing on the low - mass end , we note that the three least massive teps with well determined masses ( gj 436b , hat - p-11b and hd 149026b , * ? ? ? * ) , are also the three teps with the smallest radii ( excluding corot-3b ) and highest inferred core mass fractions . above this , we begin to see planets with a wide range of radii . the planets wasp-11/hat - p-10b @xcite , wasp-6b @xcite , hat - p-1b @xcite , ogle - tr-111b @xcite , wasp-15b @xcite , and xo-2b @xcite all have radii larger than or comparable to that of jupiter , whereas hat - p-3b @xcite has a radius that is only slightly larger than that of saturn . given the small number of teps known with @xmath17 , we can not yet say what is the empirical minimum mass of core - less , or envelope dominated , gas giant planets . to explore the possible transition from envelope dominated to core dominated planets , it is necessary to find more low - mass teps . the hungarian - made automated telescope network ( hatnet * ? ? ? * ) survey has been a major contributor to the discovery of teps . operational since 2003 , it has covered approximately 10% of the northern sky , searching for teps around bright stars ( @xmath18mag ) . hatnet operates six wide field instruments : four at the fred lawrence whipple observatory ( flwo ) in arizona , and two on the roof of the submillimeter array hangar ( sma ) of sao in hawaii . since 2006 , hatnet has announced and published 11 teps . in this work we report on the 12th such discovery . this planet is only the fourth sub - saturn mass tep announced to date , but unlike the other planets , it is of low density , and appears to be h / he dominated . the structure of the paper is as follows . in we summarize the observations , including the photometric detection , and follow - up observations . in we describe the analysis of the data , such as the stellar parameter determination ( ) , blend modeling ( ) , and global modeling of the data ( ) . we discuss our findings in . the transits of hat - p-12b were detected with the hat-5 telescope in arizona . the region around gsc 03033 - 00706 , a field internally labeled as 145 , was observed on a nightly basis between january and july of 2006 , whenever weather conditions permitted . we gathered 4205 exposures of 5 minutes at a 5.5-minute cadence , of which 2927 images were used in the final light curve of hat - p-12 . each image contained approximately 10,000 stars down to @xmath19 . for the brightest stars in the field we achieved a per - image photometric precision of 3mmag . the calibration of the hatnet frames was done utilizing standard procedures . the calibrated frames were then subjected to star detection and astrometry , as described in @xcite . aperture photometry was performed on each image at the stellar centroids derived from the 2mass catalog @xcite and the individual astrometrical solutions . the resulting light curves were decorrelated against trends using the external parameter decorrelation technique in `` constant '' mode ( epd , see * ? ? ? * ) and the trend filtering algorithm ( tfa , see * ? ? ? the light curves were searched for periodic box - like signals using the box least squares method ( bls , see * ? ? ? * ) . we detected a significant signal in the light curve of gsc 03033 - 00706 ( also known as 2mass 13573347 + 4329367 ; @xmath20 , @xmath21 ; j2000 ; @xmath22 , @xcite ) , with a depth of @xmath23mmag , and a period of @xmath24days . the dip had a relative duration ( first to last contact ) of @xmath25 , corresponding to a total duration of @xmath26 hours ( see fig . [ fig : hatnet ] ) . all hatnet candidates are subjected to thorough investigation before using more precious time on large telescopes . one of the important tools for establishing whether the transit - like feature in the light curve of a candidate is due to astrophysical phenomena other than a planet transiting a star is the cfa digital speedometer ( ds ; * ? ? ? * ) , mounted on the telescope . this yields high - resolution ( @xmath27 ) spectra with low signal - to - noise ratio sufficient to derive radial velocities with moderate precision ( roughly 0.5 - 1@xmath28 ) , and to determine the effective temperature and surface gravity of the host star . with this facility we are able to reject many types of false positives , such as f dwarfs orbited by m dwarfs , grazing eclipsing binaries , triple and quadruple star systems , or giant stars where the transit signal can not be due to a planet . we obtained @xmath29 observations of hat - p-12 with the ds . the rv measurements of hat - p-12 showed an rms residual of 0.43@xmath28 , consistent with no detectable rv variation . the spectra were single - lined , showing no detectable evidence for more than one star in the system . atmospheric parameters for the star , including the effective temperature @xmath30 , surface gravity @xmath31 ( log cgs ) , and projected rotational velocity @xmath32 consistent with zero with an asymmetric error of about 1@xmath28 , were derived as described by @xcite . the effective temperature and surface gravity correspond to a mid - k dwarf . the mean heliocentric radial velocity of hat - p-12 is @xmath33@xmath28 . given the significant transit detection by hatnet , and the positive ds results that exclude obvious false positives , we proceeded with the follow - up of this candidate by obtaining high - resolution and high s / n spectra to characterize the radial velocity variations and to determine the stellar parameters with higher precision . using the hires instrument @xcite on the keck i telescope located on mauna kea , hawaii , we obtained 22 exposures with an iodine cell , plus one iodine - free template . the observations were made on 16 nights during a number of observing runs between 2007 march 27 and 2008 september 17 . [ ht ] the width of the spectrometer slit used on hires was @xmath34 , resulting in a resolving power of @xmath35 , with a wavelength coverage of @xmath36 . the iodine gas absorption cell was used to superimpose a dense forest of @xmath37 lines on the stellar spectrum and establish an accurate wavelength fiducial ( see * ? ? ? relative rvs in the solar system barycentric frame were derived as described by @xcite , incorporating full modeling of the spatial and temporal variations of the instrumental profile . the final rv data and their errors are listed in tab . [ tab : rvs ] . the folded data , with our best fit ( see ) superimposed , are plotted in fig . [ fig : rvbis ] . in fig . [ fig : rvbis ] we also plot the relative s index . this index is computed following the prescription given by @xcite after matching each spectrum to a reference spectrum using a transformation that includes a wavelength shift and a flux scaling that is a polynomial as a function of wavelength . the transformation is determined on regions of the spectra that are not used in computing the s index . note that the relative s index has not been calibrated to the scale of @xcite . the relative s index does not show any significant variation correlated with the orbital phase ; such a correlation might have indicated that the rv variations are due to stellar activity . @xmath38 & @xmath39 & @xmath40 & @xmath41 & @xmath42 & @xmath43 & @xmath44 + @xmath45 & @xmath46 & @xmath47 & @xmath48 & @xmath49 & @xmath50 & @xmath51 + @xmath52 & & & @xmath53 & @xmath54 & @xmath55 & @xmath51 + @xmath56 & @xmath57 & @xmath58 & @xmath59 & @xmath60 & @xmath50 & @xmath51 + @xmath61 & @xmath62 & @xmath63 & @xmath64 & @xmath65 & @xmath66 & @xmath67 + @xmath68 & @xmath69 & @xmath70 & @xmath71 & @xmath60 & @xmath72 & @xmath51 + @xmath73 & @xmath74 & @xmath75 & @xmath76 & @xmath77 & @xmath50 & @xmath44 + @xmath78 & @xmath79 & @xmath80 & @xmath81 & @xmath82 & @xmath83 & @xmath84 + @xmath85 & @xmath86 & @xmath87 & @xmath88 & @xmath89 & @xmath72 & @xmath90 + @xmath91 & @xmath92 & @xmath93 & @xmath94 & @xmath95 & @xmath96 & @xmath97 + @xmath98 & @xmath99 & @xmath100 & @xmath101 & @xmath102 & @xmath103 & @xmath104 + @xmath105 & @xmath106 & @xmath107 & @xmath108 & @xmath109 & @xmath50 & @xmath97 + @xmath110 & @xmath111 & @xmath75 & @xmath112 & @xmath113 & @xmath72 & @xmath44 + @xmath114 & @xmath115 & @xmath116 & @xmath117 & @xmath118 & @xmath66 & @xmath51 + @xmath119 & @xmath120 & @xmath121 & @xmath122 & @xmath123 & @xmath124 & @xmath125 + @xmath126 & @xmath127 & @xmath128 & @xmath129 & @xmath130 & @xmath131 & @xmath44 + @xmath132 & @xmath133 & @xmath134 & @xmath135 & @xmath136 & @xmath103 & @xmath125 + @xmath137 & @xmath138 & @xmath139 & @xmath140 & @xmath141 & @xmath103 & @xmath142 + @xmath143 & @xmath144 & @xmath145 & @xmath146 & @xmath147 & @xmath131 & @xmath148 + @xmath149 & @xmath150 & @xmath139 & @xmath151 & @xmath152 & @xmath96 & @xmath90 + @xmath153 & @xmath154 & @xmath155 & @xmath156 & @xmath157 & @xmath124 & @xmath90 + @xmath158 & @xmath159 & @xmath160 & @xmath161 & @xmath162 & @xmath72 & @xmath44 + to confirm the transit signal and obtain high - precision light curves for modeling the system we conducted photometric follow - up observations with the keplercam ccd on the telescope . we observed four transit events of hat - p-12b on the nights of 2007 march 27 , 2007 april 25 , 2009 february 5 and 2009 march 6 ( fig . [ fig : lc ] ) . on 2007 march 27 , 151 frames were acquired with a cadence of 90 seconds ( 75 seconds of exposure time ) in the sloan @xmath163-band ; the observations were interrupted at mid - transit due to clouds . on 25 april 2007 , 372 frames were acquired with a cadence of 45 seconds ( 30 seconds of exposure time ) in the sloan @xmath164-band . on 5 february 2009 , 218 frames were obtained with a cadence of 70 seconds ( 30 seconds exposure time ) in @xmath164-band . finally , on 6 march 2009 , 213 frames were acquired with a cadence of 70 seconds ( 60 seconds exposure time ) in the sloan @xmath165-band . this follow - up @xmath165-band light curve was obtained to further constrain possible blend scenarios ( ) . the reduction of the images was performed as follows . after bias and flat - field calibration , we derived an initial second - order astrometric transformation between the @xmath166 brightest stars and the 2mass catalog , as described in @xcite , yielding a residual of @xmath167 pixels . the primary reason for precise astrometry is to minimize the photometric errors that would originate from the centroid errors for the individual stars on each frame . aperture photometry was then performed on the resulting fixed positions , using a series of apertures . the instrumental magnitude transformation was done in two steps : first , all magnitude values were transformed to the photometric reference frame ( selected to be the sharpest image ) , using the individual poisson noise error estimates as weights . in the second step , the magnitude fit was repeated using the mean individual light curve magnitudes as reference and the rms of these light curves as weights . in both of the magnitude transformations , we excluded from the fit the target star itself and the @xmath168@xmath169 outliers . we performed epd and tfa against trends simultaneously with the light curve modeling ( for more details , see and @xcite ) . from the series of apertures , for each night , we chose the one yielding the smallest fit rms for the light curve . this aperture conveniently fell in the middle of the aperture series . the final light curves are shown in the upper plots of fig . [ fig : lc ] , with our best fit transit light curve models superimposed ( see also ) . lrrrr @xmath170 & @xmath171 & @xmath172 & @xmath173 & i + @xmath174 & @xmath172 & @xmath172 & @xmath175 & i + @xmath176 & @xmath177 & @xmath178 & @xmath179 & i + @xmath180 & @xmath181 & @xmath182 & @xmath183 & i + @xmath184 & @xmath185 & @xmath172 & @xmath186 & i + @xmath187 & @xmath188 & @xmath178 & @xmath189 & i + @xmath190 & @xmath191 & @xmath172 & @xmath192 & i + @xmath193 & @xmath194 & @xmath172 & @xmath195 & i + @xmath196 & @xmath197 & @xmath182 & @xmath198 & i + @xmath199 & @xmath200 & @xmath182 & @xmath201 & i + we derived the initial stellar atmospheric parameters by using the template spectrum obtained with the keck / hires instrument . we used the sme package of @xcite along with the atomic - line database of @xcite , which yielded the following _ initial _ values and uncertainties ( which we have conservatively increased to include our estimates of the systematic errors ) : effective temperature @xmath202k , stellar surface gravity @xmath203(cgs ) , metallicity @xmath204}}={\ensuremath{-0.29\pm0.05}}$]dex , and projected rotational velocity @xmath205 . at this stage we could use the effective temperature and the surface gravity as a luminosity indicator , and determine the stellar parameters based on these two constraints using a set of isochrones . however , the effect of @xmath206 on the spectral line shapes is typically subtle and as a result it is generally a rather poor luminosity indicator in practice . for planetary transits , the @xmath207 normalized semi - major axis and related @xmath208 mean stellar density typically impose a stronger constraint on possible stellar models @xcite . the validity of our assumption , namely that the adequate physical model describing our data is a planetary transit ( as opposed to a blend ) , is shown later in . using the values of @xmath209 , @xmath210 $ ] , and @xmath206 from the sme analysis , and corresponding quadratic limb darkening coefficients ( @xmath211 , @xmath212 , etc . ) from @xcite , we performed a global modeling of the data ( ) , yielding a full monte - carlo distribution of @xmath213 . this was complemented by a monte - carlo distribution of @xmath209 and @xmath210 $ ] , obtained by assuming gaussian uncertainties based on the 1-@xmath169 error bars of the initial sme analysis . for each combination within the large ( @xmath214 ) set of @xmath213 , @xmath209 , and @xmath210 $ ] values , we searched the stellar isochrones of the @xcite models for the best fit stellar model parameters ( such as @xmath215 , @xmath216 , @xmath206 , etc ) . we interpolated these isochrones to the sme - based stellar metallicity of @xmath204}}= { \ifthenelse{\equal{{i}}{i}}{{\ensuremath{-0.29\pm0.05}}}{{\ensuremath{-0.36\pm0.04}}}}$ ] . the majority of the parameter combinations in the monte carlo search did not match any isochrone . in such cases ( @xmath166% of all trials ) we skipped to the next randomly drawn parameter set . at the end we derived the mean values and uncertainties of the physical parameters based on their _ a posteriori _ distribution . we note that the spread of the input stellar parameters ( based on the gaussian uncertainties ) was large compared to what the isochrones cover as a function of age , due to the very slow evolution of k dwarfs ( see fig . [ fig : iso ] ) . this is partly the reason for the 40% match ratio . we also note that the match ratio is very sensitive to changing fundamental parameters of the isochrones , such as the mixing length or the metallicity . we then repeated the sme analysis by fixing @xmath206 to the refined value of @xmath217 based on the isochrone search , and only adjusting @xmath209 , @xmath210 $ ] and @xmath218 . this second iteration yielded @xmath219k , @xmath204}}= { \ensuremath{-0.36\pm0.04}}$ ] and @xmath220@xmath28 . curiously , the new @xmath209 and @xmath210 $ ] values from this second iteration provide a somewhat inferior match with the @xcite isochrones , as compared to the initial match . possible reasons for this include i ) systematic errors in the sme analysis due to the relatively low snr of our keck spectra , ii ) increasing uncertainty in the sme analysis due to the late stellar type of the host star ( note that hat - p-12 has a temperature that is below the 4700 k cut - off for stars included in the analysis of @xcite ) , iii ) general uncertainty in the isochrones for mid - k dwarfs ( there is a well - known discrepancy between the observed and predicted mass - radius relation for k and m dwarf stars in double - lined eclipsing binaries such that the observed radii are larger than the predicted radii , though there is some evidence that this discrepancy only holds for rapidly rotating , active stars , e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? thus we accepted the initial values of @xmath209 , @xmath210 $ ] and @xmath218 as the final atmospheric parameters for this star , along with the isochrone based stellar parameters , yielding @xmath215=@xmath221@xmath5 , @xmath216=@xmath222@xmath7 and @xmath223=@xmath224@xmath225 . along with other stellar parameters , these are summarized in tab . [ tab : stellar ] . the stellar evolutionary isochrones from @xcite for metallicity @xmath210$]=@xmath226 are plotted in fig . [ fig : iso ] , with the final choice of effective temperature @xmath227 and @xmath213 marked , and encircled by the 1-@xmath169 and 2-@xmath169 confidence ellipsoids . the stellar evolution modeling also yields the absolute magnitudes and colors in various photometric passbands . we used the apparent magnitudes from the 2mass catalogue @xcite to determine the distance of the system . the magnitudes reported in the 2mass catalogue have to be converted to the cit system @xcite , in which the stellar evolution models specify the colors . the reported magnitudes for this star are @xmath228 , @xmath229 and @xmath230 ; which are equivalent to @xmath231 , @xmath232 and @xmath233 ; in the cit photometric system ( see * ? ? ? thus , the converted 2mass magnitudes yield a color of @xmath234 that is within 1-@xmath169 of the expected , isochrone - based @xmath235 . we thus relied on the 2mass @xmath236 apparent magnitude and the @xmath237 absolute magnitude derived from the above - mentioned modelling to determine the distance : @xmath238pc . the @xmath236-band was chosen because it is the longest wavelength band - pass with the smallest expected discrepancies due to molecular lines in the spectrum of this k4 dwarf . lcl @xmath227 ( k ) & & sme + @xmath204}}$ ] & & sme + @xmath32 ( @xmath28 ) & & sme + @xmath239 ( @xmath28 ) & & sme + @xmath240 ( @xmath28 ) & & sme + @xmath241 ( @xmath28 ) & @xmath33 & ds + @xmath211 & @xmath242 & sme+claret + @xmath212 & @xmath243 & sme+claret + @xmath244 & @xmath245 & sme+claret + @xmath246 & @xmath247 & sme+claret + @xmath248 & @xmath249 & sme+claret + @xmath250 & @xmath251 & sme+claret + @xmath252 ( @xmath253 ) & @xmath221 & baraffe+@xmath213+sme + @xmath254 ( @xmath255 ) & @xmath222 & baraffe+@xmath213+sme + @xmath256 ( cgs ) & @xmath257 & baraffe+@xmath213+sme + @xmath258 ( @xmath259 ) & @xmath224 & baraffe+@xmath213+sme + @xmath260 ( mag ) & 12.84 & tass + @xmath261 ( mag ) & @xmath262 & baraffe+@xmath213+sme + @xmath236 ( mag , cit ) & @xmath263 & 2mass+carpenter + @xmath264 ( mag , cit ) & @xmath265 & baraffe+@xmath213+sme + age ( gyr ) & @xmath266 & baraffe+@xmath213+sme + distance ( pc ) & @xmath238 & baraffe+@xmath213+sme + following @xcite , we explored the possibility that the measured radial velocities are not real , but are instead caused by distortions in the spectral line profiles due to contamination from a nearby unresolved eclipsing binary . a bisector analysis based on the keck spectra was done as described in 5 of @xcite . [ fig : rvbis ] shows the bisector spans ( bs ) phased with the orbital period of the planet . in calculating the bs we use the convention @xmath267 where @xmath268 is the velocity of the bisector of the cross - correlation profile at a low cross - correlation value , and @xmath269 is the velocity at a high cross - correlation value . while the bs do not show significant variations with an amplitude that is comparable to or greater than the rv variations , there does appear to be a correlation between the bs and rv measurements . applying a spearman rank - order correlation test , we find that the two variables are correlated with 98% confidence ( i.e. there is a @xmath270 false alarm probability ) . since such a correlation might indicate a blend scenario , we consider below , and rule out , the possibility that the system is a blend between a bright foreground k star and a background eclipsing binary ( ) or a hierarchical triple system ( ) . the scenarios that we consider are summarized in table [ tab : blendtypes ] . we then consider the possibility that the correlation is not astrophysical , but rather results from variations in the sky contamination of the spectra ( ) ; we conclude that this is the most likely explanation for the correlation . the high proper motion of hat - p-12 allows us to rule out one possible scenario that could potentially fit the available observations , namely a background eclipsing binary that is aligned , by chance , with the foreground k4 dwarf hat - p-12 ( we refer to this as the h , b(s - s ) model , where h stands for the foreground star hat - p-12 , and the b(s - s ) is a background eclipsing binary star system ; here b refers to the fact that the eclipsing binary is in the background rather than being associated with the star h ) . to reproduce the observed @xmath271 deep transit the background object can not be more than 4 mag fainter than hat - p-12 ( objects fainter than this would contribute less than 2.5% of the total combined light and so could not cause the transit even if they were to be completely eclipsed by an object that emits no light ) . because hat - p-12 has a high proper motion ( @xmath272@xmath273 ; * * ) it is possible to use the palomar observatory sky survey plates from 1955 ( poss - i , red and blue plates ) to view the sky at the current position of hat - p-12 ( this same technique was used for hat - p-11 , * ? ? ? * ) . between 1955 and the follow - up observations in 2009 , hat - p-12 has moved @xmath274 . [ fig : poss ] shows an image stamp from the poss - i plate compared with a recent observation from the . we can rule out a background object down to @xmath275 mag within @xmath276 of the current position of hat - p-12 . any background object must be @xmath277 mag fainter than hat - p-12 and thus could not be responsible for the observed transit . following @xcite we consider the possibility that hat - p-12 is a hierarchical triple system , consisting of two eclipsing bodies that are diluted by a third star . in the following we refer to the bright star , with properties determined from the sme analysis , as hat - p-12 . we consider three scenarios . in the first scenario we assume that the bright star hat - p-12 is uneclipsed , and that the two eclipsing components are stars with parameters constrained by common origin to fall on the same age / metallicity isochrone as hat - p-12 ( we refer to this model as the h , s - s model , where h denotes the bright star hat - p-12 and s - s denotes a physically associated eclipsing binary consisting of a brighter star s and a fainter star s ) . in the second scenario we assume that hat - p-12 is uneclipsed , that one of the eclipsing components is a fainter star and that the other component is a planet with negligible mass and luminosity compared to the star ( the h , s - p model , where s stands for the fainter star , and p is the planet ) . in the third scenario we assume that hat - p-12 is a star that is transited by a planet and that there is a fainter star diluting the observed transit ( the h - p , s model ) . these models will be compared to the fiducial model of a single star orbited by a planet ( the h - p model ) . h , b(s - s ) & eclipsing binary star diluted by unresolved , unrelated star & proper motion ( ) + h , s - s & hierarchical triple star system , two components are eclipsing & light curve fit ( ) + h , s - p & binary star system , fainter star has a transiting planet & light curve fit ( ) + h - p , s & binary star system , brighter star has a transiting planet & light curve fit ( ) + h - p & single star with a transiting planet & not excluded + for the h , s - s and h , s - p scenarios we fit the follow - up @xmath164- , @xmath163- , and @xmath165-band light curves together with the hatnet @xmath278-band light curve following the procedure described by @xcite . we include the hatnet light curve to constrain the possibility of a secondary eclipse ; to exclude points that do not contribute to the fit , we only include points that are within 1 transit duration of the start of transit ingress or end of transit egress , or within 1 transit duration of the start of secondary ingress or end of secondary egress . we use the tfa hatnet light curve and apply epd on the out - of - transit portion of the follow - up light curves . we scale the formal photometric errors on each light curve so that @xmath279 for the out - of - transit portion of the light curve . we take the magnitudes and radii of the stars from the @xcite isochrones , transforming the @xmath280 magnitudes to the sloan system using the relations from @xcite . to make a fair comparison between the blend models and the h - p model we also fit the h - p model to the light curves using the same procedure used to fit the blend - models ( see for a more detailed analysis of the h - p model used for the final parameter determinations ) . [ fig : blend - ht - lcs ] compares the best fit h , s - s , h , s - p and h - p models . the best fit h , s - s model , consisting of an eclipsing pair with masses @xmath281 and @xmath282 that is diluted by the star hat - p-12 ( @xmath283 ) , has @xmath284 with 1333 degrees of freedom . we compare this to the best fit h - p model which has @xmath285 with 1334 degrees of freedom . because the photometric noise appears to be temporally correlated ( fig . [ fig : blend - ht - lcs ] ) , formal estimates for the significance of @xmath286 between two models will overestimate the confidence with which one model can be rejected in favor of another . we therefore conduct monte carlo simulations to estimate the expected distribution of @xmath286 values under the assumption that the h - s , s model is correct , and accounting for temporal correlations in the noise . to generate light curves for the monte carlo simulations that have similar time - correlated noise as the real light curves we fourier transform the residual of each light curve from the best - fit h , s - s model , randomize the phases , inverse fourier transform it , and then add in the h , s - s model . this method forces the simulated light curves to have the same noise power - spectrum ( and hence auto - correlation function ) as the actual light curve residuals . we scale the errors of each simulated light curve to have @xmath287 in the out - of - transit portion of the light curve . the fourier transforms are carried out assuming uniform time - sampling ( a good approximation for the follow - up light curves ) ; to use the fast - fourier transform algorithm , we cyclically repeat each light curve so that the total number of points in the light curve is a power of 2 . we then fit the h - p and h , s - s models to the simulated sets of light curves and record @xmath288 for each simulation . from 1000 simulations we find a median value of @xmath289 with a standard deviation of 12.4 ; we find no instances where @xmath290 , the minimum value attained is @xmath291 . we conclude that , based on the light curves , the h , s - s model can be rejected in favor of the h - p model at the @xmath292 confidence level . for the h , s - p scenario , we find that the best - fit model consists of a star with mass @xmath293 transited by a planet with @xmath294 and diluted by the star hat - p-12 ( @xmath283 ) . this model has @xmath295 with 1333 degrees of freedom . to determine the significance of @xmath296 we repeat the monte carlo simulations , this time adopting the best - fit h , s - p model as the fiducial model . from 1000 simulations we find a median value of @xmath297 with a standard deviation of 13.9 . there are 4 simulations with @xmath298 , so we conclude that the h - p model is preferred over the h , s - p model with @xmath299 ( @xmath300 ) confidence . as described in it is possible to correct for systematic errors in the photometry by simultaneously applying epd and tfa to the light curves while fitting a physical model to them . by using a more sophisticated model of this form we are able to rule out the h , s - p model with higher confidence , and also rule out the h - p , s model . we perform the global modeling as described in incorporating three additional parameters that allow for dilution in the @xmath165- , @xmath163- and @xmath164-bands . this model effectively encompasses both the h - p , s and h , s - p models because the only h , s - p models that provide a reasonable fit to the light curve are models where the planet - bearing star has a mass that is nearly equal to the mass of the diluting star hat - p-12 . we allow the dilution factors to vary independently in the fit . we find that models with no dilution are strongly preferred , and place 1@xmath169 upper limits on the light contribution in each filter from an uneclipsed star ( `` third light '' ) of @xmath301 , @xmath302 and @xmath303 . any additional star thus makes a negligible contribution to the total light of the system . this test thus rules out both the h - p , s and h , s - p models . as shown in sections [ sec : hitrip ] and [ sec : bgeb ] , blend scenarios involving an eclipsing binary star system are inconsistent with other observations of the system . we therefore look for an explanation of the apparent bs - rv correlation shown in that does not invoke a blend . one possibility is that it is due to varying contamination from the sky spectrum . because hat - p-12 is relatively faint , the flux from the sky is non - negligible compared to the flux from the source . scattered light from the moon illuminating the sky near hat - p-12 has a solar - like spectrum which yields a peak near @xmath304 in the cross - correlation profile of the observed spectrum . the degree to which this second peak contaminates the peak from the star varies with the sky brightness and the radial velocity difference between the moon and the star . because the observer - centric velocity of hat - p-12 is always less than the velocity of the moon , an increase in the sky brightness or a decrease in the velocity difference will lead to a positive bs variation for our adopted bs sign convention . while the sky brightness is not directly measureable from the available data , in order to quantify this effect we may introduce a sky contamination factor ( scf ) given by @xmath305 where @xmath278 is the ratio of the flux in the spectrum due to the moon to the flux due to the star , @xmath306 is the observer - centric radial velocity difference between the moon and the star , @xmath307 is the width of the lorentzian function that best fits the mean cross - correlation profile , and the form for the denominator is chosen because the cross - correlation profile is well fit by a lorentzian function . we estimate @xmath278 via the relation @xmath308 where @xmath309 is the total flux received in the region of the spectrum used to compute the bs , @xmath310 is the exposure time , @xmath311 and @xmath312 are the values for the spectrum with the highest count - rate ( these are used to account for changes in the flux received from the star due to variations in the seeing or transparency ) , @xmath313 is the @xmath314-band magnitude of the star ( we take @xmath315 assuming @xmath316 for a dwarf star with @xmath317k ) , and @xmath318 is the effective magnitude of the sky due to the moon at the position of the star ( in an area of @xmath319 square arcseconds ) . to estimate @xmath318 we use the model for the sky brightness due to moonlight given by @xcite , extending it to the @xmath314 band by taking @xmath320 for the moon ( e.g. * ? ? ? * ) and @xmath321 for the extinction coefficient ( this is a typical value at the summit of mauna kea for the johnson @xmath314-band which is roughly the region of the spectrum used to calculate the bs , * ? ? ? the values for @xmath318 range from @xmath322 to @xmath323 . when the moon is below the horizon we take @xmath324 . [ fig : scf ] compares the scf to the bs values and to the orbital phase . note that we normalize the scf to have a mean value of 1.0 . there is a positive correlation between the scf and the bs . by chance , spectra taken between orbital phases 0.5 and 1 had higher sky contamination on average than those taken between orbital phases 0 and 0.5 . when points with @xmath325 are removed , the correlation between the remaining bs and rv values is no longer significant ( the correlation significance is 37% ) . we conclude that this a plausible explanation for the apparent bs - rv correlation . as a further test on this hypothesis we simulate sky contaminated spectra and measure the bs values using the same procedure as for the actual spectra . to simulate a spectrum we take @xmath326 where @xmath310 is the iodine - free template spectrum of hat - p-12 , @xmath327 is the iodine - free template spectrum of hat - p-13 scaled to have the same total flux though the @xmath314-band as @xmath310 ( hat - p-13 has @xmath328 k , and is thus a better approximation to a solar spectrum than hat - p-12 ; * ? ? ? * ) , @xmath329 and @xmath330 are the barycentric velocity corrections for the templates , @xmath331 is the barycentric velocity correction for spectrum @xmath163 , @xmath332 is the average radial velocity of hat - p-13 , @xmath278 is given by eq . [ eqn : moonbrightness ] , and @xmath333 is a function that redshifts the spectrum @xmath334 by velocity @xmath335 . figure [ fig : scfsim ] compares the scf to the bs for the simulated spectra . the simulations show a correlation between the scf and bs that is comparable to that seen in fig . [ fig : scf](a ) . this confirms that sky contamination may affect the bs values at the level that is observed . our model for the follow - up light curves used analytic formulae based on @xcite for the eclipse of a star by a planet , where the stellar flux is described by quadratic limb - darkening . the limb darkening coefficients were derived from the sme results ( ) , using the tables provided by @xcite for @xmath164- , @xmath163- , and @xmath165-bands . the transit shape was parametrized by the normalized planetary radius @xmath336 , the square of the impact parameter @xmath337 , and the reciprocal of the half duration of the transit @xmath338 . we chose these parameters because of their simple geometric meanings and the fact that these show negligible correlations ( see * ? ? ? * ) . our model for the hatnet data was the simplified `` p1p3 '' version of the @xcite analytic functions , for the reasons described in @xcite . following the formalism presented by @xcite , the rv curve was parametrized by an eccentric keplerian orbit with semi - amplitude @xmath236 , and lagrangian orbital elements @xmath339 . we assumed that there is a strict periodicity in the individual transit times . in practice , we assigned the transit number @xmath340 to the first high quality follow - up light curve gathered on 2007 march 27 . the adjusted parameters in the fit were the first transit center observed by hatnet , @xmath341 , and the last transit center observed by the telescope , @xmath342 , covering all of our measurements with the hatnet telescopes , and the telescope . we prefer using @xmath341 and @xmath342 as adjusted parameters rather than the period and epoch for the reasons discussed by @xcite and @xcite . the transit center times for the intermediate transits were interpolated using these two epochs and the @xmath343 transit number of the actual event . the model for the rv data contains the ephemeris information through the @xmath341 and @xmath342 variables @xcite . altogether , the 11 parameters describing the physical model were @xmath341 , @xmath342 , @xmath344 , @xmath337 , @xmath338 , @xmath236 , @xmath345 , @xmath346 , and three additional ones related to the instrumental configuration . these are the instrumental blend factor @xmath347 of hatnet which accounts for possible dilution of the transit in the hatnet light curve , the hatnet out - of - transit magnitude , @xmath348 , and the relative rv zero - point @xmath349 . we extended our physical model with an instrumental model that describes the systematic variations of the data . this was done in a similar fashion to the analysis presented in @xcite . basically , the hatnet photometry has been already epd- and tfa - corrected before the global modeling , so we only considered systematic corrections to the follow - up light curves . we chose the `` eltg '' method , i.e. epd was performed in `` local '' mode with epd coefficients defined for each night , and tfa was performed in `` global '' mode using the same set of stars and tfa coefficients for all nights . the underlying physical model was based on the @xcite analytic formulae , as described earlier . the five epd parameters were the hour angle ( characterizing a monotonic trend that changes linearly over time ) , the square of the hour angle , and the stellar profile parameters ( equivalent to fwhm , elongation , position angle ) . the exact functional form of the above parameters contained 6 coefficients , including the auxiliary out - of - transit magnitude of the individual events . the epd parameters were independent for all 4 nights , implying 24 additional coefficients in the global fit . for the global tfa analysis we chose 18 template stars that had good quality measurements for all nights and on all frames , implying an additional 18 parameters in the fit . we apply epd to the template star light curves using the same set of parameters as used for the hat - p-12 light curves before incoporating them in the analysis . thus , the total number of fitted parameters is 11 ( physical model ) + 24 ( local epd ) + 18 ( global tfa ) = 53 , i.e. much smaller than the number of data - points ( @xmath350 ) . the joint fit was performed as described in @xcite . we minimized @xmath351 in the parameter space by using a hybrid algorithm , combining the downhill simplex method ( amoeba , see * ? ? ? * ) with the classical linear least squares algorithm . uncertainties on the parameters were derived using the markov chain monte - carlo method ( mcmc , see * ? ? ? * ) using `` hyperplane - clls '' chains @xcite . the _ a priori _ distributions of the parameters for these chains were chosen from a generic gaussian distribution , with eigenvalues and eigenvectors derived from the fisher covariance matrix for the best fit value . the fisher covariance matrix is calculated analytically using the partial derivatives given by @xcite and @xcite . since the eccentricity of the system appeared as insignificant ( @xmath352 , @xmath353 ) , we repeated the global fit by fixing these to 0 . the best fit results for the relevant physical parameters are summarized in tab . [ tab : planetparam ] . tab . [ tab : planetparam ] also lists the rv `` jitter '' , which is a component of assumed astrophysical noise intrinsic to the star that we add in quadrature to the rv measurement uncertainties in order to have @xmath354 from the rv data for the global fit . in addition , some auxiliary parameters ( not listed in the table ) were : @xmath355 ( bjd ) , @xmath356 ( bjd ) , @xmath357@xmath358(for the keck rvs , note that this does _ not _ correspond to the true center of mass radial velocity of the system , but is only a relative offset ) . the planetary parameters and their uncertainties can be derived by the direct combination of the _ a posteriori _ distributions of the light curve , radial velocity and stellar parameters . we found that the mass of the planet is @xmath359 , the radius is @xmath360 and its density is @xmath361@xmath14 . the final planetary parameters are summarized at the bottom of table [ tab : planetparam ] . lc @xmath362 ( days ) & @xmath363 + @xmath364 ( @xmath365 ) & @xmath366 + @xmath367 ( days ) & @xmath368 + @xmath369 ( days ) & @xmath370 + @xmath207 & @xmath371 + @xmath338 & @xmath372 + @xmath344 & @xmath373 + @xmath337 & @xmath374 + @xmath375 & @xmath376 + @xmath163 ( deg ) & @xmath377 + @xmath236 ( @xmath358 ) & @xmath378 + @xmath379 & @xmath380 + @xmath381 & @xmath380 + @xmath382 & @xmath380 + rv jitter ( @xmath358 ) & @xmath383 + @xmath384 ( @xmath385 ) & @xmath386 + @xmath387 ( @xmath388 ) & @xmath389 + @xmath390 & @xmath391 + @xmath392 ( @xmath14 ) & @xmath393 + @xmath394 ( au ) & @xmath395 + @xmath396 ( cgs ) & @xmath397 + @xmath398 ( k ) & @xmath399 + @xmath400 & @xmath401 + @xmath402 ( @xmath403 ) & @xmath404 + comparing hat - p-12b to the theoretical models of @xcite , we find that the mass and radius of the planet are consistent with the 1.0 gyr , @xmath405 non - irradiated model , or with a 1.0 - 5.0 gyr , @xmath406 irradiated model ( fig . [ fig : exomr ] ) . with an equilibrium temperature of @xmath407k , hat - p-12b has an equivalent solar semi - major axis of @xmath408 au , so the irradiation received by the planet , while not insignificant , is less than what is used to calculate the irradiated model ( @xmath409 au ) . the inferred metal fraction is expected to be closer to @xmath410 than @xmath411 for the @xcite models if the correct irradiation were used . we have also compared hat - p-12b to the theoretical models of @xcite . in fig . [ fig : exomr ] we have interpolated these models to @xmath412 au , and find that the mass and radius of hat - p-12b are consistent with a 10@xmath413 core , 1 gyr model , and lie between the core - less and 10@xmath413 core , 4.5 gyr models . we conclude , therefore , that hat - p-12b is most likely a h / he dominated planet with a core of perhaps @xmath414 , and a total metal fraction of @xmath415% . this makes hat - p-12b the least massive h / he dominated gas giant planet found to date ; the previous record holder was saturn . it is interesting to compare the properties of hat - p-12b to those of saturn and hd 149026b , the two planets with known radii that have masses closest to that of hat - p-12b . measurements of the mass of hd 149026b range from @xmath416 to @xmath417 , while determinations of its radius range from @xmath418 to @xmath419 @xcite . the planet appears to have a significant core , with estimates ranging from @xmath420 to @xmath421 ( see * ? ? ? * and references therein ) , implying a high metal fraction of @xmath422 . saturn has a mass of @xmath423 @xcite , equatorial radius of @xmath424 @xcite , an estimated core mass of @xmath425 , and a total heavy element fraction of @xmath426 @xcite . although hat - p-12b is less massive than both hd 149026b and saturn , it has a larger radius than both planets . note that hat - p-12b does not have a detectable eccentricity , so its large radius may not be due to tidal heating ( in the models by * ? ? * however , close - in planets may have tidally inflated radii even with eccentricities @xmath427 ) . the large radius in comparison with saturn may be due in part to the enhanced irradiation received by hat - p-12b , and to hat - p-12b potentially having a smaller core mass than saturn . hd 149026b , on the other hand , receives more irradiation than hat - p-12b ( @xmath428 au using the parameters from * ? ? ? * ) , so the difference in radii suggests that hat - p-12b has a substantially smaller core mass and metal enhancement than hd 149026b . it is interesting to note that the inferred core mass of the three planets appears to correlate with the host star metallicity ( hat - p-12 has @xmath204}}= { \ensuremath{-0.29\pm0.05}}$ ] , the sun has @xmath204}}= 0 $ ] , and hd 149026 has @xmath204}}= 0.36 \pm 0.05 $ ] from * ? ? ? this correlation has been previously noted by @xcite and by @xcite , and is perhaps suggestive evidence for the core accretion model of planet formation ( e.g. * ? ? ? * and references therein ) . further discoveries of teps with masses comparable to or less than that of saturn are needed to determine whether or not this correlation holds . note from fig . [ fig : exomr ] that the radii of planets in this mass regime are more sensitive to the core mass than are the radii of more massive planets for which a given core mass is a smaller fraction of the total planet mass . finally , one might wonder why other planets like hat - p-12b have not been found to date ( see fig . [ fig : exomr ] ) . the significant @xmath271 transit depth of hat - p-12b is well within the range that is easily detectable for many transit surveys , and the @xmath429@xmath28rv semi - amplitude , while small , is still easily measured with high - precision rv spectrometers ( though more observations may be needed for a robust confirmation , which may slow the rate at which these planets are announced ) . we conclude that of hot gaseous planets with radii similar to jupiter , only a small fraction have masses similar to saturn such as hat - p-12b ; the majority have masses similar to jupiter . with the discovery of hat - p-12b , we estimate that the fraction is @xmath430 , with considerable uncertainty . we would like to thank the referee , peter mccullough , for several suggestions that improved the quality of this paper , and scott gaudi for a helpful discussion . hatnet operations have been funded by nasa grants nng04gn74 g , nnx08af23 g and sao ir&d grants . work of g..b . and j. johnson were supported by the postdoctoral fellowship of the nsf astronomy and astrophysics program ( ast-0702843 and ast-0702821 , respectively ) . we acknowledge partial support also from the kepler mission under nasa cooperative agreement ncc2 - 1390 ( d.w.l . , thanks the hungarian scientific research foundation ( otka ) for support through grant k-60750 . this research has made use of keck telescope time granted through noao and nasa . the digitized sky surveys were produced at the space telescope science institute under u.s . government grant nag w-2166 . the images of these surveys are based on photographic data obtained using the oschin schmidt telescope on palomar mountain and the uk schmidt telescope . the plates were processed into the present compressed digital form with the permission of these institutions . this research has made use of the simbad database , operated at cds , strasbourg , france . | we report on the discovery of hat - p-12b , a transiting extrasolar planet orbiting the moderately bright @xmath0 k4 dwarf gsc 03033 - 00706 , with a period @xmath1d , transit epoch @xmath2 ( bjd ) and transit duration @xmath3d .
the host star has a mass of @xmath4@xmath5 , radius of @xmath6@xmath7 , effective temperature k and metallicity @xmath8 = { \ifthenelse{\equal{{i}}{i}}{{\ensuremath{-0.29\pm0.05}}}{{\ensuremath{-0.36\pm0.04}}}}$ ] .
we find a slight correlation between the observed spectral line bisector spans and the radial velocity , so we consider , and rule out , various blend configurations including a blend with a background eclipsing binary , and hierarchical triple systems where the eclipsing body is a star or a planet .
we conclude that a model consisting of a single star with a transiting planet best fits the observations , and show that a likely explanation for the apparent correlation is contamination from scattered moonlight . based on this model
, the planetary companion has a mass of @xmath9@xmath10 , and radius of @xmath11@xmath12 yielding a mean density of @xmath13@xmath14 . comparing these observations with recent theoretical models
we find that hat - p-12b is consistent with a @xmath15 gyr , mildly irradiated , h / he dominated planet with a core mass @xmath16 .
hat - p-12b is thus the least massive h / he dominated gas giant planet found to date .
this record was previously held by saturn . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
the possibility that supermassive black holes ( sbhs ) inhabit the centers of many if not most galaxies , and the observed correlation between sbh masses and galactic bulge properties , has potentially a fundamental significance for our understanding of galaxy formation and evolution . the relationships between black hole and bulge properties include a loose relationship between sbh and bulge masses , @xmath5 , and an apparently much tighter one between the sbh mass and the velocity dispersion in the corresponding bulge , @xmath6 ( e.g. , ferrarese & merritt 2000 ; gebhardt et al . 2000 ; tremaine et al . 2002 ; cf . reviews by kormendy & gebhardt 2001 ; merritt & ferrarese 2001 ) . in this paper we attempt to provide a physical explanation for these relationships between sbhs and their host galaxies . our model is based on the interaction between the dark haloes of galaxies and the baryonic components settling in their midst . as baryonic matter accumulates to form the bulge and sbh , the orbital structure of the underlying gravitational potential is modified . this , in turn , affects the subsequent accumulation of gas , which is highly dissipative and therefore is sensitive to orbital geometry . consequently , halo properties will determine those of the bulge and the sbh , and this will give rise to correlations between them that are compatible with the observed ones . cuspy , triaxial haloes appear to be a natural outcome of dissipationless collapse in a cold dark matter ( cdm ) dominated universe ( e.g. , cole & lacey 1996 ) . interactions with the baryonic component during the initial stages of collapse may affect the triaxiality , making it milder but still non - negligible ( dubinsky 1994 ) . before it even becomes dominant in the central regions , a clumpy baryonic component can also level off the central cusps of dark haloes , producing harmonic cores ( el - zant , shlosman & hoffman 2001 ) . even though this may affect the equidensity contours , again making them rounder , it need not symmetrize the equipotentials ; these can remain asymmetric if triaxiality is not affected beyond some radius ( for example , a homogeneous bar has a non - axisymmetric force contribution inside its density figure ) . from an observational standpoint it appears that haloes of fully formed galaxies tend to have nearly constant density cores ( de blok & bosma 2002 ) and that residual potential axial ratios of about 0.9 in cdm haloes are plausible , _ even in present day galaxies _ ( e.g. , kuijken & tremaine 1994 ; rix & zaritsky 1995 ; rix 1995 ) . the orbital structure of the inner regions of slowly rotating , non - axisymmetric potentials with harmonic cores is dominated by box orbits ( e.g. , binney & tremaine 1987 ) , which have no particular sense of circulation . they are self - intersecting and , therefore , can not be populated with gas . dissipation causes material to sink quickly toward the only long - lived attractor available the center ( e.g. , pfenniger & norman 1990 ; el - zant 1999 ) . in the process , interaction with the triaxial harmonic core causes the baryonic material to lose most of its angular momentum . the combined system would , therefore , also be slowly rotating . thus , unless star formation terminates the collapse , the final concentration of the first baryonic material could be extremely large . the onset of star formation is expected to occur when the baryonic material becomes self - gravitating roughly speaking , when its density becomes larger than that of the halo core . this also happens to be the criterion for the destruction of the harmonic core and the emergence of loop ( or tube ) orbits , which do have a definite sense of circulation . the role of the sbh is to contribute to the emergence of these orbits in the very central region . it is the collusion between the sbh and the more extended hot baryonic component in creating the loop orbits that leads to the correlations claimed in this paper . whereas , in three dimensions , a box orbit can be represented as a superposition of three ( generally ) incommensurable radial oscillations along mutually perpendicular axes , which are efficiently attenuated by dissipation , loop orbits are best described in terms of modest radial and vertical excursions superposed on rotational motion about the center . while the vertical and radial excursions , like the oscillations characterizing box orbits , are attenuated by dissipation , the rotational motion is not efficiently dissipated among gas clouds populating such orbits ( in the same sense of circulation ) . the minimization of the radial and vertical oscillations results in closed periodic orbits that are confined to a plane . we expect the amount of gas dissipation on loop orbits to depend on their axial ratios for , again , motion on highly eccentric orbits would lead to shocks and the accompanying loss of angular momentum on a short dynamical timescale , as observed in numerical simulations of gas flows in barred galaxies ( e.g. , heller & shlosman 1994 ) . lacking a detailed model for the dissipation rate associated with gaseous motion , we will assume that there is a critical eccentricity _ below _ which the loop orbits can serve as long - lived attractors for dissipative motion . in other words , gas populating these orbits will evolve only secularly and not dynamically . fortunately , this dependence on critical eccentricity will turn out to be weak . in the following we explore , within the above framework , the formation of galactic bulges and central compact objects . we show that it is possible to deduce a well - defined linear correlation between the masses of the central sbhs and corresponding bulges in given dark matter haloes , with mass ratios comparable to those observed . we demonstrate that , if dark matter halo cores follow a faber - jackson type relation ( faber & jackson 1976 ) between their masses and velocity dispersions , then a similar relation also applies to the bulges . an @xmath4 relation between the sbh mass and the bulge velocity dispersion , which under certain assumptions can have smaller scatter than the @xmath7 relation , also follows naturally . within this framework , it is possible to make a number of testable predictions concerning the related structures of bulges , sbhs and their host haloes . we discuss these in the final section . formal aspects of the perturbation analysis applied to bulge - halo systems have been deferred to appendices a and b. preliminary results of this work have been reported by shlosman ( 2002 ) . since , at this stage , we are interested in _ generic _ dynamical phenomena related to orbital structure , the exact form of the halo potential is immaterial as long as it exhibits a harmonic core where no loop orbits can exist . the exact distribution of the baryonic material is also not crucial , except , again , for its central density distribution . if this diverges , say as @xmath8 , loop orbits will be created all the way to the center , since the bulge - halo system no longer possesses a harmonic core . in this case there is no need to form the sbh . if , on the other hand , the ( proto)-bulge has a harmonic core of its own , there will still be a nearly constant density region near the center with only box orbits unless a central point mass is present . we will assume that ` cuspy ' bulges , when they exist , are products of later evolution ( for example , a result of star formation and cold dissipationless collapse : e.g. , lokas & hoffman 2000 ) . we use a logarithmic standard form ( e.g. , binney & tremaine 1987 ) to represent the halo potential : @xmath9 where @xmath10 is the asymptotic ( in the limit @xmath11 and @xmath12 ) circular velocity , @xmath13 is the core radius and @xmath14 , @xmath15 are the potential axis ratios . we will consider the process of bulge formation to be terminated , or at least substantially slowed down , when the combined ( baryonic plus halo ) potential admits sufficiently round non - intersecting loop orbits , which permit the long - term circulation of gas without excessive dissipation . such motions necessarily take place in a symmetry plane determined by the ( orbit - averaged ) angular momentum ( e.g. , frank , king & raine 2002 ) . therefore , for our purposes , it will suffice to consider only orbits in the plane @xmath16 and to ignore the vertical dimension . we fix @xmath14 at @xmath17 ( though the effects of its variation are discussed where relevant ) and adopt a threshold axial ratio @xmath18 for orbits that can be populated with gas . the value of @xmath18 that best describes when this happens remains to be investigated , but its mere existence is what is important here . as we show below , our results are rather insensitive to the exact value of @xmath18 . the halo core mass will be taken to be @xmath19 and we define the halo density to be @xmath20 . this is conveniently close to the value of the density in the region where the potential is effectively harmonic that is , within the region where , in the absence of the bulge component , no loop orbits exist for @xmath21 . we note , however , that a bulge with a larger core radius probes a region of the halo core with smaller average density than the region probed by a bulge with a relatively small core radius . this effect will be discussed in section 4 . since our aforementioned criterion depends on the potential in a chosen symmetry plane , the exact three - dimensional mass distribution of the bulge is unimportant , as the same planar potential may arise from a variety of these . a particularly convenient form for the potential in this situation is that of miyamoto & nagai ( 1975 ) , @xmath22 this potential can approximate a disk - bulge system , with the parameters @xmath23 and @xmath24 determining the scale length and height , respectively . in the symmetry plane , therefore , the potential can arise from a range of density distributions from highly flattened to spherical ones . in general , when we refer to the `` bulge '' we will have a spherical system in mind with @xmath25 . in particular , all densities quoted are calculated under this assumption . we will comment on the effects of assuming a flattened distribution where relevant . in that case @xmath26 in eq . ( [ miyamoto ] ) is replaced by @xmath2 , the spherical bulge mass . the bulge _ core _ is defined by the radius @xmath27 and encloses the core mass @xmath28 . the bulge core density is defined as @xmath29 . the scaling relations that arise do not depend on the absolute masses and lengths , but instead , on the ratios of these quantities among the various contributions to the potential . an important property of the dynamics in our model is that _ any transformation that leaves the relative masses and lengths of the components constant will also not affect the orbital structure once scale transformations are taken into account . _ such transformations only change the time units , hence the dynamics remains invariant on a new characteristic timescale . in addition , _ in regions where the baryonic and halo components have the same mass distributions , i.e. , inside their cores , only the density ratios of these components determine their relative contributions to the potential , _ and hence the orbital structure . for the remainder of this paper , we will describe all components of the potential in terms of units scaled to the dimensions of the halo potential , eq . ( [ logarithm ] ) . we scale all distances to @xmath13 , with @xmath30 , and all masses to @xmath31 so that @xmath32 . the normalized bulge core density is then @xmath33 . the bulge potential in scaled units is @xmath34 the black hole potential , in the same units , is @xmath35 where @xmath36 . note that all orbits in the total potential @xmath37 depend only on three parameters , which may be taken to be @xmath38 , @xmath39 , and @xmath40 . in this section , we demonstrate that the arguments outlined above imply a linear relation between the sbh and bulge masses , provided that the baryonic and cdm density distributions are not completely axisymmetric and exhibit , at some stage , nearly constant density cores . fig . [ homofin ] ( the result of orbital integrations ) exhibits the axis ratios of the closed loop orbits @xmath41 as a function of the longer axis length ( @xmath42 ) . in this figure , we have fixed the scaled bulge core density , @xmath43 , and the sbh - to - bulge mass ratio , @xmath44 , but vary the scaled bulge core radius , @xmath38 . consequently , the bulge masses @xmath45 range over a factor of @xmath46 . as expected , when only the halo contributes to the potential ( dotted line ) , @xmath47 quickly drops to zero inside the halo core , since no loop orbits can exist there . ( for halo potential axis ratios smaller than the adopted value of 0.9 , i.e. , greater triaxiality , @xmath47 reaches 0 at radii that are progressively closer to the halo core radius , @xmath48 . as the baryonic component , both in the bulge and the sbh , becomes progressively more massive , correspondingly rounder loop orbits appear . loop orbits in the very central region are produced by the presence of the sbh . the latter is represented numerically by a plummer sphere with a softening scale of @xmath49 , in scaled units . near the center , where the sbh contribution is dominant , the axis ratios of the loop orbits tend to @xmath50 . to2.5 in ' '' '' the axis ratio curves exhibit two distinct minima . it is apparent that the inner minima of the curves in fig . [ homofin ] have ( nearly ) the same values of @xmath47 , but occur at different radii @xmath51 . this is a consequence of assuming a fixed ratio of sbh - to - bulge mass , @xmath40 , and a constant bulge core density @xmath52 . because the bulge and halo densities are nearly uniform in the inner regions , their relative contributions to the potential are determined by their density ratio . in the absence of the sbh contribution , @xmath47 in fig . [ homofin ] would tend to zero within the effective harmonic core of the bulge - halo system ( see eq . [ [ expo ] ] ) . when the relative density is kept constant ( and @xmath53 ) this is proportional to @xmath54 ( the effect of density variation will be examined in section [ inmin ] ) . the reason for this is that inside the nearly constant - density bulge core , the potential , which is now a superposition of two nearly harmonic potentials , is nearly harmonic ( even if the new core is less triaxial than the halo alone , e.g. , if the bulge is assumed to be spherical ) . the inner minima of the curves in fig . [ homofin ] , therefore , correspond to a transition from the region where the sbh provides the dominant contribution toward the creation of loop orbits to that where the bulge provides this contribution . therefore , the minima occur at radii where the gravitational acceleration due to the sbh is proportional to that due to the bulge+halo , @xmath55 ( since the minimum is located well within the bulge core radius ) . since @xmath40 is taken to be constant , we expect the minima to occur at radii @xmath51 . in section 2 , we have defined the critical value @xmath18 above which gas circulation can be sustained for secular timescales . here we have shown that given a critical value , @xmath18 , for the inner minimum and the relative bulge - to - halo core density ratio , a value of @xmath40 associated with this minimum is determined . however , satisfying the condition @xmath56 at the inner minimum does not imply that the same condition is satisfied at all radii within @xmath13 . until it is satisfied at the outer minimum , as well , further evolution can occur . this is briefly sketched below and outlined in more detail in section 3.4 . until the critical value of @xmath47 is reached at all radii inside the halo core , the bulge will continue to grow , since outside the bulge core @xmath47 declines again . it is clear from fig . 1 that , in contrast to that of the inner minimum , the value of @xmath47 at the outer minimum depends more sensitively on the bulge mass than on its density . the growth of the bulge , therefore , determines a _ minimal _ bulge mass fixed by the condition that @xmath57 at all radii within the halo core . if the sbh did not continue to grow along with the bulge , this growth in the bulge mass would decrease the linear correlation coefficient @xmath40 , but , as we show in section 3.4 , this leads only to near - independence of the final value of @xmath40 on @xmath18 and still keeps the values of @xmath40 within the observed range . as shown in the previous section , the creation of loop orbits with axis ratio above a given value depends , in the central region , solely on the ratio of the bulge - to - halo core density and the mass ratio @xmath58 . the actual value of the bulge mass determines only the position of the minimum ( as a fraction of the halo core radius ) , not the axis ratio at the minimum . but even the radius of the minimum is largely insensitive to the bulge mass , i.e. , @xmath59 , provided that the minimum actually exists and the bulge core density relative to the halo harmonic core density remains constant . we will now examine the bounds on @xmath0 and @xmath40 which ensure the existence of a minimum @xmath57 inside the bulge core and observe the effect of varying the bulge density . we calculate and plot @xmath60 for different choices of @xmath18 ( fig . [ compac ] ) , in the following way . first , without any loss of generality , we choose a bulge mass @xmath45 that is large enough that the outer minimum satisfies @xmath61 , for all @xmath62 . this is done in order to focus on the effects of the inner minima only . ( recall , from the previous section , that the value of the outer minimum depends on the mass of the bulge . ) fig . [ compac ] , therefore , exhibits the effect of @xmath63 and @xmath40 on the inner minima of the @xmath47 curves at @xmath64 . in order to obtain @xmath60 , we vary @xmath0 , which is some fraction @xmath40 of the chosen bulge mass . the density is varied by contracting the bulge ( by decreasing its core radius ) until the condition @xmath61 is satisfied everywhere inside the bulge core . to2.5 in ' '' '' one observes in fig . 2 that at smaller densities @xmath40 tends toward an asymptotic ( and maximal ) value associated with a given @xmath18 . we refer to this hereafter as the `` asymptotic '' regime . in the transition to this regime , along a @xmath65 curve , the value of @xmath40 is monotonically increasing . in other words , the bulge contribution toward the creation of loop orbits of a given eccentricity gradually diminishes , and is compensated by a greater contribution from the sbh component . in the process , the minimum in @xmath47 moves outward . the asymptotic regime corresponds to a situation in which the existence of loop orbits with the required elongation depends on the value of @xmath40 , irrespective of the bulge density ( that is , effectively , only on the sbh mass ) . in this limit the sbh contribution to the potential is sufficient to create loop orbits with @xmath61 at all radii within the halo core ( i.e. , at @xmath66 ) , without additional contributions from the bulge component ( cf . 3 ) . essentially , it corresponds to a rapid collapse to the center and formation of the sbh by a large fraction of the baryonic material , bypassing the formation of a bulge . this regime appears to be of academic interest only , as it implies @xmath67 all the way to the center ; it will not be discussed further . to2.5 in ' '' '' a second regime characterizes the dynamical state of the sbh - bulge - dark halo system with @xmath68 . in this `` scaling regime '' the @xmath69 curves are parallel straight lines with average slopes of about @xmath70 . this can be explained in the following manner . from eq . ( [ expo ] ) we know that the radius of the effective harmonic core of the bulge - halo system , @xmath42 , is proportional to @xmath71 . in the scaling regime , the location of the minimum in @xmath47 will be proportional to @xmath42 . now , in order to maintain the minimum at a specified value of @xmath18 , the gravitational acceleration due to the sbh must be proportional to the acceleration due to the halo ( which contains the nonaxisymmetry ) at @xmath42 . this implies @xmath72 . substituting for @xmath42 and noting that @xmath73 , we obtain @xmath74 . this is approximately what is found from figure 2 and it also holds if we change the mass of the bulge keeping the radius constant and again using eq . ( [ expo ] ) . note , furthermore , that for constant @xmath39 this relation predicts the radius of the effective harmonic core ( and the location of the minimum ) to be proportional to @xmath75 , as expected from the heuristic considerations of the previous section . while the asymptotic values of @xmath40 depend on @xmath2 , the values of @xmath40 in scaling regime do not . the transition between these two regimes can , therefore , be characterized by a sharp change in the behavior of @xmath40 , as seen from fig . 2 . we are mainly interested in the scaling region , because the onset of star formation can be tied to the baryonic component becoming self - gravitating , which corresponds roughly to the bulge core density exceeding that of the background halo , i.e. , @xmath68 . in this case , and for @xmath76 , the sbh contributes significantly to the potential only at radii @xmath77 . the @xmath47 curves , therefore , exhibit a definite inner minimum well inside the halo core . the values of @xmath78 and the initial @xmath40 , in principle , also depend on the critical eccentricity , @xmath18 , of the inner minimum , which is expected to be independent of the bulge mass , but which does depend on complex gas dynamics . if , after the sbh forms , the bulge mass falls short of the value required for creating sufficently round closed loop orbits at all radii inside the halo core , it will continue to grow until the outer minimum of the axis ratio curve also attains @xmath61 . in section 3.4 , we show that the sbh growth , if continued , becomes intermittent . in the next section , we also demonstrate that one can place constraints on the possible range of @xmath40 values by considering this constraint on the outer minimum . to obtain loop orbits rounder than a given @xmath18 _ at all radii _ @xmath79 , and not only in the central region , one in fact needs to take into account the bulge - to - halo core mass ratio and not only the density ratio . this can already be seen in fig . [ homofin ] , where all curves with sbh and bulge contributions exhibit an inner minimum with @xmath80 , while @xmath47 declines significantly as one moves further out . for low bulge masses , an additional outer minimum emerges , within @xmath81 , before @xmath47 rises again outside the halo harmonic core . only one of the @xmath47 curves , corresponding to the most massive bulge , has @xmath82 at _ all _ radii . therefore , for a given halo , there is a minimal bulge mass , @xmath83 , that is necessary to create sufficiently round loop orbits at all radii @xmath84 if we assume that the bulge core density varies at most by a factor of a few ( @xmath85 $ ] ) , @xmath86 , which is only weakly dependent on the bulge mass and density ( @xmath87 ) , varies little . the minimal bulge mass is also nearly constant ( calculations show , for example , that it varies by @xmath88% when @xmath63 changes from 1 to 2.5 ) . there is more sensitivity to the assumed @xmath18 . in fig . [ complots ] ( asterisks ) we show the bulge mass required to create loop orbits with @xmath61 at all radii outside @xmath86 , for @xmath89 . these are the masses necessary , at this density , to produce outer minima with the required values . for bulge densities @xmath90 these minima lie at radii @xmath91 . thus , unless the sbh can contribute significantly to the potential at radii comparable to @xmath13 , which seems implausible , the dominant contributions to the potential at the radii examined here should be only those due to the bulge and halo components . therefore , once the halo parameters are fixed , the creation of loop orbits with given @xmath61 outside the bulge core will depend only on the bulge parameters @xmath86 and @xmath92 . to2.5 in ' '' '' for @xmath89 , a well - defined inner minimum in the @xmath47 curve exists inside the bulge core only if @xmath93 . for larger @xmath18 , the inner minimum moves outward and merges with the outer one . systems with these properties lie outside the scaling region in fig . 2 . in this case , for the condition @xmath61 to be satisfied , the sbh has to contribute significantly at all radii no matter how massive the bulge is . ( for @xmath89 , the curve with @xmath94 in fig . 2 lies in this regime . ) furthermore , the minimal bulge density required to create loop orbits with @xmath61 , without appeal to an overmassive sbh , increases rapidly when @xmath95 ( fig . [ exmags ] ) , suggesting that the bulk of the material may form stars before the condition @xmath61 is reached everywhere within the halo core . if we demand that @xmath96 , then @xmath97 must be smaller than 0.8 in order for the model to be plausible , implying that @xmath40 must lie in the range @xmath98 . to2.5 in ' '' '' on the other hand , star formation can be somewhat delayed to higher densities . a reasonable range lies within @xmath99 . still higher densities in the bulge can be excluded based on the observed rotation curves . in this range of @xmath63 , values of @xmath100 become feasible without invoking unrealistically massive sbhs , as seen from fig . [ exmags ] . one may also reasonably assume that , due to enhanced dissipation , higher densities require larger @xmath18 to maintain long - lived gaseous motion ( detailed modeling , however , will be necessary to determine exactly how @xmath18 depends on the system parameters ) . in this situation , the range in @xmath40 that can be inferred from fig . 2 , is again about @xmath98 and compatible with the observed value of @xmath101 , which exhibits a significant scatter . the values of @xmath40 obtained so far , which should be regarded as _ initial _ values , are fixed by the inner minimum of the axis ratio curves . as argued in section 3.1 , further evolution of the bulge will occur unless sufficiently round loop orbits also exist at the outer minimum . in the next section we examine how this evolution can modify the range of @xmath40 , if at all . there are basic differences between the conditions for satisfying @xmath102 at the inner and outer minima in fig . [ homofin ] . as discussed in section 3.1 , the initial growth of the sbh is terminated after the inner loop orbits reach the critical eccentricity . the growth of the bulge , on the other hand , continues until the required @xmath18 is reached at all radii inside the halo core . since the increase of bulge mass , at roughly constant density , moves the inner minimum outward as described in section 3.1 , the sbh would have to continue growing in pace with the bulge mass if it is to maintain the critical value of @xmath40 ( which is determined only by the values of @xmath39 and @xmath18 ) . does this actually happen , and if so , how ? to2.5 in ' '' '' to see how the parallel growth of the sbh and bulge could come about , consider the solid curve in fig . [ homofin2 ] , which represents the state of the sbh / bulge system at the end of the initial stage of infall . this is equivalent to one of the curves ( say , the dash - dotted line ) in fig . 1 , except that we have chosen @xmath103 for the inner minimum , instead of 0.9 . this line has two maxima : a left - hand one at the origin , determined by the sbh , and a right - hand one , determined by the bulge core . note that the outer part of the solid curve drops below @xmath18 , implying that the bulge will continue to grow . now suppose this growth occurs _ without _ corresponding growth of the sbh ( in contrast to the case in fig . 1 , where @xmath40 is kept constant ) . if the bulge continues to grow while the sbh growth is stopped , the right - hand maximum moves further to the right , while the left - hand maximum stays the same . this opens a widening ` gap ' at the position of the inner minimum which drops below @xmath18 , allowing gas to flow again toward the radius of influence of the sbh , @xmath104 . this is illustrated by the dashed curve in fig . [ homofin2 ] , which corresponds to a factor of 10 increase in @xmath45 accompanied by a similar _ drop _ in @xmath40 . the gas is expected to accumulate in this vicinity . however , once a substantial amount of gas has collected at @xmath104 , it becomes self - gravitating and is prone to global self - gravitating instabilities . the fastest of these instabilities , @xmath105 modes or bar instability ( e.g. , bardeen 1975 ) , have been discussed in a similar context by shlosman , frank & begelman ( 1989 ) . they induce rapid ( dynamical ) gas inflow . hence we expect that the growth of the sbh at this stage will be intermittent , but because the time - averaged conditions for infall at the inner and outer maximum do not change , we expect the value of @xmath40 to stay roughly constant within the range given in fig . 2 , namely @xmath106 , depending on the value of @xmath18 . is there always enough gas to accumulate at @xmath107 in order to trigger a bar instability , due to the opening of the gap between the sbh and the bulge ? one can imagine the opposite extreme to that discussed above , in which insufficient gas enters the widening gap from outside , perhaps because star formation is efficient during the early stages of infall . let us suppose , for the sake of argument , that the sbh does _ not _ grow beyond its initial mass , as determined by the initial value of @xmath40 . how does this affect the range of final @xmath40 values ? figure 4 shows minimal bulge masses , @xmath108 , for a range of @xmath18 values . while we have no estimate for the initial bulge masses which define the initial @xmath40 in fig . 2 , it is possible to rule out very small masses , e.g. , @xmath109 , because the total baryonic mass within the halo core is taken to be about 10% . this means that for lower @xmath110 , the initial bulge mass is equal to or larger than @xmath83 estimated in section 3.3 , and , therefore , will not grow beyond its initial value . it is easy to understand this result , because one needs small baryonic masses to create loop orbits with such large eccentricities . this means that in this regime the initial value of @xmath111 ( in figures 2 and 4 ) is also its final value . alternatively , in the regime of larger @xmath112 , the bulge can grow at most by a factor of @xmath113 , reducing the initial @xmath40 by this amount . luckily , the initial values of @xmath40 for high @xmath18 lie around their high end , @xmath114 . reduction by up to two orders of magnitude brings them again to about @xmath115 . hence , whether the sbh grows after the initial stage or not does not destroy the @xmath7 correlation and does not move the values of @xmath40 outside the observed range . a corollary of this discussion is that the final value of @xmath40 appears insensitive to the value of @xmath18 , if the growth of the sbh is supressed , and depends strongly on @xmath18 , if the sbh grows in tandem with the bulge . note also that the variation of @xmath116 ( represented by the asterisks in fig . 4 ) , follows closely that of @xmath117 ( represented by filled circles ) . the ratio of these two quantities , @xmath118 is , therefore , largely independent of @xmath18 . in other words , the minimal bulge mass is predicted to be about 5 times the geometric mean between the sbh and the halo masses , provided that the sbh grows along with the bulge . to summarize , the initial value of the black hole - to - bulge mass ratio @xmath40 is fixed by the density at which the gaseous material in the inner region becomes self - gravitating and forms stars , and by the value of @xmath18 . however , unless loop orbits of sufficient eccentricity are created in the _ whole _ harmonic halo core region , the mass of the bulge will continue to grow . at the same time , the dynamical infall onto the sbh can choke , if the remaining gas mass is insufficient to cause bar instability and channel it to the sbh . in the latter case @xmath40 will decline below its initial value as the bulge grows , but will tend to approach a final value roughly independent of @xmath18 . in the opposite limit , the sbh will grow in proportion to the bulge , and @xmath40 will stay constant ( as shown by figs . 2 and 4 ) . in either limit , we assume that the process stops when the mass of the baryonic component reaches the minimal mass required to achieve loop orbits with a given @xmath61 , at all radii . for less triaxial haloes , and for a given bulge core density , lower sbh masses are required to produce loop orbits below a critical eccentricity . the minimal mass of the bulge needed to create the required loop orbits in the outer region , however , also decreases especially since , for a mildly triaxial halo , the radius at which no loop orbits exist inside the harmonic core decreases with decreasing triaxiality . thus , in the regime of mild halo triaxiality , the ratio of @xmath40 to ( normalized ) minimal bulge mass should not depend sensitively on the value of @xmath14 . we do ignore the fact that triaxiality can be a function of radius , and choose a fixed value of @xmath14 for simplicity . once orbits of sufficiently large @xmath47 are present , dissipation will reduce radial motion relative to these orbits , as well as vertical motion away from the symmetry plane defined by the angular momentum vector . infalling gas will start to populate the newly - formed , round , non - intersecting periodic loops , leading to the formation of a disk component inside the halo core . at this stage the bulge formation stops . outside the halo core , gas can accumulate at any stage of the formation process on closed loop orbits , which always exist in strongly inhomogeneous density distributions . provided that the halo triaxiality is small , these orbits will be nearly circular ( for potential axis ratio @xmath14 mildly deviating from unity @xmath119 : see , e.g. , rix 1995 ) . if the halo core radius is exceedingly large , however , it is possible that no significant disk component will form at all . for if the core is large relative to the total halo ( virial ) radius , the contracting gaseous component will end up in the core , instead of spinning up and forming an extended disk . losing most of its angular momentum to the halo , it will eventually end up as part of the bulge component . this would lead to the formation of an elliptical rather than a disk - dominated galaxy . in this context , one expects that haloes with larger core radii are host to larger spheroidal components . since it appears that , in general , more massive galaxies are usually of earlier type ( e.g. , persic , salucci & stel 1996 ) , one could deduce that more massive cores have larger core radii . this would be expected if these cores followed a faber - jackson type relation ( for example , if @xmath120 , then @xmath121 ) as tentatively suggested by observations ( burkert 1995 ; salucci & burkert 2000 ; dalcanton & hogan 2001 ) and deduced if halo cores formed via the destruction of the inner ( @xmath122 ) regions of nfw haloes ( navarro , frenk & white 1997 ) . in this case , lower mass haloes form when the universe is denser and are , therefore , more concentrated . as a consequence , the region where @xmath122 is smaller relative to the virial radius for low mass haloes ( el - zant , shlosman & hoffman 2001 ) . the existence of a faber - jackson type relation for halo cores is also required in order to reproduce the @xmath123 relation as discussed in the next section . within the above framework , the formation of the sbh is intimately tied to the bulge component , whereas the formation of the disk component takes place after the processes leading to bulge and sbh formation are essentially complete . the bulk of the disk is also expected to form at scales larger than the halo harmonic core ( but see section 5 ) . thus , the deduced correlation involving the bulge and sbh does not simply generalize to one involving the disk as well , in accordance with observations ( e.g. , kormendy & gebhardt 2001 ) . the fact that all the orbital properties discussed in this paper depend only on the relative magnitudes of the masses and spatial scales of the galactic components involved has important consequences . for the @xmath7 relationship it has the obvious implication that the derived correlation will hold for all halo masses . a more powerful prediction transpires in relation to the @xmath4 relation . for this to hold , it is necessary that the masses and velocity dispersions of halo cores are related in a similar manner . suppose that for a given halo there exist unique values for the sbh and bulge masses , and for the bulge scalelength . if a faber - jackson type relation between the halo core mass and velocity dispersion exists , a corresponding relation will exist between the bulge parameters . it also follows that a similar relation will exist between the sbh mass and the bulge ( and halo ) velocity dispersion . this can be deduced by simple scaling transformations , because the orbital properties we are interested in are all invariant with respect to spatial scale and mass transformations . this means that if we multiply the masses of the sbh , bulge and halo by some constant @xmath124 , thus effectively changing the mass units , the curves in fig . [ homofin ] will remain invariant . in the same manner , if we multiply all lengthscales by some factor @xmath125 , so that @xmath126 and @xmath127 , thus effectively changing the length unit , all curves in fig . [ homofin ] remain the same . this implies that if , for example , @xmath128 , then @xmath129 are equivalent systems in the sense decribed above ( they have the same axis ratio curves for their loop orbits with the same @xmath47 values as a function of rescaled radius ) . these systems will all follow the @xmath3 relation for the halo . in this particular case , @xmath130 implies a similar relationship between @xmath1 and @xmath2 , as well as @xmath0 . it is of course possible that , for a given halo , the masses of the sbh and bulge , as well as the bulge lengthscale , are not unique . in other words , the subset of haloes with a given @xmath31 and @xmath13 may contain bulges and sbhs with a distribution of properties . in section 3.2 we had assumed that the bulge collapse is largely terminated when its density is of the order of the halo core density . this in turn fixes @xmath40 , once @xmath18 is determined . in this section we further assume that the value of @xmath40 is not changed as a result of any subsequent bulge growth . that is , the sbh grows in tandem with the bulge ( cf . section 3.4 ) . this relaxes the assumption that the bulge masses are determined by the minimal mass required to create closed loops with @xmath61 at all radii , allowing for more massive bulges . one would like to infer to what extent variations in the bulge and sbh properies , within a given halo , affect the homology relations discussed above . we now show that when one accounts for variations of the bulge and the sbh parameters , the departure from the aforementioned relation is not dramatic . this comes about basically because the mass and core radius of the bulge are correlated , under the assumptions of our model . first , we note that the average velocity dispersion of the core of the baryonic component can be written as @xmath131 , and the average density @xmath132 . in this case @xmath133 . here @xmath134 depends on the functional form of the density distribution . its exact value is unimportant if all bulges are assumed to have the same functional form for their density distributions . henceforth we set @xmath135 . for a constant bulge - to - halo density ratio @xmath78 a relationship between bulge velocity dispersion and mass @xmath136 results . in a toy model where the halo density does not vary at all with radius a constant bulge density determines a unique @xmath40 , given @xmath18 , and an `` @xmath137 '' relation between bulge properties within a given halo arises , with index equal to 3 . considering more realistic models for the density distribution raises the index somewhat . when the core radius of the baryonic component is not very small compared to that of the halo , the density of the halo will not be strictly constant in the region of interest . as a result , the slope of the @xmath137 relationship will further increase . , where @xmath138 , the slope of @xmath123 is given by @xmath139 $ ] which stays between @xmath140 for @xmath141 , and then rises rapidly . the resulting slope of @xmath123 is some weighted average of @xmath142 , and greater than 3 . ] this occurs because larger bulges see " a smaller mean halo density , and therefore smaller bulge densities suffice to produce the same relative contribution to the potential . since @xmath143 , an inverse correlation between density and @xmath144 steepens the @xmath145 relation . ( a marginal effect is already seen in fig . 1 where the inner maxima produced by bulges with progressively larger @xmath146 also have progressively larger inner minima of @xmath47 . ) if the baryonic component giving rise to the gravitational potential is significantly flattened there will be an increase in the @xmath123 slope to 4 . in this case , the surface , rather than the volume density , will determine the gravitational field , resulting in a situation where @xmath147 . to2.5 in ' '' '' the fact that the index of the relation between @xmath2 and @xmath1 at constant bulge density is close to that of the faber - jackson relationship implies that , even if there is considerable variation in the bulge mass for a given halo , the departure from a faber - jackson relation defined by the halo parameters ( as described above ) is not too large . this is illustrated in fig . [ comsig ] , where we plot the relationship between the average velocity dispersion ( simply defined as @xmath148 ) of the baryonic component and its mass for systems having parameters @xmath40 and @xmath18 corresponding to those in fig . [ compac ] . the plots are obtained by keeping @xmath40 and @xmath18 constant and , for a given bulge mass , decreasing its characteristic radius until @xmath61 at all radii inside the halo core . to obtain physical parameters we set @xmath149 km s@xmath150 and @xmath151 kpc . the solid line in fig . [ comsig ] is actually a superposition of several lines , corresponding to different values of @xmath18 ( and @xmath40 ) , as used in fig . 2 . its average slope is around 3.2 and increases slowly toward higher dispersion velocities . these lines , however , have different end - points , determined by the minimal bulge masses associated with the different @xmath18 ( cf . section 3.3 ) which are denoted by circles in fig [ comsig ] . as the mass decreases . the result is that small hooks " ( not shown here ) appear in place of the circles in fig . [ comsig ] . in principle , this regime may be of interest , but is not considered in this paper because it occurs for a very small range of bulge masses . ] the maximum values of the bulge masses in these curves correspond to 2.5 times the halo core mass . it seems unlikely that bulges would be more massive than this . their characteristic densities or radii would have to be much larger than those of the halo , as would their contribution to the potential . in systems with disks such extreme conditions would violate the shapes of observed rotation curves . to2.5 in ' '' '' in reality , for a given halo , there is a range of possible bulge densities and associated values of the @xmath40 parameter . this , in principle , affects the normalization of the @xmath145 relation . nevertheless , the normalization turns out to be weakly dependent on @xmath78 ( @xmath152 ) . thus , variations in bulge densities within haloes of given mass and size do not significantly affect the relationship between bulge mass and velocity dispersion . in our model , the central ( bulge ) density required for the production of loop orbits with @xmath61 in the scaling regime is also only weakly dependent on @xmath40 ( as @xmath153 , cf . [ compac ] ) . the above leads to an important corollary , _ that large variation in @xmath40 will cause only small changes in _ @xmath145 , as illustrated in fig . [ varkmb ] . this relation , therefore , appears to be robust and is not heavily affected by changes in bulge and sbh parameters . note also that , for less dense bulges the slope of the lines in fig . [ varkmb ] tends to @xmath154 , especially in the limit of large values of the velocity dispersion ( which is also where most of the observations lie ) . this is due to the the effect described above massive bulges with smaller densities are more extended , and , therefore probe larger regions of the halo core . to2.5 in ' '' '' to obtain an @xmath4 relation , one has to multiply the values of the bulge masses in fig . [ comsig ] by the appropriate @xmath40 factors , arriving at an @xmath4 relation _ within a given halo . _ this introduces `` scatter '' by shifting the @xmath155 lines , as can be seen from fig . [ varkmbh ] . for any fixed value of @xmath1 , this figure reveals that the ( vertical ) scatter in @xmath0 is similar to that in @xmath40 . this leads us to an important point . in the present formulation , the @xmath145 relation is tighter than the @xmath4 one . this appears to contradict observations suggesting that @xmath4 is much tighter than the faber - jackson relation . however , this result has been obtained under the assumption that there is _ no scatter in the halo faber - jackson relation . _ a significant scatter in @xmath3 would result in a corresponding scatter in @xmath156 , which follows from the homology scaling discussed in the beginning of this section . moreover , if @xmath18 correlates with @xmath1 in such a way that gaseous sustems embedded in haloes with higher velocity dispersion require larger values of @xmath18 to remain stable , the scatter in the @xmath4 relation due to variations in @xmath40 can be reduced . for example , in fig . [ varkmbh ] , we have assumed a single halo for all the lines with different @xmath18 . if , however , lines with higher values of @xmath18 are associated with haloes with larger @xmath157 ( and thus @xmath1 ) , the constant @xmath18 lines would be shifted in such a way that the vertical distances between them at a given value of @xmath1 decrease . we illustrate this effect in fig [ shift ] , where a linear relationship is assumed between @xmath1 and @xmath18 . to2.5 in ' '' '' we suggest that the relative tightness of @xmath4 could result from a loose relation between @xmath1 and @xmath18 , coupled with scatter in @xmath3 comparable to that of the faber - jackson relation for bulges . such @xmath158 relation can result from a general trend that dissipation is increasing in more concentrated systems . testing this will require detailed modeling of the way in which the dissipation rate depends on @xmath18 , velocity and density , and is outside the scope of this paper . based on the illustrative example of fig . [ shift ] , we only claim here a plausibility of a loose relationship between @xmath18 and the system velocity dispersion . if the postulated correlation persists for haloes of different masses , a steepening of the @xmath4 relation relative to the halo ( and bulge ) mass velocity dispersion relation is expected , implying that the former should have a larger index . in the model presented here , gaseous baryonic material settles inside a mildly non - axisymmetric halo with a nearly constant density core . initially , no orbits with a definite sense of rotation exist . the first infalling baryonic material , therefore , efficiently loses its angular momentum to the core . this initial collapse terminates only when the gas becomes self - gravitating and forms stars . the increased central density concentration which is produced in this first phase , however , destroys the harmonic core , paving the way for the existence of non - intersecting closed loops with a definite sense of rotation . if the loop orbits are too eccentric , the gas will shock and depopulate them . these orbits , therefore , can not represent long - lived attractors of the dissipative motion . thus , if the baryonic component does not possess a central cusp initially , the absence of sufficiently round supporting closed loops will lead to the formation of a central mass concentration including an sbh . at larger radii the potential responsible for creating sufficiently round loop orbits is that of the extended baryonic component , in the form of a bulge . this leads to a linear relationship between the bulge and sbh masses . it is crucial that both the onset of self - gravity and the appearance of increasingly circular loop orbits are subject to the same condition , both depending on the density ratio of the collapsing gas to that of the background halo core . this limits the allowable range in densities . for bulge core densities of order those of the halo core , and for plausible values of the critical eccentricity , the value of @xmath40 lies in the range @xmath98 , which is compatible with present observations . for densities up to an order of magnitude larger , the range of values is the same , provided that the critical values of the closed loop axis ratios are larger . this can be expected if the dissipation rate along loop orbits is dependent on density a plausible assumption . the most important prediction of the model outlined above is that the bulge and sbh parameters are determined by the halo properties . in particular , relationships between sbh and bulge masses and the velocity dispersion of the bulge necessarily arise , and with the right exponents , if the haloes should also exhibit a faber - jackson type relation between their masses and velocity dispersions . moreover , within a given halo , variations in the bulge and sbh properties are not expected to destroy these relations because imposing the condition of critical eccentricity requires that the bulge and sbh masses are related to the bulge velocity dispersion _ via _ a power law , with index also close to that of the faber - jackson relationship . finally , if the faber - jackson relationship for the halo exhibits significant scatter and if , as again seems plausible , the critical eccentricities of the loops anti - correlate with the density of the system , less scatter should be present in the relationships between sbh masses and bulge velocity dispersions than the corresponding relationships between bulges masses and their own velocity dispersions the standard faber - jackson relationship . there is already tentative evidence that halo cores may indeed follow faber - jackson type relationships ( burkert 1995 ; dalcanton & hogan 2001 ) . in addition , a halo core produced by flattening out the inner region ( where @xmath8 ) of the nfw profile would produce such a relation ( el - zant , shlosman & hoffman 2001 ) . the model thus makes testable predictions concerning the relationships among the very inner regions of galaxies , their extended baryonic components , the dark matter haloes they are thought to be embedded in and the cosmology that predicts their existence . in this framework , larger and more massive halo cores produce , on average , larger and more massive bulges . disks form outside the core , or later , when the central concentration produced by the baryons destroys the core . if the core is very large , most of the baryonic material is consumed in the first phase and no significant disk forms . this effect is expected to be prominent in larger mass cores , since if these follow the faber - jackson relation , more massive haloes should have proportionally larger cores . other predictions include the requirement that the average density of the bulge in the central region should be close to that of the halo core . there is also a minimal bulge mass associated with a given core , although this varies significantly with the critical loop orbit eccentricity assumed . a number of additional consequences for galaxy formation and evolution will be discussed elsewhere . to obtain relationships between sbh masses and bulge properties , we have assumed that during the gaseous infall phase the baryonic component did not have a central density cusp . most observed bulges and ellipticals , however , do have such cusps . one then has to assume that once star formation starts , cold dissipational collapse is initiated . this would lead to a central density cusp , as it does in the case of cosmological haloes . memory of the initial state is retained via the total energy , which determines the final velocity dispersion . merritt & ferrarese ( 2001 ) have suggested that a `` self - regulating '' mechanism , related to the threshold mass necessary for the loss of triaxiality in the system , may be behind the close correlation between sbh and bulge properties . as these authors point out , however , the sbh masses required for strong chaotic behavior leading to rapid loss of triaxiality are in fact probably too large being of the order of a few percent of the mass of the system s baryonic component . this is actually of the order of the sbh mass needed to create round loop orbits at _ all _ radii inside the halo core . in our model , however , this is not assumed . instead , an additional baryonic component plays the role of creating these orbits in the outer region . the collusion between this `` bulge '' component and the sbh in destroying the harmonic core and creating a situation whereby stable gaseous motion can exist gives rise to the correlations described in this paper . the correlations obtained here are compatible with the observed ones , and with acceptable scatter , despite our lack of knowledge of the values of such parameters as @xmath18 and its variation with system properties . detailed modeling of the gas dynamics will be required to further constrain this model . it is also possible that our distinction between the dynamical role played by the sbh and that played by the bulge core is too restrictive . we have introduced this to be able to obtain quantitative results , within the model , solely on the basis of the orbital characterisitics . in general , the roles of the two components , i.e. , the sbh and the bulge , may not be too distinct formation of the sbh can take place simultaneously with a cuspy bulge . for this , the even more ambitious task of a self - consistent treatment , including gas and stellar dynamics and star formation , is required . we believe , however , that our results are generic and arise from fundamental dynamical phenomena which will manifest themselves in any formulation of galaxy formation in mildly triaxial haloes with harmonic cores . we thank our colleagues , too numerous to mention , for stimulating discussions . this work was supported in part by nasa grants nag 5 - 10823 , wku-522762 - 98 - 6 and hst go-08123.01 - 97a ( provided by nasa through a grant from the space telescope science institute , which is operated by the association of universities for research in astronomy , inc . , under nasa contract nas5 - 26555 ) to i.s . , nsf grant ast-9876887 to m.c.b . , and nsf grant ast-9720771 to j.f . arnold , v.i . 1989 , mathematical methods of classical mechanics ( new york : springer ) bardeen , j.m . 1975 , proc . a.hayli ( dordrecht : reidel ) , 297 binney , j. , tremaine , s. 1987 , galactic structure , princeton univ . press bogoliubov , n.n . , mitropolsky , y.a . 1961 , asymptotic methods in the theory of non - linear oscillations ( new york : gordon & breach ) burkert , a. 1995 , , 447 , l25 cole , s. , lacey , c. 1996 , mnras , 281 , 716 dalcanton , j.j . , hogan , c.j . 2001 , , 561 , 35 de blok , w.j.g . , bosma , a.c . 2002 , a&a , 385 , 816 de zeeuw , t. , merritt , d. 1983 , , 267 , 571 ( dzm ) dubinsky , j. 1994 , , 431 , 617 el - zant , a. 1999 , phys rep , 311 , 279 el - zant , a. , shlosman , i. , hoffman , y. 2001 , , 560 , 636 faber , s.m . , jackson , r.e . 1976 , , 204 , 668 ferrarese , l. , merritt , d. 2000 , , 539 , l9 frank , j. , king , a. , raine , d. 2002 , accretion power in astrophysics , cambridge univ . press , 3rd edition gebhardt , k. , et al . 2000 , , 539 , l13 heller , c.h . , shlosman , i. 1994 , , 424 , 84 kormendy , j. , gebhardt , k. 2001 , 20th texas symp . on relativistic astrophysics , j.c . wheeler & h. martel , eds . ( aip conf . 586 ) , 363 kuijken , k. , tremaine , s. 1994 , , 471 , 178 locas , e.l . , hoffman , y. 2000 , , 542 , l139 merritt , d. , ferrarese , l. 2001 , the central kpc of starbursts & agn : the la palma connection , j.h . knapen , j.e . beckman , j.h . , i. shlosman & t.j . mahoney , eds . ( asp : san - francisco , vol 249 ) , 335 miyamoto , m. , nagai , r. 1975 , , 27 , 533 navarro , j.f . , frenk , c.s . , white , s.d.m . 1997 , , 490 , 493 persic , m. , salucci , p. , stel , f. , 1996 , , 281 , 27 pfennifer , d. , norman , c.a . 1990 , , 363 , 391 rix , h .- w . , 1995 , iau symp . 169 , unsolved problems of the milky way , astro - ph/9501068 rix , h .- w . , zaritsky , d. 1995 , , 447 , 82 salucci , p. , burkert , a. , 2000 , , 337 , l9 shlosman , i. 2002 , agns : from central engine to the host galaxy , s. collin et al . ( asp conf . series , vol . 290 ) , in press shlosman , i. , frank , j. , begelman , m.c . 1989 , nature , 338 , 45 tremaine , s. , et al . 2002 , astro - ph/0203468 let the potential @xmath159 be a function of the cartesian coordinates in the halo symmetry plane , @xmath160 with @xmath161 . we follow de zeeuw & merritt ( 1983 , hereafter dzm ) and expand the potential in even powers of the coordinates : @xmath162 if no bulge exists , inside the halo s harmonic core only the terms quadratic in the coordinates are important . if , due to an additional component , or at the boundaries of the core , weak nonlinearity is present the series can be truncated , as above , to second order , as higher order terms are unimportant . the coefficients are given by : @xmath163 , @xmath164 , @xmath165 , @xmath166 , @xmath167 , where the derivatives are taken at @xmath168 . in terms of these one defines the auxilliary variables @xmath169 , @xmath170 , @xmath171 . the condition for stability of the loop orbits is ( see dzm ; table 2a third row ) @xmath172 we will be interested in systems that are both mildly nonlinear and mildly nonaxisymmetric : thus @xmath173 and @xmath174 . the above condition reduces to @xmath175 which , under the above conditions , is always satisfied . it is , therefore , the condition for the existence of loop orbits and not their stability that will be of interest to us . in the absence of a central mass ( sbh ) or density cusp these can not be found arbitrarily close to the center , instead there is a _ bifurcation radius _ beyond which these exist . this determines the effective core radius of the system . to second order , the condition for the existence of loop orbits is @xmath176 with @xmath177 and ( to first order ) @xmath178 , where @xmath179 is the hamiltonian . for @xmath180 ( first order potential terms dominate in eq . [ pot ] ) and @xmath181 , the above requires @xmath182 in the unperturbed case , the action variables ( e.g. , binney & tremaine 1987 ) are time - independent and ( exact ) solutions can be written in terms of these in cartesian coordinates as @xmath183 , etc . in the mildly nonlinear case the solutions of the equations of motion , averaged over a dynamical time ( denoted below by a bar " ) , approximate the true solution to first order in the ( relative amplitude of the ) perturbation and for a number of dynamical times inversely proportional to this ( dzm ; see also bogoliubov & mitropolsky 1961 ; arnold 1989 ) . in this case analogous approximate solutions can be given in terms of the corresponding action variables . in particular , for loop orbits in mildly nonlinear potentials one finds @xmath184 and @xmath185 eliminating @xmath186 one gets @xmath187 where @xmath188 , @xmath189 and @xmath190 . at bifurcation , to first order , @xmath191 where @xmath179 is given by eq . ( [ hamiltonian ] ) . these orbits are infinitely thin and represent oscillations along the @xmath192-axis , with amplitude @xmath193 which is the effective core of the bulge - halo system , as mentioned above . as one increases @xmath179 these become thicker with axis ratio @xmath194 the view we have taken in this paper is that when this ratio becomes large enough , @xmath195 , such loop orbits can support gaseous motion , and that stars formed on these orbits can constitute populations of stellar disks , thus ending the bulge formation stage . we note here that , beyond the bifurcation point , the above relation predicts a rather rapid ( as a function of radius ) transition to round loop orbits . this is confirmed by orbital integration , even though if the bulge mass is smaller than a certain minimal mass , in the fully nonlinear treatement , the orbital axis ratio can decrease again . the second minimum in axis ratio curves ( e.g. , fig . 1 ) is thus not reproduced by this perturbation analysis . we now apply the perturbation analysis to the superposition of potentials given by eqs . ( [ logarithm ] ) and ( [ bulgepot ] ) with the goal of calculating the bifurcation radii of the loops . this is the effective harmonic core radius of the bulge - halo system . since we are only interested in orbit shapes , we set @xmath196 and use scaled variables , as defined in section 2 . for the halo , we use @xmath197 where @xmath198 , corresponding to @xmath199 in eq . ( [ logarithm ] ) , parametrizes the nonaxisymmetry . for the bulge , we have @xmath200 with the above definitions for the potential we have @xmath201 @xmath202 @xmath203 and @xmath204 from equations ( [ hamiltonian ] ) , ( [ action ] ) , and ( [ ymax ] ) bifurcation happens when the value of the long axis of the loops satisfies @xmath205 substituting from the expressions above and taking the limits @xmath206 , @xmath207 , and assuming @xmath208 , we obtain @xmath209 this is the effective harmonic core radius of the bulge - halo system . | the masses of supermassive black holes ( sbhs ) show correlations with bulge properties in disk and elliptical galaxies .
we study the formation of galactic structure within flat - core _ triaxial _ haloes and show that these correlations can be understood within the framework of a baryonic component modifying the orbital structure in the underlying potential . in particular , we find that terminal properties of bulges and their central sbhs are constrained by the destruction of box orbits in the harmonic cores of dark haloes and the emergence of progressively less eccentric loop orbits there .
sbh masses , @xmath0 , should exhibit a tighter correlation with bulge velocity dispersions , @xmath1 , than with bulge masses , @xmath2 , in accord with observations , if there is a significant scatter in the @xmath3 relation for the halo . in the context of this model
the observed @xmath4 relation implies that haloes should exhibit a faber - jackson type relationship between their masses and velocity dispersions .
the most important prediction of our model is that halo properties determine the bulge and sbh parameters .
the model also has important implications for galactic morphology and the process of disk formation .
3mark iii # 1#2#3#4=.24 = .24 = .24 = .24 |
You are an expert at summarizing long articles. Proceed to summarize the following text:
recently , a number of _ chandra _ and _ xmm_-newton observations of quasars have shown local ( @xmath3 ) x - ray absorption lines ( @xcite ) . these background quasars are among the brightest extragalactic x - ray sources in the sky and some of them were used as calibration targets . the typical high ionic column densities of these x - ray absorbers ( @xmath4 @xmath5 ) imply the existence of large amounts of hot gas with temperatures around @xmath6 k. recent ultraviolet observations of the local high velocity absorbers also reveal such hot gas but at lower temperatures ( see . e.g. , @xcite ) . given the spectral resolution of _ chandra _ and _ xmm_-newton , it is still unclear where this hot gas is located : in the interstellar medium , in the galactic halo , or in the local group as the intragroup medium . in sharp contrast , so far only four targets were reported showing intervening absorption systems ( @xmath7 ) , all with low ion column densities . @xcite confirmed their first detection @xcite of an absorption system at @xmath8 , along the sight line towards pks 2155 - 304 . they found the observed column density is @xmath9 @xmath5 , and set a 3@xmath10 upper limit on the column density of @xmath11 @xmath5at the same redshift . @xcite reported the detection of two absorption systems along the sight line towards mkn 421 , but both systems showed absorption lines with column densities less than @xmath12 @xmath5 . @xcite reported a number of x - ray absorption systems towards h 1821 + 643 at 2 - 3 @xmath10 level , and @xcite reported the detection of an absorption line along the sight line towards 3c 120 . to understand the difference between the local and intervening absorption systems , we conduct a survey of a number of extragalactic targets observed with _ chandra _ and _ xmm_-newton to search for local x - ray absorption lines . in this paper , we report the result of this survey . we find that a model in which local x - ray absorbers are associated with the milky way can explain our survey results better than a model in which they are associated with the intragroup medium in the local group . recently , @xcite conducted a systematic survey of 15 nearby agns to investigate the hot x - ray absorbing gas in the vicinity of the galaxy . while their work mainly focuses on associating individual line of sight with known local structures , our work is different in that we study the generic properties of this hot gas . all the background sources are selected from the _ chandra _ and _ xmm_-newton data archives . we focus on instruments that can provide both high spectral resolution and moderate collecting area in the soft x - ray band , and this results in four different instrument combinations : ( 1 ) rgs , the reflection grating spectrometer ; ( 2 ) letg ( the low energy transmission grating ) + hrc ( the high resolution camera ) ; ( 3 ) letg + acis ( the advanced ccd imaging spectrometer ) ; and ( 4 ) hetg ( the high energy transmission grating ) + acis . the first one ( rgs ) is on board _ xmm_-newton and the last three are instruments on board _ chandra_. the instrumental resolving power is typically characterized by the line response function ( lrf ) , the underlying probability distribution of a monochromatic source . hetg - acis has the highest resolving power : the lrf of the medium energy grating ( meg ) has a full width at half maximum ( fwhm ) of @xmath13 around 20 . the other three combinations have a roughly constant resolution of @xmath14 across the relevant wavelength range . in some cases , a target has been observed with several different instrument configurations , and we select those that give the highest number of continuum counts . so far , nearly all the sources with local x - ray absorption lines show the detection of with high confidence , so in this paper we concentrate on the he@xmath1 resonance line with a rest wavelength of 21.6 . to ensure the significance of the absorption line , we select targets that have a strong continuum between 21 and 23 . specifically , we bin the spectrum to roughly half of the lrf fwhm and select targets which have at least 10 counts per bin around 21.6 . in this way , we can ensure a signal - to - noise ratio ( snr ) of at least 4 within the lrf . we test with other binning size and find essentially similar results . both _ chandra _ and _ xmm_-newton data are analyzed using standard software , i.e. , ciao 3.1 and sas 6.0 . we refer readers to @xcite for detailed data analysis procedures . the local he@xmath1 absorption lines from 8 of our 20 targets targets have been reported previously by other papers ; however , to ensure consistency in our data analysis , we re - analyze all the data sets and fit the absorption lines to obtain the equivalent width ( ew ) independently ( see column 5 of table 1 ) , and we refer the readers to the references we list in the column 7 of table 1 for spectra of detected lines . due to the complex nature of the sources in our sample and possible residual calibration effects , instead of fitting the continuum with a physically meaningful model such as an absorbed power law , we fit the continuum between 21 and 23 with a polynomial of an order between 3 to 5 , to effectively remove residual feature on scales @xmath15 . the residual is then fitted with a gaussian at @xmath16 , using isis ( interactive spectral interpretation system , see @xcite ) . while in all cases we confirm the detections ( and non - detections ) that were reported by the original authors , in some cases our measured ews are smaller than those originally reported . for example , we obtain an ew of @xmath17 m for ngc 4593 , while @xcite report a much larger value , @xmath18 m.a careful reexamination of both data analysis procedures indicates that the main discrepancy comes from the determination of the continuum level ( steenbrugge , private communication . ) nevertheless , our method is more stringent and the measured ews can be taken as conservative lower limits . some of the sources in our sample are known to have intrinsic warm absorbers . one may worry that the absorption lines detected at 21.6 could be associated with the intrinsic warm absorbers , particularly in the case of ngc 3783 where may account for part of the absorption @xcite . however , this does not have a significant impact on our conclusions . our case - by - case study of all the warm - absorbers in our sample indicates that and are the only possible sources of confusion . is unlikely because except in ngc 4051 , none of the known warm - absorbers show the outflow velocities that can compensate their redshifts . for , our study indicates that always co - exists with but with at least @xmath19 times higher column density , so a non - detection of or a detection of with column density lower than @xmath20 @xmath5would simply rule out the presence of . by looking at each case , we find that can have a contribution only for ngc 3783 and mcg6 - 30 - 15 . while can contribute at most part of the 21.6 line in ngc 3783 , the contribution to mcg6 - 30 - 15 is unknown yet . to be conservative , we therefore consider a subsample that excludes ngc 3783 and mcg6 - 30 - 15 . the sight line to 3c 273 extends into the galactic halo through the edges of radio loops i and iv , which have been attributed to supernova remnants ( see , e.g. , @xcite ) . @xcite estimated that the contribution to the column density from the supernova remnants accounts for at most 50% of the observed equivalent length . if the absorption line is unsaturated , the equivalent width of the he@xmath1 absorption line can be converted to a column density by @xcite @xmath21 however , saturation could be an important issue , as revealed by high - order transition lines discovered in the mkn 421 spectrum @xcite , the highest quality spectrum in our sample . in that case , the column density of , determined by high - order transition lines is @xmath22 @xmath5 , more than five times higher than that estimated from eq.([eq : n ] ) . with this consideration , column densities converted from eq . ( [ eq : n ] ) can serve as conservative lower limits , and we adopt @xmath23 as a fiducial value in the following discussion for simplicity . in figure [ f1 ] we show an all - sky projection of the entire sample in our survey . the green dots show the positions of those targets with detections , and the red dots show the positions of those without detections . we wish to estimate the fraction of the sky covered by gas with an equivalent width of at least ew , @xmath24 . the difficulty we face is that we have a limited sample size , and not all sources in our sample have enough counts to permit detection of an absorption line of equivalent width ew . for each source , we define the threshold equivalent width , @xmath25 , which is the minimum equivalent width that can be detected at 3@xmath10 : @xmath26 where @xmath27 is wavelength , @xmath28 is the photon flux , @xmath29 is the area of the detector , @xmath30 is the observation time , and @xmath31 is the resolving power where @xmath32 is the width of a resolution element . @xmath25 therefore depends on both the detector and the source ; as more data are gathered for a given source , its value of @xmath25 will decrease . table 1 includes values of @xmath25 for each source in our sample . to estimate @xmath24 , we consider all the sources with @xmath33 , since only these sources have enough counts to ensure detection of an absorption system with an equivalent width @xmath34 . let @xmath35 be the number of these sources , and let @xmath36 be the number of sources in this sample with detectable ovii absorption ( by definition , such sources must have @xmath37 ) . the detection fraction is then @xmath38 and in the limit of large @xmath36 it will approach @xmath24 . figure [ f2 ] shows the detection fraction as a function of the minimum detectable equivalent width , and the error bars are the standard 1@xmath10 errors for a binomial distribution . we show the error bars for the entire sample only for demonstration purposes . the detection fraction starts at @xmath3910% for the entire sample and @xmath39 5% for the subsample , corresponding to lines with @xmath40 m , and then gradually increases to 100% with increasing continuum level or decreasing ew threshold . the most striking feature is that the detection fraction approaches 100% as the sensitivity approaches @xmath41 m , which occurs at a continuum level of @xmath42 counts per bin . while it is tempting to infer that the detection fraction would be 100% if our sample had enough sensitivity , we must be cautious : given the detections of absorption lines in the five highest - quality data sets , we can conclude only that we have 90% confidence that the sky detection fraction is at least @xmath43 , or 63% while both _ chandra _ and _ xmm_-newton have unprecedented resolving power , the highest resolution we can achieve ( hetg - acis , in this case ) is about @xmath44 @xmath45 . based on the hubble flow and adopting a hubble constant of @xmath46 , this corresponds to a distance of @xmath47 mpc at @xmath48 , which makes it impossible to distinguish between a milky way - origin ( with a halo radius of @xmath49 mpc ) and a local group - origin ( with a radius of @xmath50 mpc ) for the x - ray absorbers . nevertheless , with such a large sample size , we can begin to understand the properties of these absorbers , i.e. , their size , density , temperature , spatial distribution , etc . in the previous section , we estimate the sky detection fraction @xmath51 of this hot gas . the detection fraction @xmath51 can be taken as an estimate of sky covering fraction , defined as @xmath52 . our estimation then indicates that the sky cover fraction is at least higher than 63% , and is likely close to unity . in the following , based on this estimation we ( 1 ) calculate the expected number of absorbers along lines - of - sight toward distant agns , ( 2 ) make joint analysis with x - ray emission measurements , and ( 3 ) estimate the total baryonic mass . our calculation indicates that the observed x - ray absorbers are associated with our galaxy . given the high quality of spectra in our sample , any absorbers distributed between us and background agns with similar column densities , i.e. , @xmath53 @xmath5 , would be detected with high confidence . now let us estimate the expected number of _ intervening _ absorbers with a similar column density in our sample . a key assumption in our model is that the x - ray absorbers detected at @xmath48 are not unique , i.e. , an observer located in another galaxy similar to the milky way should be able to see similar x - ray absorption . the next step is to convert the covering fraction that we presented in the previous section to the detectability of these absorbers around systems similar to the milky way or the local group , along the sight lines towards background agns . we assume that the x - ray absorbers are uniformly distributed within the halo ( of a galaxy or a group of galaxies ) that has a radius @xmath54 . let @xmath55 be the sky covering factor as observed from the center of the distribution of absorbers , which can be obtained from observation . if we view the absorbers in a different group of galaxies , the observed covering factor would be @xmath56 ( see , e.g. , @xcite ) . given the spatial density of the absorbers within the halo , we can then calculate the expected number of absorbers along the total line of sight towards all the background sources in our sample . if the spatial density of halos with radius @xmath54 is @xmath57 , the expected number is then @xmath58 where @xmath59 is the cross section and @xmath60 is the cumulative distance along the sight lines to all the targets in the sample . in reality , both the covering fraction and the available pathlength , and so @xmath61 , depend on the detection threshold @xmath25 . taking the covering fraction from eq . ( 3 ) , we can calculate @xmath61 from eq . ( 4 ) , as a function of @xmath25 ( figure [ f3 ] ) . let us consider two scenarios , in which the absorbers can be associated either with milky way - type halos or with local group - type halos . for the milky way - type halo , @xmath62 kpc , and the spatial density of halos is just the spatial density of @xmath63 galaxies , i.e. , @xmath64 @xcite . on the other hand , if the observed local absorbers are associated with the local group , @xmath54 would be @xmath50 mpc . an integral over the press - schechter mass function @xcite from @xmath65 to @xmath66 gives a halo spatial density of @xmath67 . in figure [ f3 ] , local group case and the milky way case are represented by solid and dashed lines , respectively . dark lines are for the entire sample , and red lines are for the subsample . given the fact that _ none of the targets in the sample show any intervening absorption with column densities even close to @xmath68 @xmath5 _ , clearly the observations are consistent with the association of the local x - ray absorbers with milky way - type halos , but inconsistent with a local group association . in the local group case , for the entire sample the maximum expected number of absorbers is @xmath69 if we take a threshold of @xmath70 m . the poisson probability of detecting zero if the expectation is 9.4 is just @xmath71 , at above gaussian @xmath72 level . if we take the subsample , the expected number decrease to @xmath73 , and the probability increases to @xmath74 , roughly corresponding to a gaussian @xmath72 level . we can also estimate the total baryon mass in these x - ray absorbers . let @xmath75 be the hydrogen density in an absorber and let @xmath76 be the volume filling factor of the absorbers in the halo . let @xmath77 be the typical hydrogen column density along lines of sight in which x - ray absorption is seen . averaged over the sky , the average column density is @xmath78 , so that the baryonic mass of the absorbers is @xmath79 , where @xmath80 g is the mass per h. let @xmath81 ( ) be the ionization fraction of and @xmath82 be the abundance of oxygen relative to the solar abundance ( taken as @xmath83 @xcite ) . we then have @xmath84\ ; , \ ] ] where the factor in brackets equals @xmath77 . if the absorbers were distributed throughout the local group so that @xmath85 mpc , their total mass would be @xmath86 ( ) ] @xmath87 . since the dynamical mass of the local group is @xmath88 ( see , e.g. , @xcite ) , the total baryon mass in the local group is @xmath89 for a baryon fraction of @xmath39 15% . this means the covering fraction at most can be 30% for solar metallicity . since figure [ f2 ] clearly indicates that the covering fraction can be much higher ( and probably be as high as @xmath90 ) given enough instrumental sensitivity , we conclude that a local group origin for these x - ray absorbers is unlikely because their estimated total mass would exceed the baryon mass of the local group . our conclusion is consistent with that of @xcite on the other hand , if we associate these absorbers with our galaxy , then the radius decreases by a factor of 10 - 100 and the total mass is substantially reduced . having presented evidence associating the @xmath91 x - ray absorbers with the hot gas within the milky way , we can further constrain the properties of the x - ray absorbers by assuming that the same absorbers are also responsible for at least some of the hot halo foreground observed in x - ray emission . the soft x - ray background is believed to be produced by three components : the local hot bubble ( lhb ) , the extragalactic background ( mainly from point sources ) , and a halo component . by analyzing _ rosat _ all sky survey ( rass ) data , @xcite concluded that the halo emission actually consists of two components , a hard component with @xmath92 and a soft component with @xmath93 . because the hard component is too hot to produce a substantial amount of in collisional ionization equilibrium , we assume that most of the observed resides in the soft component . let @xmath94 be the observed emission measure . since @xmath94 is based on solar abundances and is averaged over the sky , it is related to the actual emission measure , em , along a line of sight through an absorber by @xmath95 , where @xmath96 and @xmath97 and @xmath98 are the electron and proton densities . @xmath99 is the average pathlength through an absorber in pc . @xcite found an emission measure of @xmath100 for the soft component . from the absorption measurement , we have @xmath101 @xmath5 . since @xmath102 at @xmath93 , we have @xmath103 ( independent of metallicity ) , and @xmath104 kpc . this also provides independent evidence that these x - ray absorbers should be associated with our milky way instead of the intragroup medium in the local group . @xcite argued that based on a combination of emission line and absorption line measurement , the scale length of the o vii absorber should be at least @xmath105 kpc . however , their argument depended crucially on the temperature of the emitter / absorber . a shift in temperature by 0.1 to 0.2 in log space will change the scale length by a factor of 10 . they determined the temperature should be between @xmath106 k , based on the and line ratio . a recent work , based on much higher quality chandra data on mkn 421 @xcite , indicated that just from chandra measurement of the and line ratio , the temperature of the mkn 421 local absorber must be lower than @xmath107 k. such a temperature brings the scale length down to the galactic scale . we estimate this x - ray absorbing gas has a density of a few @xmath108 with temperature around @xmath6 k. our estimations are consistent with values predicted from other models such as the dynamics of the magellanic stream @xcite . @xcite suggested , based on observations of absorption in the disk and halo of the milky way and in the intergalactic medium , that these absorption systems belong to radiative cooling flow of hot gas . their predicted column density is consistent with what we find in this paper . a key question then is : how to keep this gas hot without cooling ? for gas with such a high density , the cooling time will be less than the hubble time . in collisional ionization equilibrium , the typical cooling time scale is @xmath109 with @xmath110 and @xmath111 . here , @xmath112 is the radiative cooling rate in units of @xmath113 . under the assumption that the cooling rate at @xmath114 scales as the fe abundance , the cooling rate of @xcite is @xmath115 , where the solar abundances are taken from @xcite . for the density and temperature we have estimated for the absorbers , the cooling time is much less than the hubble time , unless the metallicity is extremely low ( @xmath116 ) . to keep this relatively dense gas ( @xmath117 @xmath118 ) at temperatures around @xmath6 k without cooling down over a hubble time scale , some sort of heating mechanism must play an important role . supernova heating is a potential candidate . the cooling rate , according to the calculations above , would be @xmath119 for @xmath120 ; note that the dependence on @xmath55 , @xmath81(o vii ) , and @xmath82 has canceled out . assuming that the supernova rate is about 0.02 yr@xmath121 in our galaxy ( see , e.g. , @xcite ) , and that the energy output of each supernova explosion is about @xmath122 , the heating rate would be about @xmath123 . for @xmath124 kpc , which is consistent with our estimate for @xmath125 , only a small fraction of the supernova energy is needed to maintain the temperature of the absorbers at @xmath126 k. in this paper , we find that the sky covering fraction of this hot , -absorbing gas is at least 63% , at 90% confidence . this is based on the detections of absorption lines at @xmath127 level in the spectra of the top five high quality spectra . while we can not obtain the exact location of this hot gas within our galaxy , joint analysis with the x - ray background data indicates the scale height of this hot gas should be @xmath128 kpc . finally we conclude that , based on three independent estimations , the x - ray absorbing gas detected locally are part of the hot gas in our galaxy . future observations will provide robust test of this result . we thank julia lee for help with mcg 6 - 30 - 15 data and rik williams for help with mkn 421 data . we also thank the anonymous referee for valuable suggestions . tf was supported by the nasa through _ chandra _ postdoctoral fellowship award number pf3 - 40030 issued by the _ chandra _ x - ray observatory center , which is operated by the smithsonian astrophysical observatory for and on behalf of the nasa under contract nas 8 - 39073 . the research of cfm was supported in part by nsf grant ast00 - 98365 and by the support of the miller foundation for basic research . mgw was supported in part by a nasa long term space astrophysics grant nag 5 - 9271 . | recent _ chandra _ and _ xmm _ observations of distant quasars have shown strong local ( @xmath0 ) x - ray absorption lines from highly ionized gas , primarily he - like oxygen .
the nature of these x - ray absorbers , i.e. , whether they are part of the hot gas associated with the milky way or part of the intragroup medium in the local group , remains a puzzle due to the uncertainties in the distance .
we present in this paper a survey of 20 agns with _
chandra _ and _ xmm _ archival data .
about 40% of the targets show local he@xmath1 absorption with column densities around @xmath2 ; in particular , absorption is present in all the high quality spectra .
we estimate that the sky covering fraction of this -absorbing gas is at least 63% , at 90% confidence , and likely to be unity given enough high - quality spectra . based on ( 1 ) the expected number of absorbers along sight lines toward distant agns , ( 2 ) joint analysis with x - ray emission measurements , and ( 3 ) mass estimation , we argue that the observed x - ray absorbers are part of the hot gas associated with our galaxy .
future observations will significantly improve our understanding of the covering fraction and provide robust test of this result .
nn |
You are an expert at summarizing long articles. Proceed to summarize the following text:
light propagation in a nonlinear dielectric may be used to model various subtle effects involving quantum theory and gravity . these include lightcone fluctuations @xcite and the effects of quantum stress tensor expectation values in semiclassical gravity @xcite . a fluctuating electric field in a nonlinear material causes fluctuations of the effective speed of light of probe pulses , and is analogous to the effects of spacetime geometry fluctuations on light propagation . this analogy was developed in ref . @xcite , where the source of the fluctuations was a squeezed state of the electromagnetic field , and in ref . @xcite , where the effects of vacuum fluctuations of the electric field were investigated . in both cases , linear fluctuations of the electric field were treated , which models the lightcone fluctuations produced by active gravitational field fluctuations . these are the fluctuations of the dynamical degrees of freedom of gravity itself , as opposed to the passive fluctuations of gravity , driven by quantum stress tensor fluctuations . one of the purposes of the present paper will be to develop a model for passive spacetime geometry fluctuations . this will involve a study of the fluctuations of the time averaged squared electric field , which is of interest in its own right . a second purpose of this paper will be a further study of switched fluctuations of quantum fields . the vacuum fluctuations of quantum field operators are only meaningful if the operators have been averaged in time or in spacetime with a smooth sampling function . in the case of linear fields , such as the electric field , the associated probability distribution is gaussian . some effects of the time averaged electric field were discussed in refs . @xcite . in the latter paper , it was shown that simple arguments may be used to estimate the one loop qed corrections to potential scattering by electrons . the fluctuations of quadratic field operators , such as the stress tensor or the squared electric field , are more subtle , and are associated with non - gaussian probability distributions @xcite . these distributions typically fall more slowly than a gaussian function , increasing the probability of large fluctuations , and depend sensitively upon the choice of sampling function . it was argued in ref . @xcite that the sampling function for vacuum fluctuations in a dielectric can depend upon the geometry of the material . this idea will be further developed here , where we will consider a broader class of functions than the lorentzian function used in ref . @xcite . the outline of this paper is as follows : section [ sec : flight ] will first briefly review classical light propagation in a nonlinear material , and then address the effects of switched vacuum fluctuations of the electric field and the squared electric field . a convenient choice of switching function will be introduced in sec . [ sec : switch ] . some numerical estimates of the magnitude of the flight time fluctuations will be given in sec . [ sec : estimates ] . the probability distribution for the flight time fluctuations will be discussed in sec . [ sec : prob ] . our results will be summarized and discussed sec . [ sec : sum ] . throughout this paper we use lorentz - heaviside units with @xmath0 . a nonlinear dielectric material is one where the electric polarization vector is a nonlinear function of the electric field , and can be written as @xcite @xmath1 here repeated indices are summed upon , and @xmath2 , @xmath3 , and @xmath4 are the first , second , and third order susceptibility tensors , respectively . the second and higher order susceptibilities lead to a nonlinear wave equation for the electric field . we wish to investigate the flight time of a probe pulse propagating through a slab of optical material when second and third order coefficients of the susceptibility tensor are included . these nonlinearities of the medium couple to an external applied electric field @xmath5 , here called the background field . the electric field associated with the probe pulse is denoted by the vector @xmath6 , which we choose to be polarized in the @xmath7-direction and propagating in the @xmath8-direction , i.e. , @xmath9 . furthermore , we assume that the probe field is smaller in magnitude than the background field , but more rapidly varying . that is , @xmath10 but @xmath11 in this case , @xmath12 obeys a linearized wave equation @xcite , @xmath13 here @xmath14 is the phase velocity of the wave , which is given by @xmath15^{-1 } , \label{eq : vph}\ ] ] where @xmath16 is the refractive index of the medium measured by the probe pulse when only linear effects take place , and we define the coefficients @xmath17 equation ( [ eq : vph ] ) shows that the background field couples to the nonlinearities of the medium , affecting the velocity of the waves propagating through it . we will assume that dispersion can be ignored , so that the group velocity of a wavepacket is approximately equal to the phase velocity . in this case , the flight time of a probe pulse traveling a distance @xmath18 in the x - direction will be given by @xmath19\,dx , \label{td}\end{aligned}\ ] ] with @xmath20 here the parenthesis enclosing two indices denotes symmetrization , i.e. , @xmath21 . in writing eq . ( [ td ] ) , we have assumed that the nonlinear effects are small , so that we may taylor expand @xmath22 from eq . ( [ eq : vph ] ) to first order in @xmath23 and second order in @xmath24 . in addition , we take the integrand in eq . ( [ td ] ) to be evaluated at @xmath25 , which is the worldline of a pulse traveling at speed @xmath26 , that determined by the linear susceptibility . in this paper , we will follow ref . @xcite and study the effects of vacuum electric field fluctuations as the background field . in this case , @xmath27 becomes the quantized electric field operator , and @xmath28 defined in eq . ( [ td ] ) becomes an operator , where the term quadratic in @xmath27 is understood to be normal ordered , @xmath29 . this leads to a finite mean flight time , which in the vacuum state is , to leading order , @xmath30 our primary interest in this paper will be in the variance of the flight time , @xmath31 note that this quantity is independent of the choice of vacuum state with respect to which normal ordering is performed . a change in the state has the effect of adding a c - number , @xmath32 , to the operator @xmath28 , so that @xmath33 . it is easily verified that the right - hand - side of eq . ( [ eq : var ] ) is unchanged . a change of vacuum state can slightly change the mean time delay , @xmath34 , but does not change the variance of the flight time , which is our primary concern . however , this quantity is only finite if the field operators have been averaged with a test function . in the present context , the density profile of the slab of dielectric naturally defines a suitable function . let @xmath35 be a profile function satisfying @xmath36 now the time delay operator may be written as @xmath37\,f(x)dx \ , . \label{tdf}\end{aligned}\ ] ] the flight time variance now becomes @xmath38 . \label{dtdf}\end{aligned}\ ] ] the definition of normal ordering , @xmath39 and the use of wick s theorem lead to @xmath40 thus the flight time variance can be expressed as an integral involving the correlation functions of the electric field . this double integral is over the spacetime volume of the worldtube of the probe pulse wavepacket . this worldtube is centered upon the worldline of the middle of the wavepacket , described by @xmath41 . we will assume that the wavepacket is sufficiently localized around this worldline so that integrations over the spatial directions transverse to the @xmath8-direction may be neglected . in this limit , we are averaging the electric field and the squared electric field along the worldline of an observer comoving with the probe pulse . in the rest frame of this observer , the field operators are being averaged in time alone . once we take the coincidence limit in the transverse spatial directions , the needed electric field correlation functions for a nondispersive , isotropic material become @xcite @xmath42 ^ 2}\ , , \label{eq : exx } \\ \langle e^0_y(x , t)e^0_y(x',t')\rangle&= \langle e^0_z(x , t)e^0_z(x',t')\rangle = \frac{(\delta x)^2 + ( \delta t)^2/n_b^2}{\pi^2 \ , n_b^3\ , \left [ ( \delta t)^2/n_b^2 - ( \delta x)^2\right]^3}\ , , \label{eq : eyy } \\ \langle e_i^0({x},t)e_j^0({x}',t')\rangle & = 0,\ \ i\neq j,\end{aligned}\ ] ] where @xmath43 and @xmath44 , with @xmath45 , and @xmath46 is the refractive index measured by the background field @xmath47 . using the above results and recalling that the integrations in eq . ( [ dtdf ] ) are performed along the path of the probe pulse , given by @xmath48 , we obtain @xmath49 , \label{final2}\end{aligned}\ ] ] where @xmath50 is understood to have a small negative imaginary part . here we have defined the parameters @xmath51 and @xmath52 as @xmath53 , \\ \alpha_2 & = \frac{2{n_b}^2{n_p}^2}{\pi^4\left({n_p}^2-{n_b}^2\right)^4}\left[\mu_{xx}^2 + \left(\mu_{yy}^2+\mu_{zz}^2 + 2\mu_{zy}^2\right)\frac{\left({n_p}^2+{n_b}^2\right)^2}{\left({n_p}^2-{n_b}^2\right)^2 } % \right.-\nonumber\\&-\left . + 2\left(\mu_{xy}^2+\mu_{xz}^2\right)\frac{\left({n_p}^2+{n_b}^2\right)}{\left({n_p}^2-{n_b}^2\right)}\right ] . \label{an}\end{aligned}\ ] ] this result generalizes previous work @xcite by including the contribution from the third order nonlinear susceptibility , and by giving an expression for the flight time for a general profile function @xmath35 . we wish to choose a suitable smooth switching function that represents the transitions which occur as the probe pulse enters and exits the medium . it will be useful to have two parameters , one ( @xmath18 ) which describes the width of the slab and another ( @xmath54 ) which describes the effective length over which the nonlinearity changes smoothly as the pulse enters and exits . there are several choices for such a function . here we use a function @xmath55 defined by @xmath56 . \label{fbd}\end{aligned}\ ] ] . ] the derivative of this function with respect to @xmath8 is a sum of two lorentzian functions . figure [ figa ] presents some plots of @xmath55 for a few values of the ratio @xmath57 . the parameter @xmath58 describes the distance over which @xmath59 changes from its minimum to its maximum values , and vice versa . note that when @xmath60 we recover a step function , as expected . the integrals appearing in eq . ( [ final2 ] ) may be evaluated by contour integration , with @xmath61 , where @xmath45 . the results , and their asymptotic forms when @xmath62 , are @xmath63 if we assume @xmath62 , and use the above asymptotoic forms , we obtain @xmath64 we define the squared fractional variance in flight time of the probe field as @xmath65 the modulus of the fourier transform of @xmath55 is given by @xmath66 and its behavior is depicted in fig . [ figb ] , where we defined the dimensionless variable @xmath67 and function @xmath68 . note that @xmath69 and that @xmath70 falls exponentially as @xmath7 increases . the plot was done with the particular choice @xmath71 , for which more than @xmath72 of total area under the solid curve occurs in the range @xmath73 . recall that our approximations require ( i ) eq . ( [ eq : c1 ] ) , the dominance of the vacuum field over the probe field , ( ii ) eq . ( [ eq : c2 ] ) , which is equivalent to @xmath74 , ( iii ) a range of frequencies in which the material can be assumed free of dispersion , and ( iv ) a material which is approximately isotropic , at least for the frequencies which give the primary contribution to the background field . the rate of decay of the fourier transform @xmath75 allows us to test approximations ( ii ) and ( iii ) . the exponentially decreasing behavior of the fourier transform of this function , depicted in fig . [ figb ] , suppresses the high energy modes of the background field . is illustrated . specifically , the function @xmath68 is plotted as a function of @xmath67 for the case @xmath76 for the case @xmath71 , at least @xmath72 of the effect will occur in the range @xmath77 , which means that only wave lengths such that @xmath78 will significantly contribute . for a slab with @xmath79 , the dominant wavelengths of the background field are those with @xmath80 . shorter wavelength modes are naturally suppressed by the time averaging . furthermore , the larger contribution occurs arises from @xmath81 , which for @xmath71 , corresponds to a wavelength of @xmath82 . thus if the material is relatively free of dispersion when @xmath80 , then our assumption that @xmath46 is independent of frequency is justified . we may choose @xmath83 to satisfy eq . ( [ eq : c2 ] ) . we may also justify the assumption of the dominance of the vacuum field over the probe field , eq . ( [ eq : c1 ] ) , using essentially the same argument as was given in sec . 3.2 of ref . @xcite . the only difference is that in ref . @xcite , @xmath84 is the expectation value of the square of the averaged electric field . here it is the square root of the expectation value of the square of the averaged squared electric field , which can be obtained from eq . ( [ eq : b6 ] ) divided by @xmath85 , and is proportional to @xmath86 for @xmath87 . thus , if @xmath88 , the two quantities are of the same order , and we obtain eq . ( 30 ) of ref . @xcite as the condition that the vacuum field dominate the probe field . if @xmath89 , then the vacuum field is enhanced by the shorter switch - on and switch - off times , and it becomes easier to satisfy eq . ( [ eq : c1 ] ) . the physical reason for vacuum dominance is that many more modes contribute to the vacuum field than to the probe field . the first example we wish to study is the crystal of cadmium selenide ( cdse ) , which is a hexagonal one , point group 6 mm . this system was already investigated @xcite in the case of a lorentzian sampling function . cdse is an optical medium with nonzero second order nonlinear dielectric susceptibilities and satisfies the conditions discussed at the end of last section . this crystal has an index of refraction @xmath90 and a second order coefficient @xmath91 at a wavelength @xmath92 @xcite . now setting the wavelength of the probe field as @xmath93 , for which @xmath94 , and setting the parameter @xmath95 , we obtain from eq . ( [ final3 ] ) a fractional variance of the flight time , @xmath96 compared to the model where an idealized lorentzian distribution @xcite is used , this result shows that in the situation described by @xmath55 , with @xmath97 , the predicted effect is about 100 times stronger . this enhancement arises because the contribution to @xmath98 due to linear electric field fluctuations is proportional to @xmath99 , as may be seen from the first term on the right - hand - side of eq . ( [ frac ] ) . now we investigate a third order nonlinear optical material . silicon ( si ) is a centrosymmetric crystal ( point group m3 m ) , which means that the second order nonlinear dielectric susceptibilities are identically zero . this crystal has a third order coefficient @xmath100 at a wavelength @xmath101 @xcite , and an index of refraction @xmath102 at the same wavelength @xcite . suppose the probe wave packet has a peak wavelength of @xmath103 , for which @xmath104 @xcite . as before , using @xmath95 , we find the dominant contribution to the fractional variance of the flight time , @xmath105 note that the contribution to @xmath98 due to quadratic electric field fluctuations is proportional to @xmath106 , as may be seen from the second term on the right - hand - side of eq . ( [ frac ] ) . as expected , the effect produced by the third order coefficient tends to be smaller than that related to second order nonlinearities . it may be possible to increase the effects of quadratic fluctuations if new materials with larger third order susceptibilities can be found . in the next section , we will discuss a different type of enhancement . in the previous sections , we have been concerned with the variance of the flight time , which is in turn determined by the variance of the sampled electric field or of the squared electric field . here we wish to estimate the probability of much larger fluctuations than those described by the variance . in the case of effects produced by the second order polarizability , this probability will be very small , as the probability distribution for fluctuations of the electric field is gaussian , and hence falls very rapidly . however , flight time variations due to the third order polarizability will be associated with a more slowly decreasing probability distribution . the distributions for quadratic quantum operators have been discussed in refs . in particular , the asymptotic form for the probability distribution of the lorentzian average of the squared electric field was given in ref . the sampling function used in this paper , eq . ( [ fbd ] ) , is not lorentzian , but the magnitude of its fourier transform , eq . ( [ fourier ] ) , has the same exponential decay as in the lorentzian case . furthermore , it was argued in ref . @xcite that the decay rate of the fourier transform of the sampling function determines the asymptotic form for the probability distribution . thus it is reasonable to extrapolate the lorentzian results to the present case . here we briefly summarize the needed results from ref . let @xmath107 be the lorentzian time average of the normal ordered squared electric field operator at a given point in space , or more generally along a timelike worldline . in our problem , this will be the path of the probe wave packet . define the dimensionless variable @xmath108 where @xmath109 is the characteristic averaging time . let @xmath110 be the probability distribution for finding a given value of @xmath8 in a measurement in the vacuum state , which is normalized by @xmath111 here @xmath112 is the lower bound , the smallest value of @xmath8 which could ever be observed . note that this lower bound is negative , so measurements of @xmath113 in the vacuum state can return negative values , just as expectation values of the squared electric field in more general states can be negative . in fact , one expects most measurements in the vacuum to result in a negative value , but when the outcome is positive , it is likely to be larger in magnitude . note that a negative value of @xmath107 results in a time _ advance _ compared to the mean flight time in the material , just as positive values result in time delays . our primary interest is in the asymptotic form of @xmath110 when @xmath114 , which describes the probability of finding especially large values of the squared electric field . this asymptotic form is approximately @xmath115 where @xmath116 and @xmath117 . a striking feature of this result is the one - third power in the exponential , which causes @xmath110 to fall much more slowly than a gaussian or an exponential function . given @xmath110 , we can define the cumulative probability distribution by @xmath118 which gives the probability of finding any value greater than or equal to @xmath119 in a given measurement . if @xmath120 , we can directly integrate eq . ( [ eq : p ] ) to find @xmath121 it is shown in ref . @xcite that the second moment of @xmath110 for the squared electric field is @xmath122 so the root mean square of @xmath8 is @xmath123 . now we may use eq . ( [ eq : pcumm ] ) to find the probability of a result which exceeds a large multiple of @xmath124 . some examples are given in table [ table : probs ] . .probabilities of large squared electric field fluctuations . [ cols="^,^,^,^,^,^",options="header " , ] the same probabilities apply to the flight time delay due to vacuum squared electric field fluctuations . thus there is a probability of about @xmath125 that a given pulse will suffer a delay which is 1000 times larger than the root mean square value , given for example by eq . ( [ si ] ) . note that our discussion is rather heuristic , and these are order of magnitude estimates . in particular , we have not made a clear distinction between the squared electric field in the rest frame of the probe pulse , and that in the rest frame of the dielectric material . however , for @xmath126 , so @xmath127 to leading order , these quantities will be of the same order . another important point is that both the lorentzian function and the function @xmath55 defined in eq . ( [ fbd ] ) have tails in both directions . a more realistic choice is a function of compact support , which is strictly zero before the measurement process begins . such functions lead to even slower decrease of the probability distribution for large arguments @xcite . in this paper , we have extended previous work @xcite on analog models for lightcone fluctuations . here we have been concerned with vacuum fluctuations of the quantized electric field , and especially with fluctuations of the squared electric field . our model studies the flight time of a probe pulse through a material with a non - zero third order polarizability . vacuum fluctuations of the squared electric field lead to fractional flight time variation which can be of order @xmath128 in the example given in sec . [ sec : estimates ] . these flight time variations model the effects of the passive spacetime geometry fluctuations driven by quantum stress tensor fluctuations . we have also extended the study of the effects of temporal switching functions on the fluctuations of quadratic quantum operators . in sec . [ sec : switch ] , we discussed a specific choice of switching function , which can be relatively constant over a finite interval , and can model the density profile of the nonlinear material . the fourier transform of this function falls exponentially at a rate determined by the parameter @xmath58 , which controls the rate of rise and fall at the ends of the plateau of this function . as this parameter is decreased , increasingly higher frequency modes contribute and increase the fractional flight time variation . we were able to use the fourier transform to estimate the range of vacuum modes which contribute to the flight time variation and to test our approximation of ignoring dispersion . in sec . [ sec : prob ] , we discussed the probability of especially large fluctuations in flight time . this analog model may provide a means to study the probability of large stress tensor fluctuations , which tend to fall more slowly than a gaussian function @xcite . we estimated , for example , a probability of @xmath129 for finding a flight time delay which is at least @xmath130 times the typical delay . there are then two distinct signatures of squared electric field fluctuations . the first is the fractional flight time variation @xmath98 estimated , for example , in eq . ( [ si ] ) . the second is the pulses which undergo an especially long time delay due to a very large squared electric field fluctuation . the extent to which either can be observed in a realistic experiment is a topic for future work . one aspect of this work will be an exploration of finite duration switching , which is more realistic than functions with tails extending into the past and the future . such functions with compact support were treated in ref . @xcite , where it was shown that quadratic quantum operators averaged in time with such functions are associated with a probability distribution which falls more slowly than that for the lorentzian , eq . ( [ eq : p ] ) . this raises the possibility that large fluctuations can be more likely than was estimated in sec . [ sec : prob ] . this work was supported in part by the national science foundation under grant phy-1506066 , and by the brazilian research agencies cnpq ( grants 304486/2012 - 4 and 168274/2014 - 0 ) , fapemig ( grant etc-00118 - 15 ) , and capes . f. charra and g. gurzadyan , _ nonlinear dielectric susceptibilities _ , in springermaterials , d. f. nelson , ed . landolt - brnstein - group iii condensed matter volume 30b , sec . 6.5.2.2 , ( springer - verlag berlin heidelberg 2000 ) . | we study a model for quantum lightcone fluctuations in which vacuum fluctuations of the electric field and of the squared electric field in a nonlinear dielectric material produce variations in the flight times of probe pulses .
when this material has a non - zero third order polarizability , the flight time variations arise from squared electric field fluctuations , and are analogous to effects expected when the stress tensor of a quantized field drives passive spacetime geometry fluctuations .
we also discuss the dependence of the squared electric field fluctuations upon the geometry of the material , which in turn determines a sampling function for averaging the squared electric field along the path of the pulse .
this allows us to estimate the probability of especially large fluctuations , which is a measure of the probability distribution for quantum stress tensor fluctuations . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
network reciprocity @xcite is a general mechanism responsible for the development of spatial correlations within a viscous population , opening the possibility of persistent cooperation . several specific models have been proposed showing how these correlations are related to stable groups of cooperating individuals , whose bulk benefits of self - defense and mutual support outcompete the surface exploitation by defectors @xcite . although actual experiments have been performed @xcite , most of our knowledge comes from these simple models . in particular , a prevailing characteristic in real systems and an important ingredient for cooperation is the heterogeneous contact in systems whose interactions are given by complex @xcite or diluted networks @xcite . when we consider the prisoner s dilemma ( pd ) dynamics @xcite on a diluted lattice that , albeit heterogeneous , has only short range interactions , intermediate densities present an enhancement of cooperation @xcite and , in the presence of a small amount of noise , the optimal dilution is closely related to the ( random site ) percolation threshold for that lattice @xcite . whatever the level of heterogeneity , the contact network topology may evolve in time . although several rewiring mechanisms can be devised ( see ref . @xcite and references therein ) , this may also be accomplished when the high viscosity restriction is relaxed and the agents become mobile . mobility patterns on different scales of human activity , and their far fetched consequences , have been studied in recent decades . for example , airplane displacement and its connection with disease spread @xcite , on a global level , can be contrasted with the more local dynamics of pedestrians , crowds or traffic @xcite . of particular interest is how the observed patterns can affect the outcome of the competition between agents and , in turn , be influenced by it as well . within the evolutionary game theory framework , after several sparse , early attempts to include mobility @xcite , it was only recently that the interest in the combined effects of mobility and cooperation in the pd game had a significant increase . some level of information processing capability is required , for example , when the movement is strategy dependent @xcite or driven by payoff @xcite , success @xcite or the neighborhood composition @xcite . however , the simplest scenario is when mobility is diffusive @xcite . indeed , as hypothesized in ref . @xcite , random mobility may have evolved prior to contingent mobility , allowing bacteria to move away from each other while exploring new resources . our previous results on a lattice @xcite show that even in the framework of random , noncontingent mobility of unconditional agents , diffusion is favorable to cooperation , under rather broad conditions , if velocities are not too high . analogous conclusions , attesting the robustness of the results , were also found in off - lattice models @xcite . when diffusion occurs on a lattice and the one agent per site constraint holds , this area exclusion couples the diffusivity of the agents with the free area . this dependence on density , on the other hand , is not immediately present in off - lattice systems with point - like particles @xcite . moreover , while on a lattice the number of simultaneous interactions is limited by its coordination number , there is no such restriction on the number of point - like particles within the range of interaction in off - lattice systems ( unless it is explicitly included as in ref . a relevant question concerns the universal effects of such geometrical hindrance on the emergence and persistence of cooperation . for example , letting the average body size be a coevolving trait , there may be some evolutive pressure for not too small cooperators because , assuming random diffusion , a group of small individuals will more easily evaporate from the cluster surface . they should not be too large either and , consequently , not be able to evade defectors and avoid exploitation . analogously , for defectors , they should neither be too small in order to stay closer to their prey for longer periods , nor too large so new , more promising regions will not be explored . therefore , one intuitively expects that intermediate , optimal sizes may be beneficial to cooperation and thus be selected for . another possible interpretation for an exclusion zone around each agent is its protected region , and the resources within . whatever the interpretation , it is important to better understand the relevance of area exclusion in these games . as a first approximation , we consider an effective radius of exclusion , modeled as a hard disk . here we study , by explicitly taking into account the excluded area of the agents , the interplay between geometry , density and mobility on the capability of a simple model to sustain cooperation and the question of whether the transitions in this class of model have a geometric interpretation . although the connection between the threshold of geometric percolation , that is independent of the game dynamics , and cooperation has already been reported in refs . @xcite , we also explore the geometry of clusters of cooperators and defectors , and the connection between their critical properties and the transition between regions with and without cooperation , thus providing a geometric interpretation of these transitions . we study an off - lattice model @xcite in which the @xmath3 agents living in a square of side @xmath4 ( with periodic boundary conditions ) are characterized by an unconditional strategy ( cooperate , c , or defect , d ) and two independent geometric parameters : an interaction radius @xmath1 and a hard disk radius @xmath0 to account for excluded area . the radius @xmath1 determines the neighborhood of each agent and , as a consequence , its instantaneous contact network . the area fraction occupied by the hard disk particles is @xmath5 . we use @xmath6 as our length scale . these geometric parameters are illustrated in fig . [ fig.desenho ] . the particular case studied by meloni _ @xcite is recovered in the limit of point - like particles , @xmath2 . and the interaction radius @xmath1 . while the former sets an exclusion area around each agent , the latter defines its contact network . the characteristic length @xmath7 , as indicated on the right , is defined by dividing the available area , @xmath8 , by the area of the square box around each of the @xmath3 particles of diameter @xmath7 , that is , @xmath9.,width=264 ] initially , @xmath3 individuals with probability 1/2 of being either c or d are randomly placed in such a way that there is no overlap between any two individuals @xmath10 and @xmath11 , i.e. , their center - to - center distance @xmath12 satisfies @xmath13 . moreover , they are allowed to randomly diffuse while playing the pd game with their neighbors . two agents are considered neighbors if @xmath14 . a time step is defined as a sequence of @xmath3 attempts of diffusion and a complete , synchronous round of the pd in which each of the @xmath3 agents plays with all its neighbors . during the diffusive part , the position @xmath15 of the center of particle @xmath10 at time @xmath16 is updated if there is no overlap between particles in the final position : @xmath17 each step has a constant size , @xmath18 , and a random orientation @xmath19 drawn from a uniform distribution in the interval @xmath20 $ ] . when @xmath21 there is no mobility and we consider here that @xmath22 is small enough so that jumps over other agents do not occur . under mutual cooperation ( defection ) , both receive payoff @xmath23 ( @xmath24 ) as a reward ( punishment ) ; if one cooperates and the other defects , then the latter receives @xmath25 ( temptation ) and the former , @xmath26 . to characterize the pd , the following inequalities should hold @xmath27 and @xmath28 . in particular , we use @xmath29 , @xmath30 and @xmath31 , a common parametrization known as the weak form of the pd game . the evolution follows the finite population analog of the replicator dynamics @xcite . each individual @xmath10 , after accumulating the payoff from all combats , randomly chooses a neighbor @xmath11 with whom to compare their respective payoffs @xmath32 and @xmath33 . if @xmath34 , then @xmath10 maintains its strategy . on the other hand , if @xmath35 , @xmath10 will adopt the strategy of @xmath11 with probability proportional to the payoff difference @xmath36 where @xmath37 and @xmath38 are the number of neighbors of @xmath10 and @xmath11 , respectively . under this update rule , the total number of individuals is kept constant . most of our results are for @xmath39 , @xmath40 , @xmath41 and @xmath42 . we then check the robustness of the model by testing finite size effects with up to @xmath43 particles , as well as the dependence on @xmath25 and @xmath22 . averages are taken over 100 or more samples . two macroscopic asymptotic quantities , once averaged , are used to characterize the system : the fraction of cooperators @xmath44 ( those , among the @xmath3 agents , that cooperate ) and the fraction @xmath45 of initial conditions whose evolution ends in the absorbing state @xmath46 . their difference , @xmath47 , is a measure of the coexistence of both strategies . four regimes are present in the time evolution , as seen by the behavior of @xmath48 in fig . [ fig.temporal_evolution ] . as is often the case for this class of model , there is an initial drop in the fraction of cooperators from @xmath49 , since small cooperator clusters are easily preyed on in the beginning of the simulation . as @xmath48 approaches its minimum value at @xmath50 , fluctuations may lead to extinctions in finite size systems . away from the minimum , the surviving clusters of cooperators resume growth . these two initial regimes are quite independent of the occupied area fraction , as indicated in fig . [ fig.temporal_evolution ] by the close proximity of all curves up to @xmath51 . in the third regime , @xmath48 attains a plateau where it stays , indicating persistent coexistence of both strategies , if @xmath52 is large enough ( in the case of fig . [ fig.temporal_evolution ] , the threshold is at @xmath53 ) . when @xmath52 is below the threshold , after the stasis period on the plateau , the system enters the last regime in which cooperators take over the system , that is , @xmath54 . the time to reach this asymptotic state seems to diverge as the threshold area fraction is approached from below . , @xmath40 , @xmath55 , and @xmath41 for various area fractions @xmath52 . below @xmath53 , cooperators eventually invade the whole system . as @xmath52 approaches this threshold , the time spent close to the critical plateau at @xmath56 also increases.,width=264 ] fig . [ fig.diagram ] summarizes our most important results , showing , in the stationary regime , the average fraction of cooperators @xmath44 as a function of both @xmath57 and @xmath52 . the lines are independent measures of percolative properties that will be explained below . several regimes may be identified : two absorbing phases in which all agents eventually become either defectors ( @xmath58 , labeled ` d ' ) or cooperators ( @xmath59 , ` c ' ) and two coexistence ones ( @xmath60 and @xmath61 , ` c and d ' ) . finite systems may also present a bi - stable phase , in which all initial conditions lead to one of the absorbing states . in this case , although the average cooperativeness still obeys @xmath60 , it differs from the coexistence state since @xmath62 . however , by increasing the system size , the probability of becoming dominated by defectors goes to zero inside the c region in fig . [ fig.diagram ] and , taking this into account ( see fig . [ fig.finite_size ] ) , we already properly labeled it . we now concentrate on the results for a finite system with @xmath39 particles , and discuss the finite size effects at the end of this section . there are two limits of the diagram that have trivial results . for very large values of @xmath1 , very distant particles interact , increasing the number of contacts and decreasing the effects of spatial correlation . thus , we recover the mean field result in which all agents become defectors . this trivial region of the phase diagram was not explored . in the other limit , when @xmath63 , the hard core prevents any interaction between the agents and the fraction of cooperators remains equal to the initial one , @xmath64 and @xmath61 . this is the trivial coexistence region , labeled ` c and d ' on the left of fig . [ fig.diagram ] , above the line @xmath65 , corresponding to @xmath66 . immediately to the right of this line , interaction , albeit weak , is possible and clusters are formed . however , they are small and do not favor cooperation , therefore @xmath67 , and this all defector phase is labeled ` d ' . as we discuss below , the transitions between the other phases have geometric origins and are closely related to the percolating properties of the contact networks . and @xmath52 for @xmath39 , @xmath41 and @xmath40 . the color code is the average fraction of cooperators , @xmath44 . besides the phases corresponding to the absorbing states , ` c ' ( @xmath59 ) and ` d ' ( @xmath58 ) , there are two coexistence phases , ` c and d'.,width=302,height=207 ] [ [ geometric - percolation . ] ] geometric percolation . + + + + + + + + + + + + + + + + + + + + + + for the area fractions considered here , non - trivial cooperation first appears around @xmath68 . for small @xmath52 , the transition between phases d to c corresponds to a change in stability of the absorbing state , from the defector to the cooperator dominated phase , while for larger @xmath52 , roughly @xmath69 , the emergent phase is one in which cooperators and defectors coexist , c and d. this onset of cooperation is strongly correlated with the appearance of a _ geometric _ percolating cluster , indicated by the steep line in fig . [ fig.diagram ] at the point where the probability of finding a percolating cluster is @xmath70 . this is a purely geometric problem of disks with both an inner hard core in addition to a soft , penetrable region , and thus independent of the game dynamics . in other words , the network of contacts of the percolating cluster spans the whole length of the system . for @xmath71 , our result is consistent both with the percolation threshold obtained numerically in refs . @xcite , and the exact bounds of ref . @xcite . for finite @xmath52 , the threshold is slightly smaller than for @xmath71 , since as @xmath0 increases , the overlap between the disks decreases and percolation is attained with a smaller @xmath1 . [ [ percolating - clusters - of - defectors . ] ] percolating clusters of defectors . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + besides the clusters of particles irrespective of their strategies , we also consider the geometry of clusters composed only by defectors ( that are , in turn , intimately connected with the geometry of cooperator clusters ) , which depends on the particular strategy evolving dynamics . [ fig.perc ] shows the probability of percolation of d clusters as a function of time , @xmath72 , for @xmath73 . notice that in this region there always is a percolating geometric cluster . initially , as the fraction of cooperators decreases towards the minimum , there is a sea of defectors that obviously percolates and all curves overlap at @xmath74 . it is only when the curves of @xmath48 , for different values of @xmath52 , start to separate , around @xmath51 , that @xmath72 starts decreasing . the asymptotic probability of there being a percolating cluster of defectors attains a limiting value , as shown in the inset of fig . [ fig.perc ] , from which the threshold can be obtained . for the particular value of @xmath1 shown in this figure , when the area fraction is below the threshold at ( roughly ) 0.37 , @xmath75 and , as can be seen in fig . [ fig.temporal_evolution ] , @xmath54 . the dotted transition line in the phase diagram ( fig . [ fig.diagram ] ) is obtained in the same manner : it is the asymptotic value of @xmath52 at which @xmath75 as a function of @xmath1 . in the c region , finite size fluctuations sometimes lead to the all - d state @xmath76 , but these configurations are not taken into account for the calculation of the asymptotic value of @xmath52 . this transition line suggests an important ingredient to understand the coexistence between cooperators and defectors : only under the presence of a percolating sea of defectors is that a stable coexistence between cooperators and defectors is possible . in other words , differently from compact clusters of cooperators , isolated groups of defectors either grow and percolate or eventually become extinct . , for several area fractions @xmath52 and the same parameters of fig . [ fig.temporal_evolution ] . inset : asymptotic limit of @xmath72 as a function of @xmath52 . the linear fit indicates that the percolation probability goes to zero at @xmath53 . the small deviation seen close to this point is due to the long time of convergence.,width=302 ] [ [ finite - size - effects . ] ] finite size effects . + + + + + + + + + + + + + + + + + + + + we now discuss how the system size can affect the phase diagram , fig . [ fig.diagram ] . for finite sized systems , a small region inside the c phase has a bi - stable equilibrium , in which all initial conditions lead to an absorbing state , either @xmath77 or 1 . the all - d state is due to the fact that the population of cooperators becomes quite small during the initial drop in the first generations and , therefore , sensitive to fluctuations which occasionally cause extinctions . [ fig.finite_size ] shows @xmath78 for two different values of @xmath52 and several system sizes . for point particles @xmath79 , the absorbing all - c state region grows as the system size increases . on the other hand , for @xmath80 , the all - c region shrinks . in both cases , however , the c phase width converges to a finite value and the bi - stable region decreases as the system size increases . furthermore , the size of the c region in the limit of large systems is consistent with the transition line obtained from the percolating defector cluster analysis for @xmath39 , which explains why this phase is not homogeneously painted in fig . [ fig.diagram ] . notice that when the system is bi - stable , the average fraction of cooperators is not a good measure since it represents neither one of the final states @xcite . on the other hand , in the coexistence state , both strategies are present in the asymptotic state and , while @xmath60 , @xmath61 . for the @xmath2 case studied by meloni _ @xcite , the @xmath71 line in the phase diagram , there is no coexistence phase and the system eventually enters an absorbing state . absorbing state as a function of @xmath57 for several system sizes and two area fractions @xmath71 and 0.28 . for large values of @xmath57 , as @xmath78 increases for larger system sizes , the existence of the @xmath77 state for point particles ( @xmath71 ) is due to finite size fluctuations . for @xmath81 the behavior is the opposite and the transition becomes sharper when the system size increases.,width=302 ] [ [ robustness - against - mobility - and - temptation . ] ] robustness against mobility and temptation . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + we finally consider the robustness of our results when the mobility @xmath22 and the temptation @xmath25 are varied , fig . [ fig.mu-t ] . the top panel shows several values of @xmath22 , with the temptation fixed at @xmath40 . whatever the velocity , no cooperation is possible for @xmath82 due to the absence of a percolating geometric cluster . notwithstanding , for small mobilities , existence of cooperators is possible in a wide range of @xmath1 . when comparing the low mobility case with the one with immobile agents @xmath83 , it can be seen that there is an improvement only for low values of the @xmath57 ratio . for high values of this ratio , the curves overlap , which is expected , since the agents have a large neighborhood that is minimally perturbed by the small random movements . on the other hand , for low values of the ratio , the contact range is small and is more affected by the diffusion . random movements lead to the evaporation of cooperative clusters and , as the velocity increases , cooperation levels decrease until a threshold above which it is no longer possible . this effect of cooperation enhancement driven by a low mobility is in accordance with previous simulations on a lattice @xcite . the bottom panel of fig . [ fig.mu-t ] shows the effect of the parameter @xmath25 . as expected , as the temptation to defect increases , the fraction of cooperators decreases . for @xmath81 and several values of @xmath22 ( top ) , including @xmath21 ( empty symbols ) , and @xmath25 ( bottom ) . in both cases , @xmath39 ; for the top panel , @xmath40 while for the bottom one , @xmath41.,width=302 ] we presented numerical results for an off - lattice model of mobile agents playing the pd game while randomly moving across a closed region . this model combines ingredients found in two distinct models for such systems : the excluded area found in lattice simulations and an available continuous space . we arrive at two important results : first , if the agents are not point - like , a non - trivial coexistence phase with cooperators and defectors becomes possible ( besides the trivial one when the exclusion range is larger than the interacting radius ) ; second , it is possible to geometrically interpret , in terms of percolation , the observed transitions . an important role is played not only by geometric percolation , irrespective of the game dynamics , but also by the percolative properties of defector clusters , whose threshold depends on the details of the game . when the agents interaction is prevented by the hard core , trivial coexistence between cooperators and defectors is possible . beyond that region of the phase diagram , no cooperation is possible in the absence of a percolating geometric cluster . random mobility provides an evaporating mechanism for groups of cooperators what is detrimental to cooperation as isolated cooperators are easily preyed on . however , in the presence of a percolating clusters , it is easier for a detaching cooperator to be in contact with another cluster and be protected . when hard cores are included , movements are hindered and the agents spend some time rattling around the same region , while large displacements become less probable , what increases the correlation among agents and benefits cooperation even further . this localization is probably the mechanism responsible for the coexistence between cooperators and defectors that is not present for @xmath71 . interestingly , the presence of defectors is only possible if they form a percolating cluster and no finite cluster of defectors is stable : they either grow , merge with others and span the whole lattice or the isolated cluster becomes extinct . this is our main result : finite size cooperators and defectors , whose hard core is an effective , averaged interaction restraining their movements , are able to coexist over a broad region of the phase diagram only if defectors are organized in a interconnected cluster , a percolating sea of defectors . a similar effect was found for the public goods game played on a lattice with empty sites and no mobility @xcite . it would be interesting to investigate whether this condition for coexistence between cooperators and defectors also occurs in lattice models , where excluded area is inherent to the formulation of the problem . in this manuscript we focused on the particular homogeneous case of equal sizes and equal velocities for both cooperators and defectors . following refs . @xcite , it is essential to explore the whole @xmath84 parameter space and the dependence on the chosen dynamic rule in order to check the robustness of cooperation . furthermore , several extensions are possible . for example , velocities may not be constant @xcite and depend on the neighborhood @xcite or strategy . the hard core radius may also correlate with strategy , @xmath85 and @xmath86 for cooperators and defectors , respectively . in particular , if individuals coevolve with mutations : is there an optimal equilibrium radii ratio to which the system converges to or , instead , a permanent arms race ? what are the effects of having size dispersion ? if velocity and size coevolve along with strategies , defectors may become small and fast while cooperators become large and slow . finally , what happens if there is a fraction of fundamentalists ( both cooperators and defectors or just defectors ) whose strategies or positions never change ? in all these cases , it is important to study also the geometric properties of the interfaces between cooperator and defector clusters since these are the places where all strategy flips occur . from a more physical perspective , it would be interesting to find out , for each transition line , which is the dynamical universality class that it belongs to @xcite . these and other relevant questions are being considered . research partially supported by the brazilian agencies cnpq , capes and fapergs . jja also thanks the inct - sistemas complexos ( cnpq ) for partial support . we thank the supercomputing laboratory at universidade federal da integrao latino - americana ( lcad / unila ) , where the simulations were run , for computer time . | we study the conditions for persistent cooperation in an off - lattice model of mobile agents playing the prisoner s dilemma game with pure , unconditional strategies .
each agent has an exclusion radius @xmath0 that accounts for the population viscosity , and an interaction radius @xmath1 that defines the instantaneous contact network for the game dynamics .
we show that , differently from the @xmath2 case , the model with finite sized agents presents a coexistence phase with both cooperators and defectors , besides the two absorbing phases in which either cooperators or defectors dominate .
we provide , in addition , a geometric interpretation of the transitions between phases . in analogy with lattice models ,
the geometric percolation of the contact network ( i.e. , irrespective of the strategy ) enhances cooperation .
more importantly , we show that the percolation of defectors is an essential condition for their survival . differently from compact clusters of cooperators ,
isolated groups of defectors will eventually become extinct if not percolating , independently of their size . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
left - handed ( lh ) metamaterials are artificial , effectively homogeneous structures , featuring negative refractive index at specific frequency bands where the effective permittivity @xmath0 and permeability @xmath1 are simultaneously negative @xcite . in fact , all known realizations of lh metamaterials rely on the use of common right - handed ( rh ) elements and , thus , in a realistic situation such a composite material features both a lh and a rh behavior , in certain frequency bands . physically speaking , the difference between the two is that in the lh ( rh ) regime , the energy and the wave fronts of the electromagnetic ( em ) waves propagate in the opposite ( same ) directions , giving rise to backward- ( forward- ) propagating waves . transmission line ( tl ) theory constitutes a convenient framework for the analysis of lh metamaterials . such an analysis relies on the connection of the em properties of the medium ( @xmath0 and @xmath1 ) with the electric elements of the tl s unit cell , namely the serial and shunt impedance . as mentioned above , in practice _ composite right / left - handed ( crlh ) _ structures are quite relevant , giving rise to pertinent crlh - tl models . these models are , in fact , dynamical lattices which can be used for the description of a variety of metamaterials - based devices and systems , such as resonators , directional couplers , antennas , etc @xcite . nonlinear crlh - tls , with a serial or / and shunt impedance depending on voltages or currents , have also attracted attention . such structures may be realized by inserting diodes which mimic voltage - controlled nonlinear capacitors into resonant conductive elements ( such as split - ring resonators ) @xcite . such nonlinear crlh - tl models have been used in various works dealing , e.g. , with the parametric shielding of em fields @xcite , the long - short wave interaction @xcite , or soliton formation @xcite . experiments in nonlinear crlh - tls have also been performed ( see the review @xcite ) , and formation of bright @xcite or dark @xcite envelope solitons , described by an effective nonlinear schrdinger ( nls ) equation , was reported . notice that in earlier studies on rh - tl models it was shown that two ( or more ) solitons propagating with the same group velocity , can be described by a system of two ( or more ) nls equations @xcite ( see also @xcite for theoretical as well as experimental results ) . such coupled nls equations have been studied extensively in nonlinear optics and mathematical physics ; see , e.g. , refs . @xcite and references therein . they are well - known to give rise to a variety of vector solitons , including bright - bright , bright - dark , and dark - dark ones . in this work , we study analytically and numerically the interaction between backward- and forward - propagating solitons in a nonlinear crlh - tl . our model is a nonlinear version of a generic crlh - tl model ( see , e.g. , refs . @xcite ) : the considered nonlinear element in the unit cell of the tl is the shunt capacitor , which simulates the presence of a heterostracture barrier varactor ( hbv ) diode @xcite ( the capacitance of the hbv diode depends on the applied voltage ) . starting from the discrete lump element model of the crlh - tl , we derive a nonlinear lattice equation . first , we study the linear regime and show that , for certain frequency bands , rh- and lh - modes can propagate with the same group velocity . next , we treat the nonlinear lattice equation in the framework of the _ quasi - discrete _ ( or _ quasi - continuum _ ) approximation ( see , e.g. , @xcite and @xcite for a review ) : we thus seek for envelope soliton solutions of the nonlinear lattice model , characterized by a _ discrete carrier _ and a _ continuum envelope _ and employ an asymptotic multi - scale expansion method , to derive a system of two coupled nls equations . each of these equations describes the evolution of the envelope of a backward- ( lh- ) and a forward - propagating ( rh- ) mode . a systematic analysis of the system of the nls equations reveals the existence in certain frequency bands of three different types of vector solitons : ( a ) a backward - propagating bright soliton coupled with a forward - propagating bright soliton , ( b ) a backward - propagating bright soliton coupled with a forward - propagating dark soliton . ( c ) a backward - propagating dark soliton coupled with a forward - propagating bright soliton , and the above analytical predictions are then tested against direct numerical simulations , which are performed in the framework of the original nonlinear lattice model . the results of the simulations verify the existence of the aforementioned types of vector solitons in the full tl model , but also offer important information regarding their robustness . in particular , results of direct simulations performed for long times indicate that bright - bright solitons are the most robust among the members of the vector soliton family . indeed , the mixed ( dark - bright or bright - dark ) types are found to be less robust ; however , the dark - bright solitons in a specific frequency band , although they are deformed during their evolution , are found to be more robust than those in other bands , as well as the bright - dark solitons , which are destroyed for the same propagation time . in any case , our results indicate the existence of all three types , robustness of bright - bright solitons and partial or substantial deformation of the other types . we can thus conclude that bright - bright ( lh - rh ) , as well as dark - bright ( lh - rh ) solitons in certain frequency bands , have a better chance to be observed in experiments . the paper is organized as follows . in section ii , we introduce the nonlinear crlh - tl model and the pertinent lattice equation , and derive the system of the two coupled nls equations ( relevant details are also appended in an appendix ) . in section iii , we present analytical and numerical results for each type of vector soliton . finally , in section iv , we summarize our conclusions . we consider a generic crlh - tl , composed by both right- and left - handed elements , as shown in its unit - cell circuit shown in fig . [ fig : circuitcrlh model ] @xcite . the ( rh ) elements of this tl are the inductance @xmath2 and capacitance @xmath3 , while the lh ones are the inductance @xmath4 and capacitance @xmath5 . we assume that the tl is loaded with a nonlinear capacitance ( @xmath6 , while the capacitance @xmath7 will be assumed to be fixed and voltage independent ) . this can be implemented by proper insertion of diodes in the tl ( see , e.g. , pertinent experiments as well as theoretical work in refs . @xcite ) ; in other words , we assume that the shunt capacitor @xmath6 is nonlinear ( see details below ) . let us now consider kirchhoff s voltage and current laws for the unit - cell circuit of fig . [ fig : circuitcrlh model ] , which respectively read : @xmath8 where @xmath9 is the voltage across the capacitance @xmath7 and @xmath10 is the current across the inductor @xmath11 . the above equations , together with the auxiliary equations @xmath12 and @xmath13 , lead to the following system : @xmath14 to proceed further , we now consider a specific voltage - dependence for the nonlinear capacitance @xmath3 . here , we will assume that for sufficiently small values of the voltage @xmath15 the function @xmath16 can be approximated as follows , via a taylor expansion : @xmath17 where @xmath18 is a constant capacitance corresponding to the bias voltage @xmath19 , while @xmath20 and @xmath21 also assume constant values , depending on the particular form of @xmath22 . below , we will further discuss this approximation , in connection with the hbv diode , used in the experiments described in ref . @xcite ( similar varactor - type diodes were also used in the experiments of ref . @xcite ) . next , substituting eq . ( [ eq : taylor1 ] ) into eq . ( [ eq : eq01 ] ) and using the scale transformations @xmath23 [ where @xmath24 and @xmath25v_n $ ] , we obtain : @xmath26 where the constant parameters @xmath27 , @xmath28 and @xmath1 are given by : @xmath29 in the above expressions , @xmath30 and @xmath31 denote series and shunt frequencies , while @xmath32 denotes the characteristic frequency related to the rh part of the unit - cell circuit , respectively ; the above frequencies are defined as : @xmath33 note that if @xmath34 , i.e. , @xmath35 , then the crlh - tl is usually referred to as _ balanced _ , in the sense that the characteristic impedances of the purely lh- and rh - tl , defined as @xmath36 and @xmath37 , are equal , i.e. , @xmath38 @xcite . on the other hand , if @xmath39 , i.e. , @xmath40 , the lh part of the tl dominates , in the sense that the tl has a more pronounced lh behaviour ( the serial branch features a capacitive character while the shunt branch an inductive one ) . in the opposite case , @xmath41 , i.e. , @xmath42 , the rh part of the tl dominates and the tl has a more pronounced rh behaviour ( the serial branch features an inductive character while the shunt branch a capacitive one ) . it is now relevant to adopt physically relevant parameter values for eq . ( [ eq : model ] ) . for applications in the microwave frequency range ( e.g. , for microstrip lines @xcite or coplanar waveguide structures loaded with srrs @xcite cf . also ref . @xcite for recent work ) , typical values of the capacitances and inductances involved in the crlh structure are of the order of pf and nh , respectively . here , we will use the values @xmath43 nh , @xmath44 pf , and @xmath45 nh ; thus , the frequencies in eqs . ( [ eq : freq ] ) take the values @xmath46 ghz , @xmath47 ghz and @xmath48 ghz . on the other hand , as concerns the parameters involved with the nonlinear capacitor @xmath6 , we assume that the pertinent capacitance corresponds to a hbv diode , which is characterized by the following equation @xcite ( see also @xcite , where the same form of @xmath49 is used , but different parameter values ) : @xmath50 where @xmath51 ff/@xmath1m@xmath52 is the capacitance corresponding to bias voltage @xmath53 v , @xmath54m@xmath52 is the device area , @xmath55 v is the breakdown potential , and the exponent @xmath56 results from fitting experimental data . it is clear that , for sufficiently small @xmath57 , by taylor expanding eq . ( [ hbv ] ) one obtains eq . ( [ eq : taylor1 ] ) , where the constant parameter values involved are @xmath58 pf , @xmath59 pf / v and @xmath60 pf / v@xmath61 . to this end , the values of the normalized parameters @xmath27 , @xmath28 and @xmath1 appearing in eq . ( [ eq : model ] ) take the following values : @xmath62 below , we will use these values for the purposes of our analytical and numerical considerations ( we have checked that other values lead to qualitatively similar results ) . notice that our choice leads to @xmath40 , i.e. , we consider the case where the tl has a more pronounced lh character ; however , when considering the linear setting ( see next subsection ) , this parameter will also assume other values , corresponding to the balanced and rh - dominated behaviour as well . [ cols="^,^ " , ] figures [ fig : db13sim1 ] and [ fig : db13sim2 ] show the outcome of the simulations for a dark - bright soliton in bands i and iii ( with @xmath63 and @xmath64 ) , while figs . [ fig : db23sim1 ] and [ fig : db23sim2 ] correspond to a dark - bright soliton in bands ii and iii ( with @xmath63 and @xmath65 ) . the parameters used are @xmath66 and @xmath67 , which gives @xmath68 and @xmath69 , in bands i and iii , and @xmath70 and @xmath71 , which gives @xmath72 and @xmath73 , in bands ii and iii , respectively . in the latter case , the relatively large values of the number of particles and of @xmath74 used are motivated by the necessity of a vanishing tail for the bright component at the edges of the lattice . as seen in this set of figures , dark - bright solitons in bands ii - iii and i - iii do exist , as predicted in theory . furthermore , it is observed that the former are less robust than the latter , as seen both from their stronger deformation and the fact that they `` lose '' their solitary wave character earlier . this can be observed , e.g. , in the strong fluctuations in the evolution of the soliton center in the bottom left panel of fig . [ fig : db23sim2 ] , or perhaps most notably in the substantial modification of the wave profile upon long propagation in the top right panel of the same figure . on the other hand , the dark - bright solitons of bands i - iii seem to essentially preserve their structure even in the long evolution of fig . [ fig : db13sim2 ] . in conclusion , we have used both analytical and numerical techniques to study the existence , stability and dynamics of coupled backward- and forward - propagating solitons in a composite right / left - handed ( crlh ) nonlinear transmission line ( tl ) . the considered form of the tl was a quite generic one , finding applications to the modelling of a wide range of lh systems and devices , with `` parasitic '' rh behavior , such as resonators , antennas , directional couplers , among others @xcite . our analysis started with the derivation of a nonlinear lattice equation governing the voltage across the fundamental ( unit cell ) element of the transmission line . in the linear regime , we derived the dispersion relation for small - amplitude linear plane waves and showed that they may either propagate in a right - handed ( rh ) high - frequency region , or in a left - handed ( lh ) low - frequency region . we also identified frequency bands where rh- and lh - modes can propagate with the same group velocity . using the above result , we then investigated the possibility of nonlinearity - assisted coupling between lh- and rh - modes . this way , in order to analytically treat the nonlinear lattice equation , we used the so - called quasi - discrete approximation . the latter is a variant of the multi - scale perturbation method , which takes into regard the discreteness of the system by considering the carrier ( envelope ) of the wave as a discrete ( continuum ) function of space . employing this approach , we derived , in the small - amplitude approximation and for certain space- and time - scales , a system of two coupled nonlinear schrdinger ( nls ) equations for the unknown voltage envelope functions . this system was then used to predict the existence of coupled backward- and forward - propagating solitons , of the bright - bright , bright - dark and dark - bright type , respectively . the above existence results , as well as the propagation properties and the potential robustness of these vector solitons , were then investigated for each of the possible scenarios . this was done by means of direct numerical simulations of the full crlh - tl nonlinear lattice model , using as initial conditions the analytical forms of solitons predicted by the perturbation theory . in the simulations , apart from the evolution of the shape , we also studied the evolution of the center of mass and a power - like quantity of the various solitons . our numerical results have confirmed the existence of the various types of solitons predicted analytically , but have also revealed their distinct robustness characteristics . in particular , we found that bright - bright solitons feature a robust propagation over long times . on the other hand , as concerns solitons of the mixed - type ( namely dark - bright and bright - dark ones ) , we found that , in specific frequency bands ( bands i - iii ) , dark - bright solitons are more robust than those in other bands ( i.e. , ii - iii ) or bright - dark solitons : dark - bright solitons in bands ii - iii and bright dark solitons preserve their shape only for finite times and , for sufficiently long evolutions , they are either destroyed ( bright - dark ) or are significantly deformed ( dark - bright ) . we can thus postulate that from all types of solitons predicted analytically , bright - bright and dark - bright ones ( in bands i - iii ) are the most likely ones to be experimentally observable . in all cases , our numerical results were found to corroborate the analytical predictions , at least up to the times during which the solitary waves propagate robustly . it would be interesting to study other types of nonlinear crlh - tl lattice models modelling realistic structures composed by lh - metamaterials . in that regard , a pertinent interesting direction would be the investigation of the effects of damping and driving , which may lead to robust nonlinear waveforms which would constitute dynamical attractors in such settings . additionally , the study of higher - dimensional settings is a particularly challenging problem . in the latter context , in addition to simpler ( yet genuinely higher dimensional , or even quasi - one - dimensional ) solitary wave structures , more complex waveforms may be realizable such as vortices . the exploration of such states and their dynamical robustness will be reported in future publications . * acknowledgments . * the work of d.j.f . was partially supported by the special account of research grants of the university of athens . j.c . acknowledges financial support from the micinn project fis2008 - 04848 . pgk acknowledges support from the us - nsf via cmmi-1000337 , and the us - afosr via fa9550 - 12 - 1 - 0332 . our analytical approximation relies on the use of the quasi - continuum approximation , which is a variant of the method of multiple scales @xcite . we introduce new independent temporal variables , @xmath75 ( @xmath76 ) , and accordingly expand the time derivative operator @xmath77 as @xmath78 . next , we seek solutions of eq . ( [ eq : model ] ) in the form : @xmath79 then , we substitute eq . ( [ eq : ansatz01 ] ) into eq . ( [ eq : model ] ) and employ a continuum approximation for the envelope functions @xmath80 , i.e. , @xmath81 , where @xmath82 and @xmath83 being the lattice spacing ( the latter parameter does not appear in the results below , as one may readily rescale @xmath84 as @xmath85).furthermore , we introduce the new spatial variables @xmath86 and , thus , @xmath87 . to this end , equating coefficients of like powers of @xmath0 , we obtain the following ( first three ) perturbation equations : @xmath88=0 , \label{eq : terms3}\end{aligned}\ ] ] where the operators are given by @xmath89 we now seek for a solution of the linear problem , eq . ( [ eq : terms1 ] ) , in the form : @xmath90 where subscripts @xmath91 and @xmath92 correspond to the lh and rh frequency bands , @xmath93 is an unknown complex function , @xmath94 , while the wavenumbers @xmath95 and frequencies @xmath96 satisfy the dispersion relation provided in eq . ( [ eq : linear_disp ] ) . next , substituting eq . ( [ eq : ansatza ] ) into eq . ( [ eq : terms2 ] ) , we obtain the non - secularity condition for @xmath97 : @xmath98\frac{\partial v_j}{\partial x_1}=0 , \label{eq : secular}\ ] ] which suggests that @xmath99 , where @xmath100 , while the group velocities@xmath101 result self - consistently as @xmath102 [ cf . ( [ eq : gvelqc ] ) ] . employing eq . ( [ eq : secular ] ) , we may determine from eq . ( [ eq : terms2 ] ) , for @xmath103 , the unknown field @xmath104 : @xmath105}{{{g_3}(\omega_1+\omega_2,k_1+k_2 ) } } v_1v_2\exp(i(\theta_1+\theta_2 ) \nonumber\\ & & -\frac{2[(\omega_1-\omega_2)^{4}-\delta^2(\omega_1-\omega_2)^{2}]}{{{g_4}(\omega_1-\omega_2,k_1-k_2 ) } } v_1v_2^{*}\exp(i(\theta_1-\theta_2)\nonumber\\ & & -\sum_{j=1}^{2}f_j(x_1,x_2,\cdots , t_1,t_2,\cdots)+{\rm c.c . } , \label{eq : u2}\end{aligned}\ ] ] where functions @xmath106 are given by : @xmath107 on the other hand , functions @xmath108 can be derived at the order @xmath109 , by means of the equation : @xmath110 which leads to the result : @xmath111 to this end , we arrive at the following expression for @xmath104 : @xmath112 where @xmath113}{{{g_3}(\omega_1+\omega_2,k_1+k_2)}},\\ c_4&=&\frac{2[(\omega_1-\omega_2)^{4}-\delta^2(\omega_1-\omega_2)^{2}]}{{{g_4}(\omega_1-\omega_2,k_1-k_2)}},\\ c_{0j}&=&\frac{2\omega_j^2\delta^2}{\omega_j^4+\delta^2}. \label{eq : u2par}\end{aligned}\ ] ] finally , defining the coefficients : @xmath114 and using the variables @xmath115 and @xmath116 , we derive from the non - secularity condition at @xmath117 the coupled nls equations ( [ eq : mnls1 ] ) . m. lapine , m. gorkunov , and k. h. ringhofer , phys . e * 67 * , 065601 ( 2003 ) ; m. lapine and m. gorkunov , phys . e * 70 * , 066601 ( 2004 ) ; b. wang , j. zhou , t. koschny , and c.m . soukoulis , opt . express * 16 * , 16058 ( 2008 ) ; d. a. powell , i. v. shadrivov , and yu . s. kivshar , appl . . lett . * 95 * , 084102 ( 2009 ) . fefferman and m.i . weinstein , arxiv:1202.3839 . see also arxiv:1212.6072 . p. g. kevrekidis , d. j. frantzeskakis and r. carretero - gonzlez ( eds . ) , _ emergent nonlinear phenomena in bose - einstein condensates _ , ( springer - verlag , heidelberg , 2008 ) ; r. carretero - gonzlez , d. j. frantzeskakis , and p. g. kevrekidis , nonlinearity * 21 * , r139 ( 2008 ) . | we study the coupling between backward- and forward - propagating wave modes , with the same group velocity , in a composite right / left - handed nonlinear transmission line . using an asymptotic multiscale expansion technique ,
we derive a system of two coupled nonlinear schrdinger equations governing the evolution of the envelopes of these modes .
we show that this system supports a variety of backward- and forward propagating vector solitons , of the bright - bright , bright - dark and dark - bright type .
performing systematic numerical simulations in the framework of the original lattice that models the transmission line , we study the propagation properties of the derived vector soliton solutions .
we show that all types of the predicted solitons exist , but differ on their robustness : only bright - bright solitons propagate undistorted for long times , while the other types are less robust , featuring shorter lifetimes . in all cases ,
our analytical predictions are in a very good agreement with the results of the simulations , at least up to times of the order of the solitons lifetimes .
= 1 |
You are an expert at summarizing long articles. Proceed to summarize the following text:
a fundamental idea behind classical statistical mechanics is that we are living in a macroscopic world , which means that we are unable to deal with microstates on a molecular level . this forces us to distinguish heat from work and to accept the increase of entropy on probabilistic ground , when we formulate the laws of thermodynamics . however , the border between our macroscopic world and the microscopic world of molecules is becoming vague due to technological developments over the last century . this has led to a natural question ; i.e. , what would it mean to thermodynamics if it became possible to access and manipulate microstates just as we do macrostates ? the founders of statistical mechanics were already aware of this problem : for example , maxwell imagined an intelligent being s intervention on a molecular level in his famous thought experiment , now known as maxwell s demon @xcite , and expressed deep concern about the foundation of the second law of thermodynamics . the szilard engine has been regarded as the simplest implementation of maxwell s demon with a single - particle gas @xcite . its cycle consists of four processes ; inserting a wall into the center of the box , measuring the position of the particle , expanding the quantum gas isothermally , and finally removing the wall . considering its microscopic nature , it is tempting to translate the cycle into quantum - mechanical language as has been done in refs . and for other engines . in fact , there are more reasons than that : zurek @xcite , when he discusses jauch and baron s objection that the gas is compressed without the expenditure of energy upon inserting the wall , argues that one has to resort to quantum mechanics to understand the szilard engine @xcite . according to ref . , this ` apparent inconsistency ' is removed by quantum - mechanical considerations . therefore , in a sense , it is a matter of theoretical consistency . kim and coworkers have presented a detailed account for its quantum - mechanical cycle in refs . , with emphasis on the isothermal process . there seem to remain some subtle issues to settle , however , as shown in the debate on how to deal with the quantum tunneling effect through the wall in a three - boson case @xcite . in this work , we describe the isothermal expansion of the quantum szilard engine by considering a quantum gas confined in a one - dimensional cylinder @xcite . although some previous studies , such as refs . , assume that the wall is impenetrable , we will relax that assumption because it is , strictly speaking , experimentally infeasible . starting from the schrdinger equation , we calculate the energy levels and thereby obtain how much work can be extracted by an isothermal reversible expansion . we find the following : the amount of reversible work is significantly reduced compared to the claims in refs . and if we assume full thermal equilibrium inside the box , as is usual in describing isothermal expansion . this conclusion should not be affected by either the wall s insertion or the measurement before the isothermal expansion because thermal equilibrium does not remember its history . moreover , if we put two or three fermions in the box , the free - energy landscape is not monotonic with respect to the wall s position , so that we should sometimes _ perform _ work to expand the gas . these effects have not been explicitly discussed in previous studies . this work is organized as follows : an explanation of our basic setting in terms of the schrdinger equation is given in section [ sec : isoexp ] . a numeric calculation of the energy levels and the free energy and a comparison of a single - particle case to the two- and three - particle cases is presented in in section [ sec : work ] . this is followed by discussion of results and conclusions . consider the following one - dimensional potential landscape with size @xmath0 : @xmath1 where @xmath2 is the strength of the delta potential and @xmath3 specifies its location @xcite . we expect the schrdinger equation @xmath4 \psi(x ) = e \psi(x ) \label{eq : schrod}\ ] ] to have a solution with a sinusoidal form . the boundary conditions at @xmath5 and @xmath6 are satisfied if we set @xmath7 where the coefficients @xmath8 and @xmath9 , as well as the wavenumber @xmath10 , are assumed to be nonzero . the eigenfunction @xmath11 has the following properties : first , it is continuous over @xmath12 $ ] ; i.e. , @xmath13 with @xmath14 . second , the change in the derivative of @xmath11 with respect to @xmath15 around @xmath16 is obtained by using @xmath17 dx + \frac{2m}{\hbar^2 } \left [ \int_{pl-\epsilon}^{pl+\epsilon } e \psi(x ) dx - \int_{pl-\epsilon}^{pl+\epsilon } a \psi(x ) \delta(x - pl ) dx \right ] \right\ } = 0,\ ] ] from which it follows that @xmath18 plugging eq . ( [ eq : psi ] ) here and multiplying both sides by @xmath19 to use eq . ( [ eq : cont ] ) , we obtain @xmath20 this formula has also been derived in ref . in a different way . equation ( [ eq : k ] ) can be numerically solved by using the newton - raphson method to yield the allowed values of @xmath10 . the method works as follows : first , let us define @xmath21 , and look for its zeros to solve eq . ( [ eq : k ] ) . starting from @xmath22 , we check whether the sign of @xmath23 changes with increasing @xmath10 by a sufficiently small amount , say , @xmath24 . every time the sign changes between @xmath25 and @xmath26 , we run the following iteration starting from @xmath27 : @xmath28}{f'[k^{(j)}]},\ ] ] where @xmath29 . when @xmath30 has converged to a stationary value @xmath31 , i.e. , if @xmath32 within numerical accuracy , we take it as a solution of eq . ( [ eq : k ] ) and repeat the procedure by checking the sign of @xmath23 from @xmath33 . each of these solutions is identified with the wavenumber @xmath34 of the @xmath35th eigenmode in an ascending order . for the application of the newton - raphson method numerically , it is convenient to express the quantities in dimensionless units , so we divide eq . ( [ eq : schrod ] ) by @xmath36 , the ground state energy for @xmath37 , to obtain @xmath38 \psi(\eta ) = \epsilon \psi(\eta),\ ] ] where @xmath39 . then , the potential inside the box is expressed as @xmath40 with @xmath41 . we can also define a dimensionless wavenumber @xmath42 to rewrite eqs . ( [ eq : k ] ) as @xmath43 . a little algebra shows that the energy level is given by @xmath44^{-1},\ ] ] where the second term represents the potential energy coming from the overlap between the wavefunction and the delta peak potential . the numerical solutions for these @xmath45 and @xmath46 are depicted in fig . [ fig : level5 ] , together with the probability density plots . as a function of @xmath47 , and ( b ) the corresponding energy levels in units of @xmath48 . the other two panels show the probability densities for the three lowest eigenmodes ( c ) when the wall is in the middle , @xmath49 , and ( d ) when it has moved to @xmath50 . the strength of the potential is set as @xmath51.,title="fig:",scaledwidth=45.0% ] as a function of @xmath47 , and ( b ) the corresponding energy levels in units of @xmath48 . the other two panels show the probability densities for the three lowest eigenmodes ( c ) when the wall is in the middle , @xmath49 , and ( d ) when it has moved to @xmath50 . the strength of the potential is set as @xmath51.,title="fig:",scaledwidth=45.0% ] as a function of @xmath47 , and ( b ) the corresponding energy levels in units of @xmath48 . the other two panels show the probability densities for the three lowest eigenmodes ( c ) when the wall is in the middle , @xmath49 , and ( d ) when it has moved to @xmath50 . the strength of the potential is set as @xmath51.,title="fig:",scaledwidth=45.0% ] as a function of @xmath47 , and ( b ) the corresponding energy levels in units of @xmath48 . the other two panels show the probability densities for the three lowest eigenmodes ( c ) when the wall is in the middle , @xmath49 , and ( d ) when it has moved to @xmath50 . the strength of the potential is set as @xmath51.,title="fig:",scaledwidth=45.0% ] we are interested in how much work can be extracted by performing a reversible isothermal process with this system . let @xmath52 denote the @xmath35th energy level obtained by using the newton - raphson method with @xmath53 so that the ground - state energy is denoted by @xmath54 . for a given value of temperature @xmath55 , where @xmath56 is the boltzmann constant , we run the calculation to get @xmath57 . the partition function for a single quantum particle can then be obtained as @xmath58 the free energy is @xmath59 , accompanied by a differential form @xmath60 , where @xmath61 , @xmath62 , and @xmath63 are the entropy , pressure , and volume , respectively . the classical concepts , such as pressure and work , need careful consideration in quantum mechanics @xcite . to calculate work , we begin with the occupation probability of the @xmath35th level given by @xmath64 the change in the internal energy @xmath65 is then expressed as @xmath66 , where @xmath67 and @xmath68 are differentials of heat and work , respectively . the quantum thermodynamic work is identified with @xmath69 because heat is usually involved with changes in occupation probabilities for given energy levels whereas work is performed when the energy levels themselves change @xcite . consequently , the amount of reversible work during this process is @xmath70 , which is consistent with the classical case . in units of @xmath48 when the wall , starting from the center of the box , moves to a new position @xmath47 . the height of the wall is fixed at @xmath71 , and each curve represents a different @xmath72 in units of @xmath73 . the number of energy levels considered is kept as @xmath74 throughout the free - energy calculations.,scaledwidth=45.0% ] figure [ fig : df ] shows @xmath75 resulting from moving the wall , which was initially located at the center , @xmath49 . suppose that @xmath72 is so low that @xmath76 . when the wall is at @xmath77 , only the ground state contributes to the summation in eq . ( [ eq : z ] ) , which means that @xmath78 . when the wall is at @xmath49 , the ground - state energy equals @xmath79 , and the first excited state also contributes to the partition sum in eq . ( [ eq : z ] ) because it lies close to the ground state [ see fig . [ fig : level5](b ) ] . the partition function is , therefore , approximated as @xmath80 . the free - energy difference in this low-@xmath72 region is , thus , given as @xmath81 . when @xmath82 , for example , the free - energy difference roughly amounts to @xmath83 , which explains the size of the free - energy drop in fig . [ fig : df ] . as @xmath72 increases , a plateau develops near the center of the box because @xmath84 does not respond much to the wall s position . however , the drop in @xmath84 at the end of the process increases with increasing @xmath72 if @xmath85 in units of @xmath86 . our question is how it grows with @xmath72 . let us take @xmath87 and @xmath88 . then , it is the partition function for the initial and final wall positions can be approximated . the latter case of the final wall position is estimated as @xmath89\label{eq : correction}\\ & = & \sqrt{\frac{\pi}{4\gamma } } - \frac{1}{2},\end{aligned}\ ] ] where the summation in eq . ( [ eq : correction ] ) is a first - order correction to the integral approximation . likewise , the former case of the initial wall position gives @xmath90 = \sqrt{\frac{\pi}{4\gamma } } - 1,\ ] ] where the factor of @xmath91 in front of the summation is due to the fact that every pair of adjacent energy levels becomes degenerate in the limit of @xmath87 . this effect cancels the factor of @xmath92 inside the exponential arising from the reduced volume @xmath93 . although @xmath75 does not completely vanish due to the correction , the important point is that @xmath94 for @xmath95 is far less than @xmath91 , in contrast with the claim that @xmath96 in refs . and . because @xmath97 must be finite in any experimental situation , the particle can , in full thermal equilibrium , be observed on either side of the box , which reduces the amount of extracted work . on the other hand , the previous studies on the quantum szilard engine @xcite have assumed that the process is performed within a shorter time scale than required for tunneling through the wall and , therefore , concluded that @xmath98 without the factor of @xmath91 in front . strictly speaking , their engine undergoes an isothermal process only when partially equilibrated to maintain the confinement and extract @xmath99 . if two quantum particles are in the box , we should consider their symmetry , i.e. , whether they are fermions or bosons . without any consideration of the spin degree of freedom , the elements of the density matrix for the two particles are expressed in the bracket notation as follows : @xmath100 , where @xmath101 is the kronecker delta symbol , and we have a plus sign for bosons and a minus sign for fermions . the two - particle partition function is then obtained by taking the trace operation : @xmath102 the same applies to the three - particle case . the density - matrix elements are written as @xmath103 , and the partition function reads @xmath104 figure [ fig : sym ] shows the free - energy differences for the two- and the three - particle cases . the bosonic cases are qualitatively similar to the single - particle case . however , we see very different behavior for the fermionic cases where the pauli exclusion principle is in action : at low @xmath72 , we need to perform positive work to move the wall from the center to a certain position @xmath47 [ figs . [ fig : sym](b ) and ( d ) ] . obviously , the reason is that it costs free energy to place two fermions close to each other . in summary , we have considered an isothermal expansion process of a quantum gas , taking the tunneling effect into consideration . we have found that the amount of reversible work is smaller than @xmath105 in the high-@xmath72 region if the system is fully equilibrated at every moment of the isothermal expansion . the difference from refs . and arises because they have separated the time scale of tunneling from that of the partial equilibration that occurs on only one side of the wall . although it was not explicitly stated in previous studies , this separation may be a plausible assumption for the following reason : as we increase the potential height @xmath97 , we may well expect the time scale for tunneling to grow whereas the partial equilibration before tunneling is achieved within a finite amount of time . the quantum szilard engine will show its expected performance only between these two time scales . the question is , then , how large a value of @xmath97 one should have to ensure the separation of the time scales , which will be pursued in our future studies . 17ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty @noop * * , ( ) in @noop _ _ , ( , , ) , pp . @noop * * , ( ) @noop * * , ( ) in @noop _ _ , , vol . , ( , , ) , pp . @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop _ _ ( , , ) , pp . @noop * * , ( ) | for an understanding of a heat engine working in the microscopic scale , it is often necessary to estimate the amount of reversible work extracted by isothermal expansion of the quantum gas used as its working substance .
we consider an engine with a movable wall , modeled as an infinite square well with a delta peak inside . by solving the resulting one - dimensional schrdinger equation
, we obtain the energy levels and the thermodynamic potentials .
our result shows how quantum tunneling degrades the engine by decreasing the amount of reversible work during the isothermal expansion . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
we present a detailed study on the nature of biases in network sampling strategies to shed light on how best to sample from networks . a _ network _ is a system of interconnected entities typically represented mathematically as a graph : a set of vertices and a set of edges among the vertices . networks are ubiquitous and arise across numerous and diverse domains . for instance , many web - based social media , such as online social networks , produce large amounts of data on interactions and associations among individuals . mobile phones and location - aware devices produce copious amounts of data on both communication patterns and physical proximity between people . in the domain of biology also , from neurons to proteins to food webs , there is now access to large networks of associations among various entities and a need to analyze and understand these data . with advances in technology , pervasive use of the internet , and the proliferation of mobile phones and location - aware devices , networks under study today are not only substantially larger than those in the past , but sometimes exist in a decentralized form ( e.g. the network of blogs or the web itself ) . for many networks , their global structure is not fully visible to the public and can only be accessed through `` crawls '' ( e.g. online social networks ) . these factors can make it prohibitive to analyze or even access these networks in their entirety . how , then , should one proceed in analyzing and mining these network data ? one approach to addressing these issues is _ sampling _ : inference using small subsets of nodes and links from a network . from epidemiological applications @xcite to web crawling @xcite and p2p search @xcite , network sampling arises across many different settings . in the present work , we focus on a particular line of investigation that is concerned with constructing samples that match critical structural properties of the original network . such samples have numerous applications in data mining and information retrieval . in @xcite , for example , structurally - representative samples were shown to be effective in inferring network protocol performance in the larger network and significantly improving the efficiency of protocol simulations . in section [ sec : applications ] , we discuss several additional applications . although there have been a number of recent strides in work on network sampling ( e.g. @xcite ) , there is still very much that requires better and deeper understanding . moreover , many networks under analysis , although treated as complete , are , in fact , _ samples _ due to limitations in data collection processes . thus , a more refined understanding of network sampling is of general importance to network science . towards this end , we conduct a detailed study on _ network sampling biases_. there has been a recent spate of work focusing on _ problems _ that arise from network sampling biases including how and why biases should be avoided @xcite . our work differs from much of this existing literature in that , for the first time in a comprehensive manner , we examine network sampling bias as an _ asset to be exploited_. we argue that biases of certain sampling strategies can be advantageous if they `` push '' the sampling process towards inclusion of specific properties of interest . our main aim in the present work is to identify and understand the connections between specific sampling biases and specific definitions of structural representativeness , so that these biases can be leveraged in practical applications . * summary of findings . * we conduct a detailed investigation of network sampling biases . we find that bias towards high _ expansion _ ( a concept from expander graphs ) offers several unique advantages over other biases such as those toward high degree nodes . we show both empirically and analytically that such an expansion bias `` pushes '' the sampling process towards new , undiscovered clusters and the discovery of wider portions of the network . in other analyses , we show that a simple sampling process that selects nodes with many connections from those already sampled is often a reasonably good approximation to directly sampling high degree nodes and locates well - connected ( i.e. high degree ) nodes significantly faster than most other methods . we also find that the breadth - first search , a widely - used sampling and search strategy , is surprisingly among the most dismal performers in terms of both discovering the network and accumulating critical , well - connected nodes . finally , we describe ways in which some of our findings can be exploited in several important applications including disease outbreak detection and market research . a number of these aforementioned findings are surprising in that they are in stark contrast to conventional wisdom followed in much of the existing literature ( e.g. @xcite ) . not surprisingly , network sampling arises across many diverse areas . here , we briefly describe some of these different lines of research . * network sampling in classical statistics . * the concept of sampling networks first arose to address scenarios where one needed to study hidden or difficult - to - access populations ( e.g. illegal drug users , prostitutes ) . for recent surveys , one might refer to @xcite . the work in this area focuses almost exclusively on acquiring unbiased estimates related to variables of interest attached to each network node . the present work , however , focuses on inferring properties related to the _ network itself _ ( many of which are not amenable to being fully captured by simple attribute frequencies ) . our work , then , is much more closely related to _ representative subgraph sampling_. * representative subgraph sampling . * in recent years , a number of works have focused on _ representative subgraph sampling _ : constructing samples in such a way that they are condensed representations of the original network ( e.g. @xcite ) . much of this work focuses on how best to produce a `` universal '' sample representative of _ all _ structural properties in the original network . by contrast , we subscribe to the view that no single sampling strategy may be appropriate for all applications . thus , our aim , then , is to better understand the _ biases _ in specific sampling strategies to shed light on how best to leverage them in practical applications . * unbiased sampling . * there has been a relatively recent spate of work ( e.g. @xcite ) that focuses on constructing uniform random samples in scenarios where nodes can not be easily drawn randomly ( e.g. settings such as the web where nodes can only be accessed through crawls ) . these strategies , often based on modified random walks , have been shown to be effective for various frequency estimation problems ( e.g. inferring the proportion of pages of a certain language in a web graph @xcite ) . however , as mentioned above , the present work focuses on using samples to infer structural ( and functional ) properties of the _ network itself_. in this regard , we found these unbiased methods to be less effective during preliminary testing . thus , we do not consider them and instead focus our attention on other more appropriate sampling strategies ( such as those mentioned in _ representative subgraph sampling _ ) . * studies on sampling bias . * several studies have investigated _ biases _ that arise from various sampling strategies ( e.g. @xcite ) . for instance , @xcite showed that , under the simple sampling strategy of picking nodes at random from a scale - free network ( i.e. a network whose degree distribution follows the power law ) , the resultant subgraph sample will _ not _ be scale - free . the authors of @xcite showed the converse is true under traceroute sampling . virtually all existing results on network sampling bias focus on its negative aspects . by contrast , we focus on the _ advantages _ of certain biases and ways in which they can be exploited in network analysis . * property testing . * work on sampling exists in the fields of combinatorics and graph theory and is centered on the notion of _ property testing _ in graphs @xcite . properties such as those typically studied in graph theory , however , may be less useful for the analysis of _ real - world _ networks ( e.g. the exact meaning of , say , @xmath0-colorability @xcite within the context of a social network is unclear ) . nevertheless , theoretical work on property testing in graphs is excellently surveyed in @xcite . * other areas . * decentralized search ( e.g. searching unstructured p2p networks ) and web crawling can both be framed as network sampling problems , as both involve making decisions from subsets of nodes and links from a larger network . indeed , network sampling itself can be viewed as a problem of information retrieval , as the aim is to seek out a subset of nodes that either individually or collectively match some criteria of interest . several of the sampling strategies we study in the present work , in fact , are graph search algorithms ( e.g. breadth - first search ) . thus , a number of our findings discussed later have implications for these research areas ( e.g. see @xcite ) . for reviews on decentralized search both in the contexts of complex networks and p2p systems , one may refer to @xcite and @xcite , respectively . for examples of connections between web crawling and network sampling , see @xcite . we now briefly describe some notations and definitions used throughout this paper . [ defn : network ] @xmath1 is a _ network _ or _ graph _ where @xmath2 is set of vertices and @xmath3 is a set of edges . [ defn : sample ] a _ sample _ @xmath4 is a subset of vertices , @xmath5 . [ defn : neighborhood ] @xmath6 is the _ neighborhood _ of @xmath4 if @xmath7 . [ defn : inducedsubgraph ] @xmath8 is the _ induced subgraph _ of @xmath9 based on the sample @xmath4 if @xmath10 where the vertex set is @xmath5 and the edge set is @xmath11 . the induced subgraph of a sample may also be referred to as a _ subgraph sample_. we study sampling biases in a total of twelve different networks : a power grid ( powergrid @xcite ) , a wikipedia voting network ( wikivote @xcite ) , a pgp trust network ( pgp @xcite ) , a citation network ( hepth @xcite ) , an email network ( enron @xcite ) , two co - authorship networks ( condmat @xcite and astroph @xcite ) , two p2p file - sharing networks ( gnutella04 @xcite and gnutella31 @xcite ) , two online social networks ( epinions @xcite and slashdot @xcite ) , and a product co - purchasing network ( amazon @xcite ) . these datasets were chosen to represent a rich set of diverse networks from different domains . this diversity allows a more comprehensive study of network sampling and thorough assessment of the performance of various sampling strategies in the face of varying network topologies . table [ tab : datasets ] shows characteristics of each dataset . all networks are treated as undirected and unweighted . [ th ] .network properties . * key : * _ n= # of nodes , d= density , pl = characteristic path length , cc = local clustering coefficient , ad = average degree . _ [ cols="<,^,^,^,^,^",options="header " , ] -0.15 in in the present work , we focus on a particular class of sampling strategies , which we refer to as _ link - trace sampling_. in _ link - trace sampling _ , the next node selected for inclusion into the sample is always chosen from among the set of nodes directly connected to those already sampled . in this way , sampling proceeds by tracing or following links in the network . this concept can be defined formally . [ defn : linktracesampling ] given an integer @xmath0 and an initial node ( or seed ) @xmath12 to which @xmath4 is initialized ( i.e. @xmath13 ) , a _ link - trace sampling _ algorithm , @xmath14 , is a process by which nodes are iteratively selected from among the current neighborhood @xmath6 and added to @xmath4 until @xmath15 . _ link - trace sampling _ may also be referred to as _ crawling _ ( since links are `` crawled '' to access nodes ) or viewed as _ online _ sampling ( since the network @xmath9 reveals itself iteratively during the course of the sampling process ) . the key advantage of sampling through link - tracing , then , is that complete access to the network in its entirety is _ not _ required . this is beneficial for scenarios where the network is either large ( e.g. an online social network ) , decentralized ( e.g. an unstructured p2p network ) , or both ( e.g. the web ) . as an aside , notice from definition [ defn : linktracesampling ] that we have implicitly assumed that the neighbors of a given node can be obtained by visiting that node during the sampling process ( i.e. @xmath6 is known ) . this , of course , accurately characterizes most real scenarios . for instance , neighbors of a web page can be gleaned from the hyperlinks on a visited page and neighbors of an individual in an online social network can be acquired by viewing ( or `` scraping '' ) the friends list . having provided a general definition of _ link - trace sampling _ , we must now address _ which _ nodes in @xmath6 should be preferentially selected at each iteration of the sampling process . this choice will obviously directly affect the properties of the sample being constructed . we study seven different approaches - all of which are quite simple yet , at the same time , ill - understood in the context of real - world networks . * breadth - first search ( bfs ) . * starting with a single seed node , the bfs explores the neighbors of visited nodes . at each iteration , it traverses an unvisited neighbor of the _ earliest _ visited node @xcite . in both @xcite and @xcite , it was empirically shown that bfs is biased towards high - degree and high - pagerank nodes . bfs is used prevalently to crawl and collect networks ( e.g. @xcite ) . * depth - first search ( dfs ) . * dfs is similar to bfs , except that , at each iteration , it visits an unvisited neighbor of the most _ recently _ visited node @xcite . * random walk ( rw ) . * a random walk simply selects the next hop uniformly at random from among the neighbors of the current node @xcite . * forest fire sampling ( ffs ) . * ffs , proposed in @xcite , is essentially a probabilistic version of bfs . at each iteration of a bfs - like process , a neighbor @xmath16 is only explored according to some `` burning '' probability @xmath17 . at @xmath18 , ffs is identical to bfs . we use @xmath19 , as recommended in @xcite . * degree sampling ( ds ) . * the ds strategy involves greedily selecting the node @xmath20 with the highest degree ( i.e. number of neighbors ) . a variation of ds was analytically and empirically studied as a p2p search algorithm in @xcite . notice that , in order to select the node @xmath21 with the highest degree , the process must know @xmath22 for each @xmath20 . that is , knowledge of @xmath23 is required at each iteration . as noted in @xcite , this requirement is acceptable for some domains such as p2p networks and certain social networks . the ds method is also feasible in scenarios where 1 ) one is interested in efficiently `` downsampling '' a network to a connected subgraph , 2 ) a crawl is repeated and history of the last crawl is available , or 3 ) the proportion of the network accessed to construct a sample is less important . * sec ( sample edge count ) . * given the currently constructed sample @xmath4 , how can we select a node @xmath20 with the highest degree _ without _ having knowledge of @xmath23 ? the sec strategy tracks the links from the currently constructed sample @xmath4 to each node @xmath20 and selects the node @xmath16 with the most links from @xmath4 . in other words , we use the degree of @xmath16 in the induced subgraph of @xmath24 as an approximation of the degree of @xmath16 in the original network @xmath9 . similar approaches have been employed as part of web crawling strategies with some success ( e.g. @xcite ) . * xs ( expansion sampling ) . * the xs strategy is based on the concept of expansion from work on expander graphs and seeks to greedily construct the sample with the maximal expansion : @xmath25 , where @xmath0 is the desired sample size @xcite . at each iteration , the next node @xmath16 selected for inclusion in the sample is chosen based on the expression : @xmath26 like the ds strategy , this approach utilizes knowledge of @xmath23 . in sections [ sec : rep.reach ] and [ sec : biases.xs ] , we will investigate in detail the effect of this expansion bias on various properties of constructed samples . what makes one sampling strategy `` better '' than another ? in computer science , `` better '' is typically taken to be structural _ representativeness _ ( e.g. see @xcite ) . that is , samples are considered better if they are more representative of structural properties in the original network . there are , of course , numerous structural properties from which to choose , and , as correctly observed by ahmed et al . @xcite , it is not always clear which should be chosen . rather than choosing arbitrary structural properties as measures of representativeness , we select specific measures of representativeness that we view as being potentially useful for real applications . we divide these measures ( described below ) into three categories : degree , clustering , and reach . for each sampling strategy , we generate 100 samples using randomly selected seeds , compute our measures of representativeness on each sample , and plot the average value as sample size grows . ( standard deviations of computed measures are discussed in section [ sec : rep.seedsensitivity ] . applications for these measures of representativeness are discussed later in section [ sec : applications ] . ) due to space limitations and the large number of networks evaluated , for each evaluation measure , we only show results for two datasets that are illustrative of general trends observed in all datasets . however , full results are available as supplementary material . the degrees ( numbers of neighbors ) of nodes in a network is a fundamental and well - studied property . in fact , other graph - theoretic properties such as the average path length between nodes can , in some cases , be viewed as byproducts of degree ( e.g. short paths arising from a small number of highly - connected hubs that act as conduits @xcite ) . we study two different aspects of degree ( with an eye towards real - world applications , discussed in section [ sec : applications ] ) . * degree distribution similarity ( distsim ) . * we take the degree sequence of the sample and compare it to that of the original network using the two - sample kolmogorov - smirnov ( k - s ) d - statistic @xcite , a distance measure . our objective here is to measure the agreement between the two degree distributions in terms of both shape and location . specifically , the d - statistic is defined as @xmath27 , where @xmath28 is the range of node degrees , and @xmath29 and @xmath30 are the cumulative degree distributions for @xmath9 and @xmath8 , respectively @xcite . we compute the distribution similarity by subtracting the k - s distance from one . * hub inclusion ( hubs ) . * in several applications , one cares less about matching the _ overall _ degree distribution and more about accumulating the highest degree nodes into the sample quickly ( e.g. immunization strategies @xcite ) . for these scenarios , sampling is used as a tool for information retrieval . here , we evaluate the extent to which sampling strategies accumulate hubs ( i.e. high degree nodes ) quickly into the sample . as sample size grows , we track the proportion of the top @xmath31 nodes accumulated by the sample . for our tests , we use @xmath32 . figure [ fig : rep.degree ] shows the _ degree distribution similarity _ ( distsim ) and _ hub inclusion _ ( hubs ) for the slashdot and enron datasets . note that the sec and ds strategies , both of which are biased to high degree nodes , perform best on _ hub inclusion _ ( as expected ) , but are the _ worst _ performers on the distsim measure ( which is also a direct result of this bias ) . ( the xs strategy exhibits a similar trend but to a slightly lesser extent . ) on the other hand , strategies such as bfs , ffs , and rw tend to perform better on distsim , but worse on hubs . for instance , the ds and sec strategies locate the majority of the top 100 hubs with sample sizes less than @xmath33 in some cases . bfs and ffs require sample sizes of over @xmath34 ( and the performance differential is larger when locating hubs ranked higher than @xmath35 ) . more importantly , no strategy performs best on _ both _ measures . this , then , suggests a tension between goals : constructing small samples of the most well - connected nodes is in conflict with producing small samples exhibiting representative degree distributions . more generally , when selecting sample elements , choices resulting in gains for one area can result in losses for another . thus , these choices must be made in light of how samples will be used - a subject we discuss in greater depth in section [ sec : applications ] . we conclude this section by briefly noting that the trend observed for sec seems to be somewhat dependent upon the quality and number of hubs actually present in a network ( relative to the size of the network , of course ) . that is , sec matches ds more closely as degree distributions exhibit longer and denser tails ( as shown in figure [ fig : rep.dd ] ) . we will revisit this in section [ sec : biases.sec ] . ( other strategies are sometimes affected similarly , but the trend is much less consistent . ) in general , we find sec best matches ds performance on many of the social networks ( as opposed to technological networks such as the powergrid with few `` good '' hubs , lower average degree , and longer path lengths ) . however , further investigation is required to draw firm conclusions on this last point . + -0.01 in -0.15 in + -0.01 in -0.15 in many real - world networks , such as social networks , exhibit a much higher level clustering than what one would expect at random @xcite . thus , clustering has been another graph property of interest for some time . here , we are interested in evaluating the extent to which samples exhibit the level of clustering present in the original network . we employ two notions of clustering , which we now describe . * local clustering coefficient ( ccloc ) . * the local clustering coefficient @xcite of a node captures the extent to which the node s neighbors are also neighbors of each other . formally , the local clustering coefficient of a node is defined as @xmath36 where @xmath37 is the degree of node @xmath16 and @xmath38 is the number of links among the neighbors of @xmath16 . the average local clustering coefficient for a network is simply @xmath39 . * global clustering coefficient ( ccglb ) . * the global clustering coefficient @xcite is a function of the number of triangles in a network . it is measured as the number of closed triplets divided by the number of connected triples of nodes . results for clustering measures are less consistent than for other measures . overall , dfs and rw strategies appear to fare relatively better than others . we do observe that , for many strategies and networks , estimates of clustering are initially higher - than - actual and then gradually decline ( see figure [ fig : rep.clustering ] ) . this agrees with intuition . nodes in clusters should intuitively have more paths leading to them and will , thus , be encountered earlier in a sampling process ( as opposed to nodes not embedded in clusters and located in the periphery of a network ) . this , then , should be taken into consideration in applications where accurately matching clustering levels is important . + -0.01 in -0.15 in we propose a new measure of representativeness called _ network reach_. as a newer measure , _ network reach _ has obviously received considerably less attention than degree and clustering within the existing literature , but it is , nevertheless , a vital measure for a number of important applications ( as we will see in section [ sec : applications ] ) . _ network reach _ captures the extent to which a sample _ covers _ a network . intuitively , for a sample to be truly representative of a large network , it should consist of nodes from diverse portions of the network , as opposed to being relegated to a small `` corner '' of the graph . this concept will be made more concrete by discussing in detail the two measures of _ network reach _ we employ : _ community reach _ and the _ discovery quotient_. * community reach ( cnm and rak ) . * many real - world networks exhibit what is known as _ community structure_. a _ community _ can be loosely defined as a set of nodes more densely connected among themselves than to other nodes in the network . although there are many ways to represent community structure depending on various factors such as whether or not overlapping is allowed , in this work , we represent community structure as a _ partition _ : a collection of disjoint subsets whose union is the vertex set @xmath2 @xcite . under this representation , each subset in the partition represents a community . the task of a community detection algorithm is to identify a partition such that vertices within the same subset in the partition are more densely connected to each other than to vertices in other subsets @xcite . for the criterion of _ community reach _ , a sample is more representative of the network if it consists of nodes from more of the communities in the network . we measure _ community reach _ by taking the number of communities represented in the sample and dividing by the total number of communities present in the original network . since a community is essentially a cluster of nodes , one might wonder why we have included _ community reach _ as a measure of _ network reach _ , rather than as a measure of _ clustering_. the reason is that we are slightly less interested in the structural details of communities detected here . rather , our aim is to assess how `` spread out '' a sample is across the network . since community detection is somewhat of an inexact science ( e.g. see @xcite ) , we measure _ community reach _ with respect to two separate algorithms . we employ both the method proposed by clauset et al . in @xcite ( denoted as cnm ) and the approach proposed by raghavan et al . in @xcite ( denoted as rak ) . essentially , for our purposes , we are defining communities simply as the output of a community detection algorithm . * discovery quotient ( dq ) . * an alternative view of _ network reach _ is to measure the proportion of the network that is _ discovered _ by a sampling strategy . the number of nodes discovered by a strategy is defined as @xmath40 . the _ discovery quotient _ is this value normalized by the total number of nodes in a network : @xmath41 . intuitively , we are defining the _ reach _ of a sample here by measuring the extent to which it is one hop away from the rest of the network . as we will discuss in section [ sec : applications ] , samples with high _ discovery quotients _ have several important applications . note that a simple greedy algorithm for coverage problems such as this has a well - known sharp approximation bound of @xmath42 @xcite . however , link - trace sampling is restricted to selecting subsequent sample elements from the current neighborhood @xmath6 at each iteration , which results in a much smaller search space . thus , this approximation guarantee can be shown not to hold within the context of link - trace sampling . as shown in figure [ fig : rep.reach ] , the xs strategy displays the overwhelmingly best performance on all three measures of _ network reach_. we highlight several observations here . first , the extent to which the xs strategy outperforms all others on the rak and cnm measures is quite striking . we posit that the expansion bias of the xs strategy `` pushes '' the sampling process towards the inclusion of new communities not already seen ( see also @xcite ) . in section [ sec : biases.xs ] , we will analytically examine this connection between expansion bias and _ community reach_. on the other hand , the sec method appears to be among the least effective in reaching different communities or clusters . we attribute this to the fact that sec preferentially selects nodes with many connections to nodes already sampled . such nodes are likely to be members of clusters already represented in the sample . second , on the dq measure , it is surprising that the ds strategy , which explicitly selects high degree nodes , often fails to even come close to the xs strategy . we partly attribute this to an overlap in the neighborhoods of well - connected nodes . by explicitly selecting nodes that contribute to _ expansion _ , the xs strategy is able to discover a much larger proportion of the network in the same number of steps - in some cases , by actively sampling comparatively _ lower _ degree nodes . finally , it is also surprising that the bfs strategy , widely used to crawl and explore online social networks ( e.g @xcite ) and other graphs ( e.g. @xcite ) , performs quite dismally on all three measures . in short , we find that nodes contributing most to the expansion of the sample are unique in that they provide specific and significant advantages over and above those provided by nodes that are simply well - connected and those accumulated through standard bfs - based crawls . these and previously mentioned results are in contrast to the conventional wisdom followed in much of the existing literature ( e.g. @xcite ) . + -0.01 in + -0.01 in -0.15 in as described , link - trace sampling methods are initiated from randomly selected seeds . this begs the question : how sensitive are these results to the seed supplied to a strategy ? figure [ fig : std ] shows the standard deviation of each sampling strategy for both _ hub inclusion _ and _ network reach _ as sample size grows . we generally find that methods with the most explicit biases ( xs , sec , ds ) tend to exhibit the least seed sensitivity and variability , while the remaining methods ( bfs , dfs , ffs , rw ) exhibit the most . this trend is exhibited across all measures and all datasets . let us briefly summarize two main observations from section [ sec : rep ] . we saw that the xs strategy dramatically outperformed all others in accumulating nodes from many different communities . we also saw that the sec strategy was often a reasonably good approximation to directly sampling high degree nodes and locates the set of most well - connected nodes significantly faster than most other methods . here , we turn our attention to analytically examining these observed connections . we begin by briefly summarizing some existing analytical results . * random walks ( rw ) . * there is a fairly large body of research on random walks and markov chains ( see @xcite for an excellent survey ) . a well - known analytical result states that the probability ( or _ stationary _ probability ) of residing at any node @xmath16 during a random walk on a connected , undirected graph converges with time to @xmath43 , where @xmath44 is the degree of node @xmath16 @xcite . in fact , the _ hitting time _ of a random walk ( i.e. the expected number of steps required to reach a node beginning from any node ) has been analytically shown to be directly related to this stationary probability @xcite . random walks , then , are naturally biased towards high degree ( and high pagerank ) nodes , which provides some theoretical explanation as to why rw performs slightly better than other strategies ( e.g. bfs ) on measures such as _ hub inclusion_. however , as shown in figure [ fig : rep.degree ] , it is nowhere near the best performers . thus , these analytical results appear only to hold in the limit and fail to predict actual sampling performance . * degree sampling ( ds ) . * in studying the problem of searching peer - to - peer networks , adamic et al . @xcite proposed and analyzed a greedy search strategy very similar to the ds sampling method . this strategy , which we refer to as a degree - based walk , was analytically shown to quickly find the highest - degree nodes and quickly cover large portions of scale - free networks . thus , these results provide a theoretical explanation for performance of the ds strategy on measures such as _ hub inclusion _ and the _ discovery quotient_. * other results . * as mentioned in section [ sec : relatedwork ] , to the best of our knowledge , much of the other analytical results on sampling bias focus on _ negative _ results @xcite . thus , these works , although intriguing , may not provide much help in the way of explaining _ positive _ results shown in section [ sec : rep ] . + we now analyze two methods for which there are little or no existing analytical results : xs and sec . a widely used measure for the `` goodness '' or the strength of a community in graph clustering and community detection is _ conductance _ @xcite , which is a function of the fraction of total edges emanating from a sample ( lower values mean stronger communities ) : @xmath45 where @xmath46 are entries of the adjacency matrix representing the graph and @xmath47 , which is the total number of edges incident to the node set @xmath4 . it can be shown that , provided the conductance of communities is sufficiently low , sample expansion is directly affected by community structure . consider a simple random graph model with vertex set @xmath2 and a community structure represented by partition @xmath48 where @xmath49 . let @xmath50 and @xmath51 be the number of each node s edges pointing within and outside the node s community , respectively . these edges are connected uniformly at random to nodes either within or outside a node s community , similar to a configuration model ( e.g. , @xcite ) . note that both @xmath50 and @xmath51 are related directly to conductance . when conductance is lower , @xmath51 is smaller is @xmath52 , the total number of edges incident to @xmath53 is @xmath54 , and @xmath50 and @xmath51 are random variables denoting the inward and outward edges , respectively , of each node ( as opposed to constant values ) . then , @xmath55 and @xmath56 . if @xmath57 , then @xmath58 . ( in this example , the expectations are over nodes in @xmath53 only . ) ] as compared to @xmath50 . the following theorem expresses the link between expansion and _ community reach _ in terms of these inward and outward edges . [ thm : xsbias ] let @xmath4 be the current sample , @xmath16 be a new node to be added to @xmath4 , and @xmath59 be the size of @xmath16 s community . if @xmath60 , then the expected expansion of @xmath24 is higher when @xmath16 is in a new community than when @xmath16 is in a current community . let @xmath61 be the expected value for @xmath62 when @xmath16 is in a new community and let @xmath63 be the expected value when not . we compute an upper bound on @xmath63 and a lower bound on @xmath61 . + deriving @xmath63 : assume @xmath16 is affiliated with a current community already represented by at least one node in @xmath4 . since we are computing an upper bound on @xmath63 , we assume there is exactly one node from @xmath4 within @xmath16 s community , as this is the minimum for @xmath16 s community to be a _ current _ community . by the linearity of expectations , the upper bound on @xmath63 is @xmath64 , where the term @xmath65 is the expected number of nodes in @xmath16 s community that are both linked to @xmath16 _ and _ in the set @xmath66 . + deriving @xmath61 : assume @xmath16 belongs to a new community not already represented in @xmath4 . ( by definition , no nodes in @xmath4 will be in @xmath16 s community . ) applying the linearity of expectations once again , the lower bound on @xmath61 is @xmath67 , where the term @xmath68 is the expected number of nodes in @xmath16 s community that are both linked to @xmath16 _ and _ already in @xmath6 . + solving for @xmath51 , if @xmath60 , then @xmath69 . theorem [ thm : xsbias ] shows analytically the link between expansion and community structure - a connection that , until now , has only been empirically demonstrated @xcite . thus , a theoretical basis for performance of the xs strategy on _ community reach _ is revealed . recall that the sec method uses the degree of a node @xmath16 in the induced subgraph @xmath70 as an estimation for the degree of @xmath16 in @xmath9 . in section [ sec : rep ] , we saw that this choice performs quite well in practice . here , we provide theoretical justification for the sec heuristic . consider a random network @xmath9 with some arbitrary expected degree sequence ( e.g. a power law random graph under the so - called @xmath71 model @xcite ) and a sample @xmath5 . let @xmath72 be a function that returns the expected degree of a given node in a given random network ( see @xcite for more information on _ expected _ degree sequences ) . then , it is fairly straightforward to show the following holds . [ prop : secbias ] for any two nodes @xmath73 , + if @xmath74 , then @xmath75 . the probability of an edge between any two nodes @xmath76 and @xmath77 in g is @xmath78 where @xmath79 . let @xmath80 . then , @xmath81 since @xmath82 only when @xmath83 , the proposition holds . combining proposition [ prop : secbias ] with analytical results from @xcite ( described in section [ sec : biases.existing ] ) provides a theoretical basis for observed performance of the sec strategy on measures such as _ hub inclusion_. finally , recall from section [ sec : rep.degree.results ] that the extent to which sec matched the performance of ds on hubs seemed to partly depend on the tail of degree distributions . proposition [ prop : secbias ] also yields insights into this phenomenon . longer and denser tails allow for more `` slack '' when deviating from these expectations of random variables ( as in real - world link patterns that are not purely random ) . we now briefly describe ways in which some of our findings may be exploited in important , real - world applications . although numerous potential applications exist , we focus here on three areas : 1 ) outbreak detection 2 ) landmarks and graph exploration 3 ) marketing . what is the most effective and efficient way to predict and prevent a disease outbreak in a social network ? in a recent paper , christakis and fowler studied outbreak detection of the h1n1 flu among college students at harvard university @xcite . previous research has shown that well - connected ( i.e. high degree ) people in a network catch infectious diseases earlier than those with fewer connections @xcite . thus , _ monitoring _ these individuals allows forecasting the progression of the disease ( a boon to public health officials ) and _ immunizing _ these well - connected individuals ( when immunization is possible ) can prevent or slow further spread . unfortunately , identifying well - connected individuals in a population is non - trivial , as access to their friendships and connections is typically not fully available . and , collecting this information is time - consuming , prohibitively expensive , and often impossible for large networks . matters are made worse when realizing that most existing network - based techniques for immunization selection and outbreak detection assume full knowledge of the global network structure ( e.g. @xcite ) . this , then , presents a prime opportunity to exploit the power of _ sampling_. to identify well - connected students and predict the outbreak , christakis and fowler @xcite employed a sampling technique called _ acquaintance sampling _ ( acq ) based on the so - called friendship paradox @xcite . the idea is that random neighbors of randomly selected nodes in a network will tend to be highly - connected @xcite . christakis and fowler @xcite , therefore , sampled random friends of randomly selected students with the objective of constructing a sample of highly - connected individuals . based on our aforementioned results , we ask : can we do better than this acq strategy ? in previous sections , we showed empirically and analytically that the sec method performs exceedingly well in accumulating hubs . ( it also happens to require less information than ds and xs , the other top performers . ) figure [ fig : outdet ] shows the sample size required to locate the top - ranked well - connected individuals for both sec and acq . the performance differential is quite remarkable , with the sec method faring overwhelmingly better in quickly zeroing in on the set of most well - connected nodes . aside from its superior performance , sec has one additional advantage over the acq method employed by christakis and fowler . the acq method assumes that nodes in @xmath2 can be selected uniformly at random . it is , in fact , dependent on this @xcite . ( acq , then , is _ not _ a link - trace sampling method . ) by contrast , sec , as a pure link - trace sampling strategy , has no such requirement and , thus , can be applied in realistic scenarios for which acq is unworkable . -0.15 in recall from section [ sec : rep.reach ] that a community in a network is a cluster of nodes more densely connected among themselves than to others . identifying communities is important , as they often correspond to real social groups , functional groups , or similarity ( both demographic and not ) @xcite . the ability to easily construct a sample consisting of members from diverse groups has several important applications in marketing . marketing surveys often seek to construct stratified samples that collectively represent the diversity of the population @xcite . if the attributes of nodes are not known in advance , this can be challenging . the xs strategy , which exhibited the best _ community reach _ , can potentially be very useful here . moreover , it has the added power of being able to locate members from diverse groups with absolutely no _ a priori _ knowledge of demographics attributes , social variables , or the overall community structure present in the network . there is also recent evidence to suggest that being able to construct a sample from many different communities can be an asset in effective word - of - mouth marketing @xcite . this , then , represents yet another potential marketing application for the xs strategy . _ landmark - based methods _ represent a general class of algorithms to compute distance - based metrics in large networks quickly @xcite . the basic idea is to select a small sample of nodes ( i.e. the landmarks ) , compute offline the distances from these landmarks to every other node in the network , and use these pre - computed distances at runtime to approximate distances between pairs of nodes . as noted in @xcite , for this approach to be effective , landmarks should be selected so that they _ cover _ significant portions of the network . based on our findings for _ network reach _ in section [ sec : rep.reach ] , the xs strategy overwhelmingly yields the best _ discovery quotient _ and covers the network significantly better than any other strategy . thus , it represents a promising landmark selection strategy . our results for the _ discovery quotient _ and other measures of _ network reach _ also yield important insights into how graphs should best be explored , crawled , and searched . as shown in figure [ fig : rep.reach ] , the most prevalently used method for exploring networks , bfs , ranks low on measures of _ network reach_. this suggests that the bfs and its pervasive use in social network data acquisition and exploration ( e.g. see @xcite ) should possibly be examined more closely . we have conducted a detailed study on sampling biases in real - world networks . in our investigation , we found the bfs , a widely - used method for sampling and crawling networks , to be among the worst performers in both discovering the network and accumulating critical , well - connected hubs . we also found that sampling biases towards high expansion tend to accumulate nodes that are uniquely different from those that are simply well - connected or traversed during a bfs - based strategy . these high - expansion nodes tend to be in newer and different portions of the network not already encountered by the sampling process . we further demonstrated that sampling nodes with many connections from those already sampled is a reasonably good approximation to sampling high degree nodes . finally , we demonstrated several ways in which these findings can be exploited in real - world application such as disease outbreak detection and marketing . for future work , we intend to investigate ways in which the top - performing sampling strategies can be enhanced for even wider applicability . one such direction is to investigate the effects of alternating or combining different biases into a single sampling strategy . | from social networks to p2p systems , network sampling arises in many settings .
we present a detailed study on the nature of biases in network sampling strategies to shed light on how best to sample from networks .
we investigate connections between specific biases and various measures of structural representativeness .
we show that certain biases are , in fact , beneficial for many applications , as they `` push '' the sampling process towards inclusion of desired properties .
finally , we describe how these sampling biases can be exploited in several , real - world applications including disease outbreak detection and market research .
[ data mining ] |
You are an expert at summarizing long articles. Proceed to summarize the following text:
in this paper we establish a relation between the size of the fatou components of a semi - hyperbolic rational map , and the hausdorff dimension of the julia set . before formulating the results , we first discuss some background . a rational map @xmath0 of degree at least @xmath1 is _ semi - hyperbolic _ if it has no parabolic cycles , and all critical points in its julia set @xmath2 are _ non - recurrent_. we say that a point @xmath3 is non - recurrent if @xmath4 , where @xmath5 is the set of accumulation points of the orbit @xmath6 of @xmath3 . in our setting , we require that the julia set @xmath2 is not the whole sphere , it is connected and , in addition , there are infinitely many fatou components . let @xmath7 be the sequence of fatou components , and define @xmath8 . since @xmath2 is connected , it follows that each component @xmath9 is simply connected , and thus @xmath10 is connected . we say that the collection @xmath11 is a _ packing _ @xmath12 and we define the _ curvature distribution function _ associated to @xmath12 ( see below for motivation of this terminology ) by @xmath13 for @xmath14 . also , the _ exponent _ @xmath15 of the packing @xmath12 is defined by @xmath16 where all diameters are in the spherical metric of @xmath17 . in the following , we write @xmath18 if there exists a constant @xmath19 such that @xmath20 . if only one if these inequalities is true , we write @xmath21 or @xmath22 respectively . we denote the hausdorff dimension of a set @xmath23 by @xmath24 ( see section [ section - minkowski ] ) we now state our main result . [ theorem - main ] let @xmath0 be a semi - hyperbolic rational map such that the julia set @xmath2 is connected and the fatou set has infinitely many components . then @xmath25 where @xmath26 is the curvature distribution function of the packing of the fatou components of @xmath27 and @xmath28 . in particular @xmath29 . it is remarkable that the curvature distribution function has polynomial growth . as a consequence , we have the following corollary . [ corollary ] under the assumptions of theorem [ theorem - main ] we have @xmath30 where @xmath26 is the curvature distribution function , and @xmath15 is the exponent of the packing of the fatou components of @xmath27 . this essentially says that one can compute the hausdorff dimension of the julia set just by looking at the diameters of the ( countably many ) fatou components , which lie in the complement of the julia set . the study of the curvature distribution function and the terminology is motivated by the _ apollonian circle packings_. an apollonian circle packing is constructed inductively as follows . let @xmath31 be three mutually tangent circles in the plane with disjoint interiors . then by a theorem of apollonius there exist exactly two circles that are tangent to all three of @xmath31 . we denote by @xmath32 the outer circle that is tangent to @xmath31 ( see figure [ fig : apollonian ] ) . for the inductive step we apply apollonius s theorem to all triples of mutually tangent circles of the previous step . in this way , we obtain a countable collection of circles @xmath11 . we denote by @xmath33 the apollonian circle packing constructed this way . if @xmath34 denotes the radius of @xmath10 , then @xmath35 is the curvature of @xmath10 . the curvatures of the circles in apollonian packings are of great interest in number theory because of the fact that if the four initial circles @xmath36 have integer curvatures , then so do all the rest of the circles in the packing . another interesting fact is that if the curvatures of the circles are relatively prime integers , then there are infinitely many circles in the packing with curvature that is a prime number . for a survey on the topic see @xcite . in order to study the curvatures of an apollonian packing @xmath12 one defines the _ exponent _ @xmath15 of the packing by @xmath37 and the _ curvature distribution function _ associated to @xmath38 by @xmath39 for @xmath14 . we remark here that the radii @xmath34 are measured with the euclidean metric of the plane , in contrast to where we use the spherical metric . let @xmath9 be the open ball enclosed by @xmath10 . the _ residual set _ @xmath40 of a packing @xmath12 is defined by @xmath41 . the set @xmath40 has fractal nature and its hausdorff dimension @xmath42 is related to @xmath43 and @xmath15 by the following result of boyd . [ theorem - boyd ] if @xmath12 is an apollonian circle packing , then @xmath44 recently , kontorovich and oh proved the following stronger version of this theorem : [ theorem - kontor - oh ] if @xmath12 is an apollonian circle packing , then @xmath45 where @xmath46 . in particular , @xmath47 . apollonian.png ( 82,82 ) @xmath32 ( 55,85 ) @xmath48 ( 30,68 ) @xmath49 ( 75,60 ) @xmath50 [ fig : apollonian ] in @xcite , merenkov and sabitova observed that the curvature distribution function @xmath43 can be defined also for other planar fractal sets such as the sierpiski gasket and the sierpiski carpets . more precisely , if @xmath11 is a collection of topological circles in the plane , and @xmath9 is the open topological disk enclosed by @xmath10 , such that @xmath51 contains @xmath10 for @xmath52 , and @xmath9 are disjoint for @xmath52 , one can define the _ residual set _ @xmath40 of the _ packing _ @xmath33 by @xmath53 . a fundamental result of whyburn implies that if the disks @xmath54 , @xmath52 are disjoint with @xmath55 as @xmath56 and @xmath40 has empty interior , then @xmath40 is homeomorphic to the standard sierpiski carpet @xcite . in the latter case we say that @xmath40 is a sierpiski carpet ( see figure [ fig : test2 ] for a sierpiski carpet julia set ) . one can define the curvature of a topological circle @xmath10 as @xmath57 . then the _ curvature distribution function _ associated to @xmath12 is defined as in by @xmath58 for @xmath14 . similarly the exponent @xmath15 of @xmath12 is defined as in . in general , the limit @xmath59 does not exist , but if we impose further restrictions on the geometry of the circles @xmath10 , then we can draw conclusions about the limit . to this end , merenkov and sabitova introduced the notion of _ homogeneous planar sets _ ( see section [ section - homogeneous ] for the definition ) . however , even these strong geometric restrictions are not enough to guarantee the existence of the limit . the following theorem hints that a self - similarity condition on @xmath40 would be sufficient for our purposes . [ theorem - merenkov - sabitova ] assume that @xmath27 is a hyperbolic rational map whose julia set @xmath2 is a sierpiski carpet . then @xmath60 where @xmath26 is the curvature distribution function and @xmath15 is the exponent of the packing of the fatou components of @xmath27 . the authors made the conjecture that for such julia sets we actually have an analogue of theorem [ theorem - kontor - oh ] , namely @xmath61 , where @xmath28 . note that theorem [ theorem - main ] partially addresses the issue by asserting that @xmath47 . however , we believe that the limit @xmath62 does not exist in general for julia sets . observe that the conclusion of theorem [ theorem - main ] remains valid if we alter the metric that we are using in the definition of @xmath43 in a bi - lipschitz way . for example , if the julia set @xmath2 is contained in the unit disk of the plane we can use the euclidean metric instead of the spherical . on the other hand , the limit of @xmath63 as @xmath64 is much more sensitive to changes of the metric . the following simple example of the standard sierpiski carpet provides some evidence that the limit will not exist even for packings with very nice " geometry . the standard sierpiski carpet is constructed as follows . we first subdivide the unit square @xmath65 ^ 2 $ ] into @xmath66 squares of equal size and then remove the interior of the middle square . we continue subdividing each of the remaining @xmath67 squares into @xmath66 squares , and proceed inductively . the resulting set @xmath40 is the standard sierpiski carpet and its hausdorff dimension is @xmath68 . the set @xmath40 can be viewed as the residual set of a packing @xmath33 , where @xmath32 is the boundary of the unit square , and @xmath69 are the boundaries of the squares that we remove in each step in the construction of @xmath40 . using the euclidean metric , note that for each @xmath70 the quantity @xmath71 is by definition the number of curves @xmath10 that have diameter at least @xmath72 . thus , @xmath73 ( note that we also count @xmath32 ) . since @xmath74 , we have @xmath75 on the other hand , it is easy to see that @xmath76 , since there are no curves @xmath10 with diameter in the interval @xmath77 . thus , @xmath78 , and this shows that @xmath79 does not exist . in general , if one can show that there exists some constant @xmath80 such that @xmath81 for large @xmath3 , then the limit will not exist . [ fig : test1 ] carpet_julia.jpg ( 70,90 ) @xmath82 ( 44,48 ) @xmath9 [ fig : test2 ] we also note that in theorem [ theorem - main ] one might be able to weaken the assumption that @xmath27 is semi - hyperbolic , but the assumption that @xmath27 has connected julia set is necessary , since there exist rational maps whose fatou components are nested herman rings , and in fact in this case there exist infinitely many fatou components with large " diameter ( see ( * ? ? ? * proposition 7.2 ) ) . thus , if @xmath43 is the number of fatou components whose diameter is at least @xmath83 , we would have @xmath84 for large @xmath3 . the proof of theorem [ theorem - main ] will be given in two main steps . in section [ section - minkowski ] , using the self - similarity of the julia set we will establish relations between the hausdorff dimension of the julia set and its _ minkowski dimension _ ( see section [ section - minkowski ] for the definition ) . then in section [ section - homogeneous ] we will observe that the julia sets of semi - hyperbolic maps are homogeneous sets , satisfying certain geometric conditions ( see section [ section - homogeneous ] for the definition ) . these conditions allow one to relate the quantity @xmath63 with the _ minkowski content _ of the julia set . using these relations , and the results of section [ section - minkowski ] , the proof of theorem [ theorem - main ] will be completed . before proceeding to the above steps , we need some important distortion estimates for semi - hyperbolic rational maps that we establish in section [ section - conformal elevator ] , and we will refer to them as the _ conformal elevator_. these are they key estimates that we will use in establishing geometric properties of the julia set . similar estimates have been established for sub - hyperbolic rational maps in ( * ? ? ? * lemma 4.1 ) . the author would like to thank mario bonk for many useful comments and suggestions , and for his patient guidance . the heart of this section is lemma [ lemma - elevator ] and the whole section is devoted to proving it . let @xmath0 be a semi - hyperbolic map with @xmath85 ; in particular , by sullivan s classification and the fact that semi - hyperbolic rational maps do not have siegel disks or herman rings ( ( * ? ? ? * corollary ) ) , @xmath27 must have an attracting or superattracting periodic point . conjugating @xmath27 by a rotation of the sphere @xmath17 , we may assume that @xmath86 is a periodic point in the fatou set . furthermore , conjugating again with a euclidean similarity , we can achieve that @xmath87 , where @xmath88 denotes the unit disk in the plane . note that these operations do not affect the conclusion of theorem [ theorem - main ] , since a rotation is an isometry in the spherical metric that we used in the definition of @xmath43 , and a scaling only changes the limits by a factor . furthermore , since the boundaries @xmath10 of the fatou components @xmath9 have been moved away from @xmath86 , the diameters of @xmath10 in spherical metric are comparable to the diameters in the euclidean metric . this easily implies that the conclusion of theorem [ theorem - main ] is not affected if we define @xmath89 using instead the euclidean metric for measuring the diameters . in this section the euclidean metric will be used in all of our considerations . by semi - hyperbolicity ( see ( * ? ? ? * theorem ii(b ) ) ) and compactness of @xmath2 , there exists @xmath90 such that for every @xmath91 and for every connected component @xmath92 of @xmath93 the degree of @xmath94 is bounded by some fixed constant @xmath95 that does not depend on @xmath96 . furthermore , we can choose an even smaller @xmath97 so that the open @xmath97-neighborhood of @xmath2 that we denote by @xmath98 is contained in @xmath88 , and avoids the poles of @xmath27 that must lie in the fatou set . then @xmath27 is uniformly continuous in @xmath99 in the euclidean metric , and in particular , there exists @xmath100 such that for any @xmath101 with @xmath102 we have @xmath103 . let @xmath104 , @xmath105 be arbitrary , and define @xmath106 . since for large @xmath107 we have @xmath108 ( e.g. see ( * ? ? ? * corollary 14.2 ) ) , there exists a largest @xmath70 such that @xmath109 . by the choice of @xmath110 , we have @xmath111 . using the uniform continuity and the choice of @xmath112 , it follows that @xmath113 , thus @xmath114 we now state the main lemma . [ lemma - elevator ] there exist constants @xmath115 independent of @xmath116 ( and thus of @xmath110 ) such that : 1 . if @xmath117 is a connected set , then @xmath118 2 . @xmath119 3 . for all @xmath120 we have @xmath121 , with good distortion estimates . for _ hyperbolic _ rational maps ( i.e. , no parabolic cycles and no critical points on the julia set ) the map @xmath122 would actually be bi - lipschitz and part ( c ) of the above lemma would be true with @xmath123 instead of @xmath124 . however , in the semi - hyperbolic case , the presence of critical points on the julia set prevents such good estimates , but part ( a ) of the lemma restores some of them . in order to prepare for the proof we need some distortion lemmas . using koebe s distortion theorem ( e.g. see ( * ? ? ? * theorem 1.3 ) ) one can derive the following lemma . [ lemma - koebe ] let @xmath125 be a univalent map and let @xmath126 . then there exists a constant @xmath127 that depends only on @xmath128 , such that @xmath129 for all @xmath130 . we will be using the notation @xmath131 . we also need the next lemma . [ lemma - simply connected components ] let @xmath132 be a semi - hyperbolic rational map with @xmath133 and assume that @xmath134 is connected . then there exists @xmath135 such that for all @xmath136 , each component of @xmath137 is simply connected , for all @xmath138 . as before , by conjugating , we may assume that @xmath86 is a periodic point in the fatou set , and the julia set is far " from the poles of @xmath139 . by semi - hyperbolicity ( see ( * ? ? ? * theorem ii(c ) ) ) , for each @xmath136 and @xmath140 , there exists @xmath135 such that each component of @xmath137 has euclidean diameter less than @xmath141 , for all @xmath138 . by compactness of @xmath134 , we may take @xmath135 to be uniform in @xmath3 . we choose a sufficiently small @xmath141 such that the @xmath142-neighborhood @xmath143 of @xmath134 does not contain any poles of @xmath139 . we claim that each component of @xmath137 is simply connected . if this was not the case , there would exist an open component @xmath92 of @xmath144 , and a family of compact components @xmath145 of @xmath146 . thus @xmath147 for @xmath148 . assume that @xmath149 is the smallest such integer . note that @xmath92 intersects the julia set @xmath134 , because @xmath150 does so . hence , we have @xmath151 . since @xmath152 and @xmath92 share at least one common boundary point , it follows that @xmath153 , and in particular @xmath154 does not contain any poles of @xmath139 , i.e. , @xmath155 for all @xmath148 . by the choice of @xmath156 the set @xmath157 is a simply connected set in the @xmath141-neighborhood of @xmath134 . note that @xmath158 can not be entirely contained in @xmath159 , otherwise @xmath92 would not be a component of @xmath160 . thus , there exists some @xmath161 and a point @xmath162 . we connect the point @xmath163 to @xmath86 with a path @xmath164 , and then we lift @xmath165 under @xmath139 to a path @xmath166 that connects a preimage @xmath167 of @xmath163 to a pole of @xmath139 ( see ( * ? ? ? * lemma a.16 ) for path - lifting under branched covers ) . the path @xmath168 can not intersect @xmath92 , so it stays entirely in @xmath169 . this contradicts the fact that @xmath169 contains no poles . now we are ready to start the proof of lemma [ lemma - elevator ] . since @xmath170 , for @xmath171 we have @xmath172 , and for the component @xmath173 of @xmath93 that contains @xmath174 we have that the degree of @xmath175 is bounded by @xmath176 . lemma [ lemma - simply connected components ] implies that we can refine our choice of @xmath97 such that @xmath173 is also simply connected . let @xmath177 be the riemann map that maps the center @xmath178 of @xmath174 to @xmath179 , and @xmath180 be the translation of @xmath3 to @xmath179 , followed by a scaling by @xmath181 , so we obtain the following diagram : @xmath182{d}{\psi } & b(x,\varepsilon_0 ) \arrow{d}{\phi } \\ { \mathbb d}\arrow{r } { } & { \mathbb d}\end{tikzcd}\ ] ] diagram.jpg ( 40,70 ) @xmath122 ( 13,35 ) @xmath183 ( 80,35 ) @xmath184 ( 30,45 ) @xmath173 ( 20,60 ) @xmath174 ( 91,45 ) @xmath185 ( 72,59 ) @xmath186 ( 10,70 ) @xmath178 ( 75,70 ) @xmath3 ( 18.5,5 ) @xmath187 ( 69,5 ) @xmath188 [ fig : diagram ] the proof will be done in several steps . first we prove that @xmath187 is contained in a ball of fixed radius smaller than @xmath189 . second , we show a distortion estimate for @xmath183 , namely it is roughly a scaling by @xmath190 . in the end , we complete the proofs of ( a),(b),(c ) , using lemmas that are generally true for proper maps . we claim that there exists @xmath191 , independent of @xmath174 such that @xmath192 this will be derived from the following modulus distortion lemma . if @xmath193 is a jordan region , and @xmath194 is a connected subset of @xmath193 with @xmath195 , we denote by @xmath196 the modulus of the curve family that separates @xmath194 from @xmath197 . [ lemma - modulus ] let @xmath198 be jordan regions , and @xmath199 be a proper holomorphic map of degree @xmath200 . 1 . if @xmath201 is a jordan region with @xmath202 , and @xmath194 is a component of @xmath203 , then @xmath204 2 . if @xmath194 is a jordan region with @xmath205 , and @xmath206 , then @xmath207 a particular case of this lemma is ( * ? ? ? * lemma 5.5 ) , but we include a proof of the general statement since we were not able to find it in the literature . we first show @xmath208 . using a conformal map , we map the annulus @xmath209 to the circular annulus @xmath210 , and by composing with @xmath139 , we assume that we have a proper holomorphic map @xmath211 , of degree at most @xmath200 . we divide the annulus @xmath210 into nested circular annuli centered at the origin @xmath212 such that each @xmath213 does not contain any critical value of @xmath139 in its interior . note that @xmath214 where we denote by @xmath215 the modulus of curves that separate the complementary components of the annulus @xmath213 . we fix @xmath135 . by making the annuli @xmath213 a bit thinner , we can achieve that @xmath216 does not contain any critical value of @xmath139 , and @xmath217 let @xmath218 be a preimage of @xmath213 , so that @xmath219 are nested annuli separating @xmath194 from @xmath197 , and avoiding the critical points of @xmath139 . note that @xmath220 is a covering map of degree @xmath221 , thus @xmath222 . this implies that @xmath223 to see the first inequality , note that an admissible function @xmath128 for @xmath196 yields admissible functions @xmath224 for @xmath225 . combining and we obtain @xmath226 letting @xmath227 concludes the proof . the inequality in @xmath228 follows from poletski s inequality ( * ? ? ? * chapter ii , section 8) . since holomorphic maps are @xmath189-quasiregular , we have @xmath229 for all path families @xmath230 in @xmath193 . first we shrink the regions @xmath193 and @xmath231 as follows . consider a jordan curve @xmath232 very close to @xmath233 such that @xmath232 encloses a region @xmath234 that contains @xmath201 and all critical values of @xmath139 . then @xmath235 is a jordan region that contains @xmath194 and all critical points of @xmath139 . let @xmath236 be the family of paths in @xmath237 that connect @xmath238 to @xmath239 and avoid preimages of critical values of @xmath139 , which are finitely many . also , note that @xmath240 , where @xmath241 is the family of paths in @xmath242 that connect @xmath243 to @xmath244 , and avoid the critical values of @xmath139 . to see this , observe that any such path @xmath245 has a lift @xmath246 that starts at @xmath238 and ends at @xmath247 . using monotonicity of modulus and we have @xmath248 . if @xmath249 is the family of all paths in @xmath237 that connect @xmath238 to @xmath239 , then @xmath249 differs from @xmath230 by a family of zero modulus . the same is true for the corresponding family @xmath250 in @xmath251 . thus , we have @xmath252 . by reciprocality of the modulus and monotonicity , it follows that @xmath253 finally , observe that the path family separating @xmath194 from @xmath197 can be written as an increasing union of families separating @xmath194 from sets of the form @xmath254 , where @xmath255 gets closer and closer to @xmath193 . writing @xmath196 as a limit of moduli of such families , one obtains the desired inequality . we now return to the proof of . applying lemma [ lemma - modulus]@xmath208 to @xmath256 , and using the fact that @xmath257 along with monotonicity of modulus we obtain @xmath258 since modulus is invariant under conformal maps , we have @xmath259 if @xmath260 is such that @xmath261 then by grtzsch s module theorem ( see @xcite ) we have @xmath262 , where @xmath263 is a strictly decreasing bijection . thus , by @xmath264 is uniformly bounded below , and by monotonicity there exists @xmath265 such that @xmath266 . hence , @xmath267 , which proves . now , the version of koebe s theorem in lemma [ lemma - koebe ] yields latexmath:[\ ] ] the minkowski dimension is another useful notion of dimension for a fractal sets @xmath369 . for @xmath135 we define @xmath370 to be the maximal number of disjoint open balls of radii @xmath135 centered at points @xmath371 . we then define the upper and lower minkowski dimensions respectively as @xmath372 if the two numbers agree , then we say that their common value @xmath373 is the minkowski , or else , box dimension of @xmath363 . it is easy to see that the definition of the minkowski dimension is not affected if @xmath370 denotes instead the smallest number of open balls of radii @xmath135 centered at @xmath363 , that cover @xmath363 . the important difference between the hausdorff and minkowski dimensions is that in the hausdorff dimension we are taking into account coverings @xmath366 with different weights @xmath374 attached to each set , but in the minkowski dimension we are considering only coverings of sets with equal diameters . it easily follows from the definitions that we always have @xmath375 from now on , @xmath370 will denote the maximal number of disjoint open balls of radii @xmath376 , centered at points @xmath371 . based on the distortion estimates that we developed in section [ section - conformal elevator ] , and using results of @xcite and @xcite we have the following result that concerns the hausdorff and minkowski dimensions of julia sets of semi - hyperbolic maps . [ theorem - hausdorff measure positive ] let @xmath0 be a semi - hyperbolic rational map and @xmath377 . we have 1 . @xmath378 2 . @xmath379 3 . there exists a constant @xmath19 such that for all @xmath135 @xmath380 where @xmath370 is the maximal number of disjoint open balls of radii @xmath376 ( in the spherical metric ) , centered in @xmath2 . by considerations as in the beginning of section [ section - conformal elevator ] , we may assume that @xmath381 , and use the euclidean metric which is comparable to the spherical metric . this will only affect the constant in part ( c ) of the theorem . the parts ( a ) and ( b ) follow from ( * ? ? ? * theorem 1.11(e ) and ( g ) ) . also , if @xmath382 are disjoint balls of radius @xmath135 centered at @xmath2 then the collection @xmath383 covers @xmath2 , where @xmath384 has the same center as @xmath385 but twice the radius . thus , we have @xmath386 taking limits , and using ( a ) , we obtain @xmath387 which shows the left inequality in ( c ) . for the right inequality in ( c ) , we use the following result of falconer . [ theorem - falconer ] let @xmath388 be a compact metric space with @xmath389 . suppose that there exist @xmath390 such that for any ball @xmath391 of radius @xmath392 there is a mapping @xmath393 satisfying @xmath394 for all @xmath395 . then @xmath396 . it remains to show that this theorem applies in our case . to show the existence of @xmath183 we will carefully use the distortion estimates of lemma [ lemma - elevator ] . let @xmath343 be so small that for @xmath392 and @xmath104 the conclusions of lemma [ lemma - elevator ] are true for the ball @xmath116 . in particular , there exists @xmath340 , independent of @xmath174 , such that @xmath397 for some @xmath70 . for each ball @xmath398 , there exists @xmath138 such that @xmath399 is surjective ( e.g. see ( * ? ? ? * corollary 14.2 ) ) . compactness of @xmath2 allows us to choose a uniform @xmath138 , independent of @xmath400 . by the analyticity of @xmath401 , there exists a constant @xmath402 such that for all @xmath403 we have @xmath404(c ) , there exists @xmath405 independent of @xmath174 such that latexmath:[\[\begin{aligned } \label{theorem - falc- f^n } for @xmath120 . now , we can construct the desired @xmath407 . let @xmath408 be any right inverse of the surjective map @xmath409 . also , the inclusion allows us to define a right inverse @xmath410 of @xmath122 , restricted on a suitable subset of @xmath174 . now , let @xmath411 , and observe that by , and we have @xmath412 thus , the hypotheses of theorem [ theorem - falconer ] are satisfied with @xmath413 . let @xmath414 be a packing , as defined in the introduction , where @xmath10 are topological circles , surrounding topological open disks @xmath9 ( in the plane or the sphere ) such that @xmath415 contains @xmath10 for @xmath52 , and @xmath416 are disjoint . then the set @xmath417 is the residual set @xmath40 of the packing @xmath12 . in the following , one can use the euclidean or spherical metric , but it is convenient to consider @xmath418 as the boundary of the unbounded component of the packing @xmath12 ( see figures [ fig : apollonian ] and [ fig : test2 ] ) , and use the euclidean metric to study the other disks @xmath419 . thus , we will restrict ourselves to the use of the euclidean metric in this section . 1 . each @xmath419 is a _ uniform quasi - ball_. more precisely , there exists a constant @xmath423 such that for each @xmath9 there exist inscribed and circumscribed , cocentric circles of radii @xmath34 and @xmath424 respectively with @xmath425 2 . there exists a constant @xmath426 such that for each @xmath427 and @xmath428 there exists a circle @xmath10 intersecting @xmath429 such that @xmath430 3 . the circles @xmath10 are _ uniformly relatively separated_. this means that there exists @xmath431 such that @xmath432 for all @xmath433 . the disks @xmath419 are _ uniformly fat_. by definition , this means that there exists @xmath434 such that for every ball @xmath429 centered at @xmath9 that does not contain @xmath9 , we have @xmath435 where @xmath436 denotes the @xmath1-dimensional lebesgue measure . ( here one can use the spherical measure for packings on the sphere . ) condition @xmath437 means that the sets @xmath9 look like round balls , while @xmath438 says that the circles @xmath10 exist in all scales and all locations in @xmath40 . condition @xmath421 forbids two large " circles @xmath10 to be close to each other in some uniform manner . note that this only makes sense when @xmath439 are disjoint , e.g. in the case of a sierpiski carpet . finally , @xmath422 is used to replace @xmath421 when we are working with fractals such as the sierpiski gasket , or generic julia sets regarded as packings , where @xmath54 are not disjoint . we now summarize some interesting properties of homogeneous sets , that are not needed though for the proof of theorem [ theorem - main ] . a set @xmath440 is said to be _ porous _ if there exists a constant @xmath441 such that for all sufficiently small @xmath442 and all @xmath443 , there exists a point @xmath444 such that @xmath445 a jordan curve @xmath446 is called a @xmath351-_quasicircle _ if for all @xmath447 there exists a subarc @xmath448 of @xmath165 joining @xmath3 and @xmath449 with @xmath450 . the _ ( ahlfors regular ) conformal dimension _ of a metric space @xmath360 , denoted by @xmath451 , is the infimum of the hausdorff dimensions among all ahlfors regular metric spaces that are quasisymmetrically equivalent to @xmath360 . for more background see chapters @xmath452 and @xmath453 in @xcite . by @xmath437 , each @xmath9 contains a ball of diameter comparable to @xmath457 . thus , summing the areas of the sets @xmath9 , and noting that they are all contained in @xmath458 , we see that for each @xmath135 , there can only be finitely many sets @xmath9 with @xmath459 . we conclude that @xmath40 is locally connected ( see ( * ? ? ? * lemma 19.5 ) ) . condition @xmath438 implies that for @xmath460 , every ball @xmath429 centered at @xmath40 intersects a curve @xmath10 of diameter comparable to @xmath461 . let @xmath462 and consider the ball @xmath463 . then @xmath464 intersects a curve @xmath10 of diameter comparable to @xmath465 , and if @xmath466 is sufficiently small but uniform , then @xmath467 . thus @xmath429 contains a curve @xmath10 of diameter comparable to @xmath461 . by @xmath437 , @xmath9 contains a ball of radius comparable to @xmath457 and thus comparable to @xmath461 ( note that here we use the euclidean metric ) . hence , @xmath468 contains a ball of radius comparable to @xmath461 . this completes the proof that @xmath40 is porous . for our last assertion we will use a criterion of mackay ( * theorem 1.1 ) which asserts that a doubling metric space which is _ annularly linearly connected _ has conformal dimension strictly greater than @xmath189 . a connected metric space @xmath363 is annularly linearly connected ( abbr . alc ) if there exists some @xmath469 such that for every @xmath470 , and @xmath471 in the annulus @xmath472 there exists an arc @xmath473 joining @xmath3 to @xmath449 that lies in a slightly larger annulus @xmath474 . it suffices to show that @xmath40 is alc . the idea is simple , but the proof technical , so we only provide a sketch . let @xmath475 , and consider a path @xmath476 ( not necessarily in @xmath477 ) that joins @xmath3 and @xmath449 . the idea is to replace the parts of the path @xmath165 that lie in the complementary components @xmath9 of @xmath40 by arcs in @xmath455 and then make sure that the resulting arc stays in a slightly larger annulus @xmath474 . the assumption that the curves @xmath10 are quasicircles guarantees that the subarcs that we will use are not too large " , and condition @xmath421 guarantees that the large " curves @xmath10 do not block the way from @xmath478 to @xmath449 , since these curves are not allowed to be very close to each other . using @xmath421 , we can find uniform constants @xmath479 such that there exists at most one curve @xmath480 with @xmath481 that intersects @xmath482 . we call a curve @xmath10 _ large _ if its diameter exceeds @xmath483 , and otherwise we call it _ small_. we enlarge slightly the annulus ( maybe using a larger @xmath484 ) to an annulus @xmath485 so that @xmath486 contains all small curves @xmath10 that intersect @xmath165 . we now check all different cases . if @xmath165 meets the large @xmath480 that intersects @xmath487 , using the fact that @xmath480 is a quasicircle , we can enlarge the annulus to an annulus @xmath488 with a uniform @xmath489 , so that @xmath3 can be connected to @xmath449 by a path in @xmath490 . we call the resulting path @xmath165 . note that here we have to assume that @xmath491 , so that the path @xmath165 does not lie in the unbounded component of the packing and it passes through several curves @xmath10 on the way from @xmath3 to @xmath449 . the case @xmath492 , which occurs only when @xmath493 , is similar and in the previous argument we just have to choose a path @xmath165 that lies in @xmath494 . we still assume that @xmath495 contains all small curves @xmath10 that intersect @xmath165 . if @xmath165 meets a small @xmath10 that does not intersect @xmath496 , then we can replace the subarcs of @xmath165 that lie in @xmath9 with arcs in @xmath10 that have the same endpoints . the resulting arcs will lie in the annulus by construction . next , if @xmath165 meets a small @xmath10 that does intersect @xmath496 , we follow the same procedure as before , but now we have to choose the sub - arcs of @xmath10 carefully , so that they do not approach @xmath178 too much . this can be done using the assumption that the curves @xmath10 are uniform quasicircles . the resulting arcs will lie in a slightly larger annulus @xmath497 , where @xmath498 a is uniform constant . finally , if @xmath165 intersects a large @xmath10 which does not meet @xmath499 we can use the assumption that @xmath10 is a quasicircle to replace the subarcs of @xmath165 that lie in @xmath9 with subarcs of @xmath10 that have diameter comparable to @xmath461 . thus , a larger annulus @xmath500 will contain the arcs of @xmath10 that we obtain in this way . we need to ensure that this procedure indeed yields a path that joins @xmath3 and @xmath449 inside @xmath501 . this follows from the fact that @xmath502 . the latter fact follows from the assumption that the curves @xmath10 are uniform quasicircles , which in turn implies that each @xmath9 contains a ball of radius comparable to @xmath457 , i.e. , @xmath437 is true ( for a proof of this assertion see ( * ? ? ? * proposition 4.3 ) ) . as we will see in lemma [ lemma - n - n ] , a homogeneous set has the special property that there is some important relation between the curvature distribution function @xmath43 and the maximal number of disjoint open balls @xmath370 , centered at @xmath40 . thus , considerations about the residual set @xmath40 , which are reflected by @xmath370 , can be turned into considerations about the complementary components @xmath9 , which are comprised in @xmath43 . [ lemma - mereknov - sabitova ] assume that @xmath40 is the residual set of a packing @xmath33 that satisfies @xmath437 and @xmath421 ( or @xmath437 and @xmath422 ) . for any @xmath503 , there exist constants @xmath504 depending only on @xmath505 and the constants in @xmath506 ( or @xmath507 ) such that for any collection @xmath508 of disjoint open balls of radii @xmath442 centered in @xmath40 we have the following statements : 1 . [ lemma - m - s-1]there are at most @xmath509 balls in @xmath508 that intersect any given @xmath10 with + @xmath510 2 . [ lemma - m - s-2 ] there are at most @xmath511 curves @xmath10 intersecting any given ball in @xmath512 and satisfying + @xmath513 using this lemma one can prove a relation between the curvature distribution function @xmath514 ( using the euclidean metric ) and the maximal number @xmath370 of disjoint open balls of radius @xmath376 , centered at @xmath40 . namely , we have the following lemma . [ lemma - n - n ] assume that the residual set @xmath40 of a packing @xmath12 satisfies @xmath420 and @xmath421 or @xmath420 and @xmath422 . then there exists a constant @xmath19 such that for all small @xmath135 we have @xmath515 where @xmath505 is the constant in @xmath438 . let @xmath508 be a maximal collection of disjoint open balls of radius @xmath376 , centered at @xmath40 . for each ball @xmath516 , by condition @xmath438 there exists @xmath10 such that @xmath517 and @xmath518 . on the other hand , lemma [ lemma - mereknov - sabitova](a ) implies that for each such @xmath10 there exist at most @xmath509 balls in @xmath508 that intersect it . thus @xmath519 conversely , note that by the maximality of @xmath508 , it follows that @xmath512 covers @xmath40 . hence , if @xmath10 is arbitrary satisfying @xmath520 , it intersects a ball @xmath521 in @xmath512 . for each such ball @xmath521 , lemma [ lemma - mereknov - sabitova](b ) implies that there exist at most @xmath511 curves @xmath10 with @xmath520 that intersect it . thus @xmath522 by considerations as in the beginning of section [ section - conformal elevator ] , we assume that @xmath523 , and we will use the euclidean metric since this does not affect the conclusion of the theorem . let @xmath32 be the boundary of the unbounded fatou component , @xmath419 be the sequence of bounded fatou components , and @xmath455 . then @xmath524 can be viewed as a packing , and @xmath525 is its residual set . note , though , that the sets @xmath10 need not be topological circles in general . this , however , does not affect our considerations , since it does not affect the conclusions of lemmas [ lemma - mereknov - sabitova ] and [ lemma - n - n ] , as long as the other assumptions hold for @xmath10 and the simply connected regions @xmath9 enclosed by them . we will freely use the terminology curves " for the sets @xmath10 . by theorem [ theorem - hausdorff measure positive ] we have that the quantity @xmath526 is bounded away from @xmath179 and @xmath86 as @xmath527 , where @xmath28 . if we prove that @xmath2 is a homogeneous set , satisfying @xmath420 and @xmath422 , then using lemma [ lemma - n - n ] , it will follow that @xmath63 is bounded away from @xmath179 and @xmath86 as @xmath64 , and in particular @xmath528 which will complete the proof . julia sets of semi - hyperbolic rational maps are locally connected if they are connected ( see ( * ? ? ? * proposition 10 ) ) , and thus for each @xmath135 there exist finitely many fatou components with diameter greater than @xmath376 ( see ( * ? ? ? * problem 19-f ) ) . first we show that condition @xmath437 in the definition of homogeneity is satisfied . the idea is that the finitely many large fatou components are trivially quasi - balls , as required in @xmath437 , so there is nothing to prove here , but the small fatou components can be blown up with good control to the large ones using lemma [ lemma - elevator ] . the distortion estimates allow us to control the size of inscribed circles of the small fatou components . let @xmath529 , where @xmath530 are the constants appearing in lemma [ lemma - elevator ] . we also make @xmath531 even smaller so that for @xmath532 and @xmath104 the conclusions of lemma [ lemma - elevator ] are true . since there are finitely many curves @xmath10 with @xmath533 , for these @xmath10 there exist cocentric inscribed and circumscribed circles with radii @xmath34 and @xmath424 respectively , such that @xmath534 , for some @xmath535 . this implies that @xmath536 . if @xmath10 is arbitrary with @xmath537 , then for @xmath538 and @xmath539 , by lemma [ lemma - elevator](c ) there exists @xmath70 such that @xmath540 note that the fatou component @xmath9 is mapped under @xmath122 onto a fatou component @xmath541 . since @xmath122 is proper , the boundary @xmath10 of @xmath9 is mapped onto @xmath542 . then the above inequality can be written as @xmath543 hence , @xmath544 is one of the large " curves , for which there exists a inscribed ball @xmath545 such that @xmath546 . observe that @xmath547 . let @xmath548 be a preimage of @xmath549 under @xmath122 , and @xmath550 be the component of @xmath551 that contains @xmath552 . for each @xmath553 , by lemma [ lemma - elevator](c ) one has @xmath554 thus @xmath555 letting @xmath556 , and @xmath557 , one obtains @xmath558 , so @xmath437 is satisfied with @xmath559 . similarly , we show that condition @xmath438 is also true . let @xmath340 be the constant in lemma [ lemma - elevator](b ) and consider @xmath560 so small that the conclusions of lemma [ lemma - elevator ] are true for @xmath104 and @xmath532 . note that by compactness of @xmath2 there exists @xmath503 such that for @xmath561 and @xmath104 there exists @xmath10 such that @xmath562 and @xmath563 indeed , one can cover @xmath2 with finitely many balls @xmath564 of radius @xmath565 centered at @xmath2 , such that each ball @xmath566 contains a curve @xmath567 . this is possible because every ball @xmath566 centered in the julia set must intersect infinitely many fatou components , otherwise @xmath122 would be a normal family in @xmath566 . in particular , by local connectivity most " fatou components are small , and thus one of them , say @xmath568 , will be contained in @xmath566 . now , if @xmath429 is arbitrary with @xmath569 , we have that @xmath570 for some @xmath571 , and thus @xmath572 . since @xmath573 $ ] lies in a compact interval , easily follows , by always using the same finite set of curves @xmath574 that correspond to @xmath575 , respectively . we may also assume that @xmath576 for each of these curves . now , if @xmath577 , @xmath104 , by lemma [ lemma - elevator](b ) we have @xmath578 for some @xmath70 . by the previous , @xmath579 intersects some @xmath580 with @xmath581 , thus @xmath582 . hence , @xmath429 contains a preimage @xmath10 of @xmath544 , and by lemma [ lemma - elevator](a ) , ( c ) we obtain @xmath583 however , @xmath544 was one of the finitely many curves that we chose in the previous paragraph . this and the above inequalities impliy that @xmath584 with uniform constants . this completes the proof of @xmath438 . finally , we will prove that condition @xmath422 of homogeneity is satisfied . this follows easily from the fact that the fatou components of a semi - hyperbolic rational map are _ uniform john domains _ in the spherical metric ( * ? ? ? * proposition 9 ) . since we are only interested in the bounded fatou components , we can use instead the euclidean metric . a domain @xmath585 is a @xmath586-john domain ( @xmath587 ) if there exists a basepoint @xmath588 such that for all @xmath589 there exists an arc @xmath590 connecting @xmath591 to @xmath320 such that for all @xmath592 we have @xmath593 where @xmath594 . in our case , the bounded fatou components @xmath595 are uniform john domains , i.e. , john domains with the same constant @xmath596 . let @xmath429 be a ball centered at some @xmath9 that does not contain @xmath9 . we will show that there exists a uniform constant @xmath434 such that @xmath597 if @xmath598 , then @xmath599 , so is true with @xmath600 . otherwise , @xmath601 intersects @xmath455 at a point @xmath591 . we split in two cases : the basepoint @xmath320 satisfies @xmath602 . then consider a path @xmath603 from @xmath320 to @xmath178 , as in the definition of a john domain , such that @xmath604 for all @xmath592 . in particular let @xmath605 be a point such that @xmath606 , thus @xmath607 . since @xmath608 , we have @xmath609 , hence @xmath610 the basepoint @xmath320 lies in @xmath611 . consider a point @xmath612 close to @xmath613 such that @xmath614 . then , by the definition of a john domain for @xmath615 , we have @xmath616 . hence , @xmath617 , so @xmath618 even when the julia set of a semi - hyperbolic map is a sierpiski carpet , the uniform relative separation of the peripheral circles @xmath10 in condition @xmath421 need not be true . in fact , it is known that for such julia sets condition @xmath421 is true if and only if for all critical points @xmath620 , @xmath621 does not intersect the boundary of any fatou component ; see ( * ? ? ? * proposition 3.7 ) . recall that @xmath621 is the set of accumulation points of the orbit @xmath622 . in ( * ? ? ? * proposition 3.5 ) it is shown that if the boundaries of fatou components of a semi - hyperbolic map @xmath27 are jordan curves , then they are actually uniform quasicircles . if , in addition , they are uniformly relatively separated ( i.e. , condition @xmath421 ) , proposition [ proposition - homogeneous ] implies that @xmath623 . on the other hand , if @xmath27 is a semi - hyperbolic _ polynomial _ with connected julia set , then not all boundaries of fatou components are jordan curves . in fact , @xmath2 coincides with the boundary of a single fatou component @xmath624 which is a john domain , and is called the _ basin of attraction of @xmath86_. according to a recent result of kinneberg ( ( * ? ? ? * theorem 1.1 ) ) , boundaries of planar john domains have conformal dimension equal to @xmath189 , if they are connected . therefore , @xmath625 , in contrast to the previous case . . by theorem [ theorem - main ] there exists a constant @xmath19 such that @xmath626 for all @xmath14 . taking logarithms , one obtains @xmath627 letting @xmath628 yields @xmath629 which completes part of the proof . recall that the exponent @xmath15 of the packing of the fatou components of @xmath27 is defined by @xmath630 and it remains to show that @xmath631 . note that for @xmath632 the sum @xmath633 diverges . also , since for semi - hyperbolic rational maps there are only finitely many large " fatou components , if @xmath634 , then @xmath635 for all @xmath636 . if @xmath637 , using , one has @xmath638 this implies that @xmath639 . conversely , assume that @xmath640 . since there are only finitely many large " fatou components , we only need to take into account the sets @xmath10 with @xmath641 in the sum @xmath642 . using again we have @xmath643 hence @xmath644 , which completes the proof . | recently mereknov and sabitova introduced the notion of a homogeneous planar set . using this notion
they proved a result for sierpiski carpet julia sets of hyperbolic rational maps that relates the diameters of the peripheral circles to the hausdorff dimension of the julia set .
we extend this theorem to julia sets ( not necessarily sierpiski carpets ) of semi - hyperbolic rational maps , and prove a stronger version of the theorem that was conjectured by merenkov and sabitova . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
a _ software design pattern _ is not a finished design , it is a description or template that can be instanciated in order to be used in many different situations . in this paper , we propose _ inference system patterns _ that can be instanciated with monads or comonads in order to be used for proving properties of different effects . in order to formalize computational effects one can choose between types and effects systems @xcite , monads @xcite and their associated lawvere theories @xcite , comonads @xcite , or decorated logics @xcite . starting with moggi s seminal paper @xcite and its application to haskell @xcite , various papers deal with the effects arising from a monad , for instance @xcite . each of these approaches rely on some classification of the syntactic expressions according to their interaction with effects . in this paper we use decorated logics which , by extending this classification to equations , provide a proof system adapted to each effect . this paper presents equational - based logics for proving first order properties of programming languages involving effects . we propose two dual patterns , consisting in a language with an inference system , for building such a logic . the first pattern provides inference rules which can be interpreted in the cokleisli category of a comonad and the kleisli category of the associated monad . in a dual way , the second pattern provides inference rules which can be interpreted in the kleisli category of a monad and the cokleisli category of the associated comonad . the logics combine a three - levels effect system for terms consisting of pure terms and two other kinds of effects called observers / constructors and modifiers , and a two - levels system for strong and weak equations . each pattern provides generic rules for dealing with any comonad ( respectively monad ) , and it can be extended with specific rules for each effect . the paper presents two use cases : a language with state and a language with exceptions . for the language with state we use a comonadic semantics and we prove that the equational theory obtained is hilbert - post complete , which provides a new proof for a result in @xcite . for the language with exceptions we extend the standard monadic semantics in order to catch exceptions ; this relies on the duality between states and exceptions from @xcite . we do not claim that each effect arises either from a comonad or from a monad , but this paper only deals with such effects . intuitively , an effect which observes features may arise from a comonad , while an effect which constructs features may arise from a monad @xcite . however , some interesting features in the comonad pattern stem from the well - known fact that each comonad determines a monad on its cokleisli category , and dually for the monad pattern . more precisely , on the monads side , let @xmath0 be a monad on a category @xmath1 and let @xmath2 be the kleisli category of @xmath0 on @xmath1 . then @xmath3 can be seen as the endofunctor of a comonad @xmath4 on @xmath2 , so that we may consider the cokleisli category @xmath5 of @xmath4 on @xmath2 . the canonical functors from @xmath1 to @xmath2 and from @xmath2 to @xmath5 give rise to a hierarchy of terms : pure terms in @xmath1 , constructors in @xmath2 , modifiers in @xmath5 . this corresponds to the three translations of a typed lambda calculus into a monadic language @xcite . on the comonads side , we get a dual hierarchy : pure terms in @xmath1 , observers in @xmath2 , modifiers in @xmath5 . we instanciate these patterns with two fundamental examples of effects : state and exceptions . following @xcite , we consider that the states effect arise from the comonad @xmath6 ( where @xmath7 is the set of states ) , thus a decorated logic for states is built by extending the pattern for comonads . the comonad itself provides a decoration for the lookup operation , which observes the state , while the monad on its cokleisli category provides a decoration for the update operation . following @xcite , we consider that the exceptions effect arise from the monad @xmath8 ( where @xmath9 is the set of exceptions ) , thus a decorated logic for exceptions is built by extending the pattern for monads . the monad itself provides a decoration for the raising operation , which constructs an exception , while the comonad on its kleisli category provides a decoration for the handling operation . in fact the decorated logic for exceptions is not exactly dual to the decorated logic for states if we assume that the intended interpretation takes place in a distributive category , like the category of sets , which is not codistributive . other effects would lead to other additional rules , but we have chosen to focus on two effects which are well known from various points of view . our goal is to enligthen the contributions of each approach : the annotation system from the types and effects systems @xcite , the major role of monads for some effects @xcite , and the dual role of comonads @xcite , as well as the flexibility of decorated logics @xcite . moreover , proofs in decorated logics can be checked with the coq proof assistant ; a library for states is available there : http://coqeffects.forge.imag.fr . in this paper we focus on finite products and coproducts ; from a programming point of view this means that we are considering languages with @xmath10-ary operations and with case distinction , but without loops or higher - order functions . in a language with effects there is a well - known issue with @xmath10-ary operations : their interpretation may depend on the order of evaluation of their arguments . in this paper we are looking for languages with case distinction and with _ sequential products _ , which allows to force the order of evaluation of the arguments , whenever this is required . it is well known that ( co)monads fit very well with composition but require additional assumptions for being fully compatible with products and coproducts . this corresponds to the fact that in the patterns from section [ sec : patterns ] , which are valid for any ( co)monad , the rules for products and coproducts hold only under some decoration constraints . however , such assumptions are satisfied for several ( co)monads . this is in particular the case for the state comonad and the exceptions monad . in section [ sec : patterns ] we describe the patterns for a comonad and for a monad . the first pattern is instanciated with the comonad for state in section [ sec : states ] , and we prove the hilbert - post completeness of the decorated theory for state . in section [ sec : exceptions ] we instanciate the second pattern to the monad for exceptions . in this section we define a grammar and an inference system for two logics @xmath11 and @xmath12 , then we define an interpretation of these logics in a category with a comonad and a monad , respectively . the logics @xmath11 and @xmath12 are called _ decorated _ logics because their grammar and inference rules are essentially the grammar and inference rules for a `` usual '' logic , namely the equational logic with conditionals ( denoted @xmath13 ) , together with _ decorations _ for the terms and for the equations . the decorations for the terms are similar to the _ annotations _ of the types and effects systems @xcite . decorated logics are introduced in @xcite in an abstract categorical framework , which will not be explicitly used in this paper . the grammar of the equational logic with conditionals is reminded in figure [ fig : logeq - grammar ] . each term has a source type and a target type . as usual in categorical presentations of equational logic , a term has precisely one source type , which can be a product type or the unit type . each equation relates two parallel terms , i.e. , two terms with the same source and the same target . this grammar will be extended with decorations in order to get the grammar of the logics @xmath11 and @xmath12 . @xmath14}\mid { \mathit{in}}_{t , t,1}\mid{\mathit{in}}_{t , t,2}\mid{[\;]}_t\mid \\ \textrm{equations : } & e::= f\equiv f \\ \hline \end{array}\ ] ] the rules in figure [ fig : pattern - rules ] are _ patterns _ , in the following sense : when the boxes in the rules are removed , we get usual rules for the logic @xmath13 , which may be interpreted in any bicartesian category . when the boxes are replaced by decorations , we get a logic which , according to the choice of decorations , may be interpreted in a bicartesian category with a comonad or a monad . there may be other ways to decorate the rules for @xmath13 , but this is beyond the scope of this paper . @xmath15[c]{{$\box$}}\makebox[0pt][c]{\raisebox{2pt}{\tiny\ensuremath{\equiv}}}\,\,}}f } \qquad { \textrm{(sym ) } } \quad \dfrac{f^\box { { \,\,\makebox[0pt][c]{{$\box$}}\makebox[0pt][c]{\raisebox{2pt}{\tiny\ensuremath{\equiv}}}\,\,}}g^\box}{g { { \,\,\makebox[0pt][c]{{$\box$}}\makebox[0pt][c]{\raisebox{2pt}{\tiny\ensuremath{\equiv}}}\,\,}}f } \qquad { \textrm{(trans ) } } \quad \dfrac{f^\box { { \,\,\makebox[0pt][c]{{$\box$}}\makebox[0pt][c]{\raisebox{2pt}{\tiny\ensuremath{\equiv}}}\,\,}}g^\box { \;\;}g^\box { { \,\,\makebox[0pt][c]{{$\box$}}\makebox[0pt][c]{\raisebox{2pt}{\tiny\ensuremath{\equiv}}}\,\,}}h^\box}{f { { \,\,\makebox[0pt][c]{{$\box$}}\makebox[0pt][c]{\raisebox{2pt}{\tiny\ensuremath{\equiv}}}\,\,}}h } \\ { \textrm{(repl ) } } \quad \dfrac{f_1^\box{{\,\,\makebox[0pt][c]{{$\box$}}\makebox[0pt][c]{\raisebox{2pt}{\tiny\ensuremath{\equiv}}}\,\,}}f_2^\box\colon a\to b { \;\;}g^\box\colon b\to c } { g\circ f_1 { { \,\,\makebox[0pt][c]{{$\box$}}\makebox[0pt][c]{\raisebox{2pt}{\tiny\ensuremath{\equiv}}}\,\,}}g\circ f_2 } \qquad { \textrm{(subs ) } } \quad \dfrac{f^\box\colon a\to b { \;\;}g_1^\box{{\,\,\makebox[0pt][c]{{$\box$}}\makebox[0pt][c]{\raisebox{2pt}{\tiny\ensuremath{\equiv}}}\,\,}}g_2^\box\colon b\to c } { g_1 \circ f { { \,\,\makebox[0pt][c]{{$\box$}}\makebox[0pt][c]{\raisebox{2pt}{\tiny\ensuremath{\equiv}}}\,\,}}g_2\circ f } \\ \hline \mbox{categorical rules } \\ { \textrm{(id ) } } \quad \dfrac{a}{{\mathit{id}}_a^\box\colon a\to a } \qquad { \textrm{(comp ) } } \quad \dfrac{f^\box\colon a\to b \quad g^\box\colon b\to c } { ( g\circ f)^\box \colon a\to c } \\ { \textrm{(id - source ) } } \quad \dfrac{f^\box\colon a\to b}{f\circ { \mathit{id}}_a { { \,\,\makebox[0pt][c]{{$\box$}}\makebox[0pt][c]{\raisebox{2pt}{\tiny\ensuremath{\equiv}}}\,\,}}f } \qquad { \textrm{(id - target ) } } \quad \dfrac{f^\box\colon a\to b}{{\mathit{id}}_b\circ f { { \,\,\makebox[0pt][c]{{$\box$}}\makebox[0pt][c]{\raisebox{2pt}{\tiny\ensuremath{\equiv}}}\,\,}}f } \\ { \textrm{(assoc ) } } \quad \dfrac{f^\box\colon a\to b { \;\;}g^\box\colon b\to c { \;\;}h^\box\colon c\to d } { h\circ ( g\circ f ) { { \,\,\makebox[0pt][c]{{$\box$}}\makebox[0pt][c]{\raisebox{2pt}{\tiny\ensuremath{\equiv}}}\,\,}}(h\circ g)\circ f } \\ \hline \mbox{product rules } \\ { \textrm{(prod ) } } \quad \dfrac{b_1 \quad b_2 } { b_1{\!\times\!}b_2 \quad { \mathit{pr}}_1^\box\colon b_1{\!\times\!}b_2 \to b_1 \quad { \mathit{pr}}_2^\box\colon b_1{\!\times\!}b_2 \to b_2 } \\ { \textrm{(pair ) } } \quad \dfrac { f_1^\box\colon a \to b_1 \quad f_2^\box\colon a \to b_2 } { { \langle f_1,f_2 \rangle}^\box\colon a\to b_1{\!\times\!}b_2 } \\ { \textrm{(pair - eq ) } } \quad \dfrac{f_1^\box\colon a\to b_1 { \;\;}f_2^\box\colon a\to b_2 } { { \mathit{pr}}_1\circ{\langle f_1,f_2 \rangle } { { \,\,\makebox[0pt][c]{{$\box$}}\makebox[0pt][c]{\raisebox{2pt}{\tiny\ensuremath{\equiv}}}\,\,}}f_1 \quad { \mathit{pr}}_2\circ{\langle f_1,f_2 \rangle } { { \,\,\makebox[0pt][c]{{$\box$}}\makebox[0pt][c]{\raisebox{2pt}{\tiny\ensuremath{\equiv}}}\,\,}}f_2 } \\ { \textrm{(pair - u ) } } \quad \dfrac{f_1^\box{\!\colon\!}a\!{\!\to\!}\ ! b_1 { \;\;}f_2^\box{\!\colon\!}a\!{\!\to\!}\ ! b_2 { \;\;}g^\box{\!\colon\!}a\!{\!\to\!}\ ! b_1{\!\times\!}b_2 { \;\;}{\mathit{pr}}_1\circ g{{\,\,\makebox[0pt][c]{{$\box$}}\makebox[0pt][c]{\raisebox{2pt}{\tiny\ensuremath{\equiv}}}\,\,}}f_1 { \;\;}{\mathit{pr}}_2\circ g{{\,\,\makebox[0pt][c]{{$\box$}}\makebox[0pt][c]{\raisebox{2pt}{\tiny\ensuremath{\equiv}}}\,\,}}f_2 } { g { { \,\,\makebox[0pt][c]{{$\box$}}\makebox[0pt][c]{\raisebox{2pt}{\tiny\ensuremath{\equiv}}}\,\,}}{\langle f_1,f_2 \rangle } } \\ { \textrm{(unit ) } } \quad \dfrac{}{{\mathbb{1 } } } \qquad { \textrm{(final ) } } \quad \dfrac{a}{{\langle \ ; \rangle}_a^\box\colon a\to { \mathbb{1 } } } \qquad { \textrm{(final - u ) } } \quad \dfrac{f^\box\colon a\to { \mathbb{1}}}{f { { \,\,\makebox[0pt][c]{{$\box$}}\makebox[0pt][c]{\raisebox{2pt}{\tiny\ensuremath{\equiv}}}\,\,}}{\langle \ ; \rangle}_a } \\ \hline \mbox{coproduct rules } \\ { \textrm{(coprod ) } } \quad \dfrac{a_1 \quad a_2 } { a_1{\!+\!}a_2 \quad { \mathit{in}}_1^\box\colon a_1\to a_1{\!+\!}a_2 \quad { \mathit{in}}_2^\box\colon a_2\to a_1{\!+\!}a_2 } \\ { \textrm{(copair ) } } \quad \dfrac { f_1^\box\colon a_1 \to b \quad f_2^\box\colon a_2 \to b } { { [ f_1|f_2 ] } ^\box\colon a_1{\!+\!}a_2 \to b } \\ { \textrm{(copair - eq ) } } \quad \dfrac { f_1^\box\colon a_1 \to b { \;\;}f_2^\box\colon a_2 \to b } { { [ f_1|f_2 ] } \circ { \mathit{in}}_1 { { \,\,\makebox[0pt][c]{{$\box$}}\makebox[0pt][c]{\raisebox{2pt}{\tiny\ensuremath{\equiv}}}\,\,}}f_1 \quad { [ f_1|f_2 ] } \circ { \mathit{in}}_2 { { \,\,\makebox[0pt][c]{{$\box$}}\makebox[0pt][c]{\raisebox{2pt}{\tiny\ensuremath{\equiv}}}\,\,}}f_2 } \\ { \textrm{(copair - u ) } } \quad \dfrac{g^\box{\!\colon\!}a_1\!{\!+\!}\ ! a_2 { \!\to\!}b { \;\;}f_1^\box{\!\colon\!}a_1 { \!\to\!}b { \;\;}f_2^\box{\!\colon\!}a_2 { \!\to\!}b { \;\;}g\circ { \mathit{in}}_1 { { \,\,\makebox[0pt][c]{{$\box$}}\makebox[0pt][c]{\raisebox{2pt}{\tiny\ensuremath{\equiv}}}\,\,}}f_1 { \;\;}g\circ { \mathit{in}}_2 { { \,\,\makebox[0pt][c]{{$\box$}}\makebox[0pt][c]{\raisebox{2pt}{\tiny\ensuremath{\equiv}}}\,\,}}f_2 } { g { { \,\,\makebox[0pt][c]{{$\box$}}\makebox[0pt][c]{\raisebox{2pt}{\tiny\ensuremath{\equiv}}}\,\ , } } { [ f_1|f_2 ] } } \\ { \textrm{(empty ) } } \quad \dfrac{}{{\mathbb{0 } } } \qquad { \textrm{(initial ) } } \quad \dfrac{b}{{[\;]}_b^\box\colon { \mathbb{0}}\to b } \qquad { \textrm{(initial - u ) } } \quad \dfrac{g^\box\colon { \mathbb{0}}\to b}{g { { \,\,\makebox[0pt][c]{{$\box$}}\makebox[0pt][c]{\raisebox{2pt}{\tiny\ensuremath{\equiv}}}\,\,}}{[\;]}_b } \\ \hline \end{array}\ ] ] in the logic @xmath11 for comonads , each term has a decoration which is denoted as a superscript @xmath16 , @xmath17 or @xmath18 : a term is _ pure _ when its decoration is @xmath16 , it is an _ accessor _ ( or an _ observer _ ) when its decoration is @xmath17 and a _ modifier _ when its decoration is @xmath18 . each equation has a decoration which is denoted by replacing the symbol @xmath19 either by @xmath20 or by @xmath21 : an equation with @xmath20 is called _ strong _ , with @xmath21 it is called _ the inference rules of @xmath11 are obtained by introducing some _ conversion _ rules and by decorating the rules in figure [ fig : pattern - rules ] . when writing terms , if a decoration does not matter or if it is clear from the context , it may be omitted . * the conversion rules are : @xmath22 the conversions for terms are _ upcasting _ conversions . + we will always use them in a _ safe _ way , by interpreting them as injections . this allows to avoid any specific notation for these conversions ; an accessor @xmath23 may be converted to a modifier which is denoted @xmath24 : both have the same name although they are distinct terms ; similarly , a pure term @xmath25 may be converted to @xmath26 or to @xmath27 . an equation between terms with distinct decorations does not imply any downcasting of its members ; for instance , if @xmath28 then it does not follow that @xmath29 is downcasted to @xmath30 . the conversions for equations mean that strong and weak equations coincide on pure terms and accessors and that each strong equation between modifiers can be seen as a weak one . * all rules of @xmath13 are decorated with @xmath16 for terms and @xmath20 for equations : the pure terms with the strong equations form a sublogic of @xmath11 which is isomorphic to @xmath13 . thus we get @xmath31 , @xmath32 , @xmath33 , @xmath34 , @xmath35}^{{(0)}}$ ] . * the congruence rules for equations take all decorations for terms and for equations , with one notable exception : the replacement rule for weak equations holds only when the replaced term is pure : @xmath36 * the categorical rules hold for all decorations and the decoration of a composed terms is the maximum of the decorations of its components . * the product rules hold only when the given terms are pure or accessors and the decoration of a pair is the maximum of the decorations of its components . thus , @xmath10-ary operations can be used only when their arguments are accessors . * the coproduct rules hold only when the given terms are pure and a copair is always pure , which is the maximum of the decorations of its components . thus , case distinction can be done only for pure terms . in order to give a meaning to the logic @xmath11 , let us consider a bicartesian category @xmath37 with a comonad @xmath38 satisfying the epi requirement , i.e. , @xmath39 is an epimorphism for each object @xmath40 ( the dual assumption is discussed in @xcite ) . then we get a model @xmath41 of the decorated logic @xmath11 as follows . * the types are interpreted as the objects of @xmath37 . * the terms are interpreted as morphisms of @xmath37 : a pure term @xmath42 as a morphism @xmath43 in @xmath37 ; an accessor @xmath44 as a morphism @xmath45 in @xmath37 ; and a modifier @xmath46 as a morphism @xmath47 in @xmath37 . * the conversion from pure terms to accessors is interpreted by mapping @xmath48 to @xmath49 . the epi requirement implies that this conversion is safe . * the conversion from accessors to modifiers is interpreted by mapping @xmath50 to @xmath51 . it is easy to check that this conversion is safe . * when a term @xmath29 has several decorations ( because it is pure or accessor , and thus can be upcasted ) we will denote by @xmath29 any one of its interpretations : a pure term @xmath42 may be interpreted as @xmath43 and as @xmath52 and as @xmath47 , and an accessor @xmath44 as @xmath52 and as @xmath47 . the choice will be clear from the context , and when several choices are possible they will give the same result , up to conversions . for this reason , we will describe the interpretation of the rules only for the largest possible decorations . * the identity @xmath53 is interpreted as @xmath54 in @xmath37 ; * the composition of two modifiers @xmath55 and @xmath56 is interpreted as @xmath57 in @xmath37 . * an equation between modifiers @xmath58 is interpreted by an equality @xmath59 in @xmath37 . * a weak equation between modifiers @xmath60 is interpreted by an equality @xmath61 in @xmath37 . * the unit type is interpreted as the final object of @xmath37 and the term @xmath62 as the unique morphism from @xmath40 to @xmath63 in @xmath37 . * the product @xmath64 with its projections is interpreted as the binary product in @xmath37 and the pair of @xmath65 and @xmath66 as the pair @xmath67 in @xmath37 . * the empty type is interpreted as the initial object of @xmath37 and the term @xmath35}_a^{{(0)}}\colon { \mathbb{0}}\to a$ ] as the unique morphism from @xmath68 to @xmath40 in @xmath37 . * the coproduct @xmath69 with its coprojections is interpreted as the binary coproduct in @xmath37 and the copair of @xmath70 and @xmath71 as the copair @xmath72 } \colon a_1+a_2\to { t}b$ ] in @xmath37 . the dual of the decorated logic @xmath11 for a comonad is the decorated logic @xmath12 for a monad . thus , the grammar of @xmath12 is the same as the grammar of @xmath11 , but a term with decoration @xmath17 is now called a _ constructor_. the rules for @xmath12 are nearly the same as the corresponding rules for @xmath11 , except that for weak equations the replacement rule always holds while the substitution rule holds only when the substituted term is pure : @xmath73 in the rules for pairs and copairs , the decorations are permuted . the logic @xmath12 can be interpreted dually to @xmath11 . let @xmath37 be a bicartesian category and @xmath0 a monad on @xmath37 satisfying the mono requirement , which means that @xmath74 is a monomorphism for each object @xmath40 . then we get a model @xmath75 of the decorated logic @xmath12 , where a constructor @xmath44 is interpreted as a morphism @xmath76 in @xmath37 and a weak equation @xmath60 is interpreted as an equality @xmath77 in @xmath37 . let us consider a distributive category @xmath37 with epimorphic projections and with a distinguished object @xmath7 called the _ object of states_. we consider the comonad @xmath38 with endofunctor @xmath78 , with counit @xmath79 made of the projections @xmath80 , and with comultiplication @xmath81 which `` duplicates '' the states , in the sense that @xmath82 where @xmath83 is the projection . we call this comonad the _ comonad of state_. it is sometimes called the _ product comonad _ , and it is different from the _ costate comonad _ or _ store comonad _ with endofuntor @xmath84 @xcite . the category @xmath37 with the comonad of states provides a model of the logic @xmath11 . we can extend @xmath11 into a logic @xmath85 dedicated to the state comonad . first , because of the specific choice of the comonad @xmath78 , we can add new decorations to the rule patterns for pairs in @xmath11 , involving modifiers : there is a _ left pair _ @xmath86 of an accessor @xmath87 and a modifier @xmath88 , satisfying the first three rules in figure [ fig : state - prod ] . there are also three rules ( omitted ) , symmetric to these ones , for the _ right pair _ @xmath89 of a modifier @xmath90 and an accessor @xmath91 . the interpretation of the left pair @xmath92 is the pair @xmath93 of @xmath94 and @xmath95 . moreover , the rule expresses the fact that , when @xmath78 , two modifiers coincide as soon as they return the same result and modify the state in the same way . @xmath96_l}^{{(2)}}\colon a \to b_1\times b_2 } \\ { \textrm{(l - pair - eq ) } } & \dfrac { f_1^{{(1)}}\colon a\to b_1 \quad f_2^{{(2)}}\colon a \to b_2 } { { \mathit{pr}}_1^{{(0)}}\circ { \langle f_1,f_2 \rangle_l}^{{(2)}}{\sim}f_1^{{(1)}}\quad { \mathit{pr}}_2^{{(0)}}\circ { \langle f_1,f_2 \rangle_l}^{{(2)}}{\cong}f_2^{{(2 ) } } } \\ { \textrm{(l - pair - u ) } } & \dfrac{g^{{(2)}}{\!\colon\!}a { \!\to\!}b_1{\!\times\!}b_2 { \;\;}f_1^{{(1)}}{\!\colon\!}a{\!\to\!}b_1 \quad f_2^{{(2)}}{\!\colon\!}a { \!\to\!}b_2 { \;\;}{\mathit{pr}}_1^{{(0)}}\circ g { \sim}f_1 { \;\;}{\mathit{pr}}_2^{{(0)}}\circ g { \cong}f_2 } { g^{{(2)}}{\cong}{\langle f_1,f_2 \rangle_l}^{{(2 ) } } } \\ \hline { \textrm{(effect ) } } & \dfrac{f , g\colon a \to b \quad f{\sim}g \quad { \langle \ ; \rangle}_a\circ f { \cong}{\langle \ ; \rangle}_a\circ g } { f{\cong}g } \\ \hline \end{array}\ ] ] for each set @xmath97 of _ locations _ ( or identifiers ) , additional grammar and rules for the logic @xmath85 are given in figure [ fig : state - lookup ] . we extend the grammar of @xmath11 with a type @xmath98 , an accessor @xmath99 and a modifier @xmath100 for each location @xmath101 , and we also extend its rules . the rule asserts that two functions without result coincide as soon as they coincide when observed at each location . together with the rule it implies that two functions coincide as soon as they return the same value and coincide on each location . @xmath102 for each family of objects @xmath103 in @xmath37 such that @xmath104 we build a model @xmath105 of @xmath85 , which extends the model the model @xmath41 of @xmath11 with functions for looking up and updating the locations . the types @xmath98 are interpreted as the objects @xmath98 and the accessors @xmath99 as the projections from @xmath7 to @xmath98 . then the interpretation of each modifier @xmath106 is the function from @xmath107 to @xmath7 defined as the tuple of the functions @xmath108 where @xmath109 is the projection from @xmath107 to @xmath98 and @xmath110 is made of the projection from @xmath107 to @xmath7 followed by @xmath111 when @xmath112 . the logic we get , and its model , are essentially the same as in @xcite : thus , the pattern for a comonad in section [ sec : patterns ] can be seen as a generalization to arbitrary comonads of the approach in @xcite . since we have assumed that the category @xmath37 is _ distributive _ we get new decorations for the rule patterns for coproducts : the copair of two modifiers now exists , the corresponding decorated rules are given in figure [ fig : state - coprod ] . the interpretation of the modifier @xmath72}$ ] , when both @xmath113 and @xmath114 are modifiers , is the composition of @xmath72}\colon ( a_1\times s)+(a_2\times s)\to b\times s$ ] with the inverse of the canonical morphism @xmath115 : this inverse exists because @xmath37 is distributive . @xmath116}^{{(2)}}\colon a_1\!+\!a_2 \to b } \\ { \textrm{(copair - eq ) } } & \dfrac { f_1^{{(2)}}\colon a_1 \to b { \;\;}f_2^{{(2)}}\colon a_2 \to b } { { [ f_1|f_2 ] } \circ { \mathit{in}}_1 { \cong}f_1 \quad { [ f_1|f_2 ] } \circ { \mathit{in}}_2 { \cong}f_2 } \\ { \textrm{(copair - u ) } } & \dfrac{f_i^{{(2)}}{\!\colon\!}a_i { \!\to\!}b { \;\;}g^{{(2)}}{\!\colon\!}a_1\!{\!+\!}\!a_2 { \!\to\!}b { \;\;}g\circ { \mathit{in}}_i { \cong}f_i } { g^{{(2)}}{\cong } { [ f_1|f_2 ] } ^{{(2 ) } } } \\ \hline \end{array}\ ] ] to conclude with states , let us look at the constructions for conditionals and binary operations in the language for states . the rules in figure [ fig : state - coprod ] provide conditionals . there is no binary product of modifiers , but there is a left product of a constructor and a modifier and a right product of a modifier and a constructor . it follows that the _ left and right sequential products _ of two modifiers @xmath113 and @xmath114 can be defined , as in @xcite , by composing , e.g. , the left product of an identity and @xmath113 with the right product of @xmath114 and an identity . a major feature of this approach is that , for states , sequential products are defined without any new ingredient : no kind of strength , in contrast with the approach using the strong monad of states @xmath117 @xcite , no `` external '' decoration for equations , in contrast with @xcite . this property is due to the introduction of the intermediate notion of _ accessors _ between pure terms ( or _ values _ ) and modifiers ( or _ computations _ ) . now we use the decorated logic @xmath85 for proving that the decorated theory for states is hilbert - post complete . this result is proved in ( * ? ? ? * prop.2.40 ) in the framework of lawvere theories . here we give a proof in the decorated logic for states . this proof has been checked in coq . the logic we use is the fragment @xmath118 of @xmath85 which involves neither products nor coproducts nor the empty type ( but which involves the unit type ) . the _ theory of state _ , denoted @xmath119 , is the family of equations which may be derived from the axioms of @xmath118 using the rules of @xmath118 . more generally , a _ theory _ @xmath120 with respect to @xmath118 is a family of equations between terms of @xmath118 which is saturated with respect to the rules of @xmath118 . a theory @xmath121 is an _ extension _ of a theory @xmath120 if it contains all the equations of @xmath120 . two families of equations are called _ equivalent _ if each one can be derived from the other with the rules of @xmath118 . as in ( * prop.2.40 ) , for the sake of simplicity it is assumed that there is a single location @xmath101 , and we write @xmath122 , @xmath123 and @xmath124 instead of @xmath98 , @xmath125 and @xmath126 . then there is a single axiom @xmath127 . in addition , it is assumed that all types are _ inhabited _ , in the sense that for each type @xmath101 there exists a closed pure term with type @xmath101 . [ theo : equations ] every equation between terms of @xmath118 is equivalent to four equations between pure terms . the proof is obtained by merging the two parts of proposition [ prop : equations ] , which is proved in appendix [ app - complete ] . [ prop : equations ] 1 . [ prop : equations - acc ] every equation between accessors is equivalent to two equations between pure terms . [ prop : equations - modi ] every equation between modifiers is equivalent to two equations between accessors . roughly speaking , a theory ( with respect to some logic ) is said _ syntactically complete _ if no unprovable axiom can be added to the theory without introducing an inconsistency . more precisely , a theory with respect to the equational logic is _ hilbert - post complete _ if it is consistent and has no consistent proper extension ( * ? ? ? * definition 2.8 . ) . since we use a decorated version of the equational logic , we have to define a decorated version of hilbert - post completeness . [ defi : complete ] with respect to the logic @xmath118 , a theory @xmath120 is _ consistent _ if there is an equation which is not in @xmath120 . an extension @xmath121 of a theory @xmath120 is a _ pure extension _ if it is generated by @xmath120 and by equations between pure terms . it is a _ proper extension _ if it is not a pure extension . a theory @xmath120 is _ hilbert - post complete _ if it is consistent and has no consistent proper extension . the proof of theorem [ theo : complete ] relies on theorem [ theo : equations ] . we do not have to assume that the interpretation of the type @xmath122 is a countable set . we have assumed that @xmath97 is a singleton , but we conjecture that our result can be generalized to any set of locations , without any finiteness condition . [ theo : complete ] the theory for state is hilbert - post complete . the theory @xmath119 is consistent : it can not be proved that @xmath128 . let us consider an extension @xmath120 of @xmath119 and let @xmath129 be the theory generated by @xmath119 and by the equations between pure terms in @xmath120 . thus , @xmath129 is a pure extension of @xmath119 and @xmath120 is an extension of @xmath129 . let us consider an arbitrary equation @xmath130 in @xmath120 , according to theorem [ theo : equations ] we get a family @xmath9 of equations between pure terms which is equivalent to the given equation @xmath130 . since @xmath130 is in @xmath120 and @xmath120 is saturated , the equations in @xmath9 are also in @xmath120 , hence they are in @xmath129 . since @xmath9 is in @xmath129 and @xmath129 is saturated , the equation @xmath130 is also in @xmath129 . this proves that @xmath131 , so that the theory @xmath119 has no proper extension . let us consider a bicartesian category @xmath37 with monomorphic coprojections and with a distinguished object @xmath9 called the _ object of exceptions_. we do not assume that @xmath37 is distributive ( it would not help ) nor codistributive , because usually this is not the case . the _ monad of exceptions _ on @xmath37 is the monad @xmath0 with endofunctor @xmath132 , its unit @xmath133 is made of the coprojections @xmath134 , and its multiplication @xmath135 `` merges '' the exceptions , in the sense that @xmath136}\colon ( a+e)+e\to a+e$ ] where @xmath137 is the coprojection . it satisfies the mono requirement because the coprojections are monomorphisms . thus , the category @xmath37 with the monad of exceptions provides a model of the logic @xmath12 . the name of the decorations is adapted to the monad of exceptions : a constructor is called a _ propagator _ : it may raise an exception but can not recover from an exception , so that it has to propagate all exceptions ; a modifier is called a _ catcher_. for this specific monad @xmath132 , it is possible to extend the logic @xmath12 as @xmath138 , called the _ logic for exceptions _ , so that @xmath37 with @xmath132 can be extended as a model @xmath139 of @xmath138 . first , dually to the left and right pairs for states in figure [ fig : state - prod ] , we get new decorations to the rule patterns for copairs in @xmath12 , involving modifiers , as in figure [ fig : exc - coprod ] for the left copairs ( the rules for the right copairs are omitted ) . the interpretation of the left copair @xmath72_l}^{{(2 ) } } : a_1+a_2\to b$ ] is the copair @xmath72}:a_1+a_2+e\to b+e$ ] of @xmath140 and @xmath141 in @xmath37 . for instance , the coproduct of @xmath142 , with coprojections @xmath143 and @xmath35}_a^{{(0)}}:{\mathbb{0}}\to a$ ] , gives rise to the left copair @xmath72_l}^{{(2)}}:a \to b$ ] of any constructor @xmath144 with any modifier @xmath145 , which is characterized up to strong equations by @xmath72_l}{\sim}f_1 $ ] and @xmath72_l}{\cong}f_2 $ ] . this will be used in the construction of the @xmath146 expressions . moreover , the rule expresses the fact that , when @xmath132 , two modifiers coincide as soon as they coincide on ordinary values and on exceptions . @xmath147_l}^{{(2)}}\colon a_1\!+\!a_2 \to b } \\ { \textrm{(l - copair - eq ) } } & \dfrac { f_1^{{(1)}}\colon a_1 \to b { \;\;}f_2^{{(2)}}\colon a_2 \to b } { { [ f_1|f_2 ] _ l}^{{(2)}}\circ { \mathit{in}}_1^{{(0)}}{\sim}f_1^{{(1)}}\quad { [ f_1|f_2 ] _ l}^{{(2)}}\circ { \mathit{in}}_2^{{(0)}}{\cong}f_2^{{(2 ) } } } \\ { \textrm{(l - copair - u ) } } & \dfrac{g^{{(2)}}{\!\colon\!}a_1\!+\!a_2 { \!\to\!}b { \;\;}f_1^{{(1)}}{\!\colon\!}a_1 { \!\to\!}b { \;\;}f_2^{{(2)}}{\!\colon\!}a_2 { \!\to\!}b { \;\;}g\circ { \mathit{in}}_1 { \sim}f_1 { \;\;}g\circ { \mathit{in}}_2 { \cong}f_2 } { g^{{(2)}}{\cong } { [ f_1|f_2 ] _ l}^{{(2 ) } } } \\ \hline { \textrm{(effect ) } } & \dfrac{f , g\colon a \to b \quad f{\sim}g \quad f\circ { [ \;]}_a { \cong}g\circ { [ \;]}_a } { f{\cong}g } \\ \hline \end{array}\ ] ] for each set @xmath148 of _ exception names _ , additional grammar and rules for the logic @xmath138 are given in figure [ fig : exc - tag ] . we extend the grammar of @xmath12 with a type @xmath149 , a propagator @xmath150 and a catcher @xmath151 for each exception name @xmath152 , and we also extend its rules . the logic @xmath138 obtained performs the _ core _ operations on exceptions : the _ tagging _ operations encapsulate an ordinary value into an exception , and the _ untagging _ operations recover the ordinary value which has been encapsulated in an exception . this may be generalized by assuming a hierarchy of exception names @xcite . the rule asserts that two functions without argument coincide as soon as they coincide on each exception . together with the rule it implies that two functions coincide as soon as they coincide on their argument and on each exception . @xmath153}_{v_t}\circ{\mathtt{tag}}_r } \\ { \textrm{(local - global ) } } & \dfrac { f , g\colon { \mathbb{0}}\to b \quad \mbox{for all } t\in { \mathit{exn}}\ ; f\circ { \mathtt{tag}}_t { \sim}g\circ { \mathtt{tag}}_t } { f{\cong}g } \\ \hline \end{array}\ ] ] for each family of objects @xmath154 in @xmath37 such that @xmath155 we build a model @xmath139 of @xmath138 , which extends the model the model @xmath156 of @xmath12 with functions for tagging and untagging the exceptions . the types @xmath149 are interpreted as the objects @xmath149 and the propagators @xmath150 as the coprojections from @xmath149 to @xmath9 . then the interpretation of each catcher @xmath151 is the function @xmath157 defined as the cotuple ( or case distinction ) of the functions @xmath158 where @xmath159 is the coprojection of @xmath149 in @xmath160 and @xmath161 is made of @xmath162 followed by the coprojection of @xmath9 in @xmath160 when @xmath163 . this can be illustrated , in an informal way , as follows : @xmath164 encloses its argument @xmath165 in a box with name @xmath152 , while @xmath166 opens every box with name @xmath152 to recover its argument and returns every box with name @xmath167 without opening it : @xmath168^{{\mathtt{tag}}_t } & & * + [ f-]{a } \ar@{}[r]_(.2){t } & & * + [ f-]{a } \ar@{}[r]_(.2){t } \ar[rr]^{{\mathtt{untag}}_t } & & a \\ & & & & & * + [ f-]{a } \ar@{}[r]_(.2){r } \ar[rr]^{{\mathtt{untag}}_t } & & * + [ f-]{a } \ar@{}[r]_(.2){r } & \\ } \ ] ] since we did not assume that the category @xmath37 is codistributive we can not get products of modifiers in a way dual to the coproducts of modifiers for states . however these rules have not been used for proving the hilbert - post completeness of the theory for state . thus by duality from theorem [ theo : complete ] we get `` for free '' a result about the core language for exceptions . [ coro : complete ] the core theory for exceptions is hilbert - post complete . we have obtained a logic @xmath138 for exceptions , with the core operations for tagging and untagging . this logic provides a direct access to catchers ( the untagging functions ) , which is not provided by the usual mechanism of exceptions in programming languages . in fact the core operations remain _ private _ , while there is a _ programmer s _ language , which is _ public _ , with no direct access to the catchers . the programmer s language for exceptions provides the operations for _ raising _ and _ handling _ exceptions , which are defined in terms of the core operations . this language has no catcher : the only way to catch an exception is by using a @xmath146 expression , which itself propagates exceptions . thus , all terms of the programmer s language are propagators . this language does not include the private tagging and untagging operations , but the public @xmath169 and @xmath146 constructions , which are defined in terms of @xmath170 and @xmath171 . for the sake of simplicity we assume that only one type of exception is handled in a @xmath146 expression , the general case is treated in @xcite . the main ingredients for building the programmer s language from the core language are the coproducts @xmath172 and a new conversion rule for terms . the _ downcast _ conversion of a catcher to a propagator could have been defined in section [ sec : patterns ] for the logic @xmath11 , and dually for the logic @xmath12 ; the rule is : @xmath173 this downcasting conversion from catchers to propagators is interpreted by mapping @xmath174 to @xmath175 . it is related to weak equations : @xmath176 , and @xmath177 if and only if @xmath178 . but the downcasting conversion is _ unsafe _ : several catchers may be downcasted to the same propagator . this powerful operation turns an effectful term to an effect - free one ; since it is not required for states nor for the core language for exceptions , we did not introduce it earlier . [ defi : exc ] for each type @xmath179 and each exception name @xmath152 , the propagator @xmath180 is : @xmath181}_b^{{(0)}}\circ { \mathtt{tag}}_t^{{(1)}}\colon v_t\to b\ ] ] for each each propagator @xmath182 , each exception name @xmath152 and each propagator @xmath183 , the propagator @xmath184 is defined in three steps , involving two catchers @xmath185 and @xmath186 , as follows : @xmath187}_b^{{(0)}}\ ; ] } ^{{(1)}}\circ{\mathtt{untag}}_t^{{(2)}}\colon { \mathbb{0}}\to b \\ { \mathtt{try}}(f){\mathtt{catch}}(t{\rightarrow}g)^{{(2)}}= { [ \ ; { \mathit{id}}_b \;|\ ; { \mathtt{catch}}(t{\rightarrow}g ) \ ; ] _ l}^{{(2)}}\circ f^{{(1)}}\colon a\to b \\ { \mathtt{try}}(f){\mathtt{catch}}(t{\rightarrow}g)^{{(1)}}= { { \downarrow}}({\mathtt{try}}(f){\mathtt{catch}}(t{\rightarrow}g ) ) \colon a\to b \\ \end{array}\ ] ] this means that raising an exception with name @xmath152 consists in tagging the given ordinary value ( in @xmath149 ) as an exception and coerce it to any given type @xmath179 . for handling an exception , the intermediate expressions @xmath188 and @xmath189 are private catchers and the expression @xmath190 is a public propagator : the downcast operator prevents it from catching exceptions with name @xmath152 which might have been raised before the @xmath190 expression is considered . the definition of @xmath190 corresponds to the java mechanims for exceptions @xcite . the definition of @xmath190 corresponds to the following control flow , where ` exc ? ` means `` _ _ is this value an exception ? _ _ '' , an _ abrupt _ termination returns an uncaught exception and a _ normal _ termination returns an ordinary value ; this corresponds , for instance , to the java mechanims for exceptions @xcite . @xmath191 & & \\ & { \txt{exc?}}\ar[ld]_{y}\ar[rd]^{n } & & \\ { \boxed{\mathit{abrupt } } } & & f^{{(1)}}\ar[d ] & \\ & & { \txt{exc?}}\ar[ld]_{y}\ar[rd]^{n } & \\ & { \mathtt{untag}}_t^{{(2)}}\ar[d ] & & { \boxed{\mathit{normal}}}\\ & { \txt{exc?}}\ar[ld]_{y}\ar[rd]^{n } & & \\ { \boxed{\mathit{abrupt } } } & & g^{{(1)}}\ar[d ] & \\ & & \txt{{\boxed{\mathit{normal}}}\mbox { or } { \boxed{\mathit{abrupt } } } } \\ } \ ] ] to conclude with exceptions , let us look at the constructions for case distinction and binary operations in the programmer s language for exceptions , which means , copairs and pairs of _ constructors_. the general rules of the logic @xmath12 include coproducts of constructors ( figure [ fig : pattern - rules ] ) , which provide case distinction for all terms in the programmer s language for exceptions . but the general rules for a monad do not include binary products involving a constructor , hence they can not be used for dealing with binary operations in the programmer s language for exceptions when at least an argument is not pure . indeed , if @xmath65 is pure and @xmath192 does raise an exception , it is in general impossible to find @xmath193 such that @xmath194 and @xmath195 . however , there are several ways to formalize the fact of first evaluating @xmath113 then @xmath114 : for instance by using a strong monad @xcite , or a sequential product @xcite , or productors @xcite . the sequential product approach can be used in our framework ; it requires the introduction of a third kind of `` equations '' , in addition to the strong and weak equations , which corresponds to the usual order between partial functions : details are provided in @xcite . we have presented two patterns giving sound inference systems for effects arising from a monad or a comonad . we also gave detailed examples of applications of these patterns to the state and the exceptions effects . the obtained decorated proof system for states has been implemented in coq , so that the given proofs can be automatically verified . we plan to adapt this logic to local states ( with allocation ) in order to provide a decorated proof of the completeness theorem in @xcite . from this implementation , we plan to extract the generic part corresponding to the comonad pattern , dualize it and extend it to handle the programmer s language for exceptions . then a major issue is scalability : how can we combine effects ? within the framework of this paper , it may seem difficult to guess how several effects arising from either monads or comonads can be combined . however , as mentioned in the introduction , this paper deals with two patterns for instanciating the more general framework of decorated logics @xcite . decorated logics are based on spans in a relevant category of logics , so that the combination of effects can be based on the well - known composition of spans . [ [ acknowledgment . ] ] acknowledgment . + + + + + + + + + + + + + + + we are grateful to samuel mimram for enlightning discussions . 00 csar domnguez , dominique duval . diagrammatic logic applied to a parameterization process . mathematical structures in computer science 20 , p. 639 - 654 ( 2010 ) . jean - guillaume dumas , dominique duval , laurent fousse , jean - claude reynaud . decorated proofs for computational effects : states . accat 2012 . electronic proceedings in theoretical computer science 93 , p. 45 - 59 ( 2012 ) . jean - guillaume dumas , dominique duval , laurent fousse , jean - claude reynaud . a duality between exceptions and states . mathematical structures in computer science 22 , p. 719 - 722 ( 2012 ) . jean - guillaume dumas , dominique duval , jean - claude reynaud . cartesian effect categories are freyd - categories . journal of symbolic computation 46 , p. 272 - 293 ( 2011 ) . jean - guillaume dumas , dominique duval , jean - claude reynaud . a decorated proof system for exceptions . arxiv:1310.2338 ( 2013 ) . jean - guillaume dumas , dominique duval , burak ekici , damien pous . formal verification in coq of program properties involving the global state effect . arxiv:1310.0794 ( 2013 ) . jeremy gibbons , michael johnson . relating algebraic and coalgebraic descriptions of lenses bx 2012 . eceasst 49 ( 2012 ) . james gosling , bill joy , guy steele , gilad bracha . the java language specification , third edition . addison - wesley longman ( 2005 ) . bart jacobs . a formalisation of java s exception mechanism . esop 2001 . lncs , vol . 2028 , p. 284 - 301 springer ( 2001 ) . bart jacobs and jan rutten . an introduction to ( co)algebras and ( co)induction . in : d. sangiorgi and j. rutten ( eds ) , advanced topics in bisimulation and coinduction , p.38 - 99 , 2011 . paul blain levy . monads and adjunctions for global exceptions . mfps 2006 . electronic notes in theoretical computer science 158 , p. 261 - 287 ( 2006 ) . john m. lucassen , david k. gifford . polymorphic effect systems . popl 1988 . acm press , p. 47 - 57 . paul - andr mellis . segal condition meets computational effects . lics 2010 . p. 150 - 159 , ieee computer society ( 2010 ) . eugenio moggi . notions of computation and monads . information and computation 93(1 ) , p. 55 - 92 ( 1991 ) . eugenio moggi and sonia fagorzi . a monadic multi - stage metalanguage . fossacs 2003 , lncs , vol . 2620 , p. 358 - 374 , springer ( 2003 ) . matija pretnar . the logic and handling of algebraic effects . university of edinburgh 2010 . gordon d. plotkin , john power . notions of computation determine monads . fossacs 2002 . lncs , vol . 2620 , p. 342 - 356 , springer ( 2002 ) . gordon d. plotkin , matija pretnar . handlers of algebraic effects . esop 2009 . lncs , vol . 5502 , p. 80 - 94 , mpringer ( 2009 ) . lutz schrder , till mossakowski . generic exception handling and the java monad . amast 2004 . lncs , vol . 3116 , p. 443 - 459 , springer ( 2004 ) . sam staton . completeness for algebraic theories of local state . fossacs 2010 . lncs , vol . 6014 , p. 48 - 63 , springer ( 2010 ) . ross tate . the sequential semantics of producer effect systems . popl 2013 . acm press , p. 15 - 26 ( 2013 ) . tarmo uustalu , varmo vene . comonadic notions of computation . cmcs 2008 . entcs 203 , p. 263 - 284 ( 2008 ) . philip wadler . the essence of functional programming . popl 1992 . acm press , p. 1 - 14 the logic used in this appendix is the fragment @xmath118 of the decorated logic for states @xmath85 which involves neither products nor coproducts nor the empty type , but which involves the unit type . for the sake of simplicity it is assumed that there is a single location @xmath101 , and we write @xmath122 , @xmath196 and @xmath197 instead of @xmath98 , @xmath198 and @xmath199 . then there is a single axiom @xmath200 . in section [ sec : states ] , the proof of hilbert - post completeness in theorem [ theo : complete ] relies on proposition [ prop : equations ] , which is restated here as proposition [ prop : equations - app ] . the aim of this appendix is to prove proposition [ prop : equations - app ] . 1 . [ lemm : eqns - rule - unit ] @xmath201 2 . [ lemm : eqns - rule - val ] @xmath202 3 . [ lemm : eqns - strong ] + @xmath203 4 . [ lem : pure_mid_id ] @xmath204 5 . [ lemm : onepureacc ] @xmath205 6 . [ lemm : lkppureequal ] @xmath206 7 . [ lemm : lkpuseless ] @xmath207 1 . consequence of the observational rule with only one location . 2 . consequence of [ lemm : eqns - rule - unit ] applied to @xmath208 : indeed , from the axiom @xmath209 we get @xmath210 . 3 . from axiom @xmath209 by substitution we get @xmath211 ; thus , point [ lemm : eqns - rule - unit ] implies @xmath212 . 4 . from @xmath213 , as @xmath214 is pure , by the weak replacement we have @xmath215 . then , weak substitution with @xmath165 yields @xmath216 . 5 . we know that @xmath217 . + it follows that @xmath218 . let @xmath219 . composing with @xmath197 we get @xmath220 . using the axiom @xmath209 and the replacement rule for @xmath21 , which can be used here because both @xmath221 and @xmath214 are pure , we get @xmath222 . since weak and strong equations coincide on pure terms we get @xmath223 . let @xmath224 . by point [ lemm : onepureacc ] above we get @xmath225 , thus @xmath226 . then by point [ lemm : lkppureequal ] above we get @xmath227 . now , let us prove proposition [ prop : canonical - form ] , which says that , up to strong equations , it can be assumed that there is at most one occurrence of @xmath196 in any accessor and at most one occurrence of @xmath197 in any modifier . 1 . for each accessor @xmath228 , if @xmath165 is not pure then there is a pure term @xmath229 such that @xmath230 2 . for each modifier @xmath231 , if @xmath29 is not an accessor then there is an accessor @xmath232 and a pure term @xmath233 such that @xmath234 1 . if @xmath228 is not pure then it contains at least one occurrence of @xmath235 . thus , it can be written in a unique way as @xmath236 for some pure term @xmath229 and some accessor @xmath237 . since @xmath237 is such that @xmath238 , the result follows . if @xmath231 is not an accessor then it contains at least one occurrence of @xmath239 . thus , it can be written in a unique way as @xmath240 for some accessor @xmath241 and some modifier @xmath242 . from point [ prop : canonical - form - acc ] , we also have that @xmath243 for some pure term @xmath229 so that @xmath244 . * if @xmath113 is an accessor , the result follows with @xmath245 . * otherwise , @xmath90 contains at least one occurrence of @xmath239 . thus , it can be written in a unique way as @xmath246 for some accessor @xmath247 and some modifier @xmath248 . according to point [ prop : canonical - form - acc ] applied to the accessor @xmath249 , either @xmath249 is pure or @xmath250 for some pure term @xmath251 * * if @xmath250 then @xmath252 . the axiom @xmath253 and the replacement and substitution rules for @xmath21 ( since @xmath254 is pure ) yield @xmath255 . then it follows from point [ lemm : eqns - rule - val ] in lemma [ lemm : eqns ] that @xmath256 , and since @xmath257 we get @xmath258 . the result follows by induction on the number of occurrences of @xmath197 in @xmath29 : indeed , there is one less occurrence of @xmath197 in @xmath259 than in @xmath260 . * * if @xmath249 is pure then @xmath261 from point [ lemm : onepureacc ] in lemma [ lemm : eqns ] . thus the previous proof applies by replacing @xmath249 with @xmath262 . 1 . * if @xmath267 is pure , since @xmath268 ( because @xmath269 is an accessor ) we get @xmath270 , thus the result is obtained with @xmath271 . * otherwise , we have just proved that @xmath272 with @xmath273 , then @xmath274 and @xmath275 . if @xmath276 is an accessor , since @xmath277 we get @xmath278 , thus the result is obtained with @xmath279 . * otherwise , we have just proved that @xmath280 with @xmath281 , then @xmath282 and @xmath283 . we can now prove proposition [ prop : equations - app ] on which the hilbert - post completeness theorem relies . this proof has been checked with the coq proof assistant using the system for states of @xcite . the coq library with the inference system is available there : http://coqeffects.forge.imag.fr . the single proof of the following proposition ( roughly 16 pages in coq ) is directly available there : http://coqeffects.forge.imag.fr/hpcompletecoq.v . 1 . we prove that for any accessors @xmath290 there are three cases : 1 . either they are both pure and @xmath291 is the required equation between pure terms . 2 . either they are both accessors and it can be derived from @xmath292 that @xmath293 for some pure terms @xmath294 . 3 . or one of them is pure and the other one is an accessor and it can be derived from @xmath292 that @xmath293 and @xmath295 for some pure terms @xmath294 and @xmath296 . + we prove , moreover , that the converse also hold . 1 . as already mentioned , if @xmath297 and @xmath298 are both pure and @xmath291 is the required equation between pure terms . 2 . if neither @xmath297 nor @xmath298 is pure , then according to proposition [ prop : canonical - form ] @xmath299 and @xmath300 for some pure terms @xmath301 . * starting from the equation @xmath302 we thus get @xmath303 . then , using the assumption , for any function @xmath289 , we have that @xmath304 . now @xmath305 . this , together with the axiom @xmath306 and the replacement rule for @xmath21 ( which can be used here because both @xmath254 and @xmath307 are pure ) yield @xmath308 . as the latter are both pure terms we also have @xmath309 . * conversely , if @xmath310 then @xmath311 , which means that @xmath312 . the only remaining case is w.l.o.g . if @xmath297 is pure and @xmath298 is not . * then @xmath313 from proposition [ prop : canonical - form ] as previously and @xmath314 satisfies @xmath315 for any assumed @xmath289 . indeed from @xmath316 we get @xmath317 but , on the one hand , @xmath318 so that point [ lemm : onepureacc ] in lemma [ lemm : eqns ] gives @xmath319 with @xmath320 . on the other hand , @xmath321 so that @xmath322 . thus equation ( [ eq : fullpureacc ] ) rewrites as @xmath323 and point [ lemm : lkpuseless ] in lemma [ lemm : eqns ] yields @xmath324 thus now we also have @xmath325 . from the original equation @xmath326 we finally get @xmath327 * conversely , we start from @xmath328 and @xmath329 satisfying both equations ( [ eq : pureaccfirst ] ) and ( [ eq : pureaccsecond ] ) . then , we define @xmath330 which satisfies @xmath331 thanks to equation ( [ eq : pureaccfirst ] ) . the latter is also @xmath332 which is thus @xmath333 thanks to equation ( [ eq : pureaccsecond ] ) . the rule for states means that two modifiers coincide as soon as they return the same result and modify the state in the same way . this means that @xmath334 if and only if @xmath335 and @xmath336 . thanks to corollary [ coro : downcast ] the equation @xmath335 is equivalent to an equation between accessors . it remains to prove that the equation @xmath336 is also equivalent to an equation between accessors . + for @xmath337 , since @xmath338 , proposition [ prop : canonical - form ] says that @xmath339 for some accessor @xmath340 . thus , @xmath336 if and only if @xmath341 . let us check that this equation is equivalent to @xmath291 . + clearly if @xmath342 then @xmath341 . conversely , if @xmath343 then @xmath344 and since @xmath253 we get @xmath345 , which is the same as @xmath291 because @xmath297 and @xmath298 are accessors . + thus , @xmath336 if and only if @xmath291 , as required . | this paper presents equational - based logics for proving first order properties of programming languages involving effects .
we propose two dual inference system patterns that can be instanciated with monads or comonads in order to be used for proving properties of different effects .
the first pattern provides inference rules which can be interpreted in the kleisli category of a monad and the cokleisli category of the associated comonad . in a dual way
, the second pattern provides inference rules which can be interpreted in the cokleisli category of a comonad and the kleisli category of the associated monad .
the logics combine a 3-tier effect system for terms consisting of pure terms and two other kinds of effects called constructors / observers and modifiers , and a 2-tier system for up - to - effects and strong equations .
each pattern provides generic rules for dealing with any monad ( respectively comonad ) , and it can be extended with specific rules for each effect .
the paper presents two use cases : a language with exceptions ( using the standard monadic semantics ) , and a language with state ( using the less standard comonadic semantics ) .
finally , we prove that the obtained inference system for states is hilbert - post complete . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
boundaries and interfaces are omnipresent in the colloidal world @xcite . geometric confinement introduces anisotropy in the diffusive motion of sub - micron particles , and the presence of neighbouring walls leads to a general slow - down of brownian motion due to hydrodynamic interactions of the diffusing particle with boundaries @xcite . the central quantity in this context is the near - wall hydrodynamic mobility tensor @xmath0 which is related to the diffusion tensor @xmath1 by the fluctuation - dissipation theorem @xmath2 recent years have brought significant advancement in experimental techniques which allow to explore near - wall dynamics in more detail , including optical microscopy @xcite and scattering techniques , such as evanescent wave dynamic light scattering @xcite . the latter is now a well - established tool which has profitably been used to investigate translational @xcite and rotational diffusion @xcite of spherical colloids in dilute suspensions . due to the complex nature of the experiments , available experimental data for non - spherical particles such as dumbbells @xcite or rods are still lacking proper interpretation . it is therefore particularly important in this context to understand the nature of hydrodynamic interactions of an axially symmetric particle with a wall , and has partially motivated this work . axisymmetric particles moving close to a boundary experience an additional anisotropic drag force on top of their own friction anisotropy stemming from their non - spherical shape . this coupling leads to a complicated behaviour , observed e.g. in simulations of such particles sedimenting next to a vertical wall @xcite , with the mobility of the particle depending on its position and orientation . available predictions for the near - wall mobility of an axisymmetric particle mostly feature a slender - body approach , yielding quite complex results for general wall - particle orientations near a wall @xcite or a fluid - fluid interface @xcite , analysed in detail in the context of sedimentation in several special alignments @xcite . on the other hand , previous numerical works involve the boundary integral method @xcite , finite element method @xcite or stochastic rotation dynamics @xcite from which empirical relations are extracted . the lack of theoretical predictions for the near - wall mobility of a rod - shaped and non - slender particle in an arbitrary configuration requires the use of more precise numerical methods . a possible way is to use advanced algorithms involving bead - models which take into account lubrication when the particles come close to the interface@xcite , which are rather costly . in order to fill this gap , in this work we derive a general form of the dominant correction to the bulk friction tensor due to the presence of a nearby no - slip wall from which the mobility tensor is calculated . this allows to verify and correct earlier predictions in terms of distance and orientation of the particle . importantly , the correction is valid for all axisymmetric particles , not just slender ones , provided that their bulk hydrodynamic properties are known . our analysis leads to a convenient representation of the mobility tensor in situations when the particle is moderately far from the wall . for the characteristic length of the body @xmath3 , the relative correction scales as @xmath4 , where @xmath5 is the wall - particle distance , and the exponent @xmath6 depends on the component of the friction matrix ( translational , rotational , or coupling terms ) . we provide explicit analytical expressions for the dominant correction to bulk translational and rotational parts of the friction tensor which are the main result of the paper . we use them to calculate the corrections to the friction tensor of an axially symmetric particle explicitly in terms of @xmath5 and the particle s inclination angle @xmath7 . by inverting the friction tensor , we then calculate the near - wall mobility tensor . the paper is organised as follows . first , we introduce the notion of friction and mobility tensors for a colloidal particle in sec . [ sec1 ] . in sec . [ sec : correction ] , we sketch the idea behind the derivation of the correction , which is then given explicitly in sec . [ sec : axisymm ] for axially symmetric particles . the theoretical predictions are compared to numerical simulations for an exemplary case in sec . [ simulation ] , followed by conclusions in sec . [ conclusions ] . appendix [ multipole ] contains the details of the multipole method and a description of the simulation method . details of the derivation of the correction are given in appendix [ derivation ] . we consider a single colloidal particle immersed in an incompressible newtonian solvent of shear viscosity @xmath8 . the configuration of the system is described by the position of the centre of the particle @xmath9 and its orientation which , for an axially symmetric particle , is specified by the unit vector @xmath10 pointing along the particle s symmetry axis . on the colloidal length scales and for time scales typical e.g. for scattering experiments , inertia of the fluid and the particle can be neglected . the flow field @xmath11 around the particle is then described by the stationary stokes equations@xcite @xmath12 where @xmath13 is the force density the particle exerts on the fluid when subjected to flow , and @xmath14 stands for modified pressure field which includes the effect of gravity . the flow disturbances caused by the presence of the particle in confined geometry are affecting the motion of the particle itself . these dynamic , solvent - mediated hydrodynamic interactions ( hi ) are long - ranged and have a pronounced effect on the dynamics of colloidal systems . this flow field may be superposed with an ambient linear flow @xmath15 satisfying the homogeneous stokes equations , with the vorticity and rate of strain defined at a point @xmath16 as @xmath17 with the bar denoting the symmetric and traceless part . given the force density , one can calculate the force , torque , and symmetric dipole moment ( stresslet ) exerted by the fluid on the particle according to @xmath18 where the integrals are performed over the particle surface @xmath19 . higher - order moments are defined in an analogous way . in result of the external flow , motion is induced , and the particle gains linear and angular velocities , @xmath20 and @xmath21 , respectively . owing to linearity of the stokes equations , the force moments @xmath22 , @xmath23 and @xmath24 , are linearly related to the velocity moments via the generalised friction ( or resistance ) tensor@xcite @xmath25 above we have decomposed the generalised friction tensor into 9 sub - matrices . the indices @xmath26 and @xmath27 denote the translational and rotational parts , respectively . the tensors @xmath28 and @xmath29 describe the translation - rotation coupling , and the tensors with superscript @xmath30 describe the response of the particle to an external elongational flow . in most cases , it is sufficient to consider only the @xmath31 friction matrix @xmath32 relating the force and torque to linear and angular velocities . here , we extend the friction matrix to the symmetric dipole moment subspace , since these elements turn out to be essential for the calculation of the correction to the friction matrix in the presence of a wall . in a complementary problem , if the forces and torques are known , the particle motion may be resolved by determining the mobility tensor @xmath33 which is related to the friction tensor by inversion @xmath34 using the lorentz reciprocal theorem @xcite , one may prove the symmetry properties of the mobility tensors . in a bulk system , the mobility tensor and the friction tensor , denoted by @xmath35 and @xmath36 , respectively , do not depend on the position of the particle due to translational invariance . the situation is different if a confining boundary is present , since symmetry is broken and the hydrodynamic tensors depend both on the distance to the boundary , and on the relative orientation of the particle with respect to the surface . the friction tensors of a near - wall particle , @xmath37 , may be written as @xmath38 in the course of this work , we derive analytic formulae for the first - order approximation to @xmath39 , with the expansion parameter being @xmath40 , the ratio of the characteristic size of the particle , @xmath3 , to the wall - particle distance @xmath5 . by inverting @xmath37 from eq . , we arrive at a convenient approximation to the near - wall mobility @xmath41 . in order to determine the flow around a particle in a half - space bounded by an infinite , planar wall at @xmath42 , one has to solve the stokes equations ( [ stokes ] ) with the no - slip boundary condition @xmath43 at the surface . due to linearity , eq . can be transformed into the integral form @xmath44 for an unbounded fluid , the green s function @xmath45 is the oseen tensor @xcite @xmath46 , with @xmath47 and @xmath48 . in the presence of boundaries , the full green s tensor contains and additional part describing the flow reflected from interfaces . for a hard no - slip wall , the green s tensor has been first found by lorentz@xcite in 1907 as @xmath49 , where the wall contribution reads @xmath50 for a point force at a distance @xmath51 from a wall at @xmath42 . for a free surface , the wall - interaction part only contains the first term of the rhs of . here , @xmath52_{\alpha\beta } = { \frac{\partial } { \partial r_\beta } } { a}_\alpha $ ] and @xmath53 denotes the reflection operator which transforms any point into its mirror image with respect to the wall . the asterisk denotes the mirror image , i.e. @xmath54 . this expression has been interpreted by blake@xcite in terms of the method of images for stokes flows . the image system in this case involves three fundamental singularities : the reflection of the original stokeslet and the so - called stokeslet doublet and source doublet . however , since their amplitude is proportional to the wall - particle distance @xmath51 , asymptotically they die out with distance @xmath55 as @xmath56 . the tensor @xmath57 is often referred to as blake s tensor . it can also be recast in different forms@xcite . the idea of the derivation relies on the expansion of the blake s tensor about the line connecting the centre of the particle and its hydrodynamic image . thus the interaction between any two points of the particle may be represented by the vertical component of the blake s tensor . by considering the action of higher force moments , higher - order flows incident on the particle can be found . performing a multipole expansion of the resulting flow field@xcite , we project it onto the force multipole space and eventually find the explicit expressions for elements of the resistance matrix @xmath37 which involve the elements of the bulk friction tensor @xmath36 and the higher multipole elements of the blake s tensor . the details of the derivation are presented in appendices [ multipole ] and [ derivation ] . below we present specific results for an axially symmetric particle . it follows from symmetry properties that the bulk friction matrix of a general axisymmetric particle has a particular structure@xcite . moreover , if the particle has both axial and inversional symmetry @xmath58 ( i.e. it is rod - like ) , its bulk friction matrix in eq . simplifies , since @xmath59 , @xmath60 , @xmath61 and @xmath62 vanish , i.e. translational motion is not coupled to torque or elongational flow in the bulk case . the correction has the following form @xmath63 the form of the tensors @xmath64 , @xmath65 , @xmath66 is derived from the multipole expansion of the vertical ( axial ) component of the blake s tensor in appendix [ derivation ] . as we have mentioned in sec . [ sec1 ] , they depend on the @xmath67-components of the bulk generalised friction tensor of the particle and the particle s orientation via the inclination angle @xmath7 . explicitly , we write them in the following for an axisymmetric particle . at contact , the elements of the friction matrix diverge which is expected physically , although no lubrication effects are included in this scheme . the mobility matrix is then obtained by inversion @xmath68 , and by that we assure that the particle mobility decreases to a non - negative value at contact , but in a way different than with lubrication effects included . we note that different elements of the friction matrix behave differently with distance . the hydrodynamic effect of the wall is most pronounced for translational motion where @xmath69 . rotational motion is least affected with @xmath70 . this is in contrast with earlier numerical results due to padding and briels@xcite , whose empirical finding was that both corrections scale as @xmath71 , with a simple fitted angular dependence of the mobility components on @xmath7 . our findings provide exact expressions for the angular dependence , given as low - order polynomials in @xmath72 and @xmath73 , correcting the previous empirical formulae . the source of the discrepancy lies in the inaccuracies of the numerical methods used in ref . . they are also in complete agreement with the slender body expressions obtained by de mestre _ et al._@xcite for the cases @xmath74 provided that the slender body results for @xmath75 and @xmath76 are used as input . in order to discuss in detail the dependence of the correction terms on the orientation of the particle , we introduce two coordinate systems exploiting the symmetries of the problem , as sketched in fig . [ rodimage ] . the laboratory coordinate system ( lab ) consists of three basis vectors @xmath77 , with the @xmath78-axis normal to the wall and the normal vector @xmath79 . the particle resides in the @xmath80-plane . the rod - wall ( rw ) system is a body - fixed set of basis vectors @xmath81 , where @xmath10 is the unit vector along the long axis of the particle , @xmath82 is parallel to the wall and perpendicular to the particle axis , and @xmath83 completes the orthonormal basis . the basis vectors are then given by @xmath84 and @xmath85 . we note that @xmath86 . for an axially symmetric particle , it is convenient to use the representation of the mobility matrix in the rw frame , in which the bulk tensors @xmath87 and @xmath88 are diagonal . the structure of the near - wall tensors is identical to that given in ref . . in the body - fixed frame of reference rw , the correction tensors in eqs - may be explicitly written in terms of the inclination angle @xmath7 . in the formulae , we find the elements of the bulk friction tensor of the particle , namely the coefficients of translational and rotational friction in the directions parallel and perpendicular to the body axis , given respectively by @xmath89 and @xmath90 , where @xmath91 and : denotes double contraction . in addition , the correction terms for rotational motion and rotation - translation coupling contain the coefficient @xmath92 . as seen from eq . , it quantifies the stresslet exerted on the particle rotating with a prescribed angular velocity . these coefficients can be taken from bulk results for friction of axisymmetric particles . for ellipsoidal particles , analytical expressions are available@xcite , whereas for more complex shapes the coefficient may be determined using bead-@xcite or shell - models@xcite . now we can write the tensors in eqs explicitly in the body - fixed frame rw . for the translational part , we find the correction s angular dependence as @xmath93 with @xmath94 and @xmath95 \sin \theta\cos\theta.\end{aligned}\ ] ] we note here that since the axis perpendicular to the rod and parallel to the wall ( in the direction of @xmath96 ) is invariant with respect to the lab to rw frame transformation , the middle element of the matrix above is angle - independent . the translation - rotation coupling part reads @xmath97 finally , the rotational tensor @xmath66 in eq . is a sum of three contributions and has the form @xmath98 for completeness of the discussion , it is worth noting that it is possible to find analytically the leading order behaviour of the mobility functions by expanding the inverted friction matrix - . in this way , the dominant terms of the correction may be evaluated as functions of @xmath7 and indicate a very simple angular relations for the components , namely low - order polynomials in @xmath72 and @xmath73 . however , as we noted before , this is not an optimal strategy , since the mobility functions obtained by inversion may become negative when the particle approaches the wall ( @xmath5 is small compared to @xmath3 ) . therefore , it proves better to first calculate the wall - corrected friction tensor , and then invert it to obtain the mobility tensor . in order to assess the applicability range of the correction , we compare our theoretical result to precise numerical simulations using the hydromultipole package@xcite . as a representative test example , we consider a rod - like particle of aspect ratio @xmath99 constructed out of @xmath100 spherical beads glued together along a straight line . for the needs of demonstration , we choose one inclination angle , @xmath101 , implying the minimal contact distance @xmath102 . we plot the components of the mobility matrix calculated using the procedure outlined above , and compare them to the corresponding accurate numerical predictions . to this end , we introduce the following notation in the body - fixed re frame . taking into account the invariant properties of the rod - wall system and the lorentz symmetry , we can write the translational part as @xmath103 where @xmath104 . the matrix @xmath105 has a similar structure with elements @xmath106 , @xmath107 , @xmath108 , and @xmath109 , respectively . in both cases , the elements @xmath110 , @xmath111 and @xmath30 are even functions of @xmath112 and the elements @xmath113 are odd ones . for the @xmath114 part , we have @xmath115 the elements @xmath116 and @xmath117 are even functions of @xmath112 , while @xmath118 and @xmath119 are odd ones . by taking the transposition of the above matrix we get the @xmath120 part . the diagonal components of the translational and rotational diffusion tensor of the rod are plotted in fig . [ comparison ] . they reveal that for translational motion in this particular case the correction accurately represents the actual mobility even up to @xmath121 , and the asymptotic inverse - distance behaviour of the correction is evident . similarly , the rotational components are even less sensitive to the effect of the wall due to the rapid decay of the hi for rotational motion . for larger distances , the mobility matrix obeys the necessary symmetries , with @xmath122 and @xmath123 asymptotically . the presence of the wall introduces also non - diagonal component to the mobility tensors , which we depict in fig . [ comparison_nondiag ] . normalised by the appropriate combinations of bulk mobility coefficients , these elements are rather small . nevertheless , they numerical results are again in agreement with theoretical predictions up to quite close wall - particle distances , both for translations , and rotations . the translation - rotation coupling tensors become more significant as the particle approaches the wall , as seen from fig . [ comparison_tr ] . compared to the characteristic bulk quantities , however , they seem to play a marginal role in this case . with the derived correction , we are able to reproduce them accurately again up to @xmath124 . the comparison of the correction to numerical results in the case of a relatively long ( @xmath125 ) rod - like particle is quite favourable . indeed , we expect the correction to work even better for more slender particles , since it can be shown analytically using the slender body results for the bulk friction that the relative correction to the translational mobility tensor ( e.g. @xmath126 etc . ) decreases slowly with increasing aspect ratio as @xmath127 . comparison of the near - wall mobility of a rod of apect ratio @xmath125 at an angle @xmath128 to the wall , as predicted by the correction ( solid lines ) and precise hydromultipole numerical simulations ( data points ) . the coefficients are normalised by their corresponding bulk values , to that the all tend to unity at @xmath129 . top : diagonal elements of the translational mobility matrix in the rw frame . bottom : rotational diagonal elements . , title="fig:",scaledwidth=50.0% ] comparison of the near - wall mobility of a rod of apect ratio @xmath125 at an angle @xmath128 to the wall , as predicted by the correction ( solid lines ) and precise hydromultipole numerical simulations ( data points ) . the coefficients are normalised by their corresponding bulk values , to that the all tend to unity at @xmath129 . top : diagonal elements of the translational mobility matrix in the rw frame . bottom : rotational diagonal elements . , title="fig:",scaledwidth=50.0% ] non - diagonal components of the near - wall translational and rotational mobility tensors for the inclination angle @xmath128 . the data points are predictions of multipole simulations with lubrication included , while the solid lines are predicted by our analytical formulae for the correction . the coefficients are normalised by bulk average values of the diagonal terms , e.g. @xmath130 . , title="fig:",scaledwidth=50.0% ] non - diagonal components of the near - wall translational and rotational mobility tensors for the inclination angle @xmath128 . the data points are predictions of multipole simulations with lubrication included , while the solid lines are predicted by our analytical formulae for the correction . the coefficients are normalised by bulk average values of the diagonal terms , e.g. @xmath130 . , title="fig:",scaledwidth=50.0% ] components of the translation - rotation coupling mobility tensor @xmath131 for the inclination angle @xmath128 . the data points are predictions of multipole simulations with lubrication included , while the solid lines are predicted by our analytical formulae for the correction . the coefficients are normalised by a combination of bulk coefficients @xmath132 . the deviations are most pronounced in the coupling tensor . the overall values are , however , rather small , not exceeding 5% for @xmath133.,scaledwidth=50.0% ] we have presented a simple analytical scheme which allows for the representation of the near - wall friction and mobility tensors of a rod - like colloid close to a planar no - slip wall . the correction to bulk mobility , expressed in terms of the bulk hydrodynamic properties of the particle , is valid for general axially symmetric colloids , which need not be slender . our results show that the distance dependence varies between the types of motion in focus ( translational , rotational , and @xmath114-coupling ) . moreover , we have demonstrated by analytical formulae that near - wall friction and mobility for particles at moderate distances from the wall indeed depends on their orientation via simple polynomials in sine and cosines of the inclination angle @xmath7 , as conjectured by padding _ et al._@xcite . by that we have also verified earlier theoretical developments and recent numerical predictions@xcite . our results are in agreement with numerical calculations even in the case when @xmath134 , rendering the results practical for large and moderate wall - particle distances . ml acknowledges support from the national center of science grant no . 2012/07/n / st3/03120 . part of the research has been conducted under a david crighton fellowship awarded to ml at the university of cambridge , and within the mobility plus fellowship awarded to ml by the polish ministry of science and higher education . the idea of the multipole method relies on expressing the force densities and velocities on the surfaces of many spheres immersed in the fluid in the form of a boundary integral equation , which is then projected onto a complete set of multipolar solutions of the stokes equations . the resulting system of linear equations may then be truncated and solved numerically for a conglomerate of spheres moving together . by projecting the many - particle friction matrix obtained in this way onto the subspace of rigid body motions of the conglomerate , the friction tensor of a complex - shaped particle is found . the method has been greatly developed over the last decades , and is presented in more details , e.g. in refs . . with the use of the concept of induced forces due to bedeaux and mazur @xcite , the validity of the stokes equations ( [ stokes ] ) may be formally extended inside the particles by taking an appropriate surface distribution of the forces @xmath135 on the surfaces of the particles @xmath136 . for the stick boundary conditions , the velocities on the surfaces read @xmath137 and eq . on the surfaces of the particles takes the form @xmath138 where @xmath15 represents an ambient flow in the absence of the spheres . we now separate the second term on the rhs of into the contribution from distinct particles and the self - contribution . the self part is found by considering a single particle @xmath139 in an ambient flow @xmath140 . the force density @xmath141 it exerts on the fluid is linearly related to the relative velocity at the surface , viz . @xmath142 where the integral operator @xmath143 is called the single - particle resistance operator , or the friction kernel @xcite , and depends solely on the internal composition and surface properties of the particle @xcite . for the distinct part ( @xmath144 ) , we introduce the green s integral operator ( propagator ) @xmath145 : @xmath146({\mathbf{r } } ) & \equiv { \int_{}^ { } \!\ ! \mathrm{d } { { \mathbf{r}}'}\ , } { \mathbf{t}}({\mathbf{r}},{\mathbf{r}}')\cdot { \mathbf{f}}_j({\mathbf{r } } ' ) , \qquad { \mathbf{r}}\in\sigma_i,\end{aligned}\ ] ] which allows eq . to be written as @xmath147 the above equations can be transformed into an infinite set of algebraic equations by expanding the velocity field and induced force densities in a basic set of irreducible multipoles developed by felderhof and co - workers @xcite . for the velocity , the irreducible multipoles are linear combinations of lamb s solution of the homogeneous stokes equations@xcite . they are labelled by three numbers : @xmath148 , @xmath149 and @xmath150 . the force multipoles can be likewise be described by the labels @xmath151 . the details of the expansion , along with explicit form are given in refs . . we include the expansion coefficients of the velocities @xmath152 in an infinite - dimensional vector @xmath153 which encompasses all the velocity multipoles for all the particles . in a similar manner , we arrange the force multipole moments in the vector @xmath154 . after the multipole expansion is performed , the integral operators @xmath155 and @xmath156 become matrices and eq . is transformed into an algebraic equation @xmath157 the multipole matrix elements of @xmath155 for different particle models are given in ref . , while the elements of @xmath156 have been calculated in ref . for the case of an unbounded fluid , a fluid bounded by a free surface and a fluid bounded by a hard wall . in the friction problem , the forces acting on the particles are sought , given their velocities . upon inverting the above relation , the grand resistance matrix @xmath158 is found as @xmath159 the friction matrix defined in eq . can be found by projecting @xmath158 on the subspaces @xmath67 , corresponding to @xmath151 equal to @xmath160 , @xmath161 , and @xmath162 , respectively . for example , the force multipole @xmath163 with @xmath164 and @xmath165 has three spherical components @xmath166 corresponding to three components of the total force . in numerical computations , infinite matrices @xmath155 and @xmath156 in eq . ( [ multipoleeqset ] ) are truncated at the multipole order @xmath167 , so that only the elements with @xmath168 are considered @xcite . after such a truncation , the matrix @xmath169 is inverted , and the force multipoles are determined . to improve numerical convergence of this scheme , the obtained grand friction matrix @xmath158 in eq . ( [ 3_frictionproblem ] ) is additionally corrected for lubrication effects @xcite . the matrix @xmath158 constructed in the multipole method is not pairwise additive , and accounts fully for many - body hydrodynamic interactions . the approximation is introduced at the level of truncation of the multipoles , and its error may be controlled . the procedures outlined above have been implemented in a fortran code hydromultipole @xcite by wajnryb and collaborators . the method for calculating the near - wall hydrodynamic tensors has been laid out by cichocki _ we employ these codes to calculate the friction tensors of non - spherical particles represented by their bead - models . once the friction matrix @xmath32 is known , the mobility matrix @xmath0 is found by inversion . for a wall - bounded fluid , it follows from the form of eq . that the propagator @xmath156 can be decomposed as @xmath170 where the part @xmath171 consists of the multipole elements of the oseen tensor@xcite , while @xmath172 describes the wall contribution , the multipole matrix elements of which are calculated for a free surface and a hard wall in ref . ( see also ref . ) . in order to find the asymptotic correction to the bulk friction of a particle moving close to a wall , we employ the scattering expansion @xcite . we start from rewriting the grand resistance matrix in eq . ( [ 3_frictionproblem ] ) in the following form @xmath173 when the wall - particle distance is considerably larger than the particle itself , so we expect the wall contribution @xmath172 to be a small correction . expanding eq . yields the form of the correction to the bulk resistance matrix @xmath174 with @xmath175 being short for the bulk resistance matrix of the particle @xmath176 . further on , we evaluate the dominant terms of the correction for all the elements of the friction matrix in the @xmath177 subspace . the propagator @xmath178 connects the beads building up the particles with the beads of the image particle . consider two interacting beads @xmath179 and @xmath180 building up the conglomerate and its image , respectively , as illustrated in fig . [ rodimage ] . introducing coordinates relative to the centre of each conglomerate , we have @xmath181 and @xmath182 . hence the distance between the particles may be written as @xmath183 where we have used the fact that @xmath184 . for the wall - particle distance @xmath5 large compared to the particle size @xmath3 , and thus for @xmath185 , we may expand the distance between each pair around the direction normal to the wall . then , in leading order , the propagator takes the form @xmath186 . due to this fact , its multipole elements have the axial symmetry around the normal direction @xmath187 . in this case , the bulk grand resistance matrix @xmath175 reduces to the single - particle bulk friction matrix @xmath36 as in eq . ( [ friction_single ] ) . the relevant coordinate systems and a schematic illustration of the expansion of the distance between interacting points of the particle ( e.g. a rod ) and its image . we depict the interaction between the point @xmath179 with the image @xmath180 of point @xmath188 . for large wall - particle distances , it may be expanded around the vertical line connecting the particles centres lying at a distance @xmath189 apart , so that @xmath190 . ] the dominant correction may thus be looked upon as interaction of a particle of a given bulk friction matrix @xmath36 with an image particle via the propagator @xmath172 , which accounts for the flow reflected by the wall . the matrix elements of @xmath178 decay according to their multipole indices as @xmath191 with the multipolar indices denoted by @xmath192 . the indices @xmath193 refer to the superscript @xmath110 , while @xmath194 refer to @xmath111 . the directional tensors @xmath195 depend only on the direction of the normal vector @xmath187 . to derive the correction for an axisymmetric particle , we need the following elements @xmath196 the bar indicates the symmetric and traceless part of the respective tensors with respect to the indices in brackets . the symbol @xmath197 indicates the symmetric and traceless part in the index pair @xmath198 . the appropriate reductions read @xmath199 taking into account the symmetries of the bulk friction matrix @xmath36 of an axisymmetric particle ( i.e. the lack of @xmath114 and @xmath200 elements ) , we find explicit expression for the correction terms in eqs . - as @xmath201 where appropriate contractions of the tensors are taken . the evaluation of these expressions using eqs . and the general form of @xmath36 ( cf . ref . ) leads to the expressions - . | hydrodynamic interactions with confining boundaries often lead to drastic changes in the diffusive behaviour of microparticles in suspensions . for axially symmetric particles ,
earlier numerical studies have suggested a simple form of the near - wall diffusion matrix which depends on the distance and orientation of the particle with respect to the wall , which is usually calculated numerically . in this work ,
we derive explicit analytical formulae for the dominant correction to the bulk diffusion tensor of an axially symmetric colloidal particle due to the presence of a nearby no - slip wall .
the relative correction scales as powers of inverse wall - particle distance and its angular structure is represented by simple polynomials in sines and cosines of the particle s inclination angle to the wall .
we analyse the correction for translational and rotational motion , as well as the translation - rotation coupling .
our findings provide a simple approximation to the anisotropic diffusion tensor near a wall , which completes and corrects relations known from earlier numerical and theoretical findings .
* published in : * j. chem .
phys * 145 * , 034904 ( 2016 ) . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
the angular momentum content of a star at birth impacts on most of its subsequent evolution ( e.g. ekstrm et al . 2012 ) . the star s instantaneous spin rate and/or on its rotational history plays a central role in various processes , such as dynamo - driven magnetic activity , mass outflows and galactic yields , surface chemical abundances , internal flows and overall structure , and it may as well influences the planetary formation and migration processes . it is therefore of prime importance to understand the origin and evolution of stellar angular momentum , indeed one of the most challenging issues of modern stellar physics . conversely , the evolution of stellar spin rate is governed by fundamental processes operating in the stellar interior and at the interface between the star and its immediate surroundings . the measurement of stellar rotation at various evolutionary stages and over a wide mass range thus provides a powerful means to probe these processes . in this introductory course , an overview of the rotational properties of stars and of angular momentum evolution models is provided . in section [ tech ] , various techniques used to measure stellar rotation are described . in section [ lowmass ] , the rotational properties of solar - type and low - mass stars are reviewed . angular momentum evolution models developed for low - mass stars are discussed in section [ models ] . finally , the rotational properties of intermediate - mass and massive stars are briefly outlined in section [ highmass ] . stellar rotation can be measured through a variety of techniques . i illustrate here some of the most commonly applied ones to measure the rotation rates of non - degenerated objects . the various techniques are summarized in figure [ meas ] . abney ( 1877 ) was apparently the first to consider the effect rotation would have on a stellar spectrum . he suggested that doppler broadening of the photospheric line profiles should occur , as the light from the rotating surface goes through the entrance slit of the spectrograph . for a star with a linear equatorial velocity @xmath0 , the spectral broadening of photospheric lines amounts to @xmath1 , where @xmath2 is the inclination angle between the line of sight and the rotation axis . a star seen pole - on ( @xmath2=0 ) exhibits no doppler broadening , while a direct measurement of @xmath0 is obtained for an equator - on star ( @xmath2=90@xmath3 ) . the isorotation locus on the stellar disk , i.e. , points of the stellar surface having the same projected velocity , follows vertical stripes parallel to the rotational axis , whose wavelength shift is given by @xmath4 , where @xmath5 and @xmath6 are respectively the latitude and longitude of a point at the stellar surface . the integrated line profile of a rotating star is the sum of the intrinsic line profiles of all points on the stellar disk affected by their respective doppler shifts . to first order , it can be described as the convolution product of the intrinsic , non - rotating line profile with a `` broadening '' function given by ( cf . carroll 1933 ; gray 1973 ) : @xmath7^{1/2 } + { 1\over 2 } \pi\epsilon ) [ 1-(\delta\lambda/\delta\lambda_l)^2]\over\pi \delta\lambda_l(1-\epsilon/3)}\ ] ] where @xmath8 is the temperature- and wavelength - dependent limb - darkening coefficient . in the fourier domain , the convolution product becomes an arithmethic product , and the fourier transform of @xmath9 ) has the interesting property of having successive zeroes at frequencies inversely proportional to @xmath10 ( e.g. , dravins et al . 1990 ) , with the first zero occuring at @xmath11 . thus , even without the knowledge of the intrinsic line profile , the projected stellar velocity can be precisely derived from the location of the first and subsequent zeroes in the fourier transform of the observed profile . this powerful technique has been most succesfully applied to fast rotators ( @xmath1230 kms@xmath13 ) as their first zero occurs in the well - sampled , high s / n low frequency fourier domain . the highest @xmath14 measured so far with this technique , @xmath15600 kms@xmath13 , was reported for an o - type star in the large magellanic cloud ( dufton et al . for a few bright stars , the fourier technique may even provide an estimate of surface latitudinal differential rotation ( gray 1977 ; reiners & schmitt 2002 ) . in contrast , this method is not well suited to slowly rotating stars ( @xmath1620 kms@xmath13 ) whose first zero is usually lost in the high frequency fourier noise . a more common method used to measure the rotation rate of slow rotators is the cross - correlation analysis . instead of measuring the doppler broadening of a single line profile , this method consists in cross - correlating the observed photospheric spectrum with either a template spectrum of a star of similar effective temperature and negliglible rotation ( tonry & davies 1979 ) ( alternatively , a non - rotating model spectrum can be used ) or with a digital mask that let light go through predefined wavelength ranges corresponding to the location of major photospheric lines ( griffin 1967 ; baranne et al . the result of either process is a cross - correlation profile or function ( ccf ) whose width is proportional to @xmath14 and whose signal - to - noise ratio has been greatly enhanced thanks to the inclusion of thousands of spectral lines in its computation . the relationship between the ccf width and @xmath14 has to be properly calibrated using stars with known rotation rates ( benz & mayor 1981 , 1984 ; hartmann et al . 1986 ) . other applications of the cross - correlation technique include the derivation of accurate radial velocities ( ccf peak location ) and metallicity ( ccf area ) . more sophisticated spectroscopic techniques have also been used to measure rotation rates . the doppler imaging technique ( vogt & penrod 1983 ) and the related zeeman - doppler imaging technique ( semel 1989 ; donati et al . 1997 ) both take advantage of the relationship existing between the location of a feature at the surface of a rotating star and its position within the line profile ( khokhlova 1976 ) . as the star rotates , the signatures of stellar spots ( or magnetic components in polarized light ) move across the line profile and their monitoring allows the reconstruction of surface brightness and/or magnetic maps . the shape of the line profiles is thus periodically modulated by surface inhomogeneities , and the modulation period provides a direct measurement of the star s rotational period . furthermore , the latitudinal drift of spots on the stellar surface probes the rotational period at different latitudes , thus yielding an estimate of differential rotation at the stellar surface . specifically , the quantity @xmath17 is derived by assuming a simplified solar - like differential rotation law of the form : @xmath18 where @xmath19 is the angular velocity at the stellar equator and @xmath20 the latitude at the stellar surface . the relationships between surface differential rotation on the one hand and effective temperature , convective zone depth , and rotation rate on the other , have been investigated for solar - type and lower mass stars by , e.g. , barnes et al . ( 2005 ) and marsden et al . ( 2011 ) . for relatively nearby stars , the stellar disk may be resolved by interferometry ( e.g. kervella et al . 2004 ) . in such a case , the stellar oblateness , i.e. , the decimal part of the ratio between the equatorial to the polar radii can be measured . for rapidly - rotating stars , the stellar oblateness can be quite substantial . according to the roche model for stellar equipotential surfaces , the latitude - dependent radius of a fast - rotating star is given by : @xmath21\ ] ] where @xmath20 is the co - latitude and @xmath22 is the ratio between the star s angular velocity and the critical velocity at which centrifugal forces at the equator balance gravity , @xmath23 , where @xmath24 is the stellar mass and @xmath25 the polar radius , ekstrm et al . 2008 ) . for a star rotating at critical velocity , the equatorial radius is 1.5 times larger than the polar radius , yielding a stellar oblateness of 0.5 . a stellar oblateness with values up to 0.35 has been measured by interferometry for a handful of massive stars rotating close to break - up velocity ( cf . van belle 2012 ) . whenever interferometry can provide a fully reconstructed surface brightness map for a rapidly rotating star , the gravity darkening effect can be directly observed . von zeipel ( 1924 ) s theorem relates radiative flux to surface gravity and thus predicts that a star rotating close to break - up will be brighter at the pole than at the equator . this has actually been observed in altair ( @xmath26240 kms@xmath13 ) by monnier et al . ( 2007 ) who modeled the surface brightness map of this star to derive both the inclination angle of the rotational axis on the line of sight and its position angle on the sky plane . this illustrates the complementary power of interferometry compared to spectroscopy , as the former delivers the orientation of the angular momentum vector in space while the latter yields its ( projected ) modulus . in fact , interferometry and spectroscopy can be combined , a technique called spectro - interferometry , to measure the position angle of the rotational axis of stars whose surface is not fully spatially resolved . the method consists in measuring the position of the star s photocenter across a spectral line . each velocity channel within the line profile corresponds to one isorotation stripe at the stellar surface , parallel to the rotational axis . for instance , the far redward wing of the line profile spatially coincides with the limb of the receding hemisphere . as the photocenter is recorded across the line in successive velocity channels , its location slightly moves on the sky plane in a direction perpendicular to the rotational axis . using several interferometric baselines with different orientations , the direction of the projected rotational axis can thus be derived . this challenging technique was successfully applied by le bouquin et al . ( 2009 ) to demonstrate that the position angle of fomalhaut s rotational axis is perpendicular to the major axis of its planet - hosting debris disk . the oldest method used to measure stellar rotation consists in monitoring the visibility of magnetic spots on the stellar surface . in the western world , galileo galilei was amongst the first observers of the early 17@xmath27 century to provide an estimate of the sun s rotational period by observing the sunspots being carried across the stellar disk by the star s rotation ( casas et al . when the stellar surface is not resolved , starspots still modulate the star s luminosity in a periodic way . hence , the recording of the photometric light curve and the detection of a periodically modulated signal provide a direct estimate of the star s rotational period @xmath28 . this technique has the advantage over spectroscopy of yielding a measurement of the stellar rotation rate that is free of geometric effects and is straighforwardly converted to angular velocity , @xmath29 . however , this is at the expense of requiring intense photometric monitoring over several rotational periods and applying dedicated signal processing techniques in order to recover the periodic component of the light curve that truly corresponds to the star s rotational period ( e.g. irwin et al . 2009 ) . this technique recently flourished with the corot and kepler satellites that acquired continuous stellar light curves of exquisite precision over timescales of months to years ( e.g. affer et al . 2012 ; mcquillan et al . 2013 ) . the large number of rotational cycles recorded by these light curves allows not only the stellar rotational period to be derived with extreme accuracy but also to detect latitudinal differential rotation by traking spots located at different latitudes that have slightly different rotational periods ( e.g. mosser et al . the application of this technique is obviously most suited to magnetically active stars that exhibit starspots at their surface , i.e , usually solar - type and lower mass stars with a spectral type from late - f to m , extending even to brown dwarfs ( e.g. herbst et al . 2007 ) . the oscillation spectrum of a star encodes its rotational properties from the surface down to the deep interior . to first order , the oscillation frequencies of radial order @xmath30 , degree @xmath5 , and azimuthal order @xmath31 of a rotating star are related to the same frequencies in a non - rotating star by : @xmath32 where @xmath33 $ ] and @xmath34 ( goupil et al . 2004 ) . stellar rotation lifts the @xmath31-degeneracy of the oscillation modes of a non - rotating star by producing a rotational splitting whose amplitude is directly proportional to angular velocity . in case of uniform rotation , the rotational splitting of the modes provides a direct measurement of surface angular velocity . for more complex rotational profiles , modeling the frequency splitting with rotating stellar models offers a way to estimate the internal rotation profile of the star . this technique was first applied to the sun to recover the latitudinal and radial variations of the solar rotation rate through the convective envelope and down into the radiative core ( e.g. , schou et al . more recently , a similar approach based on the analysis of the rotational splitting of mixed pressure and gravity modes allowed deheuvels et al . ( 2012 ) to probe the internal rotation profile of a low - mass giant evolving off the main sequence , thus revealing a rapidly rotating inner core . additional information can be retrieved from the amplitude of the rotationally splitted modes . for instance , the ratio of the amplitudes of the @xmath35 mode components depends on the inclination of the rotational axis on the line of sight ( gizon & solanki 2003 ) . a first application of this technique has recently allowed chaplin et al . ( 2013 ) to demonstrate that the rotational axis of 2 transiting exoplanet hosts detected by kepler is perpendicular to the orbital plane of the planets . [ cols="<,<,<,<",options="header " , ] @xmath36 pinto et al . ( 2011 ) ; @xmath37 cf . ruciski ( 1988 ) we review in this section the rotational properties of stars with a mass less than 1.2m@xmath38 , from early studies to the most recent determinations of rotational period distributions ranging from the early pre - main sequence ( pms ) to the end of the main sequence ( ms ) . the definition of physical quantities related to stellar rotation that we use in this section are summarized in table [ def ] . solar values are listed for reference . kraft ( 1970 ) provided one of the first reviews on the rotational properties of stars on the main sequence . the main characteristics of the rotation rate distribution was a sharp break in velocity at a spectral type around f4 , i.e. , around a mass of @xmath151.2 m@xmath38 , with more massive stars having mean rotation rates of order of 100 - 200 kms@xmath13 , while lower mass stars had much lower rotational velocities of order of a few kms@xmath13 . the sharp decline of rotation rate for stars with deep convective envelopes had readily been interpreted by schatzman ( 1962 ) as the result of angular momentum loss due to magnetized winds . in this framework , all stars were born with high rotation rates , and only magnetically active stars with surface convective envelopes would undergo strong braking as angular momentum is removed from their surface by magnetized stellar winds . in magnetically active stars , the ionised outflow remains coupled to the magnetic field out to a distance where the magnetic tension becomes unable to compensate for coriolis force , i.e. : @xmath39 where b is the magnetic field intensity , @xmath40 the surface angular velocity , @xmath41 the poloidal velocity of the wind flow , and @xmath42 its density . the magnetic lever arm up to this radius yields angular momentum loss rates that are orders of magnitude larger than in the absence of magnetically - coupled winds . as shown by weber & davis ( 1967 ) , the angular momentum loss rate can be expressed as : @xmath43 where @xmath44 is the stellar angular velocity , @xmath45 the mass - loss rate , @xmath46 the alfvn radius , and @xmath47 the braking timescale . for the sun , the alfvn radius is about 30 times larger than the solar radius , which translates into a braking timescale by the magnetized wind of order of 1 gyr , i.e. , short enough to account for the slow rotation of the sun on the mid - main sequence . a spectacular confirmation of this magnetic wind braking concept came with one of the first studies of rotational evolution among main sequence stars . based on an earlier suggestion by kraft ( 1967 ) , skumanich ( 1972 ) used published measurements of the mean rotation rate of solar - type stars in 2 young open clusters , the pleiades and the hyades , and comparing them to the sun s rotation , derived his famous time - dependent velocity relationship @xmath48 , for ages between 0.1 and 5 gyr . this relationship is indeed what is asymptotically expected from the magnetic wind braking process , as shown by durney & latour ( 1978 ) . combining the following expression for mass - loss : @xmath49 where @xmath50 and @xmath51 are the density and poloidal velocity of the outflow at the alfvn radius @xmath52 , with the definition of alfvn velocity : @xmath53 where @xmath54 is the stellar magnetic field at the alfvn radius , and with the condition of magnetic flux conservation : @xmath55 and replacing the expressions above in eq . [ jdot ] , yields:@xmath56 further assuming that the poloidal velocity of the outflow at the alfvn radius reaches the escape velocity ( @xmath57 ) and that the stellar magnetic field is powered by an internal dynamo process wich scales as @xmath58 , finally yields:@xmath59 which asymptotically integrates to @xmath60 , i.e. , the skumanich relationship . while extremely satisfying conceptually , this derivation makes a number of symplifying assumptions including spherically symmetric radial magnetic field and wind , thermally - driven outflows , and linear dynamo relationship , none of which strictly apply to active young stars ( see section 4 ) . as @xmath14 measurements accumulated in the mid-80 s especially for stars located close to or on the zero - age main sequence ( zams ) at an age of about 100 myr , it became clear that , at these young ages , a large dispersion of rotation rates exists for solar - type and lower mass stars . thus , stauffer ( 1987 ) reported @xmath14 ranging from less than 10 kms@xmath13 up to more than 150 kms@xmath13 for g and k - type stars in the alpha persei ( 80 myr ) and pleiades ( 120 myr ) young open clusters , at the start of their main sequence evolution . clearly , this unexpectedly large scatter of rotation rates at zams pointed to a rotational evolution during the pre - main sequence that was far more complex than envisioned from the skumanich relationship on the main sequence . extrapolating the skumanich relationship back in time to the pre - main sequence ( pms ) , at an age of @xmath151 myr , would predict rotational velocities of order of 200 kms@xmath13 . additionally , if protostellar collapse is dominated by gravity , one should expect protostars to rotate close to their break - up velocity . it therefore came as a surprise when the first measurements of rotational velocities for solar - mass pms stars revealed that their rotation rate rarely exceeds 25 kms@xmath13 , i.e. , about a tenth of the break - up velocity ( vogel & kuhi 1981 ; bouvier et al . 1986 ; hartmann et al . 1986 ) . even deeply embedded protostars appear to exhibit quite moderate rotation , with a mean value of about 40 kms@xmath13 ( covey et al . clearly , significant angular momentum loss must occur during protostellar collapse and/or during the embedded protostellar phase of evolution to account for such low rotation rates as the stars first appear in the hr diagram ( see hennebelle , this volume ; belloche , this volume ) . like on the main sequence , higher mass pms stars exhibit larger rotational velocities than their lower - mass counterparts ( dahm et al . most of the so - called herbig ae - be stars actually have similar velocities than their ms counterpart , which suggests they lose little angular momentum during the pms , except for the precursors of the peculiar subgroup of magnetic a and b stars ( cf . alecian et al . 2013 ) . the low rotation rates of pms low - mass stars is even more surprising when considering that they accrete high specific angular momentum material from their circumstellar disk for a few million years ( hernandez et al . as shown by hartman & stauffer ( 1989 ) , a star accreting at a rate @xmath45 from its disk will gain angular momentum at a rate : @xmath61 where @xmath62 is the keplerian velocity of the disk material . it is then expected to spin up to an equatorial velocity of : @xmath63 where @xmath64 is the stellar moment of inertia and @xmath65 the break - up velocity . thus , for a mass accretion rate of a few 10@xmath66 m@xmath38yr@xmath13 lasting for about 3 myr , one expects the young star to rotate at more than half the break - up velocity . clearly , since most low - mass pms stars have much slower rotation rates , the accretion of high angular momentum material from the disk must be balanced by a process that efficiently removes angular momentum from the central star . based on a physical process thought to be at work in compact magnetized objects such as accreting neutron stars , knigl ( 1991 ) was first to suggest that the magnetic interaction between the inner disk and a young magnetized star might provide a way to remove part of the angular momentum gained from accretion . shortly after , evidence for a correlation between rotation rate and accretion was reported ( bouvier et al . 1993 ; edwards et al . 1993 ) , with accreting young stars rotating on average more _ slowly _ than non - accreting ones , thus providing a strong support to knigl s suggestion . yet , more than 20 years later , the controversy is still very much alive as to whether the magnetic star - disk interaction is efficient enough to counteract the accretion - driven angular momentum gain in young stars ( see ferreira , this volume ) . also , even though a number of recent studies appear to confirm the early evidence for a rotation - accretion connection in young stars ( e.g. rebull et al . 2006 ; cieza & baliber 2007 ; cauley et al . 2012 ; dahm et al . 2012 ; affer et al . 2013 ) , some discrepant results have also been reported ( e.g. le blanc et al . thus , while there is a general consensus for young accreting stars being somehow prevented from spinning up as they evolve towards the main sequence ( e.g. , rebull et al . 2004 ) , the underlying physical mechanism responsible for this behaviour is not totally elucidated yet . while the early studies from the 60 s to the 80 s mostly focused on the determination of projected rotational velocities , @xmath14 , large - scale photometric monitoring campaigns started in the 90 s that provided complete rotational period distributions for thousands of low - mass stars in the pms and ms stages . figure [ prot ] ( from irwin & bouvier 2009 , see references therein ) illustrates a compilation of some of these results . it shows how the distribution of rotational periods evolves from the start of the pms at about 1 myr to the mid - ms at 0.6 gyr . a number of clear evolutionary trends emerge , which have been confirmed by more recent studies . the initial distribution in the orion nebulae cluster at @xmath671 myr is quite broad . it was found to be bimodal for stars more massive than 0.3 m@xmath38 with a slow rotator peak with periods around 8 days and a fast rotator group with periods around 2 days ( herbst et al . the peak of slow rotators is usually attributed to pms stars still interacting with their disk , hence being prevented from spinning up , while the fast rotators are thought to mainly consist of stars that have already dissipated their circumstellar disks and therefore have started to spin up as they evolve towards the zams . in constrast , the rotational period distribution of very low - mass stars appears unimodal and skewed towards faster rotators . as time progresses towards the zams , which is reached in about 40 myr for a solar - mass star and 150 myr for a 0.5 m@xmath38 star , the period distributions evolve towards faster rotation , especially in the low mass domain where rotational periods at zams converge to values less than 1 day . however , for stars more massive than about 0.4 m@xmath38 , a large dispersion of rotation rates remains up to the zams . it is only later on the ms , by an age of about 0.5 gyr , that all but the lowest mass stars are significantly braked , to reach periods larger than about 10 days ( delorme et al . 2011 ; meibom et al . 2011 ) , and exhibit a tight rotation - mass relationship . at the low mass end , below 0.6 m@xmath38 , a significant dispersion still subsists at that age ( scholz et al . 2011 ; ageros et al . 2011 ) and even beyond for most late - type field dwarfs ( mcquillan et al . 2013 ) , lasting for perhaps as long as 10 gyr for the lowest mass stars ( [email protected] m@xmath38 ; irwin et al . 2011 ) . hence , the spin down timescale on the main sequence significantly increases towards lower mass objects , from a few 0.1 gyr for solar - type and low - mass stars up to a few gyr for very low - mass stars ( delfosse et al 1998 ) . these recent studies have highlighted that the angular momentum ( am ) evolution of cool stars is strongly mass dependent , both during the pms and on the ms , as can be clearly seen from fig . schematically , solar - type and low - mass stars ( 0.5 - 1.1 m@xmath38 ) have a large initial dispersion of rotational periods that subsists and even widens to the zams , and is eventually erased on the ms as all stars in this mass range are efficiently braked on a timescale of a few 0.1 gyr , thus yielding a well - defined rotation - mass sequence with little scatter . the rotational convergence of solar - type stars on the early ms has led to the developement of gyrochronology , i.e. , the measurement of stellar age from rotation rate ( e.g. , barnes 2003 ; delorme et al . 2011 ; epstein & pinsonneault 2012 ) . in contrast , very low - mass stars ( [email protected] m@xmath38 ) , while also exhibiting some dispersion of rotation rates at the start of the pms evolution , seems to all converge towards fast rotation at zams , and resume building up a large rotational scatter on a timescale of a few gyr on the ms . this different behaviour is well illustrated by the changing shape of the period - mass diagrams shown in fig [ prot ] as time goes by . going deeper into the mass spectrum , brown dwarfs ( bd s , [email protected] m@xmath38 ) rotational properties seem to mimic and extend those of very low - mass stars ( herbst et al . 2007 ; rodrguez - ledesma et al . 2009 ) , with no apparent rotational discontinuity at the stellar / substellar boundary . as a group , they tend to rotate faster than stars at all ages ( mohanty & basri 2003 ) , with a median period of order of 15 hours at young ages , and some indeed with rotational periods as short as a few hours , i.e. , reaching close to the rotational break - up ( scholz & eislffel 2004 , 2005 ) . rapid rotation is still measured for evolved bd s at an age of a few gyr , which suggests that they suffer much weaker angular momentum losses than stars ( reiners & basri 2008 ) . the wealth of new data acquired since the mid-90 s , now encompassing several thousands of rotational periods measured for cool stars over an age range covering from the start of the pms to the late - ms prompted renewed interest in the development of angular momentum evolution models . while a review of all existing models and their origin is far beyond the scope of this introductory course , we outline in this section the main physical processes that are thought to drive the rotational evolution of low - mass stars and how they are currently implemented in parametrized models of angular momentum evolution . the rotational evolution of low - mass stars is believed to be dictated by 3 main physical processes : star - disk interaction in the early pms , magnetized wind braking , and am transport in the stellar interior . we only briefly summarize these processes below , as they are reviewed in much more detailed in other contributions to this volume . camenzind ( 1990 ) and knigl ( 1991 ) were first to suggest that the low rotation rates of pms stars may result from the magnetic star - disk interaction . the picture envisioned at that time was inspired by the ghosh et al.s ( 1977 ) model developed for accreting neutron stars ( e.g. collier cameron et al . 1995 ) . while this model now does not seem to be efficient enough to apply to young stars , a number of alternatives have been proposed still relying on the inner disk interacting with a strong stellar magnetosphere . indeed , young stars are known to host strong magnetic fields ( cf . donati , this volume ) that are able to disrupt the inner disk regions and channel the accretion flow onto the star through magnetic funnels ( cf . bouvier et al . 2007 for a review ) . bessolaz et al . ( 2008 ) derived the following expression for the magnetospheric truncation radius : @xmath69 where @xmath70 is the sonic mach number at the disc midplane , @xmath71 the stellar magnetic field , and @xmath72 the mass accretion rate . for values of the parameters relevant to a young accreting solar - mass system , the magnetospheric truncation radius is located a few stellar radii above the stellar surface , a prediction borne out by observations ( e.g. najita et al . this distance is of the same order as the disk corotation radius , i.e. , the radius at which the keplerian angular velocity in the disk equals the star s angular velocity : @xmath73 the net flux of angular momentum exchanged between the star and the disk strongly depends upon whether the magnetospheric truncation radius is located within or beyond the disk corotation radius . therefore , the changing magnetic topology of solar - type stars evolving on their convective and radiative pms tracks will most likely impact their early rotational evolution ( e.g. gregory et al . 2012 ) . within this general framework , various scenarios have been developed to attempt to produce a negative net angular momentum torque onto the star , so as to explain why young stars are slow rotators in spite of both accretion and contraction . these include accretion - driven winds ( matt & pudritz 2008 ) , x - winds and their variants ( mohanty and shu 2008 ; ferreira et al . 2000 ) , and magnetospheric ejections ( zanni & ferreira 2013 ) . these models are discussed at length in j. ferreira s contribution to this volume . whether any of these processes is actually able to counteract the spin up due to accretion and contraction during the early pms is , however , unsettled . pending a satisfactory model for pms spin down , most current angular momentum evolution models assume that accreting pms stars evolve at constant angular velocity ( cf . [ ammodels ] ) . starting from the general expression of angular momentum loss due to magnetized stellar winds ( eq . 3.1 above ) , kawaler ( 1988 ) worked out a parametrized formulation that can be straightforwardly implemented in evolutionary models . following mestel ( 1984 ) , the am loss rate is given by : @xmath74^n\ ] ] where @xmath45 is the mass loss rate , @xmath75 the stellar angular velocity , @xmath76 the stellar radius , @xmath46 the alfvn radius , and the exponent @xmath30 reflects the magnetic field geometry with @xmath77 for a radial field and @xmath78 for a dipolar field . from eq . 3.2 , 3.3 , and 3.4 above , the expression of the alfvn radius is given by : @xmath79 where @xmath71 is the stellar magnetic field intensity , and @xmath51 the flow velocity at the alfvn radius . further assuming that the flow velocity at the alfvn radius is of order of the escape velocity , and adding a dynamo relationship for the generation of the stellar magnetic field of the form : @xmath80 where @xmath81 is the dynamo exponent , finally leads to : @xmath82 for an asumed linear dynamo relationship ( @xmath81=1 ) , and a magnetic topology intermediate ( in some sense ) between a radial and a dipolar field with @xmath30=1.5 , eq.4.6 simplifies to : @xmath83 which is easily implement in an evolutionary model . direct application of this prescription , however , proved to produce too strong braking for fast rotators compared to observations ( stauffer & hartmann 1987 ) . most models have therefore adopted a variant of kawaler s prescription , first proposed by chaboyer et al . ( 1995 ) , that assumes that the dynamo saturates ( @xmath84 ) above some angular velocity @xmath85 , i.e. : @xmath86 the shallower slope of the rotation - dependent angular momentum loss at high rotation , i.e. , @xmath87 instead of @xmath88 , provides a better agreement with the observation of very fast rotators at the zams . magnetic field measurements suggest that dynamo saturation occurs at a fixed rossby number @xmath89 in cool stars ( reiners et al . 2009 ; wright et al . 2011 ) , with @xmath90 where @xmath91 is the turnover convective time . as @xmath91 lengthens towards lower mass stars , @xmath92 is expected to decrease with mass , i.e. , lower mass stars suffer less angular momentum loss than solar - type ones . this rossby scaling thus naturally accounts for the longer spin down timescale of lower mass stars on the main sequence ( krishnamurthi et al . 1997 ; bouvier et al . however , sills et al . ( 2000 ) showed that very low - mass stars ( [email protected] m@xmath38 ) experience much less spin down than the extrapolation of the rossby scaling to very low masses would predict . recently , reiners & mohanty ( 2012 ) proposed a modification to kawaler s prescription , based on a more physically - funded dynamo relationship , that appears to alleviate this issue . recent mhd numerical simulations of stellar winds have considerably improved our understanding of wind - driven angular momentum loss ( e.g. aarnio et al . 2012 ; vidotto et al . 2009 , 2011 ) . based on 2d numerical simulations of mhd winds originating from stars with a dipolar magnetic field , matt et al . ( 2012 ) derived the following expression for the alfvn radius : @xmath93^{m}\ ] ] where @xmath94 is the ratio of the stellar rotation rate to the break - up velocity and @xmath95 where @xmath71 is the magnetic field strength at the stellar equator , @xmath96 the wind mass loss rate , and @xmath97 the escape velocity . the value of the constants appearing in eq . [ matt ] are derived from numerical simulations that explore the parameter space , yielding @xmath98 , @xmath99 and @xmath100 . provided the stellar magnetic field can be tied to the angular velocity through a dynamo prescription , and the wind mass loss rate can be computed as a function of stellar rotation and other fundamental stellar parameters ( e.g. cranmer & saar 2011 ) , the expression given by eq . [ matt ] for the alfvn radius can be implemented in eq . [ jdot ] to compute the amount of wind - driven angular momentum losses during stellar evolution . models using this new prescription for am losses are illustrated below ( cf . [ ammodels ] ) . as angular momentum is carried away by stellar winds at the stellar surface , several mechanisms may operate to redistribute angular momentum in the stellar interior . these range from various classes of hydrodynamical instabilites ( e.g. lagarde et al . 2012 , see also the contributions of palacios and rieutord in this volume ) , magnetic fields ( e.g. eggenberger et al . 2005 ) , and gravity waves ( talon & charbonnel 2008 ; charbonnel et al . 2013 ; see also mathis , this volume ) . the recent report of rapidly rotating cores in red giants from asterosismology ( e.g. mosser et al . 2012 ) and the discrepancy between the measured angular velocity gradient and model expectations indicate that angular momentum transport mechanisms in stellar interiors are still not totally elucidated ( e.g. eggenberger et al . 2012 ) . lacking a detailed physical modeling of the processes involved , macgregor & brenner ( 1991 ) introduced a parametrized prescription for angular momentum transport between the radiative core and the convective envelope . each region is considered as rotating uniformely but not necessarily at the same rate , as the convective envelope is slowed down . they assumed that am transport processes would act to erase angular velocity gradients at the boundary between the radiative core and the convective envelope ( the tachocline , cf . spiegel & zahn 1992 ) , on a timescale @xmath101 , the so - called core - envelope coupling timescale . to reach a state of uniform rotation on a timescale @xmath101 throughout the star , a quantity @xmath102 of angular momentum has to be exchanged between the radiative core and the convective envelope , with : @xmath103 where @xmath104 , @xmath105 and @xmath106 , @xmath107 are the moment of inertia and the angular momentum content of the convective envelope and the radiative core , respectively . then , the angular momentum evolution of the radiative core and the convective envelope can be written as : @xmath108 and @xmath109 where @xmath110 is the wind braking timescale . this prescription has been used in two - zone angular momentum evolution models ( e.g. allain 1998 ) that provide some insight into the value of @xmath101 and its dependence upon rotation rate and stellar parameters . a short coupling timescale corresponds to an efficient am redistribution and leads to solid - body rotation , while a long @xmath101 allows for strong angular velocity gradients to develop at the tachocline as the star evolves . thus , this parametrization offers some empirical guidance to identify the actual underlying physical mechanisms at work for angular momentum transport in stellar interiors based on the timescales involved . angular momentum evolution models have been developed in an attempt to reproduce the run of surface rotation as a function of time , as derived from observations for solar - type stars , low - mass and very low - mass stars . in this section , we illustrate a class of semi - empirical models that use parametrized prescriptions to implement the physical processes described in the previous section . figure [ gallet ] ( from gallet & bouvier 2013 ) illustrates the observed and modeled angular momentum evolution of solar - type stars , in the mass range 0.9 - 1.1 m@xmath38 , from the start of the pms at 1 myr to the age of the sun . the rotational distributions of solar - type stars are shown at various time steps corresponding to the age of the star forming regions and young open clusters to which they belong ( see fig.2 ) . three models are shown , which start with initial periods of 10 , 7 , and 1.4 days , corresponding to slow , median , and fast rotators . the models assume constant angular velocity during the star - disk interaction phase in the early pms ( cf . [ sdi ] ) , implement the matt et al . ( 2012 ) wind braking prescription ( cf . [ wind ] ) , as well as core - envelope decoupling ( cf . [ decoupling ] ) . the free parameters of the models are the initial periods , scaled to fit the rotational distributions of the earliest clusters , the star - disk interaction timescale @xmath111 during which the angular velocity is held constant at its initial value , the core - envelope coupling timescale @xmath101 , and the calibration constant @xmath112 for wind - driven am losses . the latter is fixed by the requirement to fit the sun s angular velocity at the sun s age . these parameters are varied until a reasonable agreement with observations is obtained . in this case , the slow , median , and fast rotator models aim at reproducing the 25 , 50 , and 90@xmath27 percentiles of the observed rotational distributions and their evolution from the early pms to the age of the sun . the models provide a number of insights into the physical processes at work . the star - disk interaction lasts for a few myr in the early pms , and possibly longer for slow rotators ( @xmath1135 myr ) than for fast ones ( @xmath1132.5 myr ) . as the disk dissipates , the star begins to spin up as it contracts towards the zams . the models then suggest much longer core - envelope coupling timescales for slow rotators ( @xmath11430 myr ) than for fast ones ( @xmath11412 myr ) . hence , once they have reached the zams , slow rotators exhibit much lower surface velocities than fast rotators but significantly larger angular velocity gradients at the tachocline . indeed , most of the initial angular momentum is hidden in the core of the slow rotators at zams . as they evolve on the early ms , wind braking eventually leads to the convergence of rotation rates for all models by an age of @xmath671 gyr , to asymptotically reach the skumanich s relationship . these models thus clearly illustrate the different rotational histories solar - type stars may experience , depending mostly on their initial period and disk lifetime . in turn , the specific rotational history a star undergoes may strongly impact on its properties , such as lithium content , even long after rotational convergence has taken place ( cf . bouvier 2008 ; randich 2010 ) . the models discussed above describe the rotational evolution of single stars while many cool stars belong to multiple stellar systems . in short period binaries ( @xmath11512 days ) , tidal interaction will enforce synchronization betweeen the orbital and rotational period ( zahn 1977 ) . clearly , the rotational evolution of the components of such systems is totally different from that of single stars , and rapid rotation is usually maintained over the whole main sequence ( zahn & bouchet 1989 ) and even beyond , like in , e.g. , the magnetically - active rapidly - rotating rs cvn systems . however , the fraction of such tight , synchronized systems among solar - type stars is low , of order of 3% ( raghavan et al . 2010 ) , so that tidal effects are unlikely to play a major role in the angular momentum evolution of most cool stars . presumably much more frequent is the occurrence of planetary systems around solar - type and low - mass stars ( e.g. mayor et al . 2011 ; bonfils et al . the frequency of hot jupiters , i.e. , massive planets close enough to their host star to have a significant tidal influence ( cf . dobbs - dixon et al . 2004 ) , is a mere 1% around fgk stars ( e.g. wright et al . . however , there is mounting evidence that the planetary formation process is quite dynamic , with gravitational interactions taking place between forming and/or migrating planets ( albrecht et al . this may lead to planet scattering and even planet engulfment by the host star . the impact of such catastrophic events onto the angular momentum evolution of planet - bearing stars has been investigated by bolmont et al . ( 2012 ) who showed it could be quite significant both during the pms and on the main sequence . models similar to those described above for solar - type stars have been shown to apply to lower mass stars , at least down to the fully convective boundary ( @xmath670.3 m@xmath38 ) , with the core - envelope coupling timescale apparently lengthening as the convective envelope thickens ( e.g. , irwin et al . 2008 ) . in the fully convective regime , i.e. , below 0.3 m@xmath38 , models ought to be simpler as the core - envelope decoupling assumption becomes irrelevant and uniform rotation is usually assumed instead throughout the star . yet , the rotational evolution of very low - mass stars actually appears more complex than that of their more massive counterparts and still challenges current models . rotational period measurements for field m - dwarfs show a bimodal distribution with a peak of fast rotators in the period range 0.2 - 10 days , and a peak of slow rotators with rotational periods ranging from 30 days to at least 150 days ( irwin et al . 2011 ) . most of the slow rotators appear to be thick disk members , i.e. , they are on average older than the fast ones that are kinematically associated to the thin disk . the apparent bimodality coud thus simply result from a longer spin down timescale of order of a few gyr , as advocated by reiners & mohanty ( 2012 ) . however , as shown in figure [ irwin ] ( from irwin et al . 2011 ) , this bimodality may not be easily explained for field stars at an age of several gyr . it is seen that the large dispersion of rotation rates observed at late ages for very low - mass stars requires drastically different model assumptions . specifically , for a given model mass ( 0.25 m@xmath38 in fig . [ irwin ] ) , the calibration of the wind - driven angular momentum loss rate has to differ by one order of magnitude between slow and fast rotators ( irwin et al . why does a fraction of very low - mass stars remain fast rotators over nearly 10 gyr while another fraction is slowed down on a timescale of only a few gyr is currently unclear . a promising direction to better understand the rotational evolution of very low mass stars is the recently reported evidence for a bimodality in their magnetic properties . based on spectropolarimetric measurements of the magnetic topology of late m dwarfs ( morin et al . 2010 ) , gastine et al . ( 2013 ) have suggested that a bistable dynamo operates in fully convective stars , which results in two contrasting magnetic topologies : either strong axisymmetric dipolar fields or weak multipolar fields . whether the different magnetic topologies encountered among m dwarfs is at the origin of their rotational dispersion at late ages remains to be assessed . as outlined in sect . [ early ] , stars more massive than 1.2 m@xmath38 have significantly larger rotation rates than solar - type and low - mass stars . in comparison to low - mass stars , more massive stars have higher initial angular momenta , shorter contraction timescales to the zams and shorter evolution timescales on the ms , they lack deep convective envelopes and strong magnetic fields ( except for peculiar sub - classes , such as ap - bp stars ) , and drive dense radiative winds . for all these reasons , their rotational evolution is expected to be quite different from that of their low - mass counterparts . we briefly review in the next sections the rotational properties of massive and intermediate - mass stars . braganca et al . ( 2012 ) have recently summarized the rotational distributions of 350 nearby o9-b6 stars in the galactic disk from @xmath14measurements . after correcting for projection effect ( i.e. , @xmath116 , cf . gaig 1993 ) , they find that the mean equatorial velocity is of order of 125 kms@xmath13and relatively uniform over the whole mass range they probe . a similar study was performed by huang & gies ( 2006 ) for 496 o9-b9 stars belonging to 19 young open clusters yielding an average equatorial velocity of 190 kms@xmath13 , i.e. , significantly higher than the mean rotation rate of massive field stars . specifically , the difference in mean velocity between cluster and field stars at high masses stems from the much smaller fraction of slow rotators observed in clusters , while the two populations have similar @xmath14 distributions above @xmath67100 kms@xmath13 . even though cluster members are on average younger than field dwarfs , meynet & maeder ( 2000 ) rotating evolutionary models predict only modest spin down for massive stars on the main sequence , with a braking rate of order of 15 - 20% for 9 - 12 m@xmath38 stars . indeed , the comparison of massive field dwarfs and cluster members over the same age range ( 12 - 15 myr ) still result in differing average velocities , thus suggestive of an intrinsic rather than an evolutionary effect ( strom et al . 2005 ) . wolff et al . ( 2007 ) further confirmed that massive stars formed in high - density regions , e.g. rich clusters , lack the numerous slow rotators seen for stars of similar masses in low - density regions and the field . these authors suggested that the density - dependent rotational distribution observed for massive stars may reflect a combination of initial conditions , e.g. , higher turbulence in massive proto - clusters yielding larger initial angular momenta , and environmental conditions , where the stronger ambient uv flux from o - type stars in rich clusters may shorten the disk lifetimes , thus minimizing the braking efficiency of the star - disk interaction during the early angular momentum evolution of massive stars . yet , it is unclear whether the disk locking scenario discussed above for low - mass pre - main sequence stars does apply to more massive stars ( cf . rosen et al . nevertheless , regardless of the actual physical processes at work , the observed relationship between the shape of the rotational distributions of massive stars and the specific properties of their birthplace seems to indicate that initial conditions have a long - lasting impact on their rotational properties . as mentionned in previous sections ( see [ pms ] ) , intermediate - mass pms stars , the so - called herbig ae - be stars , have on average much higher rotational velocities than their lower mass t tauri counterparts at an age of a few myr . wolff et al . ( 2004 ) investigated the rotational evolution of intermediate - mass pms stars ( 1.3 - 2 m@xmath38 ) as they evolve from convective to radiative tracks towards the zams . comparing the measured velocities on convective tracks to those of stars landing on the zams over the same mass range , they concluded that angular momentum is conserved in spherical shells within the star , i.e. , that the pms spin up is directly proportional to the contraction of the stellar radius . the lack of angular momentum redistribution in the predominantly radiative interiors of intermediate - mass pms stars would then imply that they develop a large degree of radial differential rotation from the center to the surface as they approach the zams . further insight into the rotational properties and evolution of intermediate - mass stars is provided by the large - scale study of zorec & royer ( 2012 ) who reported @xmath14 measurements for 2,014 b6- to f2-type stars . by tracing the evolution of rotation as a function of age , they confirm that intermediate - mass stars seem to evolve on the main sequence as differential rotators . striking differences between the rotational distribution of 1.6 - 2.4 m@xmath38 stars and that of 2.4 - 3.8 m@xmath38 stars , the former being unimodal while the latter is bimodal , remain to be understood . similarly , the complex rotational behaviour of stars over this mass range as they evolve onto the ms , with an apparent spin up during the first half of the ms evolution followed by a significant spin down during the second half , represents a real challenge for angular momentum evolution models . a specific sub - group of intermediate - mas stars , the magnetic ap - bp stars host surface magnetic fields of a few kg to a few 10 kg . this sub - group represents about 5 - 10% of the population and is known to exhibit systematic lower velocities that their non - magnetic counteparts ( abt & morrell 1995 ) . alecian et al . ( 2012 ) showed that their precursors , i.e. , the magnetic herbig ae - be stars , already are slower rotators that non - magnetic intermediate - mass pms stars , indicating that magnetic braking is already efficient during the pms for this particuliar subgroup . as the spin down continues on the ms , ap stars can reach very slow rotation indeed , with the longest rotational period ever reported amounting to 77@xmath11710 years ( leroy et al . the last decade has seen tremendous progress in the characterization of the rotational properties of stars at various stages of evolution and over the whole mass range from brown dwarfs to the most massive objects . these new observational results bring formidable constraints to the development of angular momentum evolution models . while the dominant processes thought to dictate the rotational evolution of stars are probably identified , much remains to be done to understand their detailed physics and their respective roles . the confrontation between models and observations , though much improved in recent years , still indicate a number of shortcomings related to transport processes in radiative interiors , the physics of stellar winds , and the interaction between the star and its environment . major advances are expected to arise from multi - dimensional numerical simulations of stellar interiors and stellar atmospheres , which will hopefully provide new clues to the elusive physical processes that govern the rotational evolution of stars from their birth to the last stages of their evolution . it is a pleasure to thank the organisers of the evry schatzman school for a very enjoyable week , the friendly atmosphere , and the unexpected celebration event during the school . i would also like to thank florian gallet and jonathan irwin for providing fig.3 and 4 , respectively , of this contribution , jean - baptiste le bouquin for interesting discussions on interferometry , rafael garcia and jrme ballot for guidance in asterosismology , juan zorec for providing useful references on the rotation of intermediate - mass stars , gaspard duchne for help on binary statistics , gauthier mathys for discussions on ap stars , and nadge meunier for an historical review of the first measurements of the sun s rotation , whose paternity remains controversial . aarnio , a. n. , matt , s. p. , & stassun , k. g. 2012 , apj , 760 , 9 abney , w. d. w. 1877 , mnras , 37 , 278 abt , h. a. , & morrell , n. i. 1995 , apjs , 99 , 135 affer , l. , micela , g. , favata , f. , & flaccomio , e. 2012 , mnras , 424 , 11 affer , l. , micela , g. , favata , f. , flaccomio , e. , & bouvier , j. 2013 , mnras , 430 , 1433 ageros , m. a. , covey , k. r. , lemonias , j. j. , et al . 2011 , apj , 740 , 110 albrecht , s. , winn , j. n. , johnson , j. a. , et al . 2012 , apj , 757 , 18 alecian , e. , wade , g. a. , catala , c. , et al . 2013 , mnras , 429 , 1027 allain , s. 1998 , a&a , 333 , 629 baranne , a. , mayor , m. , & poncet , j. l. 1979 , vistas in astronomy , 23 , 279 barnes , j. r. , collier cameron , a. , donati , j .- f . , et al . 2005 , mnras , 357 , l1 barnes , s. a. 2003 , apj , 586 , 464 benz , w. , & mayor , m. 1981 , a&a , 93 , 235 benz , w. , & mayor , m. 1984 , a&a , 138 , 183 bessolaz , n. , zanni , c. , ferreira , j. , keppens , r. , & bouvier , j. 2008 , a&a , 478 , 155 bolmont , e. , raymond , s. n. , leconte , j. , & matt , s. p. 2012 , a&a , 544 , a124 bonfils , x. , delfosse , x. , udry , s. , et al . 2013 , a&a , 549 , a109 bouvier , j. 2008 , a&a , 489 , l53 bouvier , j. , forestini , m. , & allain , s. 1997 , a&a , 326 , 1023 bouvier , j. , bertout , c. , benz , w. , & mayor , m. 1986 , a&a , 165 , 110 bouvier , j. , alencar , s. h. p. , harries , t. j. , johns - krull , c. m. , & romanova , m. m. 2007 , protostars and planets v , 479 bouvier , j. , cabrit , s. , fernandez , m. , martin , e. l. , & matthews , j. m. 1993 , a&a , 272 , 176 bragana , g. a. , daflon , s. , cunha , k. , et al . 2012 , aj , 144 , 130 camenzind , m. 1990 , reviews in modern astronomy , 3 , 234 carroll , j. a. 1933 , mnras , 93 , 478 casas , r. , vaquero , j. m. , & vazquez , m. 2006 , sol . phys . , 234 , 379 cauley , p. w. , johns - krull , c. m. , hamilton , c. m. , & lockhart , k. 2012 , apj , 756 , 68 chaboyer , b. , demarque , p. , & pinsonneault , m. h. 1995 , apj , 441 , 865 chaplin , w. j. , sanchis - ojeda , r. , campante , t. l. , et al . 2013 , arxiv:1302.3728 charbonnel , c. , decressin , t. , amard , l. , palacios , a. , & talon , s. 2013 , arxiv:1304.5470 cieza , l. , & baliber , n. 2007 , apj , 671 , 605 cohen , o. , drake , j. j. , kashyap , v. l. , sokolov , i. v. , & gombosi , t. i. 2010 , apjl , 723 , l64 collier cameron , a. , campbell , c. g. , & quaintrell , h. 1995 , a&a , 298 , 133 covey , k. r. , greene , t. p. , doppmann , g. w. , & lada , c. j. 2005 , aj , 129 , 2765 cranmer , s. r. , & saar , s. h. 2011 , apj , 741 , 54 dahm , s. e. , slesnick , c. l. , & white , r. j. 2012 , apj , 745 , 56 deheuvels , s. , garca , r. a. , chaplin , w. j. , et al . 2012 , apj , 756 , 19 delfosse , x. , forveille , t. , perrier , c. , & mayor , m. 1998 , a&a , 331 , 581 delorme , p. , collier cameron , a. , hebb , l. , et al . 2011 , mnras , 413 , 2218 dobbs - dixon , i. , lin , d. n. c. , & mardling , r. a. 2004 , apj , 610 , 464 donati , j .- f . , semel , m. , carter , b. d. , rees , d. e. , & collier cameron , a. 1997 , mnras , 291 , 658 dravins , d. , lindegren , l. , & torkelsson , u. 1990 , a&a , 237 , 137 dufton , p. l. , dunstall , p. r. , evans , c. j. , et al . 2011 , apjl , 743 , l22 durney , b. r. , & latour , j. 1978 , geophysical and astrophysical fluid dynamics , 9 , 241 edwards , s. , strom , s. e. , hartigan , p. , et al . 1993 , aj , 106 , 372 eggenberger , p. , maeder , a. , & meynet , g. 2005 , a&a , 440 , l9 eggenberger , p. , montalbn , j. , & miglio , a. 2012 , a&a , 544 , l4 ekstrm , s. , meynet , g. , maeder , a. , & barblan , f. 2008 , a&a , 478 , 467 ekstrm , s. , georgy , c. , eggenberger , p. , et al . 2012 , a&a , 537 , a146 epstein , c. r. , & pinsonneault , m. h. 2012 , arxiv:1203.1618 ferreira , j. , pelletier , g. , & appl , s. 2000 , mnras , 312 , 387 gaig , y. 1993 , a&a , 269 , 267 gallet , f. , & bouvier , j. 2013 , arxiv:1306.2130 gastine , t. , morin , j. , duarte , l. , et al . 2013 , a&a , 549 , l5 ghosh , p. , pethick , c. j. , & lamb , f. k. 1977 , apj , 217 , 578 gizon , l. , & solanki , s. k. 2003 , apj , 589 , 1009 goupil , m. j. , samadi , r. , lochard , j. , dziembowski , w. a. , & pamyatnykh , a. 2004 , stellar structure and habitable planet finding , 538 , 133 gray , d. f. 1973 , apj , 184 , 461 gray , d. f. 1977 , apj , 211 , 198 gregory , s. g. , donati , j .- f . , morin , j. , et al . 2012 , apj , 755 , 97 griffin , r. f. 1967 , apj , 148 , 465 hartmann , l. , & stauffer , j. r. 1989 , aj , 97 , 873 hartmann , l. , hewett , r. , stahler , s. , & mathieu , r. d. 1986 , apj , 309 , 275 hernndez , j. , hartmann , l. , megeath , t. , et al . 2007 , apj , 662 , 106 herbst , w. , bailer - jones , c. a. l. , & mundt , r. 2001 , apjl , 554 , l197 herbst , w. , eislffel , j. , mundt , r. , & scholz , a. 2007 , protostars and planets v , 297 huang , w. , & gies , d. r. 2006 , apj , 648 , 580 irwin , j. , & bouvier , j. 2009 , iau symposium , 258 , 363 irwin , j. , aigrain , s. , bouvier , j. , et al . 2009 , mnras , 392 , 1456 irwin , j. , berta , z. k. , burke , c. j. , et al . 2011 , apj , 727 , 56 irwin , j. , hodgkin , s. , aigrain , s. , et al . 2008 , mnras , 383 , 1588 kawaler , s. d. 1988 , apj , 333 , 236 kervella , p. , thvenin , f. , di folco , e. , & sgransan , d. 2004 , a&a , 426 , 297 khokhlova , v. l. 1976 , astronomische nachrichten , 297 , 203 knigl , a. 1991 , apjl , 370 , l39 kraft , r. p. 1967 , apj , 150 , 551 kraft , r. p. 1970 , spectroscopic astrophysics . an assessment of the contributions of otto struve , 385 krishnamurthi , a. , pinsonneault , m. h. , barnes , s. , & sofia , s. 1997 , apj , 480 , 303 lagarde , n. , decressin , t. , charbonnel , c. , et al . 2012 , a&a , 543 , a108 lanza , a. f. 2010 , a&a , 512 , a77 le blanc , t. s. , covey , k. r. , & stassun , k. g. 2011 , aj , 142 , 55 le bouquin , j .- b . , absil , o. , benisty , m. , et al . 2009 , a&a , 498 , l41 leroy , j. l. , bagnulo , s. , landolfi , m. , & landi deglinnocenti , e. 1994 , a&a , 284 , 174 macgregor , k. b. , & brenner , m. 1991 , apj , 376 , 204 marsden , s. c. , jardine , m. m. , ramrez vlez , j. c. , et al . 2011 , mnras , 413 , 1939 matt , s. , & pudritz , r. e. 2008 , apj , 681 , 391 matt , s. p. , macgregor , k. b. , pinsonneault , m. h. , & greene , t. p. 2012 , apjl , 754 , l26 mayor , m. , marmier , m. , lovis , c. , et al . 2011 , arxiv:1109.2497 mcquillan , a. , aigrain , s. , & mazeh , t. 2013 , arxiv:1303.6787 meibom , s. , barnes , s. a. , latham , d. w. , et al . 2011 , apjl , 733 , l9 mestel , l. 1984 , cool stars , stellar systems , and the sun , 193 , 49 meynet , g. , & maeder , a. 2000 , a&a , 361 , 101 mohanty , s. , & basri , g. 2003 , apj , 583 , 451 mohanty , s. , & shu , f. h. 2008 , apj , 687 , 1323 monnier , j. d. , zhao , m. , pedretti , e. , et al . 2007 , science , 317 , 342 morin , j. , donati , j .- f . , petit , p. , et al . 2010 , mnras , 407 , 2269 mosser , b. , baudin , f. , lanza , a. f. , et al . 2009 , a&a , 506 , 245 mosser , b. , goupil , m. j. , belkacem , k. , et al . 2012 , a&a , 548 , a10 najita , j. r. , carr , j. s. , glassgold , a. e. , & valenti , j. a. 2007 , protostars and planets v , 507 pinto , r. f. , brun , a. s. , jouve , l. , & grappin , r. 2011 , apj , 737 , 72 raghavan , d. , mcalister , h. a. , henry , t. j. , et al . 2010 , apjs , 190 , 1 randich , s. 2010 , iau symposium , 268 , 275 rebull , l. m. , wolff , s. c. , & strom , s. e. 2004 , aj , 127 , 1029 rebull , l. m. , stauffer , j. r. , megeath , s. t. , hora , j. l. , & hartmann , l. 2006 , apj , 646 , 297 reiners , a. , & basri , g. 2008 , apj , 684 , 1390 reiners , a. , & mohanty , s. 2012 , apj , 746 , 43 reiners , a. , & schmitt , j. h. m. m. 2002 , a&a , 384 , 155 reiners , a. , basri , g. , & browning , m. 2009 , apj , 692 , 538 rodrguez - ledesma , m. v. , mundt , r. , & eislffel , j. 2009 , a&a , 502 , 883 rosen , a. l. , krumholz , m. r. , & ramirez - ruiz , e. 2012 , apj , 748 , 97 rucinski , s. m. 1988 , aj , 95 , 1895 schatzman , e. 1962 , annales dastrophysique , 25 , 18 scholz , a. , & eislffel , j. 2004 , a&a , 419 , 249 scholz , a. , & eislffel , j. 2005 , a&a , 429 , 1007 scholz , a. , irwin , j. , bouvier , j. , et al . 2011 , mnras , 413 , 2595 schou , j. , antia , h. m. , basu , s. , et al . 1998 , apj , 505 , 390 semel , m. 1989 , a&a , 225 , 456 sills , a. , pinsonneault , m. h. , & terndrup , d. m. 2000 , apj , 534 , 335 skumanich , a. 1972 , apj , 171 , 565 spiegel , e. a. , & zahn , j .- p . 1992 , a&a , 265 , 106 stauffer , j. r. 1987 , cool stars , stellar systems and the sun , 291 , 182 stauffer , j. r. , & hartmann , l. w. 1987 , apj , 318 , 337 strom , s. e. , wolff , s. c. , & dror , d. h. a. 2005 , aj , 129 , 809 talon , s. , & charbonnel , c. 2008 , a&a , 482 , 597 tonry , j. , & davis , m. 1979 , aj , 84 , 1511 van belle , g. t. 2012 , a&ar , 20 , 51 vidotto , a. a. , opher , m. , jatenco - pereira , v. , & gombosi , t. i. 2009 , apj , 699 , 441 vidotto , a. a. , jardine , m. , opher , m. , donati , j. f. , & gombosi , t. i. 2011 , mnras , 412 , 351 vogel , s. n. , & kuhi , l. v. 1981 , apj , 245 , 960 vogt , s. s. , & penrod , g. d. 1983 , pasp , 95 , 565 von zeipel , h. 1924 , mnras , 84 , 665 weber , e. j. , & davis , l. , jr . 1967 , apj , 148 , 217 wolff , s. c. , strom , s. e. , & hillenbrand , l. a. 2004 , apj , 601 , 979 wolff , s. c. , strom , s. e. , dror , d. , & venn , k. 2007 , aj , 133 , 1092 wright , j. t. , marcy , g. w. , howard , a. w. , et al . 2012 , apj , 753 , 160 wright , n. j. , drake , j. j. , mamajek , e. e. , & henry , g. w. 2011 , apj , 743 , 48 zahn , j .- 1977 , a&a , 57 , 383 zahn , j .- p . , & bouchet , l. 1989 , a&a , 223 , 112 zanni , c. , & ferreira , j. 2013 , a&a , 550 , a99 zorec , j. , & royer , f. 2012 , a&a , 537 , a120 | this course reviews the rotational properties of non - degenerate stars as observed from the protostellar stage to the end of the main sequence .
it includes an introduction to the various observational techniques used to measure stellar rotation .
angular momentum evolution models developed over the mass range from the substellar domain to high - mass stars are briefly discussed . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
speculations that instantons could induce diquark condensation in low temperature but high density quark matter @xcite have revived the interest in the diquark clustering in the nucleon . it is sometimes also argued that diquark condensation may occur even at moderate densities , for example in heavy nuclei . this problem is strongly related to the question of instanton induced diquark clustering in the nucleon . indeed , the instanton - induced t hooft interaction is strongly attractive for a quark - quark pair with quantum numbers @xmath2 ( scalar diquark ) . this raises expectations that it binds a scalar diquark and is responsible for the scalar diquark - quark structure of the nucleon @xcite . this assumption is based on the iteration of the t hooft interaction in the @xmath3 s - channel . however , this picture of the quark - quark interaction in baryons is only a small part of a more general one , based on the effective meson - exchange interaction @xcite . when the t hooft interaction is first iterated in the @xmath3 t - channel it inevitably leads to goldstone boson exchange between constituent quarks , which is drastically different from the initial ( not iterated ) t hooft interaction due to the ( anti)screening effects . the latter effective meson - exchange interaction does not induce a bound scalar diquark , nor an appreciable diquark - quark clustering in nucleon . this effective meson exchange interaction is also the most attractive in @xmath4 @xmath3 pairs , but the nature of this attraction is very different from that of the t hooft interaction . this interaction , however , is not strong enough to bind the scalar diquark . when it is combined with a confining interaction it binds the diquark in the sense that there is no asymptotic state with two free constituent quarks , though the mass of the scalar diquark is a few tens of mev above the two - constituent - quark threshold . there is no significant diquark clustering in the nucleon either , because the nucleon is intrinsically a three - quark system and the fermionic - nature of the constituent quarks plays an important role . if the subsystem of quarks 1 and 2 is in the @xmath4 state then due to the antisymmetrization the quark pairs in the subsystems 1 - 3 and 2 - 3 are also partly in the @xmath4 state . this implies that a strong attraction in @xmath4 quark pair contributes in all quark subsystems simultaneously and makes the nucleon compact , but without appreciable quark - diquark clustering . this paper consists of two independent , but interrelated parts . in the first one we discuss how the instanton - induced interaction ( or some general nonperturbative gluonic interaction ) leads to the poles when it is iterated in the @xmath3 t - channel . these pole contributions have an evident meson - exchange interpretation . the latter meson - exchange interaction is drastically different from the initial ( bare ) t hooft interaction which becomes strongly enhanced in the channel of goldstone boson exchange quantum numbers . we also discuss the role of instantons in @xmath5 systems . there is no new wisdom in that the nonperturbative gluonic configurations , e.g. instantons , induce the dynamical breaking of chiral symmetry and explain the low - lying mesons . we include the latter discussion only with the purpose of showing how the nonperturbative gluonic interaction both explains mesons and at the same time leads to the effective meson exchange picture in the @xmath3 systems . through the latter it also explains the baryon spectra and the nuclear force . our discussion is rather general , and does not necessarily rely on the instanton - induced interaction picture . any nonperturbative gluonic interaction , which respects chiral symmetry and induces the rearrangement of the vacuum ( i.e. dynamical breaking of chiral symmetry ) , will automatically explain the @xmath6 mass splitting and will imply a meson - exchange picture in baryons . the second part of this paper is devoted to a detailed study of diquark clustering in the nucleon , based on the effective meson - exchange interactions in the baryons and the nucleon wave functions obtained from the solution of the semirelativistic three - body schrdinger equation . we show that there is no appreciable diquark clustering in the nucleon and that the effective meson - exchange interaction , which is adjusted to describe the baryon spectrum @xcite , does not bind the scalar diquark nor the nucleon . however , when this interaction is combined with the confining interaction , one finds a bound diquark but with a mass above the two - quark threshold and very similar in magnitude to that obtained recently in lattice qcd @xcite . nevertheless , as soon as the strength of the effective meson - exchange interaction is increased , not by a very big amount , it alone binds a nucleon , even without a confining force . while the contributions from the confining interaction to the nucleon mass are not small , the nucleon size , calculated with the confining interaction alone and in a full model that includes both confinement and effective meson exchange , is different . it is substantially smaller in the latter case , showing that there is indeed a soft interval between the scale when confinement becomes active , and the scale where chiral physics starts to work . however , for excited baryon states , which are much bigger in size , the role of confinement increases . it has been shown in recent years that a successful explanation of light and strange baryon spectroscopy , especially the correct ordering of the lowest states with positive and negative parity , is achieved if the hyperfine interaction between constituent quarks @xmath7 and @xmath8 has a short - range behaviour which reads schematically @xcite : @xmath9 where @xmath10 is a set of a flavor gell - mann matrices for @xmath11 and @xmath12 . this interaction is supplied by the short - range parts of goldstone boson exchange ( gbe ) , , @xmath13 and @xmath14 exchanges ; due to the axial anomaly the @xmath15 is not a goldstone boson , but in the large @xmath16 limit it also becomes a goldstone boson , and thus the coupling of @xmath15 to a constituent quark should be essentially different from that of octet mesons . ] vector - meson - like exchange and/or correlated two - pseudoscalar - meson - like exchange @xcite , etc . it is sometimes stated that the instanton - induced t hooft interaction in @xmath3 pairs could also provide a good baryon spectrum as it contains a flavor- and spin - dependence and , iterated in the @xmath3 s - channel , produces a deeply bound scalar diquark which makes the nucleon lighter than the @xmath17 @xcite . a similar picture of a deeply bound scalar diquark has been advocated in a generalized nambu and jona - lasinio ( njl ) model @xcite . then a baryon is constructed as an additive diquark - quark system or by solving `` relativistic diquark - quark faddeev equations '' that take into account the quark exchange between the diquark and quark - spectator @xcite . in this section we show that such a picture of baryons , based on the iteration of the local 4-fermion interaction in the @xmath3 s - channel is only a small part of a more general picture , based on the meson - exchange interaction . the reason is that when the t hooft interaction ( or generalized njl one ) is first iterated in the @xmath3 t - channel , it inevitably leads to the effective meson - exchange between constituent quarks , which is drastically different from the initial ( not iterated ) 4-fermion local interaction due to ( anti)screening effects . the difference is not only in the flavor- and spin - dependence , but sometimes also in the sign of the interaction . to demonstrate this we use a simple @xmath18 t hooft - determinant interaction for two light flavors ( u and d ) , neglecting for the simplicity the tensor coupling term , which is suppressed by the factor @xmath19 @xcite . we also assume zero masses for the current u and d quarks in this section . for our illustrative purposes such an approximation is justified . this hamiltonian reads : @xmath20 . \label{hooft}\ ] ] the _ dimensional _ strength of the interaction @xmath21 as well as the ultraviolet cut - off scale @xmath22 can be related to parameters of the instanton liquid @xcite ( the dimensionless coupling constant is given by @xmath23 ) . the interaction ( [ hooft ] ) is attractive in the scalar - isoscalar @xmath24 channel ( the first term ) , leading to chiral symmetry breaking , or , which is related , to a massive @xmath25-meson field and the constituent mass @xmath26 of quarks . this is readily obtained from the schwinger - dyson ( gap ) equation for a quark green function in the hartree - fock approximation . the interaction in the @xmath27 pseudoscalar - isovector channel is driven by the second term of ( [ hooft ] ) . it is so strong , that when it is iterated in the @xmath27 s - channel by solving the bethe - salpeter equation , see fig . 1 , it exactly compensates the @xmath28-energy , supplied by the first term in ( [ hooft ] ) , and thus there appear @xmath29 mesons with zero mass as deeply bound relativistic @xmath27 systems - nambu - goldstone bosons . the nonzero mass of the pseudoscalar mesons is brought about by the nonzero current quark mass as a perturbation , which is well illustrated by the current algebra results ( gell - mann - oakes - renner relations ) . the first two terms in the hamiltonian ( [ hooft ] ) form in fact the classical njl hamiltonian @xcite and the statement above is a theorem , proved by nambu and jona - lasinio many years ago . this scenario holds if the fixed strength of the interaction @xmath21 exceeds some critical level . in a more sophisticated derivation @xcite the strength of the interaction @xmath21 is not fixed and should be determined after one gets the chirally broken phase . the hamiltonian ( [ hooft ] ) does not contain any interaction in @xmath27 pairs with vector meson quantum numbers . so , according to the scenario above , the masses of vector mesons , @xmath1 and @xmath30 , should be approximately @xmath28 , which is well satisfied empirically . thus , it can not be overemphasized that the @xmath31 mass splitting is brought about not by the perturbative color - magnetic interaction between nonrelativistic constituent quarks system and its nambu - goldstone boson nature ( zero mass in the chiral limit ) can not be obtained from a nonrelativistic reduction of the second term in ( [ hooft ] ) used in the schrdinger equation . ] , but by the detailed balance between the first and second terms in ( [ hooft ] ) , which is determined exclusively by the demand that the gluonic interaction between current quarks must satisfy chiral @xmath32 symmetry ) is color - independent . one can , of course , rewrite this interaction using the pauli principle in terms of linear combinations of different operators , like @xmath33 , @xmath34 , @xmath35 , @xmath36 , @xmath37 , ... the presence of the @xmath36 structure in this decomposition does not mean that the interaction ( [ hooft ] ) becomes similar in its effect to the color - magnetic component of the one gluon exchange interaction , which is explicitly color - dependent . it is like an identity @xmath38 does not mean that effect of @xmath39 is contained in @xmath40 . ] . an important question , which is actively debated nowdays , is which particular _ nonperturbative _ gluonic configurations in qcd , e.g. instantons , or abelian monopoles , or other topological configurations , are intrinsically responsible for the chiral symmetry breaking . among other attractive features of the instanton - induced interaction ( [ hooft ] ) is that it automatically solves the @xmath41 problem , giving a much bigger mass to the pseudoscalar flavor - singlet ( in the present 2-flavor formulation that is isosinglet ) meson @xmath15 @xcite . this is because of the last term in ( [ hooft ] ) . only this term contributes in a pseudoscalar flavor - singlet quark - antiquark pair . since this interaction is repulsive , the @xmath15 becomes heavy , contrary to @xmath0 . we note in passing that the color - magnetic interaction can not explain this big @xmath42 mass splitting . clearly , the simple hamiltonian ( [ hooft ] ) is only some part of a more complicated physical situation . for instance , one definitely needs some additionl attractive interaction , e.g. confinement , otherwise the @xmath15 meson or vector mesons will be unbound , while the octet pseudoscalar mesons are probably not affected by the long - range confining interaction . having mentioned all the positive features of the hamiltonian ( [ hooft ] ) in the quark - antiquark system , we are now going to discuss its implications in quark - quark systems , i.e. in baryons . what is typically done is a fierz - rearrangement of the hamiltonian ( [ hooft ] ) into diquark @xmath3 channels @xcite ( or , similarly , a fierz - rearrangement of the generalized njl hamiltonian into diquark channels @xcite ) . then the diquark hamiltonian is iterated in the @xmath3 s - channel , see fig . the interaction in the scalar @xmath43 diquark turns out to be attractive and it produces a deeply bound scalar diquark . ] . however , as soon as the hamiltonian ( [ hooft ] ) is iterated first in the @xmath3 t - channel , see fig . 3 , it implies irreducible ( for the @xmath3 s - channel ) pion- and sigma - exchange interactions between quarks . this statement comes about as a theorem since the iteration of the hamiltonian ( [ hooft ] ) in @xmath3 t - channel is equivalent to its iteration in @xmath27 s - channel . clearly the set of diagrams in fig . 3 contains all the diagrams of fig . 2 , but in addition it contains many others , and the effect of these additional diagrams is so important that the physics implied by fig . 2 and fig . 3 is drastically different ) is applied in baryons in the framework of the chiral quark - soliton model @xcite . in this case quark - quark correlations through the self - consistent chiral _ mean field _ and quantization of its rotation take into account some part of the iterations in s - channel of fig . 2 and do not take into account the t - channel ladders of fig . 3 . ] . a simple example of the different physical implications is that fig . 3 suggests a long - range meson - exchange yukawa tail , which is crucial for the interaction of quarks , belonging to different nucleons , while if the picture of fig . 2 were correct the nuclear force would be absent . another evident difference is that according to the hamiltonian ( [ hooft ] ) and fig . 2 the interaction is absent in flavor - symmetric , @xmath44 , quark pairs ,- resonance and its excitations . if that were the case , the positive parity state @xmath45 , which belongs to the @xmath46 shell because of its positive parity , would be approximately @xmath47 mev above the negative parity pair @xmath48 . ] while the @xmath3 interaction of fig . 3 does not vanish in this case . less evident is that even the short - range interaction between quarks is crucially modified in fig . 3 as compared to fig . we call it the `` ( anti)screening effect '' and illustrate it below . in order to see it one should avoid the fierz - rearrangement of ( [ hooft ] ) into a diquark hamiltonian . instead , one can use the initial hamiltonian ( [ hooft ] ) , but assume that all initial , intermediate and final state @xmath49 wave functions are explicitly antisymmetric . consider the first term of ( [ hooft ] ) . in the nambu - goldstone mode of chiral symmetry a fermion field has a large dynamical ( constituent ) mass @xmath26 . using a @xmath50 expansion , one obtains that to leading order ( @xmath51 ) the first term of ( [ hooft ] ) leads to a @xmath52-function type attraction in all quark pairs allowed by pauli principle : @xmath53 the effect of the third term in ( [ hooft ] ) to the same order is @xmath54 the potentials ( [ first ] ) and ( [ third ] ) , combined together , produce @xmath55 note that at this order the second and fourth terms of ( [ hooft ] ) do not contribute . the potential ( [ comb ] ) suggests a strong attraction in the isospin - zero quark pair , and no interaction in @xmath44 quark pairs . assuming relative angular momentum @xmath56 within the @xmath57 quark pair the pauli principle implies that the spins of the quarks should be antiparallel , @xmath58 . when the theory is sensibly regularized the delta - function attraction is smeared out over the instanton size @xmath59 @xmath60 this substitution arises from a replacement of the static green function of the infinitely heavy particle in ( [ hooft ] ) @xmath61 by the green function of a particle with mass @xmath62 @xmath63 when the strength of the interaction is big enough , the potential ( [ comb])-([delta ] ) , iterated by solving the semirelativistic schrdinger equation ( i.e. when the kinetic energy operator is taken in a relativistic form ) can produce a deeply bound scalar diquark , in agreement with @xcite . indeed , when one takes the strength @xmath64 gev@xmath65 , @xmath66 , with the instanton size @xmath59 between 0.3 and 0.35 fm and the constituent mass @xmath67 mev @xcite one finds a very deeply bound diqurk . in the illustration above we have used a simplified but transparent nonrelativistic picture that adequately reflects in the present case the essential features of a more rigorous bethe - salpeter approach . sometimes the potential ( [ comb ] ) is applied to explain the hyperfine splittings in baryons @xcite . while it can generate the @xmath68 mass splitting , it fails to explain the lowest levels with positive and negative parity because it does not contain the necessary spin - isospin dependence ( [ gbe ] ) . it will become evident from the discussion below that such an interpretation of the role of instantons in baryons does not survive as soon as the wider class of diagrams in fig . 3 is considered . what happens when the first term in ( [ hooft ] ) is iterated in the @xmath3 t - channel ? the corresponding amplitude is @xmath69 where @xmath70 is the loop integral ( bubble ) with the scalar vertex which represents vacuum polarization in the scalar channel . ( [ iter ] ) defines `` running amplitude '' and a negative sign in the denominator implies its antiscreening behaviour . the expression ( [ iter ] ) is known to have a pole at @xmath71 @xcite , which can be identified with the exchange by scalar meson @xmath25 with the mass @xmath72 in the chiral limit . the coupling constant of the @xmath25-meson to constituent quark can be obtained as a residue of ( [ iter ] ) at the pole @xmath73 expanding the @xmath74 vertex in @xmath50 , one obtains to leading order @xmath75 the following well - known sigma - exchange potential @xmath76 the equivalence between the t - channel ladder of bubbles in fig . 3 beyond the @xmath25-meson pole in the t - channel and the meson - exchange diagram is achieved only when some form factor @xmath77 is inserted into the meson - quark vertex , i.e. the left hand side of eq . ( [ pole ] ) should be multiplied with @xmath78 . the form factor is to be normalized @xmath79 . in principle ( [ pole ] ) allows to obtain a functional form for such a form factor . however , it will be very far from reality because the toy model ( [ hooft ] ) does not contain confinement , which should be important for the interaction between quark and antiquark in the weakly bound system like @xmath25-meson ( note that its mass is just at the `` continuum threshold '' @xmath28 ) . in this situation the best way is to rely on our general understanding of the low - energy effective theory . both constituent quarks and chiral meson fields as well as their couplings make a sense only in the nambu - goldstone mode of chiral symmetry . when momentum transfer at the meson - quark vertex exceeds the chiral symmetry breaking scale @xmath80 ( which within all njl - like models coincides with the regularization scale @xmath22 ) the effective theory should be cut off . this cut off is accomplished by a form factor in the meson - constituent quark vertex and should be related to the internal structure of both quasiparticles . but in any case the scale parameter in this form factor should be comparable with @xmath80 . at high momenta one can use neither constituent quark nor chiral fields and original quark - gluon degrees of freedom should be used instead . approximating this form factor by @xmath81 instead of the potential ( [ sigma ] ) we arrive at @xmath82 note , that any functional form of form factor leads to a similar suppression of the potential at short range but it does not influence its long - range part which is determined exclusively by the position of the pole . if one takes a dipole form factor the suppression will be stronger . what is the fate of the third term in ( [ hooft ] ) , when it is iterated in the @xmath3 t - channel ? in this case one obtains the following amplitude @xmath83 where @xmath84 are isospin indices . the positive sign in the denominator indicates screening . for instance , at @xmath71 the strength of the interaction is reduced by the factor 2 versus a bare vertex . still , this suppression of the interaction at low momenta is not realistic , because the toy model ( [ hooft ] ) does not contain confinement and thus there is only a repulsion in the scalar - isovector quark - antiquark system . when confinement is added in the quark - antiquark pairs , there appear heavy scalar - isovector mesons and the sign of the amplitude ( [ iter2 ] ) becomes opposite at small momenta ! this low - momentum amplitude corresponds to the exchange by scalar - isovector mesons between quarks . the corresponding meson - exchange interaction is similar in form to ( [ sigfor ] ) , but with an additional factor @xmath85 . the expectation value of the operator @xmath86 in the scalar diquark is @xmath87 and thus the interaction ( [ comb ] ) is stronger by the factor 4 through the interaction ( [ third ] ) in the picture of fig . 2 . in contrast , in the picture of fig . 3 the contributions from the scalar and scalar - isovector meson exchanges tend to cancel each other . thus we see that the initial interaction is screened . this screening means that the interaction ( [ comb ] ) becomes weaken and that its isospin dependence is modified . it is trivial to check that the attraction ( [ sigfor ] ) does not lead to a bound scalar diquark with any reasonable coupling constant , sigma - meson mass , cut - off mass @xmath88 and constituent quark mass ( see discussion in the next chapter ) . the scalar - isovector meson exchange will further reduce this attraction , though the coupling constant of the scalar - isovector mesons to constituent quarks will be essentially smaller . both the scalar - isoscalar exchange and scalar - isovector exchanges between constituent quarks do not contain the flavor - spin dependence ( [ gbe ] ) which is necessary for baryon spectroscopy - exchange is known to be very important for the medium - range attraction in the nn system and it also contributes to binding nucleon . however it only has a small influence on the splittings via different radial behaviour of baryon wave functions . ] . now we shall extend our @xmath50 expansion of the hamiltonian ( [ hooft ] ) to the next - to - leading order , taking into account terms @xmath89 . the first and third terms of ( [ hooft ] ) will give at this order the spin - orbit forces , as well as some small corrections to the interactions ( [ first ] ) and ( [ third ] ) . the second and fourth terms generate , however , a flavor - spin dependent interaction . consider the second term in ( [ hooft ] ) . it gives both the spin - spin and tensor force components . we ignore below the tensor force as it is irrelevant to our simple discussion and for the @xmath56 @xmath3 pair . then : @xmath90 again , assuming that the instanton has a finite size @xmath59 , the potential ( [ second ] ) reads : @xmath91 let us now iterate the second term of ( [ hooft ] ) in the @xmath3 t - channel @xmath92 where @xmath93 is a bubble with a pseudoscalar vertex ( vacuum polarization in the pseudoscalar channel ) . there is a pole at @xmath94 in ( [ iterp ] ) in the chiral limit , which can be identified as a pion - exchange ( beyond the chiral limit it is shifted to a physical pion mass @xmath95 . ) the coupling constant of pion to constituent quark can be obtained as a residue of ( [ iterp ] ) at the pole @xmath96 thus near the pole ( [ iterp ] ) - ( [ polep ] ) represents a pion - exchange potential between quarks , which in the chiral limit , @xmath97 , at the order @xmath98 ( omitting the tensor force component ) is : @xmath99 the difference between ( [ second]-[secmod ] ) and ( [ pion ] ) is obvious : the interaction ( [ pion ] ) is much stronger . and @xmath100 where @xmath101 is a zero order function stemming from confinement . ] a source of this enhancement is also obvious : near the pole the original bare interaction @xmath102 becomes strongly reinforced . the interaction is represented by a bare vertex @xmath102 , but at @xmath103 it becomes infinitely enhanced in the channel with goldstone boson exchange quantum numbers . ] the pion pole is located just near the space - like region and thus strongly influences the quark - quark interaction at not very high momentum transfer . again , to retain the equivalence between the t - channel ladder of bubbles in fig . 3 and the pion - exchange diagram beyond the pole , one must insert a form factor into the @xmath104 vertex . the effect of this form factor is to smear out the @xmath52-type interaction in ( [ pion ] ) over the region @xmath105 . if this form factor is choosen in the form ( [ form ] ) , then one obtains @xmath106 the @xmath89 expansion of the fourth term of the hamiltonian ( [ hooft ] ) will give a result similar to second term , without , however , isospin - dependent factor and with the opposite sign . its iteration in the @xmath3 t - channel will produce screening effects as it is repulsive in the @xmath107 s - channel . when , however , this term is combined with an additional attractive interaction , e.g. confinement , it will give @xmath15 . then the iteration in the @xmath3 t - channel will imply @xmath15-exchange between quarks . the latter interaction is similar to ( [ pion ] ) , except that the factor @xmath108 is not present and that in this case there appears a yukawa part of the potential because the mass of @xmath15 is not zero in the chiral limit . the discussion above suggests that while for the picture of fig . 2 the most important interaction is ( [ comb ] ) and the interaction ( [ second ] ) is only some very small correction to it , in the case of fig . 3 the most important interaction in baryons becomes ( [ pion ] ) and the one of ( [ sigfor ] ) only plays a modest role for splittings . this is a consequence of an antiscreening . the antiscreening implies that if a typical momentum transfer in the meson - quark vertex ( which in qq systems is of the same order as momentum of quarks ) is below the chiral symmetry breaking scale , then the original ( bare ) quark - quark vertex in the pseudoscalar channel is strongly reinforced by the pole which occurs when one iterates it in the t - channel . these pole contributions represent the goldstone boson exchange interactions between constituent quarks in the nambu - goldstone mode of chiral symmetry . only at a rather high momentum transfer ( i.e. very far from the poles ) there should appear a sensitivity to the original ( bare ) quark - quark vertex . in the latter case the constituent quarks and chiral fields can not be used as effective degrees of freedom . so the crucial question is what a typical momentum transfer in the given system is . in the low - lying baryons it is below the chiral symmetry breaking scale thus justifying a use of the effective qq interactions there @xcite . we hope that the discussion above has been transparent enough to show a dramatic difference between the initial t hooft interaction , taken literally in @xmath3 system , and its implication after iteration in the t - channel , producing meson exchange between constituent quarks . in fact , what one needs for the chiral symmetry breaking is a scalar interaction between quarks . any pairwise gluonic interaction between quarks in the local approximation will necessarily contain the first and second terms of ( [ hooft ] ) with fixed relative strength . this is because of the chiral invariance . thus all our conclusions are rather general and do not rely necessarily on t hooft interaction . an important lesson is to see how this nonperturbative gluonic interaction , which induces the dynamical breaking of chiral symmetry , suggests an explanation of both the low - lying mesons and at the same time of baryons and the nuclear force through the effective meson exchange picure in @xmath3 systems . among the various applications of this idea , will be to work out how the meson - exchange interaction shifts the transition point from the chiral symmetry broken phase to the color - superconductor phase . we also mention a recent lattice study @xcite which shows directly that the hyperfine @xmath68 splitting is mostly due to the meson - exchange interaction between quarks . another indirect evidence in favor of the picture in fig . 3 versus that in fig . 2 is that after cooling ( the cooling means that all gluonic configurations , except for instantons , are removed ) the @xmath109 splitting disappears @xcite . while the cooling does not affect the initial t hooft interaction between quarks and thus the whole s - channel ladder of fig . 2 is active , it ruins the t - channel ladder of fig . the reason is that there are not enough antiquarks in the fock space after cooling as in quenched approximation they are mostly produced by different gluons , including perturbative ones , attached to valence quark lines ( z graphs ) . we start this section with a short description of the effective meson - exchange interaction model , adjusted to describe baryon spectroscopy within an exact semirelativistic 3-body formulation @xcite . the hamiltonian of ref . @xcite reads : @xmath110 @xmath111 here the relativistic form of the kinetic - energy operator is used , with @xmath112 the 3-momentum and @xmath113 the masses of the constituent quarks . the dynamical part consists of the quark - quark interaction @xmath114 the linear pairwise confining potential @xmath115 includes both the color - electric string @xmath116 with the color factor absorbed into the string tension @xmath117 as well as a constant @xmath118 , which is large and negative , and thus effectively includes all possible spin- and flavor - independent attractive interactions between quarks , e.g. @xmath25-exchange ( [ sigfor ] ) , etc . the flavor- and spin - dependent part of the above hamiltonian is @xmath119[3ex]{}+v_{\eta}(\vec r_{ij } ) \lambda_i^8 \lambda_j^8 + \frac{2}{3}v_{\eta'}(\vec r_{ij})\right ] \vec\sigma_i\cdot\vec\sigma_j,\end{aligned}\ ] ] @xmath120 with @xmath121 ( @xmath122 ) being the individual phenomenological meson masses , and @xmath123 the meson - quark coupling constants . the constituent mass of the light quarks @xmath124 was fixed in @xcite to a typical value , @xmath125 mev , implied by a simple static quark model formula for the nucleon magnetic moment . it is astonishing that the same value has been obtained in a lattice measurement @xcite . all other parameters of the above hamiltonian can be found in ref . @xcite . in light quark systems , like @xmath126 and @xmath17 , only the @xmath0-like , @xmath14-like and @xmath15-like parts of the potential ( [ voct ] ) contribute . the @xmath0-like exchange interaction is determined by the following matrix elements : @xmath127 while the @xmath14- and @xmath15-like exchanges depend only on the spin @xmath128 of a quark pair . combining all @xmath0 , @xmath14 and @xmath15 interactions one finds that the potential ( [ voct ] ) is most attractive at short distances in @xmath129 quark pair and essentially less attractive in the @xmath130 diquark system . in other possible color - antitriplet @xmath3 pairs it is repulsive . applying the hamiltonian ( [ htot ] ) in a color - antitriplet @xmath3 system , one finds a mass @xmath131 mev for a scalar diquark , @xmath2 , and a mass @xmath132 mev for an axial - vector diquark , @xmath133 . in both cases the relative orbital angular momentum is @xmath56 , so the total angular momentum coinsides with the spin of two quarks . these values are very similar to those obtained recently from the lattice `` diquark spectroscopy '' @xcite . the root - mean - square ( r.m.s . ) radius of the scalar diquark is 0.354 fm and of the axial - vector one - 0.438 fm . these radii do not include the size of the constituent quark . it is evident that the confining interaction implies a bound diquark in the sense that there are no asymptotically free constituent quarks . so it is very instructive to compare the mass of the above diquarks with the unphysical two - constituent - mass threshold , @xmath134 mev . the scalar diquark mass is a few tens of mev above the threshold , which indicates that the meson - exchange part of the interaction , including @xmath118 , does not bind a diquark without the confining interaction @xmath116 . this can also be checked explicitly . to this end we combine the spin- and isospin - dependent interaction ( [ voct ] ) with the @xmath25-exchange potential ( [ sigfor ] ) and drop the confining potential ( [ conf ] ) . the @xmath135 coupling constant is constrained to be equal to the @xmath104 one , as suggested by chiral symmetry , @xmath136 the sigma mass is taken to be @xmath137 , which is implied by the well known result for all njl - like interactions , @xmath138 . with these constraints we do not find a bound diquark with any reasonable value for @xmath139 gev . only with @xmath140 gev does a weakly bound scalar diquark appear . if one increases the @xmath141 coupling constant by a factor 1.5 , but keeps the @xmath104 coupling constant , then we obtain a bound diquark only at @xmath142 gev . thus we conclude that the meson - exchange interaction itself does not bind a diquark . the next question we address in this section is whether the meson - exchange interaction binds nucleon itself , without confinement . a - priori one can not exclude the possibility that while the diquark is unbound the three quark system will be bound because of genuine 3-body effects ( compare , e.g. , the binding energy of tritium and deuteron ) . indeed , a full model , including confinement ( [ conf ] ) , produces a nucleon mass which is below the three constituent mass threshold , @xmath143 mev . hence the positive contribution from the rising potential @xmath116 , 631 mev , is not big compared to the negative contribution from @xmath144 mev in combination with the negative contribution of the spin - dependent part of interaction , -750 mev . superficially one could thus conclude that the meson - exchange part of the hamiltonian could bind nucleon without any support from confinement . however , such an interpretation can not be taken for two reasons . firstly , we do not know which part of the negative constant @xmath118 comes from the @xmath25-exchange , and which - from the genuine color - electric confinement , because the @xmath145-shape of the gauge - invariant 3-body confining interaction can be approximated by a sum of pairwise potentials only when some additional constant contribution is added . secondly , a perfect fit of the baryon spectrum with a quality similar to that in ref . @xcite can be obtained with a constituent mass smaller than @xmath146 . so we have performed a direct calculation of the nucleon , replacing the potential ( [ conf ] ) by the @xmath25-exchange potential ( [ sigfor ] ) . we have found that for @xmath147 mev the nucleon is unbound and becomes bound at higher values of @xmath88 . if one increases the @xmath141 by a factor 1.5 , then the nucleon becomes bound for @xmath148 mev . these results indicate that while the nucleon is unbound with the meson - exchange potential parameters fixed of ref . @xcite , it could be bound as soon as a spin- and isospin - independent @xmath25-like exchange interaction and/or spin- and isospin - dependent interactions are made stronger , not by a big amount . it is evident that a description of all excited states demands the presence of confinement because all these states are much above the @xmath149 threshold . we shall use the following set of jacobi coordinates and a coupling scheme with self - evident notation : @xmath150 @xmath151 @xmath152 @xmath153 @xmath154 let @xmath155 be a projector onto a subspace with a given value of spin @xmath156 and isospin @xmath157 of the particles 1 and 2 . the probability density for finding particles 1 and 2 in a spin - isospin state @xmath158 at a relative distance @xmath159 is given by @xmath160 where @xmath161 is an antisymmetric 3-body baryon wave function . one can similarly define the probability density for finding the particle 3 at a distance @xmath162 from the center of mass of particles 1 and 2 @xmath163 then one can calculate the corresponding moments @xmath164 @xmath165 in table 1 we present the @xmath166 moments for @xmath126 and @xmath17 in two cases : ( i ) full model , ( ii ) no spin - dependent interaction at all ( i.e. only confinement is active ) . comparing the nucleon r.m.s . radius , @xmath167 = 0.304 fm , with the radius of a scalar diquark , 0.354 fm , we can deduce the role of genuine 3 body effects - they make the nucleon essentially more compact than the diquark . the empirical mean square charge radius of the proton , @xmath168 @xmath169 , consists of a few contributions : the contribution from the mean square matter radius above , the charge mean square radius of the constituent quark , the meson exchange current contribution @xcite , the proton anomalous magnetic moment contribution , etc . the rather small value of the matter radius , obtained above , is consistent with large contributions from other sources . for instance , the charge radius of the constituent quark should be mainly determined by the @xmath1-meson pole in the time - like region ( vector meson dominance ) and thus can be expected to be of the order @xmath170 fm . the r.m.s . radius of the @xmath17-resonance , @xmath171 = 0.390 fm , is larger than that of the nucleon . this result is easy anticipate since the @xmath17-resonance wave function does not contain @xmath172 components , where the potential ( [ voct ] ) is strongly attractive at short range , and thus the size of the @xmath17-resonance is determined mainly by the weak attraction in the @xmath173 quark pairs as well as by the confining interaction . the bigger size of @xmath17 has a well known experimental consequence : the @xmath174 electromagnetic form factor falls off faster than the nucleon elastic one . when the meson - exchange interaction is switched off , the nucleon matter radius becomes larger , @xmath167 = 0.442 fm . this illustrates that there is a soft gap between the scale where chiral physics starts to work and the scale where confinement is important . the crucial role of three body effects can also be seen from the comparison of the root mean square distance between quarks in the @xmath175 quark pair in the nucleon , 0.354 fm , with the same distance in a free scalar diquark , 0.708 fm . similarly , the three body effects and the antisymmetrization are responsible for the fact that the root mean square distance in the @xmath175 quark pair in the nucleon , 0.354 fm , is similar to that one in the @xmath176 subsystem in the nucleon , 0.387 fm , while the potential is very different in both cases . a comparison of the two numbers above gives an idea about how unimportant clustering is in the nucleon . it can also be seen from fig . 4 and fig . 5 where we show probability density distributions . with a pure static `` right triangle '' 3q configuration the relation between @xmath177 and @xmath178 would be @xmath179 . this relation is almost exactly satisfied with the @xmath17 wave function or with the @xmath180 wave function when the meson - exchange interaction is switched off . in the nucleon wave function there is a deviation from this relation but it is not large . we thus conclude that there is no appreciable clustering in the nucleon . what is the physical reason for the absense of a significant clustering ? the answer is that there are genuine 3-body effects and the fermi - nature of quarks do not support clustering . indeed , if the quarks , say , with numbers 1 and 2 form a pair @xmath175 , the antisymmetry of the wave function suggests that there are at the same time pairs with quantum numbers @xmath181 or @xmath182 ( along with other quantum numbers ) . thus a strong attraction acts simultaneously in all quark pairs which makes the nucleon compact but not clustered much . only a much stronger and `` sharper '' interaction in the @xmath183 diquark would lead to an appreciable clustering , but at a cost that @xmath68 splitting will become enormous . here we summarize our main conclusions . the nonperturbative gluonic interaction between quarks , e.g. instanton - induced one , which is responsible for the dynamical breaking of chiral symmetry in qcd and thus explains the @xmath6 mass splitting , iterated in the @xmath3 t - channel implies a meson - exchange picture between constituent quarks , and through the latter also explains baryons and nuclear force . this is a simple consequence of crossing symmetry : if one obtains pion as a solution of the bethe - salpeter equation in the quark - antiquark s - channel , then one inevitably obtains a pion - exchange in the quark - quark systems as a result of iterations in the @xmath3 t - channel . \2 . due to ( anti)screening effects the implications of this nonperturbative gluonic interaction in @xmath3 systems are drastically different when it is iterated only in the s - channel as compared to a more general case , when it is first iterated in the t - channel , leading to a meson exchange , and only after that iterated in the s - channel . \3 . the effective meson - exchange interaction in @xmath3 systems does not bind diquarks without an additional confining force and does not induce any appreciable clustering in the nucleon . l.ya.g . thanks d. diakonov and d.o . riska for comments . he is indebted to kek - tanashi and tokyo institute of technology nuclear theory groups for a warm hospitality . his work is supported by a foreign visiting guestprofessorship program of the ministry of education , science , sports and culture of japan . work of k.v . is supported by the otka grant no . t17298 ( hungary ) and by the us department of energy , nuclear physics division , under contract no . w-31 - 109-eng-39 . r. rapp , t. schfer , e. shuryak , m. velkovsky , phys . lett . * 81 * , 53 ( 1998 ) . m. alford , k. rajagopal , f. wilczek , phys . b422 * , 247 ( 1998 ) . t. schfer and e. shuryak , rev . phys . , * 70 * , 323 ( 1998 ) , see also references therein . l. ya . glozman and d.o . riska , physics reports * 268 * , 263 ( 1996 ) . glozman , surveys in high energy physics , * 14 * , 109 ( 1999 ) ; hep - ph/9805345 . glozman , w. plessas , k. varga , r. f. wagenbrunn , phys . rev . * d58 * , 094030 ( 1998 ) . m. he s , f. karsch , e. laermann , i. wetzorke , phys . rev . * d58 * , 111502 ( 1998 ) . u. vogl and w. weise , progr . * 27 * , 195 ( 1991 ) . r. alkofer and h. reinhard , chiral quark dynamics , chapter 5 , lecture notes in physics , m33 , springer , 1995 . n. ishii , w. bentz and k. yazaki , nucl . a587 * , 617 ( 1995 ) . d. diakonov , in : selected topics in nonperturbative qcd , proc . enrico fermi school , course cxxx , a. digiacomo and d. diakonov eds . , bologna ( 1996 ) ; hep - ph/9602375 , see also references therein . y. nambu and g. jona - lasinio , phys . rev . * 122 * , 345 ( 1961 ) ; ibid . * 124 * 246 t hooft , phys . rev . * d14 * , 3432 ( 1976 ) ; erratum : _ ibid . _ * d18 * , 2199 ( 1978 ) . d. diakonov , h. forkel , m. lutz , phys . lett . * b373 * , 147 ( 1996 ) . d. diakonov , v. petrov , p. pobylitza , nucl . phys . * b306 * , 809 ( 1988 ) . glozman , in preparation e.v . shuryak and j. rosner , phys . lett . * b218 * , 72 ( 1989 ) . v. h. blask et al , z. phys . * a337 * , 327 ( 1990 ) . glozman , hep - ph/9908207 k.f . liu et al , phys . rev . * d59 * 112001 ( 1999 ) . chu , j. grandy , s. huang , j. negele , phys . rev . * d49 * , 6039 ( 1994 ) . | it is shown that the instanton - induced interaction in qq pairs , iterated in t - channel , leads to a meson - exchange interactions between quarks . in this way
one can achieve a simultaneous understanding of low - lying mesons , baryons and the nuclear force .
the discussion is general and does not necessarily rely on the instanton - induced interaction .
any nonperturbative gluonic interaction between quarks , which is a source of the dynamical chiral symmetry breaking and explains the @xmath0 - @xmath1 mass splitting , will imply an effective meson exchange picture in baryons . due to
the ( anti)screening there is a big difference between the initial t hooft interaction and the effective meson - exchange interaction .
it is demonstrated that the effective meson - exchange interaction , adjusted to the baryon spectrum , does not bind the scalar diquark and does not induce any significant quark - diquark clustering in the nucleon because of the nontrivial role played by the pauli principle .
pacs number(s ) : 12.38.lg , 12.39.-x , 14.20.dh |
You are an expert at summarizing long articles. Proceed to summarize the following text:
spin transport represents a new branch in mesoscopic physics with several technological applications@xcite , e.g. information storage , magnetic sensors and potentially quantum computation@xcite . while most theoretical models are based on fermi liquid theory , some work has been done on strongly correlated 1d systems using luttinger liquid theory@xcite . this work has focused in the weak tunneling regime between the ferromagnet and the 1d system and found that spin transport may provide experimental evidence of spin charge separation , one of the main predictions of luttinger liquid theory that remains to be observed experimentally in an unambiguously accepted way . despite the possible technological applications and contributions to the study of spin charge separation in strongly correlated systems , very little experimental work has been carried out on spin transport in 1d systems.@xcite this work is complicated by the use of multi - walled carbon nanotubes , and explored only situations with ferromagnetic contacts with parallel or antiparallel magnetizations . early experimental work with nanotube devices was limited by poor contacts between the electrodes and the nanotube , and accordingly theoretical models focused in the weak tunneling regime . recently , liang _ et al . _ @xcite have succeeded in fabricating single - walled carbon nanotube devices with near - perfect ohmic contacts to the electrodes . a schematic representation of their experiment is presented in fig . [ scheme ] . these devices are characterized by values of the conductance as high as @xmath0 , close to the theoretically predicted higher limit@xcite of @xmath1 . at temperatures below 10k , the measured conductance exhibits approximately periodic oscillations as a function of the gate voltage . these oscillations are due to fabry - perot interference i.e. quantum interference between propagating electron waves inside the resonant cavity defined by the two nanotube - electrode interfaces . experimental geometry ( from ref . ) . a single - walled carbon nanotube is located on a silicon gate and oxide layer . the electrodes , which may be ferromagnetic , are grown on top of the nanotube . the doped silicon is used as a gate electrode to modulate the charge density . the electronic transport properties of the nanotube devices were characterized as a function of bias ( @xmath2 ) and gate ( @xmath3 ) voltages . , title="fig : " ] in order to explain their result , ref . considered a model of non - interacting electrons and used the multi - channel landauer - bttiker formalism to calculate the differential conductance as a function of the bias and gate voltages . they have found qualitative agreement between the calculated conductance and their experimental data , especially with regard to the variation of the low - bias conductance with gate voltage . on the other hand , transport experiments on carbon nanotubes@xcite have shown that electrons in nanotubes are strongly correlated and are better described by a luttinger liquid model@xcite . this implies the electrons in these systems do not exhibit fermi liquid properties but instead form collective excitations better described by charge and spin like density waves that propagate with different velocities . this luttinger liquid behavior drastically changes the charge conductance for these systems and it is interesting to know how this affects the results obtained for the particular setup used in ref . . furthermore , this setup can be generalized to the use of ferromagnetic electrodes , in order to study both charge and spin transport in 1d electron systems . in this paper we investigate the spin and charge transport properties in 1d electron systems with near - perfect contacts to ferromagnetic electrodes ( the normal metal electrodes correspond to the particular case of zero magnetization ) . we consider both the case of quantum wires , i.e. single - channel electron systems , and single - walled carbon nanotubes , but mainly focus on the latter one . we use a non - interacting stoner model to treat the ferromagnetic leads and a luttinger liquid model for the nanotube and consider the case of near - perfect contacts to the leads , therefore treating backscattering at the contacts perturbatively . in order to introduce the effect of a finite bias voltage , we use the non - equilibrium keldysh formalism . following this procedure , we obtain the conductance , spin and spin current as functions of the gate and bias voltages , the external magnetic field and the orientation of the magnetization in each lead . we study how the strong coulomb interactions affect these transport properties and find some features in the fabry - perot interference patterns that are related to spin charge separation . a single - walled carbon nanotube with long - range coulomb interactions is well described by a forward - scattering model@xcite . in this model the hamiltonian density is given by @xmath4 ^ 2 , \end{aligned}\ ] ] where the right / left moving electron operators @xmath5 have the labels @xmath6 for the band and @xmath7 for the spin projection of the electrons in the nanotube and @xmath8 is the interaction strength . the term in the square brackets corresponds to the electron density . we also consider the same problem for a single - channel quantum wire , for which there is no sub - band degeneracy and the band index can be dropped . due to the similarities between the two cases , we give explicit analytical formulas throughout the paper only for the nanotube , but will present results for the quantum wire where appropriate . the metallic leads are modeled as two semi - infinite 1d - non - interacting systems@xcite , which is obtained with a position dependent @xmath8 , constant in the wire and zero in the leads . schematic representation of the model . + the leads are modeled as 1d non - interacting electron systems , the luttinger liquid parameter is therefore @xmath9 . the nanotube ( bold line ) , on the other hand , is described by a 1d interacting system , in this case @xmath10 , which corresponds to repulsive interactions . the contacts to the leads are modeled as two weak backscattering barriers . the two backscatterers generate fabry - perot interference.,title="fig : " ] we allow for ferromagnetism in the leads using a non - interacting stoner model@xcite ( mean - field treatment of the magnetization ) . the hamiltonian density is @xmath11 with @xmath12 and @xmath13 where @xmath14 are the pauli matrices and @xmath15 is the `` exchange field '' , which is proportional to the magnetization . this is constant in each ferromagnetic lead , i.e. @xmath16 for @xmath17 and @xmath18 for @xmath19 and in ordinary paramagnetic leads @xmath20 . in this case , the total system corresponding to a nanotube between two ferromagnetic leads is described by the hamiltonian @xmath21 the total hamiltonian @xmath22 can take a form identical to the hamiltonian in the case of normal metal leads by applying the following transformation to the electron field operators separately in the left and right leads respectively @xmath23 this transformation leaves @xmath24 invariant and @xmath25 transforms into @xmath26 . we apply the usual bosonization procedure to study this model@xcite . the four electron modes are associated to four bosonic modes described by the fields @xmath27 and their duals @xmath28 via the bosonization transformation @xmath29 where @xmath30 is a short - distance cutoff . it is convenient to consider the following linear combinations of the fields@xcite : @xmath31 and @xmath32 , with @xmath33 and @xmath34 . this allows us to define the new fields @xmath35 , which corresponds to the total charge mode and is the only interacting mode , and @xmath36 , @xmath37 and @xmath38 ; with similar transformations for the @xmath39 fields . in terms of these new fields the luttinger liquid hamiltonian density ( [ h ] ) is diagonalized @xmath40 \nonumber \\ & + \sum_{i=2}^4 \frac{{v_{\mathrm{\scriptscriptstyle f}}}}{2\pi } \left [ \left(\partial_x \varphi_i \right)^2 + \left(\partial_x \theta_i \right)^2 \right]\ , \end{aligned}\ ] ] where @xmath41 is the fermi velocity , @xmath42 is the renormalized velocity due to the interactions and @xmath43 is the luttinger liquid parameter . in the inhomogeneous model @xmath43 and @xmath42 are functions of the position : @xmath9 and @xmath44 in the leads and @xmath45 and @xmath46 in the nanotube . the contacts between the leads and the nanotube are modeled by weak backscattering at the contact points , the corresponding hamiltonian density has the form @xmath47 \\ & = \sum_{m , a , b=1}^2 \sum_{\alpha=\pm 1 } \delta(x - x_m ) \left\ { \left ( u_m^{a b } + \alpha v_m^{a b } m_m^z \right ) e^ { i \left[\theta_1+\alpha\theta_2+(-1)^{a+1 } \delta_{a b } \left ( \theta_3+\alpha\theta_4 \right)+ ( -1)^{a+1 } \left ( 1-\delta_{a b } \right ) \left ( \varphi_3+\alpha \varphi_4 \right)\right ] } \right . \nonumber \\ & \ \ + v_m^{a b } \left ( m_m^x+i\alpha m_m^y \right ) e^ { i \left[\theta_1 + ( -1)^{a+1 } \delta_{a b } \theta_3 + ( -1)^{a+1 } \left ( 1 - \delta_{a b } \right ) \alpha \theta_4 + \alpha \varphi_2 + ( -1)^{a+1 } \left ( 1-\delta_{a b } \right ) \varphi_3 + ( -1)^{a+1 } \delta_{a b } \alpha \varphi_4 \right ] } + \ , { \mathrm{h.c . } } \left . \vphantom{e^ { i \theta_1 ^ 2 } } \right\ } \ , \nonumber\end{aligned}\ ] ] where @xmath48 labels the position of the contacts : @xmath49 ( @xmath50 is the nanotube length ) and @xmath51 and @xmath52 are constants proportional to the strength of the backscattering , @xmath53 and the same for @xmath52 . the backscattering terms are restricted by symmetry according to charge conservation and spin rotational symmetry around the axis of magnetization of the ferromagnetic contact . we consider only terms of the form @xmath54 because these are the most relevant in the renormalization group sense ( the scaling dimension in real space of these terms is @xmath55 , while the scaling dimension of terms of the form @xmath56 is @xmath57 ) . hence if all scattering terms are weak , these terms will dominate . it is straightforward to extend the present treatment to include the neglected interactions , though we do not attempt this here . the effect of the magnetization appears only on the backscattering term . in the case of near - perfect contacts to the electrodes , we can treat the backscattering hamiltonian @xmath58 as a perturbation to the hamiltonian @xmath59 . this procedure is described in section [ greens ] in the context of the keldysh formalism that we use in order to account for the effects of the finite bias voltage . the gate voltage introduces a term in the hamiltonian density proportional to @xmath60 , where @xmath61 is the electron density . the constant of proportionality relates the voltage applied at the gate with the voltage felt by the nanotube and therefore depends on the sample . the hamiltonian density @xmath62 becomes @xmath63 , after applying the following transformation to the @xmath64 field : @xmath65 , where we absorbed a constant of proportionality into the definition of @xmath3 for simplicity . this transformation needs to be applied to the total hamiltonian , including the backscattering term @xmath58 , which means that the gate voltage after this transformation will only contribute to the perturbation hamiltonian . the effect of the external magnetic field is introduced via a zeeman coupling term in the hamiltonian @xmath66 the contribution of the magnetic field can be transfered to the perturbation hamiltonian @xmath58 using a similar procedure to the one described above for the gate voltage . taking the z - direction as the direction of the magnetic field , i.e. @xmath67 , the zeeman hamiltonian density becomes @xmath68 and applying @xmath69 ( with @xmath70 ) , to the hamiltonian @xmath71 , it transforms as @xmath72 . the results for non - zero magnetic field are presented in appendix [ ap : magn ] . due to the finite bias voltage the distribution in this system is not in thermal equilibrium . this non - equilibrium situation is studied using the keldysh formalism ( for a review , see ref . ) . to define a non - equilibrium initial state , we assume that until some initial time @xmath73 , the system has reached quasi - equilibrium _ in the absence of impurity scattering _ @xmath74 . without impurity scattering , the total number of right- and left - moving carriers , @xmath75 , @xmath76 , are separately conserved , so that a partial equilibrium can be established with well - defined separate chemical potentials for the right and left movers . hence , the system can be described , up to this time , by a thermal distribution governed by the grand canonical hamiltonian @xmath77 where the chemical potentials in each lead are taken to be @xmath78 and @xmath79 . the right and left moving particle densities are given in the bosonization procedure by @xmath80 . then @xmath81 we emphasize that the appearance of the voltage @xmath2 in @xmath82 does not represent a physical force on the electrons , but rather parameterizes their non - equilibrium distribution . after the initial time @xmath73 the evolution of the system is governed by a different hamiltonian @xmath22 , which as deduced in section [ model ] is @xmath83 , with @xmath84 given in ( [ hll ] ) and @xmath58 in ( [ hbs ] ) . we expect on physical grounds that introducing localized scattering at the ends of the wire or nanotube reduces the current , but can not affect the non - equilibrium distribution in the reservoirs . hence , we believe that the prescription of defining the voltage @xmath2 from the initial distribution as is done using ( [ truthfull ] ) gives a faithful description of ideal leads . according to this prescription , a physical observable , represented by an operator @xmath85 , is then calculated from @xmath86 where @xmath87 the difficulty in evaluating such an expectation value is that , unlike in a conventional equilibrium calculation , the hamiltonian @xmath82 governing the initial distribution is different from @xmath22 , which governs the time evolution . thus such an expectation value can not be evaluated by equilibrium green s function techniques . instead , we take advantage of the special property of @xmath82 that the voltage couples only to @xmath88 , which are decoupled `` zero mode '' degrees of freedom . this technique has been applied a number of times before to related problems@xcite , but to our knowledge the details of its derivation have never been published . for completeness , pedagogical value , and to highlight the physical assumptions of the method , we include a thorough derivation in appendix [ ap : derivation ] . the correction to @xmath89 due to the backscattering is given by @xmath90 where @xmath91\ ] ] is the evolution operator for a system with the time - dependent hamiltonian @xmath92 . here @xmath93 is the hamiltonian in the frame co - moving with the ideal current , defined by @xmath94_{\theta_1 \rightarrow \theta_1 + v t } \ , \ ] ] with @xmath95 . note that ( because @xmath96=0 $ ] ) all the time dependence in @xmath93 is in the backscattering term , and is hence easy to handle when working perturbatively in @xmath58 . ( [ eq : expect4]-[eq : hi ] ) provide a reformulation of the transport problem which is particularly convenient for a perturbative treatment of the backscattering . note that because the voltage @xmath2 appears only within @xmath58 a direct expansion of eq . ( [ eq : expect4 ] ) in @xmath58 will involve _ equilibrium _ real - time propagators calculated with respect to @xmath84 . we develop this perturbation theory using the keldysh path integral formulation . this involves the usual trotterization of the two evolution operators @xmath97 in eq . ( [ eq : expect4 ] ) using coherent - state fields denoted @xmath98 for @xmath99 ( `` forward branch '' ) and @xmath100 ( `` backward branch '' ) for @xmath101 . further noting that @xmath84 is quadratic and @xmath58 acts only at the ends of the nanotube / wire , the fields away from @xmath102 can be integrated out to obtain the keldysh integral @xmath103 { \mathcal o}_k e^{i s_0 - i \int\ ! { \mathrm{d}}t h_{\mathrm{pert}}(t ) } \ , \ ] ] with @xmath104 - h_{\mathrm{bs } } [ \varphi^-_{i},\theta^-_{i}+\delta_{i1 } v t ] \ .\ ] ] here @xmath105 is an appropriate keldysh representation of the operator @xmath106 , which can be chosen as usual from fields lying on either the forward or backward moving branch , or any linear combination thereof see below for convenient choices . the quadratic action @xmath107 is a functional of @xmath108 , which can be determined from the fact that it must reproduce the _ equilibrium _ correlation and response functions for these fields . indeed , we do not require an explicit expression for @xmath107 , but instead give the response and correlation functions , defined by @xmath109 \rangle \ , \end{aligned}\ ] ] where we have applied the standard keldysh rotation to the fields @xmath110 . by construction @xmath111 . the green s functions involving the @xmath39 fields are defined in a similar way , replacing @xmath112 by @xmath39 in the above equations . there are also green s functions that involve both @xmath112 and @xmath39 , these are defined by @xmath113 \rangle \ , \end{aligned}\ ] ] and similar definitions for @xmath114 and @xmath115 . again , by construction @xmath116 . using the above procedure we obtain ( up to additive constants that will not contribute to the final result ) the green s functions for the @xmath64 fields : @xmath117 \nonumber \\ r^{\theta i}_{12}(t ) & = -\frac{\pi}{2 } ( 1-\alpha^2 ) \sum_{k \geq 0 } \alpha^{2k}\ , \theta ( t- ( 2k+1 ) t_v ) \ , \nonumber \\ c^{\theta i}_{11}(t ) & = -\frac{1-\alpha}{4 } \left [ \log t^2 + \phantom{\sum_{k \geq 1 } } \right . \nonumber \\ & \qquad\qquad + \left . \frac{1+\alpha}{\alpha } \sum_{k \geq 1 } \alpha^{2k}\ , \log \left|t^2-(2 k t_v)^2 \right| \right ] \ , \nonumber \\ c^{\theta i}_{12}(t ) & = -\frac{1-\alpha^2}{4 } \sum_{k \geq 0 } \alpha^{2k}\ , \log\left|t^2- [ ( 2k+1 ) t_v ] ^2 \right| \ , \end{aligned}\ ] ] where the subscripts label the position of the contacts @xmath118 ( e.g. @xmath119 ) and @xmath120 we also need the green s functions for the non - interacting modes @xmath121 , @xmath122 and @xmath123 . these are obtained from ( [ thetagreens ] ) by taking @xmath124 and replacing @xmath125 by @xmath126 , @xmath127 the green s functions for the @xmath39 fields can be obtained from those for the @xmath112 fields given in ( [ thetagreens ] ) , by replacing @xmath43 by @xmath128 , i.e. by replacing @xmath129 by @xmath130 . on the other hand , the only @xmath39 green s functions that contribute to the transport properties studied in the following sessions are those that correspond to the non - interacting modes @xmath131 , and therefore they are identical to the functions given in ( [ freegreens ] ) . in order to compute the spin transport properties in section [ spint ] , we also need the following functions for the @xmath132 and @xmath133 fields @xmath134 @xmath135 and @xmath136 . in this section we study the charge transport properties of 1d electron systems and how these are affected by the magnetization of the leads and , more importantly , the presence of the strong coulomb interactions . we use the procedure described in sections [ model ] and [ greens ] to calculate the differential conductance for these systems . this is obtained from the expectation value of the current in a nanotube , i.e. a four mode 1d electron system with the hamiltonian given in ( [ hll ] ) , as @xmath137 after a lengthy but straightforward calculation we obtain that the differential conductance @xmath138 to second order in perturbation theory is given by @xmath139\cos(v t ) \!\right\}\ ] ] with @xmath140 \ , \\ u_2= & 2 \cos ( { v_{\!g}}l ) \sum_{a b } \left [ u^{a b}_1\ , u^{a b}_2 + v^{a b}_1\ , v^{a b}_2 \ , \vec{m}_1 \cdot \vec{m}_2 \right ] \ , \nonumber\end{aligned}\ ] ] and @xmath141 and similarly for @xmath142 . for a quantum wire ( i.e. a single - channel electron system ) these are replaced by @xmath143 and the global normalization is divided by a factor of two ( since the quantum wire has two modes instead of four ) . ( [ g2 ] ) and ( [ uu ] ) are valid for zero external magnetic field , which is the case considered in this section , the equations for non - zero magnetic field are presented in appendix [ ap : magn ] . equation ( [ g2 ] ) can be easily generalized to arbitrary order in perturbation theory , but the time integrals need to be computed numerically . we present the calculated conductance to second order for three different physical models in fig . [ fig : g ] . this models correspond to ( a ) a nanotube with non - interacting electrons , i.e.taking @xmath9 , ( b ) a quantum wire with @xmath144 and ( c ) a nanotube with @xmath145 , which is a physically relevant value for single - walled carbon nanotubes@xcite . the effect of the interactions is visible in the dependence of the conductance with bias voltage , at constant gate voltage . the conductance is a quasi - periodic function of the bias voltage . at @xmath146 , for the non - interacting case , see fig . [ fig : g].(a ) , this dependence is a cosine function with period @xmath147 . for a quantum wire , [ fig : g].(b ) , there are clearly two different `` periods '' in the oscillations , these are related to the two time scales @xmath126 and @xmath148 . the existence of these two different time scales is due to the two bosonic excitations in this system : the spin excitation with velocity @xmath41 and the charge excitation with velocity @xmath42 , and is therefore an effect of spin charge separation . the same effect appears in fig . [ fig : g].(c ) , but since for the nanotube there are three non - interacting modes with velocity @xmath41 and only one mode , the total charge , with velocity @xmath42 , it is less visible than in the previous example . the most visible effect of the interactions in the nanotube , is the enhancement of the amplitude in the conductance around @xmath149 . this effect is observable in the experimental data presented in ref . . the calculated conductance at @xmath150 as a function of the bias voltage for a nanotube with different interaction strengths corresponding to @xmath151 and @xmath152 , is presented in fig . [ fig : pi2 ] . it can be seen using eqs . ( [ g2 ] ) and ( [ uu ] ) that the conductance for this value of the gate voltage only depends on the green s functions @xmath153 and @xmath154 , which depends only on @xmath125 . as a result , we can clearly see in fig . [ fig : pi2 ] that the period of the oscillations is @xmath155 , and therefore depends strongly on the interaction strength . the amplitude of this oscillations is very small except for the first oscillation , which is enough to identify this effect . at constant gate voltage : @xmath156 , the period of oscillations is @xmath157 , i.e. depends strongly on the interaction strength . in order for this effect to be clearly visible , we scaled and shifted the functions differently , therefore the values of the @xmath158 axes are not meaningful . , title="fig : " ] as for the dependence with the gate voltage , the conductance is a periodic function , which is modulated by @xmath159 , and this is the main effect of the magnetization in the leads on the conductance . in particular if @xmath160 , there is an angle between the two magnetizations for which @xmath161 vanishes for any value of the gate voltage , in this case the conductance is given also in fig . [ fig : pi2 ] . in this section we study spin transport properties , i.e. the spin density in the nanotube and the spin current generated by the magnetization in the leads . the spin density expectation value in the nanotube , calculated using bosonization and the keldysh perturbation formalism as described in sections [ model ] and [ greens ] , is given by @xmath162 for zero magnetic field , it is technically simplest to calculate @xmath163 from the bosonized form @xmath164 and then obtain the other two components by rotational invariance . for non - zero magnetic field the calculation as well as the final results are much more involved , therefore and for the sake of clarity we only present the results in appendix [ ap : magn ] . the result is @xmath165\sin(v t ) \right . \nonumber \\ & + \sin ( { v_{\!g}}l ) \left\{\vec u_2 \int\!\!{\mathrm{d}}t\ , e^{{\bm c}_{12}(t ) } \sin\!\left [ \frac{1}{2 } { \bm r}_{12}(t)\right]\sin(v t ) \right . \nonumber \\ & \left . + \vec u_3 \int\!\!{\mathrm{d}}t\ , e^{{\bm c}_{12}(t ) } \sin\!\left [ \frac{1}{2 } { \bm r}_{12}(t)\right]\cos(v t ) \right\ } \right ) \ , \end{aligned}\ ] ] with @xmath166 notice that the spin density does not depend on the position in the nanotube , hence the total spin is @xmath167 . the first term , proportional to @xmath168 , is the known non - equilibrium spin accumulation effect@xcite . it is maximum for @xmath169 , when , in the case of identical contacts , the other terms vanish . this term does not couple two backscatterers , is independent of the gate voltage , and is an increasing function of the bias voltage . it is depicted in fig . [ fig : spin](a ) , since it is the only term term that corresponds to the component of the spin in the direction of @xmath170 at @xmath146 . the second term corresponds to the component of the spin perpendicular to the plane of the magnetizations and is depicted in fig . [ fig : spin](c ) , as function of the bias and gate voltages . the third term is the only one that survives in equilibrium , i.e. at zero bias , it is due to the fact that the backscattering strengths depend on the spins of the incoming and outgoing electron relative to the direction of the magnetizations . it is maximum for @xmath171 , when again for identical contacts the other terms vanish . this is depicted in fig . [ fig : spin](b ) , at @xmath172 . these terms that couple the two backscatterers , and hence depend on the gate voltage , vary with bias voltage in a manner approximately described by a sum of two periodic functions , with `` periods '' given by @xmath147 and @xmath173 , as discussed in the previous section for the conductance . the spin current @xmath174 is as the spin density calculated from the @xmath175 component in its bosonized form ( again see result for non - zero magnetic field in appendix [ ap : magn ] ) @xmath176 it is not well - defined at the contact points because the backscattering term in the hamiltonian ( [ hbs ] ) does not conserve spin , and therefore it has different expressions in the nanotube and the leads . the spin current in the left ( @xmath177 ) and right ( @xmath178 ) leads is given by @xmath179\sin(v t ) \nonumber \\ & + \left[\pm \sin ( { v_{\!g}}l)\ \vec u_2 + \cos ( { v_{\!g}}l)\ , \vec u_3 \right ] \nonumber\\ & \qquad \times \int\!\!{\mathrm{d}}t\ , e^{{\bm c}_{12}(t ) } \sin\!\left [ \frac{1}{2 } { \bm r}_{12}(t)\right]\sin(v t ) \end{aligned}\ ] ] and in the nanotube by : @xmath180\sin(v t ) \\ & + \cos ( { v_{\!g}}l)\,\left\ { \ \vec u_2 \int\!\!{\mathrm{d}}t\ , e^{{\bm c}_{12}(t ) } \sin\!\left [ \frac{1}{2 } { \bm r}_{12}(t)\right]\cos(v t ) \right . \nonumber \\ & \quad\quad\qquad + \left . \vec u_3 \int\!\!{\mathrm{d}}t\ , e^{{\bm c}_{12}(t ) } \sin\!\left [ \frac{1}{2 } { \bm r}_{12}(t)\right]\sin(v t ) \right\ } \ . \nonumber\end{aligned}\ ] ] with @xmath181 and @xmath182 defined in ( [ us ] ) and @xmath183 similarly to the results for the spin discussed above , the first term , which only involves one backscatterer and is independent of the gate voltage , corresponds to the usual spin injection effect . it is an increasing function of the gate voltage and is maximum for @xmath184 . this can be seen in fig . [ fig : spincurr](b ) , since it is the only term that does not vanish in the direction of @xmath185 at @xmath186 . at @xmath146 , the terms proportional to @xmath187 and @xmath182 contribute equally to the component of the current in the direction of @xmath170 , the result for this case is presented in fig . [ fig : spincurr](a ) . the second term , proportional to @xmath188 corresponds to an exchange interaction between the magnetizations of the leads , mediated by the nanotube . it has opposite signs in the two leads and it is shown in fig . [ fig : spincurr](c ) . we studied the charge and spin transport properties of 1d systems , e.g. quantum wires and carbon nanotubes , focusing on the latter . we considered the case of nearly perfect ohmic contact between the 1d system and the electrodes and included the strong coulomb interaction via a luttinger liquid model . we found important effects on the transport properties of these systems that are due to the coulomb interactions . these appear in the dependence with bias voltage . in particular , the conductance is enhanced at low bias voltage , furthermore it is an oscillatory function where we can distinguish two quasi - periodic components , with periods that are related to the two velocities of the excitations of a luttinger liquid , @xmath42 and @xmath41 . this effect is therefore a direct consequence of spin and charge separation . it is clearly visible in single band quantum wires . in nanotubes , the amplitude of the higher period component is reduced by the presence of three ( as opposed to one ) neutral modes . still , we can find evidence of the two velocities @xmath42 and @xmath41 by comparing the dependence of the conductance with bias voltage for two different gate voltages ( @xmath189 and @xmath190 ) . it is perhaps worth noting that , for the case of non - magnetic leads with symmetric contacts , the conductance formula involves only two unknown parameters : the overall amplitude of the backscattered current , and the luttinger parameter @xmath43 , both of which can be simply estimated . nevertheless , a non - trivial functional dependence upon bias voltage is predicted . the spin and spin current have one component in the plane of the magnetization that does not couple the two leads and is therefore independent of the gate voltage . this term should be understood as arising from incoherent spin injection at each contact . it is a monotonic function of the bias voltage , and corresponds to the known spin accumulation ( for the spin ) and spin injection ( for the spin current ) effects . the other components that couple the two leads , and therefore depend on the gate voltage , are backscattering processes occurring with coherence between the two contacts . these oscillate with the bias voltage , in a manner approximately described as a sum of two periodic components , with periods related to the two velocities of the excitation of the luttinger liquid . the amplitude of the higher - period component is largest in a single - channel quantum wire , and somewhat suppressed in nanotubes by the sub - band degeneracy . c.s.p . was supported by fct and fse through grant praxis / bd/18554/98 . l.b . was supported by nsf through grant dmr9985255 , and by the sloan and packard foundations . thanks the deutsche forschungsgemeinschaft for support through heisenberg grant wi/1932 1 - 1 . in this appendix , we derive eq . ( [ eq : expect4 ] ) . in particular , we consider a large periodic system of size @xmath191 , where ultimately @xmath192 . we define the right / left - moving combinations @xmath193 in the system of size @xmath191 we can decompose into finite wavevector and `` zero mode '' components . in particular , for the total charge fields , we define @xmath194 where @xmath195 contains the non - zero momentum modes of the @xmath196 fields . with these definitions , the zero - mode variables form two canonically conjugate pairs : @xmath197 & = & [ n_l,\phi_l ] = i \ , \\ \nonumber [ n_r , \phi_l ] & = & [ n_l,\phi_r ] = [ n_r , n_l]=[\phi_r,\phi_l ] = 0 \ . \end{aligned}\ ] ] moreover , @xmath198 commute with @xmath195 and all fields associated with channels @xmath199 . since the interactions which transform the system from a fermi liquid into a luttinger liquid ( eq . ( [ h ] ) ) exist only for @xmath200 , they do not affect the zero - mode terms in the hamiltonian . hence one may separate @xmath201 where @xmath202 is the luttinger liquid hamiltonian , eq . ( [ hll ] ) , with the zero - mode terms subtracted , i.e. with @xmath203 and @xmath204 . we then see , using the independence of the zero mode variables , that the unitary operator @xmath205 generates the following transformation @xmath206 , hence @xmath207 where @xmath208 is an unimportant constant . inserting this into eqs . ( [ eq : expect]-[eq : pf ] ) , one obtains @xmath209 with @xmath210 and @xmath211 for the operators of interest , @xmath212 where @xmath213 is the current which would flow in an ideal nanotube in the absence of backscattering . defining @xmath214 , with @xmath215 for @xmath216 and @xmath217 for @xmath218 , one has then @xmath219 we then apply the formula @xmath220 \ , \ ] ] to arrive at eqs . ( [ eq : expect4 ] ) ff . given in the main text . the differential conductance including the magnetic field is still given by eqs . ( [ g2 ] ) and ( [ uu ] ) with only the following change in @xmath161 @xmath221\right\ } \nonumber \\ & + \sin ( b l ) \sin ( { v_{\!g}}l ) \sum_{ab } \left [ u_1^{ab } v_2^{ab } ( \vec m_2 \cdot \hat h ) + u_2^{ab } v_1^{ab } ( m_1\cdot \hat h ) \right ] \ , \end{aligned}\ ] ] where @xmath222 . the total spin of the nanotube is @xmath223 \sin { v_{\!g}}l \nonumber \\ & - u_1^{ab } u_2^{ab } \hat h l \sin b l \cos { v_{\!g}}l - v_1^{ab } v_2^{ab } \left [ \left ( ( \vec m_1 \cdot \hat h ) \vec m_2 + ( \vec m_2 \cdot \hat h ) \vec m_1 \right ) \tfrac{1}{b } \left(1 - \cos b l \right ) \right . \nonumber \\ & \qquad \qquad \qquad \left . + ( \vec m_1 \cdot \hat h)(\vec m_2 \cdot \hat h ) \hat h \left ( l \sin b l - \tfrac{2}{b } \left ( 1 - \cos b l \right ) \right ) \right ] \cos { v_{\!g}}l \biggr\ } \int_t e^{{\bm c}_{12 } } \sin \left ( \tfrac{1}{2 } { \bm r}_{12 } \right)\cos ( v t ) \nonumber \\ + & \biggl\ { \left ( u_1^{ab } v_2^{ab } ( \vec m_2 \times \hat h ) - u_2^{ab } v_1^{ab } ( \vec m_1 \times \hat h ) \right ) \tfrac{1}{b } \left ( 1 - \cos b l \right ) \cos { v_{\!g}}l + v_1^{ab } v_2^{ab } \left [ \left ( ( \vec m_1\cdot\hat h ) ( \vec m_2 \times \hat h ) - ( \vec m_2 \cdot \hat h ) ( \vec m_1 \times \hat h ) \right ) \right . \nonumber \\ & \qquad \qquad \times \left . \left ( l-\tfrac{1}{b } \sin b l \right ) + \vec m_1 \times \vec m_2 l \right ] \sin { v_{\!g}}l \biggr\ } \int_t e^{{\bm c}_{12 } } \sin \left ( \tfrac{1}{2 } { \bm r}_{12 } \right)\sin ( v t ) \biggr ) \ .\end{aligned}\ ] ] the spin current in the nanotube ( eq . [ jsnt ] ) and the leads ( eq . [ jsl ] ) is @xmath224 + u_2^{ab } v_2^{ab } ( \vec m_2 \cdot \hat h ) \hat h \left [ 1 - \cos b ( x - x_2 ) \right ] \nonumber \\ & + u_1^{ab } v_1^{ab } ( \vec m_1 \times \hat h ) \sin b ( x - x_1 ) - u_2^{ab } v_2^{ab } ( \vec m_2 \times \hat h ) \sin b ( x - x_2 ) \biggr\ } \int_t e^{{\bm c}_{11 } } \sin \left ( \tfrac{1}{2 } { \bm r}_{11 } \right ) \sin ( v t ) \nonumber \\ + \sum_{ab } & \biggl\ { \bigl [ u_1^{ab } v_2^{ab } ( \vec m_2 \cdot \hat h ) \hat h \left [ \cos b l - \cos b ( x - x_1 ) \right ] + u_2^{ab } v_1^{ab } ( \vec m_1 \cdot \hat h ) \hat h \left [ \cos b l - \cos b ( x - x_2 ) \right ] \nonumber \\ & \qquad + u_1^{ab } v_2^{ab } \vec m_2 \cos b ( x - x_1 ) + u_2^{ab } v_1^{ab } \vec m_1 \cos b ( x - x_2 ) \bigl ] \cos { v_{\!g}}l \nonumber \\ & + v_1^{ab } v_2^{ab } \bigl [ ( \vec m_1 \cdot \hat h)(\vec m_2 \cdot \hat h ) \hat h \left [ \sin bl - \sin b(x - x_1)+ \sin b(x - x_2 ) \right ] \nonumber \\ & \qquad + ( \vec m_1 \cdot \hat h ) \vec m_2 \sin b(x - x_1 ) - ( \vec m_2 \cdot \hat h ) \vec m_1 \sin b(x - x_2 ) \nonumber \\ & \qquad + u_1^{ab } u_2^{ab } \hat h \sin bl \bigr ] \sin { v_{\!g}}l \biggr\ } \int_t e^{{\bm c}_{12 } } \sin \left ( \tfrac{1}{2 } { \bm r}_{12 } \right ) \sin ( v t ) \nonumber \\ + \sum_{ab } & \biggl\ { \left [ u_1^{ab } v_2^{ab } ( \vec m_2 \times \hat h ) \sin b ( x - x_1 ) - u_2^{ab } v_1^{ab } ( \vec m_1 \times \hat h ) \sin b ( x - x_2 ) \right ] \sin { v_{\!g}}l \nonumber \\ & - v_1^{ab } v_2^{ab } \left [ ( \vec m_1\cdot\hat h ) ( \vec m_2 \times \hat h ) \left [ 1-\cos b(x - x_1 ) \right ] - ( \vec m_2 \cdot \hat h ) ( \vec m_1 \times \hat h ) \left [ 1-\cos b(x - x_2 ) \right ] \right . \nonumber \\ & \qquad \left . + \vec m_1\times \vec m_2 \right ] \cos { v_{\!g}}l \biggr\ } \int_t e^{{\bm c}_{12 } } \sin \left ( \tfrac{1}{2 } { \bm r}_{12 } \right ) \cos ( v t ) \ , \end{aligned}\ ] ] @xmath225 + u_2^{ab } v_2^{ab } ( \vec m_2 \cdot \hat h ) \hat h \left [ 1 - \cos b ( x - x_2 ) \right ] \nonumber \\ & \mp \left [ u_1^{ab } v_1^{ab } ( \vec m_1 \times \hat h ) \sin b ( x - x_1 ) + u_2^{ab } v_2^{ab } ( \vec m_2 \times \hat h ) \sin b ( x - x_2)\right ] \biggr\ } \int_t e^{{\bm c}_{11 } } \sin \left ( \tfrac{1}{2 } { \bm r}_{11 } \right ) \sin ( v t ) \nonumber \\ + \sum_{ab } & \biggl\ { \bigl [ u_1^{ab } v_2^{ab } ( \vec m_2 \cdot \hat h ) \hat h \left [ \cos b l - \cos b ( x - x_1 ) \right ] + u_2^{ab } v_1^{ab } ( \vec m_1 \cdot \hat h ) \hat h \left [ \cos b l - \cos b ( x - x_2 ) \right ] \nonumber \\ & \qquad + u_1^{ab } v_2^{ab } \vec m_2 \cos b ( x - x_1 ) + u_2^{ab } v_1^{ab } \vec m_1 \cos b ( x - x_2 ) \bigl ] \cos { v_{\!g}}l \nonumber \\ & + v_1^{ab } v_2^{ab } \bigl [ ( \vec m_1 \cdot \hat h)(\vec m_2 \cdot \hat h ) \hat h \left [ \sin bl - \sin b(x - x_1)+ \sin b(x - x_2 ) \right ] \nonumber \\ & \qquad + ( \vec m_1 \cdot \hat h ) \vec m_2 \sin b(x - x_1 ) - ( \vec m_2 \cdot \hat h ) \vec m_1 \sin b(x - x_2 ) \bigr ] \sin { v_{\!g}}l \nonumber \\ & \mp \left [ u_1^{ab } v_2^{ab } ( \vec m_2 \times \hat h ) \sin b ( x - x_1 ) + u_2^{ab } v_1^{ab } ( \vec m_1 \times \hat h ) \sin b ( x - x_2 ) \right ] \cos { v_{\!g}}l \nonumber \\ & \mp v_1^{ab } v_2^{ab } \left [ ( \vec m_1\cdot\hat h ) ( \vec m_2 \times \hat h ) \left [ 1-\cos b(x - x_1 ) \right ] - ( \vec m_2 \cdot \hat h ) ( \vec m_1 \times \hat h ) \left [ 1-\cos b(x - x_2 ) \right ] \right . \nonumber \\ & \qquad \left . + \vec m_1\times \vec m_2 \right ] \sin { v_{\!g}}l + u_1^{ab } u_2^{ab } \hat h \sin bl \sin { v_{\!g}}l \biggr\ } \int_t e^{{\bm c}_{12 } } \sin \left ( \tfrac{1}{2 } { \bm r}_{12 } \right ) \sin ( v t ) \ , \end{aligned}\ ] ] where the @xmath226 sign correspond to lead 1,2 respectively . | we study the two - terminal transport properties of a metallic single - walled carbon nanotube with good contacts to electrodes , which have recently been shown [ w. liang et al , nature 441 , 665 - 669 ( 2001 ) ] to conduct ballistically with weak backscattering occurring mainly at the two contacts .
the measured conductance , as a function of bias and gate voltages , shows an oscillating pattern of quantum interference .
we show how such patterns can be understood and calculated , taking into account luttinger liquid effects resulting from strong coulomb interactions in the nanotube .
we treat back - scattering in the contacts perturbatively and use the keldysh formalism to treat non - equilibrium effects due to the non - zero bias voltage . going beyond current experiments , we include the effects of possible ferromagnetic polarization of the leads to describe spin transport in carbon nanotubes .
we thereby describe both incoherent spin injection and coherent resonant spin transport between the two leads .
spin currents can be produced in both ways , but only the latter allow this spin current to be controlled using an external gate . in all cases , the spin currents , charge currents , and magnetization of the nanotube exhibit components varying quasiperiodically with bias voltage , approximately as a superposition of periodic interference oscillations of spin- and charge - carrying `` quasiparticles '' in the nanotube , each with its own period .
the amplitude of the higher - period signal is largest in single - mode quantum wires , and is somewhat suppressed in metallic nanotubes due to their sub - band degeneracy . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
the rather faint ( @xmath1 ) star sdss j142625.71 + 575218.3 ( referred to hereafter as sdss j1426 + 5752 ) is a fascinating object in several aspects . first , it belongs to the newly - discovered type of carbon - atmosphere white dwarfs , also known as hot dq stars ( dufour et al . 2007 , 2008a ) . these are exceedingly rare stars whose unexpected existence was revealed thanks to the availability of some of the data products that came out of the sloan digital sky survey ( e.g. , liebert et al . 2003 and eisenstein et al . dufour et al . ( 2008b ) found only nine such objects out of a total of about 10,000 white dwarfs identified spectroscopically . their preliminary atmospheric analysis revealed that all the hot dq white dwarfs fall in a narrow range of effective temperature , between about 18,000 and 24,000 k , and that they have atmospheric carbon - to - helium number ratios ranging from 1 to upward of 100 . dufour et al . suggested that these stars could be the cooled - down versions of the , so far , unique and very hot ( @xmath2 @xmath3 200,000 k ) carbon - rich pg 1159 star h1504 ( see , e.g. , werner & herwig 2006 ) and form a new family of hydrogen- and helium - deficient objects following the post - agb phase . in this scenario , residual helium would float rapidly to the surface after the pg 1159 phase of evolution , and the descendants of h1504-like stars would thus `` disguise '' themselves as helium - atmosphere white dwarfs ( of the do and db spectral types ) . this would last until convective mixing dilutes the thin outermost layer of helium in the effective temperature range where substantial subphotospheric convection due to carbon recombination develops in models of these stars . hence , a dramatic change in the atmospheres of such stars , from helium - dominated to carbon - dominated , would occur in the range of temperature where the hot dq s are actually found . further evolution would slowly restore the dominance of helium in the atmosphere of these objects as a result of diffusion . although quite a bit of work needs to be done to establish quantitatively the foundations of this scenario , the preliminary investigations of althaus et al . ( 2009 ) indicate that it is quite viable . an updated discussion of the properties of hot dq stars has been presented by dufour et al . ( 2009 ) . the second interesting development concerning sdss j1426 + 5752 was the important discovery by montgomery et al . ( 2008 ) that it is a luminosity variable . on the basis of 7.8 h of integrated light photometry on the mcdonald observatory 2.1 m otto struve telescope , these authors reported that sdss j1426 + 5752 has a light curve dominated by a single periodicity at 417.7 s with an amplitude of about 1.7% of the mean brightness of the star , accompanied by its first harmonic ( 208.9 s ) with a relatively large amplitude ( @xmath40.7% ) , and possibly also by its fourth harmonic as well ( @xmath40.3% ) . quite interestingly , they also reported that no luminosity variations were detected in five other hot dq s that they surveyed . using some theoretical arguments , montgomery et al . ( 2008 ) argued that the luminosity variations seen in sdss j1426 + 5752 and not in their other targets could be accounted for naturally in terms of pulsational instabilities . if true , this means that sdss j1426 + 5752 is the prototype of a new class of pulsating white dwarfs after the gw vir , v777 her , and zz ceti types ( and see , e.g. , fontaine & brassard 2008 for a detailed review on these pulsators ) . the hypothesis that the luminosity variations seen in sdss j1426 + 5752 are caused by pulsational instabilities associated with low - order and low - degree gravity - mode oscillations ( as in the known types of pulsating white dwarfs ) is backed by the exploratory nonadiabatic calculations carried out independently by fontaine , brassard , & dufour ( 2008 ) in parallel to the efforts of montgomery et al . ( 2008 ) . on the other hand , montgomery et al . ( 2008 ) also noted that the folded light curve of sdss j1426 + 5752 does not resemble those of pulsating white dwarfs showing nonlinearities in their light curves , but shows instead similarities with the folded pulse shape of am cvn , the prototype of the family of helium - transferring cataclysmic variables . the am cvn stars are close interacting binaries consisting of ( probably ) two helium white dwarfs with orbital periods in the range 1000@xmath53000 s ( and see the reviews of warner 1995 or nelemans 2005 for a lot more details on these challenging objects ) . in these systems , the main photometric period , almost always accompanied by several harmonics , corresponds to the beat period between the orbital period and the precession period of the slightly elliptical accretion disk around the more massive white dwarf . the dominant component of the light variability usually comes from the moving ( precessing ) optically thick accretion disk . thus , on the basis of similarities in the folded light pulses between sdss j1426 + 5752 and am cvn , montgomery et al . ( 2008 ) proposed an alternative to pulsational instabilities for explaining its luminosity variations : the possibility that it is , in fact , a new type of close interacting binary , a carbon - transferring analog of am cvn . in this scenario , the observed spectrum of sdss j1426 + 5752 would originate from an optically thick carbon - oxygen accretion disk around the more massive white dwarf component in the system . the pulse shape argument was again used recently by barlow et al . ( 2008 ) to favor the close interacting binary model after those other authors discovered two more luminosity variable hot dq s . however , counterarguments , favoring this time the pulsation model , have been put forward by dufour et al . ( 2009 ) and fontaine et al . ( 2009 ) . the third development concerning sdss j1426 + 5752 resulted from follow - up spectroscopic observations carried out by dufour et al . ( 2008c ) at the 6.5 m multiple mirror telescope ( mmt ) and at one of the 10 m keck telescopes . this was motivated by the sole availability of a rather poor sdss spectrum , insufficient for quantitative analysis . their objective was , firstly , to obtain a sufficiently good spectrum for detailed atmospheric modeling and , secondly , to search for the presence of helium as required to account for the observed pulsational instabilities according to the nonadiabatic calculations of fontaine et al . the spectral analysis of the improved spectra readily revealed the presence of a substantial amount of helium in the atmosphere of sdss j1426 + 5752 , an abundance comparable to that of carbon . this is in line with the expectations of nonadiabatic pulsation theory which requires an important `` helium pollution '' in the atmosphere / envelope of sdss j1426 + 5752 for it to pulsate at its current estimated effective temperature and surface gravity ( dufour et al . 2008b ) . in addition to this finding , an unexpected surprise came out of the follow - up spectroscopic observations of dufour et al . ( 2008c ) . indeed , it was found that the strong carbon lines seen in the optical spectrum of sdss j1426 + 5752 feature zeeman splitting , a structure that could not be seen in the original noisy sdss spectrum . the observed splitting between the @xmath6 and @xmath7 components implies a large scale magnetic field of about 1.2 mg . hence , if sdss j1426 + 5752 is really a pulsating star , it would be the first example of an isolated pulsating white dwarf with a large detectable magnetic field . as such , it would be the white dwarf equivalent of a rapidly oscillating ap ( roap ) star . the roap stars are main sequence ( or near main sequence ) magnetic a stars ( ap ) showing multiperiodic luminosity variations with periods in the range 5@xmath515 minutes caused by low - degree , high - order pressure - mode pulsational instabilities ( see , e.g. , kurtz 1990 ) . in view of the importance of sdss j1426 + 5752 , we carried out additional studies of that star . in particular , we present and discuss in this paper the results of some 106.4 h of integrated light photometry gathered at the steward observatory 1.55 m kuiper telescope . we also discuss the pros and the cons of the pulsational instabilities model versus those of the interacting binary model . this includes a detailed search for possible radial velocity variations due to rapid orbital motion using mmt spectroscopy . at the end of the exercise , we unequivocally conclude in favor of the pulsation model . our follow - up photometry was obtained with mont4k ( as in `` montral 4k@xmath84k camera '' ) , a new ccd camera designed and built at the steward observatory . mont4k is a partnership between universit de montral and the university of arizona . the instrument was designed primarily with differential time - series photometry in mind ( in the windowing mode ) , but the variety of filters available coupled with the excellent sensitivity of the chip and the large ( 9.7@[email protected]@xmath9 ) field of view make it ideal for many imaging projects . it is used at the kuiper telescope on mount bigelow near tucson . some details about the instrument can be found in randall et al . ( 2007 ) , but the interested reader will find more on the following web site : http://james.as.arizona.edu/ psmith/61inch / instrument s.html . at the outset , the relative faintness of sdss j1426 + 5752 posed a challenge for a small instrument such as the kuiper telescope , but we were pleasantly surprised at the quality of the light curves that we could actually obtain . the time - series observations were taken on dark nights through a broadband schott 8612 filter , and an effective exposure time of 67 s ( on average ) was used as a compromise between the s / n and the need to sample adequately the luminosity variations as reported by montgomery et al . altogether , we collected some 106.4 h of useful photometry over a 40 day span in the spring of 2008 . this corresponds to a formal temporal resolution of 0.29 @xmath10hz and a modest duty cycle of 11.8% . details of the observations are provided in table 1 . cccc 2008 mar 31 & 6.6517743 & 399 & 7.88 + 2008 apr 01 & 7.6417324 & 468 & 8.73 + 2008 apr 03 & 9.6413232 & 466 & 8.69 + 2008 apr 04 & 10.6382536 & 449 & 8.52 + 2008 apr 05 & 11.6310329 & 223 & 4.15 + 2008 apr 06 & 12.6288716 & 487 & 9.05 + 2008 apr 09 & 15.6271095 & 364 & 6.55 + 2008 apr 10 & 16.6864942 & 108 & 1.99 + 2008 apr 11 & 17.7342303 & 328 & 6.11 + 2008 may 01 & 37.6670997 & 389 & 7.26 + 2008 may 02 & 38.6404921 & 424 & 7.88 + 2008 may 03 & 39.6412906 & 418 & 7.80 + 2008 may 04 & 40.6406385 & 275 & 6.29 + 2008 may 08 & 44.6459373 & 406 & 7.65 + 2008 may 10 & 46.6446166 & 420 & 7.82 + figure 1 shows the 15 nightly light curves that were obtained . the original images were reduced using standard iraf reduction aperture photometry routines , except that we set the photometric aperture size individually for each frame to 2.0 times the fwhm in that image . we computed differential light curves of sdss j1426 + 5752 on the basis of three suitable comparison stars well distributed around the target . final detrending of the effects of differential extinction was made through a spline strategy . a zoomed - in view of the light curve gathered on may 2 ( 2008 ) is provided in figure 2 . again , in view of the magnitude of the target ( @xmath0 = 19.16 ) , the overall quality of the light curves is most gratifying . we attribute this to the excellent sensitivity of the ccd and our optimized data pipeline . the time - series photometry gathered for sdss j1426 + 5752 was analyzed in a standard way using a combination of fourier analysis , least - squares fits to the light curve , and prewhitening techniques ( see , e.g. , billres et al . 2000 for more details ) . figure 3 shows the fourier amplitude spectrum of the full data set in the [email protected] mhz bandpass ( upper curve ) and the resulting transforms after prewhitening of two ( middle curve ) and three significant peaks ( lower curve ) . these results first confirm the presence of a dominant oscillation with a period of 417.707 s and of its first harmonic at 208.853 s , in agreement with the report of montgomery et al . note , in this context , that we did not detect the 2nd harmonic , again in agreement with their report , but also that we could not verify the presence of the 4th harmonic of the main periodicity as suggested in the data of montgomery et al . ( 2008 ) because our relatively large sampling time led to an effective nyquist frequency of 7.46 mhz , too small for picking up that high - frequency peak . on the other hand , our results also reveal the presence of an additional significant oscillation with a period of 319.720 s as can be seen in the figure . it is also quite likely that there are many other frequency components in the light curve of sdss j1426 + 5752 , independent oscillations or rotationally - split components for instance , but the sensitivity of our observations did not allow them to be detected . the two upper curves in figure 4 provide a zoomed in view of the fourier amplitude spectrum in the vicinity of the 417.707 s peak before and after prewhitening . likewise , the two lower curves give a similar view for the 319.720 s peak . it is interesting to note that , within our measurement errors , the main peak at 417.707 s is a singlet . given that sdss j1426 + 5752 has a large scale magnetic field of some 1.2 mg , one could have expected instead the presence of @xmath11 components due to magnetic splitting according to jones et al . perhaps the multiplet components have much lower amplitudes than the main mode and are buried in the noise , or perhaps the field geometry is such that magnetic splitting can not be observed . either way , this remains an interesting curiosity . cccc [email protected] & [email protected] & [email protected] & [email protected] + [email protected] & [email protected] & [email protected] & [email protected] + [email protected] & [email protected] & [email protected] & [email protected] + we thus were able to extract three distinct harmonic oscillations in the light curve of sdss j1426 + 5752 . the basic characteristics of these oscillations are summarized in table 2 . note that the phase is relative to an arbitrary point in time ; in our case , the beginning of the first run on ut 31 march 2008 . the uncertainties on the period , frequency , amplitude , and phase of each oscillation as listed in the table were estimated with the method put forward by montgomery & odonoghue ( 1999 ) . we point out , in this context , that the uncertainties on the amplitudes and phases obtained by our least - squares fits during the prewhitening stage were virtually the same as those derived with the montgomery & odonoghue ( 1999 ) method . a basic quantity in that latter approach is the average noise level in the bandpass of interest , and we thus computed this mean value from the residual fourier transform ( the lower curve in fig . 3 ) spanning the [email protected] mhz interval . the mean noise level in that range turned out to be 0.060% of the mean brightness of the star . in comparison , if we use the shuffling technique discussed by kepler ( 1993 ) for estimating the noise level , we find an almost identical value of 0.061% . in the context of being able to discriminate between the two possibilities put forward by montgomery et al . ( 2008 ) to explain the luminosity variations of sdss j1426 + 5752 pulsations versus interacting binary our detection of a periodicity ( 319.720 s ) that is incommensurate with the periods of the main peak and its harmonics is potentially quite important ( see below ) . hence , it is essential that its presence be well established . formally , according to the results of table 2 , our detection of the 319.720 s periodicity is a 6.0 @xmath7 ( 0.288/0.048 ) result . in the more standard way , the derived amplitude is rather compared to the average noise level , in which case our detection would be seen as a 4.8 @xmath7 ( 0.288/0.060 ) result . if we divide our runs into two `` seasons '' , i.e. , from march 31 through april 11 , and then from may 1 through may 10 , we find that the 319.720 s period is present in both sets of observations ( [email protected] s and [email protected] s ) , at the level of 4.2 @xmath7 according to the standard criterion . we also verified explicitly that the 319.720 s periodicity is not present in the light curves of our comparison stars , thus ruling out a possible instrumental effect . we note finally that montgomery et al . ( 2008 ) would not have been able to pick up the 319.720 s periodicity in their data assuming that it was present in their light curve with an amplitude comparable to our detection because their sensitivity was at least a factor of two lower than what we could achieve on a smaller telescope ( 1.55 m versus 2.1 m ) at a brighter site but at the price of much longer observations ( 106.4 h versus 7.8 h ) . we investigated the stability of the amplitude and phase of each of the three frequencies we extracted from the light curve of sdss j1426 + 5752 by performing nightly measurements . a clue about possible amplitude variations on a daily timescale is first provided by figure 5 in which we show a montage of the nightly fourier amplitude spectra . considering the main peak ( 2.3950 mhz , 417.707 s ) , the figure does suggest some possible amplitude variations . however , things are a lot less suggestive for the two other peaks given the level of noise and their relatively small amplitudes . a more quantitative and standard way of measuring the nightly amplitudes ( and phases ) is to fix the periods at their values given in table 2 and simultaneously perform least - squares sine fits with these periods for each nightly run . the output are nightly amplitudes and phases with formal estimates of their uncertainties . it is interesting to point out that the formal estimates of the uncertainties on the amplitudes and phases that came out of our least - squares exercise were , again , essentially the same as those obtained through the method of montgomery & odonoghue ( 1999 ) , which we explicitly used after the fact as a verification . figure 6 summarizes our results in the case of the main periodicity found in the light curve of sdss j1426 + 5752 . note that we explicitly excluded the values obtained from the short runs gathered on april 5 and april 10 because the associated uncertainties are significantly larger than those shown in the figure . hence , the top panel in figure 6 displays the amplitudes of the 417.707 s peak along with their formal 1 @xmath7 uncertainties for 13 nightly runs . the central dotted horizontal line represents the weighted average of the nightly amplitudes and the horizontal lines above and below the average value give the 1 @xmath7 uncertainty on that value . in the case of the phase ( lower panel of fig . 6 ) , a similar procedure was used , except that the average value was shifted to zero ( since the phase is arbitrary ) and ultimately expressed in units of cycle . taken at face value , the results summarized in figure 6 do suggest some variations in amplitude and in phase over timescales of days . for instance , on march 31 , we find a `` low '' amplitude of [email protected]% for the dominant periodicity at 417.707 s , while we find a `` high '' amplitude of [email protected]% on may 2 . in this context , it may be appropriate to recall that amplitude modulation associated with the rotation of the star is typically seen in roap stars ( see , e.g. , kurtz 1990 ) . this phenomenon ( and many others ) observed in roap stars has been explained within the framework of the very successful oblique pulsator model . in that model , the pulsations align themselves along the magnetic field axis which is itself inclined with respect to the rotation axis of the star . the viewing aspect of the pulsations thus changes periodically with rotation , which produces amplitude modulation of a given mode . in the case of sdss j1426 + 5752 , we do not yet know if its large scale magnetic field is aligned with the rotation axis . although the material presented in this subsection is suggestive of possible amplitude and phase modulations with timescales of days for the dominant 417.707 s period , it is simply not possible at this stage to be certain about their reality . if , for example , we were to double the uncertainties on the derived nightly amplitudes and phases ( assuming that our least - squares approach or the method of montgomery & odonoghue 1999 underestimates the true errors by that factor ) , then we could only conclude that the nightly amplitudes and phases of the 417.707 s periodicity we extracted from the light curve of sdss j1426 + 5752 do not vary within our measurement errors . things are even worse for the lower amplitude 319.720 s and 208.853 s oscillations in that the uncertainties on their nightly amplitudes and phases prevent us from concluding with any certainty about possible modulations although there are hints of variations . here then is a classic case of `` more observations are needed '' . we have followed up on the remark made by montgomery et al . ( 2008 ) that the folded pulse shape of sdss j1426 + 5752 is different from that of pulsating white dwarfs and rather shows similarities with that of am cvn , the prototype of helium - transferring double degenerate binaries . the top panel of figure 7 shows our 106.4 h long light curve of sdss j1426 + 5752 folded on the period of 417.707 s. to reach a decent s / n , we distributed the folded amplitudes in 10 different phase bins , each containing 572 points on average . the error bars about each point in the folded light curve correspond to the errors of the mean in each bin . given that the first harmonic of the 417.707 s periodicity in the light curve has a very high amplitude , about 30% of that of the main peak ( see table 2 ) , it is not surprising that the pulse shape illustrated in the top panel of figure 7 is highly nonlinear . along with this , another striking characteristic with respect to known pulsating white dwarfs is the fact that it boosts a relatively flat maximum and a sharp minimum ( see montgomery et al . 2008 and the examples below ) . in the middle panel of figure 7 , we again display our folded light curve of sdss j1426 + 5752 , but only after having prewhitened the data of the first harmonic ( 208.853 s ) of the main periodicity . if higher order harmonics have negligible amplitudes , and if other modes do not interfere in the folding process , the pulse shape in the middle panel should be that of a perfect sinusoid with an amplitude equal to that of the 417.707 s oscillation described in table 2 . this is precisely what the dotted curve shows . taking into account the uncertainties , the match between the folded pulse shape and the template is essentially perfect . in the lower panel , we plotted the same template ( dashed curve ) corresponding to the 417.707 s sinusoid . taking properly into account the phase difference , we also plotted a sinusoid ( dotted curve ) with the defining characteristics of the 208.853 s periodicity as given in table 2 . the sum of these two sine waves gives the solid curve , the overall nonlinear pulse shape associated with the 417.707 s oscillation . we reported this model pulse shape in the upper panel of figure 7 ( now as a dotted curve ) so that a direct comparison can be made with the observations . again , within our measurement errors , the agreement is nearly perfect . one can see that the relatively flat maximum and sharp minimum in the pulse shape is due to the fact that the first harmonic of the main oscillation falls nearly in phase at the minimum and in antiphase near the maximum of the main sinusoid . this is particularly well illustrated in the lower panel of the figure . for comparison purposes , we carried out similar folding exercises using representative light curves from the archives that one of us ( g.f . ) built up over the years using lapoune at the 3.6 m canada - france - hawaii telescope ( cfht ) . lapoune is a portable three - channel photometer that uses photomultiplier tubes as detectors . the archived light curves are integrated `` white light '' data and were generally obtained with a sampling time of 10 s. they include a short ( 2.04 h ) light curve of am cvn itself , originally taken as a test as part of a multisite campaign carried out on that star ( provencal et al . 1995 ; solheim et al . 1998 ) . in a format identical to that of figure 7 , figure 8 summarizes the results of our calculations after having folded that light curve of am cvn on the period of 521.105 s , which is the dominant photometric periodicity in that star ( see provencal et al . although the model pulse shape ( dotted curve in upper panel and solid curve in lower panel ) based on the superposition of two sinusoids ( the 521.105 s oscillation and its first harmonic ) is far from perfect , it is nevertheless sufficient for illustrating the fact that am cvn does show a main pulse shape with a rounded top and a sharp bottom . the lower panel of figure 8 explains why : the first harmonic ( dotted curve ) of the 521.105 s oscillation shows a relatively large amplitude compared to that of the main component , and it tends to be in phase and in antiphase at the minimum and maximum , respectively , of that component . there is a significant offset however , between the extrema of the 521.105 s sinusoid and those of its first harmonic , and this largely explains the asymmetric shape of the pulse in this particular light curve . in contrast to this behavior , known pulsating white dwarfs with nonlinear light curves ( the large amplitude ones ) quite generally display pulse shapes that have flatter minima and sharper maxima . an example of this is provided by figure 9 which refers to the large amplitude zz ceti star pg 2303 + 242 , a pulsator with a light curve dominated by a periodicity at 712.250 s. in the present case , the model pulse shape based on only two components ( the dominant mode and its first harmonic ) does a very good job at explaining the folded light curve , as can be appreciated in the top panel of figure 9 . a second example is provided by another large amplitude zz ceti star , gd 154 , as illustrated in figure 10 . in that case , we retained also the second harmonic along with the main oscillation ( 1186.085 s ) and its first harmonic in the construction of the model pulse shape . although the top panel of the figure clearly demonstrates that the model could be improved ( gd 154 has a complicated multiperiodic light curve with higher - order harmonic terms ; see , e.g. , fig . 23 of fontaine & brassard 2008 ) , the results presented here are sufficient to make our point about the general shape of a light pulse in a large amplitude pulsating white dwarf . another interesting example is shown in figure 11 . it concerns balloon 090100001 , which is not a pulsating white dwarf at all , but instead belongs to the family of hot b subdwarf pulsators ( and see fontaine et al . 2006 for a brief review on these stars , if interested ) . the light curve of that star , the largest amplitude variable of that type currently known , is dominated by a main pulsation ( the fundamental radial mode according to van grootel et al . 2008 ) with a period of 356.194 s. our model pulse shape , based on the superposition of the main periodicity and of its first two harmonics , gives a perfect fit to the folded light curve . what is common in these examples , and what is generally true for large amplitude pulsating white warfs and hot subdwarfs , is that the harmonic components tend to be in phase at light maximum . this produces relatively sharp maxima and flat minima . in contrast , the first harmonic tends to be in antiphase with the main periodicity at light maximum in the light curve of sdss j1426 + 5752 and am cvn , and this now produces pulse shapes with rounded tops and sharp minima . montgomery et al . ( 2008 ) interpreted this as evidence that the luminosity variations observed in sdss j1426 + 5752 could be caused , not by pulsational instabilities ( their preferred possibility ) , but by photometric activity in a carbon - transferring analog of am cvn . we show below , however , that the pulsation hypothesis provides the better explanation . on the basis of our photometry , can we argue in favor of one or the other of the two possibilities put forward by montgomery et al . ( 2008 ) to account for the luminosity variations seen in sdss j1426 + 5752 ? to begin with , it is well known that flickering incoherent light bursts arising on short timescales is a telltale sign of mass transfer in an interacting binary system ( warner 1995 ) . flickering is actually observed in the cfht / lapoune light curve of am cvn as can be readily seen in figure 1 of provencal et al . ( 1995 ) where the light curve has been displayed for the first time . the sampling time was 10 s , sufficiently short to pick up some flickering . however , when degraded to a sampling time of 60 s , flickering all but disappears in the light curve of am cvn . we conclude from this that we could not have detected flickering in our photometric observations of sdss j1426 + 5752 if present , because of the large sampling time we used . hence , we can not use this as a diagnostic . on the other hand , the light curves of am cvn systems are known to be unstable and irregular . solheim et al . ( 1998 ) discuss this phenemenon in some detail in relation to their figure 2 , where the cfht / lapoune light curve ( binned in 40 s data points ) is again displayed . the authors comment that , in spite of the fact that the cfht / lapoune light curve is essentially noise - free , it is irregular compared to that of isolated pulsating white dwarfs . for instance , the troughs in the light curve are irregularly spaced in time . random flickering is blamed for this state of affairs . in contrast to this , and not withstanding the noise , the light curves we have gathered ( see fig . 1 ) have kept the same appearance and regularity over a six week period . however , it is not clear if this is really significant since flickering is not coherent over long timebases . the light curves of am cvn systems are generally dominated by a principal periodicity along with a suite of several harmonics of that dominant oscillation ( warner 1995 ) . montgomery et al . ( 2008 ) reported the detection of the 417 s oscillation in sdss j1426 + 5752 along with its first harmonic and possibly also the 4th harmonic , but not the 2nd or 3rd one . the detection of a period ideally several periods that would be incommensurate with such series of harmonically related oscillations would go against the interacting binary hypothesis and would favor the pulsations alternative . this would be the best evidence for pulsations according to barlow et al . our detection of the 319 s oscillation ( see subsection 2.2 ) therefore goes a long way in that direction . what about the pulse shape argument ? we do not know at this stage why the folded light curve of sdss j1426 + 5752 features a maximum that is flatter than its minimum , contrary to what is observed in large amplitude pulsating white dwarfs and hot subdwarfs . however , the fact that the pulse shape is unusual does not rule out by any means the possibility that pulsations are involved . in fact , nature provides us with explicit examples of isolated pulsating stars with pulse shapes that qualitatively resemble that seen in sdss j1426 + 5752 in that they display rounded maxima and sharper minima . these are some of the roap stars with the largest amplitudes that show harmonics of dominant modes in their light curves . an excellent case is that of hr 3831 which exhibits a pulse shape with a rounded top and a sharp bottom as can be seen in figure 1 of kurtz , shibahashi , & goode ( 1990 ) . the fast rise , rounded maximum , slower decline , and pointed minimum are very reminescent of what is observed in the light curve of the am cvn system cr boo in its high state as can be seen in figure 6 of warner ( 1995 ) for instance . yet , hr 3831 is a genuinely pulsating star . another example is provided by the roap star hd 99563 studied by handler et al . ( 2006 ) . as can be seen in their figure 1 ( and see also the discussion in the text ) , the pulsations of that star show again flatter light maxima than minima . the authors comment on the phasing of the harmonics with respect to the main mode in their paper . sdss j1426 + 5752 has one thing in common with roap stars , and that is a large scale magnetic field sufficiently important in both cases to disrupt the atmospheric layers and influence the pulsations there . it would therefore not surprise us if the magnetic field were responsible for the unusual pulse shape ( relative to nonmagnetic pulsating stars ) observed in the light curves of sdss j1426 + 5752 and large amplitude roap stars . be that as it may , in view of the very existence of roap stars such as hr 3831 and hd 99563 , the argument that the pulse shape of sdss j1426 + 5752 is different from those seen in ( nonmagnetic ) pulsating white dwarfs can not be held against the pulsations interpretation . dufour et al . ( 2008c ) presented follow - up spectroscopy of sdss j1426 + 5752 including , in particular , an optical spectrum obtained with the blue channel spectrograph at the mmt using a 500 line mm@xmath13 grating with a 1@xmath14 slit , resulting in a @xmath43.6 fwhm spectral resolution over a wavelength range of 3200@xmath56400 . a total of 17 individual spectra were obtained over a timebase of 3.37 h , resulting in an overall exposure time of 180 minutes . part of the final combined spectrum , taken from dufour et al . ( 2008c ) , is reproduced in figure 12 ( upper curve ) . it shows a s / n@xmath475 per pixel at 4500 . the most prominent features are cii lines which clearly show zeeman splitting in their cores . as a comparison , we also plotted the available sdss spectrum for the star ( lower curve ) . if sdss j1426 + 5752 is truly a double degenerate carbon analog of am cvn systems , then one should be able to pick up radial velocity variations associated with orbital periods in the range 1000@xmath53000 s as found in most such systems . this excludes , of course , the improbable configuration of having the orbital plane at nearly 90 degrees with respect to the line - of - sight . the sampling and the baseline of 3.37 h used during the mmt observations of dufour et al . ( 2008c ) are well suited to search for such radial velocity variations . in addition , the combination of the blue channel spectrograph and the mmt is well known to provide a very stable platform for radial velocity measurements , and one of us ( e.m.g . ) has developed over the years a large mmt program dedicated to radial velocity measurements in hot subdwarf stars . we therefore went back to the mmt spectroscopic data and searched for possible radial velocity variations using the tools and expertise developed at steward observatory . we provide some details on the procedure followed . we first reinterpolated the 17 individual spectra of sdss j1426 + 5752 onto a logarithmic wavelength scale with identical starting and ending wavelengths ( 3215@xmath56357 ) . after manually removing cosmic ray signatures from each spectrum , we fitted the continuum , divided by the fit , and subtracted 1.0 to get a flattened continuum with a mean level of zero . we median - filtered the 17 resulting spectra into a single spectrum to create a higher s / n radial velocity template . radial velocity cross - correlations of the individual continuum - removed spectra relative to the template spectrum were performed using the double precision version of iraf s fxcor task , fitting the cross - correlation peak with a gaussian . a ramp filter was used prior to the cross - correlation to select the optimal range in fourier space , with adopted values of 125 , 275 , 1480 , and 1490 , respectively , for the cut - on , full - on , cut - off , and full - off points . the high frequency noise cut - off corresponds to 2.75 pixels , which is slightly better than the instrumental resolution of 3.0 pixels , as determined by the fwhm s of the hearne comparison arc lines . the low - frequency cut - on , corresponding to 33 pixels , is slightly wider than the observed fwhm ( 30 pixels ) of the strongest absorption lines in the spectrum . the essential result that came out of this exercise is that the velocities are constant to within 7.1 km s@xmath13 ( the standard deviation ) , which is about 1/32 of the velocity resolution , and there is no sign of any velocity trend over the 3.37 h of the observations . we point out that 1/32 of a resolution element represents quite a high level of accuracy for spectra at this s / n . in comparison , the typical radial velocity semi - amplitude expected in a am cvn system is @xmath15 sin @xmath16 km s@xmath13 . this estimate uses the following representative values : @xmath17 for the mass of the helium degenerate primary , @xmath18 for the mass of the helium degenerate or semi - degenerate donor , and @xmath19 s for the orbital period , as can be inferred in the reviews of warner ( 1995 ) and nelemans ( 2005 ) and references therein . actually , if we assume that the dominant photometric period of 417 s is approximately equal to the orbital period in a putative sdss j1426 + 5752 carbon system equivalent of am cvn ( this is the case for the known systems of the type , except for am cvn itself ) , then the estimate of the velocity semi - amplitude goes up to @xmath20 sin @xmath16 km s@xmath13 . if , as in am cvn itself , the orbital period is rather approximately equal to twice the dominant photometric periodicity of 417 s ( 521 s in am cvn ) , the velocity semi - amplitude takes on the value @xmath21 sin @xmath16 km s@xmath13 . hence , unless the inclination of the orbital plane is quite low , this argues strongly against the interacting binary hypothesis as the explanation for the luminosity variations observed in sdss j1426 + 5752 . along with the fact that we do not detect radial velocity variations , the available spectroscopic data on sdss j1426 + 5752 can further be used to build up the case against the interacting binary model . in a am cvn - type system in its high state , most of the light does not come from the photosphere of the helium degenerate primary , but rather from an optically thick accretion disk orbiting around that star . according to warner ( 1995 ) , the hei lines seen in absorption have quite different profiles from those observed in isolated helium - atmosphere ( db ) white dwarfs . their profiles are asymmetric , and they vary both in shape and depth , contrary to the symmetric and stable lines seen in db white dwarfs . in this context , we wish to point out that the available spectroscopy on sdss j1426 + 5752 shows that the broad ( carbon ) absorption lines are symmetric ( see , e.g. , fig . furthermore , the spectrum appears quite stable , at least over a timescale of 20 h , which is the time lapse between the observations of sdss j1426 + 5752 carried out at the mmt and those gathered at the keck i telescope ( and see dufour et al . the spectrum does not show any sign of accretion disk activity . however , it does show clear zeeman splitting in the line cores , a feature that would presumably be washed away if sdss j1426 + 5752 were part of a interacting binary system because of the rapid orbital motion . zeeman splitting is seen in both the mmt and keck spectra , and the spacings between the @xmath6 and @xmath7 components are the same within the measurement errors in both spectra . in contrast , zeeman splitting has never been reported for am cvn systems . finally , we beg to disagree with montgomery et al . ( 2008 ) who suggested that the broad absorption lines seen in a hypothetical carbon analog of am cvn could mimic those seen in the photosphere of an isolated white dwarf . and indeed , it has already been established ( see , e.g. , odonoghue & kilkenny 1989 or warner 1995 ) , that the hei absorption lines seen in am cvn systems , although relatively broad by main sequence standards , have strengths that are more akin to those observed in log @xmath0 = 6 hot subdwarf stars than log @xmath0 = 8 db atmospheres . this reflects the fact that they are formed in the inner region of the accretion disk orbiting the white dwarf primary where the physical conditions are similar to those encountered in log @xmath0 = 6 atmospheres . in contrast , the spectral fits carried out by dufour et al . ( 2008b ) for hot dq stars totally rule out equivalent surface gravities of log @xmath0 = 6 , as the line strengths observed in these stars bear the clear signature of log @xmath0 = 8 environments , even log @xmath0 = 9 in the case of sdss j1426 + 5752 ( notwithstanding the presence of a magnetic field ) . in brief , the spectrum of sdss j1426 + 5752 can not be confused with that of the accretion disk in a hypothetical carbon analog of am cvn . we have presented an analysis based on follow - up photometric and spectroscopic observations of the faint but highly interesting star sdss j1426 + 5752 . on the photometric front , we carried out a campaign in integrated light over a baseline spanning some 40 days . we used the kuiper / mont4k combination at the steward observatory mount bigelow station near tucson . altogether , we acquired some 106.4 h of useful photometry during the campaign . our analysis of this data set confirms that the light curve of sdss j1426 + 5752 is dominated by a periodicity at 417.707 s along with its first harmonic ( 208.853 s ) as found originally , but with less accuracy , by montgomery et al . ( 2008 ) in their discovery paper . in addition , due to the higher sensitivity achieved in our campaign , we uncovered the presence of a new oscillation with a period of 319.720 s , a 4.8 @xmath7 result using the standard detection criterion ( a 6.0 @xmath7 detection at the formal level ) . the characteristics of these three oscillations are summarized in table 2 . we investigated the stability of the amplitude and phase of each of the three periodicities extracted from the light curve of sdss j1426 + 5752 . our results suggest possible variations over timescales of days , and this is particularly true for the dominant 417.707 s periodicity . however , this needs to be confirmed with higher s / n data since observing a @xmath0 = 19.16 star in integrated light photomery with a small telescope such as the kuiper remains challenging . on the basis of our current data , we can not be absolutely certain of the reality of the suggested amplitude and phase modulations in the light curve of sdss j1426 + 5752 . we followed up on the suggestion made by montgomery et al . ( 2008 ) that the luminosity variations in sdss j1426 + 5752 may not be caused by pulsational instabilities , but rather be associted with photometric activity in a carbon - transferring analog of am cvn . if true , sdss j1426 + 5752 would represent the prototype of a new class of cataclysmic variable . since their argument hinges on the shape of the light curve folded on the dominant periodicity of 417.707 s , we further exploited our 106.4 h data set in that direction . using this and archived light curves of known pulsating stars , we found , in agreement with montgomery et al . ( 2008 ) , that the folded pulse shape of sdss j1426 + 5752 is unusual compared to those of large amplitude pulsating white dwarfs and hot subdwarfs . however , in view of the existence of isolated pulsators such as the roap stars hr 3831 and hd 99563 exhibiting pulse shapes with flatter light maxima than minima ( the opposite of what is seen in large amplitude pulsating white dwarfs ) , we emphasize that the pulse shape argument _ can not _ be used to discriminate against the pulsations interpretation . since sdss j1426 + 5752 and these roap stars and others share the common property of having a magnetic field sufficiently strong to affect the pulsations in their atmospheric layers , we suggest instead that the magnetic field may be responsible for the different pulse shape as compared to those of ( nonmagnetic ) pulsating white dwarfs . on the other hand , arguments against the interacting binary hypothesis can be put forward . for instance , the light curves we gathered have shown to be quite regular and stable over at least a six week period , a behavior that is not commonly observed in am cvn systems . our discovery of a periodicity ( 319.720 s ) that is not harmonically related to the dominant oscillation of 417.707 s also goes against the interacting binary proposal . likewise , on the spectroscopic front this time , our detailed radial velocity analysis of the available mmt spectrocopy has revealed that the velocities are constant to within 7.1 km s@xmath13 over a period of 3.37 h. this is to be compared with expected velocity semi - amplitudes in the range 80@xmath5100 sin @xmath16 km s@xmath13 . furthermore , the spectrum of sdss j1426 + 5752 , unlike those of am cvn systems , exhibits well defined symmetric absorption lines and it has proven stable over at least a 20 h period . it shows sharp zeeman splitting in the line cores , indicative of the presence of a large scale magnetic field of 1.2 mg , a property never observed in a am cvn system . finally , the line strengths indicate a physical environment appropriate for log @xmath0 = 8 atmospheres , unlike that found in the inner region of an accretion disk around a white dwarf and more akin to what is found in a log @xmath0 = 6 stellar atmosphere . it should also be pointed out that the interacting binary hypothesis does not account well at all for the existence of the family of hot dq white dwarfs as a whole since , among other things , not all of them exhibit luminosity variations ( and see dufour et al . in contrast , the single - star evolutionary scenario originally proposed by dufour et al . ( 2007 ) appears quite viable as demonstrated recently by althaus et al . ( 2009 ) . also , the nonadiabatic calculations of fontaine et al . ( 2008 ; also dufour et al . 2008b ) do predict a mixture of pulsators and nonpulsators in the hot dq population . in addition , according to montgomery et al . ( 2008 ) themselves , citing the work of benz et al . ( 1990 ) , rasio & shapiro ( 1995 ) , and piersanti et al . ( 2003 ) , carbon - transferring am cvn - type systems are not expected to exist . putting all these arguments together , we conclude with high confidence that , of the two possibilities put forward by montgomery et al . ( 2008 ) to account for the luminosity variations seen in sdss j1426 + 5752 , pulsational instabilities is the correct cause . this is also the preferred solution of montgomery et al . ( 2008 ) . randall , s.k . , green , e.m . , van grootel , v. , fontaine , g. , charpinet , s. , lesser , m. , brassard , p. , sugimoto , t. , chayer , p. , fay , a. , wroblewski , p. , daniel , m. , story , s. , & fitzgerald , t. 2007 , a&a , 476 , 1317 | we present a follow - up analysis of the unique magnetic luminosity - variable carbon - atmosphere white dwarf sdss j142625.71 + 575218.3 .
this includes the results of some 106.4 h of integrated light photometry which have revealed , among other things , the presence of a new periodicity at 319.720 s which is not harmonically related to the dominant oscillation ( 417.707 s ) previously known in that star . using our photometry and available spectroscopy
, we consider the suggestion made by montgomery et al .
( 2008 ) that the luminosity variations in sdss j142625.71 + 575218.3 may not be caused by pulsational instabilities , but rather by photometric activity in a carbon - transferring analog of am cvn .
this includes a detailed search for possible radial velocity variations due to rapid orbital motion on the basis of mmt spectroscopy . at the end of the exercise ,
we unequivocally rule out the interacting binary hypothesis and conclude instead that , indeed , the luminosity variations are caused by @xmath0-mode pulsations as in other pulsating white dwarfs .
this is in line with the preferred possibility put forward by montgomery et al .
( 2008 ) . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
as the most massive gravitationally bound systems in the universe , the rate of emergence of galaxy clusters since the big bang might be expected to be among the most straightforward predictions of cosmological models . yet despite the advent of the era of precision cosmology ushered in by observations of sne and the cosmic microwave background ( cmb ) , significant uncertainty remains in the expected numbers of galaxy clusters at @xmath0 . the cmb temperature anisotropies on scales corresponding to clusters are not accurately known , leading to a range of values for @xmath11 , the rms matter fluctuation in a sphere of radius @xmath12 mpc at @xmath13 . estimates of @xmath11 vary significantly , including e.g. @xmath14 @xcite , @xmath15 @xcite , @xmath16 @xcite , @xmath17 @xcite , @xmath18 @xcite , @xmath19 @xcite , and @xmath20 @xcite . the range @xmath21 to 1 corresponds to a variation of a factor of nearly 20 in the predicted numbers of @xmath0 clusters with @xmath22 ( e.g. * ? ? ? removing this uncertainty is a major goal of upcoming sunyaev - zeldovich cluster surveys such as the sza ( sunyaev - zeldovich array ; * ? ? ? * ) , ami ( arcminute microkelvin imager ; * ? ? ? * ) , act ( atacama cosmology telescope ; * ? ? ? * ) , and spt ( south pole telescope ; * ? ? ? since by definition , and for spectroscopically confirmed candidates in [ clusterz ] . ] galaxy clusters contain an unusually high density of galaxies , they provide an efficient means of observing substantial numbers of galaxies at a common distance , offering the hope of constructing the analog of the hertzsprung - russell ( i.e. color - magnitude ) diagram for galaxy evolution . indeed , studies of the relationship between color and magnitude indicate that clusters are the habitat of galaxies with the oldest and most massive stellar populations ( e.g. @xcite ; @xcite ) , objects reasonably free of the complications associated with starbursts and dust . these studies are consistent with an extremely simple formation history for cluster galaxies , in which their stars are formed in a short burst at high redshift , and they evolve quiescently thereafter ( we use the term `` red spike model '' in referring to this scenario - see figure [ m_vs_z ] ) . studies of the near - ir luminosity functions of cluster galaxies reinforce this picture ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . with their large lookback times , therefore , high redshift galaxy clusters also provide an observational pillar for our understanding of the formation and evolution of galaxies . obtaining substantial samples of galaxy clusters at @xmath0 has proved challenging , largely because such objects are difficult to detect using only optical data . due to their greatly enhanced rate of star formation by @xmath8 , the uv emission from modest sized field galaxies overwhelms that from the intrinsically red spectra of quiesecent , early type galaxies preferentially found in clusters . the red sequence cluster survey @xcite uses the observed color - magnitude relationship in cluster galaxies to improve the contrast and has proven highly efficient to @xmath8 , but the optical colors of the red sequence become increasingly degenerate at higher redshifts , as they no longer span the rest 4000 break . @xcite describe a program to extend the red sequence technique to higher redshift using spitzer data , but it is also important to _ test _ for the existence of red sequences in @xmath0 clusters rather than preselecting for them , if possible . the contrast of high redshift clusters over the field improves at longer wavelengths ( figure [ m_vs_z ] ) , but the contrast against atmospheric emission declines , and until recently the relatively small formats of infrared detector arrays made surveying sufficient @xmath23 volume a formidable undertaking . @xcite reported the discovery of a cluster at @xmath24 in a 100 square arcminute survey to @xmath25(vega ) = 20 ( 10@xmath26 ) . but this survey required approximately 2 hours of exposure in both @xmath27 and @xmath25 per position , and 30 allocated nights of kpno 4 m time to complete . with estimates for the surface density of @xmath6 clusters at @xmath0 in the range @xmath28 per square degree @xcite , the discovery was in hindsight fortuitous . such considerations motivated a different approach , where extended sources in deep x - ray surveys lacking prominent optical counterparts were targeted for ir followup . this technique yielded confirmed clusters at @xmath29 1.23 , and 1.26 @xcite . with the arrival of _ xmm _ , x - ray surveys offer renewed promise , leading recently to the identification of galaxy clusters at @xmath30 @xcite and 1.45 @xcite . with exposure times @xmath31 ksec and a 30 arcmin field of view , a discovery rate of approximately 30 hours per candidate @xmath0 cluster above @xmath6 is expected ( assuming one such cluster per square degree , which corresponds to @xmath32 ) . searches for clusters around radio galaxies have yielded protoclusters with redshifts as high as 4.1 @xcite and possibly even 5.2 @xcite . the very large redshifts of these systems enable powerful inferences to be drawn regarding the formation of cluster galaxies , but they are less useful as probes of the cosmological growth of structure . lyman break galaxy surveys with intensive followup spectroscopy on the keck telescopes have also identified highly overdense structures at @xmath33 and 3.09 @xcite , and @xcite discuss a @xmath34 structure identified via ly-@xmath35 emission in a narrow band imaging survey . recent advancements in ir detector array formats have renewed interest in ground - based ir surveys . in one example of the state of the art , elston , gonzalez et al . ( 2006 ) use the @xmath36 pixel flamingos camera to map 4 deg@xmath2 to a 50% completeness limit of @xmath37 ( vega ) . with 2 hour exposures on the kpno 2.1 m each covering 1/10th deg@xmath2 , this leads to an expected discovery rate for high redshift clusters per useful hour of observing which is similar to _ xmm_. the ukidss ultra deep survey provides another recent example , finding 13 cluster candidates with @xmath38 in a 0.5 deg@xmath2 survey @xcite , one of which has 4 spectroscopic redshifts at @xmath39 @xcite , and @xcite find 12 candidates with @xmath40 in a 0.66 deg@xmath2 @xmath41-band survey in the cosmos field . @xcite present a system with a high density of galaxies with red optical to near - ir colors surrounding a galaxy at @xmath42 , identified in the 120 arcmin@xmath2 gemini deep deep survey . candidates drawn from surveys of less than a square degree are unlikely to include many rich clusters , however . with the launch of the _ spitzer space telescope _ in 2003 @xcite , sensitive infrared arrays free from foreground thermal emission were put into operation @xcite . a major scientific driver for the _ spitzer _ infrared array camera ( irac ) shallow survey @xcite was the detection of @xmath0 galaxy clusters . the irac shallow survey uses 90 second exposures per position and covers 8.5 deg@xmath2 , leading to an expected discovery rate of @xmath43 hours per @xmath0 cluster . here we present results from the irac shallow survey cluster search , finding 106 cluster and group candidates at @xmath0 , of which we estimate only @xmath3 are spurious . a surface density of over 10 systems per square degree at @xmath0 is higher than expected for bound systems with masses above @xmath6 for the current range of plausible @xmath11 estimates . while we present evidence that at least two of the @xmath0 clusters have masses well above @xmath6 , it is likely that our sample includes systems with masses below @xmath6 ( i.e. groups ) , and perhaps some unbound filaments viewed end - on . in the remainder of this paper , for brevity the terms clusters " and candidates " are used to refer to all such objects which meet our selection criteria , unless otherwise stated . this paper describes how the cluster sample was identified , and some of the overall photometric properties of the sample . we also provide new spectroscopic evidence supporting nine of these clusters , from @xmath44 to 1.373 . spectroscopic evidence in support of irac shallow survey selected clusters at @xmath45 1.243 , and 1.413 was presented in @xcite , @xcite , and @xcite respectively , and we provide additional previously unpublished spectroscopy on those clusters here , for completeness . @xcite discuss the clustering of the clusters , and galametz et al . ( in preparation ) report on agn incidence vs cluster - centric distance . followup imaging with _ hst _ ( go 10496 , perlmutter ; 10836 , stanford ; and 11002 , eisenhardt ) and _ spitzer _ ( go 30950 , eisenhardt ) is underway , and future papers will examine the scatter in the color - magnitude relation as a function of morphological type , the dependence of cluster galaxy size on redshift , starburst activity in clusters vs. redshift , and the dependence of mean galaxy properties on surface density . a cosmology with @xmath46 km s@xmath5 , @xmath47 , @xmath48 is assumed , and magnitudes are on the vega system ( defined in * ? ? ? * for irac ) . at @xmath49 , this means that one arcminute corresponds to a physical scale of @xmath50 kpc , peaking at 508 kpc at @xmath51 . unless otherwise specified , physical ( rather than co - moving ) scales are used throughout . the irac shallow survey @xcite was designed to maximize the number of reliable sources detected per unit time and to cover sufficient area to detect significant numbers of @xmath23 galaxy clusters . as explained in @xcite , a 30 second exposure time per pointing is close to optimum for maximizing source detections , and reliability was obtained by requiring three independent exposures separated by hours at each position . the survey covers @xmath52 8.5 deg@xmath2 and reaches an aperture - corrected 5@xmath26 depth of @xmath52 19.1 and 18.3 mag ( vega ) at 3.6 and @xmath53 m in 3 diameter apertures . it is a remarkable fact that 90 seconds of combined exposure with irac on the 85 cm _ spitzer space telescope _ provides sufficient sensitivity to detect evolving @xmath54 galaxies to @xmath55 ( figure 1 ) . the survey was carried out in the botes region of the noao deep wide - field survey ( ndwfs ; * ? ? ? * ) to allow photometric redshifts to be derived using the deep optical imaging available for this field . the ndwfs reaches 5@xmath26 point - source depths in the @xmath56 and @xmath57 bands of @xmath52 27.1 , 26.1 , and 25.4 respectively . typical exposure times per position with the mosaic-1 camera on the kpno mayall 4-m telescope were 1 2 hours in @xmath58 , 1 2 hours in @xmath59 , and 2 4 hours in @xmath57 , and the seeing ranged from 0.7 to 1.5 . the data acquisition , reduction , and catalog generation are discussed in detail by b. jannuzi et al . ( in preparation ) and a. dey et al . ( in preparation ) . this paper uses the ndwfs third data release ( dr3 ) images and sextractor catalogs which can be obtained through the noao data archive . the agn and galaxy evolution survey ( ages , c. kochanek et al . in preparation ) provides spectroscopic redshifts for @xmath60 objects ( using the version 2.0 catalog ) in the irac shallow survey . ages is highly complete for sources brighter than 15.7 mag at @xmath53 m , and also for sources brighter than @xmath61 , with many redshifts for sources up to @xmath62 , enabling excellent assessment of the photometric redshifts to @xmath63 ( figure 1 ) . deep near infrared imaging from the flamingos extragalactic survey ( flamex ) is available for half the botes region @xcite , and was used for deriving a prior on the redshift likelihood functions ( see [ sec : photz ] ) . imaging of the botes ndwfs field has also been obtained in the radio @xcite , at 24 , 70 , and @xmath64 m with the mips instrument on _ spitzer _ @xcite , in the @xmath65band @xcite , in the @xmath66 with @xmath67 , and in x - rays to a depth of 5 ksec with the acis instrument on the _ chandra x - ray observatory _ @xcite , but these data are not used in this paper . object selection for the cluster search was carried out in the @xmath1 m band , because the negative k - correction as the rest frame @xmath68 m peak shifts into this band leads to a flux which is nearly independent of redshift for @xmath69 ( figure 1 ) . object detection and photometry was carried out using sextractor @xcite in double image mode , allowing matched aperture photometry in the other irac bands . while smaller apertures maximize the depth of the survey , monte carlo simulations showed that 5 diameter aperture magnitudes are necessary to provide sufficiently reliable color measurements and photometric redshifts @xcite . this flux limit ( 5@xmath26 at @xmath53 m in 5 ) corresponds to @xmath70 @xmath71jy , or a vega based magnitude of @xmath72 . the ndwfs catalogs were also generated using sextractor , but run in single - image mode in each band . detections in the different optical bands and between the optical and irac catalogs were matched if the centroids were within @xmath73 of each other , using the closest optical source if more than one satisfied this criterion . for very extended objects ( generally at @xmath74 ) , detections in the different bands were matched if the centroids were within an ellipse defined using the second order moments of the light distribution of the object @xcite . the ndwfs data were taken over several years in variable conditions , and therefore the photometric depths vary somewhat from pointing to pointing . average 50% completeness limits for the @xmath58 , @xmath59 , and @xmath57bands were 26.7 , 25.6 , and 25.0 respectively . average 5@xmath26 flux ( magnitude ) limits in a 5aperture ( which is significantly larger than the optimum detection aperture for these data ) were measured via monte carlo simulations to be @xmath76 @xmath71jy ( @xmath77 mag ) in @xmath58 , @xmath78 @xmath71jy ( @xmath79 mag ) in @xmath59 , and @xmath80 @xmath71jy ( @xmath81 mag ) in the @xmath57band @xcite . in the space based irac shallow survey the depth is more uniform , and in the @xmath82 m band it is @xmath83 @xmath71jy ( @xmath84 mag ) , also derived from a monte carlo simulation . although the majority of objects in the @xmath1m - selected catalog are well detected at shorter wavelengths , this is not generally the case for @xmath85 red ellipticals . objects at these redshifts are often quite faint in the optical as the 4000 break is longward of the @xmath57band ( see figure 1 ) . where sources were observed but not detected , the flux was taken to be zero and a monte carlo 1@xmath26 error was adopted . this approach is optimal for photometric redshift fitting , where the non - detection provides important contraints on the galaxy spectral energy distribution ( sed ) . the photometry used for photometric redshift estimation ( [ sec : photz ] ) consists of @xmath86[4.5]$ ] data with the monte carlo photometric errors and limits noted above . the @xmath87 m and @xmath88 m bands were not used because they do not have the sensitivity to detect @xmath85 cluster @xmath54 galaxies in the irac shallow survey . while the irac shallow survey covers 8.5 @xmath89 in each band , the overlap area observed in both the 3.6 and @xmath53 m irac bands is 8.0 @xmath89 . all of this falls within the optical ndwfs botes region . however , due to haloes around bright objects , shorter observation times for regions at the edges of individual mosaic camera pointings , and residual ccd defects , 0.75 @xmath89 may have lower quality optical photometry and hence photometric redshifts . hence these regions are also excluded from the sample . finally , 14,044 stars were removed from the catalog using the sextractor stellarity index in the best seeing optical data for each location . comparison to the star count model of @xcite , as tabulated in @xcite , suggests that @xmath90 of stars were identified via this approach , leaving @xmath91 of the @xmath1 m sample as unrecognized stars . in summary , photometric redshifts were estimated for a total of 175,431 objects brighter than @xmath70 @xmath71jy ( @xmath92 ) at @xmath53 m in 5 in a 7.25 @xmath89 region . a full description of the photometric redshift methodology is given in @xcite . a summary of those aspects most relevant to cluster detection is provided here . photometric redshifts were computed using an empirical template fitting algorithm which linearly interpolates between the four coleman , wu , & weedman ( 1980 ) seds ( e , sbc , scd , and i m ) , augmented by the @xcite sb3 and sb2 starburst templates . these seds were extended to the far uv and near ir using @xcite models . these stellar photospheric models do not include emission from dust , in particular the pah features which dominate the @xmath93 m portion of the spectrum in starforming galaxies , but at @xmath0 the photometry used does not sample these rest wavelengths . in addition to the large ages ( c. kochanek et al . in preparation ) survey , there are @xmath94 spectroscopic redshifts extending to @xmath95 gleaned from several ongoing surveys in the botes field . these were used as training sets to adjust the templates and photometric zero points to improve overall redshift accuracy and reliability ( see * ? ? ? * for details ) . comparison with these spectroscopic samples shows that an rms dispersion of @xmath96 is achieved for 95% of galaxies to at least @xmath97 . subsequent follow up spectroscopy of high redshift candidate clusters ( [ sec : keck ] ; see also * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) verify this accuracy . a key output of the @xcite technique is the redshift probability function for each object , @xmath98 , derived directly from the redshift - axis projection of the full redshift sed likelihood surface . to transform these simple redshift likelihood functions into true probability distribution functions , a prior consisting of the observed redshift distribution was applied . this was measured in the flamex region using high quality @xmath99[4.5]$ ] photometric redshifts . comparison to the spectroscopic sample illustrates that the resulting distribution functions are statistically valid in the sense that integrated areas accurately represent redshift probabilities at the 1 , 2 , and 3@xmath26 levels . these @xmath98 functions are the input to the wavelet detection algorithm discussed in [ sec : detn ] and are also used in [ sec : discussion ] . note that the @xmath98 distributions shows relatively little dependence on galaxy type @xcite . the reason is that while all galaxies are selected to have at least 5 sigma detections in [ 4.5 ] , red galaxies are often only marginally detected or even undetected in our optical images , since they have very little blue light . this is particularly true at @xmath85 where the @xmath100-correction for early types is large in the optical . blue galaxies have good @xmath101[3.6][4.5 ] photometry , because they have more blue light . thus even though blue , late - type galaxy sed s have smaller breaks , they have more extensive ( useful ) photometry . these compensating effects lead to effectively type - independent photometric redshifts for @xmath85 galaxies . we employed a wavelet analysis to identify galaxy clusters within the botes region . wavelet decomposition is a commonly used technique for cluster identification in x - ray images ( for example , see * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , where it provides an effective means of identifying extended sources in the presence of contaminating point sources . in principle , a similar analysis can be used with optical and infrared data sets , using galaxies rather than x - ray photons to identify extended sources . as is well known , galaxy number counts are more susceptible to projection effects than is bremsstrahlung emission from the icm . this issue is one that must be dealt with for all optical and infrared cluster searches , and consequently most such searches for distant clusters make explicit assumptions about the properties of the distant cluster population , such as an assumed density profile ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) or the presence of a red sequence @xcite . the sdss c4 technique @xcite does not require a _ red _ sequence , but it does demand that the colors of cluster galaxies are similar to one another . a significant advantage of the botes data set is that the photometric redshifts permit such assumptions to be minimized . the full photometric redshift probability distributions @xmath98 were used to construct weighted galaxy density maps within overlapping redshift slices of width @xmath102 , stepping through redshift space in increments of @xmath103 . for each galaxy the weight in the map corresponds to the probability that the galaxy lies within the given redshift slice . weighting in this fashion de - emphasizes sources for which the redshift is poorly constrained . it is worth emphasizing that _ all _ galaxies were included in construction of the density map , and consequently cluster detection should be relatively independent of sed type and hence independent of morphology . finally , cluster detection will be insensitive to the resolution of the density maps provided that the pixel size is small compared to the angular extent of cluster cores at all redshifts . a resolution of 12@xmath104 kpc ) per pixel was used , which satisfies this criterion while being sufficiently large to keep computational overhead manageable . galaxy cluster candidates were detected within each redshift slice by convolving the density map with the wavelet kernel . we use a gaussian difference kernel of the form @xmath105 where @xmath106 kpc and @xmath107 kpc , and which crosses zero near @xmath108 mpc . the scale of the kernel is fixed in physical rather than angular units , preserving our ability to uniformly identify comparable systems at different redshifts . the precise physical values of @xmath26 are subject to refinement , but the selected values effectively isolate overdensities on the scale of clusters or groups . galaxy cluster candidates were detected in each redshift slice of these wavelet smoothed galaxy density maps using a simple peak - finding algorithm . to establish a consistent significance level for the candidates , 1000 bootstrap simulations were carried out within each redshift slice . the existing p(z ) distributions , right ascensions , and declinations were repeatedly shuffled , convolved with the wavelet kernel , and candidates detected to find the threshold corresponding to one false positive per redshift slice within the botes field . a list of detections above this significance threshold was generated for each redshift slice . because the right ascensions and declinations were shuffled independently , the true correlated background was not preserved . consequently , the contamination rate may be somewhat higher than the one false positive per redshift slice expected if we had preserved the correlated background . for the current analysis we accept the somewhat higher contamination rate in exchange for improved completeness , particularly for the highest redshift clusters . the majority of clusters are detected in multiple redshift slices ( a natural consequence of sampling in slices separated by step sizes finer than the galaxy redshift uncertainties ) . multiple detections with small separations in positions and redshift slices were considered to be a single cluster , which was assigned an estimated redshift and position corresponding to the slice with the highest statistical significance . a total of 335 candidates were found in redshift slices with centers from 0.1 to 1.9 , including 98 in slices with @xmath85 . while the detection technique is on three dimensional overdensities , it is not immune to projection effects . the expected number of clusters in each cylindrical bin of length @xmath102 and radius 1 mpc ( the radius of the wavelet detection kernel ) was computed to quantify the extent of projection . this calculation included both the random expectation due to the observed number density and the excess clusters expected due to the observed clustering of this sample @xcite . the projection rate is most significant for lower redshifts because of the large angular size corresponding to 1 mpc , affecting @xmath109 of systems at @xmath110 to @xmath111 at @xmath112 . the projection rate decreases to below @xmath113% at @xmath114 and is negligible at @xmath115 . overall we estimate that @xmath3 of the candidates may be spurious , including the contamination noted in the previous paragraph . the peak of the summed @xmath98 distribution at the cluster detection location was taken as the initial estimate of each cluster s redshift . to improve this estimate , individual objects within a 1 mpc radius of the cluster which included the cluster redshift within the @xmath116 range of their @xmath98 functions were considered candidate cluster members . a refined estimate ( @xmath117 ) of the mean cluster redshift was obtained from the peak of the summed @xmath98 distribution for these members . from this analysis , 104 candidates have @xmath118 . in table [ z1table ] and figures 2 - 13 we present 12 of the cluster candidates that were detected via the above criteria , and subsequently spectroscopically confirmed at the keck observatory to lie at @xmath85 ( see [ sec : keck ] ) . two of these , iscs j1434.1 + 3328 and iscs j1429.2 + 3357 , have @xmath119 but the spectroscopic mean redshifts are @xmath44 and @xmath120 respectively . hence we report a total of 106 clusters at @xmath0 , of which roughly 10% may be expected to arise by chance or from projection effects , for the reasons noted in [ sec : detn ] . column 1 of table 1 provides the catalog number of each cluster . the catalog numbers increase with decreasing detection significance ( [ sec : detn ] ) . column 2 is the iau designation for each cluster , based on the ( j2000 ) coordinates of the detection given in columns 3 and 4 . column 5 provides the @xmath117 value described above , and column 6 gives the mean redshift of the spectroscopically confirmed members . column 7 provides the number of photometric redshift members of the cluster , defined as galaxies within 1 mpc of the cluster center and with integrated @xmath121 in the range @xmath122 . these galaxies are used to calculate mean cluster colors in [ sec : im3p6_vs_z ] . column 8 reduces the number in column 7 by the number of galaxies which satisfy these criteria for each cluster redshift over the entire field , scaled to the 1 mpc radius area for each cluster redshift . column 9 gives the number of spectroscopically confirmed member galaxies in each cluster ( see below ) . column 10 gives the mean @xmath7 $ ] color for the photometric redshift member galaxies , and column 11 provides the sum of their luminosities in [ 4.5 ] relative to an @xmath54 value which evolves according to the `` red spike '' model shown in figure [ m_vs_z ] , corrected for the average over the field . column 12 gives the luminosity in [ 4.5 ] of the brightest photometric redshift member galaxy relative to @xmath54 . we define a @xmath0 cluster as spectroscopically confirmed if it contains at least 5 galaxies in the range @xmath122 and within a radius of 2 mpc , whose spectroscopic redshifts match to within @xmath123 km / s . the spectroscopic redshifts must also be of class a or b. class a spectra have unambiguous redshift determinations , typically relying upon multiple well detected emission or absorption features . class b spectral features are reliable but are less well detected . the radius threshold for spectroscopic membership is larger than the 1 mpc used for photometric redshift members for practical reasons : most of the @xmath0 redshifts reported here were obtained with slitmasks extending out to approximately 2 mpc from the cluster center . typically @xmath124 photometric redshift members within 1 mpc could be accommodated in the mask design , and often no redshift could be determined from the resulting spectra . the least significant cluster in table 1 is iscs j1434.5 + 3427 , which is the 327th most significant detection out of the sample of 335 , and the 100th most significant detection at @xmath0 . despite its relatively low detection significance , iscs j1434.5 + 3427 has a striking filamentary morphology ( figure [ color10.342 ] ) , and has eleven spectroscopically confirmed members ( see also * ? ? ? lcrccccccccc 152 & iscs_j1434.1 + 3328 & 14:34:10.37 & + 33:28:18.3 & 0.98 & 1.057 & 32 & 13 & 6 & 4.83 & 15 & 2.5 + 51 & iscs_j1429.2 + 3357 & 14:29:15.16 & + 33:57:08.5 & 0.98 & 1.058 & 45 & 26 & 7 & 4.80 & 34 & 4.2 + 19 & iscs_j1433.1 + 3334 & 14:33:06.81 & + 33:34:14.2 & 1.02 & 1.070 & 57 & 38 & 20 & 4.93 & 45 & 4.3 + 123 & iscs_j1433.2 + 3324 & 14:33:16.01 & + 33:24:37.4 & 1.01 & 1.096 & 31 & 11 & 6 & 4.94 & 15 & 4.6 + 17 & iscs_j1432.4 + 3332 & 14:32:29.18 & + 33:32:36.0 & 1.08 & 1.112 & 49 & 31 & 23 & 5.16 & 47 & 5.3 + 34 & iscs_j1426.1 + 3403 & 14:26:09.51 & + 34:03:41.1 & 1.08 & 1.135 & 31 & 13 & 7 & 5.18 & 26 & 4.0 + 14 & iscs_j1426.5 + 3339 & 14:26:30.42 & + 33:39:33.2 & 1.11 & 1.161 & 52 & 35 & 5 & 5.15 & 47 & 2.8 + 342 & iscs_j1434.5 + 3427 & 14:34:30.44 & + 34:27:12.3 & 1.20 & 1.243 & 27 & 12 & 11 & 5.51 & 27 & 6.0 + 30 & iscs_j1429.3 + 3437 & 14:29:18.51 & + 34:37:25.8 & 1.14 & 1.258 & 23 & 7 & 9 & 5.30 & 11 & 6.5 + 29 & iscs_j1432.6 + 3436 & 14:32:38.38 & + 34:36:49.0 & 1.24 & 1.347 & 30 & 17 & 8 & 5.65 & 31 & 4.3 + 25 & iscs_j1434.7 + 3519 & 14:34:46.33 & + 35:19:33.5 & 1.37 & 1.373 & 19 & 9 & 5 & 5.77 & 17 & 3.9 + 22 & iscs_j1438.1 + 3414 & 14:38:08.71 & + 34:14:19.2 & 1.33 & 1.413 & 25 & 15 & 10 & 5.75 & 24 & 3.5 + the extensive ages spectroscopic database ( c. kochanek et al . in preparation ) can be used to spectroscopically confirm clusters at @xmath125 , a redshift at which @xmath54 corresponds to @xmath126 ( figure 1 ) , the magnitude limit for ages spectroscopy . note that while ages is 94% complete to @xmath127 , sparse sampling is used for fainter galaxies , and currently spectra are available for @xmath128 of galaxies with @xmath129 . because of the high surface density of ages redshifts and the larger angular scale at @xmath125 , and the fact that the [ 4.5 ] flux limit samples substantially less luminous galaxies at these redshifts , two additions to the criteria for spectroscopic confirmation used at @xmath0 were imposed . to contribute to spectroscopic confirmation for cluster candidates with @xmath130 , galaxies were required to be more luminous than l * + 1 in [ 4.5 ] for a red spike evolving model . also , the average surface density of ages galaxies over the botes field which would satisfy our high redshift confirmation criteria was calculated as a function of redshift , and 5 galaxies above this field level were required for a cluster to be confirmed . of the 335 cluster candidates , 80 have @xmath130 . however seven lie near the edges of the botes field and hence were not well observed in ages . of the remaining 73 , 61 candidates are confirmed by ages spectroscopy using the criteria just described . this criterion is perhaps overly stringent : it rejects two clusters at @xmath131 with 17 or more matching ages redshifts , but with only 4 ( rather than 5 ) for galaxies brighter than l * + 1 after field correction . all of the other 10 candidates observed by ages which do not meet the confirmation criteria have @xmath132 , and half have @xmath133 . given the sparse sampling of ages at @xmath134 , which corresponds to @xmath54 at @xmath135 ( figure 1 ) , we believe this confirmation rate validates our estimate that only @xmath109 of our cluster candidates arise by chance or due to projection effects . further details on @xmath136 clusters will be provided in a. gonzalez et al . ( in preparation ) . most of the high redshift spectroscopic confirmation of iscs clusters has been obtained at keck observatory . three clusters observed with keck have been reported previously : iscs j1432.4 + 3332 ( @xmath137 ) , iscs j1434.5 + 3427 ( @xmath138 ) , and iscs j1438.1 + 3414 ( @xmath139 ) are presented in @xcite , @xcite , and @xcite , respectively . here we provide spectroscopic confirmation for an additional nine clusters at @xmath0 , as well as some new spectroscopic information for the initial three clusters . table [ specobstable ] details new observations of iscs clusters . table [ spectable ] provides properties of previously unreported , spectroscopically confirmed cluster members , in declination order for irac sources , followed by serendipitous sources . figure [ spectra ] shows three example spectra obtained in april 2007 with keck / deimos . lcrcccccccc iscs_j1434.1 + 3328 & 1.057 & deimos & 2007 apr 18@xmath14019 & 6@xmath1411800 & clear ; 08 - 20 + iscs_j1429.2 + 3357 & 1.058 & deimos & 2006 apr 26 & 4@xmath1411500 & not photometric + iscs_j1433.1 + 3334 & 1.070 & deimos & 2007 apr 19 & 3@xmath1411200 & clear ; 08 + iscs_j1433.2 + 3324 & 1.096 & deimos & 2007 apr 18@xmath14019 & 6@xmath1411800 & clear ; 08 - 20 + iscs_j1432.4 + 3332 & 1.112 & lris & 2005 jun 03 & 5@xmath1411800 & clear ; 09 ( elston et al . 2006 ) + & & focas & 2006 apr 21 & 5@xmath1411200 & + & & deimos & 2007 apr 19 & 3@xmath1411200 & clear ; 08 + iscs_j1426.1 + 3403 & 1.135 & lris & 2007 may 19 & 3@xmath1411800 & clear ; 09 + iscs_j1426.5 + 3339 & 1.161 & lris & 2006 apr 04 & 4@xmath1411800 & clear + iscs_j1434.5 + 3427 & 1.243 & focas & 2006 apr 22 & 5@xmath1411200 & + & & deimos & 2007 apr 18 & 8@xmath1411800 & likely cirrus ; 10 - 15 + iscs_j1429.3 + 3437 & 1.258 & lris & 2006 apr 05 & 3@xmath1411200 & + iscs_j1432.6 + 3436 & 1.347 & lris & 2007 may 21 & 7@xmath1411800 & mostly clear ; 08 + iscs_j1434.7 + 3519 & 1.373 & lris & 2005 jun 02 & 7@xmath1411800 & + iscs_j1438.1 + 3414 & 1.413 & focas & 2006 jun 28 & 9@xmath1411200 & + & & deimos & 2007 apr 19 & 7@xmath1411800 & clear ; 09 + we obtained deep optical slitmask spectroscopy for several clusters using the dual - beam low resolution imaging spectrograph ( lris ; * ? ? ? * ) on the 10 m keck i telescope during three observing runs between 2005 and 2007 . slitmasks generally included approximately 15 objects with photometric redshifts consistent with cluster membership and within 4 arcmin of the nominal cluster centers . additional irac 4.5@xmath71 m selected sources were included to fill out the slitmasks . slitlets had widths of 1.3 arcsec and minimum lengths of 10 arcsec . we employed the d580 dichroic which splits the light at @xmath142 between the two channels of lris . on the red side , the 400 line grating , blazed at 8500 , was used to cover a nominal wavelength range of 5800 to 9800 , varying somewhat depending on the position of a slit in the mask . for objects filling a slitlet , the spectral resolution for this instrument configuration is @xmath143 ( @xmath144 ) , as determined from arc lamp spectra . on the blue side , the 400 line grism , blazed at 3400 , provided coverage from the atmospheric cutoff ( @xmath145 ) up to the dichroic cut off . for objects filling a slitlet , the spectral resolution for this instrument configuration is @xmath146 ( @xmath147 ) . we obtained multiple exposures for each mask , usually with 1800 s per individual exposure . table [ specobstable ] details the exposure times and observing conditions . the observations were carried out with the slitlets aligned close to the parallactic angle , and objects were shifted along the long axis of the slitlets between exposures to enable better sky subtraction and fringe correction . the slitmask data were separated into individual spectra and then reduced using standard longslit techniques . the multiple exposures for each slitlet were reduced separately and then coadded . the spectra were reduced both without and with a fringe correction ; the former tends to yield higher quality object spectra at the shorter wavelengths , while the latter is necessary at the longer wavelengths . calibrations were obtained from arc lamp exposures taken immediately after the object exposures for the red side , and from arc lamp exposures taken during the afternoon for the blue side . corrections for small offsets in the wavelength calibration were obtained by inspection of the positions of sky lines in the object spectra . using longslit observations of the standard stars from @xcite obtained during the same observing runs , we achieved relative flux calibration of the spectroscopy . while slit losses for resolved sources preclude absolute spectrophotometry from the slit mask data , the relative calibration of the spectral shapes should be accurate . one dimensional spectra were extracted from the sum of all the reduced data for each slitlet for both the red and blue sides . for the targets in the high - redshift clusters , generally only the red side data proved useful . additional spectroscopy was obtained with the deep imaging multi - object spectrograph ( deimos ; * ? ? ? * ) on the 10 m keck ii telescope , a second generation instrument with significantly more multiplexing capabilities as compared to lris , albeit without the blue sensitivity . during an observing run in april 2006 , we targeted botes active galaxy candidates selected on the basis of mid - infrared colors ( e.g. , * ? ? ? * ) , but included candidate cluster members in one mask . in april 2007 , while observing host galaxies of high redshift cluster supernovae ( perlmutter et al . , in preparation ) , we also observed candidate cluster members , typically with more than one high redshift cluster candidate observed in each wide area mask . for both observing runs , the 600zd grating ( @xmath148 ; @xmath149 ) and a gg455 order - blocking filter were used . deimos data were processed using a slightly modified version of the pipeline developed by the deep2 team at uc - berkeley . although neither run was completely photometric , relative flux calibration was achieved from observations of standard stars from @xcite . followup spectroscopic observations were also obtained with the focas spectrograph @xcite on the 8.2-m subaru telescope in april and june 2006 . for these observations , we used the instrument in multi - object spectroscopy mode with 0.8 arcsec width slits , 300r grism , and so58 order - sorting filter , which provided about 15 spectra from 5800 to 10000 with a spectral resolution @xmath150 . total exposure times were 2 - 6 hours . the focas data were reduced with iraf using standard methods including a fringe correction . wavelength calibration was done using oh airglow emission lines . absolute flux was calibrated using standard star ( feige 34 , wolf 1346 , hz 44 ) spectra taken during the same nights . iscs j1434.1 + 3328 was observed with deimos on a mask optimized for observing a high redshift supernova identified in a different iscs high redshift cluster ( perlmutter et al . , in preparation ) which has not yet been spectroscopically confirmed . four candidate members of iscs j1434.1 + 3328 were targeted , and two more spectroscopic members were identified from additional slitlets targeting 4.5 @xmath71 m selected sources to fill the mask . all six confirmed members have clearly identified ca h and k absorption lines and d4000 breaks , and none of the galaxies exhibit emission features in the wavelengths covered by the deimos observations ( @xmath151 ) . iscs j1429.2 + 3357 was observed with deimos , on a slitmask optimized for observing faint , mid - infrared selected agn candidates ( e.g. , * ? ? ? eight candidate cluster members were included in the slitmask , six of which were confirmed spectroscopically to reside at @xmath152 . only the brightest galaxy , irac j142912.9 + 335808 , shows [ ] emission ; the remaining redshifts were derived on the basis of ca h and k absorption lines and/or d4000 breaks . in addition , one galaxy targeted spectroscopically as an irac - selected agn candidate , irac j142916.1 + 335537 , is a cluster member , bringing the tally to seven spectroscopically confirmed cluster members . the spectrum of this source shows strong , narrow ( 400 km s@xmath5 ) emission lines from [ ] @xmath153 , indicating that it is indeed an active galaxy . iscs j1433.1 + 3334 was observed with deimos on a mask optimized for observing a high redshift supernova identified in iscs j1432.4 + 3332 ( perlmutter et al . , in preparation ) . eight candidate members of iscs j1434.1 + 3328 were targeted , of which six were confirmed as cluster members , one was found to be slightly foreground to the cluster , and one was slightly behind the cluster . all eight candidates show clear d4000 breaks . many additional cluster members were identified on this mask , either serendipitously or as targeted irac - selected , @xmath0 galaxies , bringing the total number of spectroscopically confirmed cluster members to 20 . as seen in the bottom two panels of figure [ spectra ] , the confirmed sources show a range of spectral properties . while most show ca h and k absorption and d4000 breaks , some also show emission features likely due to either star formation or agn activity . iscs j1433.2 + 3324 was observed with deimos on the same mask as iscs j1434.1 + 3328 . five candidate members of iscs j1434.1 + 3328 were targeted , of which two were confirmed as cluster members , one was found to be foreground , and two yielded inconclusive spectra . four additional cluster members were identified in the same mask , from irac - selected sources . the two confirmed cluster members which were specifically targeted show strong ca h and k absorption and lack emission lines . the other four confirmed members all show [ ] emission , with three of the four also showing d4000 breaks . the spectroscopic observations which confirmed iscs j1432.4 + 3332 at @xmath154 are described in @xcite , but no data on individual sources was presented there . the @xcite result was based on nine spectroscopically confirmed cluster members , two of which were selected not as cluster members , but rather as mid - ir selected agn , and that data is now included in table [ spectable ] of this paper . one of the candidate members of this cluster hosted a high - redshift supernova ( perlmutter et al . , in preparation ) and was thus targeted for additional deimos and focas slitmask spectroscopy . an example deimos spectrum for this cluster is shown in the top panel of figure [ spectra ] . in total , there are now 23 spectroscopically confirmed members in the cluster . seven candidate members of this cluster were confirmed spectroscopically during our lris observations in may 2007 . all seven galaxies show red continuum emission and/or a clear d4000 break . two of the sources also show [ ] emission . four of the five spectroscopically confirmed members of this cluster show [ ] emission , a higher proportion than is typical for this program . this is possibly a selection effect ; sources with line emission are the easiest to spectroscopically confirm . all five confirmed members show breaks typical of early - type galaxies ( e.g. , d2900 and/or d4000 ) . this cluster is discussed in @xcite , where eight spectroscopic members were presented . table [ spectable ] of the current paper does not duplicate those data , and instead lists three additional spectroscopic members that have since been identified . this cluster has nine spectroscopically confirmed members . one is an optically - bright agn from the ages survey , while the rest were confirmed spectroscopically by lris . four of the keck / lris sample show [ ] emission , while the other four show only spectral breaks and absorption lines . this cluster has eight spectroscopically confirmed members , all from our keck / lris observations in may 2007 . only one of the sources shows [ ] emission ; the rest of the redshifts are on the basis of continuum breaks at 2640 and 2900 , as well as @xmath155 2800 absorption . this cluster has five confirmed members , one of which was serendipitously identified in the spectroscopy , all from our keck / lris spectroscopy in june 2005 . three of the members show [ ] , and the other two show spectral breaks characteristic of early - type galaxy spectra . this cluster was first published in @xcite , at which time it was the highest redshift galaxy cluster known . two candidate cluster members hosted supernovae in the _ hst_/acs program of perlmutter et al . ( in preparation ) , and this field thus has been the target of additional spectroscopy from subaru and keck . five new cluster members have been confirmed , listed in table [ spectable ] . data on the original five members is not duplicated from @xcite . of the five new members , one is an [ ] emitter , serendipitously identified in a slitlet targeting another source . three show only ca h and k absorption lines , with no emission lines identified . irac j143816.8 + 341440 ( 22.3 in table 3 ) is an agn , showing [ ] , [ ] , and [ ] emission lines . with 20 or more spectroscopic redshifts , it is possible to estimate cluster masses via scaling relations using the velocity dispersion . the line of sight velocity dispersion for the 20 spectroscopic member galaxies in cluster 19 ( iscs j1433.1 + 3334 ) at @xmath156 is 760 km s@xmath5 in the rest frame , and for the 23 spectroscopic member galaxies in cluster 17 ( iscs j1432.4 + 3332 ) at @xmath157 it is 734 km s@xmath5 . for these two clusters , which are among the richest in the sample , the velocity dispersion of @xmath158 km s@xmath5 corresponds to a virial mass of @xmath159 where @xmath160 is the virial radius in mpc ( e.g. equation 4 of * ? ? ? * ) . the x - ray temperature corresponding to @xmath161 km s@xmath5 is 3.9 kev ( table 4 of * ? ? ? * ) , which in turn gives a mass of @xmath162 @xcite . this is consistent with the average halo mass of @xmath163 estimated by @xcite for the sample . stellar luminosities can also be used to make a rough estimate of cluster masses , scaling to the coma cluster via the red spike model . within the @xmath164 kpc region of coma sampled by @xcite the integrated @xmath165-band luminosity to @xmath166 is @xmath167 , slightly higher than the values shown in column 11 of table [ z1table ] , but in a significantly smaller effective radius of 0.42 mpc . in their figure 1 , geller , diaferio , & kurtz ( 1999 ) show a mass for the coma cluster of @xmath168 within this radius ( for h=0.7 ) . at r=1 mpc they show about double that mass , and suggest the total mass for coma is about double this again . hence if the profile and @xmath169 for our clusters is similar to coma , allowing for red spike model evolution in @xmath170 , the @xmath171 values in 1 mpc radius provided in table 1 scale to total cluster masses of @xmath172 ( i.e. @xmath173 ) . clusters 17 and 19 are at the top of this range , again providing mass estimates consistent with those found using the velocity dispersion . more detailed exploration of this approach will require photometry which takes account of the extent to which a 5 diameter aperture fails to include all of the light from such galaxies , or includes light from multiple galaxies ( as determined from higher spatial resolution imaging ) . as noted in the introduction , out to @xmath8 substantial evidence exists which is consistent with an extremely simple formation history for cluster galaxies , in which their stars are formed in a short burst at high redshift , and they evolve quiescently thereafter ( ie a `` red spike '' model ) . the colors of luminous cluster galaxies out to @xmath8 typically fall on a tight sequence which is red relative to field galaxies at similar redshift ( the `` red sequence '' ) , with a mean color which evolves as the red spike model predicts . the most luminous red sequence galaxies are the reddest , a correlation which is attributed to higher metallicity in higher mass galaxies ( @xcite ; but see also @xcite ) . this correlation ( i.e the color - magnitude or mass - metallicity relation ) is explained by additional cycles of star formation and enrichment in more massive galaxies , as they are able to retain their gas more effectively against supernovae - driven winds @xcite . in the red spike formation paradigm , this is a natural consequence . in the context of the hierarchical merging galaxy formation models , a unique correlation between stellar mass and metallicity is a greater challenge . while the inclusion of feedback ( whether by supernovae or by agn ) in hierarchical models stops the buildup of galaxy mass ( e.g. * ? ? ? * ; * ? ? ? * ) , such models have difficulty reproducing the exact color and slope of the color - magnitude relation in clusters over the full redshift range for which it has been measured . recent work shows promise , however , as feedback behavior and other `` gastrophysical '' effects are taken more into account ( e.g. * ? ? ? if the onset of the star formation spike " is simultaneous for all cluster galaxies , and if the color - magnitude relation is caused by more protracted spikes in more massive galaxies , then as one approaches the star - forming epoch , massive galaxies might be expected to eventually become bluer , reversing the slope of the color - magnitude relation . in fact no measurable change is seen in color - magnitude slope out to @xmath8 ( e.g * ? ? ? * ; * ? ? ? * ) , which suggests either formation redshifts well before @xmath174 , or that the spikes begin earlier in more massive galaxies , perhaps ending rather than beginning simultaneously . the small and unchanging scatter of the red sequence in clusters out to @xmath175 ( e.g. * * ; * ? ? ? * ; * ? ? ? * ) also argues for synchronized or very early spikes and against a primarily age - based origin of the color - magnitude relation , since the scatter would increase by a much larger amount , and faster , than has been observed . testing to what extent red spike models remain consistent with cluster galaxy data at @xmath0 is one of the primary motivations for the present study . we begin by constructing the color - magnitude relations for the 12 spectroscopically confirmed clusters in table [ z1table ] . figures [ cmds1 ] and [ cmds2 ] present color magnitude diagrams for the clusters listed in table [ z1table ] . the @xmath7 $ ] color was selected because these filters bracket the 4000 break most tightly at @xmath0 . the symbol area is proportional to the integral of the object s redshift probability distribution over the range @xmath176 , which is the rms dispersion in individual photometric redshifts @xcite . circled symbols are spectroscopically confirmed members , while crosses indicate objects known _ not _ to be members on the basis of spectroscopy . figures [ cmds1 ] and [ cmds2 ] include objects within 1 mpc of the cluster center , with @xmath177 detections in both [ 3.6 ] and [ 4.5 ] , and with @xmath178 detections in @xmath57 ; or with spectroscopic redshifts . as noted in [ sec : phot - limits ] , the @xmath92 limit in [ 3.6 ] is 18.6 mag , so the limiting factor in object selection for figures [ cmds1 ] and [ cmds2 ] is from the @xmath92 limit of 17.8 mag in [ 4.5 ] . we have used the 0.1 gyr burst , @xmath9 red spike model shown in figure [ m_vs_z ] to calculate the corresponding [ 3.6 ] mag as a function of redshift , and this is shown in figures [ cmds1 ] and [ cmds2 ] by the vertical dotted lines . the @xmath57 limit is shown by the diagonal dotted lines in figures [ cmds1 ] and [ cmds2 ] . the vertical dashed line plots the expected l * magnitude at [ 3.6 ] for the red spike model . the sloped dashed line shows the observed @xmath179 color - magnitude slope of 0.22 for the coma cluster from the data of @xcite , normalized at the red spike model [ 3.6 ] magnitude and @xmath7 $ ] color for an l * galaxy . in this redshift range observed @xmath7 $ ] is close to rest @xmath179 . note that while the normalization evolves according to the red spike model , a constant , _ unevolving _ slope value is used . the brightest galaxy likely to be a member is typically 1 to 2 magnitudes brighter than the expected l * magnitude for the red spike model ( see table [ z1table ] ) . it is noteworthy that the brightest galaxy in the coma cluster is @xmath180 times brighter than l * in the @xmath41 band @xcite , suggesting less than a factor of two growth in stellar mass in such galaxies since @xmath181 . _ hst _ imaging should be used to assess this more carefully . because these clusters were _ not _ selected on the basis of containing a red sequence ( [ sec : detn ] ) , the fact that the highest probability cluster members tend to track the passively evolving coma cluster sequence shown by the dashed line is significant . evidently the red sequence persists in dense environments to @xmath182 , even when not used as a selection criterion . in addition there is no indication that the slope of the color - magnitude relation has changed sign , and in fact a non - evolving coma cluster slope appears consistent with the data . comparing the data for e.g. cluster 14 ( iscs j1426.5 + 3339 at @xmath183 ) vs. 29 ( iscs j1432.6 + 3436 at @xmath184 ) suggests real differences in the scatter of the color - magnitude relation do exist . the presence of luminous galaxies substantially bluer than the red sequence in several clusters , a number of them spectroscopically confirmed , is also noteworthy . these may represent a population of massive star - forming galaxies which fade onto the red sequence in rich clusters , but is not found in field surveys ( e.g. * ? ? ? it should be noted that the very low scatters reported for @xmath185 clusters in e.g. @xcite and @xcite are calculated for galaxies known to have early - type morphologies , using much deeper photometry than the survey / discovery data used here . further investigation of the color - magnitude relations is deferred to a future paper which will use deeper _ spitzer _ and _ hst _ imaging presently being obtained . although the survey data used to identify the clusters does not enable accurate photometry of individual galaxies in clusters , we can calculate mean properties for galaxies in each cluster with some confidence . in the next section we briefly explore the color evolution and color - magnitude relation of the entire cluster sample . figure [ im3p6vsz ] plots the average @xmath7 $ ] color for galaxies within 1 mpc of the cluster centers , and whose integrated redshift probability distribution in the range @xmath176 exceeds 0.3 . the mean values are calculated after iteratively clipping @xmath186 outliers . as implied by the color - magnitude diagrams for the spectroscopically confirmed clusters ( [ sec : cmds ] ) , the red spike model ( 0.1 gyr burst at @xmath9 ) provides a remarkably good fit to the empirical data , validating the survey design assumptions illustrated in figure [ m_vs_z ] . much of the increase in @xmath7 $ ] with redshift is due to the change in the rest frame wavelengths observed , and in figure [ im3p6relnevsz ] we plot the average color relative to a no - evolution model ( i.e. the k - corrected color ) . the no - evolution model is simply the red spike model at z=0 , i.e. at an age of 11.3 gyr for the stellar population . clearly the evolving red spike model is a better fit to the data than is the k - correction alone . note that the photometric redshifts ( [ sec : photz ] ) are calculated using _ non_-evolving templates , so this result is not a foregone conclusion , particularly at @xmath0 . the mean @xmath7 $ ] colors in figure [ im3p6vsz ] include galaxies down to the @xmath187jy survey limit at @xmath1 m , and therefore less luminous galaxies contribute to the mean at lower redshifts . the color - magnitude relation shows that less luminous galaxies are bluer , and massive galaxies are quite quiescent today , so a constant flux limit will lead to a systematic bias towards more luminous , massive galaxies and redder mean color with increasing redshift . on the other hand the excellent fit of the red spike model implies that a constant _ luminosity _ limit for the sample would select galaxies with smaller stellar masses at high redshift , since a constant stellar mass , passively evolving , will be more luminous as one approaches the formation redshift . to avoid this bias , in figure [ im3p6relnevsz ] we also show ( with red squares ) the mean colors for galaxies brighter than the red spike passively - evolving l * in [ 3.6 ] . it is evident that the mean color of the more luminous galaxies is systematically redder , and hence that the color - magnitude relation continues to hold out to @xmath181 . the color offset is 0.1 magnitudes , independent of redshift . the persistence of the essentially unchanged color - magnitude relation slope , with a passively evolving intercept , out to lookback times within 4 gyr of the big bang is a phenomenon that models of cluster galaxy formation must account for . the implication is that the correlation between high stellar population metallicity and stellar mass is already in place at @xmath181 , and that the star formation era remains well in the past for these galaxies . in figure [ im3p6relpevsz ] the mean @xmath7 $ ] colors are plotted with the predicted @xmath7 $ ] color for the red spike model subtracted . the mean colors are for galaxies more luminous in [ 3.6 ] than the red spike l*. note the red spike model is a good match to @xmath54 galaxy colors in figures [ im3p6vsz ] and [ im3p6relnevsz ] , so the offset in color out to @xmath8 is attributable to the color - magnitude relation . other effects may be in play . in table 1 the @xmath117 values tend to be lower than the @xmath188 values , which may lead to colors appearing @xmath189 mag redder relative to the red spike model than if @xmath188 were available for all @xmath0 clusters . the average colors of cluster members may still be redder than shown , however , as no attempt has been made to correct the average cluster colors for field contamination , and field galaxies tend to be bluer than cluster galaxies . blending issues could lead to colors which are systematically biased . given these uncertainties , the excellent agreement with the red spike model is the more remarkable . nevertheless , at @xmath0 the trend is towards colors which are increasingly redder than the red spike model , for both the full sample and the spectroscopically confirmed subset , though there is clearly a range . hence even higher formation redshifts ( as high as @xmath190 ) are favored for most @xmath0 clusters , within the context of red spike models . figure [ im3p6vsz ] shows that that there are relatively few cluster candidates in the irac sample with @xmath192 . why is this ? the red spike model fits , circumstantial evidence from bcg luminosity , and the persistence of the color - magnitude relation suggest that the stellar populations were formed and assembled well before @xmath181 . from figure [ m_vs_z ] , such clusters should be detectable by the irac shallow survey . a potential selection effect against @xmath115 galaxies is that photometric redshifts in this range have increasingly broad redshift probability distributions , due in part to insufficient depth in photometry at shorter wavelengths . simulations indicate a factor of two reduction in irac photometric error can help compensate , leading to a similar reduction in photometric redshift error in this redshift range . with the _ spitzer _ deep wide - field survey legacy program now underway ( pi d. stern ) , such data will be available in the near future . the tighter redshift probability distributions would improve the contrast of high redshift clusters over the field , allowing them to meet our detection threshold . but the lack of massive @xmath115 cluster detections may simply reflect a real decline in the space density of such objects . models for the hierarchical growth of structure predict only about 1 cluster with @xmath193 above @xmath194 for every 5 clusters with @xmath0 . full exploration of this possibility will require more careful simulation and assessment of our observational selection effects . we have identified 335 galaxy cluster and group candidates from a @xmath53 m selected sample of galaxies in the irac shallow survey . candidates were identified by searching for overdensities in photometric redshift slices , and 106 clusters are at @xmath0 . roughly 10% of these candidates may be expected to arise by chance or from projection effects . to date , 12 clusters have been spectroscopically confirmed at @xmath0 , as have 61 of the 73 clusters observed with ages at @xmath125 . for the two @xmath0 clusters with 20 or more spectroscopic members , total cluster masses of several @xmath195 are indicated , and the total mass estimated from the stellar luminosity yields comparable values . color - magnitude diagrams in @xmath7 $ ] vs. [ 3.6 ] for the @xmath0 spectroscopically confirmed clusters reveal that a red sequence is generally present , even though clusters were not selected for this . the brightest probable member galaxy ( at the spatial resolution of irac ) in the spectroscopically confirmed @xmath185 clusters remains 1 2 mag brighter than the passively - evolving @xmath54 luminosity . for the full cluster sample , the mean color of brighter galaxies within each cluster is systematically redder than the mean color of all probable cluster member galaxies , implying that the mass - metallicity relation is already in place at @xmath181 . the mean @xmath7 $ ] color of probable cluster members is well fit by a simple model in which stars form in a 0.1 gyr burst beginning at @xmath9 . at @xmath0 there is a tendency for mean cluster colors to favor formation redshifts @xmath10 , although a few are consistent with @xmath196 . this adds to the large body of evidence that galaxies in clusters were established at extremely early times . we thank mark dickinson , emily macdonald , and hyron spinrad for generously making time available on their scheduled nights for the deimos observations reported here . naoki yasuda , naohiro takanashi , yutaka ihara , kohki konishi , and hiroyuki utsunomiya assisted with observations at the subaru telescope . roberto de propris provided the integrated luminosity for coma cluster galaxies . thoughtful comments from the anonymous referee improved the presentation of this work . the irac shallow survey was executed using guaranteed observing time contributed by g. fazio , g. and m. rieke , m. werner , and e. wright . this work is based in part on observations made with the _ spitzer space telescope _ , which is operated by the jet propulsion laboratory , california institute of technology under a contract with nasa . this work made use of images and data products provided by the noao deep wide - field survey , which is supported by the national optical astronomy observatory ( noao ) . noao is operated by aura , inc . , under a cooperative agreement with the national science foundation . some of the data presented herein were obtained at the w.m . keck observatory , which is operated as a scientific partnership among the california institute of technology , the university of california and the national aeronautics and space administration . the observatory was made possible by the generous financial support of the w.m . keck foundation . some of the data presented were collected at the subaru telescope , which is operated by the national astronomical observatory of japan . the work of sas was performed under the auspices of the u.s . department of energy , national nuclear security administration , by the university of california , lawrence livermore national laboratory , under contract no . the work of kd and jm was partially supported by the director , office of science , department of energy , under grant de - ac02 - 05ch11231 . lccc + 152.1 & 15.45 & 1.01 & 1.057 + 152.2 & 15.99 & 0.91 & 1.055 + 152.3 & 17.00 & 0.96 & 1.054 + 152.4 & 17.47 & 1.04 & 1.065 + 152.5 & 16.40 & 0.99 & 1.055 + 152.6 & 16.76 & 0.96 & 1.055 + + 51.1 & 15.10 & 0.98 & 1.059 + 51.2 & 17.06 & 0.95 & 1.056 + 51.3 & 17.83 & 1.04 & 1.055 + 51.4 & 17.16 & 1.00 & 1.060 + 51.5 & 17.01 & 1.03 & 1.059 + 51.6 & 17.19 & 1.03 & 1.054 + 51.7 & 16.08 & 1.01 & 1.060 + + 19.1 & 17.49 & 0.99 & 1.075 + 19.2 & 16.67 & 0.99 & 1.075 + 19.3 & 17.04 & 1.06 & 1.076 + 19.4 & 16.00 & 1.06 & 1.066 + 19.5 & 16.89 & 1.09 & 1.067 + 19.6 & 16.47 & 0.94 & 1.065 + 19.7 & 17.70 & 1.00 & 1.063 + 19.8 & 17.15 & 1.10 & 1.079 + 19.9 & 16.75 & 0.98 & 1.074 + 19.10 & 16.30 & 1.07 & 1.066 + 19.11 & 16.21 & 1.03 & 1.066 + 19.12 & 16.66 & 1.03 & 1.075 + 19.13 & 15.44 & 1.06 & 1.064 + 19.14 & 17.03 & 1.28 & 1.076 + 19.15 & 17.51 & 1.00 & 1.065 + 19.16 & 17.01 & 1.27 & 1.069 + 19.17 & 17.36 & 1.04 & 1.066 + 19.18 & 16.27 & 0.86 & 1.063 + 19.19 & 16.68 & 2.45 & 1.074 + 19.20 & - & - & 1.067 + + 123.1 & 15.32 & 1.07 & 1.094 + 123.2 & 16.45 & 1.09 & 1.093 + 123.3 & 16.68 & 1.10 & 1.094 + 123.4 & 16.99 & 1.11 & 1.107 + 123.5 & 16.64 & 0.93 & 1.094 + 123.6 & 17.35 & 0.99 & 1.091 + + 17.1 & 16.94 & 2.74 & 1.1120 + 17.2 & 17.35 & 1.03 & 1.1108 + 17.3 & 16.48 & 1.07 & 1.111 + 17.4 & 15.97 & 1.05 & 1.115 + 17.5 & 16.31 & 1.04 & 1.111 + 17.6 & 15.85 & 1.09 & 1.110 + 17.7 & 16.04 & 1.35 & 1.121 + 17.8 & 17.57 & 1.20 & 1.116 + 17.9 & 16.87 & 1.00 & 1.11 + 17.10 & 17.31 & 1.04 & 1.1086 + 17.11 & 17.01 & 1.14 & 1.098 + 17.12 & 17.43 & 1.12 & 1.105 + 17.13 & 17.51 & 0.98 & 1.109 + 17.14 & 17.32 & 0.91 & 1.112 + 17.15 & 16.59 & 1.07 & 1.104 + 17.16 & 16.92 & 1.19 & 1.119 + 17.17 & 17.42 & 1.09 & 1.115 + 17.18 & 17.33 & 1.11 & 1.115 + 17.19 & 17.29 & 1.06 & 1.118 + 17.20 & 16.90 & 1.18 & 1.107 + 17.21 & 16.38 & 1.26 & 1.115 + 17.22 & - & - & 1.114 + 17.23 & - & - & 1.110 + + 34.1 & 16.35 & 1.03 & 1.1439 + 34.2 & 15.75 & 1.04 & 1.1301 + 34.3 & 16.01 & 1.11 & 1.1271 + 34.4 & 15.69 & 1.00 & 1.1328 + 34.5 & 16.12 & 1.18 & 1.134 + 34.6 & 16.90 & 0.91 & 1.13 + 34.7 & 17.24 & 0.89 & 1.144 + + 14.1 & 16.87 & 1.10 & 1.16 + 14.2 & 15.96 & 1.05 & 1.157 + 14.3 & 16.44 & 1.19 & 1.1631 + 14.4 & 16.64 & 1.17 & 1.1634 + 14.5 & 16.88 & 1.04 & 1.1637 + + 342.1 & 16.76 & 1.21 & 1.240 + 342.2 & 17.71 & 1.21 & 1.251 + 342.3 & - & - & 1.256 + + 30.1 & 16.65 & 1.18 & 1.245 + 30.2 & 16.39 & 1.13 & 1.26 + 30.3 & 17.24 & 1.15 & 1.2576 + 30.4 & 17.37 & 1.07 & 1.263 + 30.5 & 17.41 & 1.29 & 1.2611 + 30.6 & 16.86 & 1.08 & 1.2583 + 30.7 & 16.97 & 1.31 & 1.2582 + 30.8 & 15.19 & 0.97 & 1.2632 + 30.9 & - & - & 1.2546 + + 29.1 & 17.63 & 1.26 & 1.3559 + 29.2 & 17.61 & 1.25 & 1.35 + 29.3 & 16.39 & 1.32 & 1.35 + 29.4 & 16.06 & 1.21 & 1.3320 + 29.5 & 17.55 & 1.40 & 1.347 + 29.6 & 16.77 & 1.29 & 1.347 + 29.7 & 16.86 & 1.30 & 1.34 + 29.8 & 16.14 & 1.26 & 1.353 + + 25.1 & 17.52 & 3.33 & 1.37 + 25.2 & 16.98 & 1.53 & 1.372 + 25.3 & 16.28 & 1.34 & 1.37 + 25.4 & 15.68 & 1.33 & 1.374 + 25.5 & - & - & 1.380 + + 22.1 & 15.86 & 1.25 & 1.411 + 22.2 & 15.79 & 1.38 & 1.418 + 22.3 & 16.91 & 1.38 & 1.412 + 22.4 & 16.76 & 1.35 & 1.414 + 22.5 & - & - & 1.412 + | we have identified 335 galaxy cluster and group candidates , 106 of which are at @xmath0 , using a @xmath1 m selected sample of objects from a 7.25 deg@xmath2 region in the _ spitzer _ infrared array camera ( irac ) shallow survey .
clusters were identified as 3-dimensional overdensities using a wavelet algorithm , based on photometric redshift probability distributions derived from irac and noao deep wide - field survey data .
we estimate only @xmath3 of the detections are spurious . to date 12 of the @xmath0 candidates
have been confirmed spectroscopically , at redshifts from 1.06 to 1.41 .
velocity dispersions of @xmath4 km s@xmath5 for two of these argue for total cluster masses well above @xmath6 , as does the mass estimated from the rest frame near infrared stellar luminosity .
although not selected to contain a red sequence , some evidence for red sequences is present in the spectroscopically confirmed clusters , and brighter galaxies are systematically redder than the mean galaxy color in clusters at all redshifts .
the mean @xmath7 $ ] color for cluster galaxies up to @xmath8 is well matched by a passively evolving model in which stars are formed in a 0.1 gyr burst starting at redshift @xmath9 . at @xmath0 , a wider range of formation histories is needed , but higher formation redshifts ( i.e. @xmath10 ) are favored for most clusters . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
supersymmetric yang - mills theories in @xmath1 dimensions are described by an action of the form @xmath2 where indices @xmath3 correspond to the adjoint representation of some semisimple compact lie algebra , @xmath4 are spacetime vector indices ( spatial indices will be denoted by @xmath5 ) , and @xmath6 are spinorial spacetime indices ; furthermore , the nonabelian field strength is formed from the vector potential @xmath7 via @xmath8 @xmath9 being the gauge group structure constants , and the gauge covariant derivative is given by @xmath10 the equality of the number of physical bosonic and fermionic degrees of freedom required by supersymmetry furthermore forces the number of transversal modes of the gauge field @xmath11 , @xmath12 , to be a power of two , as is the size of the corresponding irreducible spinor representation of the lorentz algebra . one finds that it is in fact possible here to implement supersymmetry in @xmath13 . by truncation to configurations that have no spatial dependency ( i.e. dimensional reduction to zero space dimensions , see e.g.@xcite ) , one obtains supersymmetric quantum mechanics @xcite with a hamiltonian @xmath14 with @xmath15 . the @xmath0 case with gauge group @xmath16 regained a lot of popularity after initial work by de wit , hoppe , and nicolai @xcite who showed the relevance of this particular model for the description of the eleven - dimensional supermembrane ( where @xmath16 appears as a regularized version of the lie group of area - preserving membrane diffeomorphisms ) through the so - called m(atrix ) theory conjecture @xcite , which states that this hamiltonian of a system of @xmath17 @xmath18-branes of type iia string theory should give a complete , non - perturbative description of the dynamics of @xmath19-theory in the light cone frame .. furthermore , these matrix models play an important role in the ikkt model @xcite , which may provide a non - perturbative description of iib superstring theory . gauge groups of other types are also of interest here , as realizations of this model with @xmath20 and @xmath21 symmetry are given by systems of @xmath17 type - iia @xmath18-branes moving in orientifold backgrounds , cf . @xcite . one question of chief importance is that of normalizable zero energy vacuum states of these models ; this is difficult to settle as the potentials have flat valleys that extend to infinity , and hence , the corresponding ground states are at threshold . while it is exceedingly difficult to try to explicitly solve the schrdinger equation for these models , it is already of great interest to know the _ number _ of such normalizable ground states ; for example , it is of crucial importance to the matrix theory conjecture that there is exactly one such state for every @xmath17 in the models derived from @xmath16 gauged @xmath22 sym ( which just corresponds to a bound state of @xmath17 @xmath18-branes that appears as a graviton with @xmath17 units of momentum in the compactified direction ) . while this is widely believed to be the case by now , the situation is still much less clear for other gauge groups . as is nicely explained in @xcite , the number of normalizable ground states is given as the low temperature limit of the partition function @xmath23 but as the calculation of this quantity seems beyond reach for most systems of interest , it appears more promising to try to calculate the witten index @xmath24 instead ( where @xmath25 is the fermion number ) , as this should also give the number of normalizable ground states . in case of a discrete spectrum , this partition function would be @xmath26-independent , so we could as well take the limit @xmath27 , which is accessible in a perturbative calculation . for a continuous spectrum of the hamilton operator , this does not work in general , since supersymmetry still pairs bosonic and fermionic modes , but the spectral density of scattering states need not be equal . in this case , the ` boundary ' term @xmath28 in the decomposition @xmath29 can acquire a nonzero value . while the technique of splitting the integral into a ` principal contribution ' from the bulk term as well as a ` deficit contribution ' which takes the form of a boundary term @xcite works remarkably well in many situations , as one frequently finds that the boundary term is zero even if it _ a priori _ does not seem to have to be ( e.g. @xcite ) , this is not the case in the systems at hand . nevertheless , the boundary term has been calculated for @xmath30 in @xcite , and reasons ( that are based on the ( heuristic ) assumption that for the calculation of this deficit term the @xmath18-branes can effectively be treated as identical freely propagating particles , as in @xcite ) have been given in @xcite that this deficit term should be @xmath31 for the @xmath22 models derived from @xmath0 @xmath16 gauged sym , while one expects the value @xmath32 for the @xmath33 and @xmath34 models derived from @xmath35 , resp . @xmath36 @xmath16 gauged sym . employing the mass deformation method that has been developed in @xcite , kac and smilga @xcite showed via group - theoretical means that under the hypothesis that no large mass bound state becomes non - normalizable as the zero - mass limit is taken , the number of normalizable ground states should be given by certain simple partition functions : @xmath37 independent arguments that lead to the same @xmath20 and @xmath21 multiplicities have been presented in @xcite which are based on an analysis of the hilbert space of a chiral fermion that is constructed from d0 branes at an orientifold singularity . the bulk term @xmath38 is at least in principle independently accessible via a generalization @xcite of the brst deformation method which was devised by moore , nekrasov , and shatashvili @xcite in order to greatly simplify the calculation of the corresponding partition functions by adding terms to the action which break all but one supersymmetry ( so that the partition functions do not change ) sym with low - rank gauge groups that this method seemingly also works for other groups besides @xmath16 ] . concerning the boundary term , the method of @xcite has been generalized in @xcite to other gauge groups , but it was found that the expected witten index @xmath40 could not be obtained that way @xcite , indicating a failure of the assumption of the validity of the free hamiltonian approach used in @xcite . nevertheless , for all the @xmath41 cases investigated previously , the expected vacuum degeneracies support the hypothesis @xcite that the boundary term is ( with the possible exception of @xmath42 ) always a small number in the interval @xmath43 $ ] , and hence a prediction of the witten index should be possible from the bulk index alone . a first evaluation of @xmath44 for special orthogonal and symplectic groups employing the moore method has been performed in @xcite ; there as well as in further work on @xmath45 @xcite , the algebraic bulkiness of bulk witten index calculations was pointed out , and indeed , even for the @xmath45 case , going to ranks far beyond @xmath46 already required specialized term manipulation code to be written . in the following , we want to review the operational issues of the application of the moore method to the even far more involved @xmath0 case and present techniques ( some conservative , some speculative ) that allow the calculation to be taken to regular simple groups of rank 4 and 5 , and also give arguments that such a direct approach is barely feasible for simple groups of rank @xmath47 employing currently available computer technology . the heat kernel calculation @xcite of the bulk contribution to @xmath48 requires evaluation of the partition function @xmath49[a_\mu , a_\nu]+\frac{1}{2g^2}\tr\psi_\alpha[\gamma^\mu_{\alpha\beta}x_\mu,\psi_\beta]\right)\\ \displaystyle = \int\prod_{a=1}^{{\rm dim}\,g}\prod_{\mu=1}^d\frac{dx_\mu^a}{(2\pi)^{1/2 } } { \rm pf}\,(-if^{bcd}\gamma\cdot x^d ) \exp\left(\frac{1}{4g^2}\tr[a_\mu , a_\nu][a_\mu , a_\nu]\right)\\ \end{array}\ ] ] where the fermionic degrees of freedom have been integrated out , yielding a homogeneous pfaffian , cf . these partition functions @xmath50 , which come from the reduction of sym to zero dimensions ( see e.g. @xcite ) have been dubbed ` yang - mills integrals ' . the factor @xmath51 that relates it to the bulk witten index is basically the effective gauge group volume , see @xcite for details . the brst deformation technique of @xcite greatly simplifies this to the calculation of the integral @xcite @xmath52 where @xmath53 is the lie group rank , @xmath54 is the quotient of the orders of the center and the weyl group ( @xmath55 for @xmath56 , @xmath57 for @xmath58 , @xmath59 for @xmath60 , 1/12 for @xmath61 ; these are the only cases we are concerned with here ) , @xmath62 are auxiliary real quantities with @xmath63 which the end result will not depend on ( at least for sufficiently generic values of @xmath64 where no ` accidental ' merging of poles / zeroes happens ) . @xmath65 is the set of roots of the lie algebra @xmath66 . actually , as it stands these integrals do not make much sense , as we have poles on the real axis of integration . ( furthermore , they do not fall off fast enough towards complex infinity to rigorously justify closing the integration contour . ) the correct ( not yet fully justified , see @xcite ) interpretation of these integrals rather is the following algorithmic one : 1 . successively eliminate all @xmath67 ` integration variables ' in the term @xmath68 where the @xmath69 are all treated as real , and @xmath70 picks up the residues with positive imaginary part only . ( all the @xmath67 except the one being integrated out are treated as real . ) 2 . substitute @xmath71 3 . evaluate at generic values of @xmath72 . ( alternatively : simplify to find that the result does not depend on @xmath72 . ) due to the large number of factors in the denominator that occur in these formulae for all but the smallest simple groups , one has to resort to employing clever tricks to simplify the calculation , or massive computer aid ( or both ) . while the use of symbolic manipulation programs like maple or mathematica suggests itself for calculations like the ones at hand , and indeed interesting results have been obtained that way @xcite , the range to which these calculations can be carried by employing such systems in a head - on approach is quite limited , mostly due to the observation that ( unlike the @xmath35 case ) in the evaluation of @xmath73 integrals , one seems to generically encounter poles of order @xmath74 that require forming @xmath75th order derivatives of very large products , leading to an explosion of the number of terms generated . nevertheless , it is obviously important to try to obtain these values for as many gauge groups as possible in order to check the validity of various assumptions that had to be made in the calculation of witten indices , and perhaps even make further conjectures about analytical expressions , e.g. as in @xcite for @xmath35 . even if one can not do much about combinatoric explosion here , the question nevertheless arises , whether considering the conceptual simplicity of the problem one may be able to go a few steps further by trying to make use of as much of the structure of the calculation as possible in a dedicated program . while it is frequently possible to outperform general - purpose symbolic algebra packages by three orders of magnitude in term complexity with such an approach , more than one rabbit has to be pulled out of the hat in order to achieve a performance gain of one million or more , which we will see to be necessary here . as the underlying techniques are of interest for a far larger class of symbolic calculations , yet not widely known since there is ( to the author s knowledge ) hardly any literature on the relevant issues , we want to briefly present some of the fundamental concepts one should be aware of when taking symbolic algebra to its limits . incidentally , the calculation at hand is an almost perfect example to study these techniques . conventional symbolic algebra frequently ( and especially in calculations like these ) wastes most of its time generating intermediate quantities of very limited lifetime in dynamically allocated memory which has to be reclaimed soon after . in the following , we want to adhere to the convention to call this allocation of dynamic memory which will not explicitly be reclaimed _ consing_. as unnecessary consing forces frequent expensive calculations of reachability graphs of in - memory objects ( in order to identify reclaimable space in the garbage collection process ) , it should as a rule of thumb be avoided where possible . furthermore , one should keep in mind that for the presently dominant computer architecture the processor s memory interface is an important bottleneck , and hence it makes sense to try to find tight memory encodings for those pieces of data on which most of the calculation operates so that cache stalls are minimized . one particularly striking feature of the calculation at hand is that naively integrating out a single variable makes conventional term manipulation programs first allocate huge numbers of almost similar terms , which are then checked ( in an overwhelming number of cases in vain ) for possible annihilations . this can be avoided by re - structuring the calculation in such a way that instead of generating and processing large amounts of individual mostly similar ( yet different enough that cancellations become quite rare ) terms , one and the same term backbone is destructively modified to consecutively represent every single new term generated and then do further processing on this backbone wherever possible . this technique is particularly useful when it comes to processing higher - order derivatives ( where in addition we take care of performing the iterations over places of factors where to derive in such a way that re - occurring combinations of derivatives , as in @xmath76 are only generated once ) . one particular refinement for the present calculation is that one generally can hardly avoid creating new terms by substitution , except in the very last step ( i.e. integrating out the last variable ) where this is indeed feasible . since the last step is also executed most frequently , this almost buys us an extra rank for free . ( a slightly nontrivial subtlety for the @xmath0 calculation here is that we have to take care of the possibility of generating factors @xmath77 in both the numerator as well as the denominator . it may well happen that while accumulating the factors of a term , one intermediately encounters more such ` powers of zero ' in the denominator than in the numerator . ) as calculations with exact numbers cause some systems to perform excessive number consing , it is wise to try to avoid the exact rational number data type for all those parts of a calculation where one can use with impunity more limited data types that have fast direct hardware support ( like 32-bit integers ) . for our present calculation this means to represent the coefficients of linear functions as machine integers . ( clearly , one has to take care of proper handling of some ` balancing denominators ' that are generated by this somewhat artificial treatment that requires to take least common multiples at substitutions . ) for all of the problems that are reachable with present computer hardware , all of the substitutions of such terms among themselves will not lead us outside the range of these integer machine data types . a further property of the calculations under study is that when performed in the way described here , a large fraction of the contributions are zero , and in many cases , this can be detected somewhat easily without doing costly multiplications . hence , it makes sense to pre - scan a term for being zero before processing it wherever applicable . it is perhaps one of the less obvious properties of these yang - mills integrals that one can do much better than to substitute particular values for @xmath78 subject to the constraint @xmath63 for the final evaluation ; by bringing all factors to a proper lexicographical normal form ( @xmath79 contribution first if present , then @xmath80 contribution , then @xmath81 contribution , all after having substituted out @xmath82 ) , one observes to obtain also the correct final value by replacing every linear term by its leading coefficient . ; this was discovered through an intermediate programming error . ] while the largest calculations at present can not be done without employing this trick , we do a cross - check using the more conventional method wherever this is possible . this is indicated in the last column of our table of results . as a special refinement , one notes that in the final gathering of powers of linear factors , the most frequently encountered numerators and denominators are @xmath83 as well as small powers of 2 . indeed , a further noticeable speedup can be achieved by treating these factors special ( remembering one overall power of 2 as well as the resulting sign ) in order to avoid unnecessary use of exact rational arithmetics . as a basis for the implementation , the objective caml system @xcite appears as very appealing , since it is highly portable to a variety of different platforms , contains an optimizing compiler that can generate compact standalone binaries , allows a very smooth and easy two - way integration of c libraries and code ; all these qualities are highly desirable here especially since the problem at hand suggests itself to massive parallelization , perhaps by making use of donated computation time . ( while rudimentary parallelization support is present in our implementation and has proven its usefulness , we do not implement such a large - scale scheme here , mainly because it is expected that one would need a prohibitively large number of volunteers to successfully do the next rank . ) a further advantage may be that ocaml code generally is perceived as much less alien by the uninitiated than lisp code . one further noteworthy issue here is the quality of the built - in exact rational number arithmetics , since there are all but obvious huge performance differences between various implementations . while the implementation present in ocaml 3.07 is slower than the one of clisp by a factor of roughly 300 , this is not a big problem , since our techniques to reduce the use of fraction arithmetics are powerful enough ( at least when making use of the evaluation shortcut described above ) to make the amount of time spent in arithmetics comparable to the time spent in other parts of the program . clisp would perhaps hardly be a viable alternative despite its excellent arithmetics implementations , since it is only a byte - code interpreter system . as there are good reasons to make the code with which these calculations have been performed publicly available , it has been included in the arxiv.org preprint upload of this work . the following table shows our new as well as the previously known values for the bulk contributions to the witten index . numerical approximations as well as the expected values from the kac / smilga hypothesis are also given . calculation times refer to accumulated wall clock times on a stepping-9 intel pentium 4 cpu , 2.4 ghz , hyperthreading enabled / ocaml 3.07 , gcc 2.95 , linux kernel 2.4.24 student computer pool installation . ( values obtained on other systems have been re - scaled appropriately . ) boldface indicates new values . some features that deserve special attention here are the particularly large values for @xmath84 and @xmath85 that would require @xmath86 , the @xmath87 and @xmath88 values that would require @xmath89 , and the dramatic explosion of calculation time ; beyond some ` trivial ' cases , increasing the rank by one costs a factor ( as a rule of thumb ) of roughly 1000 in cpu time in the interesting regime . hence , it is probably not yet feasible to try to attack rank @xmath90 for the @xmath73 groups . as the @xmath91 bulk indices are just the expected ones , and as the @xmath73 are of the right magnitude , with denominators being powers of two ( this is not the case at all for individual summands ) , there is good reason to believe in both the validity of this approach and its implementation . however , the calculation of the @xmath92 index by using explicit values for @xmath72 seems to fail systematically for yet unknown reasons , despite a very careful analysis of the code . ( this may be related to yet another bug in the ocaml compiler ; this assumption is also fueled by the observation that part of the @xmath93 calculation causes memory violations that should not be possible at all in pure ocaml for a very specific combination of processors and optimization flags . ) all in all , the new data clearly show that for other gauge groups than @xmath16 , the issue of the number of vacuum states is not well understood , and more work has to be put into the determination of witten indices . in particular , we do not have at present a useful theory to calculate the boundary contributions , and it is well conceivable that some of the assumptions behind @xmath94 may be violated . i want to thank matthias staudacher for introducing me to this interesting problem , and also for many helpful comments and discussions , the administrative staff of cip.physik.uni-munchen.de , where some of the calculations have been performed , as well as steffen grunewald for letting me submit some calculation jobs to the geo600 merlin cluster at aei . furthermore , i want to thank the debian ocaml package maintainer , sven luther , for his for his prompt handling of bug reports , and damien doligez for fixing an ocaml compiler bug @xcite that was discovered through this work . l. alvarez - gaume , `` supersymmetry and the atiyah - singer index theorem , '' commun . * 90 * ( 1983 ) 161 . p. austing , `` yang - mills matrix theory , '' arxiv : hep - th/0108128 . m. baake , m. reinicke and v. rittenberg , `` fierz identities for real clifford algebras and the number of supercharges , '' j. math . phys . * 26 * ( 1985 ) 1070 . t. banks , w. fischler , s. h. shenker and l. susskind , `` m theory as a matrix model : a conjecture , '' phys . d * 55 * ( 1997 ) 5112 [ arxiv : hep - th/9610043 ] . m. claudson and m. b. halpern , `` supersymmetric ground state wave functions , '' nucl . b * 250 * ( 1985 ) 689 . u. h. danielsson , g. ferretti and b. sundborg , `` d - particle dynamics and bound states , '' int . j. mod . a * 11 * ( 1996 ) 5463 [ arxiv : hep - th/9603081 ] . b. de wit , j. hoppe and h. nicolai , `` on the quantum mechanics of supermembranes , '' nucl . phys . b * 305 * ( 1988 ) 545 . t. fischbacher , sven luther , + http://bugs.debian.org/cgi-bin/bugreport.cgi?bug=224417 d. friedan and p. windey , `` supersymmetric derivation of the atiyah - singer index and the chiral anomaly , '' nucl . b * 235 * ( 1984 ) 395 . r. flume , `` on quantum mechanics with extended supersymmetry and nonabelian gauge constraints , '' annals phys . * 164 * ( 1985 ) 189 . m. b. green and m. gutperle , `` d - particle bound states and the d - instanton measure , '' jhep * 9801 * ( 1998 ) 005 [ arxiv : hep - th/9711107 ] . a. hanany , b. kol and a. rajaraman , `` orientifold points in m theory , '' jhep * 9910 * ( 1999 ) 027 [ arxiv : hep - th/9909028 ] . n. ishibashi , h. kawai , y. kitazawa and a. tsuchiya , `` a large - n reduced model as superstring , '' nucl . b * 498 * ( 1997 ) 467 [ arxiv : hep - th/9612115 ] . d. kabat and p. pouliot , `` a comment on zero - brane quantum mechanics , '' phys . * 77 * ( 1996 ) 1004 [ arxiv : hep - th/9603127 ] . v. g. kac and a. v. smilga , `` normalized vacuum states in n = 4 supersymmetric yang - mills quantum mechanics with any gauge group , '' nucl . b * 571 * ( 2000 ) 515 [ arxiv : hep - th/9908096 ] . w. krauth and m. staudacher , `` yang - mills integrals for orthogonal , symplectic and exceptional groups , '' nucl . b * 584 * ( 2000 ) 641 [ arxiv : hep - th/0004076 ] . w. krauth , h. nicolai and m. staudacher , `` monte carlo approach to m - theory , '' phys . b * 431 * ( 1998 ) 31 [ arxiv : hep - th/9803117 ] . x. leroy , `` le systme caml special light : modules et compilation efficace en caml , '' inria research report nov 1995 , 2721 . url : http://pauillac.inria.fr/~xleroy/publi/caml-special-light-rr.ps.gz ; the ocaml website is http://www.ocaml.org g. w. moore , n. nekrasov and s. shatashvili , `` d - particle bound states and generalized instantons , '' commun . math . * 209 * ( 2000 ) 77 [ arxiv : hep - th/9803265 ] . v. pestun , `` n = 4 sym matrix integrals for almost all simple gauge groups ( except e(7 ) and e(8 ) ) , '' jhep * 0209 * ( 2002 ) 012 [ arxiv : hep - th/0206069 ] . m. porrati and a. rozenberg , `` bound states at threshold in supersymmetric quantum mechanics , '' nucl . b * 515 * ( 1998 ) 184 [ arxiv : hep - th/9708119 ] . s. sethi and m. stern , `` invariance theorems for supersymmetric yang - mills theories , '' adv . theor . math . phys . * 4 * ( 2000 ) 487 [ arxiv : hep - th/0001189 ] . s. sethi and m. stern , phys . b * 398 * ( 1997 ) 47 [ arxiv : hep - th/9607145 ] . a. v. smilga , `` witten index calculation in supersymmetric gauge theory , '' nucl . b * 266 * ( 1986 ) 45 [ yad . fiz . * 42 * ( 1985 ) 728 ] . m. staudacher , `` bulk witten indices and the number of normalizable ground states in supersymmetric quantum mechanics of orthogonal , symplectic and exceptional groups , '' phys . b * 488 * ( 2000 ) 194 [ arxiv : hep - th/0006234 ] . w. taylor , `` m(atrix ) theory : matrix quantum mechanics as a fundamental theory , '' rev . * 73 * ( 2001 ) 419 [ arxiv : hep - th/0101126 ] . c. vafa and e. witten , `` a strong coupling test of s duality , '' nucl . b * 431 * ( 1994 ) 3 [ arxiv : hep - th/9408074 ] . yi , `` witten index and threshold bound states of d - branes , '' nucl . b * 505 * ( 1997 ) 307 [ arxiv : hep - th/9704098 ] . | values for the bulk witten indices for @xmath0 yang - mills integrals for regular simple groups of rank 4 and 5 are calculated by employing the brst deformation technique by moore , nekrasov and shatashvili .
the results can not be reconciled with the double assumption that the number of normalizable ground states is given by certain simple partition functions given by kac and smilga as well as that the corresponding boundary term is always negative .
# 1tr#1 # 1tr#1 # 1res^+_#1 hep - th/0312262 + aei-2003 - 113 + * bulk witten indices from @xmath0 yang mills integrals * * t. fischbacher + * |
You are an expert at summarizing long articles. Proceed to summarize the following text:
a magnitude limited complete census of variable stars in nearby dwarf galaxies allows important contributions to the star formation history of these systems . measurements of some variable stars can supply improved distance determinations for the host galaxies , others will provide important constraints for the population analysis . different classes of variables can further improve the understanding of the star formation history of these system , functioning as tracers of star formation during different epochs . we expect the data set of our long term monitoring program to be especially well suited to study the contents of red long - period variables and to re - investigate the paucity of cepheids with @xmath1 days as reported by sandage & carlson ( 1985 ) . we selected a sample of six local group dwarf irregular galaxies which are visible with the 0.8 m telescope of our institute at mt . the names and additional data from the literature compilation by mateo ( 1998 ) are shown in table 1 . .names , variable star counts , absolute @xmath2-band brightness in mag , and current distance estimation in kpc for the dwarf galaxies observed in our project . the data are taken from the literature compilation by mateo ( 1995 ) . for leo a the data are from the work of dolphin et . al ( 2002 ) and from this work . [ cols="<,<,^,^,^,^,^ " , ] @xmath3 this work the observations so far were carried out in @xmath4 and @xmath2-band , sparsely sampling a three year period starting with test observations in 1999 . this part of the data set should be sensitive for long period variable stars with periods up to @xmath5 days . additional observations in @xmath4 , @xmath2 and @xmath6-band were obtained during 3 observing campaigns at the 1.23 m telescope on calar alto densely sampling three two week long periods . these observations should provide a ground for a search for variable stars with shorter periods ranging from @xmath7 days up to @xmath8 days . the acquired data were bias subtracted , flat - fielded and cosmic ray rejected . then , the images from one night were astrometrically aligned to a common reference frame and combined with individual weights proportional to their @xmath9 . for each epoch , consisting of all the stacked images of a single night , a difference image against a common deep reference frame was created using an implementation ( gssl & riffeser , 2002 , 2003 ) of the alard algorithm ( alard & lupton , 1998 ) . finally , these difference images were convolved with a stellar psf . to extract lightcurves from the reduced data , first all pixels deviating significantly ( @xmath10 ) from the reference image in a minimum number of epochs @xmath11 were flagged , utilizing the complete per - pixel error propagation of our data reduction pipeline . then , using these coordinates as input , values and associated errors are read from the difference images and the lightcurve data are assembled . to search for periodic signals in the extracted difference fluxes , a lomb ( 1976 ) algorithm using the interpretation from scargle ( 1982 ) is applied . the photometric calibration was conducted using the hst data published by schulte - ladbeck et al . for the galaxies leo a , and ugca 92 , we have a very good monitoring and a large fraction of the data passed already the pipeline . the leo a data set serves as test case : a total of 26 variable star candidates were detected . among them , we identified 16 secure long period variables ( typical average values @xmath12 , and @xmath13 period [ days ] @xmath14 ) , and we have 8 further candidates for lpvs . in addition we were able to identify two good candidates for @xmath0 cephei stars with best fitting periods of 6.4 and 1.69 days . the later candidate was previously described by dolphin et al . ( 2002 ) as c2-v58 with a period of 1.4 days . the dolphin et al . period solution fails in deriving a reliable lightcurve with our data , yet , applying our period value to their data set yields reasonable results . the phase convolved lightcurves for the two @xmath0 cephei variables are shown in figure 1 . the color magnitude diagram shown in the left panel of figure 2 is based upon the hst data published by tolstoy et al . ( 1996 ) and schulte - ladbeck et al . flagged by bigger symbols are those variables from our sample that lie inside the hst field of view , two @xmath0 cephei variables in the instability strip ( crosses ) and the candidates for long term variability ( triangles ) in the regime of the red giants . tolstoy et al . ( 1996 ) based on ground - based data found a distance modulus for leo a of 24.2 and a resulting distance of 690 kpc ( see also schulte - ladbeck et al . ) . this result got further support by the search for short periodic variables with the wiyn telescope within 3 consecutive days in dec . 2000 ( dolphin et al . our data complement this dataset for longer periods . the right hand panel of figure 2 shows the period - luminosity ( pl ) relation of the smc shifted to the distance determined by tolstoy et al . the short period variables measured by dolphin coincide with the shown pl relation . the overplotted values for the two cepheids from our survey ( crosses ) support this relation also in the regime of longer periods . we presented preliminary results for our survey for variable stars in a sample of irregular local group dwarf galaxies . for the leo a dwarf galaxy , the best analysed case so far , we already identified a total of 26 candidates for variability , 16 of these as long period variables and 2 @xmath0 cephei stars . we compared the later with the period - luminosity relation and the short period variables discussed by dolphin et al . we found , that our cepheids fully support their findings and the resulting distance estimate for leo a. this result is further in good agreement with the trgb distance ( tolstoy et al . , schulte - ladbeck et al . ) . the location of the lpvs in the color - magnitude diagram indicate that most of them are early asymptotic giant branch stars . while a complete census of these intermediate age stars is missing for most of the local group members , a proper statistic of their appearance can guide the reconstruction of the star formation history at the age of several gyr by - passing the age metalicity degeneracy inherent to color magnitude diagram studies . we like to thank drs . i. drozdovsky , c. maraston , r.e . schulte - ladbeck , and e. tolstoy for helpful discussion . we acknowledge the support of the calar alto and wendelstein staff . j. fliri and a. riffeser carried out some of our observations . the project is supported by the deutsche forschungsgemeinschaft grant ho 1812/3 - 1 and ho 1812/3 - 2 . alard , c. & lupton , r. h. , , 503 , 325 dolphin , a. e. et al . 2002 , , 123 , 3154 gssl c. a. & riffeser a. 2002 , , 381 , 1095 gssl , c. a. & riffeser , a. 2003 , asp conf . 295 , 229 lomb n. r. 1976 , , 39 , 447 mateo m. l. 1998 , , 36 , 435 sandage , a. & carlson , g. 1985 , , 90 , 1464 scargle j. d. 1982 , , 263 , 835 schulte - ladbeck r. et al . 2002 , , 124 , 896 tolstoy e. et al . 1996 , , 116 , 1244 | dwarf galaxies in the local group provide a unique astrophysical laboratory . despite their proximity some of these systems still lack a reliable distance determination as well as studies of their stellar content and star formation history .
we present first results of our survey of variable stars in a sample of six local group dwarf irregular galaxies . taking the leo a dwarf galaxy as an example we describe observational strategies and data reduction .
we discuss the lightcurves of two newly found @xmath0 cephei stars and place them into the context of a previously derived p - l relation .
finally we discuss the lpv content of leo a. |
You are an expert at summarizing long articles. Proceed to summarize the following text:
perhaps the simplest model of noise in a quantum system is that of the isotropic depolarizing channel @xmath2 where with probability probability a quantum state @xmath3 is left untouched while with probability @xmath4 it is mapped to the completely mixed state @xmath5 @xmath6 this channel results in the bloch `` ball '' being compressed isotropically by a factor of . one can imagine a slightly more complicated noise model whereby the noise compresses the bloch `` ball '' anisotropically along axes which are defined by the basis we choose to work in and it is this setting which we investigate below . the amounts by which we compress along each axis are called the * compression coefficients * and these form the components of the * compression vector*. we investigate the geometric properties of the set of all compression vectors and in particular we ask `` when does this set form a simplex ? '' these * generalized depolarizing channels * ( also called * anisotropic depolarizing channels * ) form a broad class of channels which can be realized experimentally ( see for example @xcite ) ; in the single qubit case they include the bit - flip and phase - flip channels . it is worth noting that _ any _ quantum channel ( including the use of a spin chain as a quantum channel @xcite ) can be turned into a generalized depolarizing channel as follows : alice sends one half of a maximally entangled bipartite state down the channel to bob ; the resulting shared state is a mixed bipartite state which is ( excepting very special cases ) not maximally entangled ; alice and bob can now use this shared resource to attempt teleportation @xcite of an unknown @xmath0-dimensional quantum state @xmath3 ; the teleportation protocol then acts as a generalized depolarizing channel on this state @xmath3 @xcite . this work is organized as follows : section [ sec : problem ] gives some precise definitions and describes in detail the problem we solve . sections [ sec : p - basis ] , [ sec : gm - basis ] and [ sec : hw - basis ] present the solution to this problem in the pauli , gell - mann and heisenberg - weyl bases respectively ( we prove the conjecture by dixit and sudarshan in section [ sec : p - basis ] ) whilst section [ sec : other - bases ] gives the general solution in an arbitrary basis . we then discuss changing basis in section [ sec : change - basis ] before showing how to adopt our method of solution to deal with more general channels in section [ sec : more - channels ] . we conclude in section [ sec : conclusions ] . throughout this work we let be a basis for matrices which satisfies the following conditions : 1 . @xmath7 2 . @xmath8 for all @xmath9 3 . @xmath10 for all @xmath11 abusing terminology slightly ( note that ) , we call such a basis * trace - free * and * trace - orthogonal*. we need not restrict ourselves to trace - orthonormal bases as we can simply divide by @xmath12 when necessary to normalize the basis . if @xmath3 is the density matrix of any @xmath0-dimensional quantum system , then we may write @xmath13 where we have chosen the normalization such that * if and only if @xmath3 is a pure state * if and only if @xmath3 is a mixed state where the norm of @xmath14 is defined as . for notational convenience we add an extra @xmath15 component , , which does not affect the value of @xmath16 . we call the * polarization vector * and @xmath17 the * polarization coefficients * of the state @xmath3 * with respect to the basis @xmath18*. the * bloch `` ball '' * is the set of all polarization vectors corresponding to quantum states . it is important to note that except for the single qubit case the bloch `` ball '' is not the ball of unit radius , but rather a convex subset of this ball ( this is because for some vectors lying within the unit ball are not valid polarization vectors as they do not correspond to positive states ) . we can now define a * ( generalized ) depolarizing channel with respect to the basis @xmath18 * to be a map @xmath19 which satisfies 1 . @xmath19 is a trace - preserving completely positive map 2 . @xmath19 compresses the bloch `` ball '' in the following manner : @xmath20 we call the * compression vector * of the channel @xmath19 as it specifies the amount by which @xmath19 compresses the bloch `` ball '' along each axis ; the @xmath21 are called * compression coefficients*. again , for notational convenience , we have added a @xmath15 component ( @xmath22 is the compression coefficient for , so ensures that @xmath19 is trace - preserving ) . note that for all @xmath23 . to see this , let the largest possible magnitude of the polarization coefficient @xmath17 be and let @xmath24 be a state with ( so @xmath24 lies on the boundary of the bloch `` ball '' ) . if then @xmath25 is not a state ( it lies outside the `` bloch ball '' ) , so as claimed . it is important to note that the notion of generalized depolarizing channels is highly basis dependent : we _ must _ define such channels with respect to a given basis . we make the following observation * if then hermiticity of @xmath3 tells us that ( where @xmath26is the complex conjugate of @xmath27 ) . furthermore , since @xmath28 must be a quantum state ( and is therefore hermitian ) , . and note its special cases : * if then and . ( note that @xmath21 can be negative . ) * if @xmath29 is hermitian ( ) then and . * if @xmath29 is skew - hermitian ( ) then @xmath17 is pure imaginary and . if we work in a fixed basis it is clear that for each depolarizing channel there is a unique compression vector . it is natural therefore to ask the question `` which vectors @xmath30 are valid compression vectors corresponding to depolarizing channels @xmath19 ? '' from the definition of a depolarizing channel given above it is clear that the answer to this question is `` a vector @xmath30 is a valid compression vector when the induced map @xmath19 is completely positive and ( @xmath19 is trace - preserving ) '' . it is clear from above that all compression vectors @xmath30 which induce depolarizing channels @xmath19 lie within the finite region . however we will see that in general the converse is not true : not all vectors in @xmath31 induce depolarizing channels . to help us decide which induced maps @xmath19 are completely positive we employ the * choi - jamiolkowski representation * @xmath32 where is the computational basis of our @xmath0-dimensional quantum system . we will make use of the following theorem and lemma [ thm : cj ] a map @xmath33 is completely positive if and only if @xmath34 is positive . see @xcite . [ lem : trace - cj ] if where @xmath35 and @xmath36 are operators ( i.e. matrices ) then . we can write . then @xmath37 since the basis @xmath18 is trace - orthogonal , we may write @xmath38 and also @xmath39 since @xmath40 is just a constant , a simple application of lemma [ lem : trace - cj ] allows us to calculate the choi - jamiolkowski representation of the channel @xmath19 @xmath41 using theorem [ thm : cj ] we see that @xmath30 is a compression vector corresponding to a completely positive depolarizing channel @xmath19 when the eigenvalues of @xmath34 are all non - negative . in the following sections we work in different bases and determine which vectors @xmath30 are compression vectors . in this section we restrict ourselves to a multiple qubit setting ( ) . we choose the basis @xmath18 to be formed of tensor products of the single qubit pauli matrices : let @xmath23 be the number whose base-@xmath42 representation is @xmath43 then we define the basis matrices @xmath29 by @xmath44 where are the pauli spin matrices . this basis is trace - free and trace - orthogonal . note that each @xmath29 is hermitian and so all @xmath17 and @xmath21 are real . in this case it is a simple matter to find the eigenvectors and eigenvalues of @xmath34 . since @xmath34 is an matrix ( it is a _ superoperator _ ) , we can think of it as being an operator on @xmath45 qubits ( as ) . let @xmath46 ( where ) be bell states on qubits @xmath47 and . ( note : in more usual notation we have , , and ) . observe @xmath48 which may be summarized as @xmath49 where @xmath50 it is now a straight forward matter to check that the eigenvectors of the choi - jamiolkowski representation of an induced channel @xmath19 are @xmath51 with corresponding eigenvalues @xmath52 the important thing to notice here is that each @xmath53 is a linear combination of the compression coefficients @xmath21 . since there are different eigenvectors we have found all the eigenvectors and eigenvalues of @xmath34 . we have therefore shown which vectors @xmath30 induce depolarizing channels @xmath19 namely those for which all eigenvalues of the choi - jamiolkowski representation of @xmath19 are non - negative : for all @xmath54 and @xmath55 . we are now in a position to prove a conjecture of dixit and sudarshan @xcite , which we present as the following theorem . [ thm : dix - sud ] when and we work in the pauli basis , the set of all compression vectors forms a simplex in compression space . since each @xmath53 is linear in the compression coefficients @xmath21 then the equation defines a hyperplane in compression space ( which is a real euclidean vector space of dimension ; it has one dimension for each component of the compression vector excepting the @xmath15 component which we suppress as it is identically equal to @xmath56 ) . since @xmath19 is completely positive if and only if all eigenvalues @xmath53 are non - negative , the hyperplanes must enclose precisely the set of of vectors which induce completely positive depolarizing channels @xmath19 . in particular the hyperplanes enclose a finite region of compression space and the shape of this enclosed region must therefore be a simplex . we now prove a small lemma before finding the of the simplex ( which are the depolarizing channels whose compression vectors form the vertices of the simplex ) . the eigenvalues of the choi - jamiolkowski representation of @xmath19 sum to the system dimension , @xmath0 . first recall that the sum of the eigenvalues of a matrix is simply the trace of that matrix . then @xmath57 the extremal channels are @xmath58 first note the identity @xmath59 where @xmath60 we now fix @xmath61 and prove that @xmath62 is an extremal channel . first note that @xmath62 is completely positive ( see for example @xcite ) and apply the above identity to see that @xmath63 it is clear then that @xmath62 is a depolarizing channel whose compression vector has components @xmath64 by combining equations ( [ eqn : lambda - gm ] ) and ( [ eqn : phi - g ] ) we see that @xmath65 it is clear that if , for fixed @xmath61 and for all @xmath23 , @xmath66 then . we now show that there exist @xmath67 and @xmath68 such that ( [ eqn : fg - condition ] ) holds : * when ( i.e. and for all ) then @xmath69 and so ( [ eqn : fg - condition ] ) holds when @xmath70 * when ( i.e. and for all ) then @xmath71 and so ( [ eqn : fg - condition ] ) holds when @xmath72 * when ( i.e. and for all ) then @xmath73 and so ( [ eqn : fg - condition ] ) holds when @xmath74 and @xmath75 are picked as in ( [ eqn : q_r ] ) and ( [ eqn : p_r ] ) above . we have now shown that there exist @xmath67 and @xmath68 with . since @xmath62 is completely positive then all the eigenvalues of @xmath76 are non - negative ; they sum to @xmath0 and we have found one which is equal to @xmath0 ; therefore all other eigenvalues are zero @xmath77 clearly then the compression vector @xmath30 of the map @xmath62 lies on all the hyperplanes except , and so @xmath62 must be an extremal channel . to finish the proof note that there are @xmath78 vertices of the simplex and there are @xmath78 channels of the form @xmath62 so we have found all the extremal channels . it is worth pointing out that any compression vector in the simplex can be written as a convex linear combination of the * extremal compression vectors * ( that is , the compression vectors which form the vertices of the simplex ) . this implies that any depolarizing channel can be written as a convex linear combination of the extremal channels @xmath79 conversely , any channel of this form is a depolarizing channel . note the following relationship @xmath80 for a single qubit ( ) the compression space has dimension 3 and we see that for a general channel @xmath19 @xmath81 and so we have the following correspondence between extremal channels and compression vectors ( we suppress the @xmath15 component of @xmath82 which is always equal to @xmath56 ) @xmath83 the geometry of these single qubit depolarizing channels is illustrated in figure [ fig : qubit - tetrahedron ] in the previous section we restricted the dimension of the quantum system to be so that we could employ the pauli basis for multiple qubits . in this section we choose one possible generalization of the pauli basis , namely the gell - mann basis , which allows us to study systems with arbitrary dimension . the * gell - mann * basis is defined to be @xmath84 where , and . for notational consistency we identify @xmath85 i.e. . note that this basis is trace - free and trace - orthogonal ( and reduces to the pauli basis when ) . furthermore each @xmath29 is hermitian and so all @xmath17 and @xmath21 are real . we now attempt to find the eigenvectors and eigenvalues of @xmath86 and begin by defining @xmath87 where and . then it is a simple matter to check that @xmath88 it is clear therefore that @xmath89 are eigenvectors of @xmath86 and that the corresponding eigenvalues @xmath90 are linear in the compression coefficients @xmath21 . now , @xmath86 has @xmath78 eigenvalues and we have found of them ; let the remaining @xmath0 eigenvalues be @xmath91 ( ) . in order to establish which vectors in compression space induce depolarizing channels we must now find when the remaining @xmath0 eigenvalues of @xmath92 are non - negative . since is symmetric for all @xmath23 , @xmath86 is also symmetric and consequently has real eigenvalues . by considering the matrix elements of @xmath86 carefully we see that the only non - zero entries in the two columns indexed by @xmath93 and @xmath94 ( with ) are in the two rows indexed by @xmath95 and @xmath96 . we may conjugate @xmath86 by a permutation matrix @xmath97 ( to permute the rows and columns ) to form a matrix @xmath98 which has identical eigenvalues to @xmath86 . by repeatedly conjugating by permutation matrices we can block - diagonalize @xmath86 to obtain a matrix @xmath99 which has the following structure @xmath100 & & & \\ & \ddots & & \\ & & [ 2 \times 2 ] & \\ & & & [ k(\phi_{\textbf{\textit{v } } } ) ] \\ \end{array } \right)\ ] ] there are blocks of size ( each of which has two eigenvalues , @xmath101 and @xmath102 , for some @xmath47 and @xmath103 ) and a large block which we call @xmath104 of size ( which has eigenvalues ) . by considering the characteristic equation of @xmath104 we see that @xmath105 where we have defined the * eigenvalue sums * @xmath106 to be @xmath107 the following lemma proves that all the eigenvalues are non - negative precisely when all the eigenvalue sums are non - negative . for all if and only if for all . one way is trivial : if @xmath108 then @xmath109 . we prove the converse by contradiction : assume that for all , recall that and note that for any @xmath110 ( this is essentially an inclusion - exclusion argument ) . but then we may split this sum up into two terms @xmath111 it is clear that these sums over @xmath103 odd and @xmath103 even are both non - negative since each @xmath106 is non - negative and @xmath91 appears to an even power in each term . but then implies that also , which contradicts the assumption that , and so . since this argument holds for all we are done . returning to @xmath86 we see from equation ( [ eqn : j(phi ) ] ) that each matrix element consists of a linear combination of the compression coefficients @xmath21 a property which is inherited by @xmath104 . careful consideration reveals that @xmath106 is a @xmath112-order polynomial in the compression coefficients @xmath113 it is tempting to conclude that is a @xmath112-order surface in compression space . however we must be careful : it is possible that all @xmath112-order coefficients @xmath114 ( i.e. those with for all @xmath103 ) are equal to zero , in which case the degree of the surface is strictly less than @xmath47 . for example , when the gell - mann basis reduces to the pauli basis and we have already seen in the previous section that all eigenvalues are linear in the compression coefficients ; in this case the surface is a plane and not a quadratic surface . in general then , we may only conclude that is a surface of order at most @xmath47 . we have now proved that the set of all vectors @xmath30 which induce depolarizing channels @xmath19 ( with respect to the gell - mann basis ) form a finite region in compression space which is bounded by * hyperplanes ( are given by and the remaining one is given by ) * a surface with order at most @xmath115 ( ) * * a surface with order at most @xmath0 ( ) note that the above does not tell us whether the eigenvalues are linear in the compression coefficients . it turns out that when at least one of the @xmath91 must be non - linear in the compression coefficients a fact which is proved in lemma [ lem : gm - curves ] in section [ sec : other - bases ] . whilst it is a slight disappointment that we have been unable to explicitly give expressions for all the eigenvalues of @xmath86 we can draw some solace from the fact that we have simplified the problem somewhat : in order to see if a vector @xmath30 induces a depolarizing channel we must check to see if all the eigenvalues of the choi - jamiolkowski representation @xmath86 of the induced map @xmath19 are non - negative . without the above results we would have to use a `` brute - force '' algorithm to calculate the @xmath78 eigenvalues of @xmath86 directly ; however the above results enable us to calculate of these quickly , leaving only @xmath0 to be calculated by the brute - force algorithm , which is a substantial speed - up . in this section we work in the heisenberg - weyl basis which is an alternative generalization of the single - qubit pauli basis to arbitrary dimension . it has certain advantages over the gell - mann basis ( for example , all the heisenberg - weyl basis matrices are invertible while most of the gell - mann basis matrices are singular ) . however there is a price to pay for such convenience which is that the heisenberg - weyl basis is not hermitian and so the compression space is a complex vector space . let us first define @xmath116 where is the primitive @xmath117-root of unity and we work modulo @xmath0 . note that for @xmath118 and that the inverse of @xmath119 is @xmath120 and similarly for @xmath121 . we now define the * heisenberg - weyl * basis to be @xmath122 note that this basis is trace - free and trace - orthogonal . we can now find the eigenvectors and eigenvalues of @xmath86 by defining @xmath123 then it is a simple matter to check that @xmath124 which shows us that @xmath125 are eigenvectors of @xmath86 . furthermore , there are @xmath78 distinct @xmath125 and we have therefore found all the eigenvectors of @xmath86 . relabeling the compression coefficients @xmath21 as @xmath126 ( with ) we see that the eigenvalues of @xmath86 are @xmath127 it is easy to check that and also . therefore , by the observation in the introduction , . in the preceding sections we have had one ( real ) compression coefficient for each basis matrix . in this section we have seen that most of the compression coefficients come in complex - conjugate pairs corresponding to two basis matrices which are , up to a complex multiplicative factor , hermitian conjugates of each other . note that we still have the same number of free compression parameters ( since for each pair of conjugate basis matrices there are two free parameters in the associated compression coefficient , namely the real and imaginary parts ) ; and so the compression space ( although now complex ) still has dimension ( recall that we suppress the dimension corresponding to @xmath22 as it is identically equal to @xmath56 ) . as an aside , we find the exact structure of the compression space . we use to see that precisely when and . when @xmath0 is odd the only solution is and ( i.e. ) . in this case compression space consists complex planes . when @xmath0 is even there are four solutions , namely . in this case , compression space consists of @xmath128 real axes and complex planes . we now prove that , when working in the heisenberg - weyl basis , the set of all compression vectors forms a simplex . we then find the extremal channels whose compression vectors lie at the vertices of this simplex . when we work in the heisenberg - weyl basis , the set of all compression vectors forms a simplex in compression space . it is evident from equation ( [ eqn : lambda - hw ] ) that the eigenvalues of @xmath86 are linear in the compression coefficients and so defines a hyperplane in compression space . since @xmath19 is completely positive if and only if all eigenvalues @xmath129 are non - negative , the hyperplanes must enclose precisely the set of vectors which induce completely positive depolarizing channels @xmath19 . in particular the hyperplanes enclose a finite region of compression space and the shape of this enclosed region must therefore be a simplex . the extremal channels are @xmath130 we fix @xmath47 and @xmath103 and prove that @xmath131 is extremal . using the identity we see @xmath132 and so is a depolarizing channel with compression coefficients . we now show that there exists @xmath4 and @xmath133 with . @xmath134 so picking and ensures that . as before we make use of the fact that the eigenvalues of the choi - jamiolkowski representation @xmath135 sum to the system dimension @xmath0 . clearly @xmath131 is completely positive and so the eigenvalues of @xmath135 are non - negative and sum to @xmath0 . clearly then @xmath136 therefore the compression vector corresponding to the channel @xmath131 lies on all the hyperplanes except one and so it lies at a vertex of the simplex . to finish note that there are @xmath78 vertices and @xmath78 distinct extremal channels , so we have found them all . as before ( when we were working in the pauli - basis ) we note that any compression vector in the simplex can be written as a convex linear combination of the extremal compression vectors ; therefore any depolarizing channel can be written as a convex linear combination of the extremal channels and vice - versa . having worked in specific bases in the previous sections we now work in an arbitrary trace - free , trace - orthogonal basis @xmath18 . we have seen that the set of all compression vectors is a simplex precisely when the eigenvalues of @xmath86 are linear combinations of the compression coefficients @xmath21 . we now find all bases for which this happens . [ thm : simplex ] let @xmath18 be a trace - free , trace - orthogonal basis . then the set of all compression vectors forms a simplex if and only if @xmath137 = 0 \quad \forall \alpha,\beta\in\{0,\ldots , n^2 - 1\}\ ] ] the set of all compression vectors forms a simplex if and only if all the eigenvalues of @xmath86 are linear in the compression coefficients ( in which case if @xmath138 is an eigenvalue of @xmath86 then defines a hyperplane which forms one of the sides of the simplex ) . let @xmath139 be an eigenvector of @xmath86 with eigenvalue @xmath138 . then by equation ( [ eqn : j(phi ) ] ) we see that that @xmath138 is a linear combination of the compression coefficients @xmath21 if and only if @xmath139 is a simultaneous eigenvector of each ; this occurs if and only if all the are simultaneously diagonalizable , which occurs if and only if for all @xmath23 and @xmath61 . note that the condition of theorem [ thm : simplex ] is weaker than for all @xmath23 and @xmath61 . for example , satisfies @xmath140=0 $ ] but not . furthermore , it is not hard to show that @xmath141 is equivalent to . we can now prove the outstanding result from section [ sec : gm - basis ] , which we present as the following lemma . [ lem : gm - curves ] when and we work in the gell - mann basis then the set of all compression vectors has at least one curved side . equivalently , at least one of the eigenvalues of @xmath86 is non - linear in the compression coefficients @xmath21 . it is a simple matter to check that @xmath142 \neq 0\ ] ] then apply theorem [ thm : simplex ] above . for the remainder of this section we restrict ourselves to working in a basis which is unitary : for all @xmath23 . when we do this we are able to find the extremal channels whose compression vectors lie at the vertices of the simplex . before proving this we first define @xmath143 note that @xmath144 is a maximally entangled state and so the @xmath145 are too ( as the @xmath29 are unitary ) . if for all @xmath23 and @xmath61 and if @xmath29 is unitary for all @xmath23 then @xmath145 are all the eigenstates of @xmath86 . note that @xmath146 . note also that unitarity of @xmath29 ensures that . then @xmath147 and it is easy to see that @xmath148 so @xmath145 is indeed an eigenstate of @xmath86 . finally note that there are @xmath78 eigenstates @xmath145 so we have found all the eigenstates of @xmath86 . the technical lemma above allows us to find the extremal channels of the simplex : if for all @xmath23 and @xmath61 and if @xmath29 are all unitary then the extremal channels are @xmath149 first note that @xmath150 is completely positive and trace - preserving , so the eigenvalues of @xmath151 are all positive and sum to @xmath0 . we can expand @xmath152 which shows that @xmath153 is a depolarizing channel with compression coefficients . now observe that @xmath154 and so for all . clearly then @xmath153 is an extremal channel as its compression vector lies on all the hyperplanes except . to finish , note that there are @xmath78 vertices to the simplex and there are @xmath78 extremal channels @xmath153 so we have found them all . we now investigate the effect of changing basis , which will enable us to see why in some bases the set of all compression vectors forms a simplex whilst in other bases it does not . let and be trace - free , trace - orthogonal bases . then we may write @xmath155 trace - orthogonality of both bases ensures that the change - of - basis matrix is unitary . furthermore if both bases are hermitian then @xmath156 is actually a real orthogonal matrix . if we define @xmath157 and similarly for @xmath158 , then we may write . we are now in a position to prove [ thm : qubit - simplices ] the set of single qubit ( ) compression vectors forms a simplex whenever we work in a hermitian , trace - free and trace - orthogonal basis . furthermore , the extremal channels are conjugations by the basis matrices . we know that any single qubit basis which is hermitian , trace - free and trace - orthogonal can be obtained from the pauli basis by an orthogonal change of basis . let be the pauli basis and be the new basis . then the change of basis matrix has the form @xmath159 where @xmath160 is a orthogonal matrix . now , any such matrix @xmath160 corresponds to a rotation of the bloch ball and is equivalent to a unitary conjugation where @xmath161 is a unitary matrix . any such map is completely positive and trace - preserving . depolarization with respect to the basis is the same as applying the reverse of the above rotation , followed by depolarization in the pauli basis , followed by the the above rotation . since the set of all compression vectors forms a simplex when we work in the pauli basis , the set of all compression vectors also forms a simplex when we work in the new basis . we know that in the pauli basis the extremal channels are . so when we start in the new basis and rotate to the pauli basis , the extremal channels will be ; rotating back to the basis ( and noting that ) we see that the extremal channels are . in the single qubit case it is true that any unitary change of basis @xmath156 corresponds to an orthogonal rotation of the polarization vector @xmath14 ; in higher dimensions this is not so ass the following example demonstrates [ ex : bloch - rotation ] let and let @xmath162 where is the single - qubit pauli-@xmath163 matrix . one can explicitly calculate the eigenvalues of @xmath3 ; they are each with multiplicity two , so @xmath3 is a state when . let us now apply any change of basis which maps ( that is , we change basis from the hermitian pauli basis to the hermitian gell - mann basis with a unitary change - of - basis matrix @xmath156 ) . then @xmath164 which has eigenvalues ( with multiplicity 3 ) and ( with multiplicity 1 ) . but then if then @xmath3 is a quantum state but @xmath165 is not ( as it is not positive ) . this example demonstrates that some changes of basis do not map the bloch `` ball '' on to itself . it is for this reason that the set of all compression vectors forms a simplex in some bases but not in others . when two trace - free , trace - orthogonal bases and are related via @xmath166 for all @xmath23 and some unitary matrix @xmath161 _ ( that is , the change - of - basis matrix @xmath156 maps the bloch `` ball '' on to itself ) _ , then the set of compression vectors either forms a simplex in both bases or does not form a simplex in either basis . furthermore , if the extremal channels are conjugations of basis matrices in one basis then they are conjugations of basis matrices in the other basis too . one may adapt the proof of theorem [ thm : qubit - simplices ] . ( note that the condition for all @xmath23 guarantees that the `` bloch ball '' is rotated onto itself and we avoid the problems exhibited in example [ ex : bloch - rotation ] . ) throughout the previous sections of this work we have studied depolarizing channels . our method can be extended to deal with a more general class of channels , namely those which both compress the bloch `` ball '' and then translate it . let @xmath167 be such a channel defined by its action on the polarization vector @xmath168 we call the vector the * translation vector * and the @xmath169 the * translation coefficients*. note that to ensure that @xmath167 is trace - preserving . so far we have only studied channels , but we now extend to general @xmath170 . we may expand @xmath171 if we now define @xmath172 for all @xmath23 ; and @xmath173 for and then @xmath174 recalling the fact that if a channel @xmath175 has the expression then the choi - jamiolkowski representation is we see that @xmath176 which enables us to find the choi - jamiolkowski representation of the channel @xmath167 @xmath177 in order to find which parameters @xmath178 induce completely positive channels one has to find for which parameters all the eigenvalues of @xmath179 are non - negative . ( recall that these channels are automatically trace - preserving if and ) . let us work in the pauli basis and consider a single - qubit channel @xmath167 . we fix the translation vector and find the set of all compression vectors @xmath30 . now , @xmath180 it turns out that for non - zero @xmath181 the eigenvectors of @xmath179 are no longer bell - states , but rather linear combinations of them . let @xmath182 where are real numbers , and @xmath183 ( ) are the bell - states defined in section [ sec : p - basis ] . one can easily check that @xmath184 where the eigenvalues satisfy the following relations @xmath185 which can be solved to give @xmath186 note that there are two possible values for @xmath187 and @xmath188 : one for and one for ; we have therefore found all four eigenvalues of @xmath179 . clearly for the surfaces and are curved . this reproduces the results of @xcite , which also shows that the set of all compression vectors forms an `` asymmetrically rounded tetrahedron '' . we could have worked through the above example with @xmath189 or @xmath190 non - zero but then the generic form of the eigenvectors of @xmath179 would be more complicated : with the coefficients satisfying . based on the above example we note that even when we work in a basis where the set of all compression vectors forms a simplex for , a tiny perturbation to destroys this simplex : the set of compression vectors @xmath30 forms a set with a curved boundary ; in this situation there are infinitely many extremal channels . in summary , we defined _ trace - free , trace - orthogonal bases _ for quantum states and used these to define _ depolarizing channels_. each depolarizing channel has an associated _ compression vector _ which lies in _ compression space _ which is an dimensional vector space ( we suppressed the other dimension as the first component of the compression vector is always equal to unity ) . we showed that the set of all compression vectors forms a simplex when we work in either the _ pauli basis _ or the _ heisenberg - weyl basis _ , but that it does not form a simplex when we work in the _ gell - mann _ basis . furthermore , in the pauli and heisenberg - weyl bases we found the extremal channels whose compression vectors lie at the vertices of the corresponding simplex . working in a general trace - free , trace - orthogonal basis , we showed precisely for which bases the set of compression vectors forms a simplex . when the basis matrices were all unitary we were able to find the extremal channels . we discussed the effects of changing basis and this indicated why the set of compression vectors forms a simplex in some bases but not in others . finally we generalized our methods to deal with a class of more general quantum channels . i would now like to recall a theorem proved in @xcite but first i must define a * doubly stochastic * quantum channel acting on an @xmath0-dimensional quantum system to be any completely - positive , trace - preserving channel which leaves the completely mixed state untouched : . let us now define @xmath191 to be the set of all doubly stochastic channels which act on an @xmath0-dimensional quantum system . as @xmath191 is a convex set we may define @xmath192 to be the set of extremal channels in @xmath191 . it is known that all doubly stochastic single - qubit channels are mixed unitary , but that in higher dimensions ( ) there are doubly stochastic channels which are not mixed unitary . ( they can however be expressed as an affine sum of unitary channels where are affine parameters : and . ) in particular , when and we work in any unitary basis for which the set of compression vectors forms a simplex , there exist doubly stochastic channels which do not lie within the simplex and are therefore not depolarizing channels . however , the following theorem ( proved in @xcite ) tells us that all such channels when appropriately averaged with the isotropic completely depolarizing channel @xmath194 , become mixed unitary . let @xmath175 be any doubly - stochastic quantum channel acting on @xmath0-dimensional quantum systems and let @xmath33 be any depolarizing channel whose compression vector lies within a simplex whose vertices are the compression vectors of unitary conjugation depolarizing channels . then there exists a constant such that for the following channel is mixed unitary @xmath197 | a generalized depolarizing channel acts on an @xmath0-dimensional quantum system to compress the `` bloch ball '' in directions ; it has a corresponding compression vector .
we investigate the geometry of these compression vectors and prove a conjecture of dixit and sudarshan @xcite , namely that when ( i.e. the system consists of @xmath1 qubits ) and we work in the pauli basis then the set of all compression vectors forms a simplex .
we extend this result by investigating the geometry in other bases ; in particular we find precisely when the set of all compression vectors forms a simplex . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
radiative decays of vector mesons have traditionally been a good laboratory for various tests of the quark model and su(3 ) symmetry @xcite . a recent discovery of the @xmath5 decay by the cmd-2 group @xcite has been the last link in the otherwise complete picture of radiative magnetic dipole transitions between light vector and pseudoscalar mesons . this observation was later confirmed by the snd group @xcite . both experiments suffered from a low number of observed events , resulting in large uncertainties in the determined branching ratio and making comparison to theory difficult . in this paper we report on the improved measurement of the rate of the @xmath5 decay based upon the total data sample accumulated with cmd-2 in the @xmath6-meson energy range . it includes 3.1 pb@xmath1 of data collected in 1992 1996 in our first measurement which used only photons observed in the csi barrel calorimeter , and about 11.4 pb@xmath1 collected in 1997 1998 . in addition , this analysis uses photons detected in either the csi barrel or the bgo endcap calorimeters for both data samples providing better detection efficiency than before . the general purpose detector cmd-2 operating at the high luminosity @xmath7 collider vepp-2 m in novosibirsk has been described in detail elsewhere @xcite . it consists of a drift chamber and proportional z - chamber used for trigger , both inside a thin ( 0.4 @xmath8 ) superconducting solenoid with a field of 1 t. the barrel calorimeter placed outside the solenoid consists of 892 csi crystals of @xmath9 @xmath10 size and covers polar angles from @xmath11 to @xmath12 . the energy resolution for photons is about 9% in the energy range from 50 to 600 mev . the end - cap calorimeter placed inside the solenoid consists of 680 bgo crystals of @xmath13 @xmath10 size and covers forward - backward polar angles from 16@xmath14 to 49@xmath14 and from 131@xmath14 to 164@xmath14 . the energy and angular resolution are equal to @xmath15 and @xmath16 radians respectively . the luminosity was determined from the detected @xmath17 events @xcite . since @xmath0 is a two - body decay and @xmath18 is a narrow state , the momentum of the recoil photon is fixed and approximately equals 60 mev . to study this decay we searched for the decay chain @xmath19 , @xmath20 . the photons are ordered by decreasing energy ( @xmath21 ) . in these events the softest photon must be a monochromatic recoil photon with the energy @xmath22 mev at the @xmath6 meson peak , while the energies of the harder ones range from 170 to 440 mev . the invariant mass of the two harder photons @xmath23 . the main source of background for this study is the decay mode @xmath24 giving the same final state with two charged pions and three photons via the decay chain @xmath25 , @xmath26 . here the hardest photon is monochromatic with @xmath27 mev and the invariant mass of two others is @xmath28 . this decay can be used as a monitoring process and the branching ratio @xmath29 will be calculated relative to @xmath30 . due to similar kinematics and detection efficiency dependence on detector parameters some systematic errors will cancel in such a ratio . events with two tracks and three photons were selected using the following criteria : * one vertex is found in the event * two tracks with opposite charges are reconstructed from this vertex and there are no other tracks * the angles of both tracks with respect to the beam are limited by @xmath31 to match the optimal drift chamber coverage * the number of photons detected in the csi and bgo calorimeters is three . the cluster in the calorimeter is accepted as a photon when it does not match any charged track and its energy is more than 30 mev in the csi calorimeter or more than 40 mev in the bgo calorimeter . * the distance from each track to the beam @xmath32 cm * the distance from the vertex to the interaction point along the beam direction @xmath33 cm * the space angle between the tracks @xmath34 * the angle between the tracks in the r-@xmath35 plane @xmath36 * the total energy of the charged particles ( assuming that both particles are charged pions ) @xmath37 mev . the events thus selected were subject to the kinematical reconstruction assuming energy - momentum conservation . events with good quality of the reconstruction were selected by the following criteria : * @xmath38 * the ratio of the photon energy measured in the calorimeter @xmath39 to that from the constrained fit @xmath40 is @xmath41 * @xmath42 mev vs hardest photon energy @xmath43 . a ) simulation of @xmath44 ; b ) simulation of @xmath45 at the @xmath6-meson energy ; c ) simulation of @xmath46 ; d ) experimental data.,scaledwidth=80.0% ] events surviving after all above criteria mostly come from the process @xmath47 , @xmath48 and @xmath45 , as illustrated by fig . [ fig : w1m23 ] showing the scatter plot of the invariant mass @xmath49 versus the hardest photon energy @xmath50 . the data are shown in fig . [ fig : w1m23]d . the region around @xmath51 mev and @xmath27 mev is densely populated with @xmath52 events . simulated events of this process are presented in fig . [ fig : w1m23]a . to determine the number of @xmath47 events we count the number of events inside the ellipse - like region : @xmath53 for our data this number is @xmath54 . determination of the number of @xmath55 events for simulation gives the detection efficiency @xmath56 . figure [ fig : w1m23]b presents the simulation of @xmath45 , where a densely populated region is also observed at large values of @xmath50 . comparison of these distributions with that for the data ( fig . [ fig : w1m23]d ) confirms that the dominant contribution to selected events comes from these two processes . the same distribution for the simulation of the process under study is shown in fig . [ fig : w1m23]c . for a ) simulation of @xmath44 ; b ) simulation of @xmath45 at the @xmath6-meson energy ; c ) simulation of @xmath46.,scaledwidth=80.0% ] to search for the rare decay @xmath57 we need to suppress the events from @xmath47 and @xmath58 . to this end a cut on the energy of the hardest photon is applied : @xmath59 mev . the @xmath50 distributions for the simulation of @xmath57 and background processes are shown in fig . [ fig : wr1 ] . vs softest photon energy @xmath60 . points present the simulation of @xmath61 , triangles data after all the selections.,scaledwidth=80.0% ] [ fig : finsum ] although this cut causes a decrease of efficiency for the @xmath62 decay ( see fig . [ fig : wr1]c ) , the suppression of the background processes is rather good . one more cut suppressing the background from the @xmath63 and @xmath64 decays is : @xmath65 mev . after all the cuts the scatter plot of the invariant masses for two hardest photons @xmath66 versus the weakest photon energy @xmath67 was studied . figure [ fig : finsum ] presents the data ( black triangles ) together with simulation of @xmath0 ( points ) . the simulation points show the region of the plot which should be populated by the events of @xmath57 and experimental points are densely covering this region . the lower part of the figure contains obvious background events which can be suppressed by imposing the additional cut @xmath68 mev . to determine the number of events the one - dimensional distribution of @xmath69 ( projection of the plot in fig . [ fig : finsum ] to the axis perpendicular to the correlation line ) was studied . such projection is shown in fig . [ fig : etpr]c for the data . the same projection for 10000 simulated events of @xmath70 is shown in fig . [ fig : etpr]b , and the fit of this distribution fixes the signal shape and gives the detection efficiency @xmath71 . the background distribution in this parameter determined from the data before applying the last two cuts ( @xmath59 mev and @xmath65 mev ) is shown in fig . [ fig : etpr]a . the fit of this distribution fixes the background shape . finally , the data were fit using the background shape fixed from fig . [ fig : etpr]a together with that of the signal from simulation in fig . [ fig : etpr]b . the result of the fit is @xmath72 . using the number of events from the fit , one can calculate the relative branching ratio : @xmath73 + where the values of the branching ratios of @xmath74 and @xmath75 were taken from @xcite . a separate analysis of the normalizing decay @xmath24 has recently been published @xcite , with a branching ratio of @xmath76 consistent with previous measurements @xcite and thus giving confidence in the analysis presented here . in the above calculation of the relative branching ratio common systematic errors such as luminosity determination cancel exactly , while others such as detector inefficiency and evaluation of radiative corrections cancel approximately . finally , using the value of @xmath77 from @xcite , one obtains : @xmath78 . the last error is our estimate of the systematic uncertainty . the sources of systematic errors are the following : * uncertainties in the ratio @xmath79 caused by different energy spectra for final photons and pions - 10% ; * uncertainties in the branching ratios @xmath80 , @xmath81 , @xmath82 and @xmath83 - 6.3% ; * determination of the background shape 5% ; * different resonance shape caused by different energy dependence of the phase space - 2% ; the total systematic error obtained by adding separate contributions quadratically is 13% . the results of our analysis have higher statistical significance than before since they are based on a data sample of 21 selected events compared to 6 events in our previous work @xcite and 5 events observed by snd @xcite . the obtained value of the branching ratio @xmath84 @xmath85 . + agrees with our previous result based on part of the whole data sample @xcite @xmath86 + as well as with the result of the snd group @xcite @xmath87 + and is more precise . within experimental accuracy it is also consistent with the preliminary result of cmd-2 based on other decay modes of the @xmath88 meson ( @xmath89 ) with four charged pions and two or more photons in the final state @xcite : @xmath90 . analysis of the available data sample of the produced @xmath6 mesons by both cmd-2 and snd and full use of other decay modes of the @xmath18 and @xmath88 mesons will further improve the statistical error . much larger increase can be expected from the da@xmath91ne @xmath6-factory where one plans to accumulate the number of @xmath6 mesons by at least two orders of magnitude higher than ours . let us briefly discuss theoretical predictions for the decay under study . usual methods of the description of radiative decays are based on the nonrelativistic quark model @xcite . various ways of incorporating effects of su(3 ) breaking have been suggested leading to the values of @xmath29 in the range @xmath92 @xcite . the value of the branching ratio studied in our work is also of interest for the problem of @xmath93 mixing which has been a subject of intense investigation for a long time @xcite . it is sensitive to the structure of the @xmath18 wave function or , in other words , to the contribution of various @xmath94 states as well as the possible admixture of glue in it @xcite . according to @xcite , a branching ratio @xmath95 would indicate a substantial glue component in the @xmath18 , while the expected branching ratio is less than @xmath96 for a pure gluonium . even smaller values were obtained in @xcite assuming a specific model of qcd violation . the revival of interest to the problem of the @xmath18 structure and possible contents of glue in it ( see @xcite and references therein ) was partially due to two recent observations by cleo involving the @xmath18 meson : in @xcite it was shown that the transition form factor of the @xmath18 studied in the two photon processes strongly differs from those for the @xmath75 and @xmath88 mesons and in @xcite the unexpectedly high magnitude of the rate of @xmath97 was observed . however , in a recent paper @xcite it is claimed that it is impossible to disentangle the effects of the nonet symmetry breaking and those of glue inside the @xmath18 . most of the models mentioned above are able to describe the data reasonably well in terms of some number of free parameters which , unfortunately , can not be determined from first principles . an attempt to overcome this drawback was made in @xcite where radiative decays of light vector mesons are considered in the approach based on qcd sum rules @xcite and the value @xmath98 is obtained for the branching ratio of @xmath5 decay . one can summarize that the variety of theoretical approaches to the problem of the description of the @xmath6 meson radiative decay to @xmath99 is rather broad and more theoretical insight into the problem is needed . using an almost five times bigger data sample than in the first measurement the cmd-2 group confirmed the observation of the rare radiative decay @xmath100 . the measured branching ratio is : @xmath78 . + its value is consistent with most of the theoretical predictions based on the quark model and assuming a standard quark structure of the @xmath18 . it rules out exotic models suggesting a high glue admixture @xcite or strong qcd violation @xcite . further progress in this field can be expected after the dramatic increase of the number of produced @xmath6 mesons expected at the da@xmath91ne @xmath6-factory and refinement of theoretical models of radiative decays . the authors are grateful to m.benayoun and v.n.ivanchenko for useful discussions . | a new measurement of the rare decay @xmath0 performed with the cmd-2 detector at novosibirsk is described . of the data sample corresponding to the integrated luminosity of 14.5 pb@xmath1 ,
twenty one events have been selected in the mode @xmath2 , @xmath3 .
the following branching ratio was obtained : b(@xmath4 . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
simulation studies of nambu - jona lasinio models have proven to be much more quantitative than those of other field theories @xcite . in particular , the logarithmic triviality of these models has been demonstrated , although determining logarithmic singularities decorating mean field scaling laws is a daunting numerical challenge . the reason for this success lies in the fact that when one formulates these four fermi models in a fashion suitable for simulations , one introduces an auxiliary scalar field @xmath3 in order to write the fermion terms of the action as a quadratic form . in this formulation @xmath3 then acts as a chiral order parameter which receives a vacuum expectation value , proportional to the chiral condensate @xmath4 , in the chirally broken phase . most importantly , the auxiliary scalar field @xmath3 becomes the dynamical mass term in the quark propagator . the dirac operator is now not singular for quarks with vanishing bare mass and its inversion @xcite , @xcite is successful and very fast . the algorithm for nambu - jona lasinio models is `` smart '' it incorporates a potential feature of the solution of the field theory , chiral symmetry breaking and a dynamical fermion mass , into the field configuration generator . the good features of the simulation algorithm for the nambu - jona lasinio model can be generalized to lattice qcd @xcite and qed @xcite by incorporating a weak four fermi term in their actions . these generalized models now depend on two couplings , the familiar gauge coupling and a new four fermi coupling . by choosing the four fermi coupling small we can be confident that all the dynamics resides in the gauge and fermi fields and the four fermi term just provides the framework for an improved algorithm which allows us to simulate the chiral limit of massless quarks directly . we shall find a line of spontaneously broken chiral symmetry transition points in the two dimensional coupling constant parameter space of the u(1)-gauged nambu - jona lasinio model . by simulating the model at several regions along the transition line , we will see that the theory is logarithmically trivial and that the four fermi term is irrelevant in the continuum limit . our conclusions will be supported by fits with very high confidence levels . because of the irrelevance of the pure four fermi interaction , this model will make `` textbook '' qed accessible and this paper will address the classic problem of whether qed suffers from complete charge screening . our measurements will show that the theory is logarithmically trivial and the systematics of the logarithms of triviality follow those of the nambu - jona lasinio model rather than the scalar @xmath5 model as usually assumed . simulating the @xmath6 case directly has substantial advantages , both theoretical and practical . when @xmath7 is set to zero , the theory has the exact chiral symmetry of the interaction terms in the action and this forbids chiral symmetry breaking counterterms from appearing in its effective action . this simplicity can lead to a large scaling window in the direction of the gauge or four fermi coupling in the theory s parameter space . our simulation results will support this point . however , when @xmath7 is not zero , as in most past studies of lattice qed and qcd , the effective action has no protection from dangerous symmetry breaking counterterms . in fact we will find that the scaling window of the lattice theory in the @xmath7-direction is very small and this fact is responsible for the failure of past approaches to lattice qed to address the question of triviality in a straightforward , convincing fashion . in fact , @xcite claimed non - triviality for the theory while @xcite found triviality and backed up their claim further in @xcite by calculating the sign of the beta function , which is directly relevant to the question of triviality . in addition , we shall check that the algorithm used in this work generates gauge field configurations for couplings near the chiral transition line which are free of lattice artifacts , such as monopoles @xcite and dirac strings , etc . in this paper we will present data and analyses . preliminary results have already appeared in letter form @xcite , but this article will contain new data , analyses and discussions . other applications of the use of a four fermi term to speed lattice gauge theory simulations are also under development and are being applied to qcd @xcite . it is important to note that in these applications the strength of the four fermi term is weak , so it is not responsible for chiral symmetry breaking . it just acts as scaffolding which leads to an algorithm that converges efficiently in the limit of massless quarks . the dynamics resides in the gauge and fermion field interactions . this paper is organized as follows . in the next section we present the formulation of the lattice action and discuss its symmetries and general features . in the third section we test the algorithm and tune its parameters . in the next three sections we present data and analyses over a range of gauge couplings for three choices of the irrelevant four fermi coupling on @xmath1 lattices . the irrelevance of the four fermi coupling is demonstated explicitly and equation of state fits are presented which show that the theory is logarithmically trivial with the same systematics as the nambu - jona lasinio model . the confidence levels of these fits range from approximately @xmath8 to @xmath9 percent . analyses of the order parameter s susceptibility reinforce our conclusions . in the seventh section we consider simulations at nonzero bare fermion masses in order to make contact with past work on pure lattice qed . we find that subdominant scaling terms are needed to fit the data . in other words , the usual assumption that the scaling window is wide enough to address the issue of triviality by simulating the model at nonzero fermion masses and fitting to a logarithmically improved mean field form is shown to be incorrect . in section eight we present data on lattices ranging in size from @xmath0 to @xmath2 to check that our data for the chiral condensate is not influenced significantly by finite size effects for the range of couplings used in the fits . in section nine we consider measurements of lattice monopole observables to check that they are not critical at the chiral transition points as long as the bare four fermi coupling is nonzero . in section ten we discuss the possible role of lattice artifacts in simulations of pure lattice qed and address some concerns in the literature . finally , in section eleven we suggest additional work in this field . we considered the @xmath10gauged nambu jona lasinio model with four species of fermions . the lagrangian for the continuum nambu - jona lasinio model is , @xmath11 the symmetries and other properties of @xmath12 have been discussed in @xcite and we refer the reader to that and related references for details . we will be brief here and just review a few conceptually important points . the pure nambu - jona lasinio model ( eq . 1 . with @xmath13 set to zero ) has been solved at large @xmath14 by gap equation methods @xcite , and an accurate simulation study of it has been presented @xcite . the lattice action for eq . 1 reads : @xmath15 where @xmath16 where @xmath3 is an auxiliary scalar field defined on the sites of the dual lattice @xmath17 @xcite , and the symbol @xmath18 denotes the set of the 16 lattice sites surrounding the direct site @xmath19 . the factors @xmath20 are the gauge connections and @xmath21 are the staggered phases , the lattice analogs of the dirac matrices . @xmath22 is a staggered fermion field and @xmath7 is the bare fermion mass , which will be set to 0 . note that the lattice expression for @xmath23 is non - compact in the lattice field @xmath24 , while the gauge field couples to the fermion field through compact phase factors which guarantee local gauge invariance . this point will be discussed further in sec.10 below . interesting limiting cases of the above action are the pure @xmath25 nambu - jona lasinio model ( @xmath26 ) , which has a phase transition at @xmath27 @xcite and the pure lattice qed ( @xmath28 ) limit , whose chiral phase transition is near @xmath29 for four flavors @xcite , @xcite . the pure qed ( @xmath28 ) model also has a monopole percolation transition which is probably coincident with its chiral transition at @xmath30 @xcite . past simulations of this lattice model have led to contradictory results @xcite , @xcite . since the gauged nambu - jona lasinio model can be simulated at @xmath6 for all gauge couplings , the results reported here will be much more precise and decisive than those of the pure lattice qed ( @xmath28 ) limit . we scanned the 2 dimensional parameter space ( @xmath31,@xmath32 ) using the hybrid molecular dynamics algorithm tuned for four continuum fermion species @xcite and measured the chiral condensate and monopole susceptibility as a function of @xmath31 and @xmath32 . we found that as we increased @xmath32 and moved off the @xmath28 axis , the peak of the monopole susceptibility shifted from @xmath30 at @xmath28 to @xmath33 at large @xmath34 . by contrast the chiral transition point shifted to a larger @xmath31 than the monopole percolation transition for a given value of @xmath34 and became distinct from the monopole percolation point as soon as @xmath34 became nonzero , as shown in the phase diagram , fig.1 . before turning to physically interesting measurements , we should address some technical issues concerning the algorithm . unlike the hybrid monte carlo algorithm , the hybrid molecular dynamics algorithm is not exact . the molecular dynamics equations of motion can be found in the literature @xcite . in order to evolve the noisy equations of motion and generate an ensemble of field configurations , one must choose a monte carlo time step @xmath35 @xcite . the discretization errors have been exhaustively studied and it has been shown that systematic errors in observables behave as @xmath36 @xcite . therefore , we must choose @xmath35 small enough that these systematic errors are no larger than the statistical errors we will encounter . in table 1 and fig . 2 we show the order parameter @xmath3 evaluated on a @xmath0 lattice at gauge coupling @xmath37 and four fermi coupling @xmath38 . ( we write @xmath3 here as a shorthand for @xmath39 , the expectation value of the field . this is a standard notational shortcut which , hopefully , should nt lead to confusion . ) the table shows that as long as @xmath40 the systematic error in @xmath3 is negligible . the figure shows that the theoretically expected quadratic dependence of the systematic error on @xmath36 has been confirmed numerically . the error bars quoted in the table have been obtained using the usual binning techniques , so they reflect the correlations in the measurements . the last column of the table gives the number of trajectories in each data set . a trajectory here means an interval of one monte carlo time unit of the algorithm ( for @xmath41 a trajectory consists of one hundred sweeps ) . after each trajectory a single measurement of @xmath3 was made . most of our production runs were done using @xmath42 . particularly close to the critical point where these systematiic errors are most dangerous , we checked our results with runs having @xmath43 . no problems were encountered . as stated in the introduction , we made accurate measurements on the chiral critical line for many choices of couplings ( @xmath31 , @xmath32 ) and lattice sizes ranging from @xmath0 to @xmath2 . in this section we review our data collected varying @xmath44 at fixed @xmath38 on a @xmath1 lattice . a discussion and presentation of this data has appeared in @xcite , so we will be brief . the data is presented in table 2 . the columns list the average values of @xmath3 , @xmath45 which is the longitudinal susceptibility of the order parameter @xcite , @xmath46 which is the monopole percolation order parameter and @xmath47 which is the susceptibility of the monopole order parameter @xcite . the monopole observables will be discussed later . the data for the order parameter was fit to a form which could accomodate either @xmath5 or nambu - jona lasinio triviality : @xmath48 , where the parameter @xmath49 , the critical point @xmath50 , the amplitude @xmath51 and the scale @xmath52 are determined by the fitting routine . recall that @xmath5 triviality gives @xmath53 and nambu - jona lasinio triviality gives @xmath54 . for the scaling window of gauge couplings @xmath31 between @xmath55 and @xmath56 , we found the parameters @xmath57 , @xmath58 , @xmath59 and @xmath60 with a confidence level of 34 percent . the reader should consult the figures and discussison in @xcite for more detail and perspective . as will be discussed below in sec.9 , these simulations also measured topological observables for the system s vacuum and we confirmed that monopoles and related objects were critical near the chiral transition @xmath57 , @xmath38 . ( we shall see there that the monopole percolation transition is very narrow and occurs at @xmath61 for @xmath38 . ) are other fitting forms possible for this data ? this is certainly true , of course . the point we are making , however , is that log - improved mean field theory fits the data with very high confidence levels and there are compelling theoretical reasons for it . the data and the fits support the conventional wisdom that qed is a trivial field theory and that the logarithms of triviality follow the systematics of the nambu - jona lasinio model rather than the scalar @xmath5 model . this last point is different from that usually assumed . in retrospect , it is very plausible that the nambu - jona lasinio model represents the triviality of qed better than @xmath5 , but the differences between the two models have not been emphasized or appreciated in the past . let s end our discussion of @xmath3 with some examples of other fits . simple power laws are the first ones to try . for example , a fit of the form @xmath62 is expected to work rather well with @xmath63 slightly larger than @xmath64 since the nambu - jona lasinio fits have worked so well . if we choose the range of @xmath31 to extend from @xmath65 to @xmath56 , we find @xmath66 , @xmath67 , @xmath68 , but the confidence level is very poor , ( @xmath69 ) . if we accept only a smaller range of couplings closer to the critical point , @xmath31 extending from @xmath55 to @xmath56 , the quality of the fit improves ( @xmath70 , confidence level of @xmath71 percent ) while the critical index @xmath63 rises to @xmath72 . this is the trend we find in the data : powerlaw estimates of the critical index @xmath63 increase as the range of couplings is restricted closer and closer to the critical coupling . this systematic drift in the fitting results suggests that a simple power is not an adequate representation of the full data set , but is simply mocking up the logarithm of the nambu - jona lasinio fit , which has a higher confidence level and is stable as different ranges of @xmath31 are considered . in @xcite we also analyzed the susceptibility associated with @xmath3 . in mean field theory , the singular piece of the longitudinal susceptibility @xmath73 diverges at the critical point @xmath50 as @xmath74 , @xmath75 , as @xmath76 approaches zero from above in the broken phase , and as @xmath77 in the symmetric phase @xcite . the critical index @xmath78 is exactly unity in mean field theory . in logarithmically trivial models @xmath78 remains unity , but the amplitudes @xmath79 and @xmath80 develop weak logarithmic dependences @xcite . in the two component @xmath5 model , @xmath81 , while in the @xmath25 nambu - jona lasinio model , @xmath82 @xcite , where the scale @xmath52 comes from the order parameter fit . constrained linear fits to the data @xcite produced the amplitude ratio @xmath83 . since @xmath3 varies from @xmath84 to @xmath85 over the @xmath31 range @xmath55 - @xmath56 of the scaling window in the broken phase , the logarithm in the theoretical prediction of the nambu - jona lasinio model states that @xmath86 should range from @xmath87 to @xmath88 . again , the agreement between the simulation data and theory is very good . we find no support for the approximate analytic schemes discussed in @xcite which predicted that gauged @xmath89 nambu - jona lasinio models with a four fermi term with continuous chiral symmetry are nontrivial , have powerlaw critical singularities with indices that vary continuously with the couplings @xmath31 and @xmath32 . additional simulations in sec.5 and 6 below will give strong evidence for the irrelevance of the four fermi term contrary to the results of @xcite . the reader should recall that truncated @xmath89 nambu - jona lasinio models which account for only restricted sets of feynman diagrams produce nontrivial `` theories '' with critical indices that vary continuously as @xmath31 and @xmath32 are varied . for example , this occurs if only `` rainbow graphs '' of gauged nambu - jona lasinio models are summed @xcite . some of these exercises may be relevant to technicolor model building . on the basis of the work here , however , we suspect that when fermion vacuum polarization is accounted for , one would find complete charge screening and every gauged nambu - jona lasinio model based on continuum noncompact @xmath89 gauge dynamics would be trivial for all couplings . we suspect that nontriviality and lines of nontrivial field theories are aspects of truncation procedures only . we suspect , on the basis of the present work and past triviality investigations in scalar qed @xcite , that only models with dynamics beyond continuum noncompact @xmath89 gauge fields and fermions can be nontrivial and have a renormalization group fixed point at nonzero gauge coupling . an example might be afforded by @xmath89 theories with fundamental monopoles @xcite . in this section we consider new data collected varying @xmath91 at fixed @xmath90 on a @xmath1 lattice . the purpose of this series of simulations is to 1 . to verify that the four fermi coupling is irrelevant , and 2 . to accumulate more evidence that the theory is logarithmically trivial in the sense of the nambu - jona lasinio model . the data is presented in table 3 in the same format as table 2 . in fig . 3 we show the data for the chiral condensate @xmath3 , at fixed @xmath90 and variable @xmath31 . we use the same fitting procedures as used in sec.4 : @xmath48 , where the parameter @xmath49 , the critical point @xmath50 , the amplitude @xmath51 and the scale @xmath52 are determined by the fitting routine . for the scaling window of gauge couplings @xmath31 between @xmath92 and @xmath93 , we found the parameters @xmath94 , @xmath95 , @xmath96 and @xmath97 with a confidence level of 87 percent . this excellent fit is the one shown in the figure . note that eight data points for @xmath31 between @xmath98 and @xmath99 were used in the fit while the figure has two additional points at stronger coupling . those points lie slightly below the fit , are slightly outside the scaling window and show the extent of the scaling window . we plot the eight data points between @xmath31 of @xmath98 and @xmath99 and the fit as shown in fig.4 . , @xmath100 vs. @xmath101 to illustrate the importance and numerical significance of the logarithm . the dashed line is the previous fit redrawn in this format , where we have zoomed in on the scaling window for emphasis . clearly this fit is stable to further cuts on the data set since all the data points lie on the fit . we conclude that nambu - jona lasinio triviality accommodates the lattice data at @xmath90 with very good confidence levels . this success also shows the irrelevance of the four fermi term in the lattice action : the scaling law for the order parameter is the same as that at the larger @xmath32 value although the lattice parameters , such as the location of the critical point , have changed . next , in fig.5 we show the inverse of the longitudinal susceptibility of the auxiliary field @xmath3 at fixed @xmath90 and variable @xmath31 . we follow the same procedures as used in sec.4 to analyze and plot the data here . the plot picks out a critical point @xmath102 and is consistent with the mean field value of the critical index @xmath103 . the constrained linear fits determine the amplitude ratio , @xmath104 . since @xmath3 varies from @xmath105 to @xmath106 over the @xmath31 range @xmath92 - @xmath93 of the scaling window in the broken phase , the logarithms in the theoretical prediction of the nambu - jona lasinio model for the amplitude ratio predict that @xmath86 should range from @xmath87 to @xmath88 . again , the agreement between the simulation data and theory is good , but is not comparable in quality or decisiveness to our other fits . in this section we consider new data collected varying @xmath108 at fixed @xmath107 on a @xmath1 lattice . in this case the four fermi coupling is four times stronger than the data discussed in the previous section , but far too weak to cause chiral symmetry breaking in the absence of the gauge coupling . the analysis and plots here are identical to the previous discussions of @xmath38 and @xmath90 , so we will be brief . the data is presented in table 4 in the same format as table 1 . in fig.6 we show the data for the chiral condensate @xmath3 , at fixed @xmath107 and variable @xmath31 . we use the same fitting procedures as used in sec.3 : @xmath48 , where the parameter @xmath49 , the critical point @xmath50 , the amplitude @xmath51 and the scale @xmath52 are determined by the fitting routine . for the scaling window of gauge couplings @xmath31 between @xmath109 and @xmath110 , we found the parameters @xmath111 , @xmath112 , @xmath113 and @xmath114 with a confidence level of 99.9 percent . this impressive fit is the one shown in the figure . we plot the data and the fit as shown in fig.7 , @xmath100 vs. @xmath101 to illustrate the importance and numerical significance of the logartihm . the dashed line is the previous fit redrawn in this format . the success of this fit reiterates the irrelevance of the four fermi term : the scaling law for the order parameter is the same as that at the @xmath32 values of @xmath115 and @xmath116 although the lattice parameters , such as the location of the critical point , have changed . next , in fig.8 we show the inverse of the longitudinal susceptibility of the auxiliary field @xmath3 at fixed @xmath107 and variable @xmath31 . the plot picks out a critical point @xmath117 and the constrained linear fits to the data shown in the figure produced the amplitude ratio @xmath118 . which compares well to the theoretical prediction @xmath119 . again , the agreement between the simulation data and theory is good , but is not comparable in quality or decisiveness to our order parameter fits . past simulations of lattice qed had to be done at nonzero fermion mass @xcite . the standard algorithms fail to converge in the limit @xmath120 because the lattice dirac operator becomes singular in the chiral limit @xcite , @xcite . this algorithmic problem has led to indecisive results for lattice qed because of large statistical and systematic errors . it is interesting to use the algorithm of this paper to discover , assess and clarify the problems in past work in this field . we chose to do simulations at nonzero fermion mass at the critical coupling @xmath50 determined by our fits presented in the previous section . in this way we can look for the width of the scaling window in the @xmath7-direction in a particularly simple fashion . recall that at criticality the order parameter @xmath3 should scale with the fermion mass @xmath7 , an explicit symmetry breaking parameter , as @xmath121 where the critical index @xmath122 should be @xmath123 in a logarithmically trivial theory and @xmath124 , the power of the logarithm should be @xmath125 for a @xmath5 theory and should be @xmath126 for a nambu - jona lasinio model @xcite . by accumulating data over a range of small @xmath7 values , we can look for the region where eq . 6 might apply and determine the width of the scaling window . it is important to keep the number of variables and parameters to a minimum in this sort of investigation . this is the reason we work at criticality . the critical coupling has been determined to be @xmath127 in sec.4 . the data for the order parameter and its susceptibility are shown in table 5 for @xmath7 ranging from @xmath128 to @xmath129 . note that the statistics for this data set is particularly high as smaller and smaller @xmath7 values are considered and the critical point is approached . the error bars in @xmath3 recorded there account for critical slowing down which forced us to accumulate such high statistics . the statistics are at least an order of magnitude greater than those of past studies and produce @xmath3 values with errors ranging from @xmath64 percent to @xmath130 percent . we learned in past studies of the pure nambu - jona lasinio that small @xmath7 values , typically below @xmath131 , are needed to find a scaling window @xcite . however , in this case the dynamics is controlled by the gauge coupling which alone is driving chiral symmetry breaking . the four fermi coupling is tiny and is not affecting the dynamics in a numerically significant fashion . therefore , the width of the scaling window must be determined anew from the data in table 5 . in fig . 9 we plot @xmath132 vs. @xmath7 in order to assess visually the relevance of the leading logarithm result eq the data clearly pick out the value @xmath133 for the dominant powerlaw singularity of the scaling law for very small @xmath7 values , all less than @xmath131 . however , we also see that the deviations from the mean field result are numerically significant over the entire mass range shown . in fact , they are far too large to be accommodated by a weak logarithmic scaling violation as expected in eq . 6 . in fact , a fit of that form to the data ranging from @xmath134 to @xmath135 produces a huge value for the power of the logarithm , @xmath136 , and a very small confidence level of @xmath137 percent ( @xmath138 ) . therefore , the data rule out the applicability of logarithmic improved mean field scaling to describe the data at nonzero @xmath7 except for the very smallest values of @xmath7 , @xmath139 . unfortunately , most data used to study the potential triviality of qed using the conventional action employed @xmath7 values considerably larger than @xmath140 in order to run efficiently and generate sufficient statistics . typical ranges of @xmath7 have been between @xmath131 and @xmath141 @xcite and are very sensitive to data taken with @xmath142 , @xmath143 and @xmath144 . this criticism applies to all past studies of noncompact qed , for example , @xcite . it also means that the methods of analysis introduced in @xcite do not apply to this data set because those methods require data in a scaling window , controlled by a single asymptotic form . higher precision data taken at the smallest values of @xmath7 , @xmath139 , are required apparently and , in fact , larger lattices than @xmath1 might be necessary also because of the possibility of significant finite size effects . two possible explanations for the data come to mind : 1 . perhaps the real critical point is significantly different from @xmath145 as determined by our fits at @xmath146 , or 2 . perhaps subdominant singularities in the scaling law are numerically significant over this range of @xmath7 . it is easy to rule out option 1 . ignoring logarithms , the mean field equation of state reads @xmath147 . this implies that if @xmath31 were different from @xmath50 , then @xmath132 would behave as @xmath148 , and the correction term would be large for small @xmath3 , which is just the opposite of the behavior observed in fig . 9 . now consider option 2 . if a subdominant singularity contributes to the equation of state , then at criticality the relation @xmath149 should be replaced by , @xmath150 where @xmath122 should be @xmath123 and @xmath151 should be considerably larger @xcite . this hypothesis fits the data beautifully : the curved dashed line in fig . 9 shows the fit which has a confidence level of @xmath152 percent ( @xmath153 ) . in fact this fitting form can be well approximated in a fashion that is useful for practical purposes , @xmath154 eq . 8 approximates eq . 7 because the correction to the constancy of @xmath132 is less than @xmath155 percent over the range of @xmath7 values in the figure . the fit gives @xmath156 , @xmath157 , and @xmath158 . we learn several lessons from this exercise . previous simulations of pure qed at nonzero @xmath7 could not possibly have detected the logarithms of triviality decorating mean field singularities . for the present range of @xmath7 values and lattice sizes , data at nonzero @xmath7 have contributions from subdominant critical singularities which are larger numerically than logarithmic corrections to mean field theory . since we are using a new algorithm which works in the limit of massless quarks , we should be careful to monitor finite size effects . some of our data are taken very near to critical points in order to find critical indices that control continuum limits of the lattice models . at these points the model s correlation length diverges and there are potentially dangerous finite size effects which could mimic finite temperature effects , for example . we need to check that the lattice is large enough to contain correlations larger than the lattice spacing but smaller than the system s spatial extent in order to work within a scaling window where we can extract continuum features of the field theory . in table 6 we show data for @xmath3 taken for gauge couplings @xmath31 ranging from @xmath159 to @xmath160 at fixed four fermi coupling @xmath161 for @xmath0 , @xmath1 and @xmath2 lattices . the comparison of the three data sets shows coincidence everywhere except at @xmath162 between the smallest lattice @xmath0 and the other two . @xmath162 was our closest approach to the critical point in the symmetry broken phase and it appears that our @xmath1 lattice was sufficient , given our @xmath64 percent statistical errors . reliance on a @xmath0 lattice would have failed us . in the next table we show @xmath0 data for a simulation where the four fermi coupling is fixed at @xmath38 and @xmath31 ranges from @xmath159 to @xmath163 . the data consists of @xmath3 as well as monopole observables that will be discussed in a later section . comparing the @xmath3 data here to that in table 3 , we confirm the absence of finite size effects within our statistical errors . in summary , the @xmath1 data we have used to extract scaling laws from @xmath3 measurements appears free of significant finite size effects . the significance of finite size effects depends strongly on the observable being simulated . we also checked that the longitudinal susceptibility data that was used to extract the logarithmic violations of scaling in the amplitudes was not distorted by finite size effects . since these susceptibilities are determined with much larger statistical error bars , this test was less demanding . certainly the finite size effects in @xmath45 are much larger than those in @xmath3 itself . however , since @xmath3 was determined within a fraction of a percent while the statistical uncertainty in @xmath45 was typically several percent , a @xmath1 lattice was adequate for the range of couplings used in this study . noncompact lattice qed was first studied with the goal of simulating the dynamics of @xmath89 gauge fields without the monopoles that accompany compact lattice qed @xcite . it was found , however , that even the noncompact formulation has monopole - like dislocations in its lattice formulation because of the space - time cutoff itself @xcite . these dislocations can undergo a percolation transition where long range correlations develop @xcite . because of this transition , it is not obvious that simulation results in pure noncompact lattice qed reflect the physics of textook qed in which field configurations are smooth and have no topological excitations . the formulation of noncompact lattice qed with a four fermi term is free of the issues raised in @xcite . the point is , as discussed in the section 2 above , the monopole percolation transition does not coincide with the chiral transition as long as the four fermi coupling is nonzero . therefore the gauge field vacuum is free of critical dislocations at the gauge couplings of interest , so we know that we are studying a model free of topological excitations , as we wish . let s find the monopole percolation transition in the model with a fixed four fermi coupling @xmath38 . the data for the monopole concentration @xmath46 and the associated monopole percolation susceptibility @xmath47 , both defined exactly as in @xcite , are given in table 2 . in fig.10 we plot the monopole concentration against the gauge coupling and find a percolation transition at @xmath164 . we determined in sec.4 that the chiral transition occurs at considerably weaker coupling , @xmath165 , where the monopole concentration is insignificant , as we read off fig.10 . it is also informative to confirm this conclusion by considering the monopole percolation susceptility , @xmath47 . in fig.11 we plot this susceptibility against the gauge coupling and see that it appears to diverge in the vicinity of @xmath164 ( we also confirmed this impression with powerlaw fits ) . in addition , in fig.12 we plot the longitudinal susceptibility of the chiral transition and confirm that it diverges near @xmath165 , as already determined in sec.4 . the two susceptibility peaks are cleanly separated : @xmath164 vs. @xmath165 . we end this section with a minor remark about the finite size effects observed in the monopole observables . comparing table 2 and table 7 , we see that as the monopole percolation transition s critical coupling is approached , there are numerically significant differences between the @xmath0 and the @xmath1 data sets for both the concentration @xmath46 as well as its associated susceptibility @xmath47 . as expected , the percolation susceptibility @xmath47 is strongly suppressed by the lattice size near the transition . in fact , as we have discussed elsewhere @xcite , finite size scaling of the peak of the susceptibility is an effective and accurate means to measure the percolation critical indices . it would take simulations on a series of lattice sizes to carry out such a program for this model . the only point we wish to make here , however , is that the percolation and chiral transitions are well separated inside the phase diagram fig . it is interesting ( and fortunate for the success of this project ) that the finite size effects in the chiral order parameter @xmath3 are significantly smaller than those in the monopole concentration . although the major topic in this research is the behavior of the gauged nambu - jona lasinio model for @xmath167 , we will briefly discuss the present confusing state of theory and simulations at the edge of the phase diagram @xmath166 where past simulations have been carried out . as we have already emphasized , the real problem with studies at @xmath166 is that they must be done at nonzero fermion mass away from the chiral limit and this has caused several problems : 1 . the simulations become excessively slow for small @xmath7 values because the lattice dirac operator is singular in that limit . therefore , at low values of @xmath7 where the best statistics are required , the statistics of the data sets are typically the poorest . the scaling window in the @xmath7-direction is extremely narrow , so fitting forms which only account for the leading critical behavior are inadequate and misleading . attempting to go beyond leading order critical singularities in fits leads to a vast proliferation of parameters which undermines firm conclusions . another potential problem concerning the @xmath166 edge of the phase diagram concerns lattice monopoles . recall that one motivation for inventing and studying noncompact lattice qed @xcite was to make a model free of monopoles in order to understand the relation between chiral symmetry breaking and single gluon exchange . however , hands and wensley @xcite pointed out that even the the noncompact model has monopole - like lattice dislocations because of gauge invariance of the pure gauge field piece of the action and because of the lattice cutoff itself . these authors also pointed out that these lattice monopoles experience a percolation transition as the gauge coupling becomes strong and in the case of quenched simulations , the monopole percolation transition is very close to the chiral transition experienced by light fermions @xcite . this led these authors to speculate that noncompact lattice qed might not be a sound framework for studying `` textbook '' qed at strong coupling @xcite . what does this possibility mean for this paper ? since we work at @xmath167 where the critical line of monopole percolation is distinct from the chiral transition line , these lattice artifacts are not relevant to our conclusions . we believe that we have a firm theoretical and numerical grasp of gauged nambu - jona lasinio models everywhere within the phase diagram fig.1 but not along the edge @xmath166 . how could this be ? following hands and wensley , the gauge field piece of the action eq.2 is invariant under local gauge transformations defined by the group of real numbers @xmath168 , while the fermionic piece of the action , which describes the gauge invariant hopping of the fermion around the lattice , has a gauge symmetry based on phases , @xmath89 . the cutoff theory described by the pure gauge piece of the action has monopole excitations attached by dirac strings @xcite . these are singular field configurations whose actions diverge when the lattice spacing is taken to zero . they would be of no concern if it were nt for the fact that as the coupling increases they experience a percolation transition where monopole clusters develop macroscopic dimensions . since the fermions are sensitive to monopole clusters through their @xmath89 phase , hands and wensley speculated that they could affect the chiral transition in the quenched and unquenched model . this speculation could be wrong for several reasons : 1 . the underlying gauge action is just a quadratic form , so it is a perfectly solvable free field theory . a free field theory ca nt have a phase transition , as emphasized in @xcite and 2 . percolation transitions need not affect the bulk properties of the underlying field theory . many examples of this sort can be cited . these complaints can be answered in part : 1 . the phase transition of percolation is not in local observables constructed out of the gauge fields , but rather is in nonlocal matrix elements . it is not unusual in statistical mechanics to make models where non - local matrix elements experience phase transitions when the underlying local field theory has no transition itself . condensed matter physics provides many examples of enormous practical importance including , for example , the localization - delocalization transition of single electrons in background fields of varying degrees of disorder . the chiral transition is sensitive to loops of the @xmath89 phase and is of this type . 2 . since fermions flip their chirality in the presence of monopoles , it is plausible that a percolating network of monopole - like excitations can induce chiral symmetry breaking in the bulk system . there is a possibility that the @xmath169 pure qed model has qualitatively different physics than that found anywhere within the phase diagram in fig.1 . only at the edge of the diagram would the percolating monopole - like excitations be critical where chiral symmetry is broken . only there are new degrees of freedom , percolating monopoles , relevant so only there could there be a new universality class . it might be that on the edge of the phase diagram , the chiral condensate is driven by monopole percolation and the chiral transition inherits a correlation length critical index @xmath170 from the percolating network and becomes the basis for a nontrivial field theory @xcite . it has been noticed that as the number of fermions is varied , both the chiral and monopole percolation transitions move in unison @xcite . in addition , in unquenched models , such as the four flavor model on the edge of the phase diagram fig.1 , the fermions induce @xmath89 plaquette terms into the theory s action which can support conventional lattice monopoles @xcite . we have nothing to add to the pros and cons of these qualitative arguments . we hope that the physics issues brought up here could be answered by striking out in new directions and finding approaches or arguments which are more precise and quantitative . the monopole percolation picture may contain only half truths , but some of those ideas might be testable in the context of models with real monopoles , generalizations of compact @xmath89 lattice qed @xcite , perhaps . we presented numerical evidence for the triviality of textbook qed using a new algorithm which converges for massless quarks . past simulations using the action with massive quarks but no four fermi term produced controversial results . recall that @xcite claimed non - triviality for the theory while @xcite found triviality and backed up their claim further in @xcite by calculating the sign of the beta function , which is directly relevant to the question of triviality . one could calculate the theory s renormalized couplings and their rg trajectories in the chiral limit , extending the work of @xcite to a two parameter space . both the gauge and the four fermi couplings should vanish as the reciprocal of the logarithm of the ultra - violet cutoff . as discussed in @xcite this calculation has some technical challenges specific to lattices of finite extent which necessitate the extrapolation of raw lattice data to achieve physical results . it would be worthwhile to investigate improved strategies here to avoid crude , indecisive results . the high quality of the equation of state fits in sec.4 , 5 and 6 should lead to improved determinations of the renormalized couplings because the lattice critical couplings are determined with excellent precision . one could also simulate the model with the @xmath25 chiral group replaced by a continuous group so the model would have goldstone bosons even on a coarse lattice @xcite . it would then be possible to test the approach and results of @xcite more quantitatively . it would also be interesting to generalize the results of sec.7 , that a subdominant critical singularity is needed to describe the data at nonzero @xmath7 , away from the critical coupling . in other words , fit the finite @xmath7 data points of previous investigations such as @xcite to equations of state with both a dominant and subdominant singularity and check that improved confidence levels are achieved with simple hypotheses . unfortunately , there will be a proliferation of fitting parameters in such a program , so its numerical significance might be questioned . nonetheless , it would definitely be worth consideration . such a program would also influence the determination of renormalized couplings because these calculations use critical couplings inferred from equation of state fits @xcite . finally , it would be interesting to simulate compact qed with a small four fermi term and study the interplay of monopoles , charges and chiral symmetry breaking . since the @xmath171 limit of the compact model is known to have a first order transition @xcite , generalizations of the action will be needed to find a continuous transition where a continuum limit of the lattice theory might exist . since the parameter space of the generalized model would be at least three dimensional , this interesting problem would be quite challenging . this work was partially supported by nsf under grant nsf - phy96 - 05199 . s. k. is supported by the korea research foundation . l. wishes to thank the _ ect@xmath172 _ , trento , for hospitality during the final stages of this project . the simulations were done at npaci and nersc . s. kim , a. koci@xmath173 and j.b . kogut , nucl . phys . * b429 * , 407 ( 1994 ) . phys . * b220 * , 102 ( 1983 ) . s. duane , a.d . kennedy , b.j . pendleton and d. roweth , phys . lett . * b195 * , 216 ( 1987 ) . s. duane and j.b . kogut , phys . * 55 * , 2774 ( 1985 ) . s. gottlieb , w. liu , d. toussaint , r.l . renken and r.l . sugar , phys . rev . * d35*,2531 ( 1987 ) . kogut , j .- f . lagae , and d.k . sinclair , phys . rev . * d58 * , 34004 ( 1998 ) . kogut , and d.k . sinclair , hep - lat/0005007 . s. kim , j.b . kogut and m .- lombardo , phys . * b55 * , 2774 ( 2001 ) . s. hands and r. wensley , phys . . lett . * 63 * , 2169 ( 1989 ) . b. rosenstein , b. warr and s. park , phys . * 205 * , 497 ( 1991 ) . y. cohen , s. elitzur and e. rabinovici , nucl . phys . * b220 * , 102 ( 1983 ) . v. azcoiti , g. di carlo , a. galante , a.f . grillo , v. laliena , and c.e . piedrafita , phys . lett . * b353 * , 279 ( 1995 ) ; * b379 * , 179 ( 1996 ) . kogut and k.c . wang , phys.rev . * d53 * , 1513 ( 1996 ) . e. dagotto and a. koci@xmath173 and j.b . b317 * , 271 ( 1989 ) . s. duane and j.b . . phys . * b275 * [ fs17 ] , 398 ( 1986 ) . a. koci@xmath173 , j.b . kogut and m .- b398 * , 376(1993 ) ; a. koci@xmath173 and j.b . . phys . * b422 * [ fs ] , 593 ( 1994 ) . c. itzykson and j .- m . drouffe , statistical field theory ( cambridge university press , 1989 . ) v. azcoiti , g. di carlo , a. galante , a.f . grillo , v. laliena , and c.e . piedrafita , phys.lett . * b355 * , 270 ( 1995 ) . c.n leung , s.t . love and w.a . bardeen , nucl . phys.*b327 * , 649(1986 ) . m. baig , h. fort , s. kim , j.b . kogut and d.k . phys . rev . * d48 * , r2385 ( 1993 ) ; m. baig , h. fort , j.b . kogut and s. kim . phys . rev . * d51 * , 5216 ( 1995 ) . a. koci@xmath173 , j.b . kogut and s. hands . b357 * , 467 ( 1991 ) ; s. hands and j.b . kogut , nucl . phys . * b462 * , 291 ( 1996 ) . a. koci and j. kogut , nucl . b * 422 * , 593 ( 1994 ) . r. myerson , t. banks and j.b . * b129 * , 493 ( 1977 ) . s. hands , a. koci@xmath173 and j.b . kogut . 289b * , 400 ( 1992 ) . j. bartholomew , j.b . kogut , s. h. shenker , j. sloan , j. shigemitsu , d. k. sinclair and h. w. wyld . b230 * , [ fs10 ] , 222 ( 1984 ) . m. gckeler , r. horsley , p.e.l . rakow and g. schierholz , d * 53 * , 1508 ( 1996 ) . m. gockeler , r. horsley , v. linke , p. rakow , g. schierholz and h. stuben , phys . * 80 * , 4119 ( 1998 ) . e. dagotto and j.b . kogut , phys . 59 * , 617 ( 1987 ) . cdd @xmath35 & @xmath3 & @xmath174 + @xmath175 & 0.1279(6 ) & 1200 + @xmath176 & 0.1280(5 ) & 2000 + @xmath177 & 0.1284(4 ) & 2400 + @xmath178 & 0.1293(4 ) & 2800 + @xmath179 & 0.1301(3 ) & 3500 + @xmath180 & 0.1314(2 ) & 4800 + cddddd @xmath181 & @xmath3 & @xmath45 & @xmath46 & @xmath47 & @xmath174 + @xmath182 & 0.11980(7 ) & 0.1756(5 ) & 0.97676(1 ) & 0.1245(5 ) & 930 + @xmath183 & 0.11248(8 ) & 0.1843(20 ) & 0.9566(1 ) & 0.261(1 ) & 950 + @xmath184 & 0.10438(8 ) & 0.1993(50 ) & 0.9221(1 ) & 0.552(2 ) & 1030 + @xmath185 & 0.09531(9 ) & 0.2160(50 ) & 0.8644(2 ) & 1.220(5 ) & 1010 + @xmath186 & 0.08520(8 ) & 0.2621(30 ) & 0.7669(3 ) & 3.04(2 ) & 1500 + @xmath187 & 0.0738(1 ) & 0.3230(40 ) & 0.5976(7 ) & 10.3(1 ) & 1310 + @xmath93 & 0.0674(1 ) & 0.3671(30 ) & 0.463(1 ) & 25.3(6 ) & 1012 + @xmath188 & 0.0609(1 ) & 0.4301(30 ) & 0.250(2 ) & 124.0(9 ) & 1130 + @xmath189 & 0.0537(2 ) & 0.4849(40 ) & 0.0812(8 ) & 122.4(9 ) & 2701 + @xmath190 & 0.0456(2 ) & 0.6591(40 ) & 0.0338(5 ) & 69.7(5 ) & 1120 + @xmath56 & 0.0367(2 ) & 0.9356(40 ) & 0.0192(2 ) & 43.5(2 ) & 1670 + @xmath191 & & & 0.0130(2 ) & 30.2(1 ) & 810 + @xmath192 & -0.00008(9 ) & 3.360(90 ) & 0.00798(8 ) & 19.57(5 ) & 810 + @xmath193 & 0.0002(5 ) & 1.903(80 ) & 0.00681(4 ) & 16.90(2 ) & 2518 + @xmath194 & -0.00002(3 ) & 1.316(70 ) & 0.00601(5 ) & 14.87(3 ) & 960 + @xmath195 & 0.0003(4 ) & 0.919(50 ) & 0.00544(2 ) & 13.34(1 ) & 3350 + @xmath196 & -0.0007(3 ) & 0.786(40 ) & 0.00483(4 ) & 12.07(2 ) & 820 + @xmath197 & -0.0006(3 ) & 0.625(50 ) & 0.00407(4 ) & 10.28(1 ) & 1010 + @xmath198 & -0.0003(2 ) & 0.525(10 ) & 0.00303(6 ) & 9.051(9 ) & 1070 + @xmath199 & -0.0002(2 ) & 0.484(10 ) & 0.00166(6 ) & 8.185(7 ) & 1230 + @xmath200 & 0.0002(2 ) & 0.432(10 ) & 0.00074(5 ) & 7.521(6 ) & 1150 + cddddd @xmath181 & @xmath3 & @xmath45 & @xmath46 & @xmath47 & @xmath174 + @xmath183 & 0.05201(4 ) & 0.066(2 ) & 0.9479(1 ) & 0.327(1 ) & 730 + @xmath201 & 0.04932(4 ) & 0.068(1 ) & 0.9293(1 ) & 0.487(2 ) & 770 + @xmath184 & 0.04642(5 ) & 0.075(2 ) & 0.9047(2 ) & 0.729(3 ) & 660 + @xmath202 & 0.04332(5 ) & 0.074(2 ) & 0.8729(3 ) & 1.106(5 ) & 600 + @xmath185 & 0.03996(6 ) & 0.082(2 ) & 0.8314(4 ) & 1.72(1 ) & 500 + @xmath203 & 0.03644(6 ) & 0.089(3 ) & 0.7765(5 ) & 2.81(2 ) & 640 + @xmath186 & 0.03267(5 ) & 0.096(3 ) & 0.7054(6 ) & 4.82(3 ) & 960 + @xmath204 & 0.02848(7 ) & 0.110(4 ) & 0.6091(9 ) & 9.35(9 ) & 770 + @xmath187 & 0.02401(9 ) & 0.136(4 ) & 0.4751(16 ) & 24.2(9 ) & 640 + @xmath93 & 0.01872(9 ) & 0.190(5 ) & 0.2677(29 ) & 115(4 ) & 780 + @xmath188 & 0.01250(9 ) & 0.38(1 ) & 0.08801(19 ) & 127(2 ) & 960 + @xmath189 & -0.00013(64 ) & 1.62(9 ) & 0.03625(95 ) & 74.1(9 ) & 310 + @xmath190 & -0.00027(38 ) & 0.75(9 ) & 0.02333(52 ) & 51.3(7 ) & 320 + @xmath56 & 0.00009(15 ) & 0.33(4 ) & 0.01576(26 ) & 35.6(2 ) & 490 + @xmath191 & 0.00009(9 ) & 0.23(4 ) & 0.01172(15 ) & 27.9(1 ) & 660 + @xmath205 & 0.00009(9 ) & 0.19(4 ) & 0.00937(12 ) & 22.8(8 ) & 590 + @xmath192 & 0.00010(8 ) & 0.16(2 ) & 0.00779(9 ) & 19.22(6 ) & 640 + @xmath193 & 0.00010(6 ) & 0.145(9 ) & 0.00665(6 ) & 16.63(4 ) & 910 + @xmath194 & 0.00009(6 ) & 0.133(7 ) & 0.00586(5 ) & 14.67(3 ) & 790 + cddd @xmath181 & @xmath3 & @xmath45 & @xmath174 + @xmath182 & 0.2525(2 ) & 0.39(5 ) & 1500 + @xmath183 & 0.2434(2 ) & 0.43(2 ) & 1500 + @xmath184 & 0.2339(2 ) & 0.43(5 ) & 1500 + @xmath185 & 0.2237(2 ) & 0.44(5 ) & 1500 + @xmath186 & 0.2129(2 ) & 0.48(3 ) & 1500 + @xmath187 & 0.2012(2 ) & 0.51(4 ) & 1500 + @xmath188 & 0.1885(2 ) & 0.61(3 ) & 1500 + @xmath190 & 0.1751(3 ) & 0.68(3 ) & 1500 + @xmath191 & 0.1606(3 ) & 0.84(2 ) & 1500 + @xmath192 & 0.1450(3 ) & 1.04(5 ) & 1500 + @xmath194 & 0.1281(4 ) & 1.26(4 ) & 1500 + @xmath196 & 0.1095(4 ) & 1.68(4 ) & 1500 + @xmath197 & 0.0881(5 ) & 2.23(5 ) & 1500 + @xmath206 & 0.000(6 ) & 5.66(10 ) & 1500 + @xmath207 & 0.000(7 ) & 3.68(10 ) & 1500 + @xmath208 & 0.000(7 ) & 3.17(10 ) & 1500 + @xmath209 & 0.000(6 ) & 2.61(10 ) & 1500 + @xmath210 & 0.000(4 ) & 2.33(10 ) & 1500 + @xmath211 & 0.000(5 ) & 2.31(7 ) & 1500 + @xmath212 & 0.000(5 ) & 1.80(6 ) & 1500 + cddd @xmath7 & @xmath3 & @xmath45 & @xmath174 + @xmath213 & 0.0286(1 ) & 0.927(9 ) & 3400 + @xmath214 & 0.0315(1 ) & 0.779(2 ) & 3194 + @xmath215 & 0.0340(1 ) & 0.681(4 ) & 3040 + @xmath216 & 0.0361(1 ) & 0.639(5 ) & 3397 + @xmath217 & 0.0380(1 ) & 0.582(8 ) & 3207 + @xmath218 & 0.03944(9 ) & 0.538(5 ) & 2174 + @xmath219 & 0.04247(9 ) & 0.474(5 ) & 1837 + @xmath220 & 0.04890(8 ) & 0.415(5 ) & 2521 + @xmath221 & 0.05357(8 ) & 0.377(4 ) & 1740 + @xmath222 & 0.06110(9 ) & 0.325(8 ) & 1260 + @xmath223 & 0.06688(8 ) & 0.290(7 ) & 1380 + @xmath224 & 0.07160(7 ) & 0.269(7 ) & 1400 + @xmath225 & 0.07571(8 ) & 0.255(6 ) & 960 + @xmath226 & 0.07916(7 ) & 0.240(6 ) & 1020 + @xmath227 & 0.08214(8 ) & 0.231(7 ) & 1000 + @xmath228 & 0.08480(8 ) & 0.217(6 ) & 910 + @xmath229 & 0.08724(7 ) & 0.211(6 ) & 970 + @xmath182 & 0.09636(7 ) & 0.193(3 ) & 970 + @xmath187 & 0.10237(8 ) & 0.177(3 ) & 730 + cddd @xmath181 & @xmath230 & @xmath231 & @xmath232 + @xmath182 & & 0.2525(2 ) & + @xmath183 & & 0.2434(2 ) & + @xmath184 & 0.2341(4 ) & 0.2339(2 ) & + @xmath185 & 0.2239(4 ) & 0.2237(2 ) & + @xmath186 & 0.2130(4 ) & 0.2129(2 ) & + @xmath187 & 0.2013(5 ) & 0.2012(2 ) & + @xmath188 & 0.1885(5 ) & 0.1885(2 ) & + @xmath190 & 0.1747(6 ) & 0.1751(3 ) & 0.1748(4 ) + @xmath191 & 0.1606(6 ) & 0.1606(3 ) & 0.1617(5 ) + @xmath192 & 0.1451(7 ) & 0.1450(3 ) & 0.1454(3 ) + @xmath194 & 0.1281(6 ) & 0.1281(4 ) & 0.1283(4 ) + @xmath196 & 0.1089(8 ) & 0.1095(4 ) & 0.1093(4 ) + @xmath197 & 0.0866(8 ) & 0.0881(5 ) & 0.0885(4 ) + cdddd @xmath181 & @xmath3 & @xmath46 & @xmath47 & @xmath174 + @xmath182 & 0.1203(2 ) & 0.9770(1 ) & 0.123(1 ) & 1000 + @xmath183 & 0.1129(2 ) & 0.9568(2 ) & 0.261(2 ) & 1000 + @xmath184 & 0.1047(2 ) & 0.9223(4 ) & 0.551(5 ) & 1000 + @xmath185 & 0.0956(2 ) & 0.8641(6 ) & 1.21(1 ) & 1000 + @xmath186 & 0.0854(3 ) & 0.7660(10 ) & 3.10(5 ) & 1000 + @xmath187 & 0.0741(3 ) & 0.5954(22 ) & 10.6(3 ) & 1000 + @xmath188 & 0.0610(4 ) & 0.2669(49 ) & 71.5(9 ) & 1000 + @xmath190 & 0.0449(7 ) & 0.0651(16 ) & 51.0(7 ) & 1000 + @xmath191 & & 0.0291(5 ) & 26.6(2 ) & 1000 + @xmath192 & & 0.0203(6 ) & 18.4(2 ) & 1000 + @xmath194 & & 0.0152(4 ) & 14.1(1 ) & 1000 + | by adding a small , irrelevant four fermi interaction to the action of noncompact lattice quantum electrodynamics ( qed ) , the theory can be simulated with massless quarks in a vacuum free of lattice monopoles .
simulations directly in the chiral limit of massless quarks are done with high statistics on @xmath0 , @xmath1 and @xmath2 lattices at a wide range of couplings with good control over finite size effects , systematic and statistical errors .
the lattice theory possesses a second order chiral phase transition which we show is logarithmically trivial , with the same systematics as the nambu - jona lasinio model .
the irrelevance of the four fermi coupling is established numerically .
our fits have excellent numerical confidence levels .
the widths of the scaling windows are examined in both the coupling constant and bare fermion mass directions in parameter space . for vanishing fermion mass
we find a broad scaling window in coupling which is essential to the quality of our fits and conclusions . by adding a small bare fermion mass to the action we find that the width of the scaling window in the fermion mass direction is very narrow .
only when a subdominant scaling term is added to the leading term of the equation of state are adequate fits to the data possible .
the failure of past studies of lattice qed to produce equation of state fits with adequate confidence levels to seriously address the question of triviality is explained .
the vacuum state of the lattice model is probed for topological excitations , such as lattice monopoles and dirac strings , and these objects are shown to be non - critical along the chiral transition line as long as the four fermi coupling is nonzero .
our results support landau s contention that perturbative qed suffers from complete screening and would have a vanishing fine structure constant in the absence of a cutoff .
-1 truecm -1 truecm |
You are an expert at summarizing long articles. Proceed to summarize the following text:
in ref . @xcite an @xmath7 conformally symmetric model was proposed for strong interactions at low energies , based on the observation , published in 1919 by h. weyl in ref . @xcite , that the dynamical equations of gauge theories retain their flat - space - time form when subject to a conformally - flat metrical field , instead of the usual minkowski background . confinement of quarks and gluons is then described through the introduction of two scalar fields which spontaneously break the @xmath7 symmetry down to @xmath8 and @xmath9 symmetry , respectively . moreover , a symmetric second - order tensor field is defined that serves as the metric for flat space - time , coupling to electromagnetism . quarks and gluons , which to lowest order do not couple to this tensor field , are confined to an anti - de - sitter ( ads ) universe @xcite , having a finite radius in the flat space - time . this way , the model describes quarks and gluons that oscillate with a universal frequency , independent of the flavor mass , inside a closed universe , as well as photons which freely travel through flat space - time . the fields in the model of ref . @xcite comprise one real scalar field @xmath10 and one complex scalar field @xmath11 . their dynamical equations were solved in ref . @xcite for the case that the respective vacuum expectation values , given by @xmath12 and @xmath13 , satisfy the relation @xmath14 a solution for @xmath12 of particular interest leads to ads confinement , via the associated conformally flat metric given by @xmath15 . the only quadratic term in the lagrangian of ref . @xcite is proportional to @xmath16 hence , under the condition of relation ( [ slvacua ] ) , one obtains , after choosing vacuum expectation values , a light @xmath10 field , associated with confinement , and a very heavy complex @xmath11 field , associated with electromagnetism . weak interactions were not contemplated in ref . @xcite , but one may read electroweak for electromagnetism . here , we will study the supposedly light mass of the scalar field that gives rise to confinement . the conformally symmetric model of ref . @xcite in itself does not easily allow for interactions between hadrons , as each hadron is described by a closed universe . hence , in order to compare the properties of this model with the actually measured cross sections and branching ratios , the model has been further simplified , such that only its main property survives , namely its flavor - independent oscillations . this way the full ads spectrum is , via light - quark - pair creation , coupled to the channels of two or more hadronic decay products for which scattering amplitudes can be measured . the ads spectrum reveals itself through the structures observed in hadronic mass distributions . however , as we have shown in the past ( see ref . @xcite and references therein ) , there exists no simple relation between enhancements in the experimental cross sections and the ads spectrum . it had been studied in parallel , for mesons , in a coupled - channel model in which quarks are confined by a flavor - independent harmonic oscillator @xcite . empirically , based on numerous data on mesonic resonances measured by a large variety of experimental collaborations , it was found @xcite that an ads oscillation frequency of @xmath17 agrees well with the observed results for meson - meson scattering and meson - pair production in the light @xcite , heavy - light @xcite , and heavy @xcite flavor sectors , thus reinforcing the strategy proposed in ref . @xcite . another ingredient of the model for the description of non - exotic quarkonia , namely the coupling of quark - antiquark components to real and virtual two - meson decay channels @xcite via @xmath18 quark - pair creation , gives us a clue about the size of the mass of the @xmath10 field . for such a coupling it was found that the average radius @xmath19 for light - quark - pair creation in quarkonia could be described by an flavor - independent mass scale , given by @xmath20 where @xmath21 is the effective reduced quarkonium mass . in earlier work , the value @xmath22 @xcite was used , which results in @xmath23 mev for the corresponding mass scale . however , the quarkonium spectrum is not very sensitive to the precise value of the radius @xmath19 , in contrast with the resonance widths . in more recent work @xcite , slightly larger transition radii have been applied , corresponding to values around 40 mev for @xmath24 . nevertheless , values of 3040 mev for the flavor - independent mass @xmath24 do not seem to bear any relation to an observed quantity for strong interactions . however , we will next present experimental evidence for the possible existence of a quantum with a mass of about 38 mev , which in the light of its relation to the @xmath18 mechanism we suppose to mediate quark - pair creation . moreover , its scalar properties make it a perfect candidate for the quantum associated with the above - discussed scalar field for confinement . @xcite , we made notice of an apparent interference effect around the @xmath25 threshold in the invariant - mass distribution of @xmath26 events , which we observed in preliminary radiation data of the babar collaboration @xcite . the effect , with a periodicity of about 74 mev , could be due to interference between the typical oscillation frequency of 190 mev of the @xmath27 pair and that of the gluon cloud . [ cols="^ " , ] thus , a signal with the shape of a narrow breit - wigner resonance seems to be visible on the slope of the @xmath28 resonance , though with little more than 2@xmath10 relevance . nevertheless , by coincidence or not , it comes out exactly in the expected place , namely at @xmath29 mev . unfortunately , the data @xcite do not have enough statistics to pinpoint possible higher excitations as well . so we can not relate , to a minimum degree of accuracy , the other observed deficit enhancements to the possible existence of hybrid states . from the fact that the @xmath30 has not been observed before , one must conclude that it probably does not interact at least to leading order through electroweak forces , but instead couples exclusively to quarks and gluons . the interference effect we discussed in sec . [ oscillations ] might well be explained by @xmath30 exchange between the quarks , which interferes with the natural quark oscillations . moreover , since the interference effect is smaller for light quarks than for heavier ones , it is likely that their coupling to the @xmath30 is proportional to flavor mass , as one expects from theory . the @xmath30 could very well be just a light scalar glueball , albeit much lighter than found in refs . its low mass precludes decay into hadrons , while the absence of electroweak couplings does not allow it to decay into leptons either , at least to lowest order . it may decay , though , into photons via virtual quark loops , and through photons , eventually , into @xmath0 pairs . however , the probability for such reactions to occur is extremely remote , since the coupling between the quarks and the light scalar is proportional to the quark mass @xcite . in the case of bottom quarks , we found events of the order of or less than one percent of the total . for light quarks , their mass ratio with respect to bottom quarks reduces this rate to @xmath31@xmath32 . hadrons will certainly interact with a light scalar ball of glue . for example , a proton struck by such a scalar particle may absorp it and then emit photons , or decay into a neutron and a lepton - neutrino pair . yet another possibility is that , being closed universes themselves , these scalar particles mainly collide elastically with hadronic matter . in that case , depending on their linear momentum , they may remove light nuclei from atoms . nevertheless , we do not see such processes happening around us . therefore , these light scalars are probably not abundantly present near us . however , in the early universe they may have existed , most probably inflated to hadrons under collisions . on the other hand , interactions with hadrons might have been observed in bubble - chamber experiments , where isolated protons could be the result of collisions with a light scalar particle emerging from one of the interaction vertices . light higgs fields have been considered in supersymmetric extensions of the standard model @xcite . furthermore , axions , which appear in models motivated by astrophysical observations , are assumed to have higgs - like couplings @xcite . model predictions for the branching fraction of @xmath33higgs decays , for higgs masses below @xmath34 @xcite , range from @xmath32 @xcite to @xmath35 @xcite . furthermore , the three anomalous events observed in the hypercp experiment @xcite were interpreted as the production of a scalar boson with a mass of 214.3 mev , decaying into a pair of muons @xcite . however , in ref . @xcite the babar collaboration found no evidence for dimuon decays of a light scalar particle in radiative decays of @xmath28 and @xmath36 mesons . the babar limits for dimuon decays of a light scalar particle rule out much of the parameter space allowed by the light - higgs @xcite and axion @xcite models . nonetheless , in ref . @xcite y .- j . zhang and h .- s . shao , pointed out that the transitions @xmath37higgs are not yet excluded by the lepton - universality test in @xmath38 decays studied by babar in refs . @xcite . the light scalar glueball we have discussed here seems to correspond to the lowest - order empty - universe solution of ref . @xcite for strong interactions . it has similar properties as the electroweak higgs , but now for strong interactions . quarks couple to it with an intensity which is proportional to their mass , in the same way that mass couples to gravity @xcite . in the standard model @xcite , the higgs boson and the graviton are the only particles yet to be observed , and no higgs particle for strong interactions is anticipated . however , n. trnqvist recently proposed @xcite the light scalar - meson nonet @xcite as the higgs bosons of strong interactions , while in ref . @xcite he obtained a nonzero pion mass by means of a small breaking of a relative symmetry between the electroweak and the strong interactions . a relation between the lightest scalar - meson nonet and glueballs has often been advocated by p. minkowski and w. ochs ( see e.g. ref . @xcite ) . in our view though , the light scalar mesons are dynamically generated through @xmath39 pair creation / annihilation , which mixes the quark - antiquark and dimeson sectors @xcite . in sec . [ intro ] we discussed why we expect an additional scalar particle to exist , besides the higgs boson for the electroweak sector . furthermore , we have estimated its mass based on the average radius for @xmath18 quark - pair creation , which had been extracted over the past three decades from numerous data on mesonic resonances ( see ref . @xcite and references therein ) . in sec . [ oscillations ] we recalled our results on an apparent interference effect in annihilation data , and stressed the possibility that it may stem from some internal oscillation with a frequency of about 38 mev . in sec . [ phenomenon ] we showed that missing data in the reactions @xmath0 @xmath1 @xmath40 @xmath1 @xmath3 and @xmath0 @xmath1 @xmath40 @xmath1 @xmath4 exhibit maxima at @xmath24 , @xmath41 , and @xmath42 , for @xmath43 mev . each of the results , viz.the interference effect observed in ref . @xcite , the small flavor - independent oscillations in electron - positron and proton - antiproton annihilation data , observed in ref . @xcite and summarized in fig . [ interference ] , the excess signals visible in the @xmath5 mass distributions of @xmath28 @xmath1 @xmath40 @xmath1 @xmath4 ( fig . [ mumuall2s ] ) , in @xmath36 @xmath1 @xmath40 @xmath1 @xmath4 ( fig . [ alldiff]a ) , in @xmath36 @xmath1 @xmath44 @xmath1 @xmath4 ( fig . [ alldiff]b ) , and in the @xmath0 mass distributions of @xmath0 @xmath1 @xmath3 ( fig . [ alldiff]c ) , and finally the resonance signal shown in fig . [ hybrid ] , is much too small to make firm claims . however , we observe here that all points in the same direction . indeed , the probability must be close to zero that one accidentally finds the same oscillations in four different sets of data ( refs . @xcite ) involving different flavors , statistical fluctuations at @xmath45 mev in yet another four sets of different data ( figs . [ mumuall2s ] and [ alldiff ] ) , and moreover a resonance - like fluctuation at @xmath46 mev in a further set of data ( fig . [ hybrid ] ) . furthermore , the related mass comes where predicted by our analyses in mesonic spectroscopy ( see ref . @xcite and references therein ) . in sec . [ gluons ] we discussed that , most probably , the missing signal is due to the emission of an as yet unobserved light scalar particle , while part of the excess data corroborates such an interpretation . since the corresponding particle has all the right properties , we conclude that we found first indications , of the possible existence of a higgs - like particle , namely the scalar boson related to confinement . furthermore , the data also suggest the existence of two replicas of the @xmath30 with masses that are two and three times heavier than the @xmath30 . in addition , we believe that this 38 mev boson , which we designate by @xmath30 , consists of a mini - universe filled with glue , thus forming a very light scalar glueball . furthermore , we have pinpointed the masses of possible @xmath6 hybrids , one of which shows up as an enhancement in the invariant - mass distribution of babar data , albeit with a 2@xmath10 significance at most . finally , we urge the babar collaboration to inspect their larger data set in order to settle , with higher statistics , the possible existence of the @xmath30 and the related @xmath6 hybrid spectrum . we are grateful for the precise measurements and data analyses of the babar , cdf , and cmd-2 collaborations , which made the present analysis possible . one of us ( evb ) wishes to thank drs . b. hiller , a. h. blin , and a. a. osipov for useful discussions . this work was supported in part by the _ fundao para a cincia e a tecnologia _ of the _ ministrio da cincia , tecnologia e ensino superior _ of portugal , under contract cern / fp/ 109307/2009 . m. dohse , arxiv:0706.1887 . e. van beveren and g. rupp , pos hql2010:003 ( 2010 ) [ arxiv:1011.2360 ] . e. van beveren , c. dullemond , and g. rupp , , 772 ( 1980 ) [ erratum - ibid . d * 22 * , 787 ( 1980 ) ] . e. van beveren , g. rupp , t. a. rijken , and c. dullemond , , 1527 ( 1983 ) . e. van beveren and g. rupp , arxiv:1009.1778 . e. van beveren , t. a. rijken , k. metzger , c. dullemond , g. rupp and j. e. ribeiro , , 615 ( 1986 ) [ arxiv:0710.4067 ] . e. van beveren and g. rupp , , 012003 ( 2003 ) [ arxiv : hep - ph/0305035 ] . e. van beveren and g. rupp , arxiv:1009.4097 . e. van beveren , and g. rupp , , 1620 ( 2009 ) [ arxiv:0809.1149 ] . e. van beveren and g. rupp , arxiv : hep - ph/0201006 . e. van beveren , d. v. bugg , f. kleefeld and g. rupp , , 265 ( 2006 ) [ arxiv : hep - ph/0606022 ] . e. van beveren and g. rupp , , 111501r ( 2009 ) [ arxiv:0905.1595 ] . b. aubert [ babar collaboration ] , slac - pub-13360 , babar - conf-08 - 004 , arxiv:0808.1543 . v. m. aulchenko _ et al . _ [ cmd-2 collaboration ] , , 743 ( 2005 ) [ pisma zh . fiz . * 82 * , 841 ( 2005 ) ] [ arxiv : hep - ex/0603021 ] . t. aaltonen _ [ cdf collaboration ] , , 152001 ( 2009 ) [ arxiv:0906.5218 ] . b. aubert _ et al . _ [ babar collaboration ] , , 112002 ( 2008 ) [ arxiv:0807.2014 ] . e. van beveren and g. rupp , arxiv:1009.5191 . del amo sanchez _ et al . _ [ the babar collaboration ] , , 191801 ( 2010 ) [ arxiv:1002.4358 ] . e. guido [ babar collaboration ] , arxiv:0910.0423 . b. aubert _ et al . _ [ babar collaboration ] , , 081803 ( 2009 ) [ arxiv:0905.4539 [ hep - ex ] ] . h. p. nilles , m. srednicki and d. wyler , , 346 ( 1983 ) . j. m. frere , d. r. t. jones and s. raby , , 11 ( 1983 ) . j. p. derendinger and c. a. savoy , , 307 ( 1983 ) . j. r. ellis , j. f. gunion , h. e. haber , l. roszkowski and f. zwirner , , 844 ( 1989 ) . x. g. he , j. tandean and g. valencia , , 081802 ( 2007 ) [ arxiv : hep - ph/0610362 ] . j. p. lees _ et al . _ [ the babar collaboration ] , , 151802 ( 2010 ) [ arxiv:1001.1883 [ hep - ex ] ] . j. a. bicudo , j. e. f. ribeiro and a. v. nefediev , , 085026 ( 2002 ) [ arxiv : hep - ph/0201173 ] . p. hasenfratz and j. kuti , 75 ( 1978 ) . s. ono , , 307 ( 1984 ) . e. klempt and a. zaitsev , , 1 ( 2007 ) [ arxiv:0708.4016 ] . f. iddir and l. semlala , arxiv : hep - ph/0211289 . m. r. pennington , arxiv : hep - ph/0309228 . p. minkowski and w. ochs , arxiv : hep - ph/0209225 . m. boglione and m. r. pennington , , 1998 ( 1997 ) [ arxiv : hep - ph/9703257 ] . p. minkowski and w. ochs , arxiv : hep - ph/9905250 . c. j. morningstar and m. j. peardon , , 034509 ( 1999 ) [ arxiv : hep - lat/9901004 ] . e. b. gregory , a. c. irving , c. c. mcneile , s. miller and z. sroczynski , pos * ( lat2005 ) * , 027 ( 2006 ) [ arxiv : hep - lat/0510066 ] . b. a. li , , 034019 ( 2006 ) [ arxiv : hep - ph/0510093 ] . m. r. pennington , e - conf . c070910:106 ( 2007 ) [ arxiv:0711.1435 ] . r. dermisek and j. f. gunion , , 041801 ( 2005 ) [ arxiv : hep - ph/0502105 ] . y. nomura and j. thaler , , 075008 ( 2009 ) [ arxiv:0810.5397 [ hep - ph ] ] . r. dermisek and j. f. gunion , , 111701r ( 2006 ) [ arxiv : hep - ph/0510322 ] . r. dermisek , j. f. gunion and b. mcelrath , , 051105r ( 2007 ) [ arxiv : hep - ph/0612031 ] . h. park _ et al . _ [ hypercp collaboration ] , , 021801 ( 2005 ) [ arxiv : hep - ex/0501014 ] . m. l. mangano and p. nason , , 1373 ( 2007 ) [ arxiv:0704.1719 [ hep - ph ] ] . y. j. zhang and h. s. shao , arxiv:0911.1766 [ hep - ph ] . a. einstein , , 844 ( 1915 ) . s. l. glashow , nucl . * 22 * , 579 ( 1961 ) . s. weinberg , , 1264 ( 1967 ) . a. salam , in elementary particle theory , edited by n. svartholm ( almquist and wiksells , stockholm , 1969 ) , p. 367 . s. l. glashow , j. iliopoulos and l. maiani , , 1285 ( 1970 ) . n. a. trnqvist , arxiv : hep - ph/0204215 . n. a. trnqvist , , 145 ( 2005 ) [ arxiv : hep - ph/0504204 ] . p. minkowski and w. ochs , , 283 ( 1999 ) [ arxiv : hep - ph/9811518 ] . | we present evidence for the existence of a light scalar particle that most probably couples exclusively to gluons and quarks .
theoretical and phenomenological arguments are presented to support the existence of a light scalar boson for confinement and quark - pair creation .
previously observed interference effects allow to set a narrow window for the scalar s mass and also for its flavor - mass - dependent coupling to quarks . here , in order to find a direct signal indicating its production , we study published babar data on leptonic bottomonium decays , viz.the reactions @xmath0
@xmath1 @xmath2 @xmath1 @xmath3 ( and @xmath4 ) .
we observe a clear excess signal in the invariant - mass projections of @xmath0 and @xmath5 , which may be due to the emission of a so far unobserved scalar particle with a mass of about 38 mev . in the process of our analysis
, we also find an indication of the existence of a @xmath6 hybrid state at about 10.061 gev .
further signals could be interpreted as replicas with masses two and three times as large as the lightest scalar particle . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
we first review the main results concerning energetics and entropy production in kinetic networks . consider a system interacting with a reservoir at temperature @xmath0 and with states @xmath1 that have energies @xmath2 and are occupied with probabilities @xmath3 . the system dynamics is described by a markovian master equation @xmath4,\ ] ] where @xmath5 is the net probability current from site @xmath6 to site @xmath1 . the rates @xmath7 describing the reservoir induced transitions from @xmath6 to @xmath1 satisfy local detailed balance @xmath8 where @xmath9 and @xmath10 is a non - conservative thermodynamic force pointing from @xmath6 to @xmath1 which could be induced for instance by a non - equilibrium chemical reservoir or a non - conservative mechanical force . the state energies @xmath2 may change in time , @xmath11 , due to the action of an external agent . for this generic scenario an unambiguous formulation of non - equilibrium thermodynamics ensues @xcite . the first law of thermodynamics reads @xmath12 and expresses the fact that the average energy of the system , @xmath13 , changes due to three mechanisms : the driving work corresponding to the energy transferred to the system by the external agent @xmath14 the non - conservative work corresponding to the energy transferred to the system by the non - conservative forces @xmath15 and the heat corresponding to the energy transferred from the reservoir to the system which , using , can be written as @xmath16 the second law expresses the fact that the change in the total entropy or entropy production ( i.e. the sum of the change in the system shannon entropy @xmath17 plus the change in the entropy of the reservoir @xmath18 ) is always nonnegative @xmath19 : @xmath20 where @xmath21 is the nonnegative edge entropy production expressed as a flux times a force @xcite . the total entropy production may also be rewritten as @xmath22 where @xmath23 is the nonequilibrium free energy of the system whose change can in turn be split as @xmath24 where @xmath25 \ , dt.\ ] ] note that , in the absence of non - conservative forces , @xmath26 , @xmath27 we now introduce a generic reversible pumping mechanism that transfers probability between two states in a way that is indistinguishable from the poissonian transitions of a markovian dynamics . poissonian transitions are characterized by the following properties : _ i ) _ the probability transferred during a small time interval @xmath28 is @xmath29 , where @xmath30 is the rate of the transition ; and _ ii ) _ the occurrence of a transition in a given time interval is independent from the transitions that occurred in the past . when considering a pump induced by a periodic driving of very small period @xmath28 ( this condition will be made more precise below ) and transferring a probability @xmath29 during each cycle , then the pump will mimic poissonian transitions since the transitions that occurred in a given cycle are independent of those occurring in the other ones . to be precise , consider the setup depicted in fig . [ fig1 ] a. the system is made of observable states @xmath31 ( @xmath32 not shown in the figure ) and two hidden states @xmath33 connecting states @xmath34 . the transition rates satisfy local detailed balance . the transitions between @xmath33 and @xmath34 can be turned on and off by an external agent without any expenditure of work ( this can be achieved for instance for arrhenius rates @xmath35 by instantaneously raising and lowering the energy barriers @xmath36 ) and do not involve any non - conservative forces @xmath26 . the external agent also controls the two energies @xmath37 and @xmath38 . the operations performed by the external agent are cyclic and of period @xmath28 chosen to be much shorter than any time scale between the observable states , i.e. @xmath39 for @xmath40 . we first describe the process along path @xmath41 where the energy @xmath37 and the barriers between @xmath42 and @xmath43 and between @xmath42 and @xmath44 are modulated . the protocol starts with the two barriers closed and an energy @xmath45 , and proceeds as follows ( see fig . [ fig1 ] b ) : a ) the barrier @xmath46 is opened ; b ) the energy @xmath37 is quasistatically lowered to @xmath47 ; c ) the barrier @xmath46 is closed ; d ) the energy @xmath37 is changed to @xmath48 ; e ) the barrier @xmath49 is opened ; f ) the energy @xmath37 is quasistatically restored to its initial value @xmath50 ; g ) the barrier @xmath49 is closed to complete the cycle . we note that , while the barriers can be opened or closed instantaneously , the changes in @xmath37 are carried out quasistatically to minimize the entropy production , except for step d ) where state @xmath42 is not connected with states @xmath43 and @xmath44 and @xmath37 can be changed arbitrarily fast without compromising the reversibility of the cycle . and @xmath51 with respective energies @xmath37 and @xmath38 , which are modified by an external agent in a cyclic way . the agent can also open ( solid vertical bars ) and close ( dashed vertical bars ) the barriers connecting the network states @xmath43 and @xmath44 with the pump states @xmath42 and @xmath51 . b ) protocol followed by pump @xmath42 ( upper figure ) and pump @xmath51 ( lower figure ) . we have labeled the 7 steps of the @xmath42 protocol , according to the description in the main text . notice that the superscript of the energies indicates the network state @xmath1 which is in contact with the pump state @xmath33 when @xmath52 . the pump is reversible if the changes in the energies @xmath53 are carried out quasistatically with respect to the time scale of the transitions between network states and pump states and if @xmath54 and @xmath55 are appropriately chosen ( see and ) . the opening and closing of the barriers can be done instantaneously without compromising the reversibility of the process . ] the cycle is engineered in such a way that site @xmath42 is practically empty at the beginning and at the end of the cycle . therefore , initially @xmath56 and the probability to be on state @xmath34 is denoted by @xmath57 . during step a ) an irreversible probability leak occurs from @xmath56 and @xmath58 to @xmath59 we assume @xmath60 and neglect this leak : @xmath61 and @xmath62 ( see the discussion below on the different scales of energy and time in our model ) . during step b ) , a quasistatic reversible transfer of probability from state @xmath43 to state @xmath42 is performed . because the two states are in equilibrium with respect to each other , the respective occupation probabilities after step b ) are @xmath63 since , after step b ) , the barrier @xmath46 remains closed for the rest of the cycle , @xmath64 is the probability that will be transferred from site 1 to site 2 after the cycle is completed . for the pump to mimic poisson transitions , this probability must be of the order of the duration of the cycle @xmath28 . therefore , we impose the following scaling relationship between @xmath28 and the energy @xmath47 @xmath65 where @xmath66 is a finite rate which we will soon prove to be the effective rate of transitions from state 1 to state 2 . for the transitions to be poissonian we have to further impose that @xmath67 , which amounts to impose @xmath68 . in the following , we approximate all the expressions up to first order in @xmath28 , since this is the approximation that yields a markovian dynamics ruled by an effective master equation , once the pump is coarse grained . notice also that the initial energy @xmath50 should be even bigger than @xmath47 since it must lead to @xmath69 in order to justify neglecting the irreversible leak of step a ) . using the scaling , the transferred probability from @xmath43 to @xmath42 to first order in @xmath28 reads @xmath70 we now impose the following relation on @xmath48 @xmath71 which is equivalent to @xmath72 as a result , the probabilities @xmath64 and @xmath73 are in equilibrium when the barrier @xmath49 is opened in step e ) . hence , the entropy production is zero along this step as well as along step f ) and g ) . beside the initial probability leak in step a ) , which can be made arbitrary small , the remaining steps are reversible . we also note that in all previous expressions , the shifts in @xmath58 and @xmath73 , due to transitions with other observable states @xmath32 or due to the dynamics along path @xmath74 , are of order @xmath28 . they will therefore only affect terms of second order in @xmath28 and will not prevent the entropy production of the process to vanish up to first order in @xmath28 . we now turn to evaluating the work performed by the external agent on the system when changing the energy @xmath37 along steps b ) , d ) , and f ) . the remaining steps a ) , c ) , and e ) involve no work since only the barriers are changed . as every step involving work is quasistatic and reversible , the driving work can be calculated as a difference of equilibrium free energy . we find @xmath75 \simeq -ktp_a '' \nonumber \\ & & \hspace{-0.7 cm } w_d = p''_a\left[e_a^{(2)}-e_a^{(1)}\right ] \\ & & \hspace{-0.7 cm } w_f = -p_2kt\left[\ln \left(e^{-\beta e_2}\right)-\ln \left(e^{-\beta e_a^{(2)}}+e^{-\beta e_2}\right)\right ] \simeq ktp_a '' \nonumber . \end{aligned}\ ] ] the overall work along path @xmath41 can thus be written as @xmath76 = p''_a\left[e_2-e_1-kt\ln\frac{p_1}{p_2}\right ] .\label{wa}\end{aligned}\ ] ] the l.h.s . of this equation is the change of free energy in the system due to the probability @xmath77 transferred by the pump , confirming that the entropy production due to the pumping mechanism vanishes . we now turn to the process affecting path @xmath74 . the energy @xmath38 and the barriers between @xmath51 and @xmath43 and between @xmath51 and @xmath44 are changed in a similar way as along path @xmath41 ( see fig . [ fig1 ] c ) . the analysis for this part of the protocol is analogous to that of @xmath41 , and the resulting expressions are obtained by just swapping @xmath42 and @xmath51 as well as @xmath43 and @xmath44 . by combining the results obtained along the two paths , @xmath78 and @xmath79 , we find the first important result of this paper , namely that the effective rates @xmath66 and @xmath80 satisfy a local detailed balance relation @xmath81 which , contrary to the original rates , now contains an effective non - conservative force @xmath82 pointing from @xmath43 to @xmath44 . furthermore the total work performed by the pump during a cycle is given by @xmath83+p_b''\left[e_b^{(1)}-e_b^{(2)}\right ] \nonumber \\ & & \hspace{0.3 cm } = ( p_a''-p_b'')\left[-kt\ln\frac{p_1}{p_2}+e_2-e_1\right ] \nonumber \\ & & \hspace{0.3 cm } = j_{21}\tau\,\left[-kt\ln\frac{p_1}{p_2}+e_2-e_1\right ] = \delta{\cal f}_{21}=\delta{\cal f}_{12 } , \label{wdrtot}\end{aligned}\ ] ] where we used with @xmath84 for the last equality . we now turn to the comparison between the real entropy production of the full network which includes the pumping states and the coarse grained entropy production obtained by just considering the dynamics on the observable states . for simplicity , we assume no non - conservative force besides the effective force @xmath85 emerging at the coarse grained level . examples with non - conservative forces will be provided in the applications . we consider pumping cycles of duration @xmath28 much smaller than the characteristic time of the dynamics of the coarse grained network . at the coarse grained level of description , the observed states are not driven and the only non - conservative force is the effective one induced by the pump . the total work in a cycle is therefore @xmath86 and , using , the entropy production per cycle reads @xmath87 where the sum runs over the observable states @xmath40 and @xmath88 is given by eq . with @xmath84 . on the other hand , in the full network all forces are conservative . using and , the true entropy production is given by @xmath89 when calculating the differential @xmath90 over a cycle of the pump operation , the contributions to @xmath91 from the hidden states @xmath33 vanish since they are empty at the beginning and at the end of the cycle . one finds @xmath92=\sum_{i < j}\delta{\cal f}_{ij}.\ ] ] inserting in , we get that @xmath93 comparing this result with , we observe that the link @xmath94 does not contribute to the entropy production , confirming the reversibility of the pumping mechanism . our second important result is that the true entropy production overestimates the coarse grained one : @xmath95 this result follows from comparing with using the inequality @xmath96 of special interest is the entropy production rate when the system reaches a stationary state . in this case , @xmath97 in , and the entropy production in the coarse grained network is given by the non - conservative work , whereas the real entropy production is proportional to the driving work . the respective entropy production rates are : @xmath98 \label{si } .\end{aligned}\ ] ] note that @xmath99 may vanish even for a finite current @xmath100 ( an example is provided below ) . the driving protocol that we have introduced to pump reversibly between a pair of observable states can be designed for any system with preassigned effective rates @xmath101 and operating in the stationary regime . indeed , the choice of @xmath66 and @xmath80 determines the effective force @xmath85 via , and along with the rest of the markov chain , also determines the stationary values of @xmath58 and @xmath73 . from these stationary values we set @xmath48 and @xmath102 using eq . and we set @xmath47 and @xmath103 using . it should be clear that our procedure can be easily generalized to systems with pumps located between several pairs of observable states and/or to systems with non - conservative forces besides the ones induced by the pumps . some examples of this are provided below . as a first example , we consider a system of @xmath104 states @xmath105 connected as a ring ( see fig . [ fig2 ] a ) . the states energies are all zero @xmath106 , no non - conservative forces act on the system , and a hidden pump is inserted between states @xmath43 and @xmath44 . local detailed balance implies equal rates @xmath107 for all transitions except @xmath66 and @xmath80 which satisfy @xmath108 , where @xmath85 is the effective force induced by the pump . the stationary state of the system is @xmath109j}{(w_{21}-w_{12})w } \nonumber \\ p_{n } & = & p_1+(n+1-n)\frac{j}{w } , \ \ n=2,\dots , n,\end{aligned}\ ] ] where the clockwise stationary current @xmath110 is given by @xmath111}{2nw+n(n-1)(w_{21}+w_{12})}.\ ] ] the real entropy production rate is proportional to the driving work performed by the pump and reads ( see ) @xmath112,\ ] ] whereas the coarse grained entropy production rate is given by ( see ) @xmath113 if the network transitions are much slower than the effective rates from 1 to 2 , i.e. if @xmath114 , then the coarse grained entropy coincides with the real one @xmath115 . the same is true if @xmath116 . on the other hand if @xmath117 , the real entropy production vanishes despite the fact that the network exhibits a finite dissipative current @xmath110 giving rise to an apparent entropy production @xmath118 . these results generalize to any conservative network obeying detailed balance and containing a hidden pump in the edge @xmath94 . if the rates along the normal edges are much larger than the effective rates along the pumping edge ( @xmath80 and @xmath66 ) , the whole chain will be at equilibrium with respect to the energy landscape @xmath2 , @xmath119 . then , according to , @xmath120 , whereas a finite current @xmath121 gives rise an apparent entropy production @xmath122 . this remarkable result is not in contradiction with any fundamental law of thermodynamics . the dissipationless finite current arises from the large separation of time scales : the current is finite over the time scales @xmath123 of the dynamics on the observable states but is induced quasistatically over the internal time scale of the pump . a similar phenomenon has been previously discussed in the context of adiabatic pumps @xcite . in the direction of the black arrow . the examples are : a ) a ring with a pump connecting two network states , 1 and 2 . b ) a kinetic network that produces high free energy molecules @xmath124 from low free energy molecules @xmath125 ( @xmath126 ) . c ) a ring with pumps at every link , working against a uniform force @xmath127.,title="fig:",width=275,height=249 ] [ fig2 ] we now consider an enzyme switching between two conformational states @xmath43 and @xmath44 with the same energy @xmath128 . the enzyme jumps due to two different mechanisms with respective rates @xmath129 and @xmath130 . the first is induced by a hidden pump generating an apparent effective force @xmath131 from @xmath43 to @xmath44 , and the second is induced by a chemical reaction @xmath132 such that @xmath133 ( see fig . [ fig2 ] b ) . when operating alone , both mechanisms favor their respective transition towards state @xmath44 . however , when operating simultaneously with @xmath134 , the pump can revert the spontaneous direction of the chemical reaction and thus generate high free energy molecules @xmath124 at a rate given by the ( clockwise ) stationary current @xmath135 where the stationary probabilities read @xmath136 the current may be rewritten as @xmath137 } { w^{\rm pump}_{12}+w^{\rm reac}_{12}+w^{\rm pump}_{21}+w^{\rm reac}_{21}}.\ ] ] the coarse grained and the real entropy production are obtained by adding the non - conservative work @xmath138 to eqs . and respectively : @xmath139 the real entropy production ranges from reversibility , @xmath120 , if @xmath140 , to @xmath141 if the pump transfers probability much faster than the reaction , @xmath142 . in the former case it is therefore possible to produce molecules of @xmath124 with very high efficiency since the synthetase can work at finite rate with a vanishing entropy production . as mentioned before , this does no contradict the second law of thermodynamics since the current occurs on much slower time scale than the driving . one can even show that , for fixed @xmath85 , the efficiency at maximum power ( over @xmath143 ) tends to 1 when @xmath144 . to demonstrate that the reversible behavior can be achieved for a large but reasonable separation of time scales , we numerically solved the master equation of the synthetase using arrhenius rates for the pumping rates @xmath145 , where @xmath36 . this will also help us to show in detail how to build a reversible pump to be inserted in a given network . energy units are measured in @xmath146 and time units in @xmath147 . we consider a reaction with a difference in chemical potential between species @xmath124 and @xmath125 of @xmath148 and with rates ranging between @xmath149 @xmath150 and @xmath151 6.065 . our goal is to build a pump exerting a force @xmath152 with effective rates @xmath153 and @xmath154 so that the system will produce @xmath124 molecules at a rate @xmath110 ranging between 0 ( for @xmath149 ) and 0.3 ( for @xmath155 ) , as obtained from eq . . to do so , we first set the cycle time to @xmath156 , i.e. small enough for the pump to generate poisson rates at the coarse - grained level . according to and the equivalent equation for pump @xmath74 , this together with @xmath128 fixes the energy of the hidden states @xmath42 and @xmath51 at the end of step b ) to @xmath157 and @xmath158 . we then fix the energies after step d ) according to to @xmath159 and @xmath160 which depend on the specific value of @xmath161 . for instance for @xmath162 , we get that @xmath163 and @xmath164 . finally , we set the time scale of the internal transitions in the pump by fixing the value of the open barriers @xmath165 ( @xmath166 and @xmath167 ) . in our numerical solution we open and close the barriers using linear ramps ranging from @xmath168 to @xmath169 . the protocol for the energies and barriers is depicted in the inset of figure [ figsim ] . the entropy production of the system @xmath170 obtained by full numerical integration ( black points connected by blue lines ) is depicted in fig . [ figsim ] . it approaches , but still differs from , the entropy production of the ideal reversible pump @xmath171 ( red curve ) and is clearly below the coarse - grained entropy production @xmath172 ( black curve ) . the irreversibility in the pump causing the discrepancy between @xmath170 and @xmath171 mainly occurs at the end of step b ) and the beginning of step f ) . our final example is a @xmath104-state ring with energies @xmath173 . each edge contains a hidden pump generating a force @xmath174 ( clockwise ) and is subjected to a constant external torque @xmath127 ( counterclockwise ) operating against the pumps ( see fig . [ fig2 ] c ) . if all the pumps are identical , then the stationary state is uniform @xmath175 and the ( clockwise ) current reads @xmath176.\ ] ] it is positive for @xmath177 meaning that the pumps generate a finite speed rotation against the torque . as in the previous example , the coarse - grained entropy production can be derived by adding to eq . the non - conservative work performed on the @xmath104 edges of the motor @xmath178 : @xmath179 it is a non - negative quantity which only vanishes at zero power @xmath180 . the calculation of the real entropy production is more subtle since , contrary to what happens for the synthetase , the external force @xmath127 affects the internal transitions of the pumps , @xmath46 , @xmath181 , @xmath182 , @xmath183 . in order to use the calculations for the driving work @xmath184 made in the previous section to the pump between site @xmath1 and @xmath185 , we must set @xmath186 . notice that the actual energy of site @xmath1 is zero , because the effect of the torque is borne by the external force . however , the work performed by the driving in the pump between site @xmath1 and @xmath185 is given by with @xmath186 and @xmath187 . the total driving work obtained by summing over the @xmath104 pumps is therfore @xmath188 and the real entropy production rate in the stationary regime vanishes @xmath189 remarkably , this motor is able to operate reversibly against any external torque @xmath127 . we have proposed a reversible time - dependent driving mechanism ( called reversible pump ) which can be inserted between any two states of a kinetic network . when coarse grained , this pump gives rise to a forward and backward poissonian rate between the two states . the ratio of these effective rates satisfies a local detailed balance displaying an emergent nonconservative force . remarkably , these pumps can always be engineered in such a way to operate reversibly when inserted in any steady state kinetic network . we found that contrary to common belief , the coarse grained markovian kinetics generated by our pumps exhibits an entropy production which is always larger than the real one . we exploit this fact to propose several hyper efficient " setups which produce finite currents ( and thus finite entropy production ) at the coarse grained level while the real entropy production vanishes . the origin of this surprising phenomenon is that coarse graining the driving affects the symmetry of the system under time reversal . entropy production is a measure of the probabilistic distinguishability between a process and its time reversal @xcite . to define the time - reversed process one must consider the time - reversed driving . but if the information concerning the driving is lost during the coarse graining procedure as is the case here , the time - reversal operation at the coarse grained level does not relate anymore to the real time - reversal operation at the underlying level . a similar phenomenon may occur if hidden variables which are odd under time reversal are considered , such as velocity , angular momentum , or magnetic moment @xcite . in fact , an external driving can be implemented by a large mass with a non zero initial velocity @xcite . our setup has also an intriguing relation with information engines that use the information gathered in a measurement to extract work , in the spirit of the celebrated maxwell demon . in ref . @xcite a driven kinetic scheme that works as a maxwell demon was introduced . when the demon is coarse grained , the resulting dynamics is markovian and mimics the dynamics of a chemical motor . the scheme is not able to always work reversibly and it is more restrictive than the one presented here , but the demon is able to run the motor with less entropy production than chemical fuel . in this case , the hidden states are long lifetime states ( with respect to the internal time scale of the motor ) featuring the strong correlation between the motor and the demon implied by the measurement @xcite . it would be interesting to find whether our general scheme also admits an interpretation in terms information . our pumping mechanism is based on two ingredients , the presence of time asymmetric driving ( an `` odd variables '' ) and a large separation of time scales . these can yield a dramatic enhancement of the performance of a kinetic network . it is an open question whether these two ingredients can be helpful for designing more efficient chemical motors and nanodevices or whether they are already present in protein motors and other biological processes . is supported by the national research fund , luxembourg in the frame of project fnr / a11/02 . j.m.r.p . acknowledges financial support from grant enfasis ( fis2011 - 22644 , spanish government ) . this work also benefited from the cost action mp1209 . puglisi a , pigolotti s , rondoni l , vulpiani a ( 20100 ) entropy production and coarse graining in markov processes . _ journal of statistical mechanics : theory and experiment _ p stacks.iop.org/jstat/2010/p05015 . leonard t , lander b , seifert u , speck t ( 2013 ) stochastic thermodynamics of fluctuating density fields : non - equilibrium free energy differences under coarse - graining . _ the journal of chemical physics _ 139:204109 . | we show that a reversible pumping mechanism operating between two states of a kinetic network can give rise to poisson transitions between these two states .
an external observer , for whom the pumping mechanism is not accessible , will observe a markov chain satisfying local detailed balance with an emerging effective force induced by the hidden pump . due to the reversibility of the pump
, the actual entropy production turns out to be lower than the coarse grained entropy production estimated from the flows and affinities of the resulting markov chain .
moreover , in presence of a large time scale separation between the fast pumping dynamics and the slow network dynamics , a finite current with zero dissipation may be produced .
we make use of these general results to build a synthetase - like kinetic scheme able to reversibly produce high free energy molecules at a finite rate and a rotatory motor achieving 100% efficiency at finite speed .
entropy production measures dissipation and is the key quantity to assess the performance of a kinetic process .
the efficiency of chemical motors or of biochemical processes such as metabolic cycles , replication , transcription , or proofreading , typically achieves its maximum value when the entropy production is minimal .
hill , in his classic work @xcite on the transduction of free energy in chemical reactions , provided the basic tools to calculate the entropy production of processes modeled by kinetic networks .
closely related results were also found by schnakenberg @xcite .
they showed that the entropy production in a network consists of a sum of positive edge contributions , each expressed as the product of a probability flux across the edge times an edge affinity ( or thermodynamic force ) .
their theoretical framework constitutes the basis of stochastic thermodynamics @xcite which has become central for the study of molecular machines @xcite . in many applications
the observer has only a partial access to the kinetic network .
some states are hidden and the resulting description of the system becomes `` coarse grained '' .
the thermodynamic implications of coarse graining is an active field of research @xcite .
the coarse grained entropy production is typically lower than the actual one since it misses the positive contribution of the hidden edges .
this result is true for autonomous systems if the hidden variables are even under time reversal .
it can be proved in various ways @xcite and remains true even when the coarse grained description is no longer markovian @xcite .
indications that odd hidden variables do not satisfy this result were analyzed in the context of langevin equations where velocities were coarse grained @xcite . in this paper
we prove that for systems driven by an external time - dependent force , the entropy production at a coarse grained level of description may overestimate the actual entropy production .
the driving plays the analogous role of an odd hidden variable when it is not invariant under time reversal .
we exploit this result to build hyper - efficient pumps .
to do so we consider systems with hidden states driven by an external cyclic time - dependent force generating currents between the apparent states .
the driving is based on adiabatic pumping previously introduced in the literature @xcite but gives rise in our case to a markovian dynamics at the coarse grained level .
we show that our reversible pumps can be used to generate currents again a bias with zero entropy production , without contradicting the second law of thermodynamics . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
n 157b ( henize , 1956 ) is a crab - type supernova remnant ( snr ) just 7 arcmin from the center of 30 doradus ( bode , 1801 ) in the large magellanic cloud ( lmc ) . it contains a 16-ms x - ray pulsar undetected at any other wavelength ( marshall et al . there is bright non - thermal x - ray emission with structure on arcsec scales just around the pulsar with an extended feature off to the northwest ( wang and gotthelf , 1998a , 1998b ; wang et al . there is bright non - thermal radio emission from the extended feature but not at the pulsar location ( lazendic et al . we shall call the extended emission region the pulsar wind nebula ( pwn ) . the overall struture suggests that the pulsar is moving toward the southeast . there is also extended radio emission toward the south that gives a hint of a shell , suggesting that the remnant may be in transition to becoming a composite remnant with a shell and a pulsar wind nebula . the differences in the radio and x - ray structures plus the apparent large motion of the pulsar make this snr unusual . we shall describe its properties and then discuss the implications of the data . figures 1 and 2 show the simularities of the radio and x - ray emission of the pwn component of n 157b but a striking difference toward the pulsar . the radio emission in the pulsar wind component sits on a plateau of the rest of the remnant . fine structure in the pwn appears very similar in both wavelength ranges although the radio emission extends further northwest . this structure probably represents clumpiness in the interstellar medium or in pre - explosion mass loss of the progenitor . the peak in the x - ray emission in the compact source around the pulsar is 13 times the brightness of the peak in the pwn but in the radio there is nothing seen above the residual brightness of the pwn . the pulsar lies at @xmath1 and @xmath2 ( wang and gotthelf , 1988b ) about 16@xmath3 out from the center along the se axis of the tail . the pulsar is about 1@xmath3 closer to the center of the tail than the peak of the x - ray emission . . the thin line shows the location of slices shown in figure 3 and the cross is the position of the pulsar.,width=340 ] wide centered on the radio peak and the ones through the pulsar position are 5@xmath3 wide.,width=340 ] wide centered on the radio peak and the ones through the pulsar position are 5@xmath3 wide.,width=340 ] further details of the emission are revealed by the slices presented in figures 3 and 4 . the emission from the pwn is clearly more extended in all directions at radio wavelengths than at x - ray ones . the overall symmetry is the same , however , with an approximately elliptical shape centered about a point at @xmath4 and @xmath5 . the radio emission falls off uniformly out to a semi - major axis extent of 20@xmath3 in the se - nw direction and 10@xmath3 in the ne - sw direction . there is no sign of the pulsar or any enhancement in the radio emission , @xmath6 mjy beam@xmath7 , at the pulsar s position of @xmath1 and @xmath2 ( wang and gotthelf , 1988b ) . the non - thermal x - rays around the pulsar position , on the other hand , show a strong approximately elliptical component , about 7@xmath3 @xmath8 3@xmath3 with its long axis perpendicular to the long axis of the pwn tail . wang and gotthelf ( 1998 ) suggested that this small source could be a bow - shock from the particles leaving the moving pulsar . we shall henceforth call that structure the shock region . from the inner ( nw ) edge of this shock , the x - ray emission first decreases somewhat and then increases gradually toward the radio center of the pwn but peaks 3@xmath3 before the radio and then falls sharply toward the northwest . to compare the actual brightnesses of the features , we show their spectra in figure 5 . the squares represent the integrated values for the radio emission of the entire snr . they give a spectral index , @xmath9 , of @xmath10 , where the flux density @xmath11 @xmath12 @xmath13 ( lazendic et al . 2000 ) . xmm - newton spectra , that can not resolve angular detail , show that most of the x - ray emission from the snr has a steep power - law spectrum with @xmath14 although some thermal emission is present as well ( dennerl et al . they do not give a value for the actual x - ray flux . for the pwn , the lower frequency radio data do not have sufficient resolution for a good separation of the components so we report only the 4.8- and 8.6-ghz results ( lazendic et al . the spectral index for the pwn is more uncertain because of the snr background . the value of @xmath15 could easily be the same as that of the whole snr . the error of the spectral index for this fit to only the two data points tries to take into account the uncertainty in evaluation of the background . we can not determine the radio spectrum of the shock region because it is not detected . we do show the upper limits for its flux density at the two radio frequencies . the x - ray spectra are from the paper by wang and gotthelf ( 1998b ) . their formal fits give values of @xmath16 for the pwn and @xmath17 for the bow - shock . realizing that these errors are only for the formal fits to the data , we suggest that the slopes of both components could be the same but that of the pwn can not be much steeper than that of the shock region . for analysis , we divide the snr into three parts as outlined by wang and gotthelf ( 1998a ) and discussed above : the elliptical pulsar wind nebula extending northwest from the pulsar with major and minor axes of @xmath18 pc ; the brignt shock region ( @xmath19 pc ) centered just outside the pulsar ; and the entire snr , about @xmath20 pc across , which extends well beyond the images in figures 1 and 2 . assuming that the pulsar has been moving southeastward from an explosion site at the radio peak , we can estimate its speed using the characteristic spin - down age of 5000 years ( marshall et al . 1998 ; wang and gotthelf , 1998b ) . to have moved 4 pc , the pulsar has a speed of 800 km sec@xmath7 , large but not impossible for a pulsar ( arzoumanian et al . because the pwn emission falls off in all directions from its center , including toward the pulsar , we are led to the conclusion that there was a brief period of particle injection by the pulsar , lasting up to about 1000 years after the explosion occurred . that period would account for the @xmath21 3/4 pc shift of the x - ray peak from the radio one and also allow for the apparent aging of the electrons on the outer edge of the pwn relative to the center . as higher energy electrons decay more rapidly from their synchrotron emission , those further from the pulsar should be the oldest and thus have reduced x - ray emission relative to the radio and steeper spectra ( pacini and salvati , 1973 ; reynolds and chevalier , 1984 ) . the particle injection and/or stimulation can not have lasted much longer , however , or the emission in the pwn would still be bright toward the southeast between the pwn peak and the current location of the pulsar . the emission from the shock region just around the pulsar appears to have arisen from a second injection of particles . the x - ray brightness rises so drastically just there and appears to sit on top of the decreasing pwn emission in that direction . the radio emission is below detection level . we also note that if the injection of particles has been continuous , the x - ray spectrum of the region closest to the pulsar should be flatter ( harder ) than that of the region further from the current pulsar position , but that is not the case . thus the newer particles around the pulsar must have been injected with a different spectrum or be interacting with a very different medium that those injected earlier near the center of the overall pwn . finally , the center of this shock source is slightly ( about 1/4 pc ) outside the pulsar position . these conditions lead us to believe that the object is indeed a shock region caused by the supersonic motion of the particles coming from the pulsar moving at about 800 km sec@xmath7 plus their injection speed and perhaps interacting with a reverse shock from the intitial blast wave . without a more accurate separation of the thermal component of the x - ray emission in that region to evaluate the temperature and density of the gas , we can not determine the actual sound speed but the relative motion could be enough to generate a strong shock wave and compressed magnetic fields across which to accelerate particles to even greater energies than they get from the pulsar . the pulsar may have also encountered a density enhancement which will further increase the total emission to produce the high observed x - ray brightness . indications of a clumpy medium around n 157b include many h@xmath9 filaments and a very low polarization of the radio emission , presumably caused by significant faraday rotation ( lazendic et al . the other lumps in the pwn may represent temporary increases due to previous brief episodes of acceleration near density enhancements . we do nt know the radio spectral index of the shock region so we do nt know if the energy injection in that spectral range is the same for both episodes . to get an approximate ratio of radio to x - ray luminosities , we will assume that the radio spectrum of the shock region is the same as the rest of the snr , @xmath22 . extrapolation of any value steeper than about @xmath230.3 would require less x - ray emission than observed . with the adopted spectral index , we calculate an upper limit to the radio luminosity between 10@xmath24 and 10@xmath25 hz of @xmath26 erg sec@xmath7 or about 1/2000 of the x - ray luminosity in the 0.2 4 kev band ( wang et al . this upper limit is a low ratio for either a pulsar wind alone , e.g. 0540 - 693 has a ratio of radio to x - ray luminosity of 1/120 , or for an snr with shock generated x - ray emission , e.g. ad1006 has a ratio of about 1/100 ( taken from allen et al . perhaps the combined processes may create a harder spectrum at low energies but then a sharper break at the transition between the emission in the two frequency ranges . while still qualitative , this description of multiple episodes of particle injection from the pulsar is the only one we can come up with to explain all the data . better radio resolution and sensitivity plus improved spectra in all wavelength bands , including a detection of infrared and optical continuum emission , will be important for checking these ideas . we thank paul plucinsky , steve reynolds , brian fields , you - hua chu , douglas bock , martin guerrero , and pat slane for help and ideas . this research is supported by nasa grant nag 5 - 11159 . | the supernova remnant n 157b contains a pulsar and three distinct synchrotron components with rather unusual properties .
1 ) a somewhat irregular elliptical pulsar wind nebula ( pwn ) visible in both x - ray and radio wavelengths . the nebula is quite symmetrical with an extent of about 10 @xmath0 5 parsecs but offset along the long axis by about 4 pc from the pulsar position .
it is apparently the result of a short - lived injection of energetic particles , perhaps starting at the time of explosion .
2 ) a very bright x - ray shock region located just outside the pulsar position in the edge of the pwn . this is undetected in the radio .
we attribute this to a new burst of particles from the pulsar suggesting there are multiple episodes rather than continuous injection .
3 ) the beginning of a radio synchrotron shell on the southern side of the snr where thermal x - rays appear to arise suggesting that n 157b is starting to become a composite snr . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
linear optics is a useful tool for constructing quantum circuits by utilizing optical interference @xcite . a number of quantum gates @xcite were demonstrated with linear optics . however , the operation of these gates has been non - deterministic , because deterministic operation requires nonlinear interactions between photons . the success probability of the linear optical cnot gate is at most 1/4 @xcite , for instance . therefore , it is quite difficult to implement large - scale quantum computations with linear optics such as shor s factorization algorithm@xcite , because the more quantum gates the quantum algorithm requires , the less the success probability . it is becoming more important to implement deterministic quantum operations . we have found that the quantum leader election ( qle ) protocol can be operated deterministically only with linear optics , and have demonstrated nearly deterministic operation in an experiment with two - photon interference . before considering the qle protocol , let us briefly introduce distributed computing . a distributed computing system is a set of computers connected via network channels and it offers the fault - tolerance and the scalability of the system . the system operates autonomously with a distributed algorithm@xcite that effectively coordinates the computers . in a lot of distributed algorithms , computers generate a @xmath0 , which is a kind of script that prescribes the operation or gives permission for them to access shared data . a computer does some operations based on the token and relays it to the next computer . then , the next computer receives the token and does operations , and so on . after going round the whole network , one distributed algorithm is completed . if the token is lost during the operation , a unique computer must be specified to restore the only one token , otherwise two or more tokens that tangles the operation may be generated . such a unique computer is called the leader , and the decision of the leader is known as the leader election problem @xcite . since only one leader should be chosen before performing any distributed algorithms , the leader election problem is one of the most fundamental problems in the distributed computing . many studies on the leader election problem have been done in the long history of computer science . we can summarize the leader election problem as : leader election problem : : @xcite + _ let n be the number of computers connected to the network . each computer i @xmath1 has a state @xmath2 and is initially set to identical values @xmath3 . set the unique computer @xmath4 , to @xmath5 , otherwise , @xmath6 @xmath7 _ in this problem , there are no third persons such as human beings to determine the leader ; therefore , the computers must do the same operations during the protocol . if the network is non - anonymous , ( @xmath8 for @xmath9 ) , in which all computers have an identifier ( i.e. , mac address ) , this protocol can be solved quite easily by comparing each identifier . the computer that has the largest value of identifier can be elected to the leader . however if the network is anonymous ( @xmath10 ) , in which the computers have no identifiers and they are in an identical initial state , the leader election is no longer trivial . they have to distinguish themselves and finally obtain different identifiers to distinguish one another using the same operation . in other words , they have to obtain asymmetric states from the symmetric state after identical operations . it is widely recognized @xcite that no classical algorithm can offer a deterministic solution within a finite time to this problem . the possible classical algorithms are stochastic @xcite , where each computer automatically generates an identifier at random . then , some computers generate the same identifiers with a certain probability , and leader election fails . in contrast , tani @xmath11 proposed a new quantum algorithm @xcite that could solve this problem deterministically within a finite time . the process for the quantum protocol is as follows : quantum leader election protocol : : + computers with identical quantum states ( @xmath12 ) are connected via quantum and classical communication channels . a unique leader is elected after all computers repeats the following processes for n-1 times for n parties : + 1 . entangled state generation , 2 . quantum communication , 3 . local quantum operation , and 4 . measurement and classical communication . the quantum state of each computer is identical in this protocol . after quantum communication , the computers build a quantum correlation by sharing entanglement . they can finally derive a non - identical measurement result . we have demonstrated the two - party qle protocol , which makes the implementation simple , because it only requires one execution of processes ( i ) @xmath13 ( iv ) without any ancillary qubits . and @xmath14,width=321 ] there is a schematic of the two - party qle circuit is shown in fig.[cir ] . since all parties must perform exactly the same operations , alice s setup is line symmetric to bob s . both alice and bob have two optical ports defined as the @xmath15 and the @xmath14 , respectively , and the four detectors a1 , a2 , b1 , and b2 are named according to the symmetry shown in fig.[cir ] here , the qubit is represented by the photon number , where single photon state @xmath16 represents qubit 1 and vacuum state @xmath17 represents a qubit 0 . the circuit consists of four parts ( i ) @xmath13 ( iv ) , carrying out the qle process are given as follows : [ [ entangled - state - generation ] ] entangled state generation + + + + + + + + + + + + + + + + + + + + + + + + + + + alice and bob must prepare identical symmetric state @xmath18 , which corresponds to a single photon pair emerged in their guest ports . note that photons must be indistinguishable with each other , otherwise we would contravene the identity of initial state . then single photon entangled states are generated at alice and bob by the 50:50 beam splitters ( bs1 ) as shown in fig.[cir ] . @xmath19 [ [ quantum - communication ] ] quantum communication + + + + + + + + + + + + + + + + + + + + + + these entangled states generated in ( i ) are shared by alice and bob after they exchange their guest ports @xmath20\notag\\ \xrightarrow{\text{\tiny exchanging}}\ket{\psi_{\rm ii}}\equiv & \frac{1}{2}\bigl[\ket{1_h0_g}_\text{a}\ket{1_h0_g}_\text{b}+\ket{1_h1_g}_\text{a}\ket{0_h0_g}_\text{b}\notag\\ & \hspace{1.5pc}+\ket{0_h0_g}_\text{a}\ket{1_h1_g}_\text{b}+\ket{0_h1_g}_\text{a}\ket{0_h1_g}_\text{b}\bigr].\end{aligned}\ ] ] [ [ local - quantum - operation ] ] local quantum operation + + + + + + + + + + + + + + + + + + + + + + + + a @xmath21-shift in the guest ports results in @xmath22 where @xmath23 represents the bell states , which is defined as @xmath24 $ ] and @xmath25 represents the existence of two photons in alice ( bob ) . [ [ measurement ] ] measurement + + + + + + + + + + + + the @xmath26 in eq . indicates that the state is no longer symmetric between alice and bob . then , they obtain different measurement results by distinguishing @xmath23 and @xmath27 states from the others by two - photon interference . we combine the host and the guest ports with a beam splitter ( bs2 ) as shown in fig.[cir ] . the beam splitters transform the @xmath26state into @xmath28\notag \\ & + \frac{1}{2}\bigl [ \underbrace{\ket{1_\text{a1}0_\text{a2}0_\text{b1}1_\text{b2}}}_{\uparrow \ket{\psi^+}_\text{a}\ket{\psi^-}_\text{b } } + \underbrace{\ket{0_\text{a1}1_\text{a2}1_\text{b1}0_\text{b2}}}_{\uparrow \ket{\psi^-}_\text{a}\ket{\psi^+}_\text{b}}\bigr].\label{eq4}\end{aligned}\ ] ] .expected results of leader election for all expected combinations of four detectors(a1,a2,b1 , and b2 ) : ( i ) either alice or bob received two photons . ( iii ) alice detects photon at detector a1 and bob detects it at detector b2 and vice versa . [ cols="^,^,^,^,^,<,^",options="header " , ] + the number of single counts ( i ) that we observed in the experiment can be written as @xmath29 where @xmath30 indicates the number of generated photon pairs soon after the nonlinear crystal . single counts consist of both one - photon detection , and two - photon detection and the number of two - photon detections can be written as @xmath31 + then , we can estimate the number of two - photon detections @xmath32 by using the value of @xmath33 measured in a different experiment as @xmath34 equation implies that the actual contribution of two - photon detections in single counts can be 1% . the corrected coincidence counts are plotted in fig.[baro](a ) . the theoretically predicted values from table [ table3 ] have been shown in fig.[baro](b ) for comparison . figure[baro](a ) shows good agreement with ( b ) . bars ( i ) and ( iii ) in fig.[baro ] indicate corrected two - photon detection and @xmath23 detection , respectively . in these cases leader election is successfully achieved . bars ( ii ) and ( iv ) , on the other hand , result from imperfect two - photon interference . however , bars ( ii ) can be regarded as successful events because the detection of alice and bob is different . bars ( iv ) indicates that the same detector ( a1&b1 , and a2&b2 ) clicked on alice and bob , then leader election failed . hence , the imperfect two - photon interference is responsible for the performance of the experimental qle protocol . since we focus only on the two photon events , the success probability per trial of qle circuit can be calculated by : @xmath35 where @xmath36 represent the counts of ( i)@xmath13(iv ) . to calculate the success probability from eq . , we must derive the number of two - photon detections n@xmath37 . since the n@xmath37 counts are obtained from rough estimates of losses as previously , we can not precisely calculate the success probability . therefore , we rewrite eq . using the coincidence probabilities between four detectors , derived in appendix [ app1 ] , as @xmath38\label{eq43}\\ & \hspace{3.2 cm } \gamma\equiv r / t \notag\end{aligned}\ ] ] where @xmath39 and @xmath40 correspond to the branching ratio of beam splitters and the visibility of two - photon interference . using eq . , we can evaluate performance of qle circuit with the imperfection of two - photon interference , which is characterized by visibility @xmath40 . to estimate the visibility @xmath40 , we observed the two - photon interference fringes shown in fig.[fringe ] . we recorded two patterns of coincidence counts : ( iii ) a1&b2 , and ( iv ) a2&b2 , which correspond to success and failure , respectively , as a function of the voltage applied to pzt . both coincidence counts fit the sinusoidal curves with visibility @xmath41 . hence , the success probability of leader election is calculated as [email protected]% , and the operation is therefore almost deterministic . let us examine how advantageous the experimental results were over the classical limit . to do this , we should not consider only success probability but also communication complexity . communication complexity is characterized by the costs of communications between computers to complete algorithms . it is commonly recognized in computer science that performance of distributed algorithms can be evaluated with the communication complexity . as shown in appendix [ app2 ] , the optimal classical algorithm involves repeating and comparing the one - bit random number generated by both alice and bob until they generate a different number . the expected minimum costs of communication @xmath43 , to complete leader election is calculated as 4 . the quantum leader election , on the other hand , is completed by operating the quantum circuits shown in fig . as noted in sec.ii , alice and bob can finish leader election without the classical communication . the total communication costs between alice and bob is 2-qubit quantum communication to exchange their guest ports . it is clear that ideal quantum leader election is advantageous over classical algorithms because leader election is completed deterministically only with 2-qubit quantum communication . however , the two photon interference is not perfect in the experiment . that causes a failure of qle with a finite probability ( [email protected]% per one trial ) . in the circuit depicted by fig2 , the detection pattern for alice and bob become the same ( ( iv ) a1&b1 or a2&b2 ) with certain probability . then , if alice receives a single photon at `` a1 '' , bob might receive one at `` b1 '' . she can not determine at which detector bob has actually received a single photon . since the leader must be elected uniquely , only the successful detection event ( iii ) must be _ post - selected_. therefore , alice and bob have to inform one another of their detection events using 2-bit classical communication to verify their detection events . the expected communication costs can be calculated along the flow chart given in fig.[flow ] : alice and bob check the photon number they received . if they received two photons with a probability of @xmath44 , then the leader election is completed only with 2-qubit quantum communication . otherwise , they need to verify the results with 2-bit classical communication . if their detection event is ( iii ) with a probability of @xmath45 , leader election is completed with a total of 4-bit communication , otherwise it fails with a probability of @xmath46 . the probability of failure until the @xmath47 - 1 th trial is written by @xmath48 and the amount of communication to complete leader election at the @xmath47-th trial before and after the verification process are written as @xmath49 and @xmath50 . then the expected communication costs in qle protocol @xmath51 is calculated as : @xmath52p^{(2)}_{suc}(1-p_{suc})^{k-1}\notag\\ & \hspace{1pc}+\sum_k^\infty\bigl[4(k-1)+4\bigr]p^{(1)}_{suc}(1-p_{suc})^{k-1}\label{eqq1}\end{aligned}\ ] ] we calculated eq . by substituting eq . and and the @xmath51 is ( in bits and qubits ) of quantum leader election as a function of the interference visibility @xmath40 of alice and bob s photons . , width=302 ] plotted in fig.[complot ] as a function of two - photon interference fringe visibility @xmath40 . note that without two - photon interference ( @xmath40=0 ) the @xmath51 is equivalent to the classical limit . however , as shown in fig.[complot ] even if two - photon interference is perfect ( @xmath40=1 , @xmath53=1 ) @xmath51 does not become that of the ideal case , but approaches to three . this extra one - bit is still leveraged for the verification of the detection events , because alice and bob never know that the operation is in the ideal regime . we obtain a visibility @xmath40 of 0.845@xmath54% , the expected total communication costs can be calculated as [email protected] which overcomes the classical limit . finally , we touch upon the extension of our implementation for more than three party case . in the case of multi parties , the qle protocol requires cnot operation , which is no longer deterministic with linear optics . moreover all the parties must prepare a cat state and exchange them . the probability to generate such multi photon states with spdc would be much lower and the optical circuit would be much more complicated than two party case . for the above reasons , it is difficult to implement multi party qle protocol with liner optics . we demonstrated the experimental 2-party quantum leader election . unlike conventional probabilistic quantum circuits based on linear optics , our leader election circuit can operate deterministically only with linear optics . we derived the success probability @xmath55 and expected total communication costs to complete leader election @xmath51 in terms of two - photon interference visibility in order to evaluate our experimental result . finally , we have obtained nearly deterministic results for a success probability of [email protected]% and an expected total communication costs of [email protected] , both of which improve upon the classical limit . we would like to thank dr . k. matsumoto and dr . m. hayashi for their helpful discussions , and especially k. yoshino for his kind suggestion and advice on the experiments . we calculate the coincidence probabilities between four detectors by calculating the overlap integral of field operators . the field operator just before detectors a1 , a2 , b1 , and b2 can be written as follows @xmath56 where @xmath57 and @xmath58 are the reflectance and the transmittance of four identical beam splitters ( bsa1 , bsb1 , bsa2 , and bsb2 ) , @xmath59 are the time delay of each optical paths , and @xmath60 are the field operators of down - converted photons . the field operator , @xmath61 , can be written by the fourier transform of the creation operator . @xmath62 the probability of coincidence detection between detector @xmath63 is given by @xmath64 where @xmath65 indicates the wave function of the input state , which is defined @xcite as @xmath66 where @xmath67 is the center frequencies of the pump beam . the spectral broadening of the pump and the down - converted photons are introduced as @xmath68 and @xmath69 , respectively . by combining eqs.@xmath13 , and after a straightforward algebra , we obtain the probabilities of two - fold coincidence and two - photon detection as follows : @xmath70\label{eqf}\\ p^{(2)}_{b1}=&p^{(2)}_{b2}=\gamma^2t^4\biggl[1+e^{-\frac{\delta\omega^2}{2}\delta t_b^2}\biggr]\label{eqb}\\ p^{coin}_{a1,a2}=&\gamma(1+\gamma^2)t^4\biggl[1-\tfrac{2\gamma}{1+\gamma^2}e^{-\frac{\delta\omega^2}{2}\delta t_a^2}\biggr]\label{eqa}\\ p^{coin}_{b1,b2}=&\gamma(1+\gamma^2)t^4\biggl[1-\tfrac{2\gamma}{1+\gamma^2}e^{-\frac{\delta\omega^2}{2}\delta t_b^2}\biggr]\\ p^{coin}_{a1,b1}=&(1+\gamma^4)t^4\biggl[1+\tfrac{2\gamma^2}{1+\gamma^4 } e^{-\frac{\delta \omega_p^2}{32}(\delta t_a+\delta t_b)^2}\notag\\ & \hspace{2pc}\times\hspace{1pc}e^{-\frac{\delta\omega^2}{8}(\delta t_a-\delta t_b)^2}\cos\tfrac{\omega_0}{2}(\delta t_a+\delta t_b)\biggr]\end{aligned}\ ] ] @xmath71\\ p^{coin}_{a1,b2}=&(\gamma+\gamma^3)t^4\biggl [ 1-\tfrac{2\gamma^2}{\gamma+\gamma^3}e^{-\frac{\delta \omega_p^2}{32}(\delta t_a+\delta t_b)^2}\notag\\ & \hspace{2pc}\times\hspace{1pc}e^{-\frac{\delta\omega^2}{8}(\delta t_a-\delta t_b)^2}\cos\tfrac{\omega_0}{2}(\delta t_a+\delta t_b)\biggr]\\ p^{coin}_{a2,b1}=&(\gamma+\gamma^3)t^4\biggl [ 1-\tfrac{2\gamma^2}{\gamma+\gamma^3}e^{-\frac{\delta \omega_p^2}{32}(\delta t_a+\delta t_b)^2}\notag\\ & \hspace{2pc}\times\hspace{1pc}e^{-\frac{\delta\omega^2}{8}(\delta t_a-\delta t_b)^2}\cos\tfrac{\omega_0}{2}(\delta t_a+\delta t_b)\biggr]\label{eql}\end{aligned}\ ] ] to obtain eq .@xmath13 , we introduce @xmath72 , which represent the time difference of two ports on alice and bob , and the branching ratio of beam splitters , respectively . here , we have made the following assumptions for simplicity : ( 1)@xmath73 , i.e. , the time difference is the same for alice and bob , since they are forced to do the same operations during the qle protocol . we also assumed ( 2)@xmath74 . i.e. , the spectral broadening of the photon pair is half that of pump beam ( @xmath75 ) . further , we assumed ( 3 ) @xmath76=0 , i.e. , the relative phase between the @xmath15 and the @xmath14 in alice and bob is set to 0 . therefore , the coincidence probabilities can be approximated as @xmath77,\label{eqap1}\\ p_{a1,a2}=p_{b1,b2}=&\gamma\frac{1+\gamma^2}{\left(1+\gamma\right)^4}\bigl[1-\tfrac{2\gamma}{1+\gamma^2}\nu\bigr],\\ p_{a1,b1}=&\bigl[1-\tfrac{2\gamma^2}{1+\gamma^4}\nu\bigr],\\ p_{a2,b2}=&\frac{2\gamma^2}{\left(1+\gamma\right)^4}\bigl[1-\nu\bigr],\\ p_{a1,b2}=p_{a2,b1}=&\gamma\frac{1+\gamma^2}{\left(1+\gamma\right)^4}\bigl[1+\tfrac{2\gamma^2}{\gamma+\gamma^3}\nu\bigr]\label{eqap2}\end{aligned}\ ] ] where @xmath78 implies the temporal overlapping of two photon wave packets on beam splitters . using@xmath13 we obtain the probability of two - photon detection @xmath44 and single photon detection @xmath45 as @xmath79 hence , the success probability per trial is calculated as @xmath80\ \ \ \biggl(\xrightarrow{\gamma,\nu=1}1\biggr)\label{eqap5}\end{aligned}\ ] ] how can we obtain optimal classical algorithms with minimum communication complexity ? we obtain the classical limit , @xmath43 , which corresponds to the minimum expected value of the total amount of communication to finish leader election . the classical algorithm for leader election is to generate and compare the numbers between alice and bob , and then the party who generated the larger number becomes the leader . the problem is how to generate the numbers and with how many bits to compare them . let @xmath81 and @xmath82 be the number of bits and the probability to generate `` 1 '' on each bit ( 1-@xmath82 for `` 0 '' ) , respectively ; then , the success probability of leader election per trial @xmath83 is calculated as @xmath84 then , the optimal condition is obtained by differentiating eq . as : @xmath85 by solving the eq . , the optimal probability , @xmath82 , is obtained as @xmath86 . the optimal success probability is calculated as @xmath87 hence , the expected value for the total amount of communication , @xmath88 , after @xmath81-bit comparison is written as : @xmath89 then , the classical limit is calculated as @xmath90 . 99 e. knill , r. laflamme , and g. j. milburn , nature ( london)409 , 46(2001 ) m. koashi , t. yamamoto , and n. imoto , phys . a 63,030301(r)(2001 ) j .- w . pan , s. gasparoni , r. ursin , g. weihs and a. zeilinger , nature ( london ) 423 , 417 ( 2003 ) . t. b. pittman , b. c. jacobs , and j. d. franson , phys . rev . a 64.062311(2001 ) p.w.shor , ieee computer society press p.124(1994 ) d. angluin and a. gardiner , local and global properties in networks of processors , proc . 12th acm symp . theory of computing , pp.82 - 331 , 1988 n.a.lynch , distributed algorithms , morgan kaufmann publishers , inc . m. yamashita and t. kameda , ieee _ trans . parallel distrib . _ , 7 ( 1 ) : 69 - 89 , 1996 . a. itai and m. rodeh . symmetry breaking in distributed networks . _ information and computation _ , 88(1):60 - 67 , 1990 . s. tani , h. kobayashi , and k. matsumoto , exact quantum algorithms for the leader election problem , stacs 2005 , lncs 3404 , pp . 581 - 592,2005 w. p. grice and i. a. walmsley , phys . a 56 , 1627(1997 ) j.d . franson , phys . lett , 62 , 2205(1989 ) c. k. hong , z. y. ou , and l. mandel , phys . lett 59,2044(1987 ) r. ghosh , c. k. hong , z. y. ou , and l. mandel , phys . a 34 , 3962(1986 ) | linear optics is a promising candidate to enable the construction of quantum computers . a number of quantum protocols gates based on linear optics have been demonstrated . however , it is well - known that these gates are non - deterministic and that higher order nonlinearity is necessary for deterministic operations .
we found the quantum leader election protocol ( qle ) can be operated deterministically only with linear optics , and we have demonstrated the nearly deterministic operation which overcomes classical limit . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
gamma - ray bursts ( grbs ) are brief pulses of @xmath9-ray radiation observed on average once a day at random directions in the sky . they are the brightest sources in this region of the electromagnetic spectrum , and they have been systematically studied in the past two decades ( e.g. * ? ? ? * and references therein ) . the origin of grbs is cosmological , as determined from the measurement of their redshifts ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , and in many cases , their host galaxies have been identified ( see * ? ? ? * ; * ? ? ? * and references therein ) . their distances imply the release of large amounts of energy ( @xmath10 ) in a short time - scale , which suggests that the origin of these phenomena could be associated to the accretion of matter onto a compact object @xcite . two populations of grbs are apparent from the distribution of their duration @xcite . those lasting less than @xmath11 are known as short grbs , while longer events are called long grbs ( lgrbs ) . these are more frequently observed and more precisely located , which makes their properties better known . lgrbs are always found in host galaxies with ongoing star formation activity @xcite , and some of them were observed to be associated with type ib / c core - collapse supernovae @xcite . these observations suggest that the progenitors of lgrbs are massive stars . stellar evolution models have tried to provide a consistent scenario where a lgrb can develop . the result is the so called _ collapsar model _ @xcite , in which this phenomenon is produced during the collapse of a massive star , due to the accretion of part of the envelope onto the recently formed black hole . nevertheless , the properties of lgrb progenitors ( mass , metallicity , rotation velocity , binarity , etc . ) are still a matter of discussion . lgrbs have been associated to single massive stars @xcite , perhaps in metallicity biased environments @xcite , and also to stars in binary systems @xcite . the importance of understanding the nature of the lgrb progenitors is beyond the interest of only stellar evolution and black hole formation . given their connection to massive stars and their large luminosities , lgrbs might be a powerful tool to investigate the star formation in the early universe , at redshifts for which standard tools become ineffective ( e.g. * ? ? ? the knowledge of the properties of the stellar progenitors would allow the assessment of possible biases originated when lgrbs are used as tracers of star formation . several authors suggest that the cosmic lgrb rate does not follow the star formation rate @xcite . the interpretation of their data requires the assumption of a differential evolution of the comoving lgrb rate density with respect to the comoving star formation rate ( sfr ) density . the same conclusion arises from the work of @xcite , who show that the luminosity function of lgrb hosts differs from that of core - collapse supernovae hosts , which are considered unbiased tracers of star formation . according to these authors , the lgrb host luminosity function can be reproduced by requiring lgrb progenitors to have metal abundances lower than that of the sun . an indirect procedure to investigate lgrb progenitors is to characterise the stellar populations of their host galaxies . given the connection between lgrbs and star formation , galaxy formation models can be used to assess the validity of lgrb progenitor models , comparing their predictions about the properties of these stellar populations to host galaxies observations . recent works such as that of @xcite , suggest that there is a strong relation between star formation and nuclear activity in a galaxy with its environment , at constant stellar mass . hence , the environment of host galaxies might constitute an independent probe for progenitor models . observational works on host galaxies environment are still inconclusive on whether host galaxies inhabit regions with certain characteristics @xcite . @xcite suggest that host galaxies are field galaxies , while @xcite find that a considerable fraction of their sample of 42 host galaxies shows evidence of interaction with other galaxies . the question of whether there is a connection between the occurrence of a lgrb and the local density of galaxies remains unanswered . since galaxy formation is a highly non - linear process , the properties of host galaxies predicted by different progenitor scenarios are better studied by means of cosmological numerical simulations @xcite . @xcite use hydrodynamical simulations of structure formation to identify a galaxy population whose properties reproduce the observed ones . according to these authors , the luminosity distribution of host galaxies is reproduced if galaxies are required to have high star formation efficiency . however , their simulations do not make any prediction on the properties of the lgrb progenitors . @xcite developed a monte carlo code to simulate the production of lgrbs in hydrodynamical simulations of galaxy formation , assuming that their progenitors are massive stars as proposed by the collapsar model . these authors follow the lgrb production as the structure forms and evolves in the universe . their results suggest that lgrb progenitors would be low metallicity stars ( @xmath12 ) , and hence lgrbs would be biased tracers of star formation principally at low redshift . however , their simulations explore a small volume of @xmath13 dominated by field galaxies . recently , @xcite have constructed a simulated population of host galaxies based on the semi - analytic model of galaxy formation of @xcite applied to cosmological simulations developed by @xcite . these simulations describe a larger volume of the universe ( a square box of @xmath14 in a side ) . @xcite explore three lgrb progenitor models based on the collapsar model , one of them assuming a given stellar mass value and an age cut - off and the other two also including metallicity cut - offs . these authors find that models with very low metallicity progenitors ( @xmath15 ) could explain the luminosities , colours and metallicities of the observed host galaxies . their results support previous claims of lgrbs being biased tracers of the star formation . clearly , the use of larger simulations with better resolution and more detailed descriptions of the physical processes driving the dynamical , chemical and star formation histories of galaxies allows a better modelling of the host galaxies of lgrbs . however , an important aspect that should not be disregarded is that , for a proper comparison with observed host galaxies , the effects of the detectability of these galaxies must be taken into account . and this is the main contribution of our work where we developed an observational pipeline which allows us to mimic , at least in part , biases affecting the observations . with this new tool we build an observable host galaxy catalogue which can be more fairly compared to current available observations . in this work , we develop a semi - analytical model for the host galaxies of lgrbs adopting the collapsar model for the progenitor stars . we apply it to one of the largest cosmological simulations available , the _ millennium simulation _ @xcite . the _ millennium simulation _ follows the evolution of dark matter in a box of @xmath16 in a side , hence providing a cosmologically representative volume a factor of 64 larger than that of @xcite and factor of @xmath17 larger than that of @xcite . we take the galaxy catalogue built up by @xcite , who describe the evolution of baryonic matter including star formation , active galactic nuclei and supernova feedback . using the star formation and chemical properties of galaxies in this catalogue to implement different lgrb progenitor scenarios , we determine the properties of the overall host galaxy population and its environment . we compute the detectability of the host galaxies in order to compare the predictions of these scenarios to the observations compiled by @xcite . this work is organized as following . in section [ sim ] , we briefly describe the cosmological simulation and semi - analytic galaxy catalogue used in our work . in section [ mod ] , we develop our implementation of lgrb progenitor scenarios . sections [ hgs ] and [ env ] show our results on host galaxy properties and environment , respectively . finally , section [ con ] presents our conclusions . the _ millennium simulation _ @xcite is one of the largest cosmological simulations of structure formation publicly available . it describes the evolution of the cold dark matter while the baryonic matter can be included via semi - analytic models on top of the numerical simulation . the _ millennium simulation _ describes the formation of structure tracking @xmath18 dark matter particles of mass @xmath19 distributed in a cubic region of 500 @xmath20 side , using the @xmath21 cosmogony . the adopted cosmological parameters are @xmath22 , @xmath23 , @xmath24 , @xmath25 and @xmath26 , where the hubble constant is @xmath27 with @xmath28 . these parameters are chosen in consistency with results obtained by the joint analysis of the two - degree field galaxy redshift survey ( 2dfgrs ) and the wilkinson microwave anisotropy probe ( wmap ) data @xcite . the simulation was performed using a modified version of gadget-2 hydrodynamical code @xcite . a _ friends - of - friends _ ( fof ) algorithm identified non - linear dark matter haloes within the simulation in an automatic manner . two particles belong to the same halo if their separation is less than @xmath29 times the mean separation between particles . only groups of more than @xmath30 particles are identified as haloes . this restriction sets a lower limit on the mass of a halo of @xmath31 . substructures orbiting within each virialized halo are identified applying a subfind algorithm @xcite . each dark matter halo has a central ( type 0 ) galaxy and one or more satellite galaxies . satellites were at some point central galaxies of smaller haloes which suffered a merger with the halo their currently inhabit . there are two types of satellite galaxies , type 1 galaxies are located at the centre of a subhalo associated with a fof group , while type 2 galaxies have lost their dark matter subhalo after falling onto a more massive halo . mean properties of synthetic galaxies ( e.g. sfr , stellar mass , gas mass , metallicity ) are obtained by applying semi - analytic models to the structure formation simulation @xcite . essentially , semi - analytical models describe the collapse of baryonic matter following dark matter haloes . after collapsing , the gas cools and gravitational instabilities produce episodes of star formation . to reproduce observations , models incorporate photoionization processes in the intergalactic medium , the growth of supermassive black holes during galactic mergers , supernova and agn feedback , star formation rate enhancement in mergers , and formation of heavy elements for each formed stellar population . the galaxy catalogue we use in this work is that of @xcite . there is recent evidence for an excessive reddening of type 2 galaxies @xcite which might be due to the poor physical treatment of this type of galaxies . in order to avoid spurious trends , we exclude type 2 galaxies from our analysis in section [ mod ] and section [ hgs ] . however , type 2 galaxies ought to be included in the environmental analysis performed in section [ env ] in order to correctly trace the underlying mass distribution . hence , the catalogue of @xcite provides us with the spatial galaxy distribution , their stellar masses , dark matter haloes , star formation activity , colours , luminosities and mean metallicities . regarding the latter , the public catalogue makes available the mean metallicity of the cold gas component and of the stellar population as a whole ( i.e. averaged over new and old stars in a galaxy ) . therefore , as a proxy for the metallicity of the lgrb progenitors , we take the mean metallicity of the cold gas component at the time these stars were born . because the @xcite model assumes instantaneous recycling , this metallicity is slightly higher than that of the newly born stars , but this correction is negligible compared to the uncertainties in the measured metallicities ( e.g. * ? ? ? * ) . we consider two scenarios for lgrb production , both of them based on the collapsar model @xcite . in our scenario i we take as lgrb progenitors all stars above a certain minimum mass @xmath32 , with no other restriction whatsoever . in this scenario lgrbs are unbiased tracers of star formation . accordingly , we obtain the intrinsic lgrb rate in a given galaxy @xmath33 at a particular redshift @xmath34 as @xmath35 where @xmath36 and @xmath37 are the lower and upper mass cut - offs of the imf @xmath38 given by @xcite . according to eqn . [ rateint ] , the production of lgrbs is not delayed with respect to the starburst that created the progenitor stars , which is justified because in the collapsar model only massive stars are lgrb progenitors . in our scenario ii , we assume that only massive stars ( @xmath39 ) below some metallicity threshold ( @xmath40 ) are able to produce lgrbs , as in the progenitor models of @xcite or @xcite . as explained before , we take the mean metallicity of cold gas in each galaxy as representative of the metallicity of the progenitor stars . the consequences of this hypothesis will be discussed in later sections . in this scenario , the intrinsic lgrb rate is given by eqn . [ rateint ] for galaxies @xmath33 with metallicity @xmath41 , and is null for those with @xmath42 . three realizations of this scenario were computed , adopting @xmath43 ( scenarios ii.1 , ii.2 and ii.3 , respectively ) . given a metallicity threshold , we adjusted the only free parameter of our scenarios ( @xmath32 ) to reproduce the lgrb rate measured by the batse experiment . for this purpose , we took the off - line grb search of @xcite , which detected 3475 grbs with @xmath44 during a live - time of @xmath45 ( 70 per cent of the @xmath46 that batse was active ) , scanning 67 per cent of the sky . this translates into a full - sky lgrb rate of @xmath47 above the threshold of the off - line search . the choice of this particular experiment was motivated by its good statistics and the availability of an accurate model for its detection efficiency . to obtain the observable rate predicted by a given scenario , we first compute the comoving lgrb rate density for each redshift as @xmath48 where @xmath49 is the comoving volume of the _ millennium simulation_. the full - sky observed lgrb rate is then @xmath50 where @xmath51 is the derivative of the comoving volume with respect to @xmath34 at fixed solid angle , @xmath52 is the maximum redshift of the simulation , @xmath53 the luminosity distance and @xmath54 the hubble constant at redshift @xmath34 , and @xmath55 the probability of detecting a lgrb at redshift @xmath34 with batse ( i.e. the probability that a particular lgrb has a peak flux above the experiment threshold ) . the integration was performed numerically , and the described procedure iterated over @xmath32 , until agreement with the observed rate was attained . the value of @xmath56 depends on the lgrb luminosity function and spectrum . it was computed using a monte - carlo scheme to simulate , at a given redshift , a large population of lgrbs with different luminosities and spectra . then , their photon fluxes in the batse energy band were estimated . for each lgrb in this population , a second monte - carlo procedure rejected those events which would be undetectable , taking into account the trigger efficiency of the @xcite off - line search . if the lgrb emission were isotropic , the fraction of retained lgrbs would directly give the detection probability due to the off - line search threshold . to account for the beamed emission of lgrbs ( e.g. * ? ? ? * ) , and assuming that the distribution of jet opening angles is independent of the lgrb luminosity , spectrum and redshift , we obtain @xmath55 by multiplying this fraction by the mean beaming fraction of the jets @xmath57 . the luminosity function and spectral parameters distributions were taken from @xcite . this procedure not only allows us to mimic the observational process but also has provided us with estimates of the minimum mass for the progenitor stars . the values of @xmath32 obtained , in solar masses , are @xmath58 , @xmath59 , @xmath60 , and @xmath61 for scenarios i , ii.1 , ii.2 and ii.3 , respectively ; the uncertainties reflect poissonian errors in the number of observed lgrbs ( see also table 1 ) . .main characteristics of our scenarios for the simulated host galaxies of lgrbs . column ( 1 ) gives the name of the scenario . column ( 2 ) lists the minimum stellar mass obtained for progenitor stars of lgrbs to reproduce batse observations , adopting @xcite imf . column ( 3 ) gives the cold gas maximum metallicity cut - off . column ( 4 ) shows ratio between the percentage of simulated host galaxies over the mass range @xmath62 and the corresponding value obtained from the sample of @xcite . [ cols="^,^,^,^",options="header " , ] a meaningful definition of a lgrb host galaxy in our scenarios is not as straightforward as it might seem at first sight . the nave definition of a host galaxy , at a given redshift @xmath34 , as being any galaxy @xmath33 with a lgbr rate of @xmath63 ( or identically , @xmath64 ) is not useful because the resulting host population would include galaxies with arbitrarily low lgrb rates . low rates are better understood in statistical terms , as very low probabilities of producing a lgrb per unit time . this implies that the corresponding galaxies have a low probability of being observed as host galaxies . hence , this nave definition would generate a host galaxies sample biased to low sfr galaxies . to fix this problem , the definition of a host galaxy could be based on the number of lgrbs @xmath65 produced in each galaxy during a time interval @xmath66 in its rest frame , so that host galaxies are only those with @xmath67 , as in @xcite . however this cut - off , and the resulting population , would be dependent on the more or less arbitrary choice of @xmath66 . given the above arguments , we preferred instead a probabilistic approach , defining the likelihood of a galaxy being observed as a lgrb host . this likelihood is then used as a weight to compute the properties of the observable host galaxies sample , which would be in this way comparable to the observed sample . for a galaxy to be detected as a host at least one lgrb must be detected within it by a high - energy observatory . then , the galaxy itself must be detected , usually by optical / nir telescopes . we would treat the biases introduced by each of these observations separately . to model the first bias , we compute the probability of detecting at least one lgrb within it by any of the high - energy observatories monitoring grbs . given that event detection processes follow poissonian statistics , if @xmath68 is the contribution of galaxy @xmath33 at redshift @xmath34 to the lgrb rate observed by experiment @xmath69 , the probability of observing at least one lgrb in a galaxy is @xmath70 where @xmath71 is the time interval of the lgrb search conducted by experiment @xmath69 . [ phg ] assumes that the searches by different observatories are independent . previous works estimate the intrinsic lgrb rate in a typical galaxy to be of the order of @xmath72 @xcite , hence the lgrb rate observed in the galaxy by any mission must be lower than this value . given that current searches for lgrbs have spanned a few years , @xmath73 , and @xmath74 , which means that the likelihood of a galaxy being observed as a host is proportional to its contribution to the mean _ observed _ number of lgrbs , @xmath75 where @xmath76 is the sky coverage of the experiment @xmath69 . note that , for a fixed @xmath34 , cosmological and observatory dependent factors in eqn . [ rateobsgal ] become constant , independently of the number of observatories considered , and cancel out in the normalization of the weights . this means that when computing mean hg properties such as mass or sfr as a function of @xmath34 , the first bias can be modeled simply by weighting the corresponding properties of each galaxy by its _ intrinsic _ lgrb rate . in the computation of the integrated properties of the whole population of host galaxies ( like the integrated mass distribution , for example ) , also the effects of volume variation , time dilation and detectability of the different observatories must be taken into account ( pellizza et al . in preparation ) . in this case it is better to compute first the mass distribution observed by each experiment at each redshift @xmath34 , weighting the galaxies in the corresponding snapshot by @xmath77 . second , the integration in @xmath34 can be performed for each experiment , and the resulting distribution can be normalized . this has the advantage of avoiding the use of the values of @xmath71 and @xmath76 , which are poorly known for some of the observatories that detected the lgrbs in the sample of @xcite . only @xmath78 for each experiment is needed , which is computed as described in sect . [ obsrate ] for @xmath56 . finally , these distributions can be combined into a single one by adding them , previously scaled to the number of lgrbs detected by each observatory . the relevant data for computing the detectability were taken from @xcite for batse , @xcite for _ swift _ , @xcite for _ beppo - sax _ , @xcite for _ hete-2 _ and @xcite for _ ulysses_. these experiments detected 36 of the 38 lgrbs in the sample of @xcite . for statistical reasons , only experiments that detected at least five lgrbs in the sample were considered . konus-_wind _ and _ near_-xgrs were discarded because all their lgrbs were observed also by other experiments above . to be consistent , the observed mass distribution to which the model predictions were compared was constructed from the data of @xcite by adding the individual distributions observed by each experiment . the second bias is more difficult to model . given that the search for host galaxies is guided by the discovery of the lgrbs themselves and usually done with a variety of different telescopes and detectors , in different bands of the electromagnetic spectrum and with different sensitivities , the biases introduced are unclear . galaxies with surface brightness below the detectability threshold of available instruments could introduce a bias towards low metallicity galaxies . an extra problem could be caused by dust obscuration which could affect the detection of the afterglows and produce a bias towards high metallicity galaxies @xcite . hence , at least these two effects might combine themselves to determine the detectability of a host galaxy . considering also the fact that we have only access to the public galaxy catalogue of the _ millennium simulation _ which provides only mean global properties and magnitudes , we adopt the observed integrated stellar mass distribution as a tool to apply the combined effect of observational biases to the simulated galaxy sample . we use the fact that the observed integrated stellar mass distribution has been affected by observational biases although we can not disentangle their individual effects . hence , we require the simulated lgrb hosts to reproduce the observed stellar mass distribution in order to build up the observable simulated lgrb hosts . the procedure is explained in detail in next section . note that , hereafterin , we will discuss the trends of the observable simulated hosts and the general galaxy population . the former can be compared to the current observed hosts , but , if this later sample changes due to better or different observational techniques , our simulated sample should be also consistently modified to match the new observed stellar mass distribution . the largest and most comprehensively uniform sample of host galaxy properties available at present is that compiled by @xcite . stellar masses , star formation rates , metallicities , absolute magnitudes and colours of 46 observed host galaxies up to @xmath79 were obtained by these authors comparing their spectral energy distributions to those of synthetic stellar populations , using the method described in @xcite . a key point of our models is that we require the stellar mass distribution of the predicted host galaxies to match that of the observed host galaxies . the stellar mass is adopted as the property to be reproduced by the models since it is now widely accepted that stellar mass is a more fundamental quantity for galaxies than luminosity ( e.g. * ? ? ? it could be possible that those lgrb events with no detected hosts occured in very low surface brightness galaxies and hence , with low stellar masses . as mentioned before the effects of dust could also prevent the detection of events in high metallicity ( and dusty ) galaxies . in our models these host galaxies exist ( and are part of the global galaxy population ) but the observability cut - off has been determined by using the current observed stellar mass distribution . [ mass ] shows the distribution of stellar masses of observed host galaxies constructed from the data of @xcite , together with those predicted by our scenarios . as it can be seen from fig . [ mass ] , the scenario which best reproduces the observed stellar mass distribution is that with @xmath80 ( scenario ii.3 , see also table 1 ) . lower metallicity thresholds predict lower observed stellar masses for the host galaxies , while no metallicity threshold ( scenario i ) predicts larger ones . in the sample of @xcite , we find that @xmath81 per cent of the studied galaxies have stellar masses over the range @xmath62 ( see also * ? ? ? the probability of getting a host galaxy within this stellar mass range in scenario ii.3 is 88 per cent , while for scenarios i , ii.1 and ii.2 it is 70 , 2 , and 43 per cent , respectively . then we conclude that scenario ii.3 predictions agree fairly well with observations while others fail , and therefore in the following sections , we will focus only on this scenario . we stress the fact that its predictions include the effects of host galaxies observability , hence , a proper comparison with observations can be made . in order to contribute to the understanding of the nature of lgrb host galaxies , we will also compare these predictions with the properties ( not weighted by host galaxy observability ) of both the sample of all galaxies with mean cold gas metallicities below @xmath82 ( hereafter low metallicity sample ) and the complete galaxy population of the catalogue of @xcite . as a first step towards understanding the nature of host galaxies , we analyse their stellar masses as a function of redshift . as shown in fig . [ massred ] , the mean stellar mass of the host galaxies as a function of redshift predicted by scenario ii.3 reproduces the observed mean trend quite well . from this figure , we can also see that host galaxies are , on average , more massive than galaxies in the complete galaxy population while the latter are more massive than those in the low metallicity sample . this can be understood taking into account that the host galaxy observability is a strong function of the star formation rate , and that the complete galaxy population in the catalogue of @xcite follows a mass - metallicity relationship ( e.g. * ? ? ? then , the cut - off adopted for the mean cold gas metallicity to reproduce the observed stellar mass distribution implies a cut - off in stellar mass since low metallicity galaxies are , on average , less massive than the general galaxy population ( see also fig . [ mass ] ) . however , as the observability of a host galaxy depends strongly on its star formation activity and most of the small galaxies have low star formation rates , the observable host galaxies tend to be , on average , the more massive ones among them . as a result , our observable sample tends to be populated by systems more massive than those in the low metallicity sample or in the complete galaxy population . in fig . [ gsfr ] , we display the mean sfr of host galaxies as a function of redshift predicted by scenario ii.3 , together with the corresponding mean values for the low metallicity sample and the complete galaxy population . as it can be seen , the prediction of scenario ii.3 reproduces very well the behaviour of the observed host galaxies . these have higher sfr than the mean of the complete galaxy population , and much higher than that of the low metallicity sample . the good agreement between our scenario ii.3 and observations suggests that the observed host galaxies are biased towards galaxies with stellar masses in the range @xmath83 , high star formation activity and relatively low gas metal content , compared to the mean properties of the complete galaxy population at a given @xmath34 . regarding the mean specific sfr ( ssfr ) , defined as the ratio between the sfr and the stellar mass of a given galaxy ( fig . [ gssfr ] ) , host galaxies are predicted to show ssfrs similar to those in the low metallicity sample , and higher than those in the complete galaxy population , for @xmath84 . between @xmath85 and @xmath86 , there are no significant differences in mean ssfrs among the three samples , which have larger ssfrs that low redshift galaxies . this can be understood because , at higher redshift , galaxies have larger gas reservoirs which can feed stronger star formation activity and are , on average , less chemically enriched . at low redshift , host galaxies seem to be particularly efficient at transforming gas into stars . note , however , that the mean ssfr predicted by scenario ii.3 at low redshift is half an order of magnitude lower than the mean observed ssfr . observations show that host galaxies tend to be bluer than the general population of galaxies observed at a given redshift , and fainter than a typical @xmath87 galaxy . in fact , this trend can be nicely reproduced by our scenario ii.3 , as shown in fig . [ colours ] . the luminosities and colours of the predicted host galaxies are in excellent agreement with the observations compiled by @xcite . from this figure , we can see that host galaxies are bluer than the complete galaxy population for @xmath88 , but have similar mean colours for higher redshfits . our scenario predicts that host galaxies are more luminous systems in the @xmath89-band , compared to the mean luminosity of the global galaxy population at all redshifts . -band luminosity ( upper panel ) and @xmath90 colour ( lower panel ) as a function of redshift predicted by scenario ii.3 ( dotted line ) with its standard deviation ( shaded grey band ) . open diamonds represent the observed host galaxies by @xcite , while filled circles correspond to their mean values in redshift intervals of 0.5 . for comparison , we also include the corresponding mean trends for the low metallicty sample ( dashed line ) and the complete galaxy population ( solid line).,title="fig:",scaledwidth=45.0% ] -band luminosity ( upper panel ) and @xmath90 colour ( lower panel ) as a function of redshift predicted by scenario ii.3 ( dotted line ) with its standard deviation ( shaded grey band ) . open diamonds represent the observed host galaxies by @xcite , while filled circles correspond to their mean values in redshift intervals of 0.5 . for comparison , we also include the corresponding mean trends for the low metallicty sample ( dashed line ) and the complete galaxy population ( solid line).,title="fig:",scaledwidth=45.0% ] in fig . [ metal ] , we compare the mean cold gas metallicity of the host galaxies predicted by scenario ii.3 to different metallicity estimations reported by @xcite . since at @xmath91 it is very difficult to distinguish hii regions with enough resolution , these authors generally measured the optical luminosity - weighted mean metallicity in a galaxy . for consistency , host galaxies at @xmath92 were treated by them as the rest of the sample ( i.e. integrating fluxes over the whole galaxy ) . in this figure , we have included metallicities obtained by using different indicators ( see * ? ? ? * for more details ) . in some cases , absorption lines in the optical afterglow can be used to obtain the metallicity of neutral cold gas along the line of sight of the grb ( the so - called grb - dlas ) . this is the case of 9 grb - dla systems studied by @xcite , all of them at @xmath93 . in this case , the metallicity could be associated more directly to the metallicity of the host galaxy , contrary to qso - dlas which are associated to hi clouds in the intergalactic medium . at @xmath94 , measurements of metallicity by @xcite are derived from hot gas and , in this case , the lower branch solution is preferred for the hosts . as shown in fig . [ metal ] , the predicted host galaxies metallicities are systematically higher than the grb - dla metallicities . this might indicate that lgrbs occur in regions of even lower metallicity than the mean metallicity of the cold gas of the host galaxies . in this sense , @xcite , adopting similar hypotheses to generate synthetic lgrb populations , but using full cosmological simulations where the chemical enrichment of baryons was consistently followed with redshift , claimed a metallicity threshold of @xmath95 for the progenitor stars in order to reproduce observations . more detailed information on the metallicity of individual stellar populations , which also takes into account the inhomogeneities of the interstellar medium , are needed to improve our understanding of this issue ( see @xcite and artale et al . in preparation ) . also along these lines , a work by @xcite compares the metallicity of grb hosts associated to type ic supernovae ( sn ic ) to broad - line sn ic without grb detection at @xmath96 . the metallicity of sn - grb host galaxies are estimated by computing the nebular oxygen abundance by three different methods . in this case , they directly probe the environments of the progenitors because either they were measured at that location or they correspond to metallicities of homogeneous dwarf galaxies . @xcite results are in agreement with the values obtained by @xcite and with our own results at low redshifts . we also include those measured by the r32 upper branch ( triangles ) or lower branch ( diamonds ) , and grb - dlas ( asterisks ) . for comparison , we include the corresponding mean trends for the low metallicty sample ( dashed line ) and the complete galaxy population ( solid line).,scaledwidth=45.0% ] it is still under discussion if host galaxies are always located in regions of a given characteristic local density or if , depending on @xmath34 , they inhabit different environments . recently , @xcite estimated the cross - correlation function of their simulated host galaxies finding that these systems tend to map underdense regions . observations provide somehow contradictory results . @xcite found host galaxies to inhabit field regions . lately , @xcite suggested that host galaxies might be biased towards interacting or merging galaxies . it is yet too soon for observations to provide a robust answer to these questions but theoretical predictions could positively contribute to this area . the advantage of our model is that taking into account the observability allows us to make a more robust prediction . in this section , we investigate the environment of the simulated host galaxies by means of three different estimators . the distance to the closest neighbour ( @xmath97 ) of a host galaxy provides an estimation of the possibility of a host galaxy being interacting with another galaxy . the distance to the fifth neighbour ( @xmath98 ) of a host galaxy provides an estimation of the density of the region the galaxy inhabits . finally , the central halo virial mass ( @xmath99 ) is an estimator of the global potential well to which the host galaxy is bound . in the particular case of the closest neighbour , it is well known that pair interactions can enhance the star formation activity ( e.g. * ? ? ? * ; * ? ? ? * ) and if lgrb progenitors are massive stars this would imply an increase in the probability of detecting a lgrb event . since semi - analytical codes do not model tidally - induced star formation , we modified the sfr for galaxies with their closest neighbour located at distances smaller than @xmath100 by a factor of two , motivated by observational and numerical results ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . however , this correction does not change any of the trends reported in this work . but the role of interactions should be further tested with models that properly include their effects . we estimate the cumulative distribution of these three environmental estimators ( @xmath101 ) predicted by scenario ii.3 . to quantify behaviours as a function of redshift , we determine the values @xmath102 and @xmath103 at which the corresponding cumulative fractions reach @xmath104 per cent in the case of @xmath97 and @xmath98 , and @xmath105 per cent in the case of @xmath106 . in fig . [ d1 ] ( upper panel ) , we show @xmath107 as a function of redshift according to scenario ii.3 , together with the trend for the low - metallicity sample and the complete galaxy population . at high redshift , the three samples tend to have the nearest neighbour at approximately similar distances . we only note a slight trend for host galaxies to systematically have a closer first neighbour than galaxies in the other two samples for @xmath108 . from @xmath109 , low metallicity galaxies tend to have their closest neighbour further away than either a galaxy in the complete population or a host galaxy . the latter tend to inhabit lower density regions than galaxies in the complete population . our models indicate that observable host galaxies would only have a slightly higher probability to be in a galaxy pair compared with the general galaxy population at high redshift ( i.e. @xmath104 per cent of galaxies have the first neighbour closer than @xmath110 ) . the global environment is quantified by @xmath111 . as shown in fig . [ d1 ] ( middle panel ) at high redshift ( @xmath112 ) , galaxies in the three samples reside in similar environments . for lower redshift there is a clear trend for galaxies in the low metallicity sample and host galaxies to inhabit lower density regions than galaxies in the complete galaxy population . this results agree with the numerical work of @xcite , and also with the observational findings of @xcite , taking into account that the mean redshift of lgrbs observed at that time was @xmath113 . in the case of the virial mass , as shown in fig . [ d1 ] ( lower panel ) , we find that @xmath105 per cent of observable host galaxies have haloes less massive than @xmath114 at any redshift . as the structure forms and groups and clusters aggregate hierarchically , galaxies tend to inhabit larger dark matter haloes . however , host galaxies stay within a narrower range of mass haloes . the inversion in the relation observed in this figure can be understood considering that host galaxies observability depends strongly on the star formation activity which , in turn , depends strongly on environment @xcite . hence , from @xmath85 active star forming galaxies ( i.e. those which are important generators of lgrbs ) with mean stellar masses of @xmath115 reside again in slightly smaller dark matter haloes ( fig . [ massred ] and fig . [ gsfr ] ) . note that this is not the case for the general low metallicity sample which although residing , on average , in smaller haloes than the global galaxy population , tend to systematically inhabit larger ones with decreasing redshift , as expected in a hierarchical scenario . the observability condition which is closely linked to the star formation activity produces this kind of halo downsizing scenario for host galaxies at low redshift , while the mean stellar mass remains approximately constant . ) . _ middle panel : _ median distance to the fifth neighbour ( @xmath111 ) . _ left panel : _ virial mass of the central halo at the @xmath105 percentile ( @xmath103).,title="fig:",scaledwidth=45.0% ] ) . _ middle panel : _ median distance to the fifth neighbour ( @xmath111 ) . _ left panel : _ virial mass of the central halo at the @xmath105 percentile ( @xmath103).,title="fig:",scaledwidth=45.0% ] ) . _ middle panel : _ median distance to the fifth neighbour ( @xmath111 ) . _ left panel : _ virial mass of the central halo at the @xmath105 percentile ( @xmath103).,title="fig:",scaledwidth=45.0% ] our results indicate that the observed lgrb host galaxies properties can be reproduced assuming that lgrb progenitors are massive stars and occur in galaxies with moderately low mean cold gas metallicities . the joint requirement for the synthetic lgrbs to reproduce the observations of batse and for the simulated systems that host them to reproduce the stellar mass distribution of the observed host galaxies determines the minimum mass for the progenitor star ( @xmath116 ) if a @xcite imf had been used , instead . ] and the maximum metallicity cut - off ( @xmath117 ) for the cold gas metallicity of the simulated host galaxies . our scenario ii.3 , which satisfies these conditions , succeeds at reproducing the dependence on redshift of stellar mass , luminosity , colour , sfr , ssfr and metallicity of the observed host galaxies compiled by @xcite over the redshift range @xmath118 . our main findings are : 1 . the average stellar mass of host galaxies is higher than the average stellar mass of the complete galaxy population , and remains within a narrow mass range around @xmath119 , with a very weak trend to lower mass for higher redshifts , due to the double requirement of low mean gas metallicity and high star formation rates . we note that the dispersion is approximately 0.5 dex around the @xmath120 mean stellar mass . 2 . compared to the characteristic stellar mass @xmath121 estimated by @xcite , host galaxies are low stellar mass systems ( 88 per cent of our host galaxies have stellar masses between @xmath122 ) . however , compared to the mean stellar mass of the complete galaxy population , host galaxies tend to be massive galaxies . the sfr of host galaxies is larger than that of galaxies in the complete population at all redshifts . at @xmath112 , the ssfr of observable host galaxies is comparable to the complete galaxy population , but at lower redshift it is systematically higher . as a consequence , host galaxies are , on average , bluer than the global galaxy population . the dispersion found for the properties of the observable host galaxies reflect the different histories of formation of galaxies at a given stellar mass . 4 . our results support the claims for a metallicity threshold to reproduce the properties of the current observed host galaxies . the global metallicity threshold of @xmath82 we derived from our models is an upper limit for the metallicity of the lgrb progenitor stars in the case of no dust effects . the comparison of the mean metallicity of host galaxies predicted by our model with grb - dlas observations suggests that lgrbs might be produced in stars of even lower metallicity as pointed out by other authors ( e.g. * ? ? ? * ; * ? ? ? * ) . at low redshift , our results are in agreement with sn - grb local metallicity estimations by @xcite . however , observations show a large spread in metallicity and there are also observational uncertainties that can affect its determination . hence , results should be taken with caution . 5 . for @xmath123 , host galaxies seem to be slightly more likely to be in pairs than galaxies belonging to the other samples . there is , however , a change of behaviour for @xmath124 , where the at least 50 per cent of the host galaxies seems to have their closest neighbour further away than galaxies in the complete population , but closer than galaxies in the low metallicity sample . we highlight that we are considering neighbouring galaxies of a mass above @xmath125 . this limit is set by the numerical resolution of the _ millennium simulation_. lower mass companions could also imprint morphological perturbations and trigger star formation activity ( e.g. * ? ? ? * ) but we can not pursue this analysis further on with this galaxy catalogue . 6 . regarding global environment , our model suggests that , at @xmath108 , observable host galaxies would preferentially inhabit environments of similar density to those populated by the general population and of slightly higher density than those inhabited by the low metallicity sample . towards @xmath126 , observable host galaxies tend to be progressively located in less dense environments , which becomes as subdense as the regions where low metallicity galaxies reside ( see also * ? ? ? * ) . our results suggest that observable host galaxies tend to have dark matter haloes in the range @xmath127 , regardless of redshift , and show a slight signal for halo downsizing from @xmath109 , distinguishing them from the other two samples which follow the expected halo mass growth in a hierarchical scenario . this result is consistent with observable host galaxies being systems with mean stellar masses approximately constant , regardless of the age of the universe . our results are mainly a consequence of the joint requirements to have high star formation activity to ensure observability and to reproduce the current distribution of stellar masses of observed host galaxies . however , within the current constrains , galaxies with high masses would be too metal - rich to produce lgrbs and low mass systems would have low probability of being observed . if the observed sample were modified by the incorporation of other galaxies , for example dusty hosts , then our model would need to be readjusted accordingly . we thank m.e . de rossi and gerard lemson for helping us to manage the _ millennium simulation _ , and sandra savaglio for her useful comments and suggestions . this work was partially supported by grants pict 2005 - 32342 , pict 2006 - 245 max planck , and pict 2006 - 2015 from argentine anpcyt . | in this work , we investigate the nature of the host galaxies of long gamma - ray bursts ( lgrbs ) using a galaxy catalogue constructed from the _ millennium simulation_. we developed a lgrb synthetic model based on the hypothesis that these events originate at the end of the life of massive stars following the collapsar model , with the possibility of including a constraint on the metallicity of the progenitor star . a complete observability pipeline was designed to calculate a probability estimation for a galaxy to be observationally identified as a host for lgrbs detected by present observational facililties .
this new tool allows us to build an observable host galaxy catalogue which is required to reproduce the current stellar mass distribution of observed hosts .
this observability pipeline predicts that the minimum mass for the progenitor stars should be @xmath0 m@xmath1 in order to be able to reproduce batse observations .
systems in our observable catalogue are able to reproduce the observed properties of host galaxies , namely stellar masses , colours , luminosity , star formation activity and metallicities as a function of redshift . at @xmath2 ,
our model predicts that the observable host galaxies would be very similar to the global galaxy population .
we found that @xmath3 per cent of the observable host galaxies with mean gas metallicity lower than @xmath4 have stellar masses in the range @xmath5@xmath6m@xmath1 in excellent agreement with observations .
interestingly , in our model observable host galaxies remain mainly within this mass range regardless of redshift , since lower stellar mass systems would have a low probability of being observed while more massive ones would be too metal - rich .
observable host galaxies are predicted to preferentially inhabit dark matter haloes in the range @xmath7@xmath8m@xmath1 , with a weak dependence on redshift .
they are also found to preferentially map different density environments at different stages of evolution of the universe . at high redshifts ,
the observable host galaxies are predicted to be located in similar environments as the global galaxy population but to have a slightly higher probability to have a close companion .
[ firstpage ] gamma - rays : bursts methods : numerical stars : formation galaxies : evolution , interactions . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
in 1984 , an unconditionally secure key distribution protocol using quantum resources was proposed by bennett and brassard @xcite . the scheme , which is now known as bb84 protocol drew considerable attention of the cryptography community by its own merit as it offered unconditional security , which was unachievable by any classical protocol of key distribution . however , the relevance of bb84 quantum key distribution ( qkd ) protocol and a set of other schemes of qkd were actually established very strongly in 1994 , when the seminal work of shor @xcite established that rsa @xcite and a few other schemes of classical cryptography @xcite would not remain secure if a scalable quantum computer is built . the bb84 protocol , not only established the possibility of obtaining unconditional security , but also manifested enormous power of quantum resources that had been maneuvered since then . specifically , this attempt at the unconditional security of qkd was followed by a set of protocols for the same task @xcite . interestingly , the beautiful applications of quantum mechanics in secure communication did not remain restricted to key distribution . in fact , it was realized soon that the messages can be sent in a secure manner without preparing a prior key @xcite . exploiting this idea various such schemes were proposed which fall under the category of secure direct quantum communication ( @xcite and references therein ) . the schemes for secure direct quantum communication can be categorized into two classes on the basis of additional classical communication required by the receiver ( bob ) to decode each bit of the transmitted message- ( i ) quantum secure direct communication ( qsdc ) @xcite and ( ii ) deterministic secure quantum communication ( dsqc ) @xcite . in the former , bob does not require an additional classical communication to decode the message , while such a classical communication is involved in the latter ( see @xcite for review ) . it is worth noting that in a scheme of qsdc / dsqc meaningful information flows in one direction as it only allows alice to send a message to bob in an unconditionally secure manner using quantum resources and without generation of a key . however , in our daily life , we often require two way communication ( say , when we speak on a telephone ) . interestingly , a modification of one of the first few qsdc schemes ( i.e. , ping - pong scheme @xcite ) led to a new type of protocol that allows both alice and bob to communicate simultaneously using the same quantum channel . this scheme for simultaneous two way communication was first proposed by ba an @xcite and is known as quantum dialogue ( qd ) . due to its similarity with the task performed by telephones , a scheme for qd are also referred as quantum telephone @xcite or quantum conversation @xcite scheme , but in what follows , we will refer to them as qd . due to its practical relevance , schemes of qd received much attention and several new schemes of qd have been proposed in the last decade @xcite . however , all these schemes of qd , and also the schemes of qsdc and dsqc , mentioned here are restricted to the two - party scenario . this observation led to two simple questions- ( i ) do we need a multiparty qd for any practical purpose ? and ( ii ) if answer of the previous question is yes , can we construct such a scheme ? it is easy for us ( specially for the readers of this paper and the authors of the similar papers who often participate in conferences and meet as members of various committees ) to recognize that conferences and meetings provide examples of situation where multiparty dialogue happens . specifically , in a conference a large number of participants can exchange their thoughts ( inputs , which may be viewed as classical information ) . although , usually participants of the conference / meeting are located in one place , but with the advent of new technologies , tele - conferences , webinar , and similar ideas that allow remotely located users to get involved in multiparty dialogue , are becoming extremely popular . for the participants of such a conference or meeting that allows users to be located at different places , desirable characteristics of the scheme for the conference should be as follows- ( a ) a participant must be able to communicate directly with all other participants , or in other words , every participant must be able to listen the talk / opinion delivered by every speaker as it happens in a real conference . ( b ) a participant should not be able to communicate different opinion / message to different users or user groups . ( c ) illegitimate users or unauthorized parties ( say those who have not paid conference registration fees ) will not be able to follow the proceedings of the conference . it is obvious that criterion ( c ) requires security and a secure scheme for multiparty quantum dialogue satisfying ( a)-(c ) is essential for today s society . we refer to such a scheme for multiparty secure communication that satisfies ( a)-(c ) as ascheme for quantum conference ( qc ) because of its analogy with the traditional conferences ( specially with the tele - conferences ) . the analogy between the communication task performed here and the traditional conference can be made clearer by noting that wikipedia defines conference as `` a conference is a meeting of people who confer about a topic '' @xcite . similarly , oxford dictionary describes a conference as `` a linking of several telephones or computers , so that each user may communicate with the others simultaneously '' @xcite . this is exactly the task that the proposed protocol for qc is aimed to perform using quantum resources and in a secure manner . thus , qc is simply a conference , which is an @xmath0-party communication , where each participant can communicate his / her inputs ( classical information ) using quantum resources to remaining @xmath1 participants . however , it should be made clear that it is neither a multi - channel qsdc nor a multi - channel qd scheme . to be precise , one may assume that each participant maintains private quantum channels with all other participants and uses those to communicate his / her input to others via qsdc or qd . this is against the idea of a conference , as in this arrangement , a participant may send different information / opinion to different participants , in violation of criterion ( b ) listed above . the fact that to the best of our knowledge , no such scheme for multiparty secure quantum communication exists has motivated us to introduce the notion of qc and to aim to design a scheme for the same . here it would be apt to note that although no scheme for qc is yet proposed , various schemes for other multiparty quantum communication tasks have already been proposed . for example , quantum schemes for voting @xcite , auction @xcite , and e - commerce @xcite are necessarily expected to be multiparty quantum communication schemes . interestingly , there are a few schemes for all these tasks proposed in the past ( @xcite and references therein ) . another recently discussed multiparty task isquantum key agreement ( qka ) ( @xcite and references therein ) , where the final key is generated by the contribution of all the parties involved , and a single or a few parties can not decide the final key . for instance , a multiparty qka scheme @xcite was proposed in the recent past , in which encoded qubits travel in a circular manner among all the parties . in fact , most of these multiparty quantum communication schemes , except qka , can be intrinsically viewed as a ( many ) sender(s ) sending some useful information in a secure manner to a ( many ) receiver(s ) under the control of a third party . further , all these schemes can be broadly categorized as secure multiparty quantum communication and secure multiparty quantum computation . though the line between the two is very faint to distinguish and categorize a scheme among one of them , qka and e - commerce may be considered in the former , while voting and auction fall under the latter . some efforts have also been made to introduce a notion of qc as a multiparty quantum communication task . however , earlier ideas of qc can be viewed as special cases of the notion of qc presented here and they are not sufficient to perform a conference as defined above in analogy with the definition provided in oxford dictionary and other sources . bose , vederal and knight @xcite proposed a generalized entanglement - swapping - based scheme for multiparty quantum communication that led to a set of quantum communication schemes related to qc , viz . , cryptographic conference @xcite , conference key agreement and conference call @xcite , and a scheme where many senders send their messages to single receiver via generalized superdense coding @xcite . in cryptographic conference , all parties share a multipartite entangled state . they perform measurement in the computational or diagonal basis , and the results of those measurements in which the bases chosen by all the users coincide are used to establish the secret key which will be known to all the users within the group . a similar notion of conference key agreement was used in @xcite , where a generalized notion of dense coding was used . clearly the notion of conference is weaker here , and in our version of conference such keys can be distributed easily if all the users communicate random bits instead of meaningful messages . recent success of designing the above mentioned schemes for multiparty quantum communication further motivated us to look for a scheme for qc . a two party analogue of qc can be considered as qd , where both parties can communicate simultaneously . the group theoretic structure of ba - an - type qd schemes has been discussed in ref . the group theoretic structure discussed in @xcite will be exploited here to introduce the concept of qc . further , an asymmetric counterpart of the ba - an - type qd scheme is proposed in the recent past @xcite . following which we will also introduce and briefly discuss an asymmetric qc ( aqc ) , where all the parties involved need not to send an equal amount of information . with the recent interest of quantum communication community on quantum internet @xcite and experimental realization of multiparty quantum communication schemes @xcite , the motivation for introducing a qc or aqc scheme can be established . remaining part of the paper is organized as follows . [ sec : ba - an - protocol ] is dedicated to a brief review of qd and the group theoretic approach of qd for the sake of completeness of the paper , which has been used in the forthcoming sections to develop the idea of qc . two general schemes for the task of qc have been introduced in sec . [ sec : quantum - conferencepro ] . in the next section , we have considered a few specific examples of both these schemes . the feasibility of an aqc scheme has also been discussed in sec . [ sec : examples - and - possible ] . finally , the security and efficiency of the proposed schemes have been discussed in sec . [ sec : security - analysis ] before concluding the paper in sec . [ sec : conclusion ] . [ [ section ] ] it would be relevant to mention that some of the present authors had presented the general structure of qd protocols in @xcite and established that the set of unitary operators used by alice and bob must form a group under multiplication . the group structure has also been found to be suitable for the asymmetric qd schemes @xcite , where alice and bob use encoding operations from different subgroups of a modified pauli group , like @xmath2 . this particular abelian group ( @xmath3 ) is of order 4 under multiplication and is called a modified pauli group as we neglect the global phase in the product of any two elements of this group , which is consistent with the quantum mechanics ( for detail see @xcite ) . the generalized group @xmath4 can be formed by @xmath0-fold tensor products of @xmath3 , i.e. , @xmath5 . in the original qd protocol @xcite , the encoding is done by alice and bob , respectively , using the same set of operations @xmath6 from the modified pauli group @xmath3 . the entire scheme of ba an @xcite can be summed up in the formula @xmath7 @xmath8 , where @xmath9 are the bell states . it is required that all the possible final states obtained after alice s and bob s encoding operations should remain orthonormal to each other and also with the initial state . once the initial and final states are known to both the legitimate users , they can exploit knowledge of their own encoding operation to extract each other s message . interestingly , alice and bob encode information with the same operators , say , @xmath10 for 00 , @xmath11 for 01 , @xmath12 for 10 , and @xmath13 for 11 . in this scenario , alice obtains a unique bijective mapping from the composite encoding of alice and bob ( @xmath14 ) to bob s operation ( @xmath15 ) using her unitary operation ( @xmath16 ) . this is obvious where there are only 2 parties , we may ask , is it possible to extend this scheme for qd to design a scheme for multiparty conference ? let us examine two cases with 3 parties : in case 1 : when all the parties encode the same bits say , 00 i.e. , they apply @xmath17 and @xmath18 ; and in case 2 : when one of them encodes the same bits used in case 1 , i.e. , 00 and other two will encode the similar bits but other than 00 , say 01 @xmath19 , i.e. , they apply @xmath20 and @xmath18 , respectively . in these two cases , the resultant state is always the same as what was prepared initially , and none of the parties can deterministically conclude each others encoding . in fact , there will be many such cases , hence , ba an s original protocol for qd can not be generalized directly to design a scheme for multiparty conference . to design a scheme for qc , we will use the idea of disjoint subgroups introduced by some of the present authors in the recent past @xcite . disjoint subgroups refer to subgroups , say @xmath21 and @xmath22 , of a group @xmath4 such that they satisfy @xmath23 . thus , except identity @xmath21 and @xmath22 do not contain any common element . the modified pauli group @xmath3 has 3 mutually disjoint subgroups : @xmath24 @xmath25 and @xmath26 . whenever there are more than two parties , we can encode using disjoint subgroups of operators , i.e. , each party may be allowed to encode with a unique disjoint subgroup . for example , if alice , bob and charlie want to set up a qc among them , then alice can encode using @xmath27 , bob can encode using @xmath28 and charlie can encode using @xmath29 the use of disjoint subgroups circumvents the limitations of the original two - party qd scheme and provides a unique mapping required for multiparty conversation . in what follows , we have proposed two protocols to accomplish the task of a qc scheme . [ [ section-1 ] ] here , we have designed two multiparty quantum communication schemes where prior generation of key is not required . these schemes may be used for qc , i.e. , for multiparty communication of meaningful information among the users . additionally , it is easy to observe that these schemes naturally reduce to the schemes for multiparty key distribution if the parties send random bits instead of meaningful messages . let us start with the simplest case , where @xmath30 parties send their message to @xmath31th party . this can be thought of as a multiparty qsdc . suppose all the parties decide to encode or communicate @xmath32-bit classical messages . in this case , each user would require a subgroup of operators with at least @xmath33 operators . in other words , each party would need at least a subgroup @xmath21 of order @xmath33 of a group @xmath34 . here , we would like to propose one such multiparty qsdc scheme . step 1.1 : : first party alice be given one subgroup @xmath35 to encode her @xmath32-bit information . similarly , other parties ( say bob and charlie ) can encode using subgroups @xmath36 , and @xmath37 , and so on for @xmath30th party diana , whose encoding operations are @xmath38 . + all these subgroups are pairwise disjoint subgroups , i.e. , they are chosen in such a way that @xmath39 . as the requirement for encoding operations to be from disjoint subgroups has been already established beforehand . + additionally , here we assume that all the parties do nothing ( equivalent to operator identity ) on their qubits for encoding a string of @xmath32 zeros . as identity is the common element in the set of encoding operations to be used by each party it will be convenient to consider this as a convention in the rest of the paper . step 1.2 : : nathan ( the @xmath31th party ) prepares an @xmath0-qubit entangled state @xmath40 ( with @xmath41 ) . + it is noteworthy that maximum information that can be encoded on the @xmath42-qubit quantum channel is @xmath42 bits and here @xmath30 parties are sending @xmath32 bits each . in other words , after encoding operation of all the @xmath30 parties the quantum states should be one of the @xmath43 possible orthogonal states . step 1.3 : : nathan sends @xmath44 qubits ( @xmath45 ) of the entangled state @xmath40 to alice in a secure manner , who applies one of the operations @xmath46 ( which is an element of the subgroup of operators available with her ) on the travel qubits to encode her message . this will transform the initial state to @xmath47 . subsequently , alice sends all these encoded qubits to the next user bob . step 1.4 : : bob encodes his message which will transform the quantum state to @xmath48 . finally , he also sends the encoded qubits to charlie in a secure manner . step 1.5 : : charlie would follow the same strategy as followed by alice and bob . in the end , diana receives all the encoded travel qubits and she also performs theoperation corresponding to her message to transform the state into @xmath49 . she returns all the travel qubits to nathan . step 1.6 : : nathan can extract the information sent by all @xmath30 parties by measuring the final state using an appropriate basis set . + it may be noted that nathan can decode messages sent by all @xmath30 parties , if and only if the set of all the encoding operations gives orthogonal states after their application on the quantum state , i.e. , @xmath50 are orthogonal for all @xmath51 . in other words , after the encoding operation of all the @xmath30 parties the quantum states should be a part of a basis set with @xmath43 orthogonal states for unique decoding of all possible encoding operations . this scheme can be viewed as the generalization of ping - pong protocol @xcite to a multiparty scenario , where multiple sender s can simultaneously send their information to a receiver . in a similar way , if all the senders wish to send and receive the same amount of information , then all of them can also choose to prepare their initial state @xmath40 independently and send it to all other parties in a sequential manner . subsequently , all of them may follow the above protocol faithfully to perform @xmath31 simultaneous multiparty qsdc protocols . in fact , @xmath31 simultaneous multiparty qsdc schemes of the above form will perform the task required in an ideal qc scheme . however , as each sender has to encode his secret multiple times ( @xmath52 times ) , it would allow him to encode different information in each round . though it may be advantageous in some communication schemes , where a sender is allowed to send different bit values to different receivers , but is undesirable in a scheme for qc . specifically , to stress on the relevance of a scheme that allows each sender to encode different bits to all the receivers , we may consider a situation where each party ( or a few of them ) publicly asks a question , and the receivers answer the question independently ( for an analogy think of a panel discussion in television ) . in this case , all the receivers may have different opinions ( say one may agree with some of them and may not with the remaining ) about various questions being asked . as far as a scheme for qc is concerned , protocol 1 described here would work under the assumption of semi - honesty . specifically , a semi - honest party may try to cheat , but he / she would follow the protocol faithfully . this assumption would enable us to consider that each party is encoding the same information every time . in what follows . we will establish that such an assumption is not required . specifically , in protocol 2 , we aim to design a genuine qc scheme , which does not require the semi - honesty assumption to restrict a user from sending different information to different receivers . here , we will attempt to design an efficient qc scheme , which can be thought of as a generalized qd scheme . in analogy of the original ba - an - type qd scheme , we will need the set of encoding operations for the @xmath31th party ( nathan ) . here , firstly we propose the protocol which is followed by a prescription to obtain the set of operations for @xmath31th party , assuming a working scheme designed for the protocol 1 . step 2.1 : : same as that of step 1.1 of protocol 1 with a simple modification that also provide nathan a subgroup @xmath53which enables him to encode a @xmath32-bit message at a later stage . + the mathematical structure of this subgroup will be discussed after the protocol . step 2.2 : : same as step 1.2 of protocol 1 . step 2.3 : : same as step 1.3 of protocol 1 . step 2.4 : : same as step 1.4 of protocol 1 . step 2.5 : : same as step 1.5 of protocol 1 . step 2.6 : : nathan applies unitary operation @xmath54 to encode his secret and the resulting state would be @xmath55 @xmath56 . step 2.7 : : nathan measures @xmath57 using the appropriate basis as was done in step 1.6 of protocol 1 and announces the measurement outcome . now , with the information of the initial state , final state and one s own encoding all parties can extract the information of all other parties . + it is to be noted that the information can be extracted only if the set of all the encoding operations gives orthogonal states after their application on the quantum state , i.e. , all the elements of @xmath58 are required to be mutually orthogonal for @xmath59 . in other words , after the encoding operation of all the @xmath31 parties the set of all possible quantum states should form a @xmath43 dimensional basis set . nathan s unitary operation can be obtained using the fact that the remaining @xmath30 parties have already utilized the channel capacity . hence , his encoding should be in such a way that after his encoding operation @xmath54 , the final quantum state should remain an element of the basis set in which the initial state was prepared . however , the bijective mapping between the initial and final states present in protocol 1 would disappear here . this is not a limitation . it is actually a requirement . this is so because , in contrast to protocol 1 where the initial and final states are secret , in protocol 2 , the choice of the initial state and the final state are publicly broadcasted . existence of a bijective mapping would have revealed all the secrets to eve . this condition provides us a mathematical advantage . specifically , it allows us to construct the set of unitary operations that nathan can apply . to do so we need to use the information about the disjoint subgroups of operators that are used by other parties . the procedure for construction of nathan s set of operations is described below . for simplicity , let us write the encoding operations of all the parties as follows : @xmath60 here , @xmath61 corresponds to the binary value of the decimal number @xmath62 , and it represents the classical information to be encoded by user x * * * ( listed in column 1 ) using the the operator @xmath63 ( listed in @xmath64th column in the row corresponding to the user x * * * ) . for example , to encode @xmath65 alice would use the operator @xmath66 , whereas for the same encoding bob and charlie would use @xmath67 and @xmath68 , respectively . further , we would like to note that by construction operators @xmath69 as @xmath70 is an element of the modified pauli group , and it is assumed that the encoding operations of the different users are chosen from the disjoint subgroups of the modified pauli groups in such a way that the product of operations listed in any column is identity , i.e. , @xmath71 this implies that if all the parties encode the same secret then the final state and the initial state would be the same . to illustrate this we may consider following example @xmath72 from eqs . ( [ eq : condition])-([eq : example ] ) , it is clear that the choice of encoding operations of the other users ( i.e. , @xmath73 would uniquely determine @xmath74 . further , it is assumed that the encoding operations used by different users to encode @xmath61 are selected in a particular order that ensures @xmath75 and particular choice of @xmath76 for example , this condition implies that if alice s operators satisfy @xmath77 then bob and charlie would be given the encoding operators in an order that satisfy @xmath78 and @xmath79 respectively , and the same ordering of operators will be applicable to all other users . now , using the above mentioned facts and convention , we need to establish that @xmath80 forms a group under multiplication . ( [ eq : condition])and the self reversibility of the elements @xmath70 lead to following identity- @xmath81 this may be used to establish the closure property of the group @xmath80 as @xmath82 . this is so because the pauli operators commute with each other under the operational definition of multiplication used in defining the modified pauli group . all the remaining properties of the group follows directly from the nature of pauli operators used to design @xmath83 . thus , it is established that the generalized multiparty qsdc scheme can be modified to a generalized qd scheme . it will be interesting to obtain the original ba an s qd scheme as a limiting case as follows . @xmath84 this particular case and all the discussions leave us with @xmath85 which is identical with alice s operations . in table [ tab : conference ] , we have provided a list comprising of the number of participants in the qc and the number of cbits they want to encode . the table explicitly mentions different multipartite states or quantum channels that can be utilized for the same . .[tab : conference ] various possibilities of qc scheme with a maximum number of @xmath31 parties each encoding @xmath32 bits using a group of unitary operators with at least @xmath43 elements . the quantum states suitable in each case and corresponding number of travel qubits are also mentioned . [ cols="^,^,^,^,^ " , ] the proposed qc scheme may also be extended to an asymmetric counterpart of the qc scheme , where each party may not be encoding the same amount of information . one such easiest example is a lecture , where the orator speaks most of the time while the remaining users barely speak . in such cases , the parties sending redundant bits to accommodate the qc scheme may choose an aqc scheme . to exploit the maximum benefit of such schemes a party encoding more information than others ( say alice ) should prepare ( and also measure ) the quantum state ( in other words , start the qc scheme ) . in this case , the choice of unitary operations by each party would also become relevant and alice should use a subgroup of higher order than the remaining users . for instance , in a 3-party scenario , alice may use a @xmath86 from row 2 of table [ tab : multiparty - quantum - conference . ] to encode 2 bits message , while the remaining three users may choose @xmath87 and @xmath88 , respectively . it is worth noting here that the security of the qc scheme discussed in the following section ensures the security of the aqc scheme designed here as well . further , the proposed schemes can also be easily modified to obtain corresponding schemes for controlled qc , where an additional party ( who is referred to as the controller ) would prepare the quantum channel in such a way that the qc task can only be accomplished after the controller allows the other users to do so @xcite . controlled qc can be achieved in various ways . for example , the controller may prepare the initial state and keep some of the qubits with himself , and in absence of the measurement outcome of the corresponding qubits the other legitimate parties would fail to accomplish the task @xcite . the same feat can also be achieved by the controller without keeping a single qubit with himself by using permutation of particles @xcite . thus , it is easy to generalize the proposed schemes for qc to yield schemes for controlled qc . such a scheme for controlled qc would have many applications . for example , a direct application to that scheme would be quantum telephone where the controller can be a telephone company @xcite that provides the channel to the respective users after authentication . thus , the present scheme can be used to generalize the scheme proposed in @xcite and thus to obtain a scheme for multiparty quantum telephone or quantum teleconference . additionally , the multiparty communication schemes proposed here can be reduced to schemes for secure multiparty quantum computation . interestingly , a recently proposed secure multiparty computation scheme designed for quantum sealed - bid auction task @xcite can be viewed as a reduction of the protocol 1 proposed here . therefore , we hope that the proposed schemes may also be modified to obtain solutions of various other real life problems . a qc protocol is expected to confront the disturbance attack ( or denial of service attack ) , the intercept - and - resend attack , the entangle - and - measure attack , man - in - the - middle attack and trojan - horse * * attack by implementing the bb84 subroutine strategy ( for detail see @xcite ) , which allows senders to insert decoy qubits prepared randomly in @xmath11-basis or @xmath13-basis in analogy with bb84 protocol and to reveal the traces of eavesdropping by comparing the initial states of the decoy qubits with the states of the same qubits after measured by the receivers randomly using @xmath11-basis or @xmath13-basis . in fact , quantum communication of all the qubits from one party to other , as mentioned in both the protocols ( for example , in step 1.3 ) , is performed in a secure manner . to accomplish the secure communication of message qubits using bb84 subroutine , an equal number of decoy qubits ( the number of decoy qubits are required to be equal to the number of message qubits traveling through the channel ) are inserted randomly in the string of travel qubits . on the authenticated receipt of this enlarged sequence of travel qubits , the sender discloses the positions of the decoy qubits and those qubits are then measured by the receiver randomly in @xmath11-basis or @xmath13-basis . subsequent comparison of the initial states and the measurement outcomes reveals the error rate . if the computed error rate is obtained below a tolerable limit , then the quantum communication of message qubits is considered to be accomplished in a secure manner @xcite , and the steps thereafter are followed . therefore , the above mentioned attacks on the proposed schemes can be defeated simply by adding decoy qubits and following bb84 subroutine . further , bob s intimation by alice that she has sent her qubits and bob s acknowledgment of the receipt of qubits , via an authenticated classical channel , is necessary to avoid the unwanted circumstances under which eve pretends as the desired party . there also exist some technical procedures to circumvent the trojan - horse attack ( @xcite and references therein ) . as a scheme of qc incorporates multiusers we have discussed below the security in two scenarios where ( 1 ) an outsider ( eve ) attacks the protocol , or ( 2 ) an insider ( one or some of the legitimate users ) attacks the protocol . further , all the attacks and counter measures mentioned in this section are applicable on both the schemes , unless specified . in the * entangle - and - measure attack * , eve entangles her qubit @xmath89 with the travel qubit in the channel . eve can extract the information by performing the @xmath13-basis measurement on her ancillae . to counter this attack , the decoy states @xmath90 , @xmath91 , @xmath92 and @xmath93 are randomly inserted and when they are examined for security , then eve is detected with probability @xmath94 when she attacks @xmath90 and @xmath91 states , otherwise the states remain separable for @xmath92 and @xmath93 . consequently , the total detection probability of eve is @xmath95 taking into account that the probability of generation of each decoy qubit state is @xmath96 . in the * intercept - and - resend attack * , eve prepares some fresh qubits and swaps one of her qubits with the accessible qubit in the channel when @xmath97th user sends it to @xmath98th user . thereafter , eve retrieves her qubit during their communication from @xmath98th user to @xmath99th user and obtains the encoding of @xmath98th user by performing a measurement on her qubits . this attack will also be defended by incorporating decoy qubits . however , eve may modify her strategy to measure the intercepted qubits randomly in either the computational or diagonal basis before sending the freshly prepared qubits corresponding to the measurement outcomes . it is evident that eve s measurement of the decoy qubits will produce disturbance if she measures in the wrong basis . let @xmath0 be the total number of travel qubits such that @xmath100 are decoy and message qubits each . eve intercepts @xmath44 qubits which will entail both decoy and message . without a loss of generality , we assume that half of the @xmath44 qubits are decoy and the other half are message qubits . since the security check is performed on the decoy qubits alone , we are interested in the @xmath101 decoy qubits which eve measures in her lab out of the @xmath100 decoy qubits in the channel . the fraction of qubits measured by eve out of the total decoy qubits is given by @xmath102 . from which the information gained by eve is @xmath103 this implies that @xmath104 times the correct basis will be chosen by eve . the error induced by eve is observed by alice and bob only when bob measures in the same basis as of alice and is @xmath105 . the amount of information bob receives is given by @xmath106)$ ] , where @xmath107 $ ] is the shannon binary entropy . the security is ensured until @xmath108 . one can calculate the fraction @xmath109 for secure communication with the tolerable error rate @xmath110 ( @xcite and references therein ) . eve s success probability is @xmath111 and it would decrease with the increasing value of @xmath44 as @xmath112 . * information leakage attack * is inherent in the qd schemes , and consequently , is applicable to protocol 2 proposed here as well . it refers to the information gained by eve about the encoding of the legitimate parties by analyzing the classical channel only . in brief , the leakage can be thought of as the difference between the total information sent by both the legitimate users and the minimum information required by eve to extract that information ( i.e. , eve s ignorance ) . the mathematical prescription for an average gain of eve s information is @xmath113 where @xmath114 is the total classical information all the legitimate parties have encoded ; and @xmath115 is eve s ignorance after the announcement of the measurement outcome and is averaged over all the possible measurement outcomes as @xmath116 , with the conditional entropy @xmath117 . if the party authorized to prepare and measure the quantum state selects the initial state randomly and sends it to all the remaining users by using a standard unconditionally secure protocol for qsdc or dsqc then the leakage can be avoided as it increases the @xmath115 , and thus decreases @xmath118 to zero corresponding to no leakage @xcite . * participant attack * is possible in both the schemes proposed here . * * in the first scheme , a participant can send different cbits to different members unless we assume semi - honest parties . although this scheme is advantageous in certain applications , like sealed bid auction ( where this attack is detected in post - confirmation steps ) @xcite or where each participant wants to encode different values to respective participant , but in the conference scenario where it is required that each participant encodes the same message to all other participants then this attack is prominent , and it is wise to follow the second scheme , which is free from the assumption of semi - honest parties . in the second scheme , the authorized party ( authorized to prepare and measure the quantum state ) encodes his information at the end just before performing the joint measurement and announcing the outcome . if he wants to cheat he can disclose an incorrect measurement outcome corresponding to his modified encoding once he comes to know others encoding . this action can be circumvented , and we can implement this protocol either with a trusted party or we can randomly select any two participants and run the scheme twice considering that respective party encodes same information . another solution would be that the initiator sends the hash value of his message at the beginning to all the remaining users , and if the hash value of his encoding revealed at the last do not match with that of the initially sent hash value , then he had cheated and will be certainly identified . * collusion attack * is a kind of illegal collaboration of more than one party who are not adjacent to each other , to cheat other members of a group to learn their encoding ( precisely of those who are in between them ) . the proposed schemes are circular in nature . in this type of an attack , the attackers generate an entangled state and circulate the same number of fake qubits as that of the travel qubits . the attackers at the end already possess the home photons of the fake qubits circulated by the first attacker and performs a joint measurement to learn the encoding of the participants in between them . it will be more effective if @xmath98th and @xmath119th participants collude . this is so , as both of them get the access of the travel particles at least once after knowing the secret of all the remaining parties . this attack can be averted by breaking the larger circle into @xmath120 sub - circles such that if less than @xmath120 attackers collude , they will not be able to cheat ( see @xcite for details ) . this attack and the solution are applicable in both the proposed schemes . the qubit efficiency of a quantum communication scheme is calculated as @xmath121 where @xmath122 bits of classical information is transmitted using @xmath123 number of qubits , and an additional classical communication of @xmath124 bits @xcite . in the first qc scheme , @xmath125 @xmath126 , and @xmath127 as each party sends @xmath32 bits and prepares @xmath0-qubit entangled state and @xmath44 decoy qubits in each round of quantum communication . therefore , the efficiency is calculated to be @xmath128 . similarly , the qubit efficiency of the second qc scheme among @xmath31 parties such that each party encodes @xmath32 bits can be computed by noting that in this case @xmath125 @xmath129 , and @xmath130 . here , @xmath131 as the classical communication of @xmath0 cbits is associated with the broadcast of the measurement outcome by the authorized party . thus , the qubit efficiency is obtained as @xmath132 . from the @xmath133 one can easily calculate the qubit efficiency of various possible qc schemes detailed in table [ tab : conference ] . for example , one can check that the qubit efficiency of a two party qc with each party encoding 2 bits ( which is ba an s qd protocol ) using bell state as quantum channel is 67% . similarly , the qubit efficiency for a qc scheme involving three parties sending 1 bit each with bell state as the quantum channel can be obtained as 43% . hence , we find that for the same initial state as quantum channel the efficiency decreases as the number of parties increases and / orthe number of encoded bits decreases . in summary , the notion of qc is introduced as a multiparty secure quantum communication task which is analogous with the notion of classical conference , and two protocols for secure qc are designed . the proposed protocols are novel in the sense that they are the first set of protocols for qc , as the term qc used earlier were connected to communication tasks that were not analogous to classical conference . further , it is shown that protocols proposed here can be reduced to protocols for qc proposed earlier considering much weaker notion of conference . one of the proposed protocols can be viewed as a generalization of the ping - pong protocol for qsdc , whereas the other one can be viewed as a generalization of the schemes for qd . it is noted that protocol 1 composes number of rounds of multiple - sender to single receiver secure direct communication , which accomplishes the task of qc under the assumption of semi - honesty of the users . however , this semi - honesty assumption is not required for protocol 2 , which is proposed here as multiple - sender to multiple - receiver scheme , where the task is performed in a single round . subsequently , both the proposed schemes are elaborated with the help of an explicit example . we have discussed the utility and applications of these protocols in different scenarios . specifically , the proposed schemes may be reduced to a set of multi - party qkd and qka schemes , if the parties involved in qc send random bits instead of meaningful messages . further , feasibility and significance of the controlled and asymmetric counterparts of the proposed qc schemes have also been established . the modified versions of the proposed schemes may also be found useful in accomplishing some real - life problems , whose primitive is secure multiparty computation . for example , one can employ the proposed schemes for voting among the five countries having power of veto in united nations , where it is desired that the choice of a voter is not influenced by the choice of the others . the proposed scheme can also be extended to obtain a dynamic version of qc , where a participant can join the conference once it has started and leave it before its termination . such a generalization is possible using the method introduced by some of the present authors in ref . @xcite . further , the effect of various types of markovian and non - markovian noise on the schemes proposed here can be investigated easily using the approach adopted in @xcite . security of the proposed schemes has been established against various types of insider and outsider attacks . further , the qubit efficiency analysis established that protocol 2 is more efficient than protocol 1 . further , one can easily observe that the proposed schemes are much more efficient compared to a simple minded scheme that performs the same task by using multiple two - party direct communication schemes , which will again work only under the assumption of semi - honest users . finally , we have also presented a set of encoding operations suitable with a host of quantum channels for performing the qc schemes for number of parties . this provides experimentalists a freedom to choose the encoding operations and the quantum state to be used as quantum channel as per convenience . further , experimental realization of quantum secure direct communication scheme , which can demonstrate protocols , like quantum dialogue , quantum authentication , has been successfully performed in @xcite , and it paves way for experimental realization of qc . keeping these facts in mind , we conclude this paper with a hope that the schemes proposed here and/or their variants will be realized in the near future . * acknowledgment : * ab acknowledges support from the council of scientific and industrial research , government of india ( scientists pool scheme ) . cs thanks japan society for the promotion of science ( jsps ) , grant - in - aid for jsps fellows kt and ap thank defense research & development organization ( drdo ) , india for the support provided through the project number erip / er/1403163/m/01/1603 . 10 bennett , c. h. , brassard , g. : quantum cryptography : public key distribution and coin tossing . in proceedings of the ieee international conference on computers , systems , and signal processing , bangalore , india , pp . 175 - 179 ( 1984 ) shor , p. w. polynomial - time algorithms for prime factorization and discrete logarithms on a quantum computer , in proceedings of 35th annual symp . on foundations of computer science , santa fe , ieee computer society press . ( 1994 ) thapliyal , k. , pathak , a. : applications of quantum cryptographic switch : various tasks related to controlled quantum communication can be performed using bell states and permutation of particles . quantum inf . process . * 14 * , 2599 - 2616 ( 2015 ) sharma , v. , thapliyal , k. , pathak , a. , banerjee , s. : a comparative study of protocols for secure quantum communication under noisy environment : single - qubit - based protocols versus entangled - state - based protocols . inf . process . * 15 * , 4681 ( 2016 ) | a notion of quantum conference is introduced in analogy with the usual notion of a conference that happens frequently in today s world .
quantum conference is defined as a multiparty secure communication task that allows each party to communicate their messages simultaneously to all other parties in a secure manner using quantum resources .
two efficient and secure protocols for quantum conference have been proposed .
the security and efficiency of the proposed protocols have been analyzed critically .
it is shown that the proposed protocols can be realized using a large number of entangled states and group of operators .
further , it is shown that the proposed schemes can be easily reduced to protocol for multiparty quantum key distribution and some earlier proposed schemes of quantum conference , where the notion of quantum conference was different .
keywords : quantum conference , quantum cryptography , secure quantum communication , multiparty quantum communication . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
strongly correlated electron systems in low dimensions are of fundamental interest due to their fascinating properties resulting from strong quantum fluctuations @xcite . especially in the case of the high - t@xmath20 cuprate superconductors , the role of quantum fluctuations is heavily debated . two - magnon raman scattering has been proven to be a powerful tool to study quantum fluctuations in the magnetic sector@xcite . in contrast to the well understood magnon dispersion as measured by inelastic neutron scattering@xcite , the quantitative understanding of the two - magnon line - shape in the raman response@xcite and in the optical conductivity@xcite remains an issue open to debate . interestingly , in the so - called cuprate ladder systems like sr123 or the telephone - number compounds ( sr , ca , la)@xmath2cu@xmath3o@xmath4 a prominent peak in the magnetic raman response is observed at the same energy of about @xmath21 @xmath13 as in the two - dimensional compounds@xcite . in contrast to the gapless long - range ordered two - dimensional compound , the quasi one - dimensional two - leg ladders are known to be realizations of a gapped spin liquid@xcite . because the elementary excitations above this groundstate are triplons@xcite , we call the corresponding raman response as two - triplon raman scattering . on the one hand , one may expect that the raman response is dominated by short - range , high - energy excitations , suggesting a certain similarity between ladders and planes , both being built from edge - sharing cu@xmath18 plaquettes . the peak frequencies are in fact at @xmath21 @xmath13 . on the other hand , the line shape and in particular the peak width strongly varies between different compounds . in 2d , the peak width is of the order of @xmath22 @xmath13 , in la6ca8 about @xmath23 @xmath13 , in sr123 and sr14 only @xmath24 @xmath13 . due to the observation of a very sharp two - triplon raman line in the spin liquid sr14 , gozar _ et al . _ have questioned whether the large line width in 2d and the related , heavily discussed spectral weight above the two - magnon peak can be attributed to quantum fluctuations@xcite . in the last years theoretical developments in the field of quasi one - dimensional systems , namely the quantitative calculation of spectral densities@xcite , has led to a deeper understanding of magnetic contributions to the raman response of undoped cuprate ladders . besides the usual heisenberg exchange terms the minimal magnetic model includes four - spin interactions which are 4 - 5 times smaller than the leading heisenberg couplings@xcite . the existence and the size of the four - spin interactions are consistent with theoretical derivations of generalized @xmath25-@xmath26 models from one - band or three - band hubbard models@xcite . in the present paper we show that the strong variation of the line width can be traced back to changes of the spatial anisotropy of the exchange constants . the sharp raman line in sr14 and sr123 results from @xmath27 , the increased line width in la6ca8 reflects @xmath15 , and the isotropic couping @xmath28 for the square lattice yields the much larger width observed in 2d . in fact , we obtain a quantitative description of the dominant raman peak in 2d using a toy model which mimics the 2d square lattice by the superposition of a vertical and a horizontal ladder . we thus conclude that the dominant raman peak is well described by short - range excitations . besides the dominant two - triplon peak , the large spectral weight measured at high energies remains an open problem for the cuprate ladders and planes . we review possible sources of the high - energy spectral weight which were suggested in the past , e.g. quantum fluctuations@xcite , the role of spin - phonon interaction@xcite and the triple resonance@xcite . in case of the cuprate planes no final conclusion concerning the origin of the high - energy weight can be drawn , but in the case of the cuprate ladders the spin - phonon coupling and the triple resonance can be ruled out . in raman scattering multi - particle excitations with zero change of the total spin can be measured . starting at @xmath29 from an @xmath30 ground state the singlet excitations with combined zero momentum are probed . the raman response in spin ladders has been calculated by first order perturbation theory@xcite and by exact diagonalization@xcite . in this work , raman line - shapes are presented obtained from continuous unitary transformations ( cut ) using rung triplons as elementary excitations@xcite . the results are not resolution limited because neither finite size effects occur nor an artificial broadening is necessary . for zero hole doping , the minimum model for the magnetic properties of the @xmath31 two - leg ladders is an antiferromagnetic heisenberg hamiltonian plus a cyclic four - spin exchange term @xmath32 @xcite [ eq : hamiltonian ] @xmath33 where @xmath34 denotes the rungs and @xmath35 the legs . the exchange couplings along the rungs and along the legs are denoted by @xmath8 and @xmath7 , respectively . the relevant couplings modeling sr123 and sr14@xcite are illustrated in fig . [ fig_sketch ] . there is also another way to include the leading four - spin exchange term by cyclic permutations @xcite which differs in certain two - spin terms from eq . ( [ eq : hamiltonian ] ) @xcite . both hamiltonians are identical except for couplings along the diagonals if @xmath8 and @xmath7 are suitably redefined @xcite . ) ) . the circles denote the positions of cu@xmath36 ions carrying a spin 1/2 each . the crystallographic axes are such that @xmath37 and @xmath38 for sr123 and @xmath39 and @xmath40 for sr14 . ] at @xmath29 the raman response @xmath41 is given by the retarded resolvent @xmath42 the observables @xmath43 ( @xmath44 ) for magnetic light scattering in rung - rung ( leg - leg ) polarization read in leading order @xcite [ observables ] @xmath45 the factors @xmath46 and @xmath47 depend on the underlying microscopic electronic model . it is beyond the scope of the present work to compute them . the results will be given in units of these factors squared . in this article we will only consider non - resonant raman excitation processes . we discuss which laser energy should be used in order to investigate the non - resonant regime . technically , we employ a cut to map the hamiltonian @xmath48 to an effective hamiltonian @xmath49 which conserves the number of rung - triplons , i.e. @xmath50=0 $ ] where @xmath51 @xcite . the ground state of @xmath49 is the rung - triplon vacuum . for the response function @xmath41 the observable @xmath52 is mapped to an effective observable @xmath53 by the same cut . the cut is implemented in a perturbative fashion in @xmath54 and @xmath55 . the effective hamiltonian is calculated up to high orders ( 1-triplon terms : 11@xmath56 , 2-triplon terms : 10@xmath56 order ) . the effective observable @xmath57 is computed to order @xmath58 in the 2-triplon sector . the resulting plain series are represented in terms of the variable @xmath59 @xcite where @xmath60 is the one - triplon gap . then standard pad extrapolants @xcite yield reliable results up to @xmath61 depending on the value of @xmath62 . consistency checks were carried out by extrapolating the involved quantities before and after fourier transforms . in case of inconclusive extrapolants the bare truncated series are used . we will estimate the overall accuracy below by comparing with dmrg results@xcite . the raman line shape is finally calculated as continued fraction by tridiagonalization of the effective two - triplon hamiltonian . sectors with odd number of triplons are inaccessible by raman scattering due to the invariance of the two observables @xmath63 and @xmath64 with respect to reflections about the centerline of the ladder@xcite . thus only excitations with even number of triplons matter . therefore the leading contributions to the raman response come from the 2-triplon sector . it was shown earlier that the two - triplon contribution is the dominant part of the raman response at low and intermediate energies@xcite . the role of the four - triplon contribution for the high - energy spectral weight will be discussed at the end of this work . in this part we will compare the theoretically obtained two - triplon contributions to the experimental line - shapes of the cuprate ladders sr123 and sr14 . the crystals of sr123 have been grown and measured under the same conditions as described in refs . and , while the data of sr14 have been provided by gozar _ et al._@xcite . the experimental raman line - shape depends strongly on the laser energy because resonant contributions are present . this becomes apparent in a strong anisotropy between the width of the two - triplon peak in leg and rung polarization for laser energies @xmath65 ev . the width of the two - triplon peak in leg polarization is much sharper . for laser energies @xmath66 ev , the strong anisotropy between both polarizations vanishes@xcite . it is therefore important to figure out which laser energy has to be used for the comparison between the non - resonant theory and the experiment in order to study the magnetic excitations only . the first criterion can be gained from the optical conductivity as for example given in ref . : the intensity of the two - triplon peak develops in the same way as the optical conductivity . for the non - resonant regime both , energy of the incident and scattered light , should be smaller than the charge transfer gap ( sr14 : @xmath67 ev ) . thus , we have chosen spectra with laser energies @xmath68 ev in the case of sr14 and @xmath69 ev for sr123 . luckily , the value of the optical conductivity is about 100 @xmath70@xmath13 in the sub gap regime at @xmath71 ev@xcite ( which is 1 - 2 orders of magnitude larger than for the 2d cuprates @xcite ) yielding a non vanishing intensity of the two - triplon peak . here @xmath72 denotes the energy of the two - triplon peak . the second criterion arises from the polarization dependence of the two - triplon peak . depending on the laser energy used one can observe a drastic difference in the line shape between the two polarizations@xcite . while the difference in the line shapes is large for @xmath73 ev , it does almost vanish in the case of @xmath74 ev @xcite . this fits very well to the weak polarization dependence of the purely magnetic response as described by eqs . ( [ observables ] ) : for @xmath75 the raman line - shape is identical in the rung and the leg polarization . small deviations @xmath76 as relevant for the description of sr123 and sr14 produce small deviations with respect to the symmetry between the rung and leg polarization of the ladder . these deviations can not account for the drastic change between the two polarizations as observed for @xmath73 ev @xcite . we therefore conclude that the spectra @xmath74 ev are the best choice in order to compare to a purely magnetic , non - resonant theory . now we discuss the dependence of the width of the two - triplon peak on the parameters @xmath77 and @xmath78 . in fig . [ fig : fwhm ] , the full width at half maximum ( fwhm ) of the two - triplon peak is shown . the overall uncertainty shown as error bars in fig . [ fig : fwhm ] of the extrapolated two - triplon fwhm was determined by comparing to dmrg data@xcite . let us first consider the case @xmath79 . here the two - triplon width should be identical in both polarizations . it can be clearly seen that the numerically obtained results reflect this property rather well indicating that the uncertainties in the extrapolation are small concerning the matrix elements . there is a strong dependence of the fwhm of the two - triplon peak on the parameter @xmath77 . the peak sharpens significantly when the ratio @xmath77 of the magnetic heisenberg exchanges increases ( 4 times from @xmath28 to @xmath27 ) . in the case of @xmath80 , the width depends on the polarization . in general , the width in ( xx)-polarization is larger than in ( yy)-polarization . for fixed @xmath77 the fwhm changes at maximum by a factor of two when varying @xmath81 from @xmath82 to @xmath83 . ( squares ) , @xmath84 ( triangles ) and @xmath27 ( diamonds ) as a function of the strength of the four - spin interactions @xmath85 . the orange ( grey ) symbols denote ( xx)-polarization and the blue ( black ) symbols ( yy)-polarization . the solid lines are a guide to the eye obtained by spline interpolation.,width=309 ] in figs . [ fig : comp](a - d ) , the experimental raman response of sr123 and sr14 is shown for ( xx)- and ( yy)-polarization ( red / black and cyan / grey curves ) . the spectra of sr123 were taken in the same way as described in ref . . the data of sr14 has been made available by gozar _ et al._@xcite . in addition , theoretically obtained two - triplon contributions are displayed ( orange / grey and blue / black ) . experimentally , the width of the two - triplon peak of both materials is almost identical ( @xmath86 150 @xmath13 ) . only the position of the two - triplon peak is different ( @xmath86 3140 @xmath13 for sr123 and @xmath86 3000 @xmath13 for sr14 ) which is a result of the slightly different madelung potentials of both compounds . for modeling the raman response we assume @xmath16 for both compounds . this order of magnitude was previously obtained for cuprate ladders by inelastic neutron scattering@xcite , by infrared absoption@xcite , raman response@xcite and theoretical works deriving extended low - energy heisenberg models@xcite . in order to account for the fwhm and the two - triplon peak position we determine @xmath27 and global energy scales @xmath87 @xmath13 for sr123 and @xmath88 @xmath13 for sr14 . it was previously argued that in sr14 a charge order of the chain subsystem modulates the magnetic exchange in the ladders@xcite . this opens a gap in the raman response which has a large effect on the two - triplon peak for the parameters @xmath15 and @xmath76 which are appropriate for la6ca8 . however , the effect is small for larger @xmath77-values because the induced gap opens well above the two - triplon peak at @xmath89 @xmath13 . the set of parameters used above for sr123 and sr14 describes quantitatively well the raman response as shown in tab . [ tab_plane ] and figs . [ fig : comp](a - d ) . especially both polarizations for each material can be modeled using only one set of parameters @xmath8 , @xmath77 and @xmath81 . the smaller fwhm of sr123 and sr14 compared to la6ca8@xcite can be directly explained by their larger @xmath77-values ( see fig . [ fig : fwhm ] ) . the coupling constants of sr14 are in good agreement with those obtained by ir absorption measurements@xcite . additionally , our set of parameters yields a spin gap of @xmath90 @xmath13 for sr123 and @xmath91 @xmath13for sr14 using the underlying one - triplon dispersion . the latter value is consistent with the spin gap measured by inelastic neutron scattering@xcite . k ) and sr14 ( @xmath92 k ) with the theoretically obtained two - triplon contribution . the data of sr14 have been provided by gozar _ et al._@xcite . ( a ) the red ( black ) curve denotes the ( xx)-polarization ( @xmath37 ) of sr123 with a laser excitation energy @xmath931.95 ev . the orange ( grey ) curve displays the theoretical two - triplon contribution with @xmath27 , @xmath16 and @xmath94 @xmath13 . ( b ) the red ( black ) curve denotes the ( xx)-polarization ( @xmath39 ) of sr14 with a laser excitation energy @xmath68 ev . the orange ( grey ) curve displays the theoretical two - triplon contribution with @xmath27 , @xmath95 and @xmath88 @xmath13 . ( c ) ( yy)-polarization ( @xmath38 ) for sr123 . identical parameters as in ( a ) . the cyan ( grey ) curve displays the experimental data and the blue ( black ) curve the theoretical two - triplon contribution . ( d ) ( yy)-polarization ( @xmath40 ) for sr14 . identical parameters as in ( b ) and the same colors as in ( c).,width=309 ] in this section we calculate the raman response for the undoped two - dimensional cuprate compounds using a toy model consisting of two uncoupled two - leg ladders ( see fig . [ fig_sketch_plane ] ) . this is motivated by the fact that the building blocks of ladders and planes are edge - sharing cu@xmath18 plaquettes . we expect that the raman response is dominated by short - range and high - energy excitations yielding a certain similarity between ladders and planes . indeed , the positions of the two - magnon peak in the 2d cuprates and the two - triplon peak in the cuprate ladders are found at almost the same frequency @xmath96 @xmath13 , but the fwhm of the two - dimensional compounds is a factor of 2 - 6 larger . we have shown in the last section that the fwhm of the two - triplon peak in the cuprate ladder compounds strongly varies with @xmath77 . we therefore conjecture that the larger fwhm of the two - dimensional cuprates originates from the isotropic coupling @xmath28 . there will be of course deviations at small energies resulting from the differences between a gapped two - leg ladder and the gapless excitations in the two - dimensional compounds . clearly , a magnon description would be the proper starting point to treat the long - ranged ordered antiferromagnetic state . we think , however , that a triplon picture which includes the interactions on the quantitative level can give a good description of the raman response . a similar treatment in terms of gapped quasi - particles already led to an improved agreement between theory and experiments@xcite . -direction and the other in @xmath77-direction . we approximate the two - dimensional square lattice by the sum of these two uncoupled orthogonal ladders . ] in the following we will show how to deduce the a@xmath97 and b@xmath98 raman spectra of the square lattice from those of the two - leg ladder . clearly , one should use @xmath28 because the square lattice is isotropic ( @xmath99 ) . starting from the fleury - loudon operator the observables @xmath100 ( @xmath101 ) for magnetic light scattering in b@xmath98 ( a@xmath98 ) polarization read in leading order for the two - dimensional square lattice@xcite [ observables_2d ] @xmath102 here @xmath103 ( @xmath104 ) denotes a summation over nearest - neighbors in @xmath77-direction ( @xmath105-direction ) . the parameters @xmath106 and @xmath107 depend on the underlying microscopic model and are in general not equal@xcite . we approximate the two - dimensional square lattice by a sum of two uncoupled two - leg ladders , one oriented in @xmath77-direction , the other in @xmath105-direction . the situation is sketched in fig . [ fig_sketch_plane ] . the summation over both ladder orientations will restore the square lattice symmetries . comparing eq . ( [ observables_2d ] ) with eq.([observables ] ) one readily deduces the following relations [ observables_rel ] @xmath108 between the relevant observables in the two - leg ladder and the two - dimensional square lattice . note that for @xmath109 , the raman response in the a@xmath98 polarization vanishes due to the property @xmath110@xcite . the latter point is consistent with earlier treatments of the two - dimensional raman response . but for a finite strength of the four - spin interactions @xmath81 , also the a@xmath98 polarization is finite@xcite . . ( a ) b@xmath98-polarization for @xmath79 ( cyan / grey ) , @xmath111 ( dashed ) and @xmath16 ( blue / black ) . ( b ) a@xmath98-polarization for @xmath79 ( orange / grey ) , @xmath111 ( dashed ) and @xmath16 ( red / black ) . note the different scales for the raman response in @xmath112 and @xmath113 spectra.,width=309 ] the theoretical two - triplon contribution of the b@xmath98 ( panel a ) and the a@xmath114 ( panel b ) polarization is shown in fig . [ fig:2d_a1g_b1 g ] . the parameters used are an isotropic coupling @xmath28 and a strength of the four - spin interactions @xmath79 , @xmath111 , and @xmath16 . the b@xmath98 polarization displays a symmetric two - triplon peak which is dominating the raman response . the four - spin interactions shift the whole spectrum to lower energies and decrease the total intensity . the fwhm of the two - triplon peak is approximately given by the average width of the two - triplon peaks of an isolated two - leg ladder in rung and in leg polarization . thus , the width is nearly independent of the value of @xmath81 . the a@xmath98 polarization is almost zero for vanishing @xmath81 . this again reflects the accurate extrapolation of the matrix elements . for non - zero @xmath81 , a finite a@xmath97 contribution is realized . the differences in the line - shape between a@xmath98 and b@xmath98 are a pure effect of different matrix elements . compared to the two - triplon peak in the b@xmath98 polarization the a@xmath98 polarization displays a two peak structure where the second peak is sharper and at higher energies . [ cols="^,^,^,^,^,^,^,^,^,^ " , ] in the following we will compare the theoretical two - triplon contribution to the raman response with low temperature experimental data on r@xmath0cuo@xmath18 ( @xmath115 ev)@xcite , sr@xmath0cuo@xmath0cl@xmath0 ( @xmath116 ev)@xcite , and yba@xmath0cu@xmath1o@xmath19 ( @xmath115 ev)@xcite taken from the literature . as discussed in section [ cl ] the laser energy used for the experiment is a crucial issue . analogous to the ladders one should use spectra of cuprate planes measured with laser energies below the charge gap for comparing to the purely magnetic theoretical response . but it turns out that the optical conductivity is rather low ( @xmath117 @xmath70@xmath13 ) below the charge gap @xmath118 which results in a vanishing intensity of the two - magnon peak @xcite . an analogous choice of the laser energies below the charge gap as discussed for the cuprate ladders is not possible . therefore , we used data measured with laser energies @xmath119 ev@xmath120 . at the energy @xmath119 ev the optical conductivity is quite smooth and @xmath121 ev@xcite . simultaneously @xmath119 ev coincides with the triple resonance at @xmath122 . the triple resonance theory predicts two peaks in the raman response at about @xmath123 and @xmath124 . the relative intensity of both peaks depends on the laser energy . the second peak is strongly suppressed at @xmath125 . in that sense this laser energy can be assigned to be closest to the non - resonant regime @xcite . in fig . [ fig:2d_comp ] experimental data and theoretical contributions using @xmath28 and @xmath16 are shown for both polarizations . frequencies are measured in units of @xmath26 . we first discuss the b@xmath98 polarization in fig . [ fig:2d_comp](a ) . we have chosen the global energy scale @xmath26 for all experimental curves such that the positions of the experimental two - magnon and the theoretical two - triplon peaks match . this yields quantitatively reasonable values for these compounds . in addition , we find quantitative agreement between the experimental fwhm and the theoretical fwhm of the two - triplon peak . the values of @xmath26 and the fwhm are listed in tab . [ tab_plane ] for all compounds . note that the fwhm for @xmath28 is larger than for the anisotropic case @xmath126 as discussed for the ladder compounds . clearly , there are also deviations between theory and experiment . as expected , the low - energy spectral weight in the theoretical line - shape is larger compared to the experimental curves . this is definitely a consequence of approximating the two - dimensional square lattice with quasi one - dimensional models . there is also spectral weight missing at higher energies above the two - triplon peak . possible explanations will be described below . the results for the a@xmath98 polarization ( shown in fig . [ fig:2d_comp](b ) ) are explained next . we used the _ same _ global energy scales @xmath26 for the experimental curves as determined from the b@xmath98 polarization above . in order to reproduce the maximum intensity of the experiment , we multiplied the theoretical curve from fig . [ fig:2d_a1g_b1g](b ) by a factor 5 . this implies that the microscopic parameters @xmath127 and @xmath107 are anisotropic . a possible reason for this anisotropy could be the restriction to the fleury - loudon observable . an extension of this observable to higher orders in @xmath128 ( four - spin and next - nearest neighbor two - spin terms ) gives additional contributions to @xmath129 and @xmath101@xcite . the relevance of these contributions has not been analyzed . in the experiment a broad hump is measured . we find it very promising that the theoretical contribution displays the dominant spectral weight just for these energies . however , the line - shape can not be resolved completely because the dip in the theoretical curve is not observed in the experiment . it originates from neglecting the finite life - time effects which are already present in the description of the isolated two - leg ladder@xcite being the building block of our square - lattice toy model . we conclude that at least a part of the experimental a@xmath98 polarization originates from the finite four - spin interactions . for @xmath109 , there is no purely magnetic contribution to the raman response for this polarization . a finite a@xmath98 raman response can be regarded as an evidence for the presence of sizable four - spin interactions . this follows entirely from symmetry arguments and holds true for the full two - dimensional model . at higher energies , spectral weight is missing in the theoretical contribution in an analogous fashion as in the b@xmath98 polarization . cuo@xmath18 ( @xmath115 ev , @xmath130 k ) and sr@xmath0cuo@xmath0cl@xmath0 ( @xmath116 ev , @xmath92 k ) . the raman data of nd@xmath0cuo@xmath18 and sr@xmath0cuo@xmath0cl@xmath0 are reproduced from refs . and . the experimental curves are smoothed and their zero position is shifted horizontally as indicated by the black horizontal lines . ( a ) b@xmath98-polarization : the blue ( black ) curve denotes the two - triplon contribution with @xmath28 and @xmath16 . the global energy scale @xmath26 is chosen such that experimental two - magnon and the theoretical two - triplon peak merge . this yields @xmath131 @xmath13 for nd@xmath0cuo@xmath18 ( cyan / grey ) and sr@xmath0cuo@xmath0cl@xmath0 ( dashed ) . ( b ) a@xmath98-polarization ( theory : red / black ; experiment : organge / grey ) : same notations as in ( a ) . note that the _ same _ magnetic exchange couplings @xmath26 are used . the a@xmath98-cut is multiplied by a factor 5 in comparison to the curve in fig . [ fig:2d_a1g_b1g].,width=309 ] as shown in sect . [ cl ] and in sect . [ cp ] the cut can not account for the missing high - energy spectral weight when comparing to the raman experiments . also other theories proposed previously like calculations based on spin - waves@xcite , paramagnons@xcite , jordan - wigner fermions@xcite and numerical studies@xcite were faced with the same problem . extended theories including _ ( i ) multi - particle contributions _ , _ ( ii ) spin - phonon coupling _ , _ ( iii ) two - magnon / triplon plus phonon absorption _ and _ ( iv ) triple resonance _ are necessary in order to describe the high - energy spectral weight . most of the publications deal with the two - dimensional compounds . here we will try to review these ideas and reexamine them in the light of our new results . especially the quantitative results for the cuprate ladders can give new insights in this discussion . _ ( i ) multi - particle contributions _ one open problem is the role of multi - particle contributions to the raman response , i.e. the four - magnon contribution in the case of the square lattice and the four - triplon contribution in the case of the two - leg ladder . at this stage no quantitative calculations are available . but it is known that the multi - triplon spectral weights are sizable for the two - leg ladder@xcite . the main effect of the four - spin interaction on the high - energy spectral weight is a small shift from the two - triplon to the multi - triplon channels@xcite . but this shift is not sufficient to account for the high - energy spectral weight as observed in experiments . this was also found in treatments for the two - dimensional square lattice@xcite . however , the complete magnetic infrared absorption spectrum ( including the high - energy part ) of la6ca8 can be described quantitatively by including multi - particle contributions@xcite . here @xmath81 does not play the dominant role for the high - energy spectral weight@xcite . it is therefore plausible that these contributions give a noticeable effect also on the high - energy raman response . there are also indications that the spectral weight can not be fully explained in this way . for example , the four - magnon spectral weight was shown to be negligible for the 2d square lattice@xcite . but the magnon - magnon interaction which was not treated in this calculation could enhance the high - energy spectral weight . also quantum monte carlo calculations which include all magnon contributions for the two - dimensional heisenberg model seem to explain only a part of the high - energy spectral weight@xcite . but finite size effects and inaccuracies of the analytical continuation can lead to uncertainties in determining the high - energy spectral weight . _ ( ii ) spin - phonon coupling _ the latter observations suggest that additional degrees of freedom are important . it was argued by several authors that the coupling to phonons produces a large amount of spectral weight above the two - triplon peak@xcite . in one approach the spin - phonon coupling modulates the magnetic exchange couplings with a gaussian distribution . another approach introduces a finite spin wave damping induced by the spin - phonon coupling . both scenarios produce a significantly broadened and asymmetric two - magnon peak as observed in experiments@xcite . nevertheless , the consistency of a spin - phonon coupling as suggested above with experiments is not clear . the magnitude of this coupling has to be unrealistically large in order to describe infrared absorption data@xcite . additionally , it was pointed out by freitas and singh@xcite that the temperature - dependent correlation length and the spin dynamics which agree well with purely magnetic models does not leave room for such a coupling@xcite . there are no investigations of the role of spin - phonon couplings for the case of the cuprate ladder systems . but the fwhm of the two - triplon peak can be quantitatively understood within a purely magnetic model as shown in sect . thus , we conclude that the spin - phonon coupling is not strong in the case of the cuprate ladder compounds . such a coupling leads to a broad two - triplon peak in the same way as for the two - dimensional case . this is a contradiction when considering the raman response and the infrared absorption of cuprate ladders simultaneously : on the one hand one needs a larger anisotropy between leg and rung coupling ( larger @xmath77 ) in order to sharpen the two - triplon peak in the raman response again ( see fig . [ fig : fwhm ] ) but on the other hand one can not explain the infrared absorption with an substantially increased @xmath77@xcite . a strong spin - phonon coupling is therefore in contradiction with the results obtained for cuprate ladders . this can be also seen as an indication that the same holds true for the two - dimensional compounds@xcite . _ ( iii ) two - magnon / triplon plus phonon absorption _ a third alternative explaining the high - energy spectral weight uses phonons as possible momentum sinks . here a strong spin - phonon coupling is not necessary . the idea is based on the work of lorenzana and sawatzky for infrared absorption@xcite . it is well accepted in the case of infrared absorption measurements on cuprate ladders@xcite and planes@xcite that the dominant processes are magnetic excitations which are assisted by phonons . it was realized by freitas and singh that similar processes could be important also for the raman response in cuprate planes@xcite . in an analogous fashion a two - triplon plus ( raman active ) phonon process for the raman response in cuprate ladders could be important . it can be used to transfer spectral weight above the two - triplon peak leading to an asymmetric line - shape . it is a difficult task to determine the relative strength of this process compared to the usual two - triplon scattering . _ ( iv ) triple resonance _ additionally , the triple resonance was proposed to account for the high - energy spectral weight in the two - dimensional compounds@xcite . as already stated in sec . [ cp ] , the experimental spectra of the planes are taken in the resonant regime . it is known that the triple resonance scenario yields an additional peak above the two - magnon peak . its intensity depends significantly on the energy of the incident light in accordance with experiments@xcite . in principle , the same effect is also present in ladder compounds . but for the laser energy @xmath132=1.92 ev@xmath133 considered for sr123 and sr14 the triple resonance condition is not fulfilled . due to the simplified model used for the 2d system no conclusion about the high - energy weight can be drawn from our results . because a large spin - phonon coupling and the triple resonance can be ruled out for the cuprate ladder systems , the observed high - energy spectral weight in the cuprate ladder compounds has to be explained most probably by the multi - triplon or two - triplon plus phonon contributions . the first part of this work deals with the theoretical understanding of non - resonant magnetic raman scattering on cuprate two - leg ladder compounds , namely sr123 , sr14 , and la6ca8 . therefore we applied a triplon - conserving cut on a microscopic spin - model which includes heisenberg couplings and additional four - spin interactions . we studied the two - triplon contribution to the non - resonant magnetic raman response . the dominating feature of the two - triplon contribution is the two - triplon peak which has a characteristic fwhm depending on the model parameters @xmath77 and @xmath81 . we carefully chose the experimental data closest to the non - resonant regime and compared them with our theory . the key observation we found is that the sharpness of the two - triplon peak in sr123 and sr14 in comparison to la6ca8 can be explained by the stronger anisotropy of the magnetic exchange along the rungs and legs of the ladder . indeed , the two - triplon peak width depends strongly on the parameter @xmath77 . both materials can be modeled with the parameters @xmath134 and @xmath135 but different global energy scales @xmath136 @xmath13 for sr123 and @xmath137 @xmath13 for sr14 . the parameters for sr14 are in good agreement with infrared absorption @xcite and inelastic neutron scattering@xcite experiments . we conclude that the dominating two - triplon peak of the magnetic raman response in cuprate ladders can be consistently explained within the microscopic model . the presence of a four - spin interaction of the order of @xmath138 can be viewed as a settled issue . in the second part of this article we used the results found for the two - leg ladder to describe the magnetic raman response of the undoped two - dimensional cuprate compounds in b@xmath98 and a@xmath98 polarization . the contribution to the a@xmath98 polarization is only allowed for finite four - spin interactions due to symmetry reasons . we use an isotropic coupling @xmath28 and @xmath16 for the comparison with the experimental data . convincingly , we find quantitative agreement for the two - triplon peak position _ and _ the two - triplon peak width for several compounds . additionally , a sizable spectral weight is found in the a@xmath98 polarization consistent with experiments . we conclude that the processes dominating the magnetic raman response are short - ranged . the last part deals with the missing high - energy spectral weight above the dominating two - triplon peak for the case of cuprate ladders and planes . we review possible sources of this spectral weight like multi - particle contributions , the role of spin - phonon coupling , a two - triplon plus phonon process and the triple resonance to the magnetic raman response . we deduced from our results that the high - energy spectral weight can not be explained with realistic values for the spin - phonon coupling . in summary , our calculations lead to an unified understanding of the magnetic raman response in cuprate ladder compounds within a purely magnetic model . a strong spin - phonon coupling can be excluded for these materials . additionally , we obtained a convincing quantitative description of the dominating two - magnon peak in the raman response of cuprate planes using a toy model consisting of two uncoupled two - leg ladders . this suggests that the short - ranged triplon excitations might be an alternative starting point for the description of the two - dimensional cuprate compounds . we gratefully acknowledge g.s . uhrig , m. grninger , a. gozar , a. reischl , and s. jandl for very helpful discussions . we thank a. gozar for the kind provision of experimental data on sr14 . this work is supported by the dfg in sfb 608 . j. orenstein and a.j . millis , science * 288 * , 468 ( 2000 ) . s. sachdev , science * 288 * , 475 ( 2000 ) . anderson , science * 288 * , 480 ( 2000 ) . p. knoll , c. thomsen , m. cardona , and p. murugaraj , phys . b * 42 * , r4842 ( 1990 ) . g. blumberg , p. abbamonte , m.v . klein , w.c . lee , , d.m . ginsberg , l.l . miller , a. zibold , phys . b * 53 * , r11930 ( 1996 ) . s. sugai , m. sato , t. kobayashi , j. akimitsu , t. ito , h. takagi , s. uchida , s. hosoya , t. kajitani , and t. fukuda , phys . b * 42 * , r1045 ( 1990 ) . p.e . sulewski , p.a . fleury , k.b . lyons , s - w . cheong , and z. fisk , phys . b * 41 * , 225 ( 1990 ) . sulewski , p.a . fleury , k.b . lyons , and s - w . cheong , phys . rev . lett . * 67 * , 3864 ( 1991 ) . m. yoshida , s. tajima , n. koshizuka , s. tanaka , s. uchida , and t. itoh , phys . b * 46 * , 6505 ( 1992 ) . e. manousakis , rev . mod . phys . * 63 * , 1 ( 1991 ) . kastner , r.j . birgeneau , g. shirane , and y. endoh , rev . * 70 * , 897 ( 1998 ) . clarke , a. harrison , t.e . mason , and d. visser , solid state commun . * 112 * , 561 ( 1999 ) . hayden , g. aeppli , r. osborn , a. d. taylor , t. g. perring , s .- w . cheong , and z. fisk , phys . lett . * 67 * , 3622 ( 1991 ) . r. coldea , s.m . hayden , g. aeppli , t. g. perring , c.d . frost , t.e . mason , s .- w . cheong , and z. fisk , phys . * 86 * , 5377 ( 2001 ) . perkins , j.m . graybeal , m.a . kastner , r.j . birgeneau , j.p . falck , and m. greven phys . lett * 71 * , 1621 ( 1993 ) . perkins , r.j . birgeneau , j.m . graybeal , m.a . kastner and d.s . kleinberg , phys . b * 58 * , 9390 ( 1998 ) . m. grninger , d. van der marel , a. damascelli , a. erb , t. nunner , and t. kopp , phys . b * 62 * , 12422 ( 2000 ) . j. lorenzana , j. eroles , and s. sorella , phys . lett . * 83 * , 5122 ( 1999 ) . a. gling , u. kuhlmann , c. thomsen , a. lffert , c. gross and w. assmus , phys . b * 67 * , 052403 ( 2003 ) . a. gozar , g. blumberg , b.s . dennis , b.s . shastry , n. motoyama , h. eisaki , and s. uchida , phys . lett . * 87 * , 197202 ( 2001 ) . e. dagotto and t.m . rice , science * 271 * , 618 ( 1996 ) . z.v . popovi , m.j . konstantinovi , v.a . ivanov , o.p . khuong , r. gaji , a. vietkin , and v.v . moshchalkov , phys . b * 62 * , 4963 ( 2000 ) . s. sugai and m. suzuki , phys . stat . sol . ( b ) * 215 * , 653 ( 1999 ) . schmidt and g.s . uhrig , phys . lett . * 90 * , 227204 ( 2003 ) . c. knetter , k.p . schmidt , m. grninger , and g.s . uhrig , phys . lett . * 87 * , 167204 ( 2001 ) . schmidt , c. knetter , and g.s . uhrig , europhys . lett . * 56 * , 877 ( 2001 ) . nunner , p. brune , t. kopp , m. windt , and m. grninger , phys . b * 66 * , 180404(r ) ( 2002 ) . m. grueninger , m. windt , e. benckiser , t.s.nunner , k.p . schmidt , g.s . uhrig und t. kopp , adv.sol.stat.phys . * 43 * , 95 ( 2003 ) . w. zheng , c.j . hamer , and r.r.p . singh , phys . lett . * 91 * , 037206 ( 2003 ) . schmidt , c. knetter and g.s . uhrig , phys . b * 69 * , 104417 ( 2004 ) . s. trebst , h. monien , c.j . hamer , z. weihong , and r.r.p . singh , phys . lett . * 85 * , 4373 ( 2000 ) . schmidt , c. knetter , m. grninger , and g.s . uhrig , phys . rev . lett . * 90 * , 167201 ( 2003 ) . s. brehmer , h .- j . mikeska , m. mller , n. nagaosa , and s. uchida , phys . b * 60 * , 329 ( 1999 ) . e. mller - hartmann and a. reischl , eur . j. b * 28 * , 173 ( 2002 ) . y. mizuno , t. tohyama and s. maekawa , j. phys . jpn * 66 * , 397 ( 1997 ) . schmidt and y. kuramoto , physica c * 167 * , 263 ( 1990 ) . m. takahashi , j. phys . state phys . * 10 * , 1289 ( 1977 ) . a. h. macdonald , s. m. girvin , d. yoshioka , phys . b * 41 * , 2565 ( 1990 ) . m. roger and j.m . delrieu , phys . rev . b * 39 * , 2299 ( 1989 ) . a. reischl , e. mller - hartmann , g.s . uhrig , phys . b * 70 * , 245124 ( 2004 ) singh , p.a . fleury , k.b . lyons , and p.e . sulewski , phys . lett . * 62 * , 2736 ( 1989 ) . canali and s.m . girvin , phys . b * 45 * , 7127 ( 1992 ) . y. honda , y. kuramoto , and t. watanabe , phys . b * 47 * , 11329 ( 1993 ) . sandvik , s. capponi , d. poilblanc , and e. dagotto , phys . b * 57 * , 8478 ( 1998 ) . katanin , and a.p . kampf , phys . b * 67 * , 100404(r ) ( 2003 ) . j. erole , c.d . batista , s.b . bacci , and e.r . gagliano , phys . b * 59 * , 1468 ( 1999 ) . m.j . reilly and a.g . rojo , phys . b * 53 * , 6429 ( 1996 ) . saenger , phys . b * 52 * , 1025 ( 1995 ) . w.h . weber and g.w . ford , phys . b * 40 * , 6890 ( 1989 ) . s. haas , e. dagotto , j. riera , r. merlin , and f. nori , j. appl . * 75 * , 6340 ( 1994 ) . f. nori , r. merlin , s. haas , a.w . sandvik , and e. dagotto , phys . lett . * 75 * , 553 ( 1995 ) . chubukov , and d.m . frenkel , phys . . lett . * 74 * , 3057 ( 1995 ) . f. schnfeld , a.p . kampf , and e. mller - hartmann , z. phys . b * 102 * , 25 ( 1997 ) . d.k . morr and a.v . chubukov , phys . b * 56 * , 9134 ( 1997 ) . c. jurecka , v. grtzun , a. friedrich , and w. brenig , eur . j. b * 21 * , 469 ( 2001 ) . y. natsuma , y. watabe and t. suzuki , phys . japan * 67 * , 3314 ( 1998 ) . m. matsuda , k. katsumata , r.s . eccleston , s. brehmer , and h.j . mikeska , j. appl . 87 * , 6271 ( 2000 ) . m. matsuda , k. katsumata , r.s . eccleston , s. brehmer , and h .- j . mikeska , phys . b * 62 * , 8903 ( 2000 ) . sr14 has intrinsically six holes per unit cell , but it is believed that the majority of the holes are located in the chain subsystem ( see ref . ) . n. ncker , m. merz , c. a. kuntscher , s. gerhold , s. schuppler , r. neudert , m.s . golden , j. fink , d. schild , s. stadler , v. chakarian , j. freeland , y.u . idzerda , k. conder , m. uehara , t. nagata , j. goto , j. akimitsu , n. motoyama , h. eisaki , s. uchida , u. ammerahl , and a. revcolevschi , phys . b * 62 * , 14384 ( 2000 ) . neglecting the two - spin exchange terms along the diagonals , our notation @xmath139 is given in terms of the other notation @xmath140 using cyclic permutations by @xmath141 , @xmath142 , and @xmath143 ( see ref . ) . p.a . fleury and r. loudon , * 166 * , 514 ( 1968 ) . shastry and b.i . shraiman , phys . lett . * 65 * , 1068 ( 1990 ) . c. knetter and g.s . uhrig , eur . j. b * 13 * , 209 ( 2000 ) . c. knetter , k.p . schmidt , and g.s . uhrig , j. phys . : condens . matter * 36 * , 7889 ( 2003 ) . c. knetter , k.p . schmidt , and g.s . uhrig , eur . j. b * 36 * , 525 ( 2004 ) . schmidt , c. knetter , and g.s . uhrig , acta physica polonica b * 34 * , 1481 ( 2003 ) . schmidt , h. monien , and g.s . uhrig , phys . b * 67 * , 184413 ( 2003 ) . c. domb and j.l lebowitz , _ phase transitions and critical phenomena _ , academic press , new york , vol . 13 ( 1989 ) . schmidt , phd thesis , university of cologne ( 2004 ) . a. lffert , c. gross , and w. assmus , j. cryst . growth 237 , 796 ( 2002 ) . choi , y.s . lee , t.w . noh , e.j . choi , y. bang , and y.j . kim phys . b * 60 * , 4646 ( 1999 ) . the extrapolation of the matrix elements are relatively accurate . the maximum deviation between the two - triplon widths in both polarizations is @xmath144 for @xmath145 . the one - triplon hopping elements yield a quantitative one - triplon dispersion when compared to dmrg data ( see ref . ) . so we conclude that the largest extrapolation error is due to the extrapolation of the two - triplon interaction amplitudes . to get an estimate of this error we compared the energy of the @xmath30 two - triplon bound state at finite momentum for @xmath109 obtained by dmrg with the results of the cut . then we changed the two - triplon interaction for the raman case by the maximum deviation of the bound state energies and calculated the effect on the fwhm of the two - triplon peak . in the end we added @xmath144 to account for the uncertainty of the matrix elements . the resulting uncertainties are shown as error bars in fig . [ fig : fwhm ] . calzado , c. de graaf , e. bordas , r. caballol , and j - p . malrieu , phys . b * 67 * , 132409 ( 2003 ) . m. windt , phd thesis , university of cologne ( 2003 ) . eccleston , m. uehara , j. akimitsu , h. eisaki , n. motoyama , and s .- uchida , phys . lett . * 81 * , 1702 ( 1998 ) . d. salamon , ran liu , m.v . klein , m.a . karlow , s.l . cooper , s - w . cheong , w.c . lee and d.m . ginsberg , phys . b * 51 * , 6617 ( 1995 ) hsu , phys . b * 41 * , 11379 ( 1990 ) . wang , phys . * , r13774 ( 1991 ) . m. windt , m. grninger , t. nunner , c. knetter , k.p . schmidt , g.s . uhrig , t. kopp , a. freimuth , u. ammerahl , b. bchner , and a. revcolevschi , phys . lett . * 87 * , 127002 ( 2001 ) . freitas and r.r.p . singh , phys . b * 62 * , 5525 ( 2000 ) . y. endoh , k. yamada , r.j . birgeneau , d.r . gabbe , h.p . jenssen , m.a . kastner , c.j . peters , p.j . picone , t.r . thurston , j.m . tranquada , g. shirane , y. hidaka , m. oda , y. enomoto , m. suzuki , and t. murakami , phys . b * 37 * , 7443 ( 1988 ) . s. chakravarty , b.i . halperin , and d.r . nelson , phys . b * 39 * , 2344 ( 1989 ) . and b. dabrowski , phys . rev . b * 49 * , r13295 ( 1994 ) . j. lorenzana and g.a . sawatzky , phys . lett . * 74 * , 1867 ( 1995 ) . j. lorenzana and g.a . sawatzky , phys . b * 52 * , 9576 ( 1995 ) . | an unified picture for the raman response of magnetic excitations in cuprate spin - ladder compounds is obtained by comparing calculated two - triplon raman line - shapes with those of the prototypical compounds srcu@xmath0o@xmath1 ( sr123 ) , sr@xmath2cu@xmath3o@xmath4 ( sr14 ) , and la@xmath5ca@xmath6cu@xmath3o@xmath4 ( la6ca8 ) .
the theoretical model for the two - leg ladder contains heisenberg exchange couplings @xmath7 and @xmath8 plus an additional four - spin interaction @xmath9 . within this model sr123 and sr14
can be described by @xmath10 , @xmath11 , @xmath12 @xmath13 and @xmath14 @xmath13 .
the couplings found for la6ca8 are @xmath15 , @xmath16 , and @xmath17 @xmath13 .
the unexpected sharp two - triplon peak in the ladder materials compared to the undoped two - dimensional cuprates can be traced back to the anisotropy of the magnetic exchange in rung and leg direction . with the results obtained for the isotropic ladder we calculate the raman line - shape of a two - dimensional square lattice using a toy model consisting of a vertical and a horizontal ladder .
a direct comparison of these results with raman experiments for the two - dimensional cuprates r@xmath0cuo@xmath18 ( r = la , nd ) , sr@xmath0cuo@xmath0cl@xmath0 , and yba@xmath0cu@xmath1o@xmath19 yields a good agreement for the dominating two - triplon peak .
we conclude that short range quantum fluctuations are dominating the magnetic raman response in both , ladders and planes .
we discuss possible scenarios responsible for the high - energy spectral weight of the raman line - shape , i.e. phonons , the triple - resonance and multi - particle contributions . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
the calculus on time scales is a recent field , introduced by bernd aulbach and stefan hilger in 1988 @xcite , that unifies and extends difference and differential equations into a single theory @xcite . a time scale is a model of time , and the new theory has found important applications in several fields that require simultaneous modeling of discrete and continuous data , in particular in the calculus of variations @xcite , control theory @xcite , and optimal control @xcite . other areas of application include engineering , biology , economics , finance , and physics @xcite . the present work is dedicated to the study of problems of calculus of variations on an arbitrary time scale @xmath0 . as particular cases , one gets the standard calculus of variations @xcite by choosing @xmath1 ; the discrete - time calculus of variations @xcite by choosing @xmath2 ; and the @xmath3-calculus of variations @xcite by choosing @xmath4 , @xmath5 . in section [ sec : prel : ts ] we briefly present the necessary notions and results of time scales , delta derivatives , and delta integrals . let @xmath0 be a given time scale with at least three points , @xmath6 , and @xmath7 be of class @xmath8 . suppose that @xmath9 and @xmath10 . we consider the following optimization problem on @xmath0 : @xmath11=\int_a^b l(t , q^\sigma(t),q^\delta(t ) ) \delta t \longrightarrow \min_{q\in \mathcal{d}},\ ] ] where @xmath12\cap \mathbb{t } \rightarrow \mathbb{r}^n,\ q\in \mathrm{c}^1_{rd},\ q(a)=q_a,\ q(b)=q_b\}\ ] ] for some @xmath13 , and where @xmath14 is the forward jump operator and @xmath15 is the delta - derivative of @xmath3 with respect to @xmath0 . for @xmath1 we get the classical fundamental problem of the calculus of variations , which concerns the minimization of an integral @xmath16=\int_a^b l(t , q(t),\dot{q}(t ) ) dt\ ] ] over all trajectories @xmath17 satisfying given boundary conditions @xmath18 and @xmath19 . several classical results on the calculus of variations are now available to the more general context of time scales : ( first ) euler - lagrange equations @xcite ; necessary optimality conditions for isoperimetric problems @xcite and for problems with higher - order derivatives @xcite ; the weierstrass necessary condition @xcite ; and noether s symmetry theorem @xcite . in this paper we prove a new result for the problem of the calculus of variations on time scales : we obtain in section [ sec : mr ] a time scale version of the classical _ second euler - lagrange equation _ @xcite , also known in the literature as the dubois - reymond necessary optimality condition @xcite . the classical second euler - lagrange equation asserts that if @xmath3 is a minimizer of , then @xmath20 \\ = - \partial_1 l ( t , q(t ) , \dot{q}(t))\ , , \end{gathered}\ ] ] where @xmath21 , @xmath22 , denotes the partial derivative of @xmath23 with respect to its @xmath24-th argument . in the autonomous case , when the lagrangian @xmath25 does not depend on the time variable @xmath26 , the second euler - lagrange condition is nothing more than the second erdmann necessary optimality condition : @xmath27 along all the extremals of the problem , which in mechanics corresponds to the most famous conservation law conservation of energy . for a survey of the classical optimality conditions we refer the reader to ( * ch . 2 ) . here we just recall that is one of the cornerstone results of the calculus of variations and optimal control @xcite : it has been used , for example , to prove existence , regularity of minimizers , conservation laws , and to explain the lavrentiev phenomena . main result of the paper gives an extension of to an arbitrary time scale ( theorem [ secondel ] ) : if @xmath3 is a solution of problem , then @xmath28 = - \partial_1 l(t , q^\sigma(t),q^\delta(t ) ) \ , .\end{gathered}\ ] ] as an application , we show in section [ sec : appl : nt ] how one can use the new second euler - lagrange equation to prove the noether s principle on time scales : the invariance of functional with respect to a one - parameter family of transformations implies the existence of a conserved quantity along the time scale euler - lagrange extremals ( theorem [ noether ] ) . when problem is autonomous one has invariance with respect to time translations and the corresponding noether s conservation law gives an extension of the second erdmann equation to time scales : @xmath29 in classical mechanics gives conservation of energy . the conservation law tells us that an analogous result remains valid on an arbitrary time scale . however , the role of the classical hamiltonian @xmath30 in is substituted by a time - scale hamiltonian @xmath31 in , , @xmath32 , with the new term @xmath33 on the left - hand side of depending on the graininess of the time scale . for a general introduction to the calculus on time scales we refer the reader to the book @xcite . here we only give those notions and results needed in the sequel . as usual , @xmath34 , @xmath35 , and @xmath36 denote , respectively , the set of real , integer , and natural numbers . a _ time scale _ @xmath0 is an arbitrary nonempty closed subset of @xmath37 . besides standard cases of @xmath34 ( continuous time ) and @xmath35 ( discrete time ) , many different models of time are used . for each time scale @xmath0 the following operators are used : * the _ forward jump operator _ @xmath38 , @xmath39 for @xmath40 and @xmath41 if @xmath42 ; * the _ backward jump operator _ @xmath43 , @xmath44 for @xmath45 and @xmath46 if @xmath47 ; * the _ forward graininess function _ @xmath48 , @xmath49 . if @xmath50 , then for any @xmath51 , @xmath52 and @xmath53 . if @xmath54 , then for every @xmath55 , @xmath56 , @xmath57 and @xmath58 . a point @xmath59 is called _ right - dense _ , _ right - scattered _ , _ left - dense _ or _ left - scattered _ if @xmath60 , @xmath61 , @xmath62 , or @xmath63 , respectively . we say that @xmath26 is _ isolated _ if @xmath64 , that @xmath26 is _ dense _ if @xmath65 . if @xmath66 is finite and left - scattered , we define @xmath67 otherwise , @xmath68 . let @xmath69 and @xmath70 . the _ delta derivative _ of @xmath71 at @xmath26 is the real number @xmath72 with the property that given any @xmath73 there is a neighborhood @xmath74 of @xmath26 such that @xmath75 for all @xmath76 . we say that @xmath71 is _ delta differentiable _ on @xmath77 provided @xmath72 exists for all @xmath70 . we shall often denote @xmath78 by @xmath79 if @xmath71 is a composition of other functions . the delta derivative of a function @xmath80 ( @xmath6 ) is a vector whose components are delta derivatives of the components of @xmath71 . for @xmath81 , where @xmath82 is an arbitrary set , we define @xmath83 . for delta differentiable @xmath71 and @xmath84 , the next formulas hold : @xmath85 if @xmath86 , then @xmath87 is delta differentiable at @xmath88 if and only if @xmath71 is differentiable in the ordinary sense at @xmath26 . then , @xmath89 . if @xmath54 , then @xmath90 is always delta differentiable at every @xmath55 with @xmath91 . let @xmath92 , @xmath10 . we define the interval @xmath93_{\mathbb{t}}$ ] in @xmath0 by @xmath94_{\mathbb{t}}:=\ { t \in \mathbb{t } : a\leq t\leq b\}.\ ] ] open intervals and half - open intervals in @xmath0 are defined accordingly . [ monotonia delta ] let @xmath95_{\mathbb{t}}\rightarrow\mathbb{r}$ ] be a continuous function that has a delta derivative at each point of @xmath93_{\mathbb{t}}^\kappa$ ] . then @xmath71 is increasing , decreasing , non - decreasing , and non - increasing on @xmath93_{\mathbb{t}}$ ] if @xmath96 , @xmath97 , @xmath98 and @xmath99 for all @xmath100_{\mathbb{t}}^\kappa$ ] , respectively . a function @xmath101 is called a _ delta antiderivative _ of @xmath102 provided @xmath103 in this case we define the _ delta integral _ of @xmath71 from @xmath104 to @xmath105 ( @xmath92 ) by @xmath106 if @xmath50 , then @xmath107 where the integral on the right hand side is the usual riemann integral . if @xmath108 , where @xmath109 , then @xmath110 for @xmath10 . in order to present a class of functions that possess a delta antiderivative , the following definition is introduced : a function @xmath111 is called _ rd - continuous _ if it is continuous at the right - dense points in @xmath0 and its left - sided limits exist ( finite ) at all left - dense points in @xmath0 . a function @xmath112 is _ rd - continuous _ if all its components are rd - continuous . we remark that a rd - continuous function defined on a compact interval , with real values , is bounded . the set of all rd - continuous functions @xmath112 is denoted by @xmath113 , or simply by @xmath114 . similarly , @xmath115 and @xmath116 will denote the set of functions from @xmath114 whose delta derivative belongs to @xmath114 . [ antiderivada ] every rd - continuous function has a delta antiderivative . in particular , if @xmath117 , then the function @xmath118 defined by @xmath119 is a delta antiderivative of @xmath71 . the following results will be very useful in the proof of our main result ( theorem [ secondel ] ) . [ theorems ] assume that @xmath120 is strictly increasing and @xmath121 is a time scale . 1 . ( chain rule ) let @xmath122 . if @xmath123 and @xmath124 exist for all @xmath125 , then @xmath126 2 . ( derivative of the inverse ) the relation @xmath127 holds at points @xmath128 where @xmath129 3 . ( substitution in the integral ) if @xmath130 is a @xmath114 function and @xmath131 is a @xmath116 function , then for @xmath132 , @xmath133 we say that @xmath134_{\mathbb{t } } , \mathbb{r}^n)$ ] is a local minimizer for problem ( [ problem ] ) if there exists @xmath135 such that @xmath136\leq i[y]\ ] ] for all @xmath137_{\mathbb{t } } , \mathbb{r}^n)$ ] satisfying the boundary conditions @xmath18 , @xmath19 , and @xmath138_{\mathbb{t}}^\kappa}\mid y^{\sigma}(t)-y_{\ast}^{\sigma}(t)\mid + \sup_{t \in [ a , b]_{\mathbb{t}}^\kappa}\mid y^{\delta}(t)-y_{\ast}^{\delta}(t)\mid < \delta \ , , \ ] ] where @xmath139 denotes a norm in @xmath140 . we recall now the ( first ) euler - lagrange equation as presented in @xcite . as in the introduction , we use @xmath21 to denote the partial derivative of @xmath25 with respect to the @xmath24-th variable ( or group of variables ) . [ thm : elts ] if @xmath3 is a local minimizer of , then @xmath3 satisfies the following euler - lagrange equation : @xmath141_{\mathbb{t}}^{\kappa}.\ ] ] the following theorem presents a generalization to time scales of the second euler - lagrange equation @xcite ( also known as the dubois - reymond equation @xcite ) . _ ( the second euler - lagrange equation on time scales ) _ : [ secondel ] if @xmath142 is a local minimizer of problem , then @xmath3 satisfies the equation @xmath143 for all @xmath144_{\mathbb{t}}^\kappa$ ] , where @xmath145 @xmath146 and @xmath147 . let @xmath148 be a local minimizer of functional @xmath149 in . we will prove that there exists @xmath150 , @xmath151 , that satisfies the condition @xmath152_{\mathbb{t}}^\kappa.\ ] ] if @xmath153 , then any @xmath150 satisfies condition . suppose now that @xmath154 . then there exists some @xmath155 such that @xmath156 where we suppose that @xmath157 . since @xmath158 is bounded on @xmath93_{\mathbb{t}}^\kappa$ ] , then there exist @xmath159 such that @xmath160_{\mathbb{t}}^\kappa.\ ] ] let @xmath161 where @xmath162 if @xmath163 . if @xmath164 we can choose @xmath165 such that @xmath166 . if @xmath167 we can choose @xmath165 such that @xmath168 . the map @xmath169_{\mathbb{t}}\rightarrow \mathbb{r}$ ] defined by @xmath170 is delta differentiable with @xmath171 and , by theorem [ monotonia delta ] , @xmath172 is strictly increasing on @xmath93_{\mathbb{t}}$ ] . note that @xmath173_{\mathbb{t}})$ ] is a new time scale ( because @xmath172 is continuous and @xmath93_{\mathbb{t}}$ ] is closed ) . by @xmath174 we denote the forward jump operator and by @xmath175 we denote the delta derivative on @xmath176 . let @xmath177 and define @xmath178 for @xmath179 . note that @xmath180 and @xmath181 by the chain rule and from , @xmath182 which gives @xmath183 by the derivative of the inverse applied to @xmath172 we can conclude that @xmath184 note that , since @xmath185 , then @xmath186 and @xmath187 from ( [ eq5 ] ) , ( [ eq3 ] ) , and the substitution in the integral , @xmath188 & = \int_a^b l(t , q_0^\sigma(t),q_0^\delta(t ) ) \delta t\\ & = \int_\alpha^\beta \tilde{l}(\tau,\eta_0^{\tilde{\sigma}}(\tau ) , \eta_0^{\tilde{\delta}}(\tau ) ) \tilde{\delta } \tau = : \tilde{i}[\eta_0 ] , \end{split}\ ] ] where @xmath189 for @xmath190 , @xmath191 , @xmath192 , @xmath193 and @xmath194 . let @xmath195 we remark that @xmath196 was chosen so small that the constraint @xmath197 is always satisfied for any function @xmath198 in the `` nearby '' of @xmath199 . since @xmath200 is by assumption a local minimizer of @xmath149 in @xmath201 , it follows that @xmath199 is a local minimizer of @xmath202 in @xmath203 , so it satisfies the euler - lagrange equation ( in integral form ) @xmath204 where @xmath205 is a constant vector . differentiating we obtain @xmath206 and @xmath207 using , , and ( [ eq3 ] ) we obtain @xmath208 note that @xmath209 hence , by the euler - lagrange equation ( [ eq9 ] ) , we may conclude that @xmath210 the last equality may be rewritten as @xmath211\\ = & - \left[\partial_3 l\left(t , q_0^\sigma(t),q_0^\delta(t)\right ) - \int_a^t \partial_2 l\left(s , q_0^\sigma(s),q_0^\delta(s)\right ) \delta s - c_1\right ] . \end{split}\ ] ] using the euler - lagrange equation for @xmath200 we arrive at the intended statement . if @xmath50 , then the equation ( [ 2equationel ] ) simplifies due to the fact that @xmath212 , and we obtain the classical second euler - lagrange equation ( , , @xcite ) : if @xmath3 is a local minimizer of the classical functional of the calculus of variations , then @xmath213\\ = -\partial_1 l ( t , q(t ) , \dot{q}(t))\end{gathered}\ ] ] holds for all @xmath100 $ ] . in the autonomous case , theorem [ secondel ] gives an extension of the classical second erdmann condition : if @xmath142 is a local minimizer of the problem @xmath214=\int_a^b l(q^\sigma(t),q^\delta(t ) ) \delta t \longrightarrow \min_{q\in \mathcal{d}},\ ] ] then @xmath3 satisfies equation for all @xmath144_{\mathbb{t}}^\kappa$ ] . [ example1 ] let @xmath77 be a time scale with @xmath215 , @xmath216 . consider problem ( [ problem ] ) with @xmath217 and a lagrangian @xmath25 given by @xmath218 . the second euler - lagrange equation for this problem is @xmath219 and the extremal is @xmath220 with @xmath221 let @xmath222 and consider the following problem on @xmath0 : @xmath223 = \int_0 ^ 1 \left[(q^\delta(t))^2 - 1\right]^2 \delta t \longrightarrow \min \ , , \\ q(0 ) = 0 \ , , \quad q(1 ) = 0 \ , , \\ q \in c_{rd}^1(\mathbb{t } ; \mathbb{r } ) \ , .\end{gathered}\ ] ] the euler - lagrange equation takes the form @xmath224 = \text{const}\ ] ] while the second euler - lagrange equation asserts that @xmath225 \left[1 + 3 ( q^\delta(t))^2\right ] = \text{const } \ , .\ ] ] let @xmath226 for all @xmath227 , and @xmath228 . one has @xmath229 , @xmath230 , and @xmath231 , @xmath232 . we see that @xmath233 is an extremal , , it satisfies the euler - lagrange equation . however @xmath233 can not be a solution to the problem since it does not satisfy the second euler - lagrange equation . in fact , any function @xmath3 satisfying @xmath234 , @xmath125 , is an euler - lagrange extremal . among them , only @xmath235 for all @xmath125 and those with @xmath236 satisfy our condition . this example shows a problem for which the euler - lagrange equation gives several candidates which are not the solution to the problem , while our second euler - lagrange equation gives a smaller set of candidates . moreover , the candidates obtained from our condition lead us directly to the explicit solution of the problem . indeed , the null function and any function @xmath3 with @xmath237 and @xmath236 , @xmath125 , gives @xmath238 = 0 $ ] . they are minimizers because @xmath238 \ge 0 $ ] for any function @xmath239 . let @xmath240_{\mathbb{t}}\rightarrow \mathbb{r}^n$ ] , @xmath241 , and consider a one - parameter family of infinitesimal transformations @xmath242 where @xmath243 , @xmath244_{\mathbb{t}}\times \mathbb{r}^n\rightarrow\mathbb{r}$ ] , and @xmath245_{\mathbb{t}}\times \mathbb{r}^n\rightarrow\mathbb{r}$ ] are delta differentiable functions . we assume that for every @xmath246 and every @xmath247 , the map @xmath93\ni t \mapsto \alpha(t):= t(t , q(t ) , \epsilon)\in\mathbb{r}$ ] is a strictly increasing @xmath116 function and its image is again a time scale with the forward shift operator @xmath248 and the delta derivative @xmath249 . we recall that the following holds : @xmath250 [ def . invariance ] functional @xmath149 in is said to be invariant on @xmath74 under the family of transformations if @xmath251 where , for simplicity of notation , we omit the arguments of functions @xmath252 and @xmath253 : @xmath254 , @xmath255 . note that the invariance notion presented in ( * ? ? ? * definition 5 ) implies definition [ def . invariance ] . indeed , for any subinterval @xmath256_{\mathbb{t}}\subseteq [ a , b]_{\mathbb{t}}$ ] , any @xmath257 , and any @xmath247 , one has @xmath258 from the arbitrariness of @xmath259 and @xmath260 it follows that @xmath261 and this implies @xmath262 [ thm : invariance ] functional @xmath149 in is invariant on @xmath74 under the family of transformations if and only if @xmath263 for all @xmath100_{\mathbb{t}}^\kappa$ ] and all @xmath257 , where @xmath264 since @xmath265 the definition of invariance is equivalent to @xmath266 which proves the desired result . [ example2 ] for example [ example1 ] one has invariance under the family of transformations with @xmath267 and @xmath268 , where @xmath269 and @xmath270 are arbitrary constants . in order to simplify expressions , we write @xmath271 instead of @xmath272 . similarly for the partial derivatives of @xmath25 . we recall that @xmath3 is an _ extremal _ to problem if it satisfies the euler - lagrange equation . [ noether ] if functional @xmath149 in is invariant on @xmath74 in the sense of definition [ def . invariance ] ( lemma [ thm : invariance ] ) , then @xmath273\cdot\tau(t , q)\end{gathered}\ ] ] is constant along all the extremals of problem ( [ problem ] ) . we must prove that @xmath274\end{gathered}\ ] ] is equal to zero along all the extremals of problem ( [ problem ] ) . we begin noting that @xmath275 \cdot\tau^{\sigma}(t , q)\\ & \quad + \bigl[l(t , q^{\sigma},q^{\delta})-\partial_{3}l(t , q^{\sigma},q^{\delta})\cdot q^{\delta}\\ & \qquad\quad -\partial_{1}l(t , q^{\sigma},q^{\delta})\cdot\mu(t)\bigr ] \cdot\tau^{\delta}(t , q ) \ , . \end{split}\ ] ] using the first and second euler - lagrange equations and , respectively , we conclude that @xmath276 since @xmath277 , then @xmath278 hence , @xmath279 using lemma [ thm : invariance ] we arrive at the intended conclusion . if @xmath50 , then @xmath212 and theorem [ noether ] reduces to the classical noether s theorem ( , , @xcite ) : if the classical fundamental functional of the calculus of variations is invariant , then @xmath280\cdot\tau(t , q)\ ] ] is constant along all the extremals of the problem . [ example3 ] for the problem of example [ example2 ] one has from theorem [ noether ] that @xmath281 along the extremals @xmath3 of the problem . this is indeed true : from example [ example1 ] we know that the extremals have the form @xmath220 for some constants @xmath282 ; thus , the conservation law takes the form @xmath283 . in this paper we obtain a second euler - lagrange equation and a second erdmann condition for the problem of the calculus of variations on time scales . since both necessary optimality conditions are important and extremely useful results in the calculus of variations and optimal control when @xmath284 , we claim that the present results are also useful for the development of the recent theory of the calculus of variations on time scales @xcite . as pointed out to us by richard vinter , our second euler - lagrange equation in the time scales setting seems to be useful in a framework for studying the asymptotics of time discretization . as an example of application of our main results , we give a simpler and more elegant proof to the noether symmetry theorem on time scales obtained in 2008 @xcite , which allows to obtain conserved quantities along the extremals of the problems . standard noetherian constants of motion are violated due to the presence of a new term that depends on the graininess @xmath285 of the time scale , while in the classical context @xmath53 . the importance of noether s conservation laws in the calculus of variations , optimal control theory , and its applications in engineering , are well recognized @xcite . their role on the general context of optimal control on time scales is an entirely open area of research . in particular , it would be interesting to investigate the techniques of @xcite with the recent higher - order euler - lagrange equations on time scales @xcite for a possible extension of theorem [ noether ] to variational problems on time scales with higher - order delta or nabla derivatives . the question of obtaining conserved quantities along the extremals of higher - order problems of the calculus of variations on time scales remains an interesting open question . zbigniew bartosiewicz was supported by biaystok university of technology grant s / wi/1/08 ; natlia martins and delfim f. m. torres by the r&d unit `` centre for research in optimization and control '' ( ceoc ) of the university of aveiro , cofinanced by the european community fund feder / poci 2010 . z. bartosiewicz , u. kotta , e. pawuszewicz and m. wyrwas , algebraic formalism of differential one - forms for nonlinear control systems on time scales , proc . estonian acad . phys . math . * 56 * ( 2007 ) , no . 3 , 264282 . z. bartosiewicz , e. piotrowska and m. wyrwas , stability , stabilization and observers of linear control systems on time scales , proceedings of the 46th ieee conference on decision and control new orleans , la , usa , dec . 12 - 14 , 2007 , 28032808 . p. d. f. gouveia and d. f. m. torres , automatic computation of conservation laws in the calculus of variations and optimal control , comput . methods appl . math . * 5 * ( 2005 ) , no . 4 , 387409 . arxiv : math/0509140 j. seiffertt , s. sanyal and d. c. wunsch , hamilton - jacobi - bellmam equations and approximate dynamic programming on time scales , ieee trans . . , part b : cybern . * 38 * ( 2008 ) , no . 4 , 918923 . d. f. m. torres , carathodory equivalence , noether theorems , and tonelli full - regularity in the calculus of variations and optimal control , j. math . ( n. y. ) * 120 * ( 2004 ) , no . 1 , 10321050 . arxiv : math.oc/0206230 | the fundamental problem of the calculus of variations on time scales concerns the minimization of a delta - integral over all trajectories satisfying given boundary conditions . in this paper
we prove the second euler - lagrange necessary optimality condition for optimal trajectories of variational problems on time scales . as an example of application of the main result
, we give an alternative and simpler proof to the noether theorem on time scales recently obtained in [ j. math .
anal .
appl . 342
( 2008 ) , no . 2 , 12201226 ] .
* mathematics subject classification 2000 : * 49k05 , 39a12 .
* keywords : * calculus of variations ; optimal control ; euler - lagrange , dubois - reymond , and second erdmann necessary optimality conditions ; noether s theorem ; time scales . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
the in - spiral and coalescence of binary neutron star systems is a topic of increasingly intensive research in observational and theoretical astrophysics . it is anticipated that the first direct detections of gravitational wave ( gw ) will be from compact binary mergers . binary neutron star ( bns ) mergers are also thought to produce short - hard gamma - ray bursts ( sgrb s ) @xcite . simultaneous detections of a prompt gravitational wave signal with a spatially coincident electromagnetic ( em ) counterpart dramatically increases the potential science return of the discovery . for this reason , there has been considerable interest as to which , if any , detectable em signature may result from the merger @xcite . other than sgrbs and their afterglows , including those viewed off - axis @xcite , suggestions include optical afterglows associated with the radio - active decay of tidally expelled r - process material@xcite ( though detailed calculations indicate they are faint @xcite ) , radio afterglows following the interaction of a mildly relativistic shell with the interstellar medium @xcite , and high - energy pre - merger emission from resistive magnetosphere interactions @xcite . merging neutron stars possess abundant orbital kinetic energy ( @xmath4ergs ) . a fraction of this energy is certain to be channelled through a turbulent cascade triggered by hydrodynamical instabilities during merger . turbulence is known to amplify magnetic fields by stretching and folding embedded field lines in a process known as the small - scale turbulent dynamo @xcite . amplification stops when the magnetic energy grows to equipartition with the energy containing turbulent eddies @xcite . an order of magnitude estimate of the magnetic energy available at saturation of the dynamo can be informed by global merger simulations . these studies indicate the presence of turbulence following the nonlinear saturation of the kelvin - helmholtz ( kh ) instability activated by shearing at the ns surface layers @xcite . the largest eddies produced are on the @xmath5 km scale and rotate at @xmath6 , setting the cascade time @xmath7 and kinetic energy injection rate @xmath8 at @xmath9ms and @xmath10 respectively . when kinetic equipartition is reached , each turbulent eddy contains @xmath11 of magnetic energy , and a mean magnetic field strength @xmath12 whether such conditions are realized in merging neutron star systems depends upon the dynamo saturation time @xmath13 and equipartition level @xmath14 . in particular , if @xmath15 then turbulent volumes of neutron star material will contain magnetar - level fields throughout the early merger phase . once saturation is reached , a substantial fraction of the injected kinetic energy , @xmath16 , is resistively dissipated @xcite at small scales . magnetic energy dissipated by reconnection in optically thin surface layers will accelerate relativistic electrons @xcite , potentially yielding an observable electromagnetic counterpart , independently of whether the merger eventually forms a relativistic outflow capable of powering a short gamma - ray burst . in this letter we demonstrate that the small - scale turbulent dynamo saturates quickly , on a time @xmath15 , and that @xmath17 g magnetic fields are present throughout the early merger phase . this implies that the magnetic energy budget of merging binary neutron stars is controlled by the rate with which hydrodynamical instabilities randomize the orbital kinetic energy . our results are derived from simulations of the small scale turbulent dynamo operating in the high - density , trans - relativistic , and highly conductive material present in merging neutron stars . we have carefully examined the approach to numerical convergence and report grid resolution criteria sufficient to resolve aspects of the small - scale dynamo . our letter is organized as follows . the numerical setup is briefly described in section 2 . section 3 reports the resolution criterion for numerical convergence of the dynamo completion time and the saturated field strength . in section 4 we asses the possibility that magnetic reconnection events may convert a sufficiently large fraction of the magnetic energy into high energy photons to yield a prompt electromagnetic counterpart detectable by high energy observatories including _ swift _ and _ fermi_. and @xmath18 . lower resolutions are shown in red and graduate to black with higher resolution . _ top _ : the root mean square magnetic field strength in units of @xmath19 . when a turbulent volume is resolved by @xmath20 zones , the small - scale dynamo proceeds so slowly that almost no amplification is observed in the first 1ms . _ middle _ : the magnetic energy in units of the rest mass @xmath21 shown on logarithmic axes . it is clear that the linear growth rate increases at each resolution . _ bottom _ : the kinetic energy ( upper curves ) shown again the magnetic energy ( lower curves ) again in units of @xmath21 . for all resolutions , the kinetic energy saturates in less than 1 @xmath7.,width=326 ] the equations of ideal relativistic magnetohydrodynamics ( rmhd ) have been solved on the periodic unit cube with resolutions between @xmath20 and @xmath18 . [ eqn : rmhd - system ] @xmath22 here , @xmath23 is the magnetic field four - vector , and @xmath24 is the total specific enthalpy , where @xmath25 is the total pressure , @xmath26 is the gas pressure and @xmath27 is the specific internal energy . the source term @xmath28 includes injection of energy and momentum at the large scales and the subtraction of internal energy ( with parameter @xmath29mev ) to permit stationary evolution . vortical modes at @xmath30 are forced by the four - acceleration field @xmath31 which smoothly decorrelates over a large - eddy turnover time , as described in @xcite . we have employed a realistic micro - physical equation of state ( eos ) appropriate for the conditions of merging neutron stars . it includes contributions from high - density nucleons according a relativistic mean field model @xcite , a relativistic and degenerate electron - positron component , neutrino and anti - neutrino pairs in beta equilibrium with the nucleons , and radiation pressure . for our conditions , all the components make comparable contributions to the pressure . we have also employed a simpler gamma - law eos and found close agreement for the conditions explored in this paper , indicating that the mhd effects are insensitive the eos for trans - sonic conditions . the models presented in our resolution study use the far less expensive gamma - law equation of state . all of the simulations presented in this study use the hlld approximate riemann solver @xcite , which has been demonstrated as crucial in providing the correct spatial distribution of magnetic energy in mhd turbulence @xcite . the solution is advanced with an unsplit , second - order muscl - hancock scheme . spatial reconstruction is accomplished with the piecewise linear method configured to yield the smallest possible degree of numerical dissipation . the divergence constraint on the magnetic field is maintained to machine precision at cell corners using the finite volume ct method of @xcite . full details of the numerical scheme may be found in @xcite . magnetic fields are amplified in our simulations by the small - scale turbulent dynamo . turbulent fluid motions stretch and fold the magnetic field lines , causing exponential growth of magnetic energy ( e.g. * ? ? ? this growth is attributed to the advection and diffusion of @xmath32 through the mhd induction equation ( eq . [ eqn : rmhd - system]c ) . when the magnetic field is weak ( @xmath33 ) @xmath32 evolves passively , and the turbulence is hydrodynamical . this limit is referred to as small - scale kinematic turbulent dynamo , and is well described by kazentzev s model @xcite . this model predicts that the power spectrum of magnetic energy peaks at the resistive scale @xmath34 and obeys a power law @xmath35 at longer wavelengths . the kinematic phase ends when the magnetic field acquires sufficient tension to modify the hydrodynamic motions , after which time a dynamical balance between kinetic and magnetic energy is established . numerical simulations of mhd turbulence are typically limited to magnetic prandtl numbers @xmath36 . however , neutron star material is characterized by @xmath37 , with the viscous cutoff due to neutrino diffusion occurring at around 10 cm , while the resistive scale is significantly smaller @xcite . however , the disparity between true and simulated magnetic prandlt number does not influence our conclusions . this is because dynamos are generically easier to establish in the high pm regime than the small @xcite . , together with the empirical model ( equation [ eqn : magnetic - fit ] ) with best - fit parameters . the horizontal dashed line indicates the magnetic energy , @xmath38 at the dynamo completion . from left to right , the vertical dashed lines mark the end of the startup , kinematic , and saturation phases.,width=326 ] we use an initially uniform , pulsar - level ( @xmath39 g ) seed magnetic field . this field is sub - dominant to the kinetic energy by 10 orders of magnitude , so that the initial field amplification is expected to be well - described by kazentsev s model . indeed , we find that during this phase the power spectrum of magnetic energy follows @xmath35 ( fig . [ fig : pspec - time - devel - res ] ) , peaking at around 10 grid zones , which we identify as the effective scale of resistivity . the saturation process begins at ever - earlier times with increasing numerical resolution . this reflects the fact that during the kinematic phase , magnetic energy exponentiates on a time scale controlled by shearing at the smallest scales . in numerically converged runs , full saturation occurs with @xmath40 and is characterized by scale - by - scale super - equipartition , with @xmath41 at all but the largest scale . the same driven turbulence model was run through magnetic saturation at resolutions @xmath20 , @xmath42 , @xmath43 , @xmath44 , @xmath45 , and @xmath46 . another model at @xmath18 was run through the end of the kinematic phase , but further evolution was computationally prohibitive . [ fig : energy - growth ] shows the development of @xmath47 , @xmath48 , and @xmath49 as a function of time at each resolution . we find that sufficiently resolved runs ( @xmath50 ) attain _ mean _ magnetic field strengths of @xmath51 g within two large eddy rotations . all models with resolutions @xmath52 eventually attain mean fields of @xmath53 g . the saturated field strength increases until resolution @xmath45 . we find that the kinematic growth rate is higher at each higher resolution , while the time - scale for the non - linear saturation converges at @xmath45 to roughly five large - eddy rotation times , or about 0.5ms for the physical parameters of binary neutron star mergers . lcc @xmath54 & startup time - scale & none , artifact @xmath55 & hydrodynamic cascade fully developed & @xmath7 @xmath56 & kinematic growth time - scale & none , @xmath57 @xmath58 & end of kinematic phase & @xmath59 @xmath60 & non - linear saturation time - scale & @xmath61 , @xmath45 @xmath38 & saturated magnetic energy & @xmath62 , @xmath46 [ tab : magnetic - fit ] in order to quantitatively describe the time development of magnetic energy @xmath63 , we describe it with an empirical model , @xmath64 where the 6 parameters ( summarized in table [ tab : magnetic - fit ] ) are obtained by a least - squares optimization . [ fig : magnetic - fit ] shows the empirical model given in equation [ eqn : magnetic - fit ] applied to a representative run at @xmath44 . the first phase , @xmath65 is a startup transient , and lasts until the hydrodynamic cascade is fully developed at @xmath66 . the kinematic dynamo phase lasts between @xmath55 and @xmath58 , during which the magnetic energy exponentiates on the time scale @xmath56 . at @xmath58 , the smallest scales reach kinetic equipartition and the growth rate slows . in the final stage , @xmath48 asymptotically approaches @xmath38 on the time - scale @xmath60 . we define the dynamo completion time @xmath13 as @xmath67 . [ fig : convergence ] shows the best - fit @xmath38 , @xmath13 , and @xmath56 as a function of the resolution . the magnetic energy @xmath38 at dynamo completion shows signs of converging to a value of @xmath62 by resolution @xmath46 . the time scale @xmath60 on which the magnetic energy asymptotically approaches @xmath38 is consistently @xmath68 at different resolutions . the dynamo completion time @xmath13 is numerically converged at @xmath69 by @xmath45 . the best - fit kinematic growth time follows a power law in the resolution , @xmath70 . this is consistent with the value of @xmath71 expected if the dynamo time is controlled by shearing at the smallest scale , the cascade is kolmogorov ( i.e. @xmath72 ) , and the viscous cutoff @xmath73 occurs at a fixed number of grid zones . in that case , @xmath74 . ( _ blue _ ) and the dynamo completion time @xmath13(_green _ ) defined as @xmath75 . _ bottom _ : convergence study of the best - fit model parameter @xmath38 expressed as the ratio of magnetic to kinetic energy @xmath76 . the converged value of the volume - averaged @xmath77 . nevertheless , at intermediate wavelengths @xmath78 . as shown in figure [ fig : pspec - time - devel - res ] , the largest scales remain kinetically dominated . this indicates the suppression of coherent magnetic structure formation near the integral scale of turbulence.,width=326 ] the time development of kinetic and magnetic energy power spectra has been studied for a single run with resolution @xmath46 . we present three - dimensional , spherically integrated power spectra with the dimensions of @xmath79 , defined as @xmath80\right|^2}\\ p_m(k_i ) & = \frac{1}{\delta k_i}\sum _ { { \mathbf k } \in \delta k_i } { \left|\mathcal{f } _ { { \mathbf k } } \left [ { \mathbf b } /\sqrt{8\pi}\right]\right|^2 } \end{aligned}\ ] ] [ eqn : pspec ] where the newtonian versions of kinetic and magnetic energy are appropriate since the conditions are only mildly relativistic . the definitions in equations [ eqn : pspec ] satisfy @xmath81 for @xmath82 and @xmath83 . figure [ fig : pspec - time - devel - res ] shows the power spectrum of kinetic and magnetic energy at various times throughout the growth and saturation of magnetic field . during the kinematic phase , the kinetic energy has a power spectrum @xmath84 consistent with the kolmogorov theory for incompressible hydrodynamical turbulence , while @xmath85 consistent with kazenstev s model . @xmath86 maintains the same shape , but exponentiates in amplitude at the time scale @xmath56 which is controlled by shearing at the resistive scale . according to kazentsev s model , @xmath86 should peak at the resistive scale . this is consistent with the observed peak in the magnetic energy at roughly 10 grid zones , the same scale at which we observe the viscous cutoff . this is also consistent with @xmath87 expected from the numerical scheme employed . when the magnetic energy at the resistive scale surpasses the level of the kinetic energy at that scale , @xmath86 changes shape . the equipartition scale @xmath88 moves into the inertial range , and migrates to larger scale until full saturation occurs with @xmath89 . the movement of @xmath88 to larger scale is associated with the formation of coherent and dynamically substantial magnetic structures of increasing size . the time - dependence has been suggested to be @xmath90 @xcite . in the fully saturated state , the magnetic field is in scale - by - scale super - kinetic equipartition throughout the inertial range , with @xmath91 . the largest scales remain kinetically dominated so that the numerically converged saturation level is @xmath92 . in this letter we have determined the time scale and saturation level of the small - scale turbulent dynamo operating in the conditions of binary neutron star mergers . we have presented numerically converged simulations showing that magnetic fields are amplified to the @xmath93 g level within a small fraction of the merger dynamical time ( @xmath15 ) , independently of the seed field strength . if hydrodynamical instabilities create fluctuating velocities on the order of @xmath94 as indicated by global simulations , then each @xmath95 turbulent volume dissipates @xmath96 erg of magnetic energy per 0.1 ms . if @xmath97 represents the fraction of the merger remnant that contains such turbulence , the magnetic energy dissipated during the merger is at least @xmath98 a fraction of that dissipation will occur through magnetic reconnection in optically thin surface layers , supplying relativistic electrons which synchrotron radiate in the merger remnant magnetosphere . if 5% of that magnetic energy dissipation creates radiation in the 15 - 150 kev band , then the fluence at 200 mpc would be @xmath3 , potentially rendering most merging neutron stars in the advanced ligo and virgo detection volumes detectable by _ swift _ bat . if so , then merging neutron stars are accompanied by a prompt electromagnetic counterpart , independently of whether a later merger phase yields a collimated outflow capable of powering a short gamma - ray burst . we suggest that merger flares may be present in the current sample of short grbs and may be roughly isotropic on the sky since they are seen to distances where the cosmological matter distribution becomes homogeneous . searches for merger flares should seek to identify short flares , not unlike soft - gamma repeaters , among the short burst population . if mergers also produce short grbs short - hard grbs , then merger flares may constitute a precursor component of the emission . the presence of strong magnetic fields may also aid in the ejection of neutron - rich material from surface layers of the merger remnant , possibly enhancing the enrichment of inter - stellar medium by r - process nuclei @xcite . enhanced production of r - process nuclei also increases the likelihood of em detection by radio - active decay powered afterglows , or `` kilonovae '' @xcite . finally , it has been shown that magnetic fields will significantly influence the gravitational wave signature and remnant disk mass , if they exist at the @xmath99 g level @xcite . such strong fields are unlikely in older neutron star binaries , but our results suggest they may be revived , albeit at small scales , during the merger . to have significant influence , those fields would have to fill a considerable fraction of the merger volume . as we have shown in this letter , the overall magnetic energy budget is controlled by the prevalence ( @xmath97 ) and vigor ( @xmath100 ) of the turbulent volumes . this fact motivates the use of higher resolution global simulations aimed at measuring @xmath97 and @xmath100 . this research was supported in part by the nsf through grant ast-1009863 and by nasa through grant nnx10af62 g issued through the astrophysics theory program . resources supporting this work were provided by the nasa high - end computing ( hec ) program through the nasa advanced supercomputing ( nas ) division at ames research center . | the simultaneous detection of electromagnetic and gravitational wave emission from merging neutron star binaries would aid greatly in their discovery and interpretation . by studying turbulent amplification of magnetic fields in local high - resolution simulations of neutron star merger conditions ,
we demonstrate that magnetar - level ( @xmath0 g ) fields are present throughout the merger duration .
we find that the small - scale turbulent dynamo converts 60% of the randomized kinetic energy into magnetic fields on a merger time scale .
since turbulent magnetic energy dissipates through reconnection events which accelerate relativistic electrons , turbulence may facilitate the conversion of orbital kinetic energy into radiation .
if @xmath1 of the @xmath2 erg of orbital kinetic available gets processed through reconnection , and creates radiation in the 15 - 150 kev band , then the fluence at 200 mpc would be @xmath3 , potentially rendering most merging neutron stars in the advanced ligo and virgo detection volumes detectable by _ swift _ bat . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
we address the modeling and simulation of elastoplastic solids in multidomain environments including also mechatronic multibody systems with nonsmooth dynamics , such as vehicles , robots and processing machinery . fast multidomain simulation is useful for concept design exploration , development of control algorithms and for interactive realtime simulators , e.g. , for operator training , human - machine interaction studies and hardware - in - the - loop testing . realizing such simulators require integrating many subsystems of different types and complexity into a single multidomain dynamics model . if the subsystems are loosely coupled , the full system simulation can be realised by means of co - simulation @xcite , but this is not the general case . for the sake of computational performance , coarse grain models are often used , with rigid and flexible bodies coupled by kinematic constraints for modeling of joints and differential algebraic equation ( dae ) models for electronics , hydraulics and powertrain dynamics @xcite . at coarse timescales the dynamics need to be treated as nonsmooth , allowing discontinuous velocities of rigid bodies undergoing impacts or frictional stick - slip transitions and instantaneous propagation of impulses through the system . this requires strong coupling through consistent mathematical formulation and coherent numerical treatment of the full system in order to achieve good stability and computational efficiency . current techniques for co - simulation does not support that . the theory and numerical methods for nonsmooth multidomain mechanics is covered in the reference @xcite . the framework in @xcite is employed for constraint regularisation and stabilisation based on a discrete variational approach with constraints introduced as the stiff limit of energy and dissipation potentials . the developed method is applicable to many different systems but one is primarily explored , namely the interaction between ground vehicles and deformable terrain @xcite . current solutions enable simulation of complex vehicle models with realtime performance but not including dynamic terrain models firmly based on solid mechanics in three dimensions . usually , empirical terrain models of bekker - wong type are used @xcite . for more accurate offline simulations with fine temporal resolution , on the other hand , many solutions exist for elastoplastic solids coupled with tire models @xcite but scarcely with more complex multibody systems and not for realtime or faster simulation . the main contribution in this paper , illustrated in figure [ fig : illustration ] , is a formulation and numerical method for simulation of elastoplastic solids as a nonsmooth multidomain multibody system on descriptor form @xcite . this enables numerical integration with large timesteps that potentially exceed current solutions and provide realtime performance or better . the regularisation and damping terms regulate the elasticity and viscous damping of deformations according to linear elasticity theory . an associative perfectly plastic drucker - prager model is employed using an elastic predictor - plastic corrector strategy to detect yielding and compute the plastic flow . in its basic form this model does not yield in hydrostatic compression in contrast to many real materials . many soils yield under hydrostatic compression by failure in the microscopic structures whereby air and fluid is released . therefore the drucker - prager model is extended to a capped version @xcite . the dynamics of other subsystems and their potentially strong coupling is treated within the same multibody dynamics framework using a variational time integrator . each timestep involve solving a block - sparse mixed linear complementarity problem mlcp , and can be efficiently integrated with tailored solvers . a meshfree method @xcite is chosen in order to handle large deformations without need for remeshing @xcite and , for future development , support fracturing and transitions to viscous flow or to granular media represented by contacting discrete elements . the displacement gradient and strain tensor is approximated by the method of moving least squares ( mls ) @xcite . matrices and vectors are represented in bold face in capital and lower case , @xmath0 and @xmath1 , respectively . latin superscripts indicate the index of a specific body @xmath2 , where @xmath3 is the total number of bodies in the system . greek subscripts indicate a specific scalar component of a vector or matrix ( may be multidimensional ) . the einstein summation convention is used where repeated indices imply summation over them , e.g. , representing matrix - vector multiplication as @xmath4 . subscripts @xmath5 or @xmath6 indicate a specific component of a vector or matrix assuming a cartesian coordinate system . as an example of these notations the position vector of a body @xmath7 is denoted @xmath8 . the relative position of particle @xmath9 to @xmath7 is @xmath10 . the @xmath11 component of this vector is @xmath12 . for rigid bodies , the position vector include also the rotation of the body . the system position vector is @xmath13 $ ] . the gradient is denoted @xmath14 and @xmath15 . the dot notation is used for the time derivative @xmath16 . the lagrangian of a constrained mechanical system is @xmath17 where @xmath18 is the kinetic energy , @xmath19 is the potential energy and @xmath20 is the rayleigh dissipation function . the system state vector @xmath21 is constrained by scleronomic holonomic constraints @xmath22 , with jacobian @xmath23 , and nonholonomic pfaffian constraints @xmath24 with corresponding lagrange multipliers @xmath25 and @xmath26 . to ensure numerical stability it is common to regularise the constraints . this may be done by treating them as the limit of strong potentials and dissipation functions @xmath27 and @xmath28 with @xmath29 @xcite . in fact , any stiff force can be transformed into a regularised constraint through a legendre transform @xcite @xmath30 this transform the euler - lagrange equations of motion into @xmath31 where @xmath32 are the explicit forces from weak smooth potentials . this constitute a system of dae of index 1 , which are easier to integrate numerically than the corresponding higher index dae in the absence of regularisation and stabilisation . a term for dissipation of motion orthogonal to the holonomic constraint surface can be added to improve the convergence . also this dissipation can be physically based by introducing it as a legendre transform of a rayleigh dissipation function @xmath34 , with damping parameter @xmath35 . this modifies eq . to @xmath36 the compliance and damping factors @xmath37 , @xmath35 and @xmath38 are not restricted to being scalar or diagonal . in what follows these are assumed to be matrices . variational integration @xcite provides a systematic approach to derive time integration schemes with good properties , e.g. , momentum preservation and symplecticity . rather than discretising the euler - lagrange equations of motion directly , the lagrangian and principle of least action is defined in time - discrete form . employing semi - implicit euler discretisation and linearising the constraint as @xmath39 leads to the following scheme @xmath40 @xmath41 where @xmath42 , @xmath43 and regularisation and stabilisation matrices @xmath44 , @xmath45 and @xmath46 . . is a linear system of @xmath47 equations . the matrix on the left hand side is block - sparse , positive definite and non - symmetric . the regularisation appearing as the diagonal perturbation matrices @xmath48 and @xmath49 are needed for handling otherwise ill - posed or ill - conditioned problems , e.g. , systems with constraint degeneracy and large mass ratios . the stabilisation terms @xmath50 on the right hand side counteract constraint violations , e.g. , sudden and large contact penetrations at impacts or small numerical constraint drift . the presented stepping scheme , referred to as spook , has been proved to be linearly stable @xcite and numerical simulations suggest a large domain of nonlinear stability . the combined effect of the regularisation and stabilisation terms is to bring elastic and viscous properties to motion orthogonal to the constraint surface @xmath51 , e.g. , for modeling elasticity in mechanical joints . the parameters @xmath37 and @xmath38 need not be chosen arbitrarily , as for the conventional approaches to constraint regularisation and stabilisation , but can be based on physics models using parameters that can be derived from first principles , found in literature or be identified by experiments . this becomes straightforward when the regularisation is introduced by potential energy as quadratic functions in @xmath52 , i.e. , @xmath53 . this has been exploited in previous work to constraint based modeling of lumped element beams @xcite , wires @xcite , meshfree fluids @xcite and granular material @xcite . in discrete time , some of the dynamics is best treated as _ nonsmooth _ @xcite , meaning that the velocity may change discontinuously to fulfil inequality constraints and complementarity conditions . introducing such conditions is an efficient way of modeling contacts , dry friction , joint limits , electric circuit switching , motor and actuator dynamics and control systems for discrete time with large step size . the alternative would be to use a fine enough temporal resolution where the dynamics appear smooth and may be modeled by daes or ordinary differential equations alone . in general , this is an intractable approach for full system simulations and interactive applications . the nonsmooth dynamics can be formally treated as _ differential variational inequalities _ @xcite . the discrete equations of motion take the form of a _ mixed linear complementarity problem _ , mlcp , when nonsmooth dynamics is included @xmath54 where @xmath55 corresponds to the left hand side of equation , @xmath56 is the negated right hand side and @xmath57 are slack variables . the terms @xmath58 and @xmath59 correspond to the upper and lower limits on the solution vector @xmath60 , respectively . the original linear system of equations is recovered by assigning @xmath61 to the upper and lower limits , i.e. , no limit . rigid body contacts are handled by the signorini - coulomb law for unilateral dry frictional contacts . this states that if the non - penetration constraint , @xmath62 , is violated then the normal contact velocity and constraint force are complementary in ensuring separation , @xmath63 , and the tangential friction force , acting to maintain zero slip , is bound to be on or inside the coulomb friction cone , @xmath64 . the latter may be linearised by approximating the cone with a polyhedral or a box . impacts are treated post facto in a separate _ impact stage _ succeeding the update of velocities and positions . at the impact stage an impulse transfer is applied enabling discontinuous velocity changes , i.e. , from @xmath65 to @xmath66 . newtons impact law , @xmath67 , is used with restitution coefficient @xmath68 in conjunction with preserving all other constraints , @xmath69 . this amounts to solving the same mlcp but with @xmath70 and @xmath71 in the right hand side of equation . a fixed timestep approach is preferred when aiming for fast full systems simulations with many nonsmooth events with inequality or complementarity conditions , e.g. , involving thousands or millions of dynamic contacts , switching in electric , hydraulic or control systems , or elastic material undergoing fracture or plastic failure . using variable timestep and exact event location become computationally intractable and may fail by occurrence of zeno points . in a multidomain simulation with strongly coupled subsystems the nonsmooth dynamics propagate instantly throughout the full system . the dynamics in the elastoplastic solid may be directly and strongly coupled with the internal dynamics of a mechatronical system . the nonsmooth multidomain dynamics approach with physics based constraint regularisation and stabilisation presented here , provides a general framework for building and efficiently solving the dynamics on mlcp form . each subsystem take the same generic form of saddle - point matrix as the full system in eq . . considering a system with two subsystems @xmath72 and @xmath73 , the full coupled system takes the following form @xmath74 \bm{h}_{a } & 0 & -\bm{g}^t_{{ab}_a}\\ 0 & \bm{h}_{b } & -\bm{g}^t_{{ab}_b}\\ \hline \bm{g}_{{ab}_a } & \bm{g}_{{ab}_b } & \bm{\sigma}_{ab } \end{bmatrix } \begin{bmatrix } \bm{z}_{a } \\ \bm{z}_b \\ \bm{\lambda}_{ab } \end{bmatrix } \label{eq : self_similar}\end{aligned}\ ] ] where @xmath75,\bm{\sigma}_{ab}$ ] are the multiplier , jacobian and regularisation of the subsystem coupling . the solid is assumed to sustain geometric large deformations . venant - kirchoff elasticity model is used in combination with the drucker - prager plasticity model @xcite . the material strain is expressed by the green - lagrange strain tensor @xmath76 where @xmath77 is the displacement field mapping a reference coordinate @xmath1 to displaced position @xmath78 . observe that the green - lagrange strain tensor transforms under large rotations and no co - rotation procedure is needed . it is convenient to use the voigt notation for representing the strain tensor on vector format , @xmath79^t$ ] , and the stress tensor @xmath80^t$ ] such that the linear constitutive law reads @xmath81 , with stiffness matrix @xmath82 where @xmath83 and @xmath84 are the first and second lam parameters which are related to the young s modulus and poisson s ratio , @xmath85 and @xmath86 , of a material as @xmath87 and @xmath88 . the corresponding strain energy density is @xmath89 plastic deformation occur when the stress of a material reaches its yield strength , @xmath90 , in which case the material undergo plastic flow if the load is increased further . the choice of yield function , @xmath91 , is based on the type of material being modeled . to represent the permanent plastic deformation , the strain tensor is decomposed into an elastic and a plastic component , @xmath92 . in ideal plasticity the plastic deformation occurs instantly according to a flow rule @xmath93 , where @xmath94 is the plastic potential and @xmath95 is the plastic multiplier . if @xmath96 the model is said to be associative and otherwise non - associative . the plastic flow last as long as it is positive , @xmath97 , incrementally reducing the stress @xmath98 , until it reaches the elastic regime , @xmath99 . this constitute a nonlinear complementarity problem known as karush - kuhn - tucker conditions @xmath100 the plastic multiplier is computed from the constitutive law , which in the plastic flow phase is @xmath101 , where the elastoplastic tangent stiffness matrix is @xmath102 predictor - corrector algorithms are conventionally used to integrate the plastic flow . in the absence of plastic hardening or softening , the plastic deformation , plastic multiplier and total stress can be computed easily using the radial return algorithm @xcite summarised in algorithm [ alg : radial_return ] . the assumed plasticity model is a capped drucker - prager model , following dolarevic @xcite , with a compaction variable @xmath103 . the yield function @xmath104 is a piecewise function consisting of the drucker - prager , tension and compression cap functions according to @xmath105 where @xmath106 is the first invariant of the stress tensor and @xmath107 is the second invariant of the deviatoric stress tensor @xmath108 . the expressions for the tension cap @xmath109 and compression cap @xmath110 are given in appendix a and the drucker - prager surface is defined as @xmath111 where @xmath112 is the internal friction angle and @xmath113 the cohesion parameter such that @xmath114 and @xmath115 . the capped yield surface is illustrated in figure [ fig : gapfunction ] . the tension cap is fix and is used to regularise the plastic flow behaviour in the corner region . the compression surface cap , on the other hand , is dynamic and the maximum hydrostatic pressure , @xmath103 , is used as main variable . the conventional drucker - prager model is a common model for the plastic deformation dynamics of soils , e.g. wet or dry sand . these materials are weak under tensile stress ( @xmath116 ) and become stronger under compression ( @xmath117 ) where it may support large shear stresses ( @xmath118 ) . the capped drucker - prager model is a generalisation that include also plastic compaction that may occur in many soils . the compaction mechanism may be failure of the individual grains whereby air or water is displaced from the soil . the compaction saturates at a maximum level , where all voids are vanished . the compaction dynamics is modeled by a variable cap on compressive side of the drucker - prager yield surface , intersecting the hydrostatic axis at @xmath119 . the compaction hardening law is chosen @xmath120 where @xmath121 is the maximum volume compaction , @xmath122 is the hardening rate and @xmath123 the initial cap position where compressive failure first occur . observe that when the plastic volume compaction @xmath124 approaches @xmath121 , the cap variable @xmath103 goes to infinity , the material does not compact further plastically and behave like the standard drucker - prager model . the elastoplastic material parameters are summarised in table [ tab : elastoplastic ] . expressions for the detailed shape of the compression cap and the derivatives of the yield functions are found in appendix a. .elastoplastic model parameters [ cols= " < , > " , ] the simulation result is compared with elasticity theory that for st . venant - kirchoff materials imply the following exact relations between applied load and uniform deformation @xmath125 where @xmath126 is the load pressure on the boundary , @xmath127 @xmath128 are the young s and bulk modulus , and @xmath129 and @xmath130 are the stretch ratios @xmath131 of the test - cube with the deformed side - length @xmath132 , initial rest length @xmath133 and cross - sectional area @xmath134 . the simulation results for @xmath135 pa and @xmath136 are presented in figure [ fig : elastic_verification ] . for large deformations ( @xmath137 % ) the simulated result agrees with the exact solution to an accuracy of @xmath138 % for uniaxial stretch and @xmath139 % for hydrostatic compression . the error decrease with size of the deformation and with finer resolution . observe that for infinitesimal deformations @xmath140 , eq . - approximates to the well - known expressions from hooke s law . increasing the stiffness or resolution further require smaller time - step for numerical stability . ) and uniaxial stretch ( @xmath141 ) with @xmath142 pa , @xmath143 and @xmath144 ( circles ) and @xmath145 ( triangles ) . the analytical solution is represented with a solid curve . , scaledwidth=45.0% ] in order to understand the nature of the error the strain field under hydrostatic compression is analysed , see figure [ fig : elastic_hydrostatic_error ] . the strain field is not uniform through the material as expected but deviate as much as @xmath146 % and most near the boundary . enforcing a uniform deformation , by initial particle displacements , produce a perfectly uniform strain and stress field . we thus conclude that the effect is caused by the way that boundary conditions are enforced . see , section [ sec : conclusions ] for a further discussion on this . through the cross - section of the elastic test cube under hydrostatic compression to @xmath147.,scaledwidth=49.0% ] the errors in the strain distribution propagate to the stress and ultimately to the plastic behaviour also . in particular the deviation from nonuniform strain will lead to a non - vanishing stress deviator in deformations where it is expected to be zero . the mean strain deviatior and its standard deviation in the case of hydrostatic compression is displayed in figure [ fig : strain_deviator ] . ^{1/2}$ ] during hydrostatic compression.,scaledwidth=49.0% ] the drucker - prager cap model is tested under hydrostatic and triaxial loading and unloading , following the tests made in @xcite with @xmath148 pa,@xmath149 , @xmath150 kpa , @xmath151 , @xmath152 , @xmath153 mm@xmath154/n . the first test examines the relation between stress and strain while undergoing plastic deformation on the compression cap , see figure [ fig : hydrostatic_plastic ] . the test produce a permanent deformation of the cube by @xmath155 compared to @xmath156 in @xcite . as can be expected this deviation is due to the deviation of uniform strain , observed in section [ sec : elastic_verification ] , that lead to non - vanishing deviatoric stress . investigating the evolution of the stress invariants , @xmath157 and @xmath158 , in the yield space , it is clear that the compression cap is not reached precisely on the hydrostatic axis and thus yield for a smaller of @xmath157 . the plastic shear is also not uniform which affect the permanent deformation . in the triaxial test , a hydrostatic pressure of @xmath159 kpa is first established . the load along one axis is then increased up to a level where the material yields and while pressure is kept fix . the material is finally unloaded . the simulated evolution of the deviatoric stress as function of strain is displayed in figure [ fig : triaxial_plastic_dp ] for a drucker - prager material and in figure [ fig : triaxial_plastic ] for a capped drucker - prager material . the simulated drucker - prager material is found to yield at a critical stress of @xmath160 kpa . this is in precise agreement with the analytical prediction from equation ( [ eq : dp ] ) with hydrostatic pressure of @xmath159 kpa . for the capped drucker - prager the yield plateau is lower and less pronounced . the difference is expected since the stress reach the compression cap surface before the drucker - prager surface . the result show a weak dependency on resolution @xmath161 and @xmath162 . the triaxial test can be used to determine the value of the plastic hardening parameter @xmath122 , whereas the maximum compaction parameter , @xmath121 , can be determined from the hydrostatic compression test . dynamic contact handling is demonstrated by dropping a beam to rest on two thin cylinders and then let if deformed by pressing a larger cylinder down on the beam . the result for an elastic and elastoplastic beam are displayed in figure [ fig : beam ] . ( green ) to @xmath163 ( red).,scaledwidth=45.0% ] in order to demonstrate multidomain capability with the method , two example systems are simulated . the first system is an articulated terrain vehicle with tracked bogies driving over a deformable terrain . images from simulation of a bogie and of a full vehicle are shown in figure [ fig : demo ] . videos from simulations are available as supplementary material at ` http://umit.cs.umu.se/elastoplastic/ ` . the vehicle weighs 4 ton and consist of roughly 200 rigid bodies including a front and rear chassis , bogie frames , wheels and track elements . these are interconnected with kinematic constraints to model chassis articulation , bogie and wheel axes , and tracks covering the wheeled bogies . the vehicle drivetrain consist of a engine , torque converter , drive shafts and differentials that distributes the engine power to front and rear part and further between left and right bogies to the wheels . the terrain is modeled as a static trimesh and a rectangular ditch with elastoplastic material . the elastoplastic material parameters are set to @xmath164 mpa , @xmath165 , @xmath166 , @xmath167 kpa , @xmath168 , @xmath169 kpa , @xmath170 mpa , @xmath171 , @xmath172 , @xmath173 kg / m@xmath174 , which represents a weak and soft forest terrain . the tracked bogie and vehicle create a rutting with permanent deformation and the rut depth can be measured . in the full vehicle simulation the solid is discretised into @xmath175 pseudo - particles . the coupled system of vehicle and terrain thus have roughly 7000 degrees of freedom and roughly the same number of constraint equations . with @xmath176 ms timestep the simulation was of the order @xmath177 slower than realtime on a conventional desktop computer and using a single core . it should be emphasised that this measure is based on prototype code with little effort on optimizing it . also the use of a direct solver for the terrain can be questioned . both the model uncertainty and spatial discretisation error are many orders in magnitude beyond the accuracy delivered by the solver . the second demonstration example is a dynamic cone penetration test where a cylindrical weight is dropped repeatedly on a cone measuring its penetration depth . this is one common way of measuring the mechanical properties of terrain . the penetration depth of the cone in simulated for two different cases . the first case is a homogeneous material and the second is material with an embedded rock , represented by a rigid body , as illustrated by figure [ fig : plastic_cone ] . the colours indicated the magnitude of displacement . the simulated cone penetration is presented in figure [ fig : plastic_cone_cm ] . the resistance is higher in the ground with an embedded rock , also before the cone actually come in contact with the rock . a meshfree elastoplastic solid model is made compatible with nonsmooth multidomain dynamics . the solid appear as a system of constrained particles in a multibody system on the same footing as articulated rigid multibodies and power transmission systems . the particles carry field variables , e.g. , the stress and strain tensor , approximated using the moving least squares method . this method provides a continuous field description throughout the solid . the dynamic interaction between the deformable solid , rigid multibodies and other geometric boundaries is modeled by unilateral contact constraints with dry friction . the full system can thus be processed using the same numerical integrator and solver framework without introducing additional coupling equations with unknown parameters and impulsive behaviour can automatically be transmitted instantly through the full and strongly coupled system . this enables fast and stable simulations of complex mechatronic systems with , or interacting with , elastoplastic materials . demonstration is made with a tracked terrain vehicle driving over deformable terrain using the capped drucker - prager model and a cone penetrometer test on terrain with and without embedded rock . the jacobian of the deformation constraint is derived and the explicit form is given in eq . ( [ eq : strainjacobiansfinal ] ) . it can be factored to a constant term that can be pre - computed and multiplied with current particle displacements . the computational bottleneck of the simulation lies in solving the block sparse mixed linear complementarity problem ( [ eq : mlcp ] ) . with dedicated hardware , parallel factorisation algorithms or using iterative solver for a more approximate solution of the terrain dynamics the performance is expected to be increased by several order in magnitude . exploring this is left for future work . the results of numerical experiments of uniform elastic deformations , presented in figure [ fig : elastic_verification ] and [ fig : elastic_hydrostatic_error ] , reveal that the solution deviate from what is expected from analytical solutions . for example , the confining pressure in hydrostatic compression of a cube discretised by @xmath178 particles is underestimated by roughly @xmath139 % when compressed up to @xmath179 % . the error decrease with reduced time - step and finer spatial discretisation . the errors are presumably due to the application of mls to the solid dynamics equations on strong form . when the problem involve traction boundary conditions , meshfree collocation methods suffer from poor accuracy and instability @xcite . the errors are not located solely to the boundaries but influence also the deformations further in the material , as seen in figure [ fig : elastic_hydrostatic_error ] , and cause the second principal deviatoric stress invariant to deviate from zero in hydrostatic pressure . this affect the plastic behaviour also . the permanent plastic deformation after loading is found to differ by the order of @xmath139 % and sometimes the material has residual stress that may cause artefacts although none have been observed . nevertheless , the developed method is applicable to many type of studies where this level of accuracy is acceptable and can not be increased further anyway because of difficulties in characterising the mechanical properties of the solid material . this deficiency can be avoided by either using a mixed strong and weak form formulation @xcite or applying correction terms of higher order on the boundary particles @xcite . future work should focus on this issue to improve the accuracy of the model . another area of improvement is to integrate the plastic yield and flow computations with the mixed complementarity problem for the constraint forces and velocity update . this will enable integration with larger time - step for strongly coupled problems than the current predictor - corrector method allow . the project was funded in part by the kempe foundation grant jck-1109 , ume university and supported by algoryx simulation . the authors are thankful fruitful discussions and ideas from prof . urban bergsten at the swedish university of agriculture , komatsu forest and olofsfors ab . this supplements section [ sec : constraint_elastoplastic ] with further details of the capped drucker - prager yield surfaces in eq . ( [ eq : dp])-([eq : caps ] ) illustrated in figure [ fig : gapfunction ] . we follow the smooth cap model of dolarevic and ibrahimbegovic @xcite with some corrections to the compression cap . the three yield functions are @xmath180 ^ 2}{r_{\textsc{\tiny c}}^{2}}+ j_2 - b\left ( \kappa \right)^2 \label{eq : c_cap}\end{aligned}\ ] ] the center , radius and intersection point of the tension cap are @xmath181 where the drucker - prager cone angle @xmath182 in the pressure - shear plane , @xmath183 is the tension cap cutoff and @xmath184^{1/2}$ ] . the compression cap intersection point is @xmath185 and @xmath186 and @xmath187 are the center and main radius of the compression cap ellipse @xcite . the stress gradient of the yield functions are @xmath188 lacoursire c. regularized , stabilized , variational methods for multibodies . in dag fritzson peter bunus and claus fhrer , editors , _ the 48th scandinavian conference on simulation and modeling ( sims 2007 ) , 30 - 31 october , 2007 , gteborg ( sr ) , sweden _ , linkping electronic conference proceedings , pages 4048 . linkping university electronic press , december 2007 . taheri s , sandu c. , taheri s. , pinto e. , gorsich d. a technical survey on terramechanics models for tire terrain interaction used in modeling and simulation of wheeled vehicles journal of terramechanics 57 , 1-22 , 2015 . | a method for simulation of elastoplastic solids in multibody systems with nonsmooth and multidomain dynamics is developed .
the solid is discretised into pseudo - particles using the meshfree moving least squares method .
the particles carry strain and stress tensor variables that are mapped to deformation constraints and constraint forces .
the discretised solid model thus fit a unified framework for nonsmooth multidomain dynamics for realtime simulations including strong coupling of rigid multibodies with complex kinematic constraints such as articulation joints , unilateral contacts with dry friction , drivelines and hydraulics .
the nonsmooth formulation allow for impulses , due to impacts for instance , to propagate instantly between the rigid multibody and the solid .
plasticity is introduced through an associative perfectly plastic modified drucker - prager model .
the elastic and plastic dynamics is verified for simple test systems and the capability of simulating tracked terrain vehicles driving on a deformable terrain is demonstrated . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
recent experiments@xcite on conductance fluctuations and weak - localization effects in quantum dots have stimulated theoretical work@xcite on phase - coherent conduction through cavities in which the classical electron motion can be regarded as chaotic . if the capacitance of the quantum dot is large enough , a description in terms of non - interacting electrons is appropriate ( otherwise the coulomb blockade becomes important@xcite ) . for an isolated chaotic cavity , it has been conjectured and confirmed by many examples that the statistics of the hamiltonian @xmath2 agrees with that of the gaussian ensemble of random - matrix theory.@xcite if the chaotic behavior is caused by impurity scattering , the agreement has been established by microscopic theory : both the gaussian ensemble and the ensemble of hamiltonians with randomly placed impurities are equivalent to a certain non - linear @xmath12-model.@xcite transport properties can be computed by coupling @xmath13 eigenstates of @xmath2 to @xmath1 scattering channels.@xcite since @xmath14 this construction introduces a great number of coupling parameters , whereas only a few independent parameters determine the statistics of the scattering matrix @xmath0 of the system.@xcite for transport properties at zero temperature and infinitesimal applied voltage , one only needs to know @xmath0 at the fermi energy @xmath15 , and an approach which starts directly from the ensemble of scattering matrices at a given energy is favorable . following up on earlier work on chaotic scattering in billiards,@xcite two recent papers@xcite have studied the transport properties of a quantum dot under the assumption that @xmath0 is distributed according to dyson s circular ensemble.@xcite in refs . [ barangerm ] and [ jpb ] the coupling of the quantum dot to the external reservoirs was assumed to occur via ballistic point contacts ( or `` ideal leads '' ) . the extension to coupling via tunnel barriers ( non - ideal leads ) was considered in ref . [ brouwerb ] . in all cases complete agreement was obtained with results which were obtained from the hamiltonian approach.@xcite this agreement calls for a general demonstration of the equivalence of the scattering matrix and the hamiltonian approach , for arbitrary coupling of the quantum dot to the external reservoirs . it is the purpose of this paper to provide such a demonstration . a proof of the equivalence of the gaussian and circular ensembles has been published by lewenkopf and weidenmller,@xcite for the special case of ideal leads . the present proof applies to non - ideal leads as well , and corrects a subtle flaw in the proof of ref . [ lewenkopfweidenmueller ] for the ideal case . the circular ensemble of scattering matrices is characterized by a probability distribution @xmath16 which is constant , that is to say , each unitary matrix @xmath0 is equally probable . as a consequence , the ensemble average @xmath17 is zero . this is appropriate for ideal leads . a generalization of the circular ensemble which allows for non - zero @xmath17 ( and can therefore be applied to non - ideal leads ) has been derived by mello , pereyra , and seligman,@xcite using a maximum entropy principle . the distribution function in this generalized circular ensemble is known in the mathematical literature@xcite as the poisson kernel , @xmath18 here @xmath19 is the symmetry index of the ensemble of scattering matrices : @xmath20 or @xmath21 in the absence or presence of a time - reversal - symmetry breaking magnetic field ; @xmath22 in zero magnetic field with strong spin - orbit scattering . ( in refs . [ mellopereyraseligman ] and [ melloleshouches ] only the case @xmath20 was considered . ) one verifies that @xmath23 for @xmath24 . ( [ mainres ] ) was first recognized as a possible generalization of the circular ensemble by krieger,@xcite for the special case that @xmath17 is proportional to the unit matrix . in this paper we present a microscopic justification of the poisson kernel , by deriving it from an ensemble of random hamiltonians which is equivalent to an ensemble of disordered metal grains . for the hamiltonian ensemble we can use the gaussian ensemble , or any other ensemble to which it is equivalent in the limit @xmath8.@xcite ( the microscopic justification of the gaussian ensemble only holds for @xmath8 . ) for technical reasons , we use a lorentzian distribution for the hamiltonian ensemble , which in the limit @xmath8 can be shown to be equivalent to the usual gaussian distribution . the technical advantage of the lorentzian ensemble over the gaussian ensemble is that the equivalence to the poisson kernel holds for arbitrary @xmath9 , and does not require taking the limit @xmath8 . the outline of this paper is as follows : in sec . [ sec3 ] the usual hamiltonian approach is summarized , following ref . [ vwz ] . in sec.[sec2 ] , the lorentzian ensemble is introduced . the eigenvalue and eigenvector statistics of the lorentzian ensemble are shown to agree with the gaussian ensemble in the limit @xmath8 . in sec . [ sec4 ] we then compute the entire distribution function @xmath16 of the scattering matrix from the lorentzian ensemble of hamiltonians , and show that it agrees with the poisson kernel ( [ mainres ] ) for arbitrary @xmath9 . in sec . [ sec5 ] the poisson kernel is shown to describe a quantum dot which is coupled to the reservoirs by means of tunnel barriers . we conclude in sec . the hamiltonian approach@xcite starts with a formal division of the system into two parts , the leads and the cavity ( see fig . [ fig1]a ) . the hamiltonian of the total system is represented in the following way : let the set @xmath25 represent a basis of scattering states in the lead at the fermi energy @xmath15 ( @xmath26 ) , with @xmath1 the number of propagating modes at @xmath15 . the set of bound states in the cavity is denoted by @xmath27 ( @xmath28 ) . we assume @xmath9 . the hamiltonian @xmath29 is then given by@xcite @xmath30 form a hermitian @xmath3 matrix @xmath2 , with real ( @xmath20 ) , complex ( @xmath31 ) , or real quaternion ( @xmath22 ) elements . the coupling constants @xmath32 form a real ( complex , real quaternion ) @xmath33 matrix @xmath34 . the @xmath35 scattering matrix @xmath36 associated with this hamiltonian is given by @xmath37 for @xmath7 the matrix @xmath0 is respectively unitary symmetric , unitary , and unitary self - dual . usually one assumes that @xmath2 is distributed according to the gaussian ensemble , @xmath38 with @xmath39 a normalization constant and @xmath5 an arbitrary coefficient which determines the density of states at @xmath15 . the coupling matrix @xmath34 is fixed . notice that @xmath40 is invariant under transformations @xmath41 where @xmath42 is orthogonal ( @xmath20 ) , unitary ( @xmath31 ) , or symplectic ( @xmath22 ) . this implies that @xmath16 is invariant under transformations @xmath43 , so that it can only depend on the invariant @xmath44 . the ensemble - averaged scattering matrix @xmath17 can be calculated analytically in the limit @xmath8 , at fixed @xmath1 , @xmath15 , and fixed mean level spacing @xmath45 . the result is@xcite @xmath46 it is possible to extend the hamiltonian ( [ hamham ] ) to include a `` background '' scattering matrix @xmath47 which does not couple to the cavity.@xcite the matrix @xmath47 is symmetric for @xmath48 and can be decomposed as @xmath49 , where the matrix @xmath50 is orthogonal and @xmath51 is real and diagonal . in the limit @xmath52 , the average scattering matrix @xmath17 is now given by@xcite @xmath53 lewenkopf and weidenmller@xcite used this extended version of the theory to relate the gaussian and circular ensembles , for @xmath54 and @xmath24 . their argument is based on the assumption that eq.([sbarsupsymext ] ) can be inverted , to yield @xmath44 and @xmath47 as a function of @xmath17 . then @xmath55 is fully determined by @xmath56 ( and does not require separate knowledge of @xmath44 and @xmath47 ) . under the transformation @xmath57 ( with @xmath42 an arbitrary unitary matrix ) , @xmath17 is mapped to @xmath58 , which implies @xmath59 for @xmath24 one finds that @xmath16 is invariant under transformations @xmath60 , so that @xmath16 must be constant ( circular ensemble ) . there is , however , a weak spot in this argument : equation ( [ sbarsupsymext ] ) can _ not _ be inverted for the crucial case @xmath24 . it is only possible to determine @xmath44 , not @xmath47 . this is a serious objection , since @xmath47 is not invariant under the transformation @xmath57 , and one can not conclude that @xmath23 for @xmath24 . we have not succeeded in repairing the proof of ref . [ lewenkopfweidenmueller ] for @xmath24 , and instead present in the following sections a different proof ( which moreover can be extended to non - zero @xmath17 ) . a situation in which the cavity is coupled to @xmath61 reservoirs by @xmath61 leads , having @xmath62 scattering channels ( @xmath63 ) each , can be described in the framework presented above by combining the @xmath61 leads formally into a single lead with @xmath64 scattering channels . scattering matrix elements between channels in the same lead correspond to reflection from the cavity , elements between channels in different leads correspond to transmission . in this notation , the landauer formula for the conductance @xmath65 of a cavity with two leads ( fig . [ fig1]b ) takes the form @xmath66 [ sec2 ] for technical reasons we wish to replace the gaussian distribution ( [ gaussens ] ) of the hamiltonians by a lorentzian distribution , @xmath67 where @xmath5 and @xmath6 are parameters describing the width and center of the distribution , and @xmath39 is a normalization constant independent of @xmath5 and @xmath6 . the symmetry parameter @xmath19 indicates whether the matrix elements of @xmath2 are real [ @xmath20 , lorentzian orthogonal ensemble ( @xmath68oe ) ] , complex [ @xmath31 , lorentzian unitary ensemble ( @xmath68ue ) ] , or real quaternion [ @xmath22 , lorentzian symplectic ensemble ( @xmath68se ) ] . ( we abbreviate `` lorentzian '' by a capital lambda , because the letter @xmath69 is commonly used to denote the laguerre ensemble . ) the replacement of ( [ gaussens ] ) by ( [ lorens ] ) is allowed because the eigenvector and eigenvalue distributions of the gaussian and the lorentzian ensemble are equal on a fixed energy scale , in the limit @xmath8 at a fixed mean level spacing @xmath45 . the equivalence of the eigenvector distributions is obvious : the distribution of @xmath2 depends solely on the eigenvalues for both the lorentzian and the gaussian ensemble , so that the eigenvector distribution is uniform for both ensembles . in order to prove the equivalence of the distribution of the eigenvalues @xmath70 ( energy levels ) , we compare the @xmath61-level cluster functions @xmath71 for both ensembles . the general definition of the @xmath72 s is given in ref . [ mehta ] . the first two @xmath72 s are defined by @xmath73 the brackets @xmath74 denote an average over the ensemble . the cluster functions in the gaussian ensemble are known for arbitrary @xmath61,@xcite for the lorentzian ensemble we compute them below . from eq . ( [ lorens ] ) one obtains the joint probability distribution function of the eigenvalues , @xmath75 we first consider the case @xmath76 , @xmath77 . we make the transformation @xmath78 the eigenvalues @xmath79 of the unitary matrix @xmath0 are related to the energy levels @xmath80 by @xmath81 the probability distribution of the eigenphases follows from eqs.([lorense ] ) and ( [ eigenphaseigenval1 ] ) , @xmath82 this is precisely the distribution of the eigenphases in the circular ensemble . the cluster functions in the circular ensemble are known.@xcite the @xmath61-level cluster functions @xmath83 in the lorentzian ensemble are thus related to the @xmath61-level cluster functions @xmath84 in the circular ensemble by @xmath85 for @xmath86 one finds the level density @xmath87 independent of @xmath88 . for @xmath89 one finds the pair - correlation function @xmath90 eq . ( [ pairlor ] ) holds for @xmath31 . the expressions for @xmath91 are more complicated . the @xmath61-level cluster functions for arbitrary @xmath5 and @xmath6 can be found after a proper rescaling of the energies . ( [ kappae1 ] ) generalizes to @xmath92 the large-@xmath13 limit of the @xmath72 s is defined as @xmath93 for both the gaussian and the lorentzian ensembles , the mean level spacing @xmath45 at the center of the spectrum in the limit @xmath8 is given by @xmath94 . therefore , the relevant limit @xmath95 at fixed level spacing is given by @xmath8 , @xmath96 , @xmath94 fixed for both ensembles . equation ( [ lorclufunc ] ) allows us to relate the @xmath97 s in the lorentzian and circular ensembles , @xmath98 it is known that the cluster functions @xmath99 in the circular ensemble are equal to the cluster functions @xmath100 in the gaussian ensemble.@xcite equation ( [ clusterlc ] ) therefore shows that the lorentzian and the gaussian ensembles have the same cluster functions in the large-@xmath13 limit . the technical reason for working with the lorentzian ensemble instead of with the gaussian ensemble is that the lorentzian ensemble has two properties which make it particularly easy to compute the distribution of the scattering matrix . the two properties are : + * property 1 : * if @xmath2 is distributed according to a lorentzian ensemble with width @xmath5 and center @xmath6 , then @xmath101 is again distributed according to a lorentzian ensemble , with width @xmath102 and center @xmath103 . + * property 2 : * if the @xmath3 matrix @xmath2 is distributed according to a lorentzian ensemble , then every @xmath35 submatrix of @xmath2 obtained by omitting @xmath104 rows and the corresponding columns is again distributed according to a lorentzian ensemble , with the same width and center . + the proofs of both properties are essentially contained in ref . [ hua ] . in order to make this paper self - contained , we briefly give the proofs in the appendix . the general relation between the hamiltonian @xmath2 and the scattering matrix @xmath0 is given by eq . ( [ sheq ] ) . after some matrix manipulations , it can be written as @xmath105 we can write the coupling matrix @xmath34 as @xmath106 where @xmath42 is an @xmath3 orthogonal ( @xmath20 ) , unitary ( @xmath31 ) , or symplectic ( @xmath22 ) matrix , @xmath107 is an @xmath35 matrix , and @xmath108 is an @xmath33 matrix with all elements zero except @xmath109 , @xmath110 . substitution into eq . ( [ sheq ] ) gives @xmath111 where we have defined @xmath112 . we assume that @xmath2 is a member of the lorentzian ensemble , with width @xmath5 and center @xmath113 . then the matrix @xmath114 is also a member of the lorentzian ensemble , with width @xmath5 and center @xmath15 . property 1 implies that @xmath115 is distributed according to a lorentzian ensemble with width @xmath116 and center @xmath117 . orthogonal ( unitary , symplectic ) invariance of the lorentzian ensemble implies that @xmath118 has the same distribution as @xmath119 . using property 2 we then find that @xmath120 [ being an @xmath35 submatrix of @xmath121 is distributed according to the same lorentzian ensemble ( width @xmath122 and center @xmath123 ) . we now compute the distribution of the scattering matrix , first for a special coupling , then for the general case . first we will consider the special case that @xmath124 is proportional to the unit matrix . the relation ( [ snn ] ) between the @xmath0 and @xmath120 is then @xmath125 thus the eigenvalues @xmath126 of @xmath120 and @xmath79 of @xmath0 are related via @xmath127 since transformations @xmath128 ( with arbitrary orthogonal , unitary , or symplectic @xmath35 matrix @xmath42 ) leave @xmath129 invariant , @xmath16 is also invariant under @xmath130 . so @xmath16 can only depend on the eigenvalues @xmath79 of @xmath0 . the distribution of the @xmath131 s is [ cf . ( [ lorense ] ) ] @xmath132 from eqs . ( [ eigenphaseigenval ] ) and ( [ eigenvaltildeh ] ) we obtain the probability distribution of the @xmath133 s , [ ke ] @xmath134 eq . ( [ ke ] ) implies that @xmath16 has the form of a poisson kernel , @xmath135 the average scattering matrix @xmath17 being given by @xmath136 now we turn to the case of arbitrary coupling matrix @xmath137 . we denote the scattering matrix at coupling @xmath137 by @xmath0 , and denote the scattering matrix at the special coupling ( [ wspecial ] ) by @xmath47 . the relation between @xmath0 and @xmath47 is @xmath138 where we abbreviated [ stransformw ] @xmath139 the symmetry of the coupling matrix @xmath137 is reflected in the symmetry of the @xmath140 matrix @xmath141 which is unitary symmetric ( @xmath20 ) , unitary ( @xmath31 ) or unitary self - dual ( @xmath22 ) . the probability distribution @xmath142 of @xmath47 is given by eq . ( [ p0distr ] ) . the distribution @xmath143 of @xmath0 follows from [ probrel ] @xmath144 where the jacobian @xmath145 is the ratio of infinitesimal volume elements around @xmath47 and @xmath0 . this jacobian is known,@xcite @xmath146 after expressing @xmath47 in terms of @xmath0 by means of eq . ( [ s0froms ] ) , we find that @xmath16 is given by the same poisson kernel as eq . ( [ p0distr ] ) , but with a different @xmath17 , @xmath147 in the limit @xmath8 at fixed level spacing @xmath148 , eq . ( [ sbarlorensarbw ] ) simplifies to @xmath149 the extended version of the hamiltonian approach which includes a background scattering matrix @xmath47 can be mapped to the case without background scattering matrix by a transformation @xmath150 ( @xmath20 ) , @xmath151 ( @xmath31 ) , or @xmath152 ( @xmath153 ) , where @xmath42 and @xmath39 are unitary matrices.@xcite ( @xmath154 is the transposed of @xmath42 , @xmath155 is the dual of @xmath42 . ) the poisson kernel is covariant under such transformations,@xcite i.e. it maps to a poisson kernel with @xmath156 ( @xmath20 ) , @xmath157 ( @xmath31 ) , or @xmath158 ( @xmath22 ) . as a consequence , the distribution of @xmath0 is given by the poisson kernel for arbitrary coupling matrix @xmath34 and background scattering matrix @xmath47 . this proves the general equivalence of the poisson kernel and the lorentzian ensemble of hamiltonians . the circular ensemble of scattering matrices is appropriate for a chaotic cavity which is coupled to the leads by means of ballistic point contacts ( `` ideal '' leads ) . in this section we will demonstrate that the generalized circular ensemble described by the poisson kernel is the appropriate ensemble for a chaotic cavity which is coupled to the leads by means of tunnel barriers ( `` non - ideal '' leads ) . the system considered is shown schematically in fig . we assume that the segment of the lead between the tunnel barrier and the cavity is long enough , so that both the @xmath35 scattering matrix @xmath47 of the cavity and the @xmath159 scattering matrix @xmath160 of the tunnel barrier are well - defined . the scattering matrix @xmath47 has probability distribution @xmath161 of the circular ensemble , whereas the scattering matrix @xmath160 is kept fixed . we decompose @xmath160 in terms of @xmath35 reflection and transmission matrices , @xmath162 the @xmath35 scattering matrix @xmath0 of the total system is related to @xmath47 and @xmath160 by @xmath163 this relation has the same form as eq . ( [ sfroms0 ] ) . we can therefore directly apply eq . ( [ probrel ] ) , which yields @xmath164 hence @xmath0 is distributed according to a poisson kernel , with @xmath165 . in conclusion we have established by explicit computation the equivalence for @xmath9 of a generalized circular ensemble of scattering matrices ( described by a poisson kernel ) and an ensemble of @xmath3 hamiltonians with a lorentzian distribution . the lorentzian and gaussian distributions are equivalent in the large-@xmath13 limit . moreover , the gaussian hamiltonian ensemble and the microscopic theory of a metal particle with randomly placed impurities give rise to the same non - linear @xmath12-model.@xcite altogether , this provides a microscopic justification of the poisson kernel in the case that the chaotic motion in the cavity is caused by impurity scattering . for the case of a ballistic chaotic cavity , a microscopic justification is still lacking . the equivalence of the poisson kernel and an arbitrary hamiltonian ensemble can be reformulated in terms of a central limit theorem : the distribution of a submatrix of @xmath101 of fixed size @xmath1 tends to a lorentzian distribution when @xmath8 , independent of the details of the distribution of @xmath2 . a central limit theorem of this kind for @xmath166 has previously been formulated and proved by mello.@xcite this work was motivated by a series of lectures by p. a. mello at the summerschool in les houches on `` mesoscopic quantum physics '' . discussions with c. w. j. beenakker , k. frahm , p. a. mello , and h. a. weidenmller are gratefully acknowledged . this research was supported by the `` stichting voor fundamenteel onderzoek der materie '' ( fom ) and by the `` nederlandse organisatie voor wetenschappelijk onderzoek '' ( nwo ) . the two proofs given below are adapted from ref . [ hua ] . the matrix @xmath2 and its inverse @xmath101 have the same eigenvectors , but reciprocal eigenvalues . therefore , property 1 of the lorentzian ensemble is proved by showing that the distribution of the eigenvalues of @xmath101 is given by eq . ( [ lorense ] ) , with the substitutions @xmath167 and @xmath168 . this is easily done , @xmath169 \nonumber \\ & = & { 1 \over v } \lambda^{m(\beta m + 2 - \beta)/2 } \prod_{i < j } \left|e_i e_j ( e_i^{-1 } - e_j^{-1 } ) \right|^{\beta } \prod_i \left [ \left({\lambda^2 + ( e_i - \varepsilon)^2}\right)^{-(\beta m + 2 - \beta)/2 } e_i^{2 } \right ] \nonumber \\ & = & { 1 \over v } \lambda^{m(\beta m + 2 - \beta)/2 } \prod_{i < j } \left|e_i^{-1 } - e_j^{-1}\right|^{\beta } \prod_i \left({\lambda^2 e_i^{-2 } + ( 1 - \varepsilon e_i^{-1})^2}\right)^{-(\beta m + 2 - \beta)/2 } \nonumber \\ & = & { 1 \over v } \tilde \lambda^{m(\beta m + 2 - \beta)/2}\prod_{i < j } \left|e_i^{-1 } - e_j^{-1}\right|^{\beta } \prod_i \left({\tilde \lambda^2 + ( e_i^{-1 } - \tilde \varepsilon)^2}\right)^{-(\beta m + 2 - \beta)/2}.\end{aligned}\ ] ] in order to prove property 2 , we may assume that after rescaling of @xmath2 we have @xmath76 , @xmath77 . first consider @xmath170 . in this case , one can write @xmath171 where @xmath65 is the @xmath35 submatrix of @xmath2 whose distribution we want to compute , @xmath172 is a vector , with real ( @xmath20 ) , complex ( @xmath31 ) , or real quaternion elements ( @xmath22 ) , and @xmath173 is a real number . for the successive integrations over @xmath173 and @xmath172 we need two auxiliary results . first , for real numbers @xmath174 , @xmath175 , @xmath176 such that @xmath177 and @xmath178 , and for real @xmath179 we have @xmath180 second , if @xmath181 is a @xmath182-dimensional vector with real components , and if @xmath183 , then @xmath184 since @xmath185 is a quadratic function of @xmath173 , the integral over @xmath173 can now be carried out using eq . ( [ lemma1eq ] ) . the result is : @xmath186 next , we integrate over @xmath172 . we may choose the basis for the @xmath172-vectors so that @xmath187 is diagonal , with diagonal elements @xmath188 . after rescaling of the @xmath172-vectors to @xmath189 one obtains an integral similar to eq . ( [ lemma2eq ] ) , with @xmath190 . the final result is @xmath191 property 2 now follows by induction . notice that eq . ( [ finproof2 ] ) allows us to determine the normalization constant @xmath39 , @xmath192 c. m. marcus , a. j. rimberg , r. m. westervelt , p. f. hopkins , and a. c. gossard , phys . 69 * , 506 ( 1992 ) ; c. m. marcus , r. m. westervelt , p. f. hopkins , and a. c. gossard , phys . b * 48 * , 2460 ( 1993 ) ; chaos * 3 * , 643 ( 1993 ) . m. w. keller , o. millo , a. mittal , d. e. prober , and r. n. sacks , surf . sci . * 305 * , 501 ( 1994 ) . a. m. chang , h. u. baranger , l. n. pfeiffer , and k. w. west , phys . lett . * 73 * , 2111 ( 1994 ) . j. p. bird , k. ishibashi , y. aoyagi , t. sugano , and y. ochiai , preprint . r. a. jalabert , a. d. stone , and y. alhassid , phys . lett . , 3468 ( 1992 ) . h. u. baranger , r. a. jalabert , and a. d. stone , phys . rev.lett . * 70 * , 3876 ( 1993 ) ; chaos * 3 * , 665 ( 1993 ) . v. n. prigodin , k. b. efetov and s. iida , phys . lett . * 71 * , 1230 ( 1993 ) . [ pei ] [ barangerm ] h. u. baranger and p. a. mello , phys.rev . lett . * 73 * , 142 ( 1994 ) . [ jpb ] r. a. jalabert , j .- l . pichard , and c. w. j. beenakker , europhys . * 27 * , 255 ( 1994 ) . z. pluhr , h. a. weidenmller , j. a. zuk , and c. h. lewenkopf , phys . lett . * 73 * , 2115 ( 1994 ) . p. w. brouwer and c. w. j. beenakker , phys . b * 50 * , 11263 ( 1994 ) . [ brouwerb ] e. r. mucciolo , v. n. prigodin , and b. l. altshuler , preprint ( cond - mat/9406048 ) . o. bohigas , m. j. giannoni , and c. schmit , phys . rev.lett . * 52 * , 1 ( 1984 ) ; o. bohigas in _ chaos and quantum physics _ , edited by m .- giannoni , a. voros , and j. zinn - justin ( north - holland , amsterdam , 1991 ) . m. v. berry , proc . r. soc . london a * 400 * , 229 ( 1985 ) . k. b. efetov , adv . phys . * 32 * , 53 ( 1983 ) . j. j. m. verbaarschot , h. a. weidenmller , and m. r. zirnbauer , phys . rep . * 129 * , 367 ( 1985 ) . [ vwz ] s. iida , h. a. weidenmller , and j. zuk , phys . rev . lett . , 583 ( 1990 ) ; ann . * 200 * , 219 ( 1990 ) . a. altland , z. phys . b. * 82 * , 105 ( 1991 ) . c. h. lewenkopf and h. a. weidenmller , ann . * 212 * , 53 ( 1991 ) . [ lewenkopfweidenmueller ] r. blmel and u. smilansky , phys . , 477 ( 1988 ) ; _ ibid . _ * 64 * , 241 ( 1990 ) ; u. smilansky , in _ chaos and quantum physics _ , edited by m .- giannoni , a. voros , and j. zinn - justin ( north - holland , amsterdam , 1991 ) . f. j. dyson , j. math . phys . * 3 * , 140 ( 1962 ) . [ mehta ] m. l. mehta , _ random matrices _ ( academic , new york , 1991 ) . p. a. mello , p. pereyra , and t. h. seligman , ann . ( n.y . ) * 161 * , 254 ( 1985 ) . [ mellopereyraseligman ] p. a. mello , in _ mesoscopic quantum physics _ , edited by e. akkermans , g. montambaux , j .- l . pichard , and j. zinn - justin ( north - holland , amsterdam , to be published ) . [ melloleshouches ] [ hua ] l. k. hua , _ harmonic analysis of functions of several complex variables in the classical domains _ ( amer . math . soc.,providence , 1963 ) . t. j. krieger , ann . * 42 * ( 1967 ) , 375 . g. hackenbroich and h. a. weidenmller , preprint ( cond - mat/9412088 ) . c. mahaux and h. a. weidenmller , _ shell - model approach to nuclear reactions _ ( north - holland , amsterdam , 1969 ) . h. nishioka and h. a. weidenmller , phys . b * 157 * , 101 ( 1985 ) . [ nishiokaweidenmueller ] [ friedmanmello ] w. a. friedman and p. a. mello , ann . phys . * 161 * , 276 ( 1985 ) . | we consider the problem of the statistics of the scattering matrix @xmath0 of a chaotic cavity ( quantum dot ) , which is coupled to the outside world by non - ideal leads containing @xmath1 scattering channels .
the hamiltonian @xmath2 of the quantum dot is assumed to be an @xmath3 hermitian matrix with probability distribution @xmath4^{-(\beta m + 2 - \beta)/2}$ ] , where @xmath5 and @xmath6 are arbitrary coefficients and @xmath7 depending on the presence or absence of time - reversal and spin - rotation symmetry .
we show that this `` lorentzian ensemble '' agrees with microscopic theory for an ensemble of disordered metal particles in the limit @xmath8 , and that for any @xmath9 it implies @xmath10 is the ensemble average of @xmath0 .
this `` poisson kernel '' generalizes dyson s circular ensemble to the case @xmath11 and was previously obtained from a maximum entropy approach .
the present work gives a microscopic justification for the case that the chaotic motion in the quantum dot is due to impurity scattering . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
in the past two decades quantum computation has attracted growing interest , encouraged by the development of tools for the manipulation of single quantum objects as well as by several remarkable theoretical findings @xcite . different systems have been proposed as candidates for quantum computing ; they are based , for instance , on cavity - laser atoms , bose - einstein condensates , or nmr techniques ( see e.g. @xcite ) . at the same time , a number of quantum algorithms have been designed and some have been shown to be even exponentially faster than their best classical counterparts @xcite . in particular , quantum search algorithms , although able to achieve `` only '' a polynomial speedup , have been proved to be very promising and of widespread use in quantum computation @xcite . one of the best known quantum search algorithms is attributed to grover @xcite . the algorithm can find a target within an unsorted database made up of @xmath0 items using @xmath1 queries . due to its broad range of applications and its ability to be effectively used as a subroutine @xcite , grover s algorithm has been thoroughly investigated and a number of different implementations have been proposed ( see e.g. @xcite ) . recently , implementations based on continuous - time @xcite as well as on discrete - time quantum walks @xcite have been introduced : given that the application of random walks in classical algorithms provided significant advantages for approximations and optimization , one is strongly motivated to study quantum walks as algorithmic tools . at the current stage the aim is not only to achieve similar computational improvements , but also to understand the capabilities of quantum computations . here we focus on the approach pioneered by farhi and gutmann @xcite and further developed by childs and goldstone @xcite , based on continuous - time quantum walks ( ctqws ) . this implementation provides several important advantages : the algorithm does not need auxiliary storage space and it makes it possible to take into account the geometrical arrangement of the database ( the latter being either a physical position space or an efficiently encoded hilbert space @xcite ) . indeed , while previous studies just considered the cases of the translationally invariant @xmath2 ( named @xmath3-dimensional periodic lattices by childs and goldstone @xcite and called hypercubic lattices in statistical physics , the term which we adopt here ) and of complete graphs @xcite , here we extend the investigations to the case of generic structures and analyze how geometrical parameters [ e.g. , ( fractal ) dimension or ( average ) coordination number ; see later in this article ] affect the dynamics of the ctqw . in @xcite it was shown that quantum searches based on ctqws recover the optimal quadratic speedup on complete graphs and on high - dimensional hypercubic lattices ( with dimension @xmath4 ) , while for low - dimensional ( @xmath5 ) lattices ctqws can not outperform their classical counterpart . hence , it may appear that @xmath6 works as a `` critical dimension '' , separating highly performing structures from poorly performing ones . we will show , both analytically and numerically , that the dimension of the substrate is not sufficient for getting a sharp transition from the ground to the first excited state ; we also take into account the success probability @xmath7 , that is , the probability of finding the quantum walker at the target site @xmath8 at time @xmath9 , given as initial state the equally weighted superposition @xmath10 : an efficient quantum walk gives rise to a success probability close to @xmath11 already at very small times @xmath9 . in particular , we will take into account several kinds of structures : translationally invariant structures , such as @xmath3-dimensional hypercubic lattices , complete graphs , fractals with low fractal dimension like dual sierpinski gaskets and t - fractals , hierarchical structures as cayley trees , and structures with ( fractal ) dimensions larger than four , such as cartesian products between euclidean lattices and dual sierpinski gaskets ( for the precise definitions of these structures and of the fractal dimension considered here , see later in this article ) . in this way we are able to show that for translationally invariant structures with high dimensions , @xmath7 displays sharp peaks , while for fractals or low - dimensional structures the peaks are low and broad so that the quantum walk is not particularly effective in the sense that there exists only a low probability @xmath7 for any @xmath9 . moreover , for any structure , we evidence interference phenomena which give rise to a non - monotonic time dependence of the @xmath7 ; such effects can be significant and must be properly taken into account when considering applications . interestingly , the ctqw hamiltonian used in the quantum search also describes , in solid - state physics , the dynamics of a tight - binding particle in the presence of static , substitutional impurities . in this context our results show that in regular , highly - connected geometries ( such as the high - dimensional tori ) the probability of finding the moving particle at the impurity site is ( quasi- ) periodic in time and that the localization can be very effective . our article is structured as follows . in sec . [ sec : quantum_search ] we first review basic principles concerning grover s search and we explain how it can be implemented by means of ctqws . in sec . [ ssec : inho_graphs ] we describe the structures used as substrate for the ctqw . then , in sec . [ sec : phase_transition ] we present several analytical results , later corroborated and deepened in sec . [ sec : numerics ] , where our numerical results are shown . in sec . [ sec : success ] we focus on the success probability . finally , sec . [ sec : conclusions ] contains our conclusions and discussion . in appendix [ appe1 ] , appendix [ appe2 ] and appendix [ appe3 ] we report the details of our analytical calculations . grover s search algorithm @xcite is meant to solve the unsorted search problem under the assumption that there exists a computational oracle working as a black - box function able to decide whether a candidate solution is the true solution . hence , the oracle knows which is the target among the @xmath0 entries . the task is to find a target @xmath8 using the fewest calls to the oracle . while the classical algorithm requires exhaustive searches implying @xmath12 queries , grover s algorithm is able to find @xmath8 using @xmath1 queries , giving rise to a quadratic speed up @xcite . a short outline of the idea behind grover s algorithm in the presence of a single marked target is as follows : first of all , one associates each of the @xmath0 index integers with a unique orthonormal vector @xmath13 in an @xmath0-dimensional hilbert space . then , one chooses as initial state @xmath14 which is delocalized over the entire set of states @xmath15 with equal weights at every site @xmath16 . this is the least biased initialization one can arrange , given the available information ( since each of the @xmath0 nodes is in principle equally likely to be the target index , the initial state is prepared as an equally weighted superposition of all @xmath0 indices ) . now , to perform the search , one needs to make the state @xmath17 evolve into a state that has almost all the amplitude concentrated in just the @xmath18 component . in such a way a single final measurement will find the system in the state @xmath19 , hence revealing the identity of the target index . more precisely , we need an evolution operator whose repeated application makes the amplitude of @xmath18 grow with the number of iterations @xcite . this task was originally accomplished within the standard paradigm for quantum computation @xcite , namely using a discrete sequence of unitary logic gates , while in the last years several different implementations have been introduced in which the state of the quantum register evolves continuously under the influence of a driving hamiltonian @xcite . in particular , here we follow the approach developed by childs and goldstone @xcite which relies on ctqw @xcite . as already mentioned , due to their versatility , ctqws allow to model any discrete database . in fact , a generic discretizable database can be represented by a graph @xmath20 made up of a set of nodes @xmath21 , each corresponding to a different item , and of a set of links @xmath22 joining nodes pairwise in such a way that the topology of the graph mirrors the arrangement of the database . the graph @xmath23 can be algebraically described by the adjacency matrix @xmath24 whose entry @xmath25 equals one if nodes @xmath26 and @xmath27 are connected , otherwise it is zero ( also for the diagonal elements ) . from @xmath24 one can directly calculate the degree matrix @xmath28 , which is a diagonal matrix with elements @xmath29 , where @xmath30 is the coordination number ( or degree ) of the @xmath26-th node , that is , the number of its nearest neighbors . ctqws on a graph are defined by the laplacian matrix @xmath31 and obey the following schrdinger equation for the transition amplitude @xmath32 from state @xmath33 to state @xmath34 @xcite : @xmath35 where the hamiltonian is just given by @xmath36 , therefore , the time is dimensionless and given in units of the coupling elements @xmath37 . the formal solution for @xmath32 can be written as @xmath38 and its magnitude squared provides the quantum mechanical transition probability @xmath39 . notice that @xmath40 is symmetric and non - negative definite ; its ground state , corresponding to eigenvalue @xmath41 , is given by @xmath42 . due to the unitary time - evolution generated by @xmath43 , ctqws are symmetric under time - inversion , which precludes @xmath44 from attaining equipartition at long times . this is different from the behavior of classical continuous - time random walks . moreover , ctqws keep memory of the initial conditions , as exemplified by the occurrence of ( quasi- ) revivals @xcite . now , the `` oracle hamiltonian '' is given by @xmath45 whose ground state , with energy @xmath46 , is just the marked item @xmath47 ; all other states have energy zero @xcite . then , the hamiltonian @xmath43 governing the time evolution of the quantum walk is @xmath48 where @xmath49 is a tunable parameter with units of inverse time , hence dimensionless here . the _ success probability _ @xmath50 is defined as ( see also @xcite ) @xmath51 i.e. , as the transtion probability to be at time @xmath9 at the target @xmath8 when starting in the delocalized state @xmath52 . here we study its dependence on time @xmath9 and on the parameter @xmath49 and we especially aim to evidence the existence of an optimization parameter @xmath53 , possibly depending on time , which maximizes @xmath50 . notice that , having fixed @xmath49 , the time @xmath9 can be measured in terms of the number of queries of the discrete grover oracle @xcite . before proceeding , it is worth introducing the geometrical structures on which we are focusing , first the dual sierpinski gasket ( dsg ) , the t - fractal ( tf ) , and the cayley tree ( ct ) ; examples of these structures are shown in fig . [ fig : frattali]a , fig . [ fig : frattali]b , and fig . [ fig : frattali]c , respectively . notice that these structures differ significantly from those analyzed previously : while hypercubic lattices with periodic boundary conditions ( toroids ) are translationally invariant , the aforementioned structures are not . the dsg , tf and ct can be built iteratively ; at the @xmath54-th iteration we have the fractal of generation @xmath54 ( see e.g. @xcite ) . the dsg and the tf are examples of exactly decimable fractals for which the fractal dimension , @xmath55 , and the spectral dimension , @xmath56 , are exactly known . while the fractal dimension relates the number of nodes inside a sphere to the radius of the sphere @xcite , the spectral dimension is obtained from the scaling of the eigenmodes of a given structure ( phonons for lattices , fractons for fractal substrates ) @xcite , most simply seen in the probability of a random walker to be ( still or again ) at the original site @xcite . here , we have namely @xmath57 and @xmath58 for the dsg , and @xmath59 and @xmath60 for the tf . we recall that for the usual , translationally invariant lattices the spectral and fractal dimensions coincide with the euclidean dimension @xmath3 , namely @xmath61 for the chain , @xmath62 for the square lattice and so on . on the other hand , on fractal structures @xmath56 often replaces @xmath3 when dealing with dynamical and thermodynamical properties @xcite . also , for fractals @xmath63 and @xmath55 is smaller than the euclidean dimension of the embedding space , that is , 2 for dsg and tf . the ct is no fractal in the classical sense , since its growth with increasing generation is exponential . however , the ct is built in a hierarchical manner , analogous to the tf . we also notice that both , the tf and the ct , are trees ( and are hence devoid of loops ) and exhibit a large number @xmath64 of end nodes . examples of fractal structures considered here ; in general the star indicates the position of the trap . panel @xmath65 : dual sierpinski gasket of generation @xmath66 and volume @xmath67 ; due to symmetry , the three corners are equivalent . panel @xmath68 : t - fractal of generation @xmath69 and volume @xmath70 ; due to the symmetry all outmost sites ( at distance @xmath71 from the central node ) are equivalent . panel @xmath72 : cayley tree of generation @xmath73 and volume @xmath74 ; all of the @xmath75 outmost sites are equivalent.,title="fig:",width=207,height=181 ] examples of fractal structures considered here ; in general the star indicates the position of the trap . panel @xmath65 : dual sierpinski gasket of generation @xmath66 and volume @xmath67 ; due to symmetry , the three corners are equivalent . panel @xmath68 : t - fractal of generation @xmath69 and volume @xmath70 ; due to the symmetry all outmost sites ( at distance @xmath71 from the central node ) are equivalent . panel @xmath72 : cayley tree of generation @xmath73 and volume @xmath74 ; all of the @xmath75 outmost sites are equivalent.,title="fig:",width=207,height=181 ] examples of fractal structures considered here ; in general the star indicates the position of the trap . panel @xmath65 : dual sierpinski gasket of generation @xmath66 and volume @xmath67 ; due to symmetry , the three corners are equivalent . panel @xmath68 : t - fractal of generation @xmath69 and volume @xmath70 ; due to the symmetry all outmost sites ( at distance @xmath71 from the central node ) are equivalent . panel @xmath72 : cayley tree of generation @xmath73 and volume @xmath74 ; all of the @xmath75 outmost sites are equivalent.,title="fig:",width=207,height=181 ] as we will show in the following , ctqw on the dsg , the tf and the ct can display a low probability @xmath76 for any @xmath9 and @xmath49 when compared with the case of the translationally invariant lattices , especially when the dimension of the lattice becomes larger than @xmath66 . therefore one can ask whether the probabilities @xmath50 can be improved by adopting ( fractal or hierarchical ) substrates which display a large spectral dimension and a large coordination number . for instance , we can build up such structures by combining the dsg and euclidean lattices by means of cartesian products . in general , the cartesian product of two graphs @xmath77 and @xmath78 is a graph @xmath79 with the vertex set @xmath80 , and such that two nodes @xmath81 are adjacent if @xmath82 and @xmath83 , or @xmath84 and @xmath85 . it has been shown @xcite that the spectral dimension @xmath56 on the product graph @xmath23 is then the sum of the corresponding dimensions of the two initial graphs @xmath86 and @xmath87 . we combine in this way the dsg with the chain @xmath88 , with the square lattice @xmath89 and with the cubic lattice @xmath90 , to obtain more complex structures displaying spectral dimensions approximately equal to @xmath91 , @xmath92 and @xmath93 , respectively . finally , it should be underlined that , dealing with such structures , the location of the target , i.e. the node @xmath8 , also ( quantitatively ) matters . in the numerical analysis of sec . [ sec : numerics ] we place the target on a peripheral site , which means , without loss of generality , the apex for the dsg , the leftmost site for the tf and an outmost site for the ct ( see fig.[fig : frattali ] ) . we expect that a peripheral position for the target site does not correspond to an optimal situation and , since a priori the target position is unknown , this choice prevents an overestimation of the probability of finding the target . let us consider the hamiltonian of eq . ( [ eq : hamiltonian ] ) and denote the corresponding set of eigenstates and eigenvalues by @xmath94 and @xmath95 , respectively . now , for large @xmath49 the contribution of @xmath96 to @xmath43 is negligible and the ground state @xmath97 is close to @xmath52 . on the other hand , as @xmath98 the ground state is close to @xmath99 since the weight of @xmath40 is small and , from perturbation theory , we expect that @xmath42 is close to @xmath100 , i.e. to the first excited state of @xmath43 @xcite . as pointed out by childs and goldstone @xcite , there exists an intermediate range of @xmath49 where , for complete graphs and hypercubic lattices with dimension @xmath4 , @xmath17 switches from the ground state @xmath101 to the first excited state @xmath100 ; in the very same region of @xmath49 the target state @xmath18 switches from a state with large overlap with @xmath100 to the ground state @xmath102 . therefore , by varying @xmath49 , we can find a particular value for @xmath49 for which the hamiltonian @xmath43 can evolve the state @xmath17 into a state close to the target state @xmath19 . the existence and the narrowness of such a range for @xmath49 are crucial for a large success probability . as anticipated , the ctqw under study can yield effective results if the hamiltonian @xmath43 is able to rotate the state @xmath42 to a state with a large overlap with @xmath18 . for this to occur a first condition which needs to be fulfilled is that there exists an intermediate range of @xmath49 over which the state @xmath42 has a substantial overlap with both @xmath102 and @xmath100 . in this subsection the occurrence of such a condition is investigated analytically for an arbitrary structure , following arguments similar to those exploited in @xcite for @xmath3-dimensional hypercubic lattices ; the details of the calculations are given in appendixes a and b while here we just report the main results . first of all , we define @xmath103 and @xmath104 , the @xmath105-th eigenstate and eigenvalue of the laplacian @xmath40 , respectively . in the basis of the eigenstates @xmath103 the target site @xmath19 can be written as @xmath106 where @xmath107 . as derived in appendix [ appe1 ] , once latexmath:[$\xi_j \equiv \sum_{k \neq 0 } @xmath42 with the ground state @xmath102 or with the first non - degenerate excited state @xmath100 is ( depending on @xmath49 ) limited from below by the same bound . we find namely @xmath109 , and also @xmath110 for @xmath111 . now , we define the ( size dependent ) critical value @xmath112 as that value of @xmath49 for which latexmath:[\[\label{eq : intersection } satisfied , i.e. , the @xmath49-value for which the projection of state @xmath10 onto the ground and the first excited state has the same magnitude . this is particularly interesting in the limit @xmath114 , when a level crossing from state @xmath102 to state @xmath100 can occur . in fact , we find that if @xmath115 $ ] converges ( at least in the limit @xmath116 ) to @xmath41 as @xmath49 approaches @xmath117 ( from different sides ) , then @xmath118 and @xmath119 both approach @xmath11 ( see eqs . ( [ eq : s_0_short ] ) and ( [ eq : s_l_short ] ) ) , namely a transition occurs at @xmath120 . notice , however , that the condition for this to occur is in general non - trivial as @xmath117 and @xmath121 both depend on @xmath0 . in appendix [ appe1 ] we find a sufficient condition in the laplacian spectrum and , in particular , in appendix [ appe2 ] we prove that such a condition holds for the dsg for which @xmath104 is exactly known @xcite ; in this case we find that @xmath122 and @xmath123 where @xmath124 is a parameter depending on the given network . in general , @xmath125 ; for hypercubic lattices @xmath126 , regardless of their dimension , while for the dsg we can numerically estimate @xmath124 as being @xmath127 ( see appendix [ appe1 ] ) . therefore , we expect ( for dsg ) @xmath112 to be , approximately : @xmath128 this result is consistent up to logarithmic corrections with the critical points found in @xcite for the linear chain and the square lattice , namely @xmath129 and @xmath130 , respectively . finally , we point out that the critical parameter @xmath131 provides interesting information in the context of quantum adiabatic computation @xcite : @xmath131 represents a threshold below which we can expect @xmath18 to have a large overlap with the ground state . we now consider the three examples of low - dimensional inhomogeneous structures described previously , for which the overlaps of the initial state @xmath17 and of the target state @xmath19 with @xmath132 and @xmath133 are shown in figs . [ fig : ds_ext]-[fig : ct_ext ] for different generations @xmath54 , as a function of the parameter @xmath49 . these plots evidence that there exists an intermediate range of @xmath49 where the state @xmath17 changes from having a large overlap with the first excited state to having a large overlap with the ground state . in the same region of @xmath49 the overlap @xmath134 is significant for structures of small size ( top panels in figs . [ fig : ds_ext]-[fig : ct_ext ] ) , while it is still very small when the size is large ( bottom panels in figs . [ fig : ds_ext]-[fig : ct_ext ] ) . this is vastly distinct from the situation found for hypercubic lattices @xcite where close to @xmath131 the overlap @xmath134 is significant . ( color online ) overlaps for a dsg of generation @xmath135 ( up ) and @xmath136 ( bottom ) as a function of ( the dimensionless ) @xmath49 , see text for details.,title="fig:",width=302,height=264 ] + ( color online ) overlaps for a dsg of generation @xmath135 ( up ) and @xmath136 ( bottom ) as a function of ( the dimensionless ) @xmath49 , see text for details.,title="fig:",width=302,height=264 ] ( color online ) overlaps for a t - fractal of generation @xmath135 ( up ) and @xmath136 ( bottom ) as a function of @xmath49 . the symbols are as in fig . [ fig : ds_ext].,title="fig:",width=302,height=264 ] + ( color online ) overlaps for a t - fractal of generation @xmath135 ( up ) and @xmath136 ( bottom ) as a function of @xmath49 . the symbols are as in fig . [ fig : ds_ext].,title="fig:",width=302,height=264 ] ( color online ) overlaps for a ct of generation @xmath135 ( up ) and @xmath137 ( bottom ) as a function of @xmath49 . the symbols are as in fig . [ fig : ds_ext].,title="fig:",width=302,height=264 ] + ( color online ) overlaps for a ct of generation @xmath135 ( up ) and @xmath137 ( bottom ) as a function of @xmath49 . the symbols are as in fig . [ fig : ds_ext].,title="fig:",width=302,height=264 ] by comparing the plots obtained for the dsg ( fig . [ fig : ds_ext ] ) , the tf ( fig . [ fig : t_ext ] ) , and the ct ( fig . [ fig : ct_ext ] ) , we notice that @xmath131 depends sensitively on the underlying topology . in fact , going from dsg to ct and then to tf we notice an amplification of the transition region , which is for tf most spread out , requiring relatively large values of @xmath49 in order for @xmath42 to have a large overlap with the ground state . such effects can be ascribed to the absence of loops and to the existence for tf of a large number of peripheral sites scattered throughout the whole structure which give rise to localization effects @xcite . we now calculate @xmath131 according to eq . ( [ eq : intersection ] ) and for several values of @xmath54 ; numerical data and relative best fits are depicted in fig . [ fig : gamma_crit ] . for the dsg , numerical points are best fitted by the function @xmath138 , in very good agreement with the analytical findings . in fact , according to our analytical investigations , @xmath131 scales with the size of the database like @xmath139 ( see eq . ( [ eq : gammacrit_short ] ) ) , where , for the dsg , @xmath140 ( @xmath124 is taken to equal @xmath141 ) . let us now consider the case of the tf : from numerical data the best fit turns out to be @xmath142 . interestingly , this result is still in very good agreement with the analytical approximation of eq . ( [ eq : gammacrit_short ] ) . in fact , for the tf the exponent gets @xmath143 ( where , again , @xmath124 is taken to equal @xmath141 , consistently with the estimates given in appendix [ appe1 ] ) . such a consistency might suggest that eq . ( [ eq : gammacrit_short ] ) is valid not only for the dsg but for any ( exactly decimable ) fractal with @xmath144 . as for the ct , not being a fractal , eq . ( [ eq : gammacrit_short ] ) does not hold . indeed , we find that the value of @xmath131 corresponding to eq . ( [ eq : intersection ] ) grows linearly with the generation of the fractal , namely logarithmically with @xmath0 . in fig . [ fig : gamma_crit ] numerical data are fitted by the curve @xmath145 ( notice the semilogarithmic plot ) . ( color online ) estimate of @xmath131 for dsg ( @xmath146 ) , tf ( @xmath147 ) and ct ( o ) in a semi - logarithmic scale . the continuous lines represent the best fits.,width=302,height=264 ] apart from this , the plots shown in figs . [ fig : ds_ext]-[fig : ct_ext ] look rather similar . in particular , for networks of large sizes @xmath148 decays with @xmath49 more rapidly than @xmath149 . hence , the range of @xmath49 over which the transition occurs is wide , analogously to what happens on low - dimensional hypercubic lattices ( see @xcite ) . as shown in the next section , this has important effects on the behaviour of the success probability and suggests that a sharp transition is associated with an effective search algorithm . in order to sketch the role of the position of a target we show in the inset of fig . [ fig : ct_ext ] for the ct of @xmath137 the case of a target placed on a nearest neighbor of the central node . we see that the transition region is shifted towards lower values of @xmath49 . this means that a more central placement of the target is improving the probability for the ctqw to reach the target . analogous results were also found for dsg and for tf . thus , limiting our focus to peripheral nodes will prevent us from overestimating the success probabilities . we now consider fractal structures exhibiting large spectral dimension ; in particular , we focus on fractals obtained from cartesian products , such as dsg @xmath150 @xmath88 , dsg @xmath150 @xmath89 and dsg @xmath150 @xmath90 , as introduced in the previous section . again , we place the target on a `` peripheral site '' , i.e. on a minimally connected site ; this displays the coordination number @xmath73 , @xmath151 and @xmath152 , for dsg @xmath150 @xmath88 , dsg @xmath150 @xmath89 and dsg @xmath150 @xmath90 , respectively . we calculate for these structures the overlaps of the initial state @xmath17 and of the target state @xmath18 with @xmath133 and @xmath132 ; results for dsg @xmath150 @xmath89 ( @xmath153 ) and for dsg @xmath150 @xmath90 ( @xmath154 ) are shown in fig . @xmath155 . [ fig : cartesian ] @xmath89 structure given by the cartesian product of a dsg of generation @xmath156 and a square lattice of size @xmath157 ; bottom panel : overlaps for a dsg @xmath150 @xmath90 obtained from a dsg of generation @xmath156 and a cubic lattice of size @xmath158 . the lines are as in fig . [ fig : ds_ext].,title="fig:",height=264 ] @xmath89 structure given by the cartesian product of a dsg of generation @xmath156 and a square lattice of size @xmath157 ; bottom panel : overlaps for a dsg @xmath150 @xmath90 obtained from a dsg of generation @xmath156 and a cubic lattice of size @xmath158 . the lines are as in fig . [ fig : ds_ext].,title="fig:",height=264 ] as stressed at the beginning of this section , these plots provide some information about the sharpness of the transition from state @xmath42 to the state @xmath133 and from state @xmath159 to the state @xmath133 . however , around @xmath131 also a significantly large overlap @xmath160 is required . here , the transitions are still rather smooth although , by increasing @xmath56 , the region of @xmath49 values , over which the curves representing @xmath161 , @xmath162 and @xmath163 intersect , is shrinking . on the other hand , the overlaps between @xmath18 and the first excited state @xmath100 are negligible for all values of @xmath49 . we now turn to the success probability @xmath50 , eq . ( [ eq : success_prob ] ) , and we investigate numerically its dependence on @xmath9 and on @xmath49 . because of its dependence on time , @xmath50 carries more information than the previously discussed overlaps . we first analyze the case of complete graphs and of @xmath3-dimensional hypercubic lattices ( for which the time dependences of @xmath50 have already been determined for special choices of @xmath49 in ref . @xcite ) before turning to the dsg , the tf and the ct . ( color online ) contour plot of the success probability @xmath164 as a function of ( the dimensionless ) time @xmath9 and of @xmath49 for the complete graph of size @xmath165 ( top ) and @xmath166 ( bottom ) . one can notice that for @xmath167 , namely @xmath168 ( top ) and @xmath169 ( bottom ) , @xmath50 has a period @xmath170 and @xmath171 , respectively.,title="fig:",width=321,height=264 ] ( color online ) contour plot of the success probability @xmath164 as a function of ( the dimensionless ) time @xmath9 and of @xmath49 for the complete graph of size @xmath165 ( top ) and @xmath166 ( bottom ) . one can notice that for @xmath167 , namely @xmath168 ( top ) and @xmath169 ( bottom ) , @xmath50 has a period @xmath170 and @xmath171 , respectively.,title="fig:",width=321,height=264 ] we start our analysis from the complete graph @xmath172 for which , as shown in @xcite , at @xmath173 the ground state changes sharply from @xmath52 to @xmath99 . this transition takes place at @xmath174 . due to the special topology of @xmath172 , we are able to calculate @xmath50 exactly , obtaining @xmath175,\ ] ] see appendix [ appe3 ] for details . in fig . [ fig : complete ] we show @xmath164 for the complete graph with @xmath165 and @xmath166 . we evaluated the figures both numerically , by first diagonalizing @xmath43 in eq . ( [ eq : success_prob ] ) and projecting on the states @xmath18 and @xmath42 and also by making use of eq . ( [ eq : success_completo ] ) . the results are numerically indistinguishable . from fig . [ fig : complete ] we see that , around the values @xmath176 and @xmath177 for @xmath165 and @xmath166 respectively , @xmath50 reaches values very close to @xmath11 . in fact , analyzing eq . ( [ eq : success_completo ] ) ( see appendix [ appe3 ] ) , one finds that @xmath50 attains its maximal value of @xmath11 for @xmath167 and for @xmath178 , where @xmath179 . therefore , @xmath180 . furthermore , due to the fact that the period between maxima is @xmath181 for @xmath182 , it follows that the ctqw takes @xmath1 queries to find the target , in agreement with previous results @xcite . on the other hand , the exact dependence on @xmath9 and on @xmath49 also allows to highlight the oscillating behaviour of @xmath50 . this means that , although we properly select @xmath183 , the result for the walk depends sensitively also on @xmath9 . in particular , for @xmath184 , the success probability is minimal and equals @xmath185 ( which also corresponds to the absolute minimum ) . for the hypercubic lattices the overlaps of the states @xmath52 and @xmath99 with @xmath133 and @xmath132 have already been analyzed in @xcite ; here we display analogous plots ( but for larger sizes ) in order to compare them with the corresponding success probability @xmath50 . ( color online ) @xmath73-dimensional torus with linear size @xmath186 ; top : overlaps ( symbols are as in fig . [ fig : ds_ext ] ) ; bottom : contour plot of the success probability @xmath164 as a function of @xmath9 and of @xmath49 ; the dashed white line represents @xmath187.,title="fig:",width=302,height=264 ] ( color online ) @xmath73-dimensional torus with linear size @xmath186 ; top : overlaps ( symbols are as in fig . [ fig : ds_ext ] ) ; bottom : contour plot of the success probability @xmath164 as a function of @xmath9 and of @xmath49 ; the dashed white line represents @xmath187.,title="fig:",width=321,height=264 ] in fig . [ fig : torus5 ] we consider the case of a @xmath73-dimensional torus ( i.e. a five - dimensional cubic lattice with periodic boundary conditions ) of linear size @xmath188 : the transition at @xmath189 is very clear ( see the top panel ) and the success probability is sharply peaked just at @xmath190 ( see the bottom panel ) . away of the critical point @xmath131 the success probability quickly decays as a function of @xmath49 : it is just in the region of largest overlap between the initial state @xmath42 and the target state @xmath99 ( namely around @xmath131 ) that one expects an optimal success probability . again , we notice that @xmath50 oscillates in time ; for @xmath186 and @xmath191 the success probability ranges from about @xmath41 to about @xmath192 . moreover , for a given time @xmath9 , @xmath50 decays very fast as @xmath193 increases . for instance , @xmath194 . as a result , the computational procedure for this structure can be very efficient , provided that the parameter @xmath49 can be sensitively controlled . ( color online ) @xmath195-dimensional square torus of linear size @xmath196 . top : overlaps ( symbols are as in fig . [ fig : ds_ext ] ) ; bottom : success probability as a function of time and @xmath49 ; the dashed white line represents @xmath197.,title="fig:",width=302,height=264 ] ( color online ) @xmath195-dimensional square torus of linear size @xmath196 . top : overlaps ( symbols are as in fig . [ fig : ds_ext ] ) ; bottom : success probability as a function of time and @xmath49 ; the dashed white line represents @xmath197.,title="fig:",width=321,height=264 ] for hypercubic lattices of dimension @xmath6 , @xmath198 and @xmath199 ( only the latter case is depicted in fig . [ fig : torus2 ] ) the peaks get more and more broadened and are of smaller magnitude . notice that the low peaks obtained are in agreement with the analytical results found in @xcite which predict for the @xmath195-dimensional torus a vanishing success probability for @xmath200 . we now turn to the analysis of the success probability for the dsg , the tf and the ct , represented in figures [ fig : ampl_ds ] , [ fig : ampl_t ] and @xmath201 , respectively ; the upper panels display the situation for small , the lower panels for larger networks . first of all , we notice that , as previously found for regular lattices @xcite , also for dgs , tf and ct , @xmath50 exhibits peaks which are lower and lower as the size @xmath0 is enlarged . moreover , for small sizes ( upper panels ) @xmath202 , namely the maxima for the success probability occur for values of @xmath49 which are approximately equal to @xmath131 . on the other hand , for large sizes ( lower panels ) , in the temporal range considered here , @xmath131 provides an upper bound for @xmath182 . however , the most striking feature which emerges from the comparison between the contour plot of the success probability for translationally invariant structures ( see figs . [ fig : torus5 ] and [ fig : torus2 ] ) and for fractal / hierarchical structures ( see figs . [ fig : ampl_ds]-@xmath201 ) is that for the latter @xmath7 is much more rough and broadened . ( color online ) contour plot of @xmath50 for the dsg of generation @xmath135 ( top ) and @xmath136 ( bottom ) ; the `` critical '' values @xmath131 are represented by the dashed line : @xmath203 and @xmath204 ( see fig . [ fig : ds_ext]).,title="fig:",width=321,height=264 ] ( color online ) contour plot of @xmath50 for the dsg of generation @xmath135 ( top ) and @xmath136 ( bottom ) ; the `` critical '' values @xmath131 are represented by the dashed line : @xmath203 and @xmath204 ( see fig . [ fig : ds_ext]).,title="fig:",width=321,height=264 ] ( color online ) contour plot of @xmath50 for the tf of generation @xmath135 ( top ) and @xmath136 ( bottom ) ; notice that @xmath205 and @xmath206 ( see fig . [ fig : t_ext]).,title="fig:",width=321,height=264 ] ( color online ) contour plot of @xmath50 for the tf of generation @xmath135 ( top ) and @xmath136 ( bottom ) ; notice that @xmath205 and @xmath206 ( see fig . [ fig : t_ext]).,title="fig:",width=321,height=264 ] [ fig : ampl_ct ] for the ct of generation @xmath135 ( top ) and @xmath137 ( bottom ) ; notice that @xmath207 and @xmath208 ( see fig . [ fig : ct_ext]).,title="fig:",width=321,height=264 ] for the ct of generation @xmath135 ( top ) and @xmath137 ( bottom ) ; notice that @xmath207 and @xmath208 ( see fig . [ fig : ct_ext]).,title="fig:",width=321,height=264 ] now , it is worth comparing the @xmath73-dimensional torus of fig . [ fig : torus5 ] , the square torus of fig . [ fig : torus2 ] and the ct of fig . @xmath201 since all three are of comparable size @xmath0 . from the computational point of view the @xmath73-dimensional torus corresponds to the best situation : @xmath50 is sharp and reaches its maximum value around @xmath192 after approximately @xmath209 unit steps ; the ct displays a success probability around @xmath210 after @xmath211 time steps ; the @xmath195-dimensional torus corresponds to an ineffective candidate situation : after @xmath212 time steps the peak is still less than @xmath213 . in order to compare also to structures with spectral dimensions larger than four , we show in fig . @xmath214 the success probabilities @xmath50 for the cartesian products of a dsg of generation @xmath135 and a square lattice of size @xmath157 as well as a cubic lattice of size @xmath157 . although the maxima of the success probabilities are in both cases larger than the ones for the structures with low dimensions , they show still a fairly unregular pattern . this is in contrast to the highly regular structure of the 5-dimensional torus , see fig . [ fig : torus5 ] . [ fig : ampl_cp_dsgl ] for the cartesian product of a dsg of generation @xmath135 and a square lattice of size @xmath157 ( top ) and a cubic lattice of size @xmath157 ( bottom).,title="fig:",width=321,height=264 ] for the cartesian product of a dsg of generation @xmath135 and a square lattice of size @xmath157 ( top ) and a cubic lattice of size @xmath157 ( bottom).,title="fig:",width=321,height=264 ] finally , we stress that , as a result of interference phenomena , @xmath50 oscillates with time . this has some important consequences : although we can determine and set the optimal @xmath182 , the probability of the ctqw reaching the target depends on the instant of time at which it is calculated . in particular , oscillations are `` faster '' for systems of smaller size ; for example for the complete graph we find a period @xmath215 ( see eq . ( [ eq : success_completo ] ) ) , namely @xmath216 , while for the @xmath73-dimensional torus the numerical analysis makes it possible to estimate a period @xmath217 which grows exponentially with the lattice size @xmath218 ; for @xmath186 we get @xmath219 ( see fig . [ fig : torus5 ] ) . in this work we considered ctqws mimicking grover s quantum search problem ; we especially focused on how the topology of the space over which the walk takes place affects the position and sharpness of the transition of the ground state . previous studies @xcite highlighted that for translationally invariant structures , such as the hypercubic lattices , the quantum walk can be highly efficient for sufficiently high dimensions , i.e. @xmath4 . however , here we evidence that on generic graphs the dimension does not represent the key geometric parameter ; indeed , both the ( average ) coordination number and the fact that the structure is translationally invariant or not determine the sharpness of the transition . in fact , on the one hand , a high coordination reduces the distance among nodes and increases the possibility of interference effects , on the other hand , ( in the absence of a target ) translational invariance prevents the emergence of localization effects @xcite . in particular , we considered the success probability @xmath220 ( here @xmath52 and @xmath99 are the initial and the target state , respectively ) as a function of the computation time @xmath9 and of a properly tunable parameter @xmath49 entering the hamiltonian . we showed that for highly dimensional ( @xmath4 ) translationally invariant structures ( @xmath73-dimensional torus ) there exists a narrow range of @xmath49 around a specific value @xmath131 where the ground state @xmath102 undergoes a transition from having a large overlap with @xmath52 to a state with a large overlap with @xmath99 . this corresponds to a sharply peaked success probability : @xmath220 displays a set of maxima just at @xmath131 ; the first one is reached after @xmath1 queries . conversely , for structures with low coordination number and/or fractal or hierarchical topology - such as the cubic ( @xmath90 ) and square ( @xmath89 ) tori , the dsg , the tf , the ct and the cartesian products dsg @xmath150 @xmath89 and dsg @xmath150 @xmath90 - the transition from the initial state to the target state takes place over a wider region of @xmath49 around the value @xmath131 . as a consequence of such a spread - out transition , the success probability displays broadened peaks whose locations depend on time . therefore , for the non - translationally invariant structures considered here , even with large fractal dimension ( @xmath221 ) , the large success probabilities found for high - dimensional periodic lattices are not recovered . these results , in agreement with previous findings @xcite , highlight a possible connection between the sharpness of the transition occurring at @xmath131 and the efficiency of the search algorithm . a mathematical , rigorous proof stating whether a sharp transition is a necessary condition for a good algorithm , which is beyond the aim of this article , could provide a very useful tool for further investigations on quantum search algorithms . for the dsg we also proved that @xmath131 scales like @xmath222 ( @xmath125 ) ; interestingly our results suggest that such a scaling might be generalized to all ( exactly decimable ) fractals with spectral dimension @xmath223 . apart from the deterministic fractals considered here , it will be extremely interesting to also consider disordered structures such as percolation clusters and random graphs characterized by a degree distribution @xmath224 . these networks display a tunable average degree @xmath225 , which , in principle , can take values ranging from @xmath41 for totally disconnected networks up to @xmath0 for completely connected networks . according to the results discussed here , we expect that for a sufficiently large @xmath225 and for a sufficiently peaked @xmath224 the transition from the ground state @xmath17 to a state with significant overlap with @xmath19 occurs sharply and that , consequently , the ctqw speeds up . support from the deutsche forschungsgemeinschaft ( dfg ) and the fonds der chemischen industrie is gratefully acknowledged . ea thanks the italian foundation `` angelo della riccia '' for financial support . let us denote by @xmath226 and by @xmath227 the sets of eigenstates and of eigenvalues of the laplacian @xmath40 , respectively . on the basis of the eigenstates the state @xmath19 localized at the target site can be written @xmath228 now , let us consider the complete hamiltonian @xmath43 and the corresponding eigenvalue equation for the state labeled @xmath65 : @xmath229 as shown in @xcite , when one sets @xmath230 , it is possible to write @xmath231 here we notice that eq . [ eq : eigenvalue1 ] , and therefore also eq . [ eq : normalize ] , hold when the eigenvalue @xmath232 is _ non - degenerate _ ( for example , this condition is not fulfilled if we place the trap on the central node of the tf and ct ) . equations ( [ eq : eigenvalue1 ] ) and ( [ eq : normalize ] ) allow then to express the overlap of a given state @xmath233 ( with non - degenerate @xmath232 ) with @xmath234 as @xmath235 where @xmath236 and , in particular , @xmath237 ( see @xcite for more details ) . it is worth underlining that eq . ( [ eq : overlap ] ) found in @xcite for hypercubic lattices actually holds for all structures . the topological details are contained in @xmath232 and @xmath238 . we now proceed with the calculations without making any assumptions on the topology of the database . first , by using eq . ( [ eq : expansion ] ) and @xmath239 , we rewrite @xmath240 as @xmath241 whose derivative is @xmath242 for a generic index @xmath105 , @xmath243 ; while the lower bound is clear , the upper bound derives from the fact that the eigenstate @xmath99 can not correspond to any laplacian eigenstate . for euclidean lattices bloch s theorem makes it possible to write @xmath244 so that @xmath245 , for any @xmath105 . on the other hand , for a generic connected structure , a priori , one can only write @xmath246 , as a consequence of the fact that the laplacian eigenstate corresponding to the smallest eigenvalue @xmath247 is just @xmath248 . this suggests a proper restriction of the previous upper bound : @xmath249 , with @xmath250 and @xmath125 , both depending on the particular topology chosen . in particular , for euclidean structures , @xmath251 and @xmath126 and one expects that the more inhomogeneous the topology , the larger @xmath124 . by means of numerical calculations we can estimate @xmath124 : for not too small dsg and tf we find that @xmath252 ( see fig . @xmath253 ) . [ fig : alfa ] for the dsg and the tf.,title="fig:",height=226 ] now , before going on , we define the quantity @xmath254 , which will be useful in the following : @xmath255^j } \leq c n^{\alpha } \sum_{k \neq 0 } \frac{1}{[\mathcal{e}(k)]^j } \equiv c n^{\alpha } \zeta_j,\ ] ] where we set @xmath256^j \equiv \zeta_j$ ] and used the upper bound @xmath257 ; for hypercubic lattices @xmath258 which can be approximated by an integral @xcite . using eqs . ( [ eq : overlap ] ) and ( [ eq : f_deriv_gen ] ) the overlap of @xmath259 with the ground state turns out to be @xmath260^{-1 } \\ \nonumber & > & 1 - n e_0 ^ 2 \sum_{k \neq 0 } \frac{|a_k|^{2}}{(\gamma \mathcal{e}(k ) + |e_0|)^2 } \\ & > & 1 - \frac{n e_0 ^ 2}{\gamma^2 } \sum_{k \neq 0 } \frac{|a_k|^2}{[\mathcal{e}(k)]^2 } , \end{aligned}\ ] ] where in the first inequality we used that the sum is positive , while in second inequality we used that both @xmath104 and @xmath261 are positive . from eq . ( [ eq : s_0_preli ] ) and eq . ( [ eq : xi_j ] ) , we have @xmath262 in general , as @xmath49 is varied , @xmath263 is bounded as @xmath264 and the bounds can be improved by exploiting the following @xmath265 where for the lower bound we used that the first sum appearing in eq . ( [ eq : e_0 ] ) is positive and that @xmath266 . now , from eq . ( [ eq : s_0 ] ) and eq . ( [ eq : bound_e0 ] ) it follows straightforwardly that @xmath267 following analogous arguments we find for the first non - degenerate excited state labelled as @xmath11 , being @xmath268 non degenerate , @xmath269 and @xmath270 from which we get @xmath271 notice that in this case @xmath111 due to the fact that @xmath272 . now , by comparing eq . ( [ eq : bound_ground ] ) and eq . ( [ eq : bound_any ] ) we can evidence the existence of a critical value @xmath131 such that when @xmath49 approaches @xmath131 the ground state switches from @xmath273 to @xmath132 , at least in the limit @xmath200 . in fact , if we take @xmath274 , then in eq . ( [ eq : bound_ground ] ) and eq . ( [ eq : bound_any ] ) we can set @xmath275 , with @xmath276 and @xmath277 respectively , obtaining @xmath278 therefore , recalling that @xmath279 , the condition @xmath280 , as @xmath281 is sufficient for @xmath282 . otherwise stated , as @xmath49 approaches @xmath131 from above ( @xmath283 ) or from below ( @xmath277 ) , the overlap of the ground state with @xmath133 and with @xmath132 , respectively , is close to @xmath11 . the laplacian spectrum @xmath104 for the dsg is exactly known @xcite and we can therefore calculate exactly the quantities @xmath284 and @xmath285 , obtaining estimates for the critical parameter @xmath131 . at a given generation @xmath54 , the spectrum includes the eigenvalue @xmath69 with degeneracy @xmath286 , the eigenvalue @xmath73 with degeneracy @xmath287 and the eigenvalues @xmath288 and @xmath289 stemming from the eigenvalue @xmath290 , belonging to previous generation and both carrying degeneracy @xmath291 . for each eigenvalue @xmath290 the next - generation eigenvalues @xmath292 are defined according to @xmath293 now , it follows directly from eq . ( [ eq : iteration ] ) that any couple @xmath294 , @xmath295 sum up as @xmath296 and applying this result iteratively to all the couples making up the spectrum we get @xmath297 as for @xmath285 we can implement a similar iterative procedure , by noticing that @xmath298 therefore @xmath285 is made up of two terms stemming from the first and second order contributions , which are , respectively , @xmath299 and @xmath300 subtracting eq . ( [ eq : xi2_dsg ] ) and eq . ( [ eq : xi1_dsg ] ) we finally get @xmath301 ^ 2 } = \frac{1}{900 } ( -13 -14\times 3^g + 21 \times 5^g + 6 \times 25^g).\ ] ] now , recalling that @xmath302 and that the spectral dimension of the dsg is @xmath303 , we can write @xmath304 and obtain expressions for @xmath284 and @xmath285 as a function of just the volume @xmath0 and the spectral dimension @xmath56 of the substrate : @xmath305 @xmath306 where the asymptotic expressions hold for large @xmath0 . we notice that as @xmath200 , @xmath284 and @xmath285 appearing in eqs . ( [ eq : xi1_n ] ) and ( [ eq : xi2_n ] ) satisfy the condition @xmath307 found in appendix [ appe1 ] . more precisely , @xmath308 and this is sufficient for @xmath309 and @xmath310 to converge to zero as @xmath311 . in particular , from eq . ( [ eq : xi1_n ] ) and eqs . ( [ eq : bound_ground ] ) and ( [ eq : bound_any ] ) , the state @xmath42 is expected to switch from the ground to the first excited state at @xmath312 as explained in sec . [ sec : phase_transition ] , the expressions found here for @xmath284 and @xmath285 are consistent with @xmath313^j \sim n^{2j / d}$ ] obtained in @xcite for lattices . therefore , and as suggested by the numerical data discussed in sec . [ sec : numerics ] , it is plausible that eqs . ( [ eq : xi1_n ] ) , ( [ eq : xi2_n ] ) and , above all , the expression for the critical parameter @xmath131 in eq . ( [ eq : gammacrit ] ) can be extended to arbitrary structures of spectral dimension @xmath144 . let us start from the definition of success probability given in eq . ( [ eq : success_prob ] ) . now , the propagator @xmath314 , can be calculated as @xmath315 and the success probability can be rewritten as @xmath316 hence , in order to calculate the success probability @xmath50 we need to find the elements corresponding to the @xmath8-th column of the @xmath317-th power of the hamiltonian . for complete graphs , @xmath318 displays a high degree of symmetry which allows its exact calculation ( see for example @xcite for a similar calculation where the hamiltonian is provided by the adjacency matrix ) . without loss of generality we can fix @xmath319 so that the hamiltonian @xmath43 is @xmath320 and it is easy to see that , regardless of @xmath317 , @xmath321 , with @xmath322 . therefore , eq . ( [ eq : success_prob_1 ] ) can be rewritten as @xmath323 now , our task is to calculate @xmath324 and @xmath325 for which we need the entries @xmath326 and @xmath327 of the @xmath317-th power of the hamiltonian ; for better readability we set @xmath328 and @xmath329 . thus , from eq . ( [ eq : hamiltonian_mat ] ) we derive the following recursive relations : @xmath330 and @xmath331 from their combination we get @xmath332 whose solution is @xmath333,\ ] ] with @xmath334 and @xmath335 . from eq . ( [ eq : recursive_1 ] ) and eq . ( [ eq : a1va ] ) , @xmath336 is also explicitly defined : @xmath337,\ ] ] now , according to eq . ( [ eq : success_prob_1 ] ) , we can calculate @xmath324 as @xmath338 \\ \nonumber & = & \frac{\gamma}{b } \left \ { \exp{\left[-it(a+b)/2 \right ] } -\exp{\left[-it(a - b)/2 \right ] } \right\}\\ & = & \frac{\gamma}{b } \exp{-i t a/2 } ( -2 i \sin{\frac{tb}{2 } } ) . \end{aligned}\ ] ] analogous calculations lead to @xmath339 hence , by inserting eqs . ( [ eq : a1v ] ) and ( [ eq : avv ] ) into eq . ( [ eq : success_prob_2 ] ) , and performing the summation @xmath340,\end{aligned}\ ] ] which provides the exact success probability for the complete graph of size @xmath0 as a function of @xmath49 and of @xmath9 . we can notice that @xmath50 is strictly larger than zero and that it can be equal to @xmath11 when both @xmath341 , with @xmath342 and @xmath343 are satisfied . the latter condition holds for @xmath167 , just consistent with @xcite : there it is found that for @xmath167 the walk rotates the state from @xmath52 to @xmath99 and that the gap between the corresponding energies is smallest . for @xmath167 the first condition reads @xmath344 . hence , for the exact @xmath49 and at the right time the success probability is unitary , but for larger volumes the right times get sparser . we also notice that when @xmath0 is large and @xmath345 , eq . ( [ eq : success_prob_3 ] ) can be simply rewritten as @xmath346 from eq . ( [ eq : success_prob_3 ] ) we deduce that for a given time @xmath347 , @xmath50 decreases as @xmath348 gets larger . chuang and m.a . nielsen , _ quantum computation and quantum information _ , cambridge university press , cambridge , 2000 l.k . grover , phys . . lett . * 79 * , 325 ( 1997 ) c. weiss and m. holthaus , europhys . lett . * 59 * , 486 ( 2002 ) d. daems and s. gurin , phys . rev . lett . * 99 * , 170503 ( 2007 ) p.w . shor , _ proceedings of the 35th ieee symposium on foundations of computer science _ ( ieee , los alamos , 1994 ) , a. ambainis , _ quantum search algorithms _ , sigact news * 35*(2 ) , 22 ( 2004 ) a.m. childs and j. goldstone , phys . rev . a * 70 * , 042312 ( 2004 ) l.k . grover , phys . lett . * 79 * , 325 ( 1997 ) j. roland and n.j . cerf , phys . a * 68 * , 062312 ( 2003 ) s. aaronson and a. ambainis , in _ proceedings of the 44th ieee symposium on foundations of computer science _ ( ieee , los alamos , 2003 ) , e. farhi and s. gutmann , phys . a * 58 * , 915 ( 1998 ) a.m. childs and j. goldstone , phys . a * 70 * , 022314 ( 2004 ) j. roland and n.j . cerf , phys . a * 65 * , 042308 ( 2002 ) a. tulsi , phys . a * 78 * , 012310 ( 2008 ) j.p . dowling , nature * 439 * , 919 ( 2006 ) a.m. childs , e. farhi and s. gutmann , quantum inf . process . * 1 * , 35 ( 2002 ) j. kempe , contemp * 44 * , 307 ( 2003 ) a. ambainis , j. kempe and a. rivosh , _ soda 05 : proceeding of the sixteenth annual acm - siam symposium of discrete algorithms _ , 1099 ( 2005 ) n. shenvi , j. kempe and k.b . whaley , phys . a * 67 * , 052307 ( 2003 ) c.h . bennett , e. bernstein , g. brassard and u. vazirani , siam j. comput . * 26 * , 1510 ( 1997 ) e. farhi and s. gutmann , phys . a * 57 * , 2403 ( 1998 ) e. farhi , j. goldstone , s. gutmann and m. sipser , e - print quant - ph/0001106 i. sh . averbukh and n.f . perelman , phys . a * 139 * , 449 ( 1989 ) o. mlken and a. blumen , phys . e * 71 * , 016101 ( 2005 ) d.h . dunlap and v.m . kenkre , phys . b * 34 * , 3625 ( 1986 ) m.g . cosenza and r. kapral , phys . a * 46 * , 1850 ( 1992 ) k. hattori , t. hattori and h. watanabe , prog . . phys.suppl . * 92 * , 108 ( 1987 ) a. blumen and a. jurjiu , j. chem . phys . * 116 * , 2636 ( 2002 ) e. agliari , a. blumen and o. mlken , j. phys . a * 41 * , 445301 ( 2008 ) b. khang and s. redner , j. phys . a * 22 * , 887 ( 1989 ) d. ben - avraham and s. havlin , _ diffusion and reactions in fractals and disordered systems _ , cambridge university press , cambridge , 2000 ; s. redner , _ a guide to first - passage processes _ , cambridge university press , cambridge , 2001 . s. alexander , j. bernasconi , w. r. schneider , and r. orbach , rev . phys . * 53 * , 175 ( 1981 ) . s. alexander and r. orbach , j. physique lett . * 43 * , 625 ( 1982 ) . g.e . santoro and e. tosatti , j. phys . a * 39 * , r393 ( 2006 ) w. woess , _ random walks on infinite graphs and groups _ , cambridge university press , cambridge , 2000 . | we study continuous - time quantum walks mimicking the quantum search based on grover s procedure .
this allows us to consider structures , that is , databases , with arbitrary topological arrangements of their entries .
we show that the topological structure of the database plays a crucial role by analyzing , both analytically and numerically , the transition from the ground to the first excited state of the hamiltonian associated with different ( fractal ) structures .
additionally , we use the probability of successfully finding a specific target as another indicator of the importance of the topological structure . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
in most countries , learning microscopic origin of nature is considered to be an important topic in science education . however , there are not so many student experiments or demonstrations which exhibit the existence of individual atom in direct and intuitive manner . the cloud chamber experiment is a rare exception ; it provides most direct and intuitive way to convince students existence of microscopic particles . beautiful tracks of particles draw audience s attention and interest . various chambers have been developed and used in classroom @xcite . most of simple chambers seem to be based on the results of needles and nielsen @xcite and cowan @xcite , which use a block of dry ice and a beaker filled with ethanol vapour . the great simplicity of their chamber enables students to make diy chamber at home or classroom . such a chamber works well if one uses a radioactive source . from spring 2008 , one of the authors ( s.z . ) has been working on classroom experiment program using cloud chamber . an one - day experiment course had been given for local junior high school students . from spring 2011 , some students ( rest of the authors ) have joined this project as a part of super science high school ( ssh ) activity . the typical setting of a chamber is shown in figure [ typicalchamber ] . a chamber is not so large ; smallest example is a glass laboratory dish with radius 5 cm and height 1 cm . a standard chamber we have used is a round glass container with radius 8.5 cm and 8.0 cm high @xcite . such chambers require a _ radioactive source . _ it is very hard to find particle tracks of background radiation in this kind of chamber . one will find few tracks in a minute only in very dark room . in japan , people s attention to radioactivity is increased much after the fukushima daiichi nuclear disaster following the tohoku earthquake and tsunami on 11 march 2011 . large part of public unease or fear about radioactivity is due to insufficient education about radioactivity , since curricula of primary and secondary schools lack serious and extensive study of radioactivity . after the tohoku earthquake , people have been nervous about radioactivity . in particular , parents do not wish their children to join scientific activity using radioactive source , even if it is relatively safe one such as a gas mantle with thorium . therefore , we think it is very important to develop a sensitive cloud chamber which _ does not require any radioactive source . _ the chamber should be simple , small , portable and cheap since our aim is educational applications such as student experiment course or a workshop for citizens in tohoku area . in this article , we would like to present a construction of a cloud chamber works well without radioactive sources . there is only minor modification from a typical chamber shown in this section , but its performance is remarkable . we also present a result of a performance test . the construction of our cloud chamber is shown in figure [ fig : chamber ] . a major difference from a common chamber is use of a black anodised aluminium heat sink as a bottom plate instead of a metal plate . its dimension is 134 mm @xmath0 134 mm @xmath0 20 mm , weight is 622 g , and a side with fins is placed downward . as we see later , the heat sink greatly improves performance of our chamber . our method has an additional advantage that the heat sink plays a role of supporting base of the chamber which prevents direct contact of side wall with refrigerant . clear walls without a bottom and a top are placed on the heat sink . walls and top are made of square , 5 mm thick acrylic plates . top plate is removable . a piece of black felt e is placed along three sides of a box . upper half of it is soaked with ethanol . total amount of ethanol is about 10 g for each experiment . light from pc projector is introduced from a face without a piece of black felt . all of the above instruments are placed on a shallow styrofoam tray b. liquid nitrogen can be directory poured into this box to this tray . powder of dry ice can be used also . in this case , a heat sink is placed above powdered dry ice . in the placement mentioned above , tracks of particles can be observed from above . figure [ fig : working ] is a photo of our cloud chamber in operation . a pc projector is slanted to obtain fine view of particle tracks . in closing this section , we would like to mention that our result is not new . an advantage of our chamber is good performance in spite of its smallness and simpleness . in fact , various examples of sensitive chambers have been known in japan . very large and expensive chambers for display are available in some science museums @xcite . a glass chamber presented in the beginning of this section is also available as a product by rado ltd . a use of a small heat sink with liquid nitrogen is first presented in @xcite . another example of a sensitive chamber is given in @xcite . surprisingly , the very simple construction presented in the previous section is enough to see many particle tracks without any radioactive sources . external electric field is also unnecessary . we would like to describe operation of a cloud chamber . to begin with , liquid nitrogen was poured to the styrofoam box . our lab was at room temperature , so the temperature of the bottom heat sink rises as liquid nitrogen evaporates . during the observation , this change of the temperature was measured by a thermocouple . a result is shown in figure [ fig : bottomtemp ] . obtained data fits well to a quadratic curve . particle tracks begin to appear in few minutes . here we present our result with a video @xcite uploaded to youtube . figure [ fig : alphabeta ] is captured image from the video . most thin and wiggy tracks , such as the left picture of figure [ fig : alphabeta ] frequently seen in the chamber , can be considered as those of beta particles . we can see long tracks since the bottom plate is cooled uniformly . some of long tracks are slightly curved even though no external magnetic field is imposed . we think it is due to multiple collision with atoms in the air . an alpha particle also can be observed as a short , thick line such as the right picture in figure [ fig : alphabeta ] . one can see that many droplets are formed along a track and fall down toward bottom of the chamber . although many tracks are seen in the video , there should be no influence of fukushima daiichi nuclear disaster , since in our city ( yokote , akita , japan ) has never observed apparent rise of air dose rate since 11th march . obtaining of high quality video is important both for scientific analysis and publication on the web . the later is especially important in for education , since lower quality videos found in the web are not enough to tell the beauty and excitement that people must feel in this experiment . we find that a digital slr camera ( in our case , eos kiss x3 ) is very useful for this purpose . it has larger ccd and better lens required for high quality video . in our experiment , a camera is directly put on the top plate as in figure [ fig : camera ] , but use of rigid is better if available . a qualitative evaluation of the performance of a chamber will be useful for development of new cloud chamber and finding better working condition . in fact , some test runs tells us that numbers of particle tracks seems to differ for each run . using a recorded video @xcite , we counted all tracks seen in the chamber regardless of its length , width and shape . therefore alpha particles , beta particles and other particles are not distinguished . figure [ fig : countpermin ] shows a relation between count per 10 seconds and elapsed time . we observe that , although fluctuation is large , the count is independent of the bottom temperature until it reaches to ` critical ' value . our observation shows the fall of the count begins around @xmath1 degrees . , @xmath2 and @xmath3 degrees respectively . black and white are converted . , scaledwidth=90.0% ] while a number of particle tracks does not change below critical temperature , a number of background droplets affects visibility of particle tracks . many droplets are seen at lower temperature as shown in the left of figure [ fig : droplets ] . contrast of whole image is reduced at such low temperature . we also found that alpha particle tracks become thinner at lower temperature , as mentioned in @xcite . on the other hand , it becomes hard to identify particle tracks at high temperature . therefore , the best temperature range for observation should be middle region in figure [ fig : countpermin ] . we expect our cloud chamber will be useful for various application to education , for example , a student research project or demonstrations in public . in particular , it will be important to measure numbers of particle tracks in a city where dose rate is still high . also not examined in this paper , the high quality video enables us various computer analysis such as qualitative evaluation of amount of condensation or automatic counting of particle tracks . such kind of analysis will be useful for a student s research projects . we would like to thank hiroki kanda of tohoku university for useful comments at the beginning of our project , ichiro itoh of yokote seiryo gakuin h.s . for providing a voltage multiplier , ryoko yamaishi for support in lab . we also thank kenichiro aoki of keio university for careful reading of the manuscript . this work is supported by saito kenzo honour fund and the super science high school funding from japan science and technology agency . 99 andy foland s cloud chamber page http://w4.lns.cornell.edu/~adf4/cloud.html , how to build a cloud chamber ! http://www.youtube.com/watch?v=pewtysxftqk , how to make a diffusion cloud chamber http://www.youtube.com/watch?v=pvcdaa_vvvg needels t s and neilsen c e 1950 a continuously sensitive cloud chamber rev . 21 976 cowan e w 1950 a continuously sensitive cloud chamber rev . 21 991 rado ltd . http://www.kiribako-rado.co.jp/goods-m.html mori y 1994 let s see positorn in a cloud chamber using liquid nitorogen toray science education award 26 18 , hayashi h 2007 a sensitive cloud chamber in magnetic field as a teaching material of atomic physics j. phys . 55 297 sensitive cloud chamber http://www.youtube.com/watch?v=hcv3fdz1rfk y mori http://www.youtube.com/watch?v=mwm-wo7dokw http://www.youtube.com/watch?v=r9swciiqwvm slatis h 1957 on diffusion cloud chambers nucl . instrum . 1 213 | we present a sensitive diffusion cloud chamber which does not require any radioactive sources . a major difference from a commonly used chamber
is use of a heat sink as its bottom plate .
a result of a performance test of the chamber is given . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
deep surveys of the extragalactic sky have been made at x - ray ( hasinger et al . 1998 ; cowie et al . 2001 ) , optical ( lilly et al.1996 ; madau et al . 1996 ; steidel et al . 1999 ) , infra - red ( goldschmidt et al . 1997 ; rowan - robinson et al . 1997 ; elbaz et al . 1999 ) , sub - mm ( hughes et al . 1998 ; barger et al . 1999 ; blain et al . 1999 ; barger et al . 2000 ; scott et al . 2000 ) , as well as radio wavelengths ( windhorst et al . 1984 ; windhorst et al . 1985 ; condon & mitchell 1984 ; donnelly et al . 1987 ; fomalont et al . 1991 ; windhorst et al.1993 ; richards et al . 1998 ; richards et al . 1999 ; prandoni et al.2001 ) . much of the interest in these surveys of weak objects concern the evidence for star formation and agn evolution in galaxies in the early universe , at redshifts in the range 1 to 3 ( condon & yin 1990 ; rowan - robinson et al . 1993 ; cram 1998 ; mobasher et al . 1999 ; steidel et al . 1999 ; haarsma et al . 2000 ) . the radio observations are unique in that at radio wavelengths it is possible to peer through the gas and dust that obscures the nuclear regions of galaxies at other wavelengths . the radio observations also have sufficient angular resolution to distinguish between emission that is driven by star formation and that driven by agn . although star forming activity in the early universe is perhaps most readily observed at sub - millimeter wavelengths , current sub - millimeter instruments do not have sufficient angular resolution to avoid confusion due to blending of nearby sources , so high resolution radio observations are needed to uniquely identify optical counterparts . moreover , because of the large negative k - correction , dusty galaxies observed at sub - mm wavelengths all have high redshifts , whereas the radio observations are sensitive to a wide range of redshift and a mix of star - burst and agn activity . as the most sensitive radio telescope available for these kinds of observations , the very large array ( vla ) has been used for many deep surveys @xcite . deep radio source surveys have also been made with the australian telescope compact array @xcite , and with the westerbork synthesis radio telescope @xcite , but with less sensitivity and poorer resolution than possible with the vla . these observations show that below levels of about 10 mjy , the number of radio sources increases more rapidly than the number between 1 mjy and 500 mjy and are composed of a different population of sources . in this paper we report on the results of new 8.4 ghz vla surveys which cover two fields , each of diameter @xmath12 to the 8% sensitivity level of the on - axis position . one field , located at @xmath13 + 43@xmath1 , which we will designate as the sa13 field , was one of the hubble space telescope medium deep survey ( mds ) key projects , observed in 1992 - 3 @xcite . these optical observations , which were made before hst refurbishment , reach a limiting sensitivity of i=25.5 mag . the extensive complementary vla observation detected sources as faint as @xmath3jy at 8.4 ghz . a preliminary account of the hst identifications in this field @xcite , hereinafter designated as paper i , was based on the first half of these observations which were made at lower resolution , and included only radio sources located in the small 2.5 ( arcmin)@xmath14 field of view of the hst . we have also used these same vla observations of this field to examine the small scale fluctuations in the cosmic microwave background radiation @xcite . this field has also been observed with the vla at 1.4 ghz , covering a much larger area , with additional kpno optical imaging @xcite , and with the merlin / vlbi national facility of the university of manchester at 1.4 ghz @xcite a second field , located at @xmath15 + 50@xmath1 and designated at the hercules field , was previously imaged with wfpc2 in a 48-orbit exposure , and reaches a 10-@xmath16 point source sensitivity of about 27.7 mag in b and v and 26.8 in i @xcite . twelve known objects in this field , including three radio - weak agn , have spectroscopic redshifts z near 2.4 ( pascarelle et al . 1996a ; pascarelle et al . 1996b ; pascarelle et al . 1998 ; pascarelle et al . 2001 ) . a wfpc2 image made with a medium band redshifted ly@xmath17 filter indicates the presence of a number of other compact galaxy candidates with redshifts near 2.4 . the vla image of this field reached a limiting flux density of only @xmath6jy , but provides improved statistics for radio sources above this level . in @xmath182 of this paper we describe the vla observations , calibrations and imaging for the two fields . the radio and optical parameters for the sources are given in @xmath183 , and detailed radio / optical images with discussions are given in @xmath184 , along with discussions of the identifications . in addition , using data from five deep 8.4-ghz surveys , we discuss the spectral properties , the source density and the optical properties of the micro - jansky radio sources . further discussion is given in the final section . the sa13 field was selected from among the hst medium deep survey fields to be devoid of strong radio sources , which would limit the sensitivity of a deep radio survey . this field , centered at @xmath19 and @xmath20 , epoch j2000 , was observed with the vla for a total of 190 hours , of which we obtained 159 hours of good data ( more detail given below ) . these data included 84 hours in the most compact d - configuration ( maximum baseline 1 km ) observed between 1993 october and 1994 january , and 75 hours in the c - configuration ( maximum baseline 3 km ) observed between 1994 november and 1995 january . the earlier discussion in paper i of the optical counterparts in this field was based on the 1993/1994 d - configuration observations , but with the preliminary new c - configuration data . the hercules field was observed for 12 hours in the vla c - configuration in 1996 february . the center of this field is at @xmath21 and @xmath22 , epoch j2000 . the observations were made at two frequencies , 8.415 ghz and 8.465 ghz , each with dual circular polarization with a bandwidth of 50 mhz . at this frequency the full - width half - power ( fwhp ) beamwidth of the vla 25-m antennas is @xmath23 . we examined the area out to the 8% power level , at a radius of @xmath24 from the field center . this corresponds to a solid angle of @xmath25 sr for each field . each observing session lasted 8 - 10 hours and was split into 30-minute segments which included 27 minutes on the survey field and 3 minutes on a nearby calibrator source . for the sa13 field , we used the unresolved calibrator source , j1244 + 4806 ( @xmath26 , @xmath27 ; j2000 ) , and for the hercules field , the calibrator source j1658 + 4749 ( @xmath28 , @xmath29 ; j2000 ) . the position grids for the surveys are tied to these calibrators , both of which have a position known to better than @xmath30 . the amplitude calibration was derived from observations of 3c286 made once per day ; we assumed a flux density of 5.19 jy at 8.44 ghz . the data from each observing session were edited for occasional short periods of interference , excessive noise in individual correlators or other technical problems , antenna shadowing , and inclement weather conditions . in the sa13 field , less than one percent of the data was lost to interference and technical problems , but we had to discard about 30 hours of data due to poor weather , primarily snow , during 4 of the 27 observing sessions , each of 6 to 8 hours duration . both fields were imaged and deconvolved with the aips tasks imagr using a pixel separation @xmath31 over a field of view of @xmath32 in order to detect radio sources well outside of the primary beam field of view which could seriously affect the image quality @xcite . the visibility data were weighted ( aips terminology of robust=1 ) to obtain high signal to noise with low sidelobe levels and good resolution . for the sa13 field which contained both c and d configuration data , the full - width at half - power ( fwhp ) resolution was @xmath33 . we also made an image with a resolution of @xmath34 to help resolve possible source blends . for the hercules field with only c - configuration data , the resolution was @xmath35 . the measured rms noise in the sa13 field was @xmath2jy , which makes this the most sensitive radio image yet obtained . for the hercules field the rms noise was @xmath36jy . the strongest source in the sa13 field has a measured map flux density of @xmath37jy . this is less than one hundred times the thermal receiver noise limit , so dynamic range limitations are not an important factor . the hercules field contained a known ` bright ' radio source of 6 mjy at the field center , 53w002 @xcite , which was the target of the deep hst study @xcite . artifacts from the 6-mjy source limited the dynamic range of the image , but self - calibration with a solution interval of 3 min had sufficient signal to noise to improve the calibration , and produce an image was signal - to - noise limited the completeness limit of a survey is that level above which there is near certainty that any source is real . as discussed in previous vla papers on radio surveys with comparable resolution , crowdedness and field of view , a source with a peak flux density on the image @xmath38-@xmath16 has a probability of @xmath39 of being a true detection @xcite . this percentage includes the effects of receiver noise and side - lobe contamination from faint sources . the most negative peak flux density on the image is also a good indication of the detection limit @xcite . we found that detection levels of @xmath3jy and @xmath40jy level for the sa13 and hercules fields , respectively , met both of the above criteria . we also confirmed 19 of the 20 sources detected in paper i using similar criteria . in order to search for possible extended radio sources with low surface brightness , we convolved the images to @xmath41 resolution for sources above the detection level of the highest resolution image , but above that for a lower resolution image . one such extended source was found in the sa13 field ( source 27 in table 1 ) . the radio images for the entire sa13 and the hercules fields are shown in figures 1 and 2 , respectively . in figure 3 we show the image of the central part of the sa13 field where the radio sensitivity is greatest . the figure shows the excellent quality of the radio image , the precise alignment of most of the radio / optical identifications and the obvious faintness of the identifications for some of the sources . the sources labeled with a number are given in table 1 . seven weaker radio sources which are coincident with an optical object are labeled with a letter and given at the end of table 1 . parameters for the 50 sources labeled in the sa13 field and the 10 sources labeled in the hercules field have entries listed in tables 1 and 2 , respectively . thirty - four sources in the sa13 field were found in the complete sample with a peak flux density greater than @xmath3jy and within @xmath24 of the field center where the sensitivity drops to 8% of the on - axis sensitivity . sixteen additional sources , not in the complete sample , are also included in the table . all instrumental and resolution effects used to obtain the source parameters have been described elsewhere @xcite . table 1 is organized as follows : columns 1 and 2 show the source numbers used in this paper and in paper i , respectively , to identify the sources more conveniently . column 3 gives the source name . an asterisk before the name indicates the fifteen sources which are * not * in the complete sample . column 4 shows the total sky flux density and error estimate after correction for all instrumental and resolution effects including the primary beam attenuation of the telescopes . column 5 shows the signal - to - noise on the @xmath33-resolution image and column 6 gives the deconvolved source angular size or limit in arcseconds . the right ascension and declination , with rms errors are given in columns 7 and 8 , respectively . the remaining four columns list the optical identification type and integrated magnitude in three bands . further descriptions of the optical data are given in the next section . eight faint radio sources , which are below our formal completeness level but have probable identifications with optical counterparts , are listed at the end of table 1 , and labeled @xmath42 through @xmath43 . the probability that these radio sources is real is less than 30% based on the radio flux density alone . however , their near coincidence with an optical counterpart increases their reality to 80% ; hence , only one of the eight sources is likely to be bogus @xcite . there is good agreement between the source list in table 1 and that in paper i based on about one - half as much data , and at lower resolution . of the 20 sources above the previous detection level of @xmath44jy , 16 are in the complete sample of table 1 . the four remaining sources from paper i are just below our @xmath3jy completion limit ; three of these objects are identified with galaxies . the source list for the hercules field is given in table 2 . the resolution is @xmath35 and the completeness limit is @xmath6jy . ten sources are listed ; six of these are in the complete sample . columns 1 through 7 are similar in content to that in table 1 . only two of the radio sources have been identified from an hst image which have been described elsewhere @xcite ; the optical information is given in column 8 . the identification of radio sources in the sa13 field is based on two sets of optical material . the hst - mds wf / pc images observed with two filters : v - band ( 5420 ) and i - band ( 8920 ) , respectively were used for the identification of radio sources near the center of the sa13 radio field . the calibration and processing of this optical data was discussed in paper i. the identification limits are v=26.8 mag and i=25.8 mag with fwhp resolution of @xmath45 . for radio sources outside the mds field of view , identifications were made with the kpno 4-m telescopes with b - band ( 4420 ) and i - band ( 8920 ) filters . the limiting magnitudes here were b=26.0 and i=25.5 mag , with seeing of about fwhm @xmath46 . the observational details are discussed elsewhere @xcite . the i - magnitude scale of the kpno images was calibrated to agree with that of the mds , with an estimated error of 0.2 mag . the b - magnitude scale of the kpno images were calibrated using the extrapolation of the mds i- and v - magnitude scale , assuming a power law extrapolation of the intensity of the the flat - spectrum ( blue ) objects . this extrapolation is accurate to an estimated error of 0.4 mag . the precise registration of the kpno and the mds images with the radio grid was determined from the alignment of the high quality radio / optical coincidences to an accuracy of @xmath45 . only zero and first order corrections to the optical image were needed . confidence in the registration of the images at this accuracy is important for the proper interpretation of the identifications . the integrated optical magnitudes for all identifications in the sa13 field are given in table 1 . the galaxy type is given in column 9 and the b- , v- and i - magnitudes are given in columns 10 , 11 and 12 , respectively . the galaxy types are : g = galaxy of unknown morphology ; g?=uncertain identification of a very faint object ; ef = empty field ; g / b = binary galaxies or closely spaced galaxies , with the magnitude given for the objects containing the radio emission ; q = quasar : ell = elliptical galaxy ; sp = spiral galaxy . in figures 4a and 4b we show postage - stamp plots for all 42 radio sources in the sa13 field listed in table 1 . in contrast to figure 3 , the green contour lines in figures 4a and 4b show the optical i - band distribution , while the location of the position of the radio source is indicated by the small black ellipses in each plot . each field is @xmath47 on a side and centered at the position of the radio source . most of the radio sources are unresolved ( typically @xmath48 ) and the black ellipses indicates the 1- and 2-@xmath16 limits for the position of the radio centroid . all sources have a kpno i - band image fwhm @xmath46 seeing and the 22 sources nearest the field center also have an mds i - band image with fwhp @xmath45 resolution . ( see paper i for a discussion of the removal of the chromatic aberration from the pre - refurbished hst images . ) 0.3truein * comments on the sa13 identifications : * * * no 1 . j131157 + 423910 : * detected only on the i - band kpno image with i=23.5 mag . the optical identification is too faint to be classified . * * no 2 . j131157 + 423610 : * empty field . possible detection of an i=26 mag object . * * no 3 . j131203 + 424030 : * detected only on the i - band kpno image with i=24.3 mag . the identification is too faint to be classified . * j131203 + 423331 : * identified with an isolated galaxy , somewhat extended in pa=@xmath49 . * * no 5 . j131205 + 423851 : * empty field . * * no 6 . j131207 + 423945 : * empty field . * * no 7 . j131209 + 424217 : * identified with an isolated galaxy , possibly extended in pa=@xmath50 . * j131211 + 424053 : * the radio emission is identified with the southern pair of two circular galaxies separated by @xmath51 . the identified galaxy has ( b - i ) color = 3.7 mag and a size of about @xmath52 . the northern galaxy , also circular , has about the same angular size , but is not unusually red . its radio flux density is @xmath53jy , but lies below the formal detection level . * * no 9 . j131213 + 423333 : * a weak radio source , not in the complete sample , which is identified with a bright galaxy with a faint companion @xmath52 to the north . another elliptically - shaped galaxy about @xmath34 south is extended towards the radio source and may be interacting with the bright galaxy . because of the faintness of the radio source , the radio - optical position offset is uncertain , * * no 10 . j131213 + 424129 : * identified with a bright galaxy with an extensive , blue halo . * j131213 + 423706 : * a faint smudge is present on both the kpno and mds images with i@xmath54 mag . * * no 12 . j131214 + 423821 : * a faint radio source below the completeness level is identified with a galaxy . both this galaxy and a fainter galaxy , about @xmath51 nw , are elongated in the same direction . the radio emission is extended about @xmath51 , about the size of the galaxy disk . the light profile of the galaxy is exponential , suggesting two interacting galaxies @xcite . * * no 13 . j131214 + 423730 : * identified with a faint blue galaxy . it is too faint to be classified further . * * no 14 . j131215 + 423702 : * identified with a bright , distorted galaxy with z=0.322 ( paper i ) . there may be a bridge of optical emission between this galaxy and a fainter galaxy @xmath51 to the west . the galaxy shows a bulge population and a strong @xmath55 profile . it is relatively blue in color and shows narrow emission lines . @xcite . * * no 15 . j131215 + 423901 : * identified with a quasar with z=2.561 ( paper i ) . the radio source is slightly extended with an angular size of @xmath51 . * j131216 + 423921 : * empty field associated with the bright radio source . another faint radio source , about @xmath56 south with a flux density of @xmath57jy ( source b in table 1 ) , is probably identified with a i=24 galaxy . the two sources are probably unrelated . * * no 17 . j131217 + 423912 : * identified with the brighter of a pair of two overlapping galaxies . the fainter galaxy , @xmath5 to the nw is 1.5 mag fainter and is clearly distorted . the velocity profile of the galaxy is exponential @xcite and the galaxy is very red . * * no 18 . j131217 + 423930 : * empty field . * * no 19 . j131218 + 423807 : * a faint radio source , not in the complete sample , is probably identified with a faint 25 mag galaxy or galaxies . a brighter 23-mag galaxy about @xmath34 to the west appears unrelated . * j131218 + 423843 : * identified with the brighter pair of two nearly overlapping galaxies . the fainter galaxy , @xmath5 to the se is 1.5 mag fainter and is is more distorted than the brighter galaxy , which is very red . * j131219 + 423608 : * identified with a distorted galaxy . the radio source is extended @xmath58 in position angle @xmath59 . from the higher resolution observations at 1.4 ghz @xcite , the radio emission is composed of two small - diameter sources whose positions are indicated by the two black ellipses in the image . since the two radio components lie in about the same orientation as the optical object , they are probably both associated with the galaxy . the cause of this non - nuclear radio emission is unknown . the radio objects may be a small compact double source , resembling a fr-1 morphology , or they maybe associated with individual galactic objects far from the galactic center . * * no 22 . j131219 + 423631 : * the higher resolution observations at 1.4 ghz suggest that this extended radio source is probably composed of two separated radio sources @xcite . the fainter radio component to the nw is coincident with a bright , elliptically - shaped galaxy . the brighter radio component to the se is unidentified with a limit of i=26 mag . we believe these two radio components are unrelated , although the unidentified source could be associated with a galactic component in the halo of the bright galaxy . the light profile of the galaxy is exponential , indicating a disk - dominated galaxy @xcite . * * no 23 . j131220 + 423923 : * empty field . however , this radio sources is only @xmath60 west of the bright 18-m galaxy associated with j131221 + 423923 ( no . 27 ) @xcite . the question arises if this radio source is associated with a galactic component in the halo of this bright galaxy ? * * no 24 . j131220 + 423704 : * empty field . * * no 25 . j131220 + 424029 : * empty field . * * no 26 . j131220 + 423535 : * empty field . * * no 27 . j131221 + 423923 : * identified with an 18-m barred spiral galaxy with z=0.302 ( paper i ) . the radio source is about @xmath60 in size , contained within the disk of the galaxy . the galaxy disk is distorted ; it is blue in color and there are narrow emission lines @xcite . * * no 28 . j131221 + 423722 : * identified with an 18-m , edge - on , barred spiral galaxy with z=0.180 ( paper i ) . the galaxy has a relatively blue color and shows weak emission lines . more details are given elsewhere @xcite . * * no 29 . j132121 + 423827 : * identified with the middle optical peak in a chain of three or four galaxies comprising one of the most distorted optical regions associated with a detected radio source . the light profile of the galaxy is exponential , indicating a disk galaxy . @xcite . * * no 30 . j131222 + 423813 : * identified with a quasar with z=2.561 , i.e. the same redshift for j131315 + 423901 ( no 15 ) . they are probably in the same distant cluster or group of objects . with a separation of @xmath61 they are unlikely to be a gravitational - lens pair . * * no 31 . j131223 + 423908 : * identified with the brighter of a pair of galaxies , separated by @xmath33 . the brighter galaxy is somewhat distorted and the fainter galaxy is slightly extended towards the brighter . the radio emission is also slightly extended , although well within the disk of the brighter galaxy . the velocity profile of the galaxy is exponential and its color is very red @xcite . * * no 32 . j131223 + 423712 : * identified with the brighter of a pair of overlapping galaxies , separated by @xmath52 . two other overlapping galaxies lie @xmath60 to the east , and all four may be connected by a faint optical bridge . the radio emission is coincident with the galactic nucleus of the bright galaxy and is @xmath62 in angular size . higher resolution observations at 1.4 ghz @xcite give a size of @xmath46 in pa=@xmath59 . this radio source may be associated with an interacting system of galaxies . the redshift of both bright galaxies is 0.401 ( paper i ) . the galaxy containing the radio source has little significant bulge population , contains weak emission lines and is rather red @xcite . * * no 33 . j131223 + 423525 : * identified with a bright , somewhat elliptical galaxy with a large distorted halo . * * no 34 . j131224 + 423804 : * a faint radio source , not in the complete sample , which is identified with a faint object or objects . however , the identification is too faint to be further categorized . * * no 35 . j131225 + 424656 : * possibly identified with a i=25 mag object , although the radio / optical offset is larger than expected . * * no 36 . j131225 + 424103 : * a relatively bright radio source which lies @xmath52 south of the i=22 mag ir galaxy and @xmath34 north of an i=20 mag asymmetric galaxy . the radio source may be associated with a faint extension south from the ir galaxy with i=25.5 mag . the faint optical extensions associated with the two bright galaxies indicate they are probably interacting with each other . * * no 37 . j131225 + 423941 : * a bright radio source which may be associated with an object just at the detection limit . it is @xmath46 north of an i=23.5 mag galaxy . this radio source lies near the mds field edge . * * no 38 . j131226 + 424227 : * identified with an i=18 mag , slightly elliptical galaxy . the radio source is not in the complete sample . * * no 39 . j131227 + 423800 : * identified with a i=18mag edge - on galaxy which is @xmath60 sse of an i=17 mag elliptical galaxy ( the contours are distorted near this galaxy ) . these two galaxies overlap , although there is no appreciable distortion ; hence , their interaction is not certain . the light profile of the galaxy is exponential and its color is unusually red , possibly a dusty , disk galaxy @xcite . * * no 40 . j131232 + 424038 : * identified with an i=24.8 mag object which may be extended in the east - west direction . the identification is too faint to be further classified . * j131236 + 424027 : * this radio source is displaced in pa=@xmath63 from the nucleus of this i=18 mag galaxy . it is not in the complete radio sample so the four - sigma radio / optical offset might be spurious . the galaxy is elliptical and symmetric with little distortion . however , the galaxy orientation is also in pa=@xmath63 , which somewhat supports the radio / optical offset . * * no 42 . j131239 + 423911 : * identified with an i 24.5 mag galaxy which is too faint to be further classified . * * weak source a. j131213 + 423946 : * identified with an i=20.0 mag circular galaxy . another galaxy with i=22.5 mag lies @xmath34 to the south - west . * * weak source b. j131213 + 423826 : * identified with a faint galaxy which is too faint to be further classified . * * weak source c. j131215 + 423913 : * identified with a diffuse , extended galaxy . * * weak source d. j131216 + 423920 : * radio source is @xmath52 east of an i=23.8 mag , somewhat extended , galaxy . * * weak source e. j131219 + 423932 : * radio source near an i=24.0 mag galaxy . brighter galaxies lie @xmath52 to the south - west and @xmath51 to the north - east . * * weak source f. j131221 + 423836 : * companion galaxy lies @xmath51 east - north - east . * * weak source g. j131225 + 423742 : * bright elliptical galaxy with i=18.0 mag . radio source may be extended in the same direction . * * weak source h. j131225 + 423907 : * identified with a slightly extended i=22.5 mag galaxy . we have gathered data from five deep vla integrations at 8.4 ghz in order to study the radio spectral properties , the optical properties and the source count of the micro - jansky radio population . parameters for the five surveys made over the past decade are summarized in table 3 , which includes references to lower frequency radio data and optical data as well . column 5 gives the integration time in each field , and column 6 gives the complete survey detection level . column 7 , 8 and 9 show the total number of sources listed in each survey , the number of sources in the complete sample above the detection level , and the number of sources expected based on the best source count described below ; respectively . although the five catalogs contain a total of 201 sources , only 89 of them are in a complete flux - limited 8-ghz sample , and only these are listed in table 4 . the table contains the source properties that are discussed in this section . the last column indicates the detection of x - ray emission from the sources in the hubble and sa13 fields @xcite accurate spectral indices of the 89 sources have been determined by comparing our results to observations at frequencies lower than 8.4 ghz ( see references at the end of table 3 ) . only seven of them are lower limits ( one in the sa13 field and six in the hubble fields ) . in figure 5 we show four histogram comparison of the spectral index distributions associated with various radio and optical properties . in figure 5a we show the spectral index distribution for all 89 sources and separately for the 34 sources in the sa13 field . there is no significant difference between the two distributions . in the total distribution , the spectral range of the seven sources which were not detected above twice the 1.4-ghz noise level upper limits on the spectral index , are indicated by the arrows in the histogram . since the observations at the various frequencies were made at different times , source variability could affect the spectral index calculation , but variability of micro - jansky sources is not common @xcite . about one - quarter of the sources have a spectral index in the range @xmath64 . the distribution drops precipitously at lower spectral indices ( steeper spectra ) , but more slowly for high spectral indices ( flatter spectra ) . the median value is about @xmath65 . only a few objects have an inverted spectral index with @xmath66 , although some sources with only a 1.4 ghz flux density upper limit could fall in this category . the spectral distribution shows a higher proportion of steep spectrum objects than reported from less sensitive , 8.4-ghz micro - jansky surveys where the median spectral index was @xmath67 @xcite . the reason for this difference is illustrated in figure 5b where the spectral distribution is shown for 46 sources with s@xmath68jy and for 43 weaker sources with s@xmath69jy . _ the lower flux density sources are more likely to have a steep spectral index than the high flux density sources . _ the likelihood that the distribution of two samples chosen randomly would differ by at least this much is only 18% . this change explains why a flatter average spectrum was found in the less sensitive , earlier , 8.4 ghz surveys . possible causes of the change of spectral properties below @xmath6jy are given in @xmath186 . figure 5c shows that those objects which are fainter than i=25.5 mag are predominantly steep radio spectrum objects . although there are only 17 sources in this faint galaxy sample ; no object has a spectral index flatter than @xmath70 and most are steeper than @xmath71 . figure 5d is a plot of the spectral index distribution for the 41 galaxies which have measured colors . as a discriminatory of the color for the sa13 field we used the ( b - i ) magnitudes ; for the hubble field we used the [ i-(h+k)/2 ] magnitudes . in this sample of 41 sources , 19 galaxies are significantly reddened and their spectral index distribution is also shown . the slight difference in the radio spectral index between the two groups suggests that fewer of reddened objects have a flat spectrum . this trend was also found from vla observations at 5 ghz and 1.4 ghz and from the canadian - france redshift survey @xcite . this correlation , although weaker than that in figure 5c , is compatible since reddened objects would tend to have relatively faint i - band emission . nine of the sources in table 4 are identified with x - ray sources @xcite . two are in the sa13 field . in contrast , seven x - ray identifications have been made in the hubble field and these associations show no trends in radio spectral index or optical counterpart , and are identified with galaxies of various types . the density of sources ( source count ) is generally specified as @xmath72 where @xmath73 is the number of sources per solid angle per flux density interval @xmath74 at a flux density @xmath75 . for a flat space which is uniformly filled with radio sources and ignoring the attenuation effects of redshift , the slope @xmath76 . in order to determine the two parameters , @xmath77 and @xmath78 , from the 89 source sample determined from five surveys with different sensitivities , we used a modification of the maximum likelihood method ( mlm ) originally developed for a uniformly obtained sample @xcite . for a range of test models , we chose reasonable parameters , @xmath77 and @xmath78 , and calculated the relative probability that this model would produce the observed flux density distribution of the 89 sources , using the appropriate constraints for each survey . the best fit model was the one with the highest probability , and the error estimate was determined by the set of models that were at least 37% ( 1/e ) as likely as the best model . the number of micro - jansky sources , @xmath73 ( arcmin@xmath79jy@xmath80 ) , is given by @xmath81 the corresponding integral count , @xmath82 ( arcmin@xmath83 is @xmath84 where n is the number of sources above a flux density @xmath75 in micro - jansky . the value , 40 @xmath9jy , is the mean flux density of the sources , and in this form , the two parameters @xmath77 and @xmath78 are nearly orthogonal . above @xmath85jy the source density is 0.1 ( arcmin)@xmath11 , or @xmath86 sources sr@xmath80 . at the minimum detection level of these surveys , @xmath3jy , there are 0.64 sources ( arcmin)@xmath11 , or @xmath87 sources sr@xmath80 . extrapolation to weaker flux density with the same slope gives 6 sources ( arcmin)@xmath11 above @xmath88jy , corresponding to an average source separation of @xmath89 . the density of sources is still far from the natural confusion limit when sources begin to blend with each other . if @xmath90 is the proportion of sky covered with emission , significant blending occurs for @xmath91 . if we assume an average source size of @xmath52 . then then @xmath92 . extrapolating to @xmath88jy and @xmath93jy gives @xmath94 and @xmath95 , respectively . unless the average angular size of most sources is less than @xmath96 , radio source blending will become a problem below about @xmath97jy . recent results from merlin , however , shows that some faint radio sources contain a significant part of their emission within an angular size of @xmath4 @xcite . the plot of the _ integral _ source count at 8.4 ghz is shown in figure 6 . the ordinate scale has been normalized to the euclidean integral slope of @xmath98 . at any point in the plot , the ordinate value times s@xmath99 gives the number of sources ( arcmin)@xmath11 with a flux density greater than @xmath100jy . the count derived from the mlm calculation , describe above , is shown by the solid line . this line extends to @xmath88jy , although we can not exclude that the slope of the counts change below our detection limit of @xmath3jy . the plotted points at the micro - jansky levels have been obtained by gridding the data in table 4 into five flux density bins . the results of this binned data are shown in table 5 . the mlm fit is clearly compatible with the gridded data , as it should be . the number of sources in each of the five bins listed in column 4 agrees reasonably well with the number predicted by the mlm fit , given in column 5 . we have added in the count of sources at 8.4 ghz at high flux densities @xcite . there are no extensive surveys of sources at 8.4 ghz of intermediate flux density sources between one and eight milli - janskys . the source count at 8.4 ghz derived at micro - jansky levels is consistent with that at the milli - jansky and jansky levels . extrapolation of the micro - jansky count falls slightly under the milli - jansky points , the excess being produced by the rapid evolution of the brightest radio sources in the sky . the dotted line shows the best fit count obtained from the lynx and cepheus fields only @xcite which contained only 20 sources to a detection level of @xmath101jy . the ` old ' count and the ` new ' count differ at the 1.4-@xmath16 level near @xmath85jy . there may be several reasons for the discrepancy : first , the difference may be real and reflect the anisotropic distribution of sources in the sky on scales of tens of arcmin . such differences between the number of sources among radio fields have already been noted by others at various frequencies ( windhorst et al . 1985 ; windhorst et al . 1993 ; hopkins et al . 1998 ; hopkins et al . 1999 ; richards 2000 ) . second , the optical and radio selection criteria for the various fields were not the same . the hubble and sa13 fields were deliberately selected to be radio quiet above 0.5 mjy , whereas the hercules field contained a known radio source of about 10 mjy . the optical constraints for the hdf selection ( no relatively bright objects ) may also be relevant . however , since most of the radio sources and statistics come from sources which are in the 10 to 200 @xmath9jy flux density range at 8 ghz , the systematic effect of the avoidance of radio fields with sources stronger than 500 @xmath9jy is unclear . third , the lynx and cepheus fields were observed with lower resolution than the hubble and sa13 fields . although the effects of source blending were considered for the lynx and cepheus fields @xcite , the blending may be more important than originally thought . the average angular size of the sources in these two fields were also larger than in other fields , and perhaps a consequence of the blending of weak sources . regardless of the reasons for the somewhat different count of sources in the lynx and cepheus fields , the new results are based on more than four times the number of sources found in five different areas and with higher resolution . when extrapolated to @xmath88jy , the new counts give a density of @xmath102 ( arcmin)@xmath11 , compared with the previous value of @xmath103 ( arcmin)@xmath11 . this difference is about 1.8-@xmath16 and should not be regarded as significant . the average sky brightness at 8.4 ghz , contributed by sources with flux densities above @xmath3jy is @xmath104k . extrapolating to weaker sources , the sky brightness will increase as s@xmath105 , assuming a constant slope @xmath78 . using @xmath106 , the estimated sky brightness from discrete sources increases by only a factor of 2 for a decrease of 1000 in flux density . hence , the sky brightness at 8.4 ghz contributed by discrete sources down to the nano - jansky level is unlikely to exceed 0.001 k unless there is a new population of weaker sources , allowing the slope @xmath78 to become more negative ( i.e. steeper ) than @xmath107 . figure 7 shows the distribution of the i - magnitudes of the 63 sources in the complete samples in the sa13 field plus the hubble deep and flanking fields . the density of sources with radio identifications and optical magnitudes between i=17.5 and 20.5 mag is about 30% of the density of all galaxies within this magnitude range @xcite . for galaxies fainter than i=20.5 mag the distribution remains relatively constant per magnitude interval to i=26.0 mag , with about 20% of the sources unidentified below this magnitude . this distribution is similar to that found in other investigations of the identification of weak radio sources @xcite . analysis of several hst fields shows that 34%@xmath1089% of all galaxies brighter than i=23.3 mag are in binary systems . this is much higher than the 7% found for the galaxy population brighter than i=15 mag , which comprise the local population @xcite . thus , galaxy interaction appears to be a strong function of cosmological time , and such galaxy interactions may be the main cause of the strong evolution of radio sources with redshift @xcite . this connection between galaxy interaction and radio emission is supported in these data . for the complete sample of 63 radio sources in the sa13 and hubble fields , 37 are identified with galaxies brighter than i=23.3 mag . seventeen of these identifications are classified as _ g / b _ meaning that at least one other galaxy lies within @xmath60 of the radio galaxy , possibly indicating galaxy interactions . thus , 46% of these radio identifications are associated with multiple systems . this percentage is somewhat larger than the 34% of galaxies that are found in multiple systems below @xmath109 mag , and suggests that the radio emission is even more snhanced by galaxy interactions . a more complete analysis of the optical properties of the radio sources in the sa13 and hubble fields is given elsewhere @xcite . we have reported the results from two new 8.4 ghz vla surveys of micro - jansky radio sources . in one of the fields , sa13 , the rms noise is @xmath2jy and we have detected 49 sources of which 34 are in a complete sample with a flux density above @xmath3jy . six sources were detected above @xmath6jy in the hercules field . in tables 1 and 2 we present the radio catalog for these two fields . comments and a detailed radio / optical comparison are given for 42 sources in the sa13 field using deep hst and kpno images . the results from these two fields double the number of 8.4-ghz radio sources now known at micro - jansky levels . in the second part of this paper , we have compiled a list of 89 sources in a flux density complete sample from five deep vla surveys at 8.4 ghz . this source list is dominated by observations of the sa13 field ( detection level of @xmath3jy ) and the hubble deep field plus the flanking fields ( detection level of @xmath110jy ) . we have used other optical and radio data in order to determine the properties of these weak radio sources . the count of radio sources follows an approximate power - law with a slope @xmath111 from 10 mjy to less than @xmath112jy , with good continuity to the count at the jansky level ( see figure 6 ) . there are no significant differences in the number of sources detected in each field , as shown by the comparison of columns 8 and 9 in table 3 , which indicate the number of sources detected versus the number expected from the best fit to the count . the average sky brightness temperature contributed by the micro- and nano - jansky radio sources is extrapolated to be less than 0.001 k as long as the slope @xmath78 does not become steeper than @xmath107 . most of the sources are unresolved with @xmath33 or even @xmath34 resolution . for the 63 sources in the hubble and sa13 samples , 15 sources ( 24% ) are larger than about @xmath51 , and only four sources are larger than @xmath60 . thus , the angular size of the micro - jansky sources at 8.4 ghz appears statistically somewhat smaller than the angular size at 1.4 ghz @xcite . the median spectral index of these 15 extended sources is @xmath113 , steeper than than our general sample and consistent with an increasing angular size with lower frequency . the correlation of large angular size with steeper spectral index is seen for all radio sources , regardless of their flux density . the combination of emission from opaque small - diameter components and large components showing an aging synchrotron ( steep ) spectrum probably cause this difference in the correlation of angular size and spectral index at the two different frequencies . higher resolution observations are needed in order to resolve the majority of the micro - jansky radio sources @xcite . the radio sources are identified with a variety of galaxy types . only two of the radio sources are identified with quasars and none with stars . we estimate that about 30% of all galaxies brighter than i=21 mag can be detected at the @xmath112jy level at 8.4 ghz . for fainter magnitudes , the radio source identifications are spread relatively evenly , per magnitude interval , with about 20% of the radio sample fainter than i=26 mag . the centroid of the radio emission is nearly always located within @xmath4 of the galaxy nucleus . the few exceptions in the sa13 field are : j131220 + 423923 is an empty field which is displaced @xmath60 from an i=18-mag galaxy ; j131219 + 423608 contains two radio components each displaced about @xmath52 on opposite sides of a i=22 mag galaxy nucleus and is possibly a small double radio source ( see notes ) ; j131225 + 424103 is displaced @xmath52 south of an i=22 mag galaxy ( see notes ) ; j131239 + 423911 is displaced @xmath96 from the galaxy centroid , but well within the disk component . the likelihood that a micro - jansky radio source will be associated with a galaxy , between i=18 to 22 mag with at least one neighbor closer than @xmath60 , is nearly 50% . this percentage is somewhat larger than the 34% of all such galaxies which are in groups . this suggests that radio source luminosity and/or density evolution is enhanced somewhat within galaxy groups which may be interacting . the spectral index distribution of the micro - jansky radio sources at 8.4 ghz shows one clear trend ; the spectral index distribution steepens for sources below @xmath6jy ( see figure 5b ) . somewhat less conclusively , the spectral index distribution may also steepen for sources with optical counterparts fainter than i=25.5 mag ( see figure 5c ) . the change in the source count and spectral properties of radio sources fainter than 1 mjy is well - documented . below this level , star - forming galaxies begin to dominate the source population over agn . the cause for a further change in spectral properties less than @xmath6jy may be associated with the different evolution of agn and star - burst galaxies , but a definitive answer must await for additional observations of the higher energy photons emitted from these fainter radio sources , and measurement of their radio size and internal complexity . both the agn and the star - burst phenomena produce a wide range in spectral index , and can not be used as a good discriminator between the two types . a steepening of the spectral index below @xmath6jy could arise in several ways : ( 1 ) the opaque emission ( always with flat spectrum ) from agn components in weak sources may be a smaller proportion of the total emission ; ( 2 ) the star - burst phenomenon may produce steeper radio spectra for the weak sources , possibly because of synchrotron aging associated with larger magnetic fields in the weaker , distant sources . the national radio astronomy observatory is a facility of the national science foundation , operated under cooperative agreement by associated universities , inc . the space telescopes science institute is operated by associated universities for research in astronomy , inc . , under contract to nasa . rbp was supported in part by nsf grant ast96 - 16971 and by the keck northeast astronomy consortium . ear was supported by a hubble fellowship . 1 & & j131157@xmath114423910 & 109.3 @xmath108 8.1 & 14.8 & @xmath115 & 13 11 57.499 @xmath108 0.02 & 42 39 10.16 @xmath108 0.21 & g & @xmath11626.0 & @xmath117 & @xmath118 23.5 + 2 & & j131157@xmath114423630 & 196.6 @xmath108 10.2 & 23.2 & @xmath115 & 13 11 57.903 @xmath108 0.01 & 42 36 30.15 @xmath108 0.14 & ef & @xmath11626.0 & @xmath117 & @xmath119 25.5 + 3 & & * j131203@xmath114424030 & 213.2 @xmath108 8.7 & 36.7 & @xmath115 & 13 12 03.092 @xmath108 0.01 & 42 40 30.68 @xmath108 0.10 & g & @xmath11725.5 & @xmath117 & @xmath118 24.3 + 4 & & j131203@xmath114423331 & 886.1 @xmath108 60.9 & 15.6 & @xmath115 & 13 12 03.508 @xmath108 0.02 & 42 33 31.72 @xmath108 0.20 & g & @xmath11724.0 & @xmath117 & @xmath118 22.2 + 5 & & j131205@xmath114423851 & 12.4 @xmath108 2.6 & 5.0 & @xmath120 & 13 12 05.596 @xmath108 0.06 & 42 38 51.15 @xmath108 0.60 & ef & @xmath11626.0 & @xmath117 & @xmath119 25.5 + 6 & & j131207@xmath114423945 & 14.0 @xmath108 2.8 & 5.4 & @xmath121 & 13 12 07.645 @xmath108 0.06 & 42 39 45.87 @xmath108 0.56 & ef & @xmath11626.0 & @xmath117 & @xmath119 25.5 + 7 & & j131209@xmath114424217 & 93.4 @xmath108 17.8 & 5.8 & @xmath121 & 13 12 09.068 @xmath108 0.05 & 42 42 17.49 @xmath108 0.52 & g & @xmath11725.5 & @xmath117 & @xmath118 23.4 + 8 & & j131211@xmath114424053 & 717.0 @xmath108 21.9 & 183.0 & @xmath115 & 13 12 11.029 @xmath108 0.01 & 42 40 53.66 @xmath108 0.05 & g / b & @xmath11724.5 & @xmath117 & @xmath118 20.8 + 9 & & * j131213@xmath114423555 & 12.2 @xmath108 2.6 & 4.7 & @xmath120 & 13 12 13.303 @xmath108 0.07 & 42 35 55.76 @xmath108 0.64 & g / b & @xmath11721.7 & @xmath117 & @xmath118 19.5 + 10 & & j131213@xmath114424129 & 48.4 @xmath108 5.6 & 9.2 & @xmath121 & 13 12 13.446 @xmath108 0.04 & 42 41 29.44 @xmath108 0.33 & ell & @xmath11721.9 & @xmath117 & @xmath118 18.7 + 11 & & j131213@xmath114423706 & 9.4 @xmath108 1.7 & 5.6 & @xmath121 & 13 12 13.858 @xmath108 0.06 & 42 37 06.50 @xmath108 0.53 & g ? & @xmath117 & @xmath11626.8 & @xmath118 25.5 + 12 & 1 & * j131214@xmath114423821 & 6.5 @xmath108 1.6 & 4.3 & @xmath122 & 13 12 14.518 @xmath108 0.08 & 42 38 21.86 @xmath108 1.02 & g / b & @xmath11723.7 & @xmath11723.4 & @xmath118 21.8 + 13 & 2 & * j131214@xmath114423730 & 6.2 @xmath108 1.6 & 4.4 & @xmath120 & 13 12 14.666 @xmath108 0.09 & 42 37 30.96 @xmath108 1.13 & g & @xmath117 & @xmath11726.3 & @xmath118 25.2 + 14 & 3 & j131215@xmath114423702 & 13.4 @xmath108 1.7 & 7.8 & @xmath121 & 13 12 15.130 @xmath108 0.04 & 42 37 02.70 @xmath108 0.39 & g / b & @xmath11721.4 & @xmath11720.7 & @xmath118 19.7 + 15 & 4 & j131215@xmath114423901 & 26.6 @xmath108 1.8 & 14.9 & @xmath123 & 13 12 15.280 @xmath108 0.02 & 42 39 01.16 @xmath108 0.21 & q & @xmath11718.6 & @xmath11718.4 & @xmath118 17.8 + 16 & 5 & j131216@xmath114423921 & 29.5 @xmath108 2.0 & 15.4 & @xmath121 & 13 12 16.083 @xmath108 0.02 & 42 39 21.47 @xmath108 0.20 & ef & @xmath117 & @xmath11626.8 & @xmath119 25.8 + 17 & 6 & j131217@xmath114423912 & 20.1 @xmath108 1.8 & 11.7 & @xmath115 & 13 12 17.181 @xmath108 0.03 & 42 39 12.15 @xmath108 0.26 & g / b & @xmath11722.5 & @xmath11722.2 & @xmath118 20.0 + 18 & 7 & j131217@xmath114423930 & 15.5 @xmath108 1.9 & 8.6 & @xmath121 & 13 12 17.601 @xmath108 0.04 & 42 39 30.49 @xmath108 0.35 & ef & @xmath117 & @xmath11626.8 & @xmath119 25.8 + 19 & 8 & * j131218@xmath114433907 & 6.4 @xmath108 1.6 & 4.3 & @xmath120 & 13 12 18.311 @xmath108 0.09 & 42 39 07.79 @xmath108 1.07 & g ? & @xmath117 & @xmath11626.8 & @xmath118 25.2 + 20 & 9 & j131218@xmath114423843 & 25.0 @xmath108 1.7 & 14.8 & @xmath124 & 13 12 18.437 @xmath108 0.02 & 42 38 43.92 @xmath108 0.21 & g / b & @xmath11724.0 & @xmath11723.1 & @xmath118 20.3 + 21 & & j131219@xmath114423608 & 26.4 @xmath108 2.4 & 11.4 & @xmath125 & 13 12 19.838 @xmath108 0.03 & 42 36 08.91 @xmath108 0.27 & g & @xmath11723.8 & @xmath117 & @xmath118 22.1 + 22 & 10 & j131219@xmath114423831 & 11.6 @xmath108 1.6 & 6.7 & @xmath126 & 13 12 19.932 @xmath108 0.05 & 42 38 31.39 @xmath108 0.45 & g & @xmath11723.0 & @xmath11722.2 & @xmath118 20.4 + 23 & 12 & j131220@xmath114423923 & 14.5 @xmath108 1.9 & 7.7 & @xmath121 & 13 12 20.016 @xmath108 0.04 & 42 39 23.97 @xmath108 0.39 & ef ? & @xmath117 & @xmath11626.8 & @xmath119 25.8 + 24 & 11 & j131220@xmath114423704 & 13.9 @xmath108 1.7 & 8.4 & @xmath121 & 13 12 20.185 @xmath108 0.04 & 42 37 04.02 @xmath108 0.36 & ef & @xmath117 & @xmath11626.8 & @xmath119 25.8 + 25 & & j131220@xmath114424029 & 33.7 @xmath108 3.0 & 11.5 & @xmath115 & 13 12 20.236 @xmath108 0.03 & 42 40 29.67 @xmath108 0.26 & ef & @xmath11626.0 & @xmath117 & @xmath119 25.5 + 26 & & j131220@xmath114423535 & 20.4 @xmath108 3.0 & 6.7 & @xmath121 & 13 12 20.871 @xmath108 0.05 & 42 35 35.33 @xmath108 0.45 & ef & @xmath11626.0 & @xmath117 & @xmath119 25.5 + 27 & 13 & j131221@xmath114423923 & 11.5 @xmath108 1.9 & 5.4 & @xmath124 & 13 12 21.109 @xmath108 0.06 & 42 39 23.49 @xmath108 0.55 & sp & @xmath11720.2 & @xmath11719.2 & @xmath118 18.0 + 28 & 15 & j131221@xmath114423722 & 10.0 @xmath108 1.7 & 5.9 & @xmath121 & 13 12 21.397 @xmath108 0.05 & 42 37 22.91 @xmath108 0.51 & sp & @xmath11720.0 & @xmath11719.0 & @xmath118 17.8 + 29 & 16 & j131221@xmath114423827 & 18.5 @xmath108 1.7 & 10.2 & @xmath121 & 13 12 21.839 @xmath108 0.03 & 42 38 27.55 @xmath108 0.30 & g / b & @xmath117 & @xmath11724.3 & @xmath118 22.7 + 30 & 17 & j131222@xmath114423813 & 10.4 @xmath108 1.6 & 5.6 & @xmath127 & 13 12 22.428 @xmath108 0.06 & 42 38 13.47 @xmath108 0.54 & q & @xmath11721.3 & @xmath11721.0 & @xmath118 19.6 + 31 & 18 & j131223@xmath114423908 & 19.3 @xmath108 2.0 & 9.2 & @xmath128 & 13 12 23.298 @xmath108 0.04 & 42 39 08.35 @xmath108 0.33 & g / b & @xmath11722.6 & @xmath11721.6 & @xmath118 19.3 + 32 & 19 & j131223@xmath114423712 & 27.7 @xmath108 2.0 & 15.2 & @xmath115 & 13 12 23.693 @xmath108 0.02 & 42 37 12.09 @xmath108 0.20 & g / b & @xmath11721.3 & @xmath11720.6 & @xmath118 18.8 + 33 & & j131223@xmath114423525 & 65.5 @xmath108 4.1 & 17.7 & @xmath115 & 13 12 23.999 @xmath108 0.02 & 42 35 25.57 @xmath108 0.18 & ell & @xmath11722.4 & @xmath117 & @xmath118 18.9 + 34 & 20 & * j131224@xmath114433804 & 6.2 @xmath108 1.6 & 4.2 & @xmath120 & 13 12 24.133 @xmath108 0.09 & 42 38 04.89 @xmath108 1.02 & g & @xmath117 & @xmath11725.5 & @xmath118 24.0 + 35 & & j131225@xmath114423656 & 12.9 @xmath108 2.1 & 6.5 & @xmath121 & 13 12 25.201 @xmath108 0.05 & 42 36 56.66 @xmath108 0.47 & g ? & @xmath11626.8 & @xmath117 & @xmath118 25.0 + 36 & & j131225@xmath114424103 & 34.9 @xmath108 4.8 & 6.7 & @xmath121 & 13 12 25.228 @xmath108 0.05 & 42 41 03.44 @xmath108 0.45 & g ? & @xmath11626.0 & @xmath117 & @xmath118 25.5 + 37 & 23 & j131225@xmath114423941 & 200.3 @xmath108 6.5 & 79.3 & @xmath115 & 13 12 25.743 @xmath108 0.01 & 42 39 41.58 @xmath108 0.06 & g / b & @xmath11626.5 & @xmath117 & @xmath118 25.5 + 38 & & * j131226@xmath114424227 & 318.3 @xmath108 25.3 & 13.0 & @xmath115 & 13 12 26.287 @xmath108 0.03 & 42 42 27.43 @xmath108 0.24 & ell & @xmath11720.9 & @xmath117 & @xmath118 17.7 + 39 & 24 & j131227@xmath114423800 & 25.6 @xmath108 2.3 & 11.6 & @xmath115 & 13 12 27.532 @xmath108 0.03 & 42 38 00.23 @xmath108 0.26 & g / b & @xmath11722.1 & @xmath11722.0 & @xmath118 19.8 + 40 & & j131232@xmath114424038 & 165.7 @xmath108 8.9 & 20.5 & @xmath120 & 13 12 32.797 @xmath108 0.02 & 42 40 38.42 @xmath108 0.15 & g & @xmath11626.0 & @xmath117 & @xmath118 24.8 + 41 & & * j131236@xmath114424027 & 54.6 @xmath108 11.5 & 4.8 & @xmath115 & 13 12 36.044 @xmath108 0.07 & 42 40 27.66 @xmath108 0.63 & g & @xmath11720.2 & @xmath117 & @xmath118 18.3 + 42 & & j131239@xmath114423911 & 66.2 @xmath108 12.7 & 5.1 & @xmath115 & 13 12 39.705 @xmath108 0.06 & 42 39 11.32 @xmath108 0.60 & g & @xmath11625.0 & @xmath117 & @xmath118 24.5 + a & & * j131213@xmath114423932 & 4.2 @xmath108 1.7 & 2.5 & @xmath129 & 13 12 13.521 @xmath108 0.26 & 42 39 32.16 @xmath108 1.33 & g & @xmath117 & @xmath117 & @xmath118 20.0 + b & & * j131213@xmath114423826 & 4.8 @xmath108 1.6 & 2.7 & @xmath129 & 13 12 13.583 @xmath108 0.22 & 42 38 26.68 @xmath108 1.13 & g & @xmath117 & @xmath117 & @xmath118 24.5 + c & & * j131215@xmath114423913 & 6.7 @xmath108 1.6 & 3.8 & @xmath120 & 13 12 15.847 @xmath108 0.12 & 42 39 13.22 @xmath108 0.70 & g & @xmath117 & @xmath117 & @xmath118 23.2 + d & & * j131216@xmath114423920 & 4.8 @xmath108 1.6 & 2.8 & @xmath130 & 13 12 16.521 @xmath108 0.25 & 42 39 20.40 @xmath108 2.20 & g & @xmath117 & @xmath117 & @xmath118 23.8 + e & & * j131219@xmath114423932 & 5.0 @xmath108 1.9 & 2.9 & @xmath130 & 13 12 19.092 @xmath108 0.22 & 42 39 32.20 @xmath108 1.80 & g / b & @xmath117 & @xmath117 & @xmath118 24.0 + f & & * j131221@xmath114423836 & 4.8 @xmath108 1.8 & 2.8 & @xmath127 & 13 12 21.794 @xmath108 0.13 & 42 38 36.96 @xmath108 1.40 & g & @xmath117 & @xmath117 & @xmath118 22.4 + g & & * j131225@xmath114423742 & 5.5 @xmath108 1.8 & 3.3 & @xmath131 & 13 12 25.065 @xmath108 0.17 & 42 37 42.33 @xmath108 1.50 & g & @xmath117 & @xmath117 & @xmath118 18.0 + h & & * j131225@xmath114423907 & 5.0 @xmath108 1.8 & 3.0 & @xmath129 & 13 12 25.549 @xmath108 0.15 & 42 39 07.01 @xmath108 1.60 & g & @xmath117 & @xmath117 & @xmath118 22.5 + 1 & j171349@xmath114501610 & 1136 @xmath108 63 & 20.3 & @xmath115 & 17 13 49.077 @xmath108 0.02 & 50 16 10.49 @xmath108 0.16 + 2 & j171354@xmath114501547 & 153 @xmath108 22 & 6.7 & @xmath121 & 17 13 54.160 @xmath108 0.05 & 50 15 47.70 @xmath108 0.45 + 3 & * j171407@xmath114501547 & 41 @xmath108 8 & 4.9 & @xmath120 & 17 14 07.032 @xmath108 0.06 & 50 15 47.76 @xmath108 0.61 + 4 & j171411@xmath114501602 & 37 @xmath108 7 & 5.0 & @xmath120 & 17 14 11.951 @xmath108 0.06 & 50 16 02.42 @xmath108 0.60 & obj 18 , 23.5 m g / b + 5 & j171414@xmath114501530 & 6482 @xmath108 20 & 0.9 & @xmath132 & 17 14 14.754 @xmath108 0.00 & 50 15 30.46 @xmath108 0.05 & 53w002 , 22.1 m g / b + 6 & j171415@xmath114501535 & 49 @xmath108 7 & 8.5 & @xmath120 & 17 14 15.219 @xmath108 0.01 & 50 15 35.45 @xmath108 0.15 + 7 & j171416@xmath114501817 & 495 @xmath108 22 & 30.4 & @xmath115 & 17 14 16.771 @xmath108 0.01 & 50 18 17.10 @xmath108 0.11 + 8 & * j171424@xmath114501339 & 62 @xmath108 13 & 4.7 & @xmath120 & 17 14 24.613 @xmath108 0.07 & 50 13 39.78 @xmath108 0.64 + 9 & * j171436@xmath114501329 & 194 @xmath108 42 & 4.5 & @xmath120 & 17 14 36.222 @xmath108 0.07 & 50 13 29.56 @xmath108 0.67 + 10 & * j171442@xmath114501640 & 505 @xmath108 99 & 4.9 & @xmath120 & 17 14 42.629 @xmath108 0.07 & 50 16 40.09 @xmath108 0.62 + cepheus & 1988 - 89 & 03 10 00 & 80 00 00 & 30 & 20.3 & 36 & 6 & 10.2 & 1,2,3 + lynx & 1989 - 90 & 08 41 40 & 44 45 00 & 63 & 12.8 & 46 & 14 & 17.1 & 1,4 + hercules & 1996 & 17 14 15 & 50 15 30 & 12 & 35.0 & 10 & 6 & 5.5 & 0 + sa13 & 1994 - 95 & 13 12 17 & 42 38 05 & 159 & 7.5 & 50 & 34 & 30.9 & 0,5,6 + hubble & 1996 - 97 & 12 36 49 & 62 12 58 & 139 & 9.0 & 60 & 29 & 25.2 & 7,8,9 + lrrlrc j030755 + 801008 & 1221 & @xmath1331.0 & & @xmath117 + j030852 + 801409 & 449 & @xmath1340.2 & & @xmath117 + j030902 + 800955 & 54 & @xmath1340.2 & & @xmath117 + j030924 + 801159 & 34 & @xmath1330.4 & & @xmath117 + j031005 + 800824 & 52 & @xmath1330.2 & & @xmath117 + j031101 + 800943 & 293 & @xmath1330.6 & & @xmath117 + j084120 + 444453 & 58 & @xmath1330.5 & & @xmath117 + j084121 + 444318 & 289 & @xmath1330.7 & & @xmath117 + j084126 + 444648 & 62 & @xmath1330.6 & & @xmath117 + j084133 + 444544 & 20 & @xmath1330.7 & & @xmath117 + j084134 + 444409 & 50 & @xmath1330.2 & & @xmath117 + j084135 + 444442 & 21 & @xmath1330.8 & & @xmath117 + j084136 + 444443 & 32 & @xmath1330.6 & & @xmath117 + j084141 + 444509 & 32 & @xmath1330.6 & & @xmath117 + j084141 + 444452 & 35 & @xmath1331.0 & & @xmath117 + j084142 + 444613 & 110 & @xmath1330.6 & & @xmath117 + j084142 + 444718 & 115 & @xmath1330.2 & & @xmath117 + j084144 + 444534 & 62 & @xmath1330.8 & & @xmath117 + j084150 + 444545 & 31 & @xmath1331.1 & & @xmath117 + j084155 + 444640 & 78 & @xmath1330.9 & & @xmath117 + j123632 + 621105 & 22 & @xmath1350.1 & sp & @xmath11720.1 + j123634 + 621212 & 56 & @xmath1330.7 & g / b & @xmath11719.3 + j123634 + 621240 & 53 & @xmath1330.7 & g / b & @xmath11723.5 + j123637 + 621135 & 18 & @xmath1350.2 & sp & @xmath11718.2 + j123640 + 621010 & 29 & @xmath1330.4 & g & @xmath11725.0 + j123641 + 621142 & 19 & @xmath1330.3 & g / b & @xmath11722.7 + j123642 + 621331 & 80 & @xmath1330.9 & g ? & @xmath11725.8 & yes + j123642 + 621545 & 54 & @xmath1330.5 & g & @xmath11722.0 + j123644 + 621249 & 10 & @xmath1350.4 & g / b & @xmath11721.9 & yes + j123644 + 621133 & * 599 & @xmath1330.3 & ell & @xmath11720.5 & yes + j123646 + 621448 & * 25 & @xmath1330.8 & g / b & @xmath11726.0 + j123646 + 621445 & 13 & @xmath1331.0 & g & @xmath11723.9 + j123646 + 621404 & 190 & @xmath1180.0 & sp & @xmath11720.0 & yes + j123649 + 621313 & 22 & @xmath1330.7 & g / b & @xmath11721.8 & yes + j123651 + 621030 & 26 & @xmath1330.7 & sp & @xmath11720.7 + j123651 + 621221 & 17 & @xmath1330.7 & g & @xmath11727.8 & yes + j123652 + 621444 & 185 & @xmath1180.1 & ell & @xmath11719.4 + j123653 + 621139 & 15 & @xmath1330.8 & sp & @xmath11722.3 + j123655 + 621311 & 12 & @xmath1350.2 & g / b & @xmath11723.1 & yes + j123657 + 621455 & 15 & @xmath1350.2 & g & @xmath11722.8 + j123700 + 620908 & * 67 & @xmath1330.9 & g & @xmath11726.3 + j123701 + 621146 & * 30 & @xmath1330.7 & g / b & @xmath11725.4 + j123707 + 611408 & * 29 & @xmath1330.3 & g & @xmath11726.4 + j123708 + 621056 & * 26 & @xmath1330.4 & sp & @xmath11720.4 + j123708 + 621246 & 20 & @xmath1350.1 & off & @xmath11724.4 + j123711 + 621331 & * 31 & @xmath1330.7 & g / b & @xmath11723.1 + j123716 + 621512 & * 145 & @xmath1330.4 & g / b & @xmath11720.4 + j123721 + 621129 & 677 & @xmath1330.3 & ef & @xmath11627.0 + j123725 + 621128 & * 613 & @xmath1331.2 & g & @xmath11724.1 + j131157 + 423910 & 109 & @xmath1330.3 & g & @xmath11723.5 + j131157 + 423630 & 197 & @xmath1330.6 & ef & @xmath11625.5 + j131203 + 423331 & 886 & @xmath1330.3 & g & @xmath11722.2 + j131205 + 423851 & 12 & @xmath1330.7 & ef & @xmath11625.5 + j131207 + 423945 & 14 & @xmath1330.7 & ef & @xmath11625.5 + j131209 + 424217 & 93 & @xmath1330.4 & g & @xmath11723.4 + j131211 + 424053 & 717 & @xmath1330.6 & g / b & @xmath11720.8 + j131213 + 424129 & 48 & @xmath1180.0 & ell & @xmath11718.7 + j131213 + 423706 & 9 & @xmath1330.8 & g ? & @xmath11725.5 + j131215 + 423702 & 13 & @xmath1330.4 & g / b & @xmath11719.7 + j131215 + 423901 & * 27 & @xmath1330.5 & q & @xmath11717.8 & yes + j131216 + 423921 & 30 & @xmath1330.9 & ef & @xmath11625.8 + j131217 + 423912 & 20 & @xmath1330.6 & g / b & @xmath11720.0 + j131217 + 423930 & 16 & @xmath1330.9 & ef & @xmath11625.8 + j131218 + 423843 & * 25 & @xmath1330.9 & g / b & @xmath11720.3 + j131219 + 423608 & * 26 & @xmath1330.9 & g & @xmath11722.1 + j131219 + 423831 & * 12 & @xmath1330.8 & g & @xmath11720.4 + j131220 + 423923 & 15 & @xmath1330.7 & ef & @xmath11625.8 + j131220 + 423704 & 14 & @xmath1331.0 & ef & @xmath11625.8 + j131220 + 424029 & 34 & @xmath1330.3 & ef & @xmath11625.5 + j131220 + 423535 & 20 & @xmath1330.6 & ef & @xmath11625.5 + j131221 + 423923 & * 12 & @xmath1330.8 & sp & @xmath11718.0 + j131221 + 423722 & 10 & @xmath1330.9 & sp & @xmath11717.8 + j131221 + 423827 & 19 & @xmath1330.8 & g / b & @xmath11722.7 + j131222 + 423813 & 10 & @xmath1330.5 & q & @xmath11719.6 & yes + j131223 + 423908 & * 19 & @xmath1330.8 & g / b & @xmath11719.3 + j131223 + 423712 & 28 & @xmath1330.8 & g / b & @xmath11718.8 + j131223 + 423525 & 66 & @xmath1330.1 & ell & @xmath11718.9 + j131225 + 423656 & 13 & @xmath1350.1 & g ? & @xmath11725.0 + j131225 + 424103 & 35 & @xmath1330.7 & g ? & @xmath11725.5 + j131225 + 423941 & 200 & @xmath1330.7 & ef & @xmath11625.5 + j131227 + 423800 & 26 & @xmath1330.9 & g / b & @xmath11719.8 + j131232 + 424038 & 166 & @xmath1330.8 & g & @xmath11724.8 + j131239 + 423911 & 66 & @xmath1330.6 & g & @xmath11724.5 + j171349 + 501610 & 1138 & @xmath118 & & @xmath117 + j171354 + 501547 & 153 & @xmath118 & & @xmath117 + j171411 + 501602 & 37 & @xmath118 & g / b & @xmath11723,5 + j171414 + 501530 & 6480 & @xmath1331.1 & g / b & @xmath11722.1 + j171415 + 501535 & 49 & @xmath118 & & @xmath117 + j071416 + 501817 & 495 & @xmath118 & & @xmath117 + crcrrc 7.5 to 19.0 & 13.7 & 65 & 21 & 26.6 & @xmath136 + 9.0 to 30.0 & 24.5 & 174 & 20 & 15.6 & @xmath137 + 30.0 to 65.0 & 44.9 & 317 & 20 & 24.8 & @xmath138 + 65.0 to 600.0 & 195.0 & 558 & 21 & 29.4 & @xmath139 + @xmath140 & 1675.0 & 653 & 7 & 3.1 & @xmath141 + | we present the results from two radio integrations at 8.4 ghz using the vla .
one of the fields , at 13@xmath0 + 43@xmath1 ( sa13 field ) , has an rms noise level of @xmath2jy and is the deepest radio image yet made .
thirty - four sources in a complete sample were detected above @xmath3jy and 25 are optically identified to a limit of i=25.8 , using our deep hst and ground - based images .
the radio sources are usually located within @xmath4 ( typically 5 kpc ) of a galaxy nucleus , and generally have a diameter less than @xmath5 .
the second field at 17@xmath0 + 50@xmath1 ( hercules field ) has an rms noise of @xmath6jy and contains 10 sources .
we have also analyzed a complete flux density - limited sample at 8.4 ghz of 89 sources from five deep radio surveys , including the hubble deep and flanking fields as well as the two new fields .
half of all the optical counterparts are with galaxies brighter than i=23 mag , but 20% are fainter than i=25.5 mag .
we confirm the tendency for the micro - jansky radio sources to prefer multi - galaxy systems .
the distribution of the radio spectral index between 1.4 and 8.4 ghz peaks at @xmath7 ) , with a median value of @xmath8 .
the average spectral index becomes steeper ( lower values ) for sources below @xmath6jy , and for sources identified with optical counterparts fainter than i=25.5 mag .
this correlation may suggest that there is an increasing contribution from star - burst galaxies compared to active galactic nuclei ( agns ) at lower radio flux densities and fainter optical counterparts .
the differential radio count between 7.5 and 1000 @xmath9jy has a slope of @xmath10 and a surface density of 0.64 sources ( arcmin)@xmath11 with flux density greater than @xmath3jy . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
complex states of matter like spin liquids are suspected to exist in quantum spin models with frustration due to geometry or due to the nature of the spin - spin interaction @xcite . spin liquids are complicated quantum many - body states that exhibit significant entanglement of their wave functions without symmetry breaking , and could also exhibit emergent quantum phenomena within their low - energy excitation spectra . classical computation , such as exact diagonization and quantum monte carlo simulation , or conventional theories based on local order parameters fail to describe these systems without bias . for example , exact diagonalization studies are limited to small size lattices and hence usually have strong finite - size effects , while quantum monte carlo simulations can suffer from the sign problem or have a large computational expense to describe long - range interactions and hence can not reach the low temperatures needed to see the predicted exotic phases . feynman proposed that one could use controlled quantum - mechanical systems with few quantum gates to simulate many - body problems @xcite as an useful quantum computation before achieving universal quantum computation . in recent years , there has been significant success in trying to achieve this goal by quantum simulation of desired spin models through analogous cold atom systems @xcite . we focus here on one platform for performing analog quantum computation , the simulation of interacting quantum spins via manipulation of hyperfine states of ions in a linear paul trap @xcite although many ideas presented here can be generalized to adiabatic quantum state computation in the two dimensional penning trap as well @xcite . in the paul trap systems , clock states of the ions ( states with no net @xmath0-component of angular momentum ) are the pseudospin states , which can be manipulated independently by a pseudospin - dependent force driven by laser beams . the lasers couple the pseudospin states to the lattice vibrations of the trapped ions , which leads to effective spin - spin interactions when the phonon degrees of freedom are adiabatically eliminated @xcite based on the idea of geometric phase gate @xcite or mlmer - srensen gate @xcite . theoretically , the analog ion - trap simulators can be described as nonequilibrium driven quantum systems with both spin and phonon degrees of freedom . sufficiently small systems can be treated numerically in an exact fashion by truncating the phonon basis and taking into account all possible quantum states in the solution of the time - dependent schrdinger equation . experimentally , ion traps have been used to simulate the transverse - field ising model with a small number of ions @xcite based on simulated quantum annealing @xcite ( see ref . for a review ) . it has been known experimentally that moderate phonon creation is commonplace ( on the order of one phonon per mode ) @xcite , even when the system is cooled to essentially the phonon ground state prior to the start of the simulation . in addition , the role phonons play are intrinsic and essential for the mediated spin - spin interaction in trapped ion systems especially in the presence of noncommuting magnetic field hamiltonian in addition to the spin hamiltonian of interest . therefore , an understanding of the role phonons play in the spin simulator is crucial to understanding its accuracy . the organization of this paper is as follows . in sec . ii , we describe the microscopic hamiltonian for the ion - trap - based simulators and then show how one can factorize the time - evolution operator into a pure phonon term , a coupled spin - phonon term , a pure spin - spin interaction term , and a complicated term that primarily determines the degree of entanglement of the spins . next we use adiabatic perturbation theory to determine how adiabatic state evolution can be used to reach a complicated , potentially spin - liquid - like ground state , and detail under what circumstances the evolution is not adiabatic ( diabatic ) . in sec . iii , we show numerical comparison studies in various relevant circumstances based on a direct integration of the time - dependent schrdinger equation , including both spin and phonon degrees of freedom ( the latter in a truncated basis ) . in sec . iv , we conclude with discussions and possible experimental limitations and improvements . when @xmath1 ions are placed in a linear paul trap @xcite with harmonic trapping potentials , they form a nonuniform ( wigner ) lattice , with increasing interparticle spacing as one moves from the center to the edge of the chain . the ions vibrate in all three spatial dimensions about these equilibrium positions @xcite with @xmath2 normal modes . two hyperfine clock states ( relatively insensitive to external magnetic field fluctuations because the @xmath0-component of total angular momentum is zero ) in each ion will be the pseudospins ( and are split by an energy difference @xmath3 ) . hence , the bare hamiltonian @xmath4 including the pseudospin and motional degrees of freedom for the ion chain is given by @xmath5 where @xmath6 is the pauli spin matrix at the @xmath7th ion site and the second term is the phonon hamiltonian @xmath8 with the phonon creation operator of the normal mode @xmath9 along the three spatial directions @xmath10 . the notation @xmath11 refers to the pseudospin orientation in the bloch sphere . the @xmath12th spatial component of the @xmath7th ion displacement operator @xmath13 is related to the @xmath12th phonon normal mode amplitude ( unit norm eigenvector of the dynamical matrix ) @xmath14 and the @xmath12th phonon creation and annihilation operator via @xmath15 $ ] with @xmath16 the mass of the ion and @xmath17 the normal - mode frequency . a laser - ion interaction is imposed to create a spin - dependent force on the ions by using bichromatic laser beams to couple these clock states to a third state via stimulated raman transitions @xcite . effectively , this process is equivalent to an off - resonant laser coupling to the two clock states by a small frequency detuning @xmath18 determined by the frequency difference of the bichromatic lasers . the ions are crystallized along the easy axis ( @xmath19-axis ) of the trap with hard axes in the @xmath20- and @xmath21-directions where the transverse phonons lie . then coupling the raman lasers in the transverse direction minimizes effects of ion heating and allows for an identical spin axis for each ion @xcite . by accurate control of the locked phases of the blue detuned and red detuned lasers with similar rabi frequencies , an effective laser - ion hamiltonian @xcite along the spin direction @xmath22 can be engineered in the lamb - dicke limit @xmath23 : @xmath24 in which the effective rabi frequency @xmath25 is generated by one effective blue - detuned beam and one red - detuned beam simultaneously ( refer to appendix a for details ) . in experiments , one uses adiabatic quantum state evolution to evolve the ground state from an easily prepared state to the desired complex quantum state that will be studied . for spin models generated in an ion trap , it is easy to create a fully polarized ferromagnetic state along the @xmath0-direction via optical pumping , and then apply a spin rotation ( for instance , with a pulsed laser ) to reorient the ferromagnetic state in any direction ( usually chosen to be the @xmath26-direction ) . then , if one introduces a hamiltonian with a magnetic field in the direction of the polarized state , it is in the ground state of the system . by slowly reducing the magnitude of the field and turning on the spin hamiltonian of interest , one can adiabatically reach the ground state of the hamiltonian ( at least in principle ) . hence , one has an additional zeeman term with a spatially uniform time - dependent effective magnetic field @xmath27 coupled to the different pauli spin matrices as @xmath28 ( the magnetic field has units of energy here , since we absorbed factors of the effective magnetic moment into the definition of * b * , and it can also be expressed in units of frequency if we use units with @xmath29 ) . note that we are using an unconventional sign for the coupling to the magnetic field , since this is the sign convention often used in the ion - trapping community . in this case , the ground state of the magnetic field hamiltonian has the spins aligned opposite to that of the field , while the highest - energy state has them aligned with the field . the magnetic field @xmath27 is made in the @xmath26-direction by directly driving a resonant radio - frequency field with frequency @xmath30 between the two hyperfine states to implement the spin flips @xcite or by indirect raman coupling through a third state to effectively couple the two hyperfine states @xcite . the full hamiltonian is then @xmath31 . we solve for the quantum dynamics of this time - dependent hamiltonian by calculating the evolution operator as a time - ordered product @xmath32 $ ] and operating it on the initial quantum state @xmath33 . for the adiabatic evolution of the ground state , we start our system in a state with the spins aligned along the magnetic field and the system cooled down so that there are no phonons at time @xmath34 : @xmath35 . this means we will be following the highest excited spin state of the system , as described in more detail below . while it is also possible to examine incoherent effects due to thermal phonons present at the start of the simulation , we do not do that here , and instead focus solely on intrinsic phonon creation due to the applied spin - dependent force . in time - dependent perturbation theory , one rewrites the evolution operator in the interaction picture with respect to the time - independent part of the hamiltonian . this procedure produces an exact factorized evolution operator @xmath36 which is the first step in our factorization procedure [ the first factor is called the phonon evolution operator @xmath37 . ] [ note that @xmath38 is time independent and it is multiplied by the factor @xmath39 in the exponent . ] the second factor is the evolution operator in the interaction picture , which satisfies an equation of motion given by @xmath40u_i(t , t_{0})$ ] , with @xmath41\mathcal{h}_{li}(t)\exp[-i \mathcal{h}_{ph}(t - t_0)/\hbar]$ ] since @xmath42=0 $ ] . the only difference between @xmath43 and @xmath44 is that the phonon operators are replaced by their interaction picture values : @xmath45 $ ] and @xmath46 $ ] . we now work to factorize the evolution operator further . motivated by the classic problem on driven harmonic oscillators @xcite , we factorize the interaction picture evolution operator via @xmath47\bar{u}(t , t_0)$ ] with @xmath48 defined by @xmath49 ( we call the factor on the left the phonon - spin evolution operator @xmath50 $ ] and the one on the right is the remaining evolution operator ) . the key step in this derivation is that the multiple commutator satisfies @xmath51,v_i(t^{\prime\prime})]=0 $ ] . this fact greatly simplifies the analysis below . the equation of motion for the remaining evolution operator @xmath52 satisfies @xmath53 in which the operator @xmath54 is given by the expression @xmath55e^{-\frac{i}{\hbar}w_{i}(t)}.\ ] ] the operator @xmath54 can then be expanded order by order as @xmath56 \\ \label{eq : main_b } e^{\frac{i}{\hbar}w_{i}(t)}\mathcal{h}_{b}(t)e^{-\frac{i}{\hbar}w_{i}(t ) } & = & \sum_{j=1}^{n}\big\ { { \bf b}(t)\cdot\hat \sigma_{j}+\\ & & \stackrel{\underbrace{\frac{i}{\hbar } [ w_{i}(t),{\bf b}(t)\cdot\hat \sigma_{j } ] + \ldots}\big\}}{\rm{residual \ terms } } , \nonumber \\ e^{\frac{i}{\hbar}w_{i}(t)}i\hbar\partial_{t}e^{-\frac{i}{\hbar}w_{i}(t)}&= & v_{i}(t)\nonumber\\ & + & \frac{1}{2}\frac{i}{\hbar}[w_{i}(t),v_{i}(t)]\end{aligned}\ ] ] where we used the facts that @xmath57 and @xmath51,v_i(t^{\prime\prime})]=0 $ ] . explicit calculations then yield @xmath58 a_{\alpha \nu } + h. c. \big \ } , % \nonumber\end{aligned}\ ] ] in which @xmath59 is the lamb - dicke parameter for the phonon mode @xmath60 with the @xmath12th component of the laser momentum @xmath61 . when the terms in eq . ( [ eq : main_b ] ) vanish , virtually excited phonons will be shown to play no role on the spin - state probabilities as a function of time , but in the presence of a transverse field , due to the noncommuting nature of quantum operators , phonon creation can significantly affect the spin - state probabilities . this fact has not been considered in detail before , and involves one of the most important results of our work , as detailed below . with a vanishing transverse magnetic field , the hamiltonian @xmath62 can be greatly reduced to the spin - only hamiltonian @xmath63 $ ] . because the spin operators @xmath64 in @xmath65 and @xmath66 commute , one can exactly derive the following ising spin hamiltonian @xcite @xmath67 then the expression for the spin exchange interaction @xmath68 is @xmath69,\ ] ] which can be uniformly antiferromagnetic ( @xmath70 ) or ferromagnetic ( @xmath71 ) for the instantaneous ground state of the hamiltonian @xmath72 when the laser detuning @xmath18 is detuned close to center of mass mode frequency @xmath73 . however , the interaction @xmath70 can also be inhomogeneous and frustrated when the laser is detuned in between phonon modes with details depending on the properties of the nearby phonon modes @xmath74 , @xmath75 , and @xmath76 . the mlmer - srensen gate @xcite was originally proposed to disentangle phonon effects from the spins in ion - trap quantum computing . it was discovered that because the phonons are harmonic , one could operate on the spins in such a way that the phonon state is unmodified after the gate operation ( irrespective of the initial population of phonons ) . but one needs to keep in mind that this gate has no transverse field present , which can modify it because the transverse field operator does not commute with the ising hamiltonian . we begin our discussion by assuming that the laser is closely detuned to the transverse center of mass mode with angular frequency @xmath73 ( @xmath77 ) and the addressing laser intensity for each ion is uniform and moderate ( @xmath78 ) . in this situation , the time dependent term with the frequency @xmath79 in the interaction @xmath66 can be neglected . therefore , the interaction @xmath80 and the operator @xmath81 , which are proportional to the collective spin operator @xmath82 , can be reduced to the following forms @xmath83 , \\ w_{i}(t ) & = & \frac{\hbar \eta_{cm}\omega } { 2\sqrt{n}(\omega_{cm}-\mu)}s_{x } \nonumber \\ & \times & { \big[(1-e^{-i(\omega_{cm}-\mu)t})a_{cm } + h. c.\big ] } \label{eq : molmer}\end{aligned}\ ] ] where @xmath84 is the lamb - dicke parameter for the center of mass mode and @xmath85 is chosen . there are two parameter regimes where phonon effects disappear . in the weak - coupling regime @xmath86 , the operator @xmath81 almost vanishes and the time evolution operator @xmath87 is solely determined by the spin - only hamiltonian because the phonon dynamics are adiabatically eliminated . any extra phonon state redistribution takes a long time to be experimentally observable and therefore phonon effects are under control . outside the weak - coupling regime , one can also prevent phonon effects by preparing spin states determined by the spin hamiltonian @xmath72 at a particular waiting time interval @xmath88 . the idea is to choose the time interval @xmath89 such that the operator is periodic with integer @xmath90 cycles so that @xmath91 . therefore , the initial phonon state at the start of the simulation will be revived at the time intervals @xmath92 as can be clearly seen from eq . ( [ eq : molmer ] ) when the we start with the phonon ground state @xmath93 for example , but it is generally true for any occupancy of the phonon states . in the presence of a transverse magnetic field , the ideas of the mlmer - srensen gate are modified . while it is tempting to claim that the residual spin - phonon terms in the magnetic field are irrelevant in the lamb - dicke limit @xmath94 , it is difficult to quantify this if the residual terms are relevant in the presence of the time dependent magnetic field @xmath95 which can be large in magnitude . in fact , phonon effects often modify the time evolution of the spin states when a transverse field is present . however , one can say that in cases where the integral of the field over time is small ( which occurs when the field is small , or when it is rapidly ramped to zero ) or when the @xmath96 field lies along the @xmath97-direction only ( or vanishes ) , the residual terms in eq . ( [ eq : main_b ] ) are irrelevant . for large detuning ( weak - coupling regime ) , where @xmath98 or @xmath99 , the residual terms are always higher - order perturbations with respect to the leading transverse magnetic field term @xmath95 in the course of the quantum simulation . on those occasions , one can also consider the system as described by only the quantum ising spin model in a transverse magnetic field . in general though , we need to determine how large the residual terms are , which often can only be done with numerical calculations . we illustrate this for a number of different cases below . the residual terms are @xmath100\mathcal{h}_b(t)\exp[-iw_i(t)/\hbar]-\mathcal{h}_b(t).\ ] ] the equation of motion for @xmath101 can be written as @xmath102+\mathcal{h}_b(t)+\mathcal{h}_{res}(t)\right \}\nonumber\\ & \times&\bar u(t , t_0).\end{aligned}\ ] ] we perform the final factorization by writing @xmath103 where the spin - evolution operator satisfies @xmath104/2\hbar+\mathcal{h}_b(t)\}u_{spin}(t , t_0)$ ] and the entangled evolution operator satisfies @xmath105 the spin evolution operator @xmath106 becomes @xmath107= \mathcal{t}_{t}\exp\left[-\frac{i}{\hbar}\int_{t_{0}}^{t}dt ' h_{spin}(t')\right ] , \ ] ] which is the third factor for the evolution operator of the ising model in a transverse field and we define @xmath72 in the exponent . the spin exchange terms @xmath108 as given in eq . ( [ eq : exchange ] ) include a time - independent exchange interaction between two ions @xmath109 and a time - dependent exchange interaction @xmath110 . the time - independent term can be thought of as the effective static spin - spin hamiltonian that is being simulated , while the time - dependent terms can be thought of as diabatic corrections , which are often small in current experimental set - ups , but need not be neglected . for simplicity , we set the initial time @xmath111 . the entanglement evolution operator @xmath112 is a complicated object in general , but it simplifies when one can approximate the operator @xmath112 as @xmath113 for the special situations discussed at the end of the last subsection . in general , this evolution operator involves a coupling of spins to phonons in all directions and has a very complicated time dependence . if one evaluates the first few terms of the series for the time - ordered product , one finds it involves multispin interactions , spin - phonon coupling , and spin - exchange interactions in all spatial directions . but the net weight of all of the terms is governed by the integral of the magnetic field over time , so if that integral is small , then this factor will also be small . therefore , the adiabatic elimination of phonons based on mlmer - srensen gate @xcite can be justified only in the case of a vanishing transverse magnetic field . with a constant magnetic field , the entanglement between spins and phonons can be periodic so that phonon effects can continue to be nulled at integer multiples of the appropriate period @xcite . but such a procedure would be more complicated than the standard gate , and is not relevant for adiabatic state creation simulations , so we wo nt discuss it further here . from a mathematical standpoint , because the entanglement evolution operator is on the far right of the factorization , it s main effect is to modify the state from an initial spin state in direct product with the phonon vacuum to a state that will typically involve some degree of entanglement between phonon and spin degrees of freedom . we can use this factorization to show that in cases where the spin - entanglement evolution operator can be approximated by the unit operator , then phonons have no observable effects on the probability of product states ( regardless of the number of coherent phonons created during the simulation ) , so this result is similar in spirit to the original mlmer - srensen gate , but is different because it holds in the presence of a transverse field and requires no special times for periodic variations to recur . to do this , we need one final identity . we further factorize the entangled phonon - spin evolution operator @xmath114 $ ] into the product @xmath115 $ ] @xmath116\exp[-(1/2)\sum_{\nu}\gamma_{x \nu}(t){\gamma}_{x \nu}^ { * } ( t)]$ ] with the spin operator defined to be @xmath117 and its complex conjugate is @xmath118 , while the function @xmath119 satisfies @xmath120 $ ] . at this stage , we have factorized the evolution operator into four main terms , each term being an evolution operator evolving the system from time @xmath34 to time @xmath121 . we have explicit values for the first three factors , but the last term ( the entanglement evolution operator ) can be quite complicated ; we have also described situations where the exponent of that term is small and can be neglected . in this case , the probabilities to observe any of the @xmath122 product states with a quantization axis along the ising axis ( @xmath123 for the @xmath1 ionic spins ) is unaffected by the presence of an arbitrary number of real excited phonons ( which are excited by the phonon - spin evolution operator ) . using the fundamental axiom of quantum mechanics , the probability @xmath124 to observe a product spin state @xmath125 starting initially from the phonon ground state @xmath93 and not measuring any of the final phonon states involves the trace over all possible final phonon configurations @xmath126 where @xmath127 is any initial spin state ( it need not be a product state ) and @xmath128 denotes the number of phonons excited in the @xmath9th mode in the @xmath20-direction . the operator in the matrix element entangles the phonons and the spins , so we evaluate the matrix element in two steps : ( 1 ) first we evaluate the phonon part of the operator expectation value , and then ( 2 ) we evaluate the spin part . note that since the pure phonon factor of the phonon evolution operator @xmath129 $ ] is a phase factor , it has no effect on the probabilities when evaluated in the phonon number operator basis , so we can drop that factor . next , the term @xmath130 $ ] gives 1 when operating on the phonon vacuum to the right , so it can be dropped . we are thus left with three factors in the evolution operator . one involves exponentials of the phonon creation operator multiplied by spin operators ( and is essentially a coherent - state excitation for the phonons with the average phonon excitation number determined by the spin state being measured ) , one involves products of spin operators that resulted from the factorization of the coupled phonon - spin evolution operator factor , and one is the pure spin evolution factor @xmath131 . the two remaining factors that appear on the left involve only @xmath132 spin operators , and hence the product state basis is an eigenbasis for those operators . this fact allows us to directly evaluate the expression in eq . ( [ eq : prob ] ) . we expand the evolution of the initial state at time @xmath121 in terms of the product - state basis @xmath133 with @xmath134 denoting each of the @xmath135 product state basis vectors and @xmath136 is a ( complex ) number . using the fact that the product states satisfy the eigenvalue equation @xmath137 with eigenvalues @xmath138 ( for @xmath139 ) or @xmath140 ( for @xmath141 ) , we arrive at the expression for the probability @xmath142}\ ] ] @xmath143 .\ ] ] we used the matrix element @xmath144 @xmath145 in the derivation . the summations become exponentials , which exactly cancel the remaining exponential term and finally yield @xmath146 , which is what we would have found if we evaluated the evolution of the spins using just the spin evolution operator @xmath131 and ignoring the phonons altogether . hence , _ the coherently excited phonons have no observable effects on the probability of product states for the transverse - field quantum ising model when we can neglect the entanglement evolution operator . _ if we do not measure the probability of product states , then the terms from the coupled spin - phonon evolution operator remain spin operators , and one can show that the probabilities are changed by the phonons . in other words , it is because the spin - phonon evolution operator is diagonal in the product space basis for phonons and spins that allows us to disentangle the phonon and spin dynamics . in cases where this can not be done , we expect the phonon and spin dynamics to remain entangled . in other words , phonons have observable effects on any spin measurements which introduces spin operators away from the _ ising quantization axis _ such as most entanglement witness operator measurements . finally , we may ask what does the entanglement evolution operator do to the system ? it is difficult to find any simple analytic estimates of the effect of this term , but it acts on the initial state which has the spins aligned along or opposite to the magnetic field and has no phonons . during the evolution of that operator , new terms will be created which involve entanglement of spin states with states that have created phonons . if the amplitude of those extra terms is small , they will not have a large effect , but if it is not , then one has no other recourse but to examine the full problem numerically , which is what we do next . first we examine a perturbation - theory treatment , and then we consider the full numerical evolution of the system . one may have noticed that the spin evolution operator was not the evolution of a static ising spin model . there were additional time - dependent factors in the evolution operator which arose from the additional time dependence of the exchange operators that was inherited by the phonons when they were `` adiabatically '' removed from the problem . in this section , we use adiabatic perturbation theory ( reviewed in appendix b ) to analyze the effect of those extra time - dependent terms on the spin evolution of the system . in an adiabatic quantum simulation , one initially prepares the system in a certain pure state @xmath147 of the initial hamiltonian @xmath148 with the occupation @xmath149 and the probability amplitudes in all other states vanishing [ @xmath150 . thereafter , the probability amplitudes to be excited into the other states can be approximated by @xmath151},\ ] ] for later times , as long as the transition amplitudes @xmath152 are much smaller than one during the time evolution . this is the main expression we will use to evaluate the diabatic effects due to the time - dependent exchange interactions @xmath153 . here @xmath154 are the instantaneous eigenstates of the spin hamiltonian @xmath72 with a static exchange interaction @xmath155 and @xmath156 are the corresponding dynamic phases given by the integrals @xmath157 , and we assume there are no degeneracies in the instantaneous spectrum as a function of time . let us briefly describe the experimental protocol for a typical trapped ion quantum simulator restricted to the spin - only hamiltonian @xmath158 defined in eq . ( [ eq : hspin ] ) . the system is initially prepared in a spin - polarized state @xmath159 along ( or opposite to ) the direction of the transverse magnetic field @xmath160 by optical pumping followed by a @xmath161 spin rotation . the spin - only hamiltonian is then turned on with a maximum effective transverse magnetic field @xmath162 followed by an exponential ramping down of the magnetic field to a final value @xmath163 at time @xmath121 ( @xmath164 is the exponential ramping time constant for the decay of the magnetic field ) . after evolving to time @xmath121 , the projection of the spin states along the @xmath97-axis of the ising hamiltonian is taken to find the probability to be in a particular spin state at time @xmath121 ( in actual experiments another @xmath161 pulse is applied to rotate the @xmath97-axis to the @xmath0-axis where the measurement is made ) . if the system is perfectly adiabatic during the evolution , the outcome of the quantum state would be the highest excited state of the ising hamiltonian @xmath165 if the simulation starts out in the highest excited state of the magnetic field hamiltonian @xmath166 at time @xmath167 , which corresponds to the spins aligned along the @xmath26-axis . this procedure is theoretically identical to the ground state passage of the spin polarized state @xmath168 to the ground state of the negative of the ising hamiltonian @xmath169 with the system hamiltonian being modified as @xmath170 @xcite . in a typical trapped ion quantum simulator , the frequency @xmath18 is sufficiently far from any phonon frequencies such that the condition @xmath171 holds to avoid the heating of the system away from the initial phonon vacuum state during the simulation . in addition , the maximum magnetic field strength is much larger than the time independent exchange interactions @xmath172 to ensure the system initially starts in an eigenstate of the initial hamiltonian . to optimize the adiabaticity of the simulation , the ramping time constant @xmath164 for the magnetic field has to be chosen to be much greater than the largest characteristic time scale of the system , which is shown below to be the minimum of the inverse of the frequencies @xmath173 . we now discuss the effects of the time - dependent exchange interaction @xmath110 . for concreteness , we will follow the highest energy state , starting from the spin state aligned along the direction of the magnetic field . starting with the expression for the transition probability amplitude @xmath174 in eq . ( [ eq : excitations ] ) , we find the dominant diabatic transition is to the state with the minimum energy difference @xmath175 with the initial spin polarized state @xmath159 , assuming the matrix element in the numerator does not depend too strongly on @xmath176 , which is true when @xmath177 . at the initial time @xmath178 , where the ising couplings @xmath179 can be shown to always vanish , all of the spin states with one spin flipped along the @xmath26-axis are degenerate . this degeneracy will be broken by the ising hamiltonian @xmath180 at finite time @xmath181 . due to spin - spin interaction in @xmath165 , the states along the y axis of the bloch sphere called @xmath182 , have nonzero matrix components @xmath183 with the lowest energy gap @xmath184 with respect to the initial spin state @xmath185 . to approximately evaluate the transition amplitude @xmath186 from the initial state to the two spin - flipped states @xmath182 , we do not actually need to know the state @xmath182 . the only relevant information we need is that it is one of the two spin - flipped states which tells us what the denominator is . hence , we can approximate @xmath187 using similar reasoning , we approximate @xmath188 as @xmath189 in which @xmath190 is the magnetic angular frequency . the operator @xmath191 consists of modes with frequencies @xmath192 , @xmath193 , and @xmath194 with the time derivative @xmath195 given by @xmath196 @xmath197.\ ] ] @xmath198 the last approximate expression is derived by keeping the contribution from the slow mode @xmath193 and dropping the high frequency modes @xmath192 , and @xmath194 because the detuning @xmath199 is closely detuned to certain phonon modes in the quantum simulation . as a consequence , the probability amplitude @xmath200 induced by a single phonon mode is given by the expression @xmath201\langle m|\sigma_{j}^{x}\sigma_{j'}^{x}|n\rangle f(t).\ ] ] the function @xmath202 can be approximated in experiments [ when @xmath193 is much larger than @xmath203 at slow ramping @xmath204 as @xmath205.\ ] ] we therefore reach the conclusion that the probability amplitude @xmath206 is given by @xmath207 @xmath208.\ ] ] we note that diabatic effects manifested in @xmath209 due to time - dependent ising couplings grow exponentially in time as @xmath210 signifying that the theory is only accurate for short times . to suppress the diabatic effects , the criterion that has to hold for all phonon modes @xmath9 is @xmath211 based on this expression , when the laser is closely detuned to one of the phonon resonance frequencies @xmath74 , the transition probability between states caused by @xmath212 becomes large ( diabatic ) . in addition , a stronger magnetic field is required to suppress the diabatic transitions with smaller detuning @xmath18 . this is supported by the following numerical discussion in section iii c. notice that the above expression should be a reasonable estimation as long as the condition @xmath213 applies at time @xmath121 after the beginning of the quantum simulation . we can estimate the maximal time @xmath214 for which @xmath213 holds . the cutoff @xmath215 is set by @xmath216 where @xmath217 is the absolute value for the maximum exchange interaction @xmath218 between the spins . as a result , the cut - off time @xmath214 is proportional to the ramping time constant @xmath164 with a logarithmic factor given approximately by @xmath219.\ ] ] in the parameter regime where @xmath220 , in which our theory holds , @xmath214 can be extended somewhat beyond the ramping time constant @xmath164 . based on our numerical discussion , the diabatic effects are largest when the magnetic field is ramped through the transition from paramagnetic state to other targeted spin states , which is also accompanied by larger phonon creations due to the shrinkage of the spin gap near the transition . in this section , we focus on showing the circumstances where quantum emulators can or can not be described by the transverse - field ising model with high fidelity . since our goal is to understand under what circumstances the effect of the phonons is small , we consider different cases for the time - evolution of the system including various detunings and initial transverse magnetic field strengths . to isolate different effects , we compare two spin - only models in the presence of the ramping magnetic fields with the theoretically exact spin - phonon model based on numerical diagonization . the first is the _ ideal spin model _ which considers the evolution of the system with a static ising model ( spin - exchange coefficients are the time - averaged exchange coefficients ) and a time - dependent magnetic field . while one might think this is a purely adiabatic model , it has some diabatic effects , since the fully polarized state is _ not _ generically the ground state of the ising plus magnetic field hamiltonian because the ( static ) ising exchange interactions are nonzero at the initial time . hence , one can invoke a sudden approximation to the system initially , and find that the initial state is a superposition of different energy eigenstates . in addition , the magnetic field varies in time and hence can cause additional diabatic effects due to its derivative with respect to time . the second is the _ effective spin model _ which involves , essentially , evolution of the spin system according to the spin evolution operator only in eq ( [ eq : hspin ] ) . hence it has the static ising hamiltonian , the time - dependent ising interactions and the time - varying transverse magnetic field . this model can have its schrdinger equation solved in a spin - basis only , since all phonon effects are neglected except virtual phonon excitations . the third model is the _ exact spin - phonon model _ , where we evolve the system according to the original spin - phonon hamiltonian expanded in the lamb - dicke limit [ eq . ( [ eq : phonon ] ) ] . the only approximation used in this last model is the cutoff for the phonons . the strategy we use is to numerically integrate the schrdinger equation using a direct product basis which involves a spin state in direct product with a phonon state . we do this because the hamiltonian only connects states that differ by plus or minus one phonon number , and hence is block sparse in this basis . the spin states are chosen to include all possible ising spin states for the number of ions in the trap . the phonon basis is chosen to have a cutoff of a maximal phonon excitation . the maximal cutoff is always chosen to be larger than the average occupancy of the phonons in each normal mode of the ion chain . of course , we expect more phonons to be excited into the phonon modes closest to the beatnote frequency of the lasers , so the cutoffs that are chosen will vary from one normal mode to another . for example , we often find we can set the phonon cutoff to be one for some of the phonon modes far from the driving frequency of the spin - dependent force . to facilitate our discussion , we define the root - mean square average @xmath221 of the fully connected ising interaction for @xmath1 ions as @xmath222 in which the static ising interaction @xmath223 is given by the static term in eq . ( [ eq : exchange ] ) and the integer indices @xmath224 both range from @xmath225 to @xmath1 . let us discuss the symmetry of the spin - only system first , which is relevant for the exact diagonization of the spin - only hamiltonian . there is one spatial inversion symmetry ( @xmath226 ) in the ion chain , since the equilibrium ion positions are distributed symmetrically about the origin in the trap and all phonon modes involve symmetric or antisymmetric displacements of corresponding ion positions . there is also a spin reflection symmetry ( @xmath227 , @xmath228 , and @xmath229 ) in the spin - only models with a transverse magnetic field @xmath230 . this spin - reflection symmetry preserves all commutation relations of the spin operators and leaves the hamiltonian invariant . therefore , there are four symmetry sectors for the eigenstates of the spin model ( even space , even spin ; even space , odd spin ; odd space , even spin ; and odd space , odd spin ) . if the static ising couplings are all negative ( positive ) , the spin ground state is ferromagnetic ( antiferromagnetic ) and the highest spin eigenstate is the opposite , namely antiferromagnetic ( ferromagnetic ) . we will focus on a detuning to the blue of the center - of - mass phonon mode . in this case , all spin exchange couplings are positive and the ground state is antiferromagnetic , while the highest excited state is ferromagnetic . we will examine the adiabatic state evolution of the highest eigenstate . with all the respected discrete symmetries , we can construct the symmetric and antisymmetric ferromagnetic states of the spin - only hamiltonian as @xmath231 or @xmath232 which is in the ( even , even ) or ( even , odd ) sectors , respectively . the experimental protocol is to prepare the system initially in a spin polarized state @xmath233 , [ which is the highest eigenstate of the transverse magnetic field @xmath234 , by optical pumping and a coherent spin rotation and then to gradually turn off the magnetic field with an exponential ramp @xmath235 while keeping the spin - dependent laser force in the @xmath97-direction on during simulation time through stimulated raman transitions between the spin states . according to adiabatic evolution , if the quantum state is initially prepared in the highest eigenstate of the field - only hamiltonian @xmath230 , the outcome of the quantum simulation will adiabatically follow the corresponding highest eigenstate of the hamiltonian ( ising spin hamiltonian ) , if there are no level crossings ( which does not occur in this system ) . in the case with positive static ising coupling , the ferromagnetic highest energy eigenstate is the symmetrical ferromagnetic entangled state ( the so - called ghz state ) @xmath231 , when @xmath236 . there are two intrinsic errors which can impede the quantum simulation in trapped ions . the first is diabatic effects which occur primarily when either parameters in the hamiltonian are changed too rapidly in time , or when energy gaps in the instantaneous eigenvalue spectrum become to small . the second is the error induced by phonons in the presence of time - dependent transverse magnetic fields . for example , the phonon - spin hamiltonian does not have spin - reflection symmetry because it is linear in the @xmath22 operators , and hence the spin - phonon interaction breaks this @xmath237 symmetry . one consequence of this is to couple the symmetric and antisymmetric ferromagnetic states which is likely to reduce the spin entanglement of the ghz state . ( other errors such as phonon decoherence effects due to spontaneous emission are not considered here . ) current experiments use atomic cycling transitions to measure the spin state of the ion ( which clock state the ion is in ) , and do not measure the phonons excited in the system . hence , the experimental observables are the probability @xmath238 of a spin - polarized state after tracing out phonons in the tensor product of the spin - phonon hilbert space @xmath239 as mentioned above when discussing eq . ( [ eq : prob ] ) . if one performs rotations about the bloch sphere prior to making the measurement of the probabilities , then one can also measure a number of different spin - entanglement witness operators . a spin - entanglement witness operator ( for a target entanglement state ) @xcite is a mathematically constructed observable that has a negative expectation value when the system is entangled . no witness operator can measure general entanglement , but instead a witness operator is constructed to measure a specific type of spin entanglement . for example , the witness operator @xmath240 for an @xmath1-ion chain can be constructed as @xcite @xmath241\ ] ] with the stabilizing spin operators expressed in terms of the pauli spin operators by @xmath242 notice that the target spin polarized state in this paper is along the ising @xmath97 axis in the bloch sphere instead of the @xmath0 axis . therefore , we modified the original expression @xcite to our problem by the transformation @xmath243 and @xmath244 . based on the above construction , ghz state entanglement measurements can be detected by the observable @xmath245 with the density matrix @xmath246 constructed by pure states or mixed states during the quantum simulation . for a perfect ghz state entanglement , one can show that the entanglement witness operator satisfies @xmath247 ( refer to appendix c ) . any deviation from perfect ghz entanglement would lead to a value greater than @xmath140 . note that this is one of the few cases of a witness operator where the degree of entanglement is correlated with the magnitude of the expectation value of the witness operator . the systems we consider range from @xmath248 to @xmath249 which is far from the thermodynamic limit . the quantum phase transition ( qpt ) due to the discontinuity of the ground - state wave function in the thermodynamic limit @xmath250 only manifests itself as a state avoiding crossing in the energy spectrum , which is adiabatically connected to the qpt at large @xmath1 . the system parameters and the higher set of transverse phonon modes , which belongs to the higher branch of two transverse motional degrees of freedom , for different numbers of ions @xmath1 are summarized in table i. the trapping parameters are given by the aspect ratio and the cm mode frequency @xmath73 along the transverse ( tight ) axis . the axial ( easy ) trapping frequency is given by the product of the aspect ratio and the cm mode frequency @xmath73 . the choice of these parameters comes from trap parameters and typical operating regimes of the ion - trap experiment at the university of maryland . most results are robust with moderate changes of parameters and our choices do not intimate that fine tuning of parameters is needed to achieve the results we show . .parameter set i [ cols="^,^,^,^",options="header " , ] [ tab : parameters2 ] ( left panel ) and ( b ) @xmath251 ( right panel ) . the horizontal axis is the laser detuning scaled by the transverse cm mode frequency @xmath73 . the vertical axis is the transverse magnetic field @xmath252 scaled by the root - mean - square average of ising coupling @xmath221 . the blue area represents the one - kink phase and the red area indicates the ferromagnetic phase . the range of the detuning @xmath18 ( in units of @xmath73 ) is shown between the second phonon mode and the third phonon mode . the value of the order parameter @xmath253 ( varying from @xmath140 to @xmath254 ) is described by the color scale to the right of the figure . , title="fig : " ] + in fig . [ fig : fig8 ] , we numerically map out the time dependence of the probability @xmath253 for _ ideal spin models_. the nexponentially ramped magnetic field @xmath255 is chosen with different initial values @xmath256 ( scaled by @xmath221 as determined by the detuning @xmath18 and the rabi angular frequency @xmath257 ) . the value of the rabi angular frequency @xmath258 is chosen so that it is safely within the weak field regime @xmath259 near the central region of the phase diagram for all phonon modes . the total simulation time @xmath89 is chosen so that it is proportional to the inverse of @xmath260 . we select the exponential ramping time constant @xmath164 for the exponential reduction of the magnetic field @xmath255 to be one - fifth of the experimental simulation time ( @xmath261 ) . by comparing ( a ) to ( b ) [ or ( c ) to ( d ) ] , we observe that diabatic effects are greatly suppressed when the exponential ramping time constant @xmath164 is large enough so that the transition to the closest excited state in energy is negligibly small . this effect shows up as much deeper colors dictating the order parameter @xmath253 in the ferromagnetic states and the kink states when @xmath255 approaches zero on the vertical axis , as illustrated in subplots ( b ) and ( d ) . the diabatic effects also show up clearly as a slow oscillation in the probabilities @xmath253 at larger @xmath262 before the simulation ends along the vertical axis in subplots ( a ) and ( c ) . we also notice some fast background oscillations in @xmath253 covering the entire phase diagram in cases ( c ) and ( d ) . this effect is due to the fact that the time derivative of the dynamic phase @xmath263 between @xmath264 states is roughly stationary in time ( as analyzed with adiabatic perturbation theory ) . for short exponential ramping time constants @xmath164 , one can not see the noticeable interference pattern between these states because of a random phase cancelation along the path of the state evolution in time . in panels ( b ) and ( d ) the main difference is a reduction of the period of the background interference pattern , which is shorter in panel ( b ) ( larger magnetic field ) . calculated for the ideal spin model with diabatic effects included . the horizontal axis is the laser detuning @xmath18 scaled by transverse cm frequency @xmath265mhz . the vertical axis is the instantaneous transverse magnetic field @xmath255 scaled by the root - mean - square average of the spin couplings @xmath221 ( note the range changes for different panels ) . the rabi angular frequency @xmath257 and the dimensionless lamb - dicke parameter for the center of mass mode @xmath84 are selected to be @xmath258 and @xmath266 respectively . ( a ) @xmath267 , ( b ) @xmath268 , ( c ) @xmath269 , and ( d ) : @xmath270 . , title="fig : " ] + what are the phonon effects on the corresponding ferromagnetic to kink phase diagram ? we show our calculations for the @xmath271 case in fig . [ fig : fig9 ] . phonon creation has serious effects on the phase diagram . in the cases with fast ramping time constants [ ( a ) and ( c ) ] , the ferromagnetic states are destabilized and appear only with small probability . for slow ramping time constants [ cases ( b ) and ( d ) ] , the fm domain disappears near the leftmost phonon mode due to large phonon creation as the phonon is being more resonantly driven . but the kink state domain reduces only slightly near the rightmost phonon mode . as a consequence , phonons restrict the available parameter space to observe the fm to kink phase diagram but do not rule out the possibility of observing the phase as long as the exponential ramping time constant @xmath164 is long enough . in the current numerical simulation we show , the exponetial ramping time constant @xmath164 is roughly on the order of a few milliseconds ( close to feasibility in current experiments ) . one may suspect that phonons can ruin the stability of the fm state when the number of ions scales up because the fm domain shrinks in size and moves closer to the leftmost phonon mode , and if we are too close to the phonon mode , phonon creation ruins the chance to see the fm state . one can try to increase the experimental simulation time and reduce the rabi frequency , but doing this too much eventually runs into coherence issues or problems from spontaneous emission . with the same cases as in fig [ fig : fig8 ] . the phase diagrams are calculated with the exact spin - phonon hamiltonian . the horizontal axis is the laser detuning @xmath18 scaled by transverse cm mode frequency @xmath73 . the vertical axis is the instantaneous transverse magnetic field @xmath255 scaled by the root - mean - square average of spin - spin coupling @xmath221 . ( a ) : @xmath267 , ( b ) : @xmath268 , ( c ) : @xmath269 , and ( d ) : @xmath270 . , title="fig : " ] + in fig . [ fig : fig10 ] , we show the @xmath251 case for the _ ideal spin model_. the behavior of the fm - kink phase diagram is similar to the @xmath271 case in fig . [ fig : fig8 ] except the boundary of fm states and kink states is shifted toward lower detunings @xmath18 . as a consequence , the fm state domain ( deep red area ) occupies the region where the detuning is close to the leftmost phonon mode and the kink state domain ( deep blue area ) occupies most of the detuning region . however , the phase diagram for low magnetic field @xmath272 is very close to the adiabatic phase diagram ( see the right panel of fig . [ fig : fig7 ] ) when the initial magnetic field @xmath256 is large and the ramping time constant @xmath164 is long as shown in fig . [ fig : fig10 ] ( b ) and fig . [ fig : fig10 ] ( d ) ] except for the background interference patterns that were described above . one also notices that there is much less diabatic effects at low @xmath255 due to the fact that the smallest spin excitation gap is larger near the central area of the phase diagram . when we add phonon effects , we might expect the phase diagram to only deviate when we are detuned close to a phonon line , but the situation is much worse for @xmath251 , as shown in fig . [ fig : fig11 ] . the kink phase ( deep blue zone ) exists for a wide range of ramping and onset magnetic fields @xmath256 as shown in all cases . however , the ferromagnetic domain ( red zone ) disappears even for slow ramping [ like the exponential ramping time constant @xmath273 msec in cases ( c ) and ( d ) ] . this does not rule out the possibility of observing the fm phase for even longer ramping time constants @xmath164 ( or smaller rabi angular frequency @xmath257 ) but a ramping time constant @xmath273 msec is already well beyond what is used in current ion - trap experiments where @xmath164 is usually less than one millisecond . this problem gets worse for larger @xmath1 , and already for @xmath274 the fm - kink phase diagram appears to be impossible to observe . this arises in part due to the fact that the spin gap closes exponentially fast with the system size @xcite . as a result , one needs to dramatically reduce the diabatic effects to see the transition . in addition , phonon effects also make it hard to see the transition by not allowing the detuning to move too close to either phonon line , and thereby misses significant regions where the fm phase is stable . as calculated for the _ ideal spin model_. the horizontal axis is the laser detuning @xmath18 scaled by transverse mode trapping frequency @xmath265mhz . the vertical axis is the instantaneous transverse magnetic field @xmath255 scaled by the root - mean - square average of spin - spin coupling @xmath221 . the rabi angular frequency @xmath257 and the dimensionless lamb - dicke parameter for the cm mode @xmath84 are @xmath275 and @xmath266 , respectively . ( a ) @xmath267 , ( b ) @xmath268 , ( c ) @xmath269 , and ( d ) @xmath270 . , title="fig : " ] + and the corresponding parameter set as in fig . [ fig : fig10 ] . the horizontal axis is the laser detuning @xmath18 scaled by the transverse cm frequency @xmath73 . the vertical axis is the instantaneous transverse magnetic field @xmath255 scaled by the root - mean - square average of spin - spin coupling @xmath221 . ( a ) @xmath267 , ( b ) @xmath268 , ( c ) @xmath269 , and ( d ) @xmath270 . , title="fig : " ] + we have examined a number of different issues related to the importance of phonons in analog quantum simulation of the transverse field ising model . we show when the spin - phonon entanglement operator @xmath276 can be approximated by @xmath225 ( a longitudinal magnetic field along the ising axis , or a vanishing transverse magnetic field ) , one can show that the phonons do not affect the probability to measure the spins in product states in the direction of the ising interaction , but they can reduce the entanglement of the spin eigenstates . surprisingly , in cases when the operator @xmath276 can not be approximated by @xmath225 , the effect of the phonons is often to make the system look more like a static ising spin hamiltonian plus a time - varying transverse magnetic field . this result holds primarily when laser detuning is blue of the cm mode , and hence corresponds to a ferromagnetic case when one looks at the highest excited state . we emphasize that the common belief based on the geometric phase gate , in which phonon effects can be suppressed by choosing the period to be the inverse of close detuning from a phonon mode due to periodic spin - phonon entanglement dynamics , is no longer valid in a finite decaying transverse magnetic field . our work shows that one must consider phonon effects in most ion - trap spin simulator experiments especially when the spin - spin interaction is highly frustrated . in cases when laser is detuned blue of the transverse center of mass mode , phonons are beneficial to make the system look more and more like the static spin hamiltonian being emulated ( at the expense of reducing spin - spin entanglement ) . in cases when spin interactions are frustrated with multiple phonon modes stimulated , the phonons can work to suppress the true adiabatic spin phases from having a high fidelity or even invalidate the spin phases completely . generically the phonon effects beyond adiabatic elimination are remarkable when the detuning lies close to at least one of the phonon frequencies , and hence on average more than one phonon per mode is excited . in conclusion , large laser detuning is essential to suppress phonon coherent population while it also causes the shrinkage of spin excitation gaps in the adiabatic spin simulation . alternative adiabatic quantum simulation schemes which do not create noticeable phonon occupation , while maintaining large spin excitation gaps would be desirable . we acknowledge interesting discussions with kihwan kim , rajibul islam , wes campbell , emily edwards , simcha korenblit , zhe - xuan gong , chris monroe , and l .- m . duan . joseph wang thanks marcos rigol for sharing computational resources . this work was supported under aro grant number w911nf0710576 with funds from the darpa ole program . j. k. f. acknowledges support of the mcdevitt bequest at georgetown . we describe how different laser - ion interactions are employed to ultimately simulate effective spin models . the spin - dependent dipole force along any direction of the equatorial plane of the bloch sphere and an effective transverse magnetic field are created by using multiple laser beams and the optical - dipole interaction between the ions and phase - locked lasers . we start with a reference raman beam that has frequency @xmath277 and then superpose another perpendicular raman beam ( that has frequencies in a frequency comb @xmath278 ) . the lasers use off - resonant raman coupling through dipole - allowed excited states of the ion to generate an effective spin - phonon interaction . take the ytterbium ion @xmath279 as an example : the qubit states are the clock states @xmath280 and @xmath281 , formed from the hyperfine states of the @xmath282 valence electron and the spin one - half nucleus . these states have no linear zeeman effect , and hence are less prone to background magnetic fields . the hyperfine state @xmath281 is denoted as the spin - up state and the state @xmath280 is the spin - down state in the z - axis of the pseudospin bloch sphere . the energy - level spacing is in the gigahertz range , so one could , in principle , directly make transitions that flip the spins from up to down and vice - versa by stimulated emission and absorption processes acting on the hyperfine states . but , it is common to instead generate these spin - flip transitions via off - resonant raman coupling to a third state to suppress incoherent spontaneous emission effects . to do this we need two laser beams with different frequencies which can be detuned away from the energy level spacing between the clock states and each of the frequencies are chosen to be far away from dipole allowed resonant transitions in each ion . we denote the two beams wavevectors and frequencies by @xmath283 respectively . by adiabatic elimination @xcite of dipole - allowed excited states through the raman procedure for ion @xmath7 , one can write down the interaction for an ion as @xmath284,\ ] ] where @xmath25 is the effective rabi frequency of the stimulated - raman transition , @xmath285 and @xmath286 are the effective momentum and energy of the photons , respectively , @xmath287 is the controlled phase shift between the two laser beams , and the pseudospin flip operators are @xmath288 . the full hamiltonian involves the sum of this term plus the clock state energy level difference @xmath289 multiplied by the @xmath0-component spin operator . now we go to the interaction picture with respect to the clock state energy level difference @xmath289 , @xmath290h_{li } \exp[-\frac{i\omega_{0}}{2}\sigma_{j}^{z}t]\ ] ] at time @xmath121 . with the photon energy difference @xmath291 comparable to the clock - state energy splitting @xmath289 , only terms with slow modes ( rotating wave approximation ) are kept and we arrive at the following hamiltonian relevant for our discussion : @xmath292 + h.c . , \ ] ] in which the slow mode angular frequency is given by @xmath293 and @xmath287 is the static phase shift between the laser beams . the coupling of the reference raman beam with photon frequency @xmath277 and the blue - detuned photon with frequency @xmath294 ( @xmath199 ) in the second beam leads to an effective blue - detuned beam with the frequency difference @xmath295 as given by the hamiltonian @xmath296 @xmath297 where @xmath296 is the interaction with the blue detuned ( @xmath298 ) laser that has a beatnote frequency @xmath199 and @xmath299 is the wavevector difference of the two interfering laser beams that generate the raman coupling , @xmath300 is the ion position operator with the equilibrium ion position @xmath301 at site @xmath7 . similarly , the coupling of the photon from the reference beam with the photon in the red - detuned beam with frequency @xmath302 leads to the effective red - detuned laser with the frequency difference @xmath303 given by the hamiltonian @xmath304 @xmath305 employing a superposition of multiple frequency components and adiabatically eliminating the dipole allowed excited states @xcite allows one to show that the interaction of laser beams with ions consists of interactions between the reference beam @xmath277 and the other frequencies . as a result , after the summations in eqs . ( a4 ) and ( a5 ) , one arrives at the following expression : @xmath306,\ ] ] in which hermiticity of the rabi frequency is used , and the static phases are @xmath307 and @xmath308 . in the lamb - dicke limit , we have @xmath309}\approx 1+i\delta{\bf k}\cdot\delta { \bf \hat{r}}_{j}$ ] , and the hamiltonian @xmath310 is reduced to @xmath311+h.c.\ ] ] the first term only induces resonant carrier transitions in the pseudospin sector without coherent phonon excitations . the second term induces first ( red or blue ) side - band transitions with the change of one phonon occupation number at each phonon mode as can be seen by replacing the displacement operator @xmath312 by phonon creation and annihilation operators @xmath313 . the spin - dependent force pointing along the azimuthal angle @xmath314 in the equator of bloch sphere is then derived from the phonon side bands as @xmath315 in which the spin phase is given by @xmath316 , the relation @xmath317 is used , and the spin orientation is given by @xmath318 . the expression for the spin - dependent force can be justified by keeping the phases @xmath319 locked . take a transverse phonon mode scheme for example , in which @xmath320 , with @xmath321 . the spin - orientation @xmath322 can be locked along the @xmath97-axis in bloch sphere when the phase difference @xmath323 is maintained . this can be achieved by passing the second beam through an acousto - optic modulator ( aom ) maintaining the phase difference between the frequency components @xmath324 to be out of phase . as one can tell from the dependence of the spin phase @xmath325 on @xmath326 , it is not sensitive to transverse phonon excitations ( coherent or thermal ) in contrast to the sensitivity it has to the longitudinal phonon modes . this is why most state - of - art trapped ion quantum spin simulators couple to transverse phonon modes . one should note that there is a fast oscillating term in the transverse magnetic field @xmath327 due to carrier transitions . this term causes very fast oscillations of low amplitude which are averaged over during the time of an experiment , so we neglect them here . let us now consider how to generate a slow effective transverse magnetic field by using two continuous raman beams with frequencies @xmath277 , and @xmath328 , with phase difference @xmath287 , and wavevector difference @xmath299 . starting from eq . ( a6 ) but with a different effective rabi frequency @xmath329 for the resonant beam with @xmath330 @xmath331+\phi}}+h.c.,\ ] ] we choose the lasers to be out of phase ( @xmath332 ) so that the side - band terms vanish within the lamb - dicke expansion @xmath333}\approx 1+i\delta{\bf k}\cdot \delta{\bf r}_{j}(t)$ ] . the effective transverse magnetic field can then be derived by direct substitution as @xmath334 in which the transverse magnetic field is given by @xmath335 when the phase shift @xmath287 is equal to @xmath336 and @xmath337 is the pauli spin operator ( we will be working in a nontraditional pauli spin matrix representation , where @xmath22 is diagonal , @xmath338 is real , and @xmath339 is imaginary ) . hence , the transverse magnetic field @xmath340 can have its amplitude changed as a function of time by adjusting the laser intensity in the mode that has its frequency equal to @xmath328 with an aom . the time - dependent schrdinger equation for the evolution of the wave function @xmath341 is ( we drop the spin subscript on the hamiltonian ) @xmath342 since the hamiltonian is always hermitian , we introduce instantaneous eigenfunctions @xmath343 with the instantaneous eigenenergies defined by @xmath344 the time - dependent wave function @xmath341 can then be expanded in terms of the orthonormal eigenbasis @xmath343 as @xmath345 in which the coefficients @xmath346 are the time - dependent quantum amplitudes projected onto the instantaneous eigenbasis @xmath343 . therefore , the equation of motion for the expansion coefficients @xmath347 can be derived by direct substitution into the schrdinger equation in eq . ( [ eq : schrodinger ] ) which becomes @xmath348=e_{m}(t)c_{m}(t),\ ] ] after using the orthonormality relation @xmath349 for the instantaneous eigenfunctions . one can further relate the matrix elements @xmath350 to the matrix elements @xmath351 . simply take the time derivative of eq . ( [ eq : instantneous ] ) and project onto the state @xmath352 , to show @xmath353 for @xmath354 [ this derivation assumes the instantaneous energy spectrum has no states that are degenerate with @xmath355 . in the adiabatic approximation , the transition matrix elements @xmath351 between different instantaneous eigenstates are assumed to be so small they can be neglected . in this limit , the system simply follows the instantaneous eigenstates @xmath343 without transitions between different instantaneous eigenstates . in general , transitions between eigenstates @xmath343 should be considered to determine the corrections to the adiabatic state evolution . choose the gauge @xmath356 with the phase @xmath357 . the probability amplitude @xmath174 induced by the diabatic transitions can be solved by integration with respect to time in eq . ( [ eq : coefficient ] ) as @xmath358},\ ] ] in which @xmath359 and @xmath360 are the dynamic phases accumulated by the states @xmath352 and @xmath343 . in an adiabatic quantum simulation , one initially prepares the system in a certain pure state @xmath147 of the initial hamiltonian @xmath148 with the occupation @xmath149 and the probability amplitudes in all other states vanishing [ @xmath150 . therefore , the probability amplitudes to be excited into the other states can be approximated by @xmath361},\ ] ] for later times , as long as the transition amplitudes @xmath152 are much smaller than one during the time evolution . this is the main expression we will use to evaluate the diabatic effects due to the time - dependent exchange interactions @xmath153 . the observable for the measurement of ghz state entanglement is given by @xmath362 with the density matrix @xmath246 constructed from either pure states or mixed states after tracing over the phonons . one can explicitly show that a pure ghz state @xmath363 leads to an entanglement measure @xmath364 to be @xmath140 . the entanglement measure @xmath364 is independent of the basis ; therefore , it is natural to choose spin polarized states along the ising @xmath97-axis in our discussion . based on the ghz state @xmath365 , the corresponding density matrix @xmath366 in the ising product state representation is given by therefore , the nonzero matrix elements in @xmath368 are among the fully ferromagnetic states and each has an expectation value equal to @xmath369 . as a consequence , one only needs to know the @xmath370 matrix elements among the fm states to evaluate the ghz state entanglement @xmath364 . following the definition of the witness operator @xmath370 @xmath371\ ] ] with the stabilizing spin operators @xmath372 expressed in terms of the pauli spin operators by @xmath373 the nonzero matrix elements for the global spin flipping operator @xmath374 amongst the @xmath1 ion fm states is as a consequence , the only nonzero matrix elements for the witness operator @xmath370 are the ones between two fm states which become @xmath383 hence , the measurement @xmath364 for the pure @xmath384 state is characterized by the value @xmath140 as can be seen by the following manipulations @xmath385 99 l. balents , nature * 464 * , 199 ( 2010 ) . o. i. motrunich and m. p. a. fisher , phys . b * 75 * , 235116 ( 2007 ) . c. n. varney , k. sun , v. galitski , and m. rigol , phys 107 * , 077201 ( 2011 ) . r. feynman , int . * 21 * , 467 ( 1981 ) . s. lloyd , science * 273 * , 2073 ( 1996 ) l. buluta and f. nori , science * 326 * , 108 ( 2009 ) . j. simon , w. s. bakr , r. ma , m. e. tai , p. m. preiss , m. greiner , nature * 472 * , 307 ( 2011 ) . j. w. britton , b. c. sawyer , a. keith , c .- c . joseph wang , j. k. freericks , h. uys , m. j. biercuk , and j. j. bollinger , nature * 484 * , 489 ( 2012 ) . k. kim , m .- s . chang , s. korenblit , r. islam , e. e. edward , j. k. freericks , g .- d . lin , l .- m . duan and c. monroe , nature * 465 * , 590 ( 2010 ) . k. kim , m .- s . chang , r. islam , s. korenblit , l .- m . duan and c. monroe , phy . lett . * 103 * , 120502 ( 2009 ) . r. islam , e. e. edwards , k. kim , s. korenblit , c. noh , h. carmichael , g .- d.lin , l .- m . duan , c .- c . joseph wang , j. k. freericks , and c. monroe , nature commun . * 2 * , 1374 ( 2011 ) . a. friedenauer , h. schmitz , j. t. glueckert , d. porras , and t. schaetz , nat . phys . * 4 * , 757 ( 2008 ) . c. monroe , d. m. meekhof , b. e. king , w. m. itano , and d. j. wineland , phys . * 75 * , 4714 ( 1995 ) . d. porras and j. i. cirac , phys . lett . * 92 * , 207901 ( 2004 ) . shi - liang zhu , c. monroe , and l .- m . duan , phys . * 97 * , 050505 ( 2006 ) . d. leibfried , b. demarco , v. meyer , d. lucas , m. barrett , j. britton , w. m. itano , b. jelenkovic , c. langer , t. rosenband , d . j. wineland , nature * 422 * , 414 ( 2003 ) a. srensen and k. mlmer , phys . rev . a * 62 * , 022311 ( 2000 ) . a. das and b. k. chakrabarti , rev . phys . * 80 * , 1061 ( 2008 ) . k. kim , s. korenblit , r. islam , e. e. edwards , m. s. chang , c. noh , h. carmichael , g .- d . lin , l .- m . duan , c .- c . wang , j. k. freericks , and c. monroe , new j. phys . * 3 * , 105003 ( 2011 ) . m. johanning , a. f. varon , c. wunderlich , j. phys . b : at . mol . . phys . * 42 * , 154009 ( 2009 ) . d. f. v. james , appl . b * 66 * , 181 ( 1998 ) ; c. marquet , f. schmidt - kaler and d. f. v. james , appl . phys . b * 76 * , 199 ( 2003 ) . p. j. lee , k .- a . brickman , l. deslauriers , p. c. haljan , duan , and c. monroe , j. opt . b * 7 * , s371 ( 2005 ) . d. j. wineland , c. monroe , w. m. itano , d. leibfried , b. e. king , and d. m. meekhof , j. res . technol . * 103 * , 259 ( 1998 ) . kurt g@xmath386ttfried , _ quantum mechanics : vol . 1 , fundamentals _ , ( new york , benjamin , 1966 ) . g. t@xmath387th and o. g@xmath388hne , phys . lett . * 94 * , 060501 ( 2005 ) ; phys . rep . * 474 * , 1 ( 2009 ) . lin , c. monroe , and l .- m . duan , phys . * 106 * , 230402 ( 2011 ) . | linear paul traps have been used recently to simulate the transverse field ising model with long - range spin - spin couplings .
we study the intrinsic effects of phonon creation ( from the initial phonon ground state ) on the spin - state probability and spin entanglement for such quantum spin simulators .
while it has often been assumed that phonon effects are benign because they play no role in the pure ising model , they can play a significant role when a transverse field is added to the model .
we use a many - body factorization of the quantum time - evolution operator of the system , adiabatic perturbation theory and exact numerical integration of the schrdinger equation in a truncated spin - phonon hilbert space followed by a tracing out of the phonon degrees of freedom to study this problem .
we find that moderate phonon creation often makes the probabilities of different spin states behave differently from the static spin hamiltonian . in circumstances in which phonon creation is minor ,
the spin dynamics state probabilities converge to the static spin hamiltonian prediction at the cost of reducing the spin entanglement .
we show how phonon creation can severely impede the observation of kink transitions in frustrated spin systems when the number of ions increases .
many of our results also have implications for quantum simulation in a penning trap .
= 1 |
You are an expert at summarizing long articles. Proceed to summarize the following text:
-5 mm recall that for a suitably regular function @xmath0 on the unit disc @xmath1 we can apply integration by parts / stoke s formula twice to obtain for @xmath2 , @xmath3 where @xmath4 so actually we re integrating @xmath5 . in the presence of singularities things continue to work . for example suppose @xmath6 is a holomorphic map of complex spaces and @xmath7 a metricised effective cartier divisor on @xmath8 , with @xmath9 , and @xmath10 , where @xmath11 is the tautological section , then we obtain , @xmath12 obviously it s not difficult to write down similar formulae for not necessarily effective cartier divisors , meromorphic functions , drop the condition that @xmath9 provided @xmath13 , extend to ramified covers @xmath14 , etc . , but in all cases what is clear is , * facts 1.3 . * _ ( a ) if @xmath8 is compact then @xmath15 is very close to being positive if @xmath16 , e.g. in the particular hypothesis preceding ( 1.2 ) , @xmath17 _ \(b ) there is no such principle for the usual area function @xmath18 except in extremely special cases such as @xmath19 ample . equally the essence of the study of curves on higher dimensional varieties lies in understanding their intersection with divisors , and , of course , the principle that a curve not lying in a divisor intersects it positively is paramount to the discussion . consequently the right notion of intersection number for non - compact curves is the so - called characteristic function defined by either side of the identity ( 1.2 ) . on the other hand intersection number and integration are interchangeable in algebraic geometry , and whence we will write , * notation 1.4 . * _ let @xmath20 be a @xmath21 form on @xmath1 then for @xmath2 , @xmath22 _ the more traditional characteristic function notation is reserved for the current associated to a map , i.e. * definition 1.5 . * _ let @xmath6 be a map of complex spaces then for @xmath2 , we define , _ @xmath23 evidently in many cases one works with forms which are not quite smooth , so there are variations on the definition . in any case in order to motivate our intersection formalism let us pause to consider , -5 mm the basic theorem in the study of subvarieties of a projective variety is grothendieck s existence and properness of the hilbert scheme , or if one prefers a sequence of subvarieties of bounded degree has a convergent subsequence . of course families of smooth curves do not in general limit on smooth curves but rather semi - stable ones , and as such we must necessarily understand convergence of discs in the sense of gromov [ g ] , i.e. * definition 2.1 . * _ a disc with bubbles @xmath24 is a connected @xmath25-dimensional complex space with singularities at worst nodes exactly one of whose components is a disc @xmath1 and such that every connected component of @xmath26 is a tree of smooth rational curves . _ for @xmath27 and @xmath28 the corresponding tree of rational curves , and provided @xmath29 we can extend our integral ( 1.4 ) to this more general situation by way of , @xmath30 while if @xmath31 is a map then we have a graph , @xmath32 an appropriate formulation of gromov s compactness theorem is then , * fact 2.2 . * _ let @xmath33 be the space of maps from discs with bubbles into a projective variety @xmath8 topologised by way of the hausdorff metric on the graphs then for @xmath34 any function and @xmath35 compact the set , @xmath36 where @xmath37 is a metricised ample divisor , is compact . _ under more general hypothesis on @xmath8 , 2.2 continues to hold , but in the special projective case one has an essentially trivial proof thanks to the ubiquitous jensen formula , cf . [ m ] v.3.1 . equally although the appearance of automorphisms looks like an unwarranted complication they are necessitated by , * remarks 2.3 . * \(a ) the possibility of bubbling at the origin . \(b ) the defect of positivity for the intersection number as per ( 1.3)(a ) . observe moreover that the introduction of @xmath33 and its precise relation to @xmath38 are both necessary , and easy respectively , i.e. * fact 2.4 . * ( [ m ] v.3.5 ) _ let @xmath39 be such that the bounded subsets in the sense of @xmath40 are relatively compact then @xmath41 . moreover assuming that @xmath8 is not absurdly singular then @xmath42 iff @xmath8 contains no rational curves . _ one can equally generalise this to a log , or quasi - projective situation by introducing a divisor @xmath19 , whose components @xmath43 should be @xmath44-cartier at which point the appropriate variation thanks to a lemma of mark green , [ gr ] , is , * fact 2.5 . * _ @xmath45 iff @xmath46 and @xmath47 do not contain any affine lines . _ in particular @xmath48 is relatively compact in @xmath38 if and only if @xmath49 is compact and the boundary is _ mildly hyperbolic _ in the sense that @xmath50 does not contain affine lines . the latter question is purely algebraic and closely related to the log minimal model programme . in the case of foliations by curves an even more delicate result holds since as brunella has observed , [ b ] , the equivalence of @xmath51 with hom for invariant maps into the orbifold smooth part of a foliated variety is itself equivalent to the said foliated variety being a minimal model . -5 mm bloch s famous dictum , `` nihil est in infinito quod non fuerit prius in finito '' , might thus be translated as , * question 3.1 . * suppose for a projective variety @xmath8 , or more generally a log variety @xmath52 there is a zariski subset @xmath53 of @xmath46 through which every non - trivial map @xmath54 must factor then do we have _ hyperbolicity modulo _ @xmath53 , i.e. is it the case that a sequence @xmath55 in @xmath56 not affording a convergent subsequence in @xmath57 must be arbitrarily close ( in the compact open sense ) to @xmath58 . in the particular case that 2.5 is satisfied we can replace @xmath59 by @xmath60 and ask for _ complete hyperbolicity modulo _ @xmath53 , but outside of surfaces ( 2.5 ) seems difficult to guarantee . regardless in his thesis brody , [ br ] , provided an affirmative answer for both @xmath53 and @xmath19 empty by way of his reparameterisation lemma which was subsequently extended by green to the case of @xmath53 empty and every @xmath61 not containing holomorphic lines . bearing in mind the singular variant of green s lemma implicit in 2.5 , which for example makes it applicable to stable families of curves , it would appear that the unique known case not covered by the methods of brody and green was a theorem of bloch himself , [ bl ] , i.e. @xmath624 planes in general position@xmath63 , and its subsequent extension by cartan to @xmath64 , [ c ] . however , even here , a moment s inspection shows that 2.5 holds , so one knows a priori that there can be no bubbling , and whence complete hyperbolicity in the sense of 3.1 trivially implies so - called normal convergence modulo the diagonal hyperplanes , and the correct structure is obscured . now an extension of the reparameterisation lemma to cover 3.1 would be by far the most preferable way forward , since the non - existence of holomorphic lines is an essentially useless qualitative statement without the quantitative information provided by the convergence of discs . nevertheless we can vaguely approximate a reparameterisation lemma thanks as ever to jensen s formula . specifically consider as given , * data 3.2 . * _ _ * a @xmath44-cartier divisor @xmath65 on a log - variety @xmath52 . * a sequence @xmath66 which neither affords a convergent subsequence nor is arbitrarily close to @xmath67 . in light of ( b ) we can choose convergent automorphisms @xmath68 , such that @xmath69 is bounded away from @xmath67 , and given , modulo subsequencing , the convergence of the @xmath70 we may as well suppose this . moreover for each @xmath71 we can normalise the current @xmath72 of 1.5 by its degree with respect to an ample divisor @xmath73 , which we ll denote by @xmath74 and take a weak limit for a suitable subset @xmath75 of @xmath76 to obtain a current @xmath77 . in addition 3.2(b ) also tells us that for some fixed @xmath78 , the degrees of the @xmath55 at @xmath79 go to infinity , and whence by ( 1.1 ) and ( 1.2 ) * pre - fact 3.3 . * _ for @xmath80 , @xmath81 is a positive harmonic current such that , @xmath82 for all effective divisors @xmath83 supported in @xmath67 . _ what is somewhat less trivial , but once more the key is jensen s formula , is , * fact 3.3(bis ) . * ( [ m ] v.2.4 ) _ subsequencing in @xmath75 as necessary , then for @xmath80 outside of a set of finite hyperbolic measure ( i.e. @xmath84 ) @xmath81 is closed . _ obviously there are various choices involved but whenever we re dealing in the context of countably many projective varieties they can all be rendered functorial , up to a constant , with respect to push forward . the constant itself only causes a problem should it be zero which is usually what one wants to prove anyway , and as such the notation @xmath85 is relatively unambiguous , and represents in a vague sense a parabolic limit of the sequence @xmath55 . -5 mm applications of course require some knowledge of intersection numbers , and quite generally even for a compact curve @xmath86 there is very little that one can say in general beyond , * observation 4.1 . * _ let @xmath87 be the derivative @xmath88 in the notation of ega ) with @xmath89 the tautological bundle then , @xmath90 _ this is of course the riemann - hurwitz formula if @xmath91 , and there s an equally trivially log - variant where on the right hand side we have to throw in the number of points in the intersection with the boundary @xmath19 counted without multiplicity the special case of @xmath92 being mason s `` @xmath93 '' theorem for polynomials . the correct generality for best possible applications is to work with log - smooth deligne - mumford stacks ( or alternatively just orbifolds since the inertia tends to be irrelevant ) , however for simplicity let s stick with log - smooth varieties and metricise @xmath94 by way of a complete metric @xmath95 on @xmath52 , which in turn leads to a mildly singular metricisation @xmath96 of the tautological bundle . supposing for simplicity that @xmath9 with @xmath97 unramified at the origin then jensen s formula yields , * observation 4.2(bis ) . * _ notations as above , _ @xmath98 combining the concavity of the logarithm and once more jensen s formula , but this time for @xmath99 for any norm on the boundary divisor @xmath19 , immediately yields in the notations of 3.3 , * fact 4.3 . * _ let @xmath100 be the current associated to the logarithmic derivative of a sequence @xmath101 with @xmath102 not arbitrarily close to @xmath19 and which does not afford a convergent subsequence then outwith a set of finite hyperbolic measure , _ @xmath103 the so - called tautological inequality 4.3 is well adapted for applications to convergence of discs ( note incidentally that it s implicit to the formulation that a smooth metric on the bundle @xmath94 is being employed ) . nevertheless for more delicate questions such as quantifying degenerate / non - convergent behaviour . there is a wealth of information in ( 4.2 ) that is lost in the coarser corollary . indeed even using the concavity of the logarithm distorts a very delicate term measuring the ` ramification at @xmath104 ' , i.e. the distorsion of the boundary from it s length in the poincar metric , which is closely related to the difficulty of extracting an isoperimetric inequality from a knowledge of hyperbolicity in the sense of 3.1 . while from the still deeper curvature point of view , 4.2 is simply a doubly integrated tautological schwarz lemma , since by definition metricising @xmath105 by way of a metric @xmath106 of curvature @xmath107 is equivalent to a lower bound of the left hand side of the form , @xmath108 for all infinitely small , and whence all in the large , possible discs . while on the subject of curvature and isoperimetric inequalities , a variant specific to dimension 1 replaces the current @xmath109 implicitly hidden in ( 4.2 ) by the current associated to the boundary of a simply connected region , i.e. * variant 4.4 . * _ suppose @xmath110 , let @xmath111 be simply connected , @xmath112 isomorphisms and put @xmath113 then specific to dimension 1 , @xmath114 is closed and may be written , @xmath115 for @xmath73 an ample divisor of degree 1 . now apply jensen s formula to recover an integrated form of ahlfor s isoperimetric inequality , and the five island s theorem . _ returning to varieties and divisors it s still possible to employ ( 4.2 ) to get integrated isoperimetric inequalities for more general situations that preserve some 1-dimensional flavour , i.e. discs which are invariant by foliations by curves , with _ canonical foliation singularities _ ( with the obvious definition of that notion which is functorial with respect to the ideas ) and which do not pass through the singularities . the latter hypothesis which is reasonable for the study of the leafwise variation of the poincar metric is however somewhat restrictive for other applications , and is probably unnecessary as suggested by the essentially optimal inequality of [ m ] v.4.4 for foliations on surfaces which employs ( 4.2 ) to a very large number of monoidal transformations in the foliation singularities . regardless here is a genuinely 2-dimensional theorem , * theorem 4.5 . * ( [ m ] v.5 ) _ let @xmath52 be a smooth logarithmic surface with @xmath116 big ( e.g. log - general type , and @xmath117 ) then there is a proper zariski subset @xmath53 of @xmath46 such that @xmath46 is complete hyperbolic ( in the sense of @xmath118 et sequel ) modulo @xmath53 . _ indeed one can even optimally quantify ( cf . op . the degeneration of the kobayashi metric ( which is evidently continuous and non - zero off @xmath53 ) around @xmath53 . amusingly the theorem only covers @xmath625 planes in general position@xmath63 , although it s a good exercise in the techniques ( cf . op . v.4 ) to prove bloch s theorem too , at which point a rather small sequence of blow ups replaces all of the original estimation . in any case ( 4.5 ) should only be seen as a stepping stone which in order of ascending difficulty leaves open the following questions , viz , * concluding remarks 4.6 . * for concreteness take a smooth algebraic surface @xmath8 of general type with @xmath119 ( otherwise the following should be understood in terms of higher jets , but not for anything more general than a surface ) then , * do we have an isoperimetric inequality with appropriate degeneration along the subset @xmath53 of ( 4.5 ) . * is the kobayashi metric negatively curved . * for each @xmath120 and @xmath121 a tangent direction at @xmath122 , is there a unique up to the usual action of @xmath123 pointed disc with maximal tangent in the direction @xmath121 , and if so does it continue to be so along its image , i.e. is there a continuous ( off @xmath53 ) connection whose geodesics are the discs defining the kobayashi metric . -5 mm in closing it is a pleasure to thank m. brunella for introducing me to the unit disc and bubbling , together with m. gromov for continuing this education , but above all ccile without whom the reader could never have got this far . | we consider the algebro - geometric consequences of integration by parts .
4.5 mm * 2000 mathematics subject classification : * 32 , 14 . * keywords and phrases : * jensen s formula . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
linear regression for interval - valued data has been attracting increasing interests among researchers . see @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , for a partial list of references . however , issues such as interpretability and computational feasibility still remain . especially , a commonly accepted mathematical foundation is largely underdeveloped , compared to its demand of applications . by proposing our new model , we continue to build up the theoretical framework that deeply understands the existing models and facilitates future developments . in the statistics literature , the interval - valued data analysis is most often studied under the framework of random sets , which includes random intervals as the special ( one - dimensional ) case . the probability - based theory for random sets has developed since the publication of the seminal book of @xcite . see @xcite for a relatively complete monograph . to facilitate the presentation of our results , we briefly introduce the basic notations and definitions in the random set theory . let @xmath1 be a probability space . denote by @xmath2 or @xmath3 the collection of all non - empty compact subsets of @xmath4 . in the space @xmath3 , a linear structure is defined by minkowski addition and scalar multiplication , i.e. , @xmath5 @xmath6 and @xmath7 . a natural metric for the space @xmath3 is the hausdorff metric @xmath8 , which is defined as @xmath9 where @xmath10 denotes the euclidean metric . a random compact set is a borel measurable function @xmath11 , @xmath3 being equipped with the borel @xmath12-algebra induced by the hausdorff metric . for each @xmath13 , the function defined on the unit sphere @xmath14 : @xmath15 is called the support function of x. if @xmath16 is convex almost surely , then @xmath17 is called a random compact convex set . ( see @xcite , p.21 , p.102 . ) the collection of all compact convex subsets of @xmath4 is denoted by @xmath18 or @xmath19 . when @xmath20 , the corresponding @xmath19 contains all the non - empty bounded closed intervals in @xmath0 . a measurable function @xmath21 is called a random interval . much of the random sets theory has focused on compact convex sets . let @xmath22 be the space of support functions of all non - empty compact convex subsets in @xmath19 . then , @xmath22 is a banach space equipped with the @xmath23 metric @xmath24^{\frac{1}{2}},\ ] ] where @xmath25 is the normalized lebesgue measure on @xmath14 . according to the embedding theorems ( see @xcite , @xcite ) , @xmath19 can be embedded isometrically into the banach space @xmath26 of continuous functions on @xmath14 , and @xmath22 is the image of @xmath27 into @xmath26 . therefore , @xmath28 , @xmath29 , defines a metric on @xmath27 . particularly , let @xmath30=[x^c - x^r , x^c+x^r]\ ] ] be an bounded closed interval with center @xmath31 and radius @xmath32 , or lower bound @xmath33 and upper bound @xmath34 , respectively . then , the @xmath35-metric of @xmath36 is @xmath37 and the @xmath35-distance between two intervals @xmath36 and @xmath38 is @xmath39^{\frac{1}{2}}\\ & = & \left[\left(x^c - y^c\right)^2+\left(x^r - y^r\right)^2\right]^{\frac{1}{2}}.\end{aligned}\ ] ] existing literature on linear regression for interval - valued data mainly falls into two categories . in the first , separate linear regression models are fitted to the center and range ( or the lower and upper bounds ) , respectively , treating the intervals essentially as bivariate vectors . examples belonging to this category include the center method by @xcite , the minmax method by @xcite , the ( constrained ) center and range method by @xcite , and the model m by @xcite . these methods aim at building up model flexibility and predicting capability , but without taking the interval as a whole . consequently , their geometric interpretations are prone to different degrees of ambiguity . take the constrained center and range method ( ccrm ) for example . adopting the notations in @xcite , it is specified as @xmath40 where @xmath41 and @xmath42 . it follows that @xmath43 ^ 2+\left[\beta_1^r\left(x_i^r - x_j^r\right)\right]^2.\end{aligned}\ ] ] because @xmath44 in general , a constant change in @xmath45 does not result in a constant change in @xmath46 . in fact , a constant change in any metric of @xmath36 as an interval does not lead to a constant change in the same metric of @xmath38 . this essentially means that the model is not linear in intervals . in the second category , special care is given to the fact that the interval is a non - separable geometric unit , and their linear relationship is studied in the framework of random sets . investigation in this category began with @xcite developing a least squares fitting of compact set - valued data and considering the interval - valued input and output as a special case . precisely , he gave analytical solutions to the real - valued numbers @xmath47 and @xmath48 under different circumstances such that @xmath49 is minimized on the data . the pioneer idea of @xcite was further studied in @xcite , where the @xmath35-metric was extended to a more general metric called @xmath50-metric originally proposed by @xcite . the advantage of the @xmath50-metric lies in the flexibility to assign weights to the radius and midpoints in calculating the distance between intervals . so far the literature had been focusing on finding the affine transformation @xmath51 that best fits the data , but the data are not assumed to fulfill such a transformation . a probabilistic model along this direction kept missing until @xcite , and simultaneously @xcite , proposed the same simple linear regression model for the first time . the model essentially takes on the form of @xmath52 with @xmath53 and @xmath54 , c\in\mathbb{r}$ ] . this can be written equivalently as @xmath55 it leads to the following equation that clearly shows linearity in @xmath19 : @xmath56 some advances have been made regarding this model and the associated estimators . @xcite derived least squares estimators for the model parameters and examined them from a theoretical perspective . @xcite established a test of linear independence for interval - valued data . however , many problems still remain open such as biases and asymptotic distributions , as anticipated in @xcite . this paper presents a continuous development addressing some issues and open problems in the direction of model ( [ mod - gil ] ) . first , we relax the restriction of model ( [ mod - gil ] ) that the hukuhara difference @xmath57 must exist ( see @xcite ) and generalize the univariate model to the multiple case . we also give analytical least squares ( ls ) solutions to the model parameters . second , we show that our model and ls estimation together accommodate a decomposition of the sums of squares in @xmath19 analogous to that of the classical linear regression . third , we derive explicit formulas of the ls estimates for the univariate model , which exist with probability going to one . the ls estimates are further shown to be asymptotically unbiased . a simulation study is carried out to validate our theoretical findings , as well as compare our model to ccrm . finally , we apply our model to a climate data set to illustrate the applicability of our model . the rest of the paper is organized as follows : section 2 formally introduces our model and the associated ls estimators . then , the sums of squares and coefficient of determination in @xmath19 are defined and discussed . section 3 presents the theoretical properties of the ls estimates for the univariate model . the simulation study is reported in section 4 , and the real data application is presented in section 5 . we give concluding remarks in section 6 . technical proofs and useful lemmas are deferred to the appendices . we consider an extension of model ( [ mod - gil ] ) to the form @xmath58 where @xmath59=[-c , c]$ ] , @xmath60 . it is equivalently expressed as @xmath61 this leads to the following center - radius specification @xmath62 where @xmath63 , @xmath64 , and the signs @xmath65 " correspond to the two cases in ( [ mod - cases ] ) . define @xmath66 our model is specified as @xmath67 where @xmath68 , @xmath69 $ ] , @xmath70 , and @xmath71 . to model the outcome intervals @xmath72 $ ] by @xmath73 interval - valued predictors @xmath74 $ ] , @xmath75 ; @xmath76 , we consider the multivariate extension of ( [ mod-1 ] ) : @xmath77 which leads to the following center - radius specification @xmath78 where @xmath68 , @xmath69 $ ] , @xmath79 , and @xmath80 . we have assumed @xmath81 and @xmath82 are independent in this paper to simplify the presentation . the model that includes a covariance between @xmath81 and @xmath82 can be implemented without much extra difficulty . least squares method is widely used in the literature to estimate the interval - valued regression coefficients ( @xcite , @xcite , @xcite ) . it minimizes @xmath83 on the data with respect to the parameters . denote @xmath84 then the sum of squared @xmath35-distance between @xmath85 and @xmath86 is written as @xmath87\\ & = & \sum_{i=1}^{n}\left[\left(b+\sum_{j=1}^{p}a_jx_{j , i}^c - y_i^c\right)^2+\left(\sum_{j=1}^{p}\left|a_j\right|x_{j , i}^r+\mu - y_i^r\right)^2\right].\end{aligned}\ ] ] therefore , the lse of @xmath88 is defined as @xmath89 let @xmath90 be the sample covariances of the centers and radii of @xmath91 and @xmath92 , respectively . especially , when @xmath93 , we denote by @xmath94 and @xmath95 the corresponding sample variances . in addition , define @xmath96 as the sample covariances of the centers and radii of @xmath91 and @xmath38 , respectively . then , the minimization problem ( [ def - ls ] ) is solved in the following proposition . [ prop : ls_solu ] the least squares estimates of the regression coefficients @xmath97 , if they exist , are solution of the equation system : @xmath98 and then , @xmath99 are given by @xmath100 the variance of a compact convex random set @xmath36 in @xmath4 is defined via its support function as @xmath102 where the expectation is defined by aumann integral ( see @xcite , @xcite ) as @xmath103 see @xcite . for the case @xmath20 , it is shown by straightforward calculations that @xmath104,\\ & & \text{var}(x)=\text{var}\left(x^c\right)+\text{var}\left(x^r\right).\end{aligned}\ ] ] this leads us to define the sums of squares in @xmath105 to measure the variability of interval - valued data . a definition of the coefficient of determination @xmath101 in @xmath105 follows immediately , which produces a measure of goodness - of - fit . the total sum of squares ( sst ) in @xmath27 is defined as @xmath106.\ ] ] the explained sum of squares ( sse ) in @xmath27 is defined as @xmath107.\ ] ] the residual sum of squares ( ssr ) in @xmath27 is defined as @xmath108.\ ] ] the coefficient of determination ( @xmath101 ) in @xmath27 is defined as @xmath109 where @xmath110 and @xmath111 are defined in ( [ def : sst ] ) and ( [ def : ssr ] ) , respectively . analogous to the classical theory of linear regression , our model ( [ mmod-1**])-([mmod-2 * * ] ) together with the ls estimates ( [ def - ls ] ) accommodates the partition of @xmath110 into @xmath112 and @xmath111 . as a result , the coefficient of determination ( @xmath101 ) can also be calculated as the ratio of @xmath112 and @xmath110 . the partition has a series of important implications of the underlying model , one of which being that the residual @xmath113/@xmath114 and the predictor @xmath115 are empirically uncorrelated in @xmath116 . [ thm : ss ] assume model ( [ mmod-1**])-([mmod-2 * * ] ) . let @xmath117 and @xmath118 in ( [ exp - c])-([exp - r ] ) be calculated according to the ls estimates @xmath119 in ( [ def - ls ] ) . then , @xmath120 it follows that the coefficient of determination in @xmath19 is equivalent to @xmath121 it is possible to get negative values of @xmath122 by its definition ( [ exp - r ] ) . theorem [ thm : pred - adjust ] gives an upper bound of the probability of this unfortunate event . if the model largely explains the variability of @xmath123 , @xmath124 should be very small and so is this bound . then , the rare cases of negative @xmath122 can be rounded up to 0 since @xmath118 is nonnegative . otherwise , if most of the variability of @xmath123 lies in the random error , the probability of getting negative predicts may not be ignorable , but it is essentially due to the insufficiency of the model and a different model should be pursued anyway . [ thm : pred - adjust ] consider model ( [ mmod-1**])-([mmod-2 * * ] ) . let @xmath86 be defined in ( [ exp - c])-([exp - r ] ) . then , @xmath125 in this section , we study the theoretical properties of the lse for the univariate model ( [ mod-1**])-([mod-2 * * ] ) . applying proposition [ prop : ls_solu ] to the case @xmath126 , we obtain the two sets of half - space solutions , corresponding to @xmath127 and @xmath128 , respectively , as follows : @xmath129 and @xmath130 the final formula for the ls estimates falls in three categories . in the first , there is one and only one set of existing solution , which is defined as the lse . in the second , both sets of solutions exist , and the lse is the one that minimizes @xmath131 . in the third situation , neither solution exists , but this only happens with probability going to @xmath132 . we conclude these findings in the following theorem . [ thm : ls_solu ] assume model ( [ mod-1**])-([mod-2 * * ] ) . let @xmath133 be the least squares solution defined in ( [ def - ls ] ) . if @xmath134 , then there exists one and only one half - space solution . more specifically , + * i. * if in addition @xmath135 , then the ls solution is given by @xmath136 * ii . * if instead @xmath137 , then the ls solution is given by @xmath138 otherwise , @xmath139 , and then either both of the half - space solutions exist , or neither one exists . in particular , + * iii . * if in addition @xmath140 , then both of the half - space solutions exist , and @xmath141 * iv . * if instead @xmath142 , then the ls solution does not exist , but this happens with probability converging to 0 . unlike the classical linear regression , ls estimates for the model ( [ mod-1**])-([mod-2 * * ] ) are biased . we calculate the biases explicitly in proposition [ prop : ls_exp ] , which are shown to converge to zero as the sample size increases to infinity . therefore , the ls estimates are asymptotically unbiased . [ prop : ls_exp ] let @xmath143 be the least squares solution in theorem [ thm : ls_solu ] . then , @xmath144,\end{aligned}\ ] ] @xmath145.\end{aligned}\ ] ] [ thm : ls_consist ] consider model ( [ mod-1**])-([mod-2 * * ] ) . assume @xmath146 and @xmath147 . then , the least squares solution @xmath143 in theorem [ thm : ls_solu ] is asymptotically unbiased , i.e. @xmath148 as @xmath149 . we carry out a systematic simulation study to examine the empirical performance of the least squares method proposed in this paper . first , we consider the following three models : + * model 1 : @xmath150 , @xmath151 , @xmath152 , @xmath153 , @xmath154 ; * model 2 : @xmath155 , @xmath151 , @xmath152 , @xmath153 , @xmath156 ; * model 3 : @xmath150 , @xmath151 , @xmath157 , @xmath153 , @xmath154 ; where data show a positive correlation , a negative correlation , and a positive correlation with a negative @xmath25 , respectively . a simulated dataset from each model is shown in figure [ fig : sim - data ] , along with its fitted regression line . + to investigate the asymptotic behavior of the ls estimates , we repeat the process of data generation and parameter estimation 1000 times independently using sample size @xmath158 for all the three models . the resulting 1000 independent sets of parameter estimates for each model / sample size are evaluated by their mean absolute error ( mae ) and mean error ( me ) . the numerical results are summarized in table [ tab : sim ] . consistent with proposition [ prop : ls_exp ] , @xmath159 tends to underestimate @xmath47 when @xmath160 and overestimate @xmath47 when @xmath128 . this bias also causes a positive and negative bias in @xmath161 , when @xmath160 and @xmath128 , respectively . similarly , a positive bias in @xmath162 is induced by the negative bias in @xmath163 . all the biases diminish to 0 as the sample size increases to infinity , which confirms our finding in theorems [ thm : ls_consist ] . .evaluation of parameter estimation [ cols="^ , > , > , > , > , > , > , > " , ] [ tab : sim - com ] in this section , we apply our model to analyze the average temperature data for large us cities , which are provided by national oceanic and atmospheric administration ( noaa ) and are publicly available . the three data sets we obtained specifically are average temperatures for 51 large us cities in january , april , and july . each observation contains the averages of minimum and maximum temperatures based on weather data collected from 1981 to 2010 by the noaa national climatic data center of the united states . july in general is the hottest month in the us . by this analysis , we aim to predict the summer ( july ) temperatures by those in the winter ( january ) and spring ( april ) . figure [ fig : real - data ] plots the july temperatures versus those in january and april , respectively . the parameters are estimated according to ( [ eqn : lse-1])-([eqn : lse-3 ] ) as @xmath164 denote by @xmath165 , @xmath166 , and @xmath167 , the average temperatures in a us city in january , april , and july , respectively . the prediction for @xmath167 based on @xmath165 and @xmath166 is given by @xmath168 the three sums of squares are calculated to be @xmath169 therefore , the coefficient of determination is @xmath170 finally , the variance parameters can be estimated as @xmath171 thus , by theorem [ thm : pred - adjust ] , an upper bound of @xmath172 on average is estimated to be @xmath173 which is very small and reasonably ignorable . we calculate @xmath174 for the entire sample and all of them are well above @xmath132 . so , for this data , although @xmath175 and it is possible to get negative predicted radius , it in fact never happens because the model has captured most of the variability . the empirical distributions of residuals are shown in figure [ fig : real - residual ] . both distributions are centered at 0 , with the center residual having a slightly bigger tail . we have rigorously studied linear regression for interval - valued data in the metric space @xmath176 . the new model we introduces generalizes previous models in the literature so that the hukuhara difference @xmath177 needs not exist . analogous to the classical linear regression , our model together with the ls estimation leads to a partition of the total sum of squares ( ssr ) into the explained sum of squares ( sse ) and the residual sum of squares ( ssr ) in @xmath176 , which implies that the residual is uncorrelated with the linear predictor in @xmath176 . in addition , we have carried out theoretical investigations into the least squares estimation for the univariate model . it is shown that the ls estimates in @xmath176 are biased but the biases reduce to zero as the sample size tends to infinity . therefore , a bias - correction technique for small sample estimation could be a good future topic . the simulation study confirms our theoretical findings and shows that the least squares estimators perform satisfactorily well for moderate sample sizes . 99 artstein z , vitale , ra . a strong law of large numbers for random compact sets . annals of probability . 1975;5:879882 . aumann rj . integrals of set - valued functions . j. math . 1965;12:112 . billard l , diday e. regression analysis for interval - valued data . in : data analysis , classification and related methods , proceedings of the seventh conference of the international federation of classification societies ( ifcs00 ) . springer , belgium ; 2000.p.369374 . billard l , diday e. symbolic regression analysis . in : classification , clustering and data analysis , proceedings of the eighth conference of the international federation of classification societies ( ifcs02 ) . springer , poland ; 2002.p.281288 . billard l. dependencies and variation components of symbolic interval - valued data . in : selected contributions in data analysis and classification . springer , berlin heidelberg ; 2007.p.312 . blanco - fernndez a , corral n , gonzlez - rodrguez g. estimation of a flexible simple linear model for interval data based on set arithmetic . computational statistics & data analysis . 2011;55:25682578 . blanco - fernndez a , colubi a , gonzlez - rodrguez g. confidence sets in a linear regression model for interval data . journal of statistical planning and inference . 2012;142:13201329 . carvalho fat , lima neto ea , tenorio , cp . a new method to fit a linear regression model for interval - valued data . lecture notes in computer sciences . 2004;3238 : 295306 . cattaneo megv , wiencierz a. likelihood - based imprecise regression . international journal of approximate reasoning . 2012;53:11371154 . diamond p. least squares fitting of compact set - valued data . j. math . 1990;147:531544 . gil ma , lopez mt , lubiano ma , and montenegro m. regression and correlation analyses of a linear relation between random intervals . 2001;10,1:183201 . gil ma , lubiano ma , montenegro m , lopez mt . least squares fitting of an affine function and strength of association for interval - valued data . metrika . 2002;56 : 97111 . gil ma , gonzlez - rodrguez g , colubi a , montenegro m. testing linear independence in linear models with interval - valued data . computational statistics & data analysis . 2007;51:30023015 . gonzlez - rodrguez g , blanco a , corral n , colubi a. least squares estimation of linear regression models for convex compact random sets . advances in data analysis and classification . 2007;1:6781 . hrmander h. sur la fonction dappui des ensembles convexes dans un espace localement convexe . arkiv fr mat . 1954;3:181186 . hukuhara m. integration des applications mesurables do nt la valeur est un compact convexe . funkcialaj ekvacioj . 1967;10:205223 . kendall dg . foundations of a theory of random sets . in : harding ef and kendall dg ( ed ) stochastic geometry . john wiley & sons , new york ; 1974 . krner r. a variance of compact convex random sets . institut fr stochastik , bernhard - von - cotta - str . 2 09599 freiberg ; 1995 . krner r. on the variance of fuzzy random variables . fuzzy sets and systems . 1997;92:8393 . krner r , nther w. linear regression with random fuzzy variables : extended classical estimates , best linear estimates , least squares estimates . information sciences . 1998;109 : 95118 . lyashenko nn . limit theorem for sums of independent compact random subsets of euclidean space . journal of soviet mathematics . 1982;20:21872196 . lyashenko nn . statistics of random compacts in euclidean space . journal of soviet mathematics . 1983;21:7692 . manski cf , tamer t. inference on regressions with interval data on a regressor or outcome . 2002;70:519546 . matheron g. random sets and integral geometry . john wiley & sons , new york ; 1975 . molchanov i. theory of random sets . springer , london ; 2005 . lima neto ea , carvalho fat . centre and range method for fitting a linear regression model to symbolic interval data . computational statistics & data analysis . 2008 ; 52:15001515 . lima neto ea , carvalho fat . constrained linear regression models for symbolic interval - valued variables . computational statistics & data analysis . 2010;54:333347 . radstrm h. an embedding theorem for spaces of convex sets . differentiating @xmath131 with respect to @xmath25 , @xmath48 , and @xmath178 , @xmath76 , respectively , and setting the derivatives to zero , we get @xmath179 equations ( [ lse-1])-([lse-2 ] ) yield @xmath180 equations ( [ eqn : lse-1 ] ) are obtained by plugging ( [ lse-4])-([lse-5 ] ) into ( [ lse-3 ] ) , and equations ( [ eqn : lse-2])-([eqn : lse-3 ] ) follow from ( [ lse-4])-([lse-5 ] ) . this completes the proof . according to definitions ( [ def : sst])-([def : ssr ] ) , @xmath181\nonumber\\ & = & sse+ssr+2\sum_{i=1}^{n}\left[\left(y_i^c-\hat{y}_i^c\right)\left(\hat{y}_i^c-\overline{y^c}\right ) + \left(y_i^r-\hat{y}_i^r\right)\left(\hat{y}_i^r-\overline{y^r}\right)\right]\nonumber\\ & = & sse+ssr+2\sum_{i=1}^{n}\left[\left(y_i^c-\hat{y}_i^c\right)\hat{y}_i^c+\left(y_i^r-\hat{y}_i^r\right)\hat{y}_i^r\right].\label{ss : eqn-1}\end{aligned}\ ] ] the last equation is due to ( [ lse-1])-([lse-2 ] ) . further in view of ( [ exp - c])-([exp - r ] ) and ( [ lse-3 ] ) , we have @xmath182\\ & = & \sum_{i=1}^{n}\left[\left(y_i^c-\hat{y}_i^c\right)\sum_{j=1}^{p}a_jx_{j , i}^c+\left(y_i^r-\hat{y}_i^r\right)\sum_{j=1}^{p}|a_j|x_{j , i}^r\right]\\ & = & \sum_{j=1}^{p}a_j\sum_{i=1}^{n}\left[\left(y_i^c-\hat{y}_i^c\right)x_{j , i}^c+\left(y_i^r-\hat{y}_i^r\right)sgn(a_j)x_{j , i}^r\right]\\ & = & 0.\end{aligned}\ ] ] this together with ( [ ss : eqn-1 ] ) completes the proof . notice that @xmath183 an application of markov s inequality completes the proof . parts * i * , * ii * and * iii * are obvious from proposition [ prop : ls_solu ] . part * iv * follows from lemma [ lem : cov_r ] in appendix ii . we prove the cases @xmath127 and @xmath128 separately . to simplify notations , we will use @xmath184 throughout the proof , but the expectation should be interpreted as being conditioned on @xmath36 . + * case i : @xmath127*. + from lemma [ lem : cov_est ] , we have @xmath185 + \sum_{i < j}(x_i^r - x_j^r ) \left [ ( y_i^r - y_j^r)-a(x_i^r - x_j^r ) \right]}{\sum_{i < j}(x_i^c - x_j^c)^2+\sum_{i < j}(x_i^r - x_j^r)^2}\\ = & \frac{\sum_{i < j}(x_i^c - x_j^c)(\lambda_i-\lambda_j)+\sum_{i < j}(x_i^r - x_j^r)(\eta_i-\eta_j ) } { \sum_{i < j}(x_i^c - x_j^c)^2+\sum_{i < j}(x_i^r - x_j^r)^2}.\\\end{aligned}\ ] ] this immediately yields @xmath186 similarly , @xmath187 } { \sum_{i < j}(x_i^c - x_j^c)^2+\sum_{i < j}(x_i^r - x_j^r)^2},\ ] ] and consequently , @xmath188 notice now @xmath189\\ & & -\left[\int_{\{\hat{a}=a^-\}}(a^+-a ) \mathrm{d}\mathbb{p}+\int_{\{\hat{a}=a^-\}}(a^--a ) \mathrm{d}\mathbb{p } \right]\nonumber\\ & = & e\left(a^+-a\right)-\int_{\{\hat{a}=a^-\}}(a^+-a^- ) \mathrm{d}\mathbb{p}\nonumber\\ & = & -e(a^+-a^-)i_{\{\hat{a}=a^-\}}\label{eqn-3}.\end{aligned}\ ] ] here , equation ( [ eqn-3 ] ) is due to ( [ eqn-1 ] ) . recall that @xmath190 } { \sum_{i < j}(x_i^c - x_j^c)^2+\sum_{i < j}(x_i^r - x_j^r)^2},\label{a+-a-}\end{aligned}\ ] ] since @xmath191 . therefore , @xmath192 } { \sum_{i < j}(x_i^c - x_j^c)^2+\sum_{i < j}(x_i^r - x_j^r)^2}\right\}i_{\{\hat{a}=a^-\}}\nonumber\\ & = & -\frac{2\sum_{i < j}\left[|a|(x_i^r - x_j^r)^2p(\hat{a}=a^-)+(x_i^r - x_j^r)e(\eta_i-\eta_j)i_{\{\hat{a}=a^-\}}\right ] } { \sum_{i < j}(x_i^c - x_j^c)^2+\sum_{i < j}(x_i^r - x_j^r)^2}\nonumber\\ & = & -\frac{2\sum_{i < j}(x_i^r - x_j^r)^2p(\hat{a}=a^-)}{\sum_{i < j}(x_i^c - x_j^c)^2+\sum_{i < j}(x_i^r - x_j^r)^2}\nonumber\\ & = & -\frac{2as^2(x^r)}{s^2(x^c)+s^2(x^r)}p(\hat{a}=a^-).\label{rst-1}\end{aligned}\ ] ] similar to the preceding arguments , @xmath193 recall again that @xmath194}{s^2\left(x^c\right)+s^2\left(x^r\right)}.\label{a++a-}\end{aligned}\ ] ] it follows that @xmath195 * case ii : @xmath128 * + in this case , we have @xmath196}{s^2(x^c)+s^2(x^r)},\\ & a^--a = \frac{\sum_{i < j}(x_i^c - x_j^c)(\lambda_i-\lambda_j)-\sum_{i < j}(x_i^r - x_j^r)(\eta_i-\eta_j)}{s^2(x^c)+s^2(x^r)}.\\\end{aligned}\ ] ] these imply @xmath197 similar to the case of @xmath127 , we obtain @xmath198 these , together with ( [ a+-a- ] ) and ( [ a++a- ] ) , imply , @xmath199 the desired result follows from ( [ rst-1 ] ) , ( [ rst-2 ] ) , ( [ rst-3 ] ) and ( [ rst-4 ] ) . from ( [ b+ ] ) and ( [ b- ] ) , @xmath200 similarly , from ( [ mu+ ] ) and ( [ mu- ] ) , @xmath201 hence , the desired result follows by proposition [ prop : ls_exp ] and lemma [ lem : sign - consist ] in the appendix . [ lem : cov_r ] assume model ( [ mod-1**])-([mod-2 * * ] ) and @xmath202 . then @xmath203 . consequently , @xmath204 with probability converging to 1 . according to ( [ mod-2 * * ] ) , @xmath205-e\left(x^r\right)e\left(|a|x^r+\eta_1\right)\nonumber\\ & = & |a|e\left(x^r\right)^2+\mu e\left(x^r\right)-|a|\left[e\left(x^r\right)\right]^2-\mu e\left(x^r\right)\nonumber\\ & = & |a|\text{var}\left(x^r\right)\nonumber\\ & \geq & 0,\label{cov_true}\end{aligned}\ ] ] provided that @xmath202 . by the slln , @xmath206 ( [ cov_sample ] ) together with ( [ cov_true ] ) completes the proof . to prove , @xmath209\\ & = n\sum_{i=1}^nx_i^vy_i^v-(\sum_{i=1}^nx_i^v)(\sum_{i=1}^ny_i^v)=n^2s\left(x^v , y^v\right).\\\end{aligned}\ ] ] follows by replacing @xmath210 with @xmath211 and @xmath210 with @xmath212 in the above calculations . [ lem : sign - consist ] assume model ( [ mod-1**])-([mod-2 * * ] ) . assume in addition that @xmath146 and @xmath147 . let @xmath133 be the least squares solution defined in ( [ def - ls ] ) . then @xmath213 as @xmath149 . we prove the case @xmath127 only . the case @xmath128 can be proved similarly . under the assumption that @xmath127 , @xmath214 and consequently , @xmath215 . according to theorem [ thm : ls_solu ] , the only other circumstance under which @xmath216 is when @xmath217 and @xmath218 simultaneously . it is therefore sufficient to show that @xmath219 notice @xmath220\\ & & + \frac{1}{n}\sum_{i=1}^{n}\left[\left(a^+x_i^r+\mu - y_i^r\right)^2-\left(a^-x_i^r+\mu - y_i^r\right)^2\right]\\ : = & & \frac{1}{n}\left(i+ii\right).\end{aligned}\ ] ] the first term @xmath221\\ & = & \sum_{i=1}^{n}\left[\left(a^{+}-a\right)^2\left(x_i^c-\overline{x^c}\right)^2+\left(\lambda_i-\overline{\lambda}\right)^2 -2\left(a^{+}-a\right)\left(x_i^c-\overline{x^c}\right)\left(\lambda_i-\overline{\lambda}\right)\right]\\ & & -\sum_{i=1}^{n}\left[\left(a^{-}-a\right)^2\left(x_i^c-\overline{x^c}\right)^2+\left(\lambda_i-\overline{\lambda}\right)^2 -2\left(a^{-}-a\right)\left(x_i^c-\overline{x^c}\right)\left(\lambda_i-\overline{\lambda}\right)\right]\\ & = & \left[\left(a^{+}-a\right)^2-\left(a^{-}-a\right)^2\right]\sum_{i=1}^{n}\left(x_i^c-\overline{x^c}\right)^2\\ & & -2\left(a^{+}-a^{-}\right)\sum_{i=1}^{n } \left(x_i^c-\overline{x^c}\right)\left(\lambda_i-\overline{\lambda}\right)\\ & = & \left(a^{+}-a^{-}\right)\left[\left(a^{+}+a^{-}-2a\right)\sum_{i=1}^{n}\left(x_i^c-\overline{x^c}\right)^2 -2\sum_{i=1}^{n}\left(x_i^c-\overline{x^c}\right)\left(\lambda_i-\overline{\lambda}\right)\right].\end{aligned}\ ] ] from this , and the assumption that @xmath217 , we see that @xmath222 is equivalent to @xmath223 on the other hand , @xmath224\sum_{i=1}^{n}\left(x_i^c-\overline{x^c}\right)^2 -\sum_{i=1}^{n}\left(x_i^c-\overline{x^c}\right)\left(\lambda_i-\overline{\lambda}\right)\\ & & = \left[\frac{\sum_{i < j}\left(x_i^c - x_j^c\right)\left(\lambda_i-\lambda_j\right)}{\sum_{i < j}\left(x_i^c - x_j^c\right)^2+\sum_{i < j}\left(x_i^r - x_j^r\right)^2 } -a\frac{s^2\left(x^r\right)}{s^2\left(x^c\right)+s^2\left(x^r\right)}\right]\sum_{i=1}^{n}\left(x_i^c-\overline{x^c}\right)^2\\ & & -\sum_{i=1}^{n}\left(x_i^c-\overline{x^c}\right)\left(\lambda_i-\overline{\lambda}\right)\\ & & = \frac{\sum_{i=1}^{n}\left(x_i^c-\overline{x^c}\right)^2}{\sum_{i < j}\left(x_i^c - x_j^c\right)^2+\sum_{i < j}\left(x_i^r - x_j^r\right)^2 } \sum_{i < j}\left(x_i^c - x_j^c\right)\left(\lambda_i-\lambda_j\right)\\ & & -\sum_{i=1}^{n}\left(x_i^c-\overline{x^c}\right)\left(\lambda_i-\overline{\lambda}\right ) -a\frac{s^2\left(x^r\right)}{s^2\left(x^c\right)+s^2\left(x^r\right)}\sum_{i=1}^{n}\left(x_i^c-\overline{x^c}\right)^2\\ & & = \frac{\sum_{i=1}^{n}\left(x_i^c-\overline{x^c}\right)^2}{\sum_{i < j}\left(x_i^c - x_j^c\right)^2+\sum_{i < j}\left(x_i^r - x_j^r\right)^2 } \left[n\sum_{i=1}^{n}\left(x_i^c-\overline{x^c}\right)\left(\lambda_i-\overline{\lambda}\right)\right]\\ & & -\sum_{i=1}^{n}\left(x_i^c-\overline{x^c}\right)\left(\lambda_i-\overline{\lambda}\right ) -a\frac{s^2\left(x^r\right)}{s^2\left(x^c\right)+s^2\left(x^r\right)}\sum_{i=1}^{n}\left(x_i^c-\overline{x^c}\right)^2\\ & & = \sum_{i=1}^{n}\left(x_i^c-\overline{x^c}\right)\left(\lambda_i-\overline{\lambda}\right ) \left[\frac{s^2\left(x^c\right)}{s^2\left(x^c\right)+s^2\left(x^r\right)}-1\right]\\ & & -a\frac{s^2\left(x^r\right)}{s^2\left(x^c\right)+s^2\left(x^r\right)}\sum_{i=1}^{n}\left(x_i^c-\overline{x^c}\right)^2\\ & & = -\frac{s^2\left(x^r\right)}{s^2\left(x^c\right)+s^2\left(x^r\right ) } n\left[as^2\left(x^c\right)+s\left(x^c , \lambda\right)\right],\end{aligned}\ ] ] where @xmath225 denotes the sample covariance of the random variables @xmath31 and @xmath226 , which converges to @xmath132 almost surely by the independence assumption . therefore , @xmath227\nonumber\\ & & \to c_1<0\label{eqn : consist-2}\end{aligned}\ ] ] almost surely , as @xmath149 . + by the similar calculation , we have that the second term @xmath228\nonumber\\ & & \to c_2<0\label{eqn : consist-3}\end{aligned}\ ] ] almost surely , as @xmath149 . ( [ eqn : consist-2 ] ) and ( [ eqn : consist-3 ] ) together imply that @xmath229 this completes the proof . | it has been some time since interval - valued linear regression was investigated . in this paper , we focus on linear regression for interval - valued data within the framework of random sets .
the model we propose generalizes a series of existing models .
we establish important properties of the model in the space of compact convex subsets of @xmath0 , analogous to those for the classical linear regression .
furthermore , we carry out theoretical investigations into the least squares estimation that is widely used in the literature . a simulation study is presented that supports our theorems .
finally , an application to a climate data set is provided to demonstrate the applicability of our model . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
it remains an open question whether massive neutrinos are majorana particles , whose antiparticles are themselves @xcite . the final answer to this fundamental question will tell us whether the lepton number is conserved or not in nature , and help us explore the origin of neutrino masses . currently , the most promising way to answer if massive neutrinos are their own antiparticles is to observe the @xmath0 decays @xmath19 , where @xmath20 and @xmath21 stand respectively for the atomic and mass numbers of a nuclear isotope @xmath22 @xcite . over the last few decades , a great number of dedicated experiments have been carried out to search for this kind of decays @xcite . so far , we have not observed any positive signals , and a lower bound on the half - life of the implemented nuclear isotope can be drawn from experimental data . the gerda phase - i experiment @xcite has disproved the signals of @xmath23 decays claimed by the heidelberg - moscow experiment @xcite , and the joint lower bound from all the previous @xmath24-based experiments on the half - life turns out to be @xmath25 at the @xmath26 confidence level @xcite . for @xmath27-based experiments , a combined analysis of the exo-200 @xcite and kamland - zen phase - i data @xcite gives rise to a lower bound @xmath28 at the @xmath26 confidence level . more recently , kamland - zen announced their phase - ii result @xcite , and improved the lower bound to @xmath29 at the @xmath26 confidence level with both phase - i and phase - ii data . if neutrino mass ordering is inverted ( i.e. , @xmath30 ) , the next - generation @xmath23 experiments with a few tons of target mass will be able to discover a remarkable signal in the near future @xcite . the schechter - valle theorem @xcite states that a clear signal of @xmath23 decays will unambiguously indicate a finite majorana mass of neutrinos , if neither a fine - tuning among parameters nor a cancellation among different contributions is assumed . decay rate remains nonzero as the nuclear medium effects on quarks may break any intricate cancellation . ] obviously , this theorem signifies the physical importance of searching for @xmath23 decays experimentally . the quantitative impact of the schechter - valle theorem has already been studied by duerr , lindner and merle in ref . @xcite , where it is found that the majorana neutrino masses implied by the schechter - valle theorem are too small to explain neutrino oscillations . explicitly , assuming one short - range operator to be responsible for @xmath23 decays , they find that current experimental lower bounds on the half - lives of @xmath23-decaying isotopes indicate an upper bound on the majorana neutrino mass @xmath31 , where @xmath32 denotes the effective neutrino mass term associated with @xmath33 for @xmath34 . in this paper , we reexamine this problem , and obtain an upper bound @xmath35 that agrees with the above result from ref . @xcite on the order of magnitude . furthermore , we generalize the analysis of @xmath0 decays to that of the lnv rare decays of @xmath15 , @xmath13 and @xmath36 mesons . for instance , we obtain @xmath37 , @xmath38 and @xmath39 from current upper bounds on the lnv rare decays of @xmath36 mesons . the radiative majorana neutrino masses related to other lnv decays are also tabulated . therefore , we confirm the conclusion from ref . @xcite that although the schechter - valle theorem in general implies a tiny majorana neutrino mass , we have to explore other mechanisms to generate the observed neutrino masses at the sub - ev level . the remaining part of this work is organized as follows . in sec . 2 , we recall the calculation of majorana neutrino masses from the four - loop diagram mediated by the effective operator , which is also responsible for the @xmath0 decays . the generalization to the lnv meson decays is performed in sec . 3 , where the corresponding majorana masses are computed . finally , we summarize our main conclusions in sec . 4 . in this section , we present a brief review on the calculation of majorana neutrino masses radiatively generated from the operator that leads to the @xmath0 decays , following ref . @xcite closely . such a calculation can be readily generalized to the case of majorana neutrino masses induced by the lnv meson decays , as shown in the next section . at the elementary - particle level , the @xmath0 decays can be expressed as @xmath40 , where the up quark @xmath41 , the down quark @xmath42 and the electron @xmath43 are all massive fermions . if the @xmath0 decays take place , they can be effectively described by the lnv operator @xmath44 , in which the chiralities of charged fermions have been omitted and will be specified later . as already pointed out by schechter and valle @xcite , this operator will unambiguously result in a majorana neutrino mass term @xmath45 . the relevant feynman diagrams are given in fig . [ fig:0n2b ] . it is worthwhile to notice that quark and charged - lepton masses are indispensable for the schechter - valle theorem to be valid , as emphasized in ref . @xcite . in the standard model ( sm ) , only left - handed neutrino fields participate in the weak interactions , so the electron masses can be implemented to convert the right - handed electron fields into the left - handed ones , which are then coupled to left - handed neutrino fields via the charged weak gauge boson @xmath46 . this does make sense , since the chirality of electrons in the operator @xmath47 can in general be either left - handed or right - handed . for the same reason , quark masses are also required to realize the hadronic charged - current interactions in the sm . in this case , the operator @xmath48 in fig . [ fig:0n2b](a ) can be attached to the left - handed neutrinos through two propagators of @xmath46 , leading to the neutrino self - energy diagram in fig . [ fig:0n2b](b ) . assuming that @xmath0 decays are mediated by short - range interactions , one can write down the most general lorentz - invariant lagrangian that contains various point - like operators as follows @xcite @xmath50 where @xmath51 and @xmath52 denote respectively the fermi constant and the proton mass , and @xmath53 ( for @xmath54 ) are effective coupling constants . in eq . ( [ eq : operators ] ) , the hadronic currents are defined as @xcite @xmath55 ( 1 \pm \gamma_5^ { } ) d \ ; , \end{aligned}\ ] ] while the leptonic currents are given by @xmath56 ( 1 \pm \gamma_5^ { } ) e^{\rm c } \ ; , \end{aligned}\ ] ] where @xmath57 with @xmath58 is the charge - conjugated electron field . according to @xmath59 and the fact that fermion fields are grassmann numbers , one can immediately verify that the tensor leptonic current @xmath60 automatically vanishes . different chiralities of hadronic and leptonic currents in eqs . ( [ eq : hadronic ] ) and ( [ eq : leptonic ] ) should be distinguished by the left- and right - handed projection operators @xmath61 . for instance , we have defined @xmath62 , and similarly for the other types of currents in eqs . ( [ eq : hadronic ] ) and ( [ eq : leptonic ] ) , in which the corresponding subscripts l " or r " are omitted without causing any confusions . in this connection , the effective coupling constants @xmath53 should also be regarded as @xmath63 ( for x , y , z = l , r ) , which are carrying the superscripts for different chiralities of hadronic and leptonic currents . given one of the five operators in eq . ( [ eq : operators ] ) , one can set an upper limit on their coupling @xmath64 by assuming that it is responsible for the @xmath65 decay and saturates the experimental lower bound on the half - life , as done in ref . recently , some of those limits have been recalculated in ref . @xcite , using more recent results for the nuclear matrix elements . the effective coupling constants for the operators @xmath66 and @xmath67 have been found to be @xmath68 and @xmath69 , respectively . having obtained these couplings , we are then ready to evaluate the induced neutrino mass by inserting the dimension - nine effective operators into the butterfly diagram , as depicted in fig . [ fig:0n2b ] . the authors of ref . @xcite have demonstrated that the operator @xmath66 leads to a vanishing neutrino mass term via the butterfly diagram , while the other one @xmath70 does lead to a tiny majorana neutrino mass , which will be revisited below . now that the operator @xmath71 is responsible for the @xmath0 decays , the radiatively induced majorana mass term for electron neutrinos can be extracted from the self - energy in fig . [ fig:0n2b](b ) by setting the external momentum to zero as @xcite , @xmath72 where @xmath73 is the weak gauge coupling , and @xmath74 and @xmath75 are the up - quark , down - quark and electron masses , respectively . in addition , the loop integral is given by @xmath76 ^ 2_{}$ ] with @xmath77 ( q_1 ^ 2 - m_d^2 ) ( k_1 ^ 2-m_{w}^2 ) } \ ; , \end{aligned}\ ] ] where @xmath78 is the @xmath79-boson mass , @xmath80 and @xmath81 stand for the four - momenta of internal particles running in the loop and can be easily identified via the integrand on the right - hand side of eq . ( 5 ) and from fig . [ fig:0n2b](b ) . to evaluate this integral , we employ the technique for massive two - loop diagrams in ref . @xcite and arrive at @xmath82 with @xmath83 as usually introduced in the dimensional regularization and @xmath6 being the renormalization scale . the relevant function reads as @xcite @xmath84 [ ( p+q)^2 + m_k^2 ] } \\ & = & \pi^4 ( \pi m_i^2)^{n-4 } \frac{\gamma(2-\frac{1}{2}n)}{\gamma(3-\frac{1}{2}n ) } \int_0 ^ 1 { \rm d}x \int_0 ^ 1 { \rm d}y ~ [ x ( 1-x)]^{\frac{1}{2}n-2 } y ( 1-y)^{2-\frac{1}{2}n } \nonumber \\ & & \times \left\ { \frac{(y^2 \kappa^2 + \eta^2)\gamma(5-n)}{[y(1-y)\kappa^2 + y + ( 1-y)\eta^2 ] ^{5-n } } + \frac{n}{2 } \frac{\gamma(4-n)}{[y(1-y)\kappa^2 + y + ( 1-y)\eta^2 ] ^{4-n } } \right \},\end{aligned}\ ] ] with @xmath85 as usual , the integral is expanded with respect to @xmath86 in the limit of @xmath87 and the ultraviolet divergences can be separated as inverse powers of @xmath88 . since the loop integral involves the divergent terms proportional to @xmath89 and @xmath90 , we need to keep the terms up to @xmath91 in @xmath92 , namely , @xmath93 so as to obtain all the finite parts of @xmath94 . in our numerical calculations , we have adopted the renormalization scale of @xmath95 , which is a characteristic scale of typical energy transfer in nuclear processes . the other implemented parameters can be found in table [ tb : constants ] . in the scheme of minimal subtraction , we finally get the induced neutrino mass from eq . ( [ eq : sigmap ] ) as latexmath:[\[\begin{aligned } which agrees with the result @xmath31 from ref . @xcite on the order of magnitude . since this mass is extremely small , one has to implement other mechanisms to account for neutrino masses . in this sense , the main conclusion in ref . @xcite is still valid that the schechter - valle theorem is qualitatively correct , but quantitatively irrelevant for the neutrino mass - squared differences required for neutrino oscillation experiments . | the schechter - valle theorem states that a positive observation of neutrinoless double - beta ( @xmath0 ) decays implies a finite majorana mass term for neutrinos when any unlikely fine - tuning or cancellation is absent . in this note , we reexamine the quantitative impact of the schechter - valle theorem , and find that current experimental lower limits on the half - lives of @xmath0-decaying nuclei have placed a restrictive upper bound on the majorana neutrino mass @xmath1 radiatively generated at the four - loop level . furthermore , we generalize this quantitative analysis of @xmath0 decays to that of the lepton - number - violating ( lnv ) meson decays @xmath2 ( for @xmath3 , @xmath4 = @xmath5 or @xmath6 ) .
given the present upper limits on these rare lnv decays , we have derived the loop - induced majorana neutrino masses @xmath7 , @xmath8 and @xmath9 from @xmath10 , @xmath11 and @xmath12 , respectively .
a partial list of radiative neutrino masses from the lnv decays of @xmath13 , @xmath14 and @xmath15 mesons is also given .
* majorana neutrino masses from neutrinoless double - beta decays and lepton - number - violating meson decays * * jun - hao liu @xmath16 * + @xmath17institute of high energy physics , chinese academy of sciences , beijing 100049 , china + @xmath18center for high energy physics , peking university , beijing 100871 , china pacs number(s ) : 11.30.fs , 12.15.lk , 14.60.lm |
You are an expert at summarizing long articles. Proceed to summarize the following text:
strong and rapid variability is a key observational feature of systems driven by the accretion of material onto a central compact object . the power spectral density ( psd ) of emission from such objects is characterized by broadband noise across several decades in frequency . this broadband component is generally well described by a broken - powerlaw form , however the physics that governs such breaks is poorly understood . additionally , the psd can feature distinct and resolved peaks indicative of quasi - periodic oscillations ( qpos ) in the emission . the broadband noise structure displays many interesting characteristics beyond the psd . light curves show log - normality in their structure and a strong linear relationship between the root - mean square ( rms ) variability and the flux of the source ( uttley & mchardy 2001 ; negoro & mineshige 2002 ; gaskell 2004 ) . studies of frequency dependent coherence between energy bands in the time - series data indicates that the emission is coherent at low frequencies then becomes incoherent at high frequencies ( e.g. cygnus x-1 , see nowak et al . 1999 for details ) . consequently , many sources also show frequency - dependent phase / time lag behavior between emissions at different wavebands . this was first clearly demonstrated in accreting stellar mass black holes which often displayed a constant time lag at low frequencies and constant phase at high frequencies ( nowak et al . qualitatively similar behavior is now seen in active galactic nuclei ( agn , vaughan , fabian & nandra 2003 ) and accreting non - magnetic white dwarfs ( scaringi et al . 2013 ) . the broadband variability is often attributed to inward propagating fluctuations driven by stochasticity in the angular momentum transport mechanism ( lyubarski 1997 , hereafter l97 ) . within the framework of the standard shakura & sunyaev ( 1973 ) disk model in which the internal `` viscous '' shear stresses are proportional to the pressure ( @xmath0 ) , l97 attempted to explain this broadband flicker - noise by introducing a small stochastic perturbation into the @xmath1-parameter . this perturbation generated broadband fluctuations in the accretion ( @xmath2 ) . however , the analysis of l97 was ultimately of a linearized form of the disk equation . in fact , many of the above features , most notably the rms - flux relationship , are not formally captured by this model since linearization means that all perturbations are applied to the base state and do not build upon each other multiplicatively . still , it was suggested by uttley & mchardy ( 2001 ) that the log - normality and rms - flux relation are a result of propagating fluctuations , with the intuitive but formally unproven statement that such features would arise from an analysis of the disk equations with non - linear fluctuations ( see also uttley et al . 2005 ) . this problem has also been examined by kelly et al . ( 2011 , hereafter k11 ) using the tools of stochastic differential equations . specifically , they studied the use of the stochastic ornstein - uhlenbeck ( ou ) process to model the observed light curves and power spectra of systems with accretion - driven variability . this was motivated by the fact that a solution of the stochastically perturbed linearized disk equation is a linear combination of ou processes which , in the case where the stochastic driving terms are brownian motion , are simply damped random walks ( also see wood et al . 2001 , titarchuk et al . 2007 ) . k11 found that a linear combination of only one or two such processes was sufficient to well describe the data , and that they could reproduce breaks in the observed power - spectra . in this paper , we construct numerical realizations of _ non - linear _ fluctuations in a simple one - dimensional viscous disk . we show that these disk models produce mass accretion rates and luminosities that feature a linear rms - flux relationship and approximate log - normality . we also examine the fourier space relations between the mass accretion rates and dissipations at different radii , one possible proxy for light curves in different energy / wavelength bands . we find coherence between radii for frequencies less than the viscous frequency . however , we fail to cleanly reproduce the frequency dependence of the observed time - lags . therefore , this toy non - linear model provides some theoretical underpinning for attempts to understand the rms - flux relationship but further developments of the model are required to capture the full behavior of real systems . this work is complimentary to the analysis of the propagating fluctuation model performed by ingram & van der klis ( 2013 ) , however they begin from a statistical model for the mass accretion whereas we base our analysis on the non - linear disk equations . this paper is structured as follows : section 2 outlines our modifications to the standard viscous disk model and numerical setup , section 3 presents the results of our simulations , and section 4 discusses the results of the previous section and provides suggestions for future work . we consider accretion disks in the framework of a simple 1d viscous disk ( e.g. , see pringle 1981 ) . we operate within the thin disk approximation which assumes that the material in the disk is narrowly confined in a common orbital plane . this ensures that the vertical scale of the disk ( h ) is much less than the radial scale ( r , @xmath3 ) . consequently , we must also assume that the disk is radiatively efficient . if this were not the case , viscous dissipation would cause the disk to heat to temperatures of order the virial temperature and hence puff up into a quasi - spherical configuration , violating our thin disk assumptions . furthermore , also required to be consistent with the thin - disk assumption , we assume a fluid element s orbit is essentially keplerian in nature , with a very small radial velocity responsible for the actual accretion . while an obvious application of this model is to black hole accretion disks , we neglect relativistic effects and assume a newtonian gravitational potential ( @xmath4 ) . the canonical equation for such a disk ( in units where @xmath5 ) is the non - linear diffusion equation given in pringle ( 1981 ) as @xmath6,\label{eq : basiceqn}\end{aligned}\ ] ] where @xmath7 is the distance from the central engine , @xmath8 is the surface density of the disk material , and @xmath9 is the effective shear viscosity of the disk . as we discuss in detail below , @xmath10 will be taken as a stochastic variable with some ( prescribed ) stationary base state @xmath11 to which we add spatially - uncorrelated but temporally - correlated stochastic fluctuations . we now wish to make solutions to eqn . ( [ eq : basiceqn ] ) more computationally convenient with this eventual application in mind . first , we introduce the change of variables @xmath12 . equation ( [ eq : basiceqn ] ) then becomes @xmath13 multiplying both sides of equation [ eq : mixeq ] by the term @xmath14 ( which has no @xmath15 dependence ) while defining @xmath16 and @xmath17 , equation ( 2 ) can be written as @xmath18 which is the form of the disk evolution equation that serves as the basis for our numerical solution . we are primarily concerned with the local mass accretion rate across , and viscous dissipation at , each radius . the instantaneous mass accretion rate is @xmath19 which in our new variables translates to @xmath20 we can perform a similar manipulation of the dissipation , @xmath21 which gives @xmath22 the total disk luminosity is then given by @xmath23 ( 2 \pi r ) dr = \int_{\text{disk } } \frac{9 \psi}{x^4 } dx.\end{aligned}\ ] ] we now assume a stochastic perturbation in the viscous parameter of the form @xmath24 . here , @xmath25 is the fluctuation relative to the baseline level @xmath26 , and physically represents the turbulent fluctuations in the maxwell stresses of a magnetohydrodynamic accretion disk . unlike previous treatments ( l97 and k11 ) , we do not assume that @xmath27 is small . to proceed , we need to specify the nature of the random process @xmath27 . we employ assumptions similar to those of l97 . specifically , we suppose that the fluctuations are spatially uncorrelated ( likely to be valid for @xmath28 ) but have temporal correlations on some timescale @xmath29 . thus , we impose the condition that @xmath30 when @xmath31 . the characteristic timescale imposes the condition that @xmath32 when @xmath33 . a specific realization of these conditions can be generated by an ornstein - uhlenbeck ( ou ) process . the ou process , models the evolution of a statistical variable of interest as it responds to an additive input noise with an exponential decay to its mean ( see kelly et al . 2009 , k11 ) . this evolution is governed by a stochastic differential equation of the form @xmath34 where @xmath35 is the characteristic frequency , @xmath36 is the mean of the process , @xmath37 is the driving noise amplitude , and @xmath38 denotes a wiener process . in the typical case where @xmath38 is brownian motion , @xmath39 is gaussian white noise . equation . [ eq : ou ] then simply represents a damped random walk . equation [ eq : ou ] can be made physically relevant for our disk with the appropriate choice of parameters . we assume the mean in the fluctuations is zero ( since the average component of the viscosity is already described via the @xmath40 term . ) . next , we take guidance from l97 and , for most of the work presented here , assume that the relevant timescale for the fluctuations is the local viscous timescale such that @xmath41 , where @xmath42 . we take @xmath39 to be gaussian white noise with zero mean and unity standard deviation . to ensure that the rms amplitude of the fluctuations @xmath27 are independent of radius we note that when @xmath39 has zero mean the variance is given by k11 as @xmath43 . therefore , we must take @xmath37 as proportional to the square root of the local accretion time . we normalize this factor to the accretion time at the inner edge of the disk yielding @xmath44 . the final form of the ou equation is then : @xmath45 which satisfies all of the physically relevant requirements for the distribution of fluctuations . we need one additional restriction on the viscous fluctuations in order to make our mathematical model well - posed . if , due to a downward fluctuation , the overall viscosity @xmath10 becomes negative over an extended region of the disk , the basic diffusion equation is unstable and singularities will develop in the solution . thus , we impose a positivity criterion whereever and whenever the solution of eqn . [ eq : ou2 ] results in @xmath46 ( and hence @xmath47 ) , we peg @xmath48 . lastly , when evolving @xmath49 , we scale the amplitude of @xmath27 by a fiducial value of 0.5 . this is done to control the effect of high sigma spikes on the stability of the solution . we performed our simulations using an explicit numerical integration algorithm . the simulation was run on a spatial grid with @xmath50 and @xmath51 with @xmath52 . we chose a fiducial value for the baseline viscosity @xmath53 . simulations were run with a time - step computed to safely satisfy the necessary cfl stability conditions , typically @xmath54 . the initial condition on @xmath55 was chosen such that the disk began in a steady state with unity accretion rate , i.e. @xmath56 . lastly , we imposed dirchlet boundary conditions @xmath57 and @xmath58 . in order to protect the simulation from stochastic effects at the outer boundary we established a buffer zone between @xmath59 and @xmath60 where @xmath61 the simulation was run for a duration of @xmath62 , and the state of the disk was recorded with a cadence of @xmath63 . our primary goal is to examine whether the phenomenology of the broadband noise in real sources can be reproduced by our toy model . consequently , we focus our analysis on the dissipation of the disk as this most closely corresponds to the observational emissions of real astrophysical disks . we integrate the dissipation across the entire disk at each time to produce the total time - dependent luminosity of the disk . this light curve can be seen in figure [ fig : lightcurve ] . qualitatively , it resembles the light curve for an accreting source . log - normality can be visually inferred by observing the decreased amplitude of fluctuations during dips compared with the flares . the luminosity distribution is shown in figure [ fig : distribution ] . we fit this distribution with a standard log - normal function of the form @xmath64,\end{aligned}\ ] ] where @xmath65 is the luminosity and the fitting parameters @xmath36 and @xmath66 are the mean and standard deviation respectively . we found best fit parameters of @xmath67 and @xmath68 . it must be noted that we also found a reasonable , albeit visually slightly worse , fit using a normal distribution with @xmath69 and @xmath70 . the best - fit lines are plotted in figure [ fig : distribution ] . we investigated the existence of an rms - flux relationship in our data using the integrated luminosity as a function of time . the resulting light curve was divided into 600 equal segments of @xmath71 . for a black hole of 10@xmath72 this corresponds to @xmath73s per segment . the mean luminosity @xmath74 was computed for each segment . we then computed the rms fluctuation in each segment as @xmath75 where @xmath76 is the luminosity at the @xmath77 point in the segment . the resulting rms - flux plot can be seen in figure [ fig : rmsflux ] . due to the nature of the stochastic noise in the simulation it is necessary to bin the resulting rms - flux data . the data were linearly binned into 50 segments and then averaged over each bin . the linear nature of the binned data is immediately apparent . following the prescription of uttley & mchardy ( 2001 ) we fit the binned data with a line of the form @xmath78 where @xmath79 and @xmath80 are fitting constants . we obtained a fit to the data with @xmath81 and @xmath82 . the fitted line is shown on fig . [ fig : rmsflux ] to trace the data qualitatively well . the fact that our simulation produces a linear result is an important confirmation of the log - normality of the light curve . furthermore , this directly supports the phenomenological model that the multiplicative effect of numerous inward propagating fluctuations are the cause of the observed log - normality . we next wish to understand how our light curves behave in frequency space . we divided the total integrated disk luminosity into 36 segments , each of length 819200 m ( @xmath83 40 seconds at 10 @xmath72 ) . the fast fourier transform was taken for each segment . we used these fourier segments , denoted @xmath84 , to construct the power spectrum density ( psd ) of the @xmath77 segment as per the standard prescription @xmath85 where the use of angle brackets denotes an averaging over segments . the resulting power spectrum across the inner boundary is shown in figure [ fig : psd ] . it can be seen that the psd is consistent with a system dominated by broadband noise . there are no qpos , as expected given the simple nature of the model . we attempted to fit the psd to a broken power - law of the form @xmath86 and @xmath87 , where we take the threshold frequency to be a free parameter of the model . the best - fit returned @xmath88 and @xmath89 at a break frequency of @xmath90 . the slope is likely not consistent with the @xmath91 expectation put forth by l98 because we are not dealing with a strict @xmath92 accretion disk . however , while the power - law dependence is flatter than that indicated by observational data , this indicates a clear break in the power spectrum as expected . the break frequency is comparable to the viscous frequency for the inner regions of the disk ( i.e. @xmath93 ) . given that the dissipation is dominated by the inner regions it is reasonable to expect the psd to obey these timescales . a key aspect of the propagating fluctuation model is the behavior of the fluctuations as they propagate inward from the outermost radii . therefore , it is important to analyze to what extent various disk radii are capable of communicating and at what frequencies this communication may occur . this can be accomplished through the use of the coherence function . from nowak et al . ( 1999 ) we define the coherence function , @xmath94 , as follows : consider two time series @xmath95 and @xmath96 with fourier transforms @xmath97 and @xmath98 . the coherence function between @xmath95 and @xmath15 is then @xmath99 where , again , the angle brackets indicate an averaging over segments of the data . it then becomes clear that the coherence function is merely the modulus of the averaged cross - spectrum between two time series normalized by the averaged power spectra of the individual series . consequently , the coherence function takes on values between zero and unity . in the case of our data we take the dissipation at different radii as our time signals . we can choose the inner radius as the first time series and compute its coherence function with all of the other radii in order to produce a coherence map , @xmath100 . such an approach can be thought of as being analogous to the common practice of studying the coherence function between different wavebands . if we assume the emission is locally a blackbody then the natural temperature gradient in the disk will cause emission from two radii to occur at different wavebands . the resulting coherence map for the disk as shown in figure [ fig : coherencemap_visc ] . there is clear structure in the map as radii become coherent on roughly the viscous frequency or less . this provides some indication of the timescales on which radii communicate information about the perturbations . we should note that coherence does not necessary arise because the radii are globally in perfect phase . rather , it indicates that a linear transformation exists between the time series in the time domain , i.e. , @xmath101 where @xmath102 is the transfer function ( nowak et al . our coherence map confirms the notion of l97 that , at a given radius , fluctuations are passed down to smaller radii only if they fall below the local viscous frequency . if two radii of the disk have time - coherent dissipations , this implies the existence of well - defined ( frequency - dependent ) phase shifts between the dissipation time - series . more precisely , if we write the cross - spectrum between two time series as a complex number then we have @xmath103 where @xmath104 is the frequency - dependent fourier phase shift . the calculation of the phase shift produces values in the domain from @xmath105 $ ] with the assumption that @xmath106 as @xmath107 . we can now scale the phase by the frequency yielding the frequency - dependent time lag : @xmath108 . given the nature of our model , the fourier time lags represent the physical reprocessing time of the fluctuations as they are diffused in the disk . to illustrate the behavior of the phases and time lags , fig . [ fig : phase_1_2 ] shows these quantities derived from the dissipation at @xmath109 and @xmath110 ( corresponding to @xmath111 and @xmath112 ) . from the coherence map ( fig . [ fig : coherencemap_visc ] ) , we can see that these two radii are extremely coherent at low frequencies ( @xmath113 for @xmath114 ) , and then the coherence gradually falls to low values ( @xmath115 ) over a span of about 1.5 decades of frequency around the viscous frequency . consistent with this , at low frequency there is little scatter in the phases from the individual segments . the mean phase gradually increases magnitude as frequency increases in a manner consistent with approximately equal time - lag . we can see that at frequencies around the local viscous frequency ( @xmath116 ) the scatter in the phase increases ( corresponding to the decreasing coherence ) . at frequencies above @xmath117 , the scatter in phase becomes large and phases get wrapping , becoming almost randomly scattered between @xmath118 and @xmath119 at @xmath120 . the sharp decrease in the averaged phase shift above this frequency is entirely an artifact of averaging incoherence processes with a phase wrapped cross spectrum argument . in order to counteract some effects of phase wrapping we binned the data at the level of the cross - spectrum . the phases and lags were then computed from these binned data . looking at the time lag information across the same radii we see that the binned lag data shows a lag which is small for very low frequencies , increases in magnitude to @xmath121 at @xmath122 and then very slowly declines in magnitude to @xmath123 before coherence is lost . given our convention , the sense of this lag is that the inner ( @xmath109 ) dissipation profile lags the dissipation further out , expected from inwardly propagating fluctuations . at frequencies above the local viscous frequency the lag begins to weakly rise before the radii become incoherent . by comparison , studies of the fourier lag in cygnus x-1 have shown lags with a strong frequency dependent decline ( @xmath124 and hence consistent with approximately constant phase shift ; nowak 1999 ) . this is clearly not visible in the dissipation data . it is interesting to note that the lag times are much less than the naively computed local viscous timescale , @xmath125 . when looking at the innermost radii it is important to check for the possible effects of the computational boundary on our analysis . we repeated the fourier analysis , this time computing the coherence between the dissipation at radii @xmath110 and @xmath126 . the viscous frequency between these two radii is @xmath127 . we see in fig . [ fig : phase_2_5 ] an interesting pattern of time lags , with slowly decreasing lags below the viscous frequency , and negligible lags ( approximately constant phase ) above the viscous frequency . coherence is completely lost ( resulting in essentially randomly distributed phases ) for @xmath128 . this pattern of time lags has tantalizing similarities to the behavior seen in cygnus x-1 and other accreting sources . the above analysis considers the behavior of the dissipation across the entire disk . we also investigated the local behavior of key observables ( e.g. light curves , histograms , psds , and the rms - flux relationship ) at the specific radii @xmath129 , @xmath126 and @xmath130 . we focused on the inner radii as that is the region of the disk where the dissipation is most dominant . qualitatively , we observe the same behavior at each radii as the disk shows globally . we observe dissipation light curves and histograms with indications of log - normal behavior , psds that show a clear break , and a linear rms - flux relationship . while we have considered the observational features of our simulated disk it is important to consider other physical processes . given its dynamical importance , there is motivation to study the behavior of @xmath2 in our simulations . repeating the above analysis for @xmath2 we found many of the same behavioral properties . there is clear log - normality in the light curve along with the expected linear rms - flux relationship . the psd is composed entirely of broadband noise and shows no break at the viscous frequency at the inner radius . the most striking differences between the two datasets are the coherence map and subsequent phase / lag information . the coherence map shows the same functional form as the as the dissipation , however @xmath2 appears to become coherent only at lower frequencies when compared to the dissipation ( fig . [ fig : coherencemap_mdot ] ) . we can understand this by examining the phases / lags at two illustrative radii ; again we choose @xmath109 and @xmath110 . we see in fig . [ fig : phase_mdot ] , for @xmath131 , there is a constant time lag . however , this lag is substantially larger than that found for the dissipations , @xmath132 as opposed to the @xmath133 for the dissipations . it is clear from fig . [ fig : phase_mdot ] that the corresponding larger phase shift starts to phase wrap well before there is a loss of coherence . thus , the coherence function @xmath134 is suppressed at intermediate frequencies by phase wrapping well before there is a genuine loss of communication between the radii . for completeness , we examine the fourier relationship between the dissipation and mass accretion rate . this could be astrophysically relevant if , for example , the hard x - ray emission from an accreting black hole was tied closely to mass accretion rate through the innermost circular stable orbit but the soft x - rays were thermal emission from a dissipative disk . looking at the coherence map , we see that the regions of high coherence again trace the viscous frequency with regions becoming coherent at frequencies similar to those seen for @xmath2 ( figure [ fig : coherencemap_mdot_diss ] ) . interestingly , there is evidence of convergence to a constant phase of @xmath135 ( which is also phase wrapped to @xmath118 ) in this phase plot ( figure [ fig : phase_mdot_diss ] ) . while the viscous timescale is one natural choice as the characteristic frequency for the driving perturbations it may be illuminating to consider other possibilities . the only other relevant quantity in our toy model is the dynamical frequency , @xmath136 , assuming keplerian orbits . we ran a second simulation with the characteristic frequency of the ou process set to the dynamical frequency . analysis of the resulting data showed a disk that was strongly inconsistent with the aforementioned phenomenologies . while the simulation was able to produce a light curve with log - normality and a linear rms - flux relationship , the disk was not able to produce reasonable behavior in frequency space . the coherence map for both the dissipation and the mass accretion rate show large regions of little to no coherence , while the regions of high coherence are concentrated at low frequencies or along the inner boundary . as an example , the coherence map for the dissipation is show in figure [ fig : coherencemap_dyn ] . there is no hint of any meaningful structure in these plots . due to lack of general coherence we do not expect to see any frequency - dependent phase behavior apart from at the lowest possible frequencies . the phase and lag plots are shown in figure [ fig : phase_dyn ] . we do note that there is a weak trend in the binned data above the viscous frequency likely due to a combination of noise and diffusion effects in the disk . we have presented one of the first numerical attempts to understand the propagating fluctuation " model of lyubarskii ( 1997 ) in the context of non - linear accretion disk models . while the observational effects of this model have long been intuited from detailed studies of observational data , many key phenomenological features are not rigorously attributable to the linear model . one of the most striking features is the linear relationship between the rms flux - variability and the average flux . we have shown that our toy model , which starts with the basic disk equations , is capable of reproducing this relationship . going beyond the rms - flux relationship , we can examine the behavior of our model disk in fourier space . we find that the time - series of the dissipations at two different radii have temporal coherence up to frequencies comparable to , or slightly greater than , the viscous frequency relating the two radii . across most of this range of coherence frequencies , the relationship between the dissipations appears to be one of constant time lag , with the inner radius lagging the outer radius by about a tenth of a viscous timescale . at higher frequencies there is some hint , seen most clearly when comparing radii away from the inner computational boundary , for a constant phase regime ( with a time - lag that correspondingly declines with increasing frequency ) , but this only lasts over about a factor of @xmath137 in frequency before all coherence is lost . this frequency behavior in real sources has been well studied observationally . for example , detailed studies of cygnus x-1 with the _ rossi x - ray timing explorer ( rxte _ ; nowak et al . 1999 ; poutanen 2000 ; gilfanov et al . 2000 ; uttley et al . 2011 ) reveal coherence between the soft and hard x - ray bands at frequencies below the viscous frequency , possible regions of constant time - lag , and extended regions of approximately constant phase shift . our model is able to reproduce the coherence behavior , but does not show extensive regions of constant phase . nowak et al . 1999 also noted that black hole x - ray binaries show stepped " lag behavior , with the appearance of several sub - components each with their own constant lag . it was thought that these steps in lag were due to the lorentzian sub - components used to model the psd . it may then be the case that our observed constant lag behavior is the result of a single lorentzian component seen when comparing specific pairs of radii ( i.e. @xmath109 and @xmath110 ) , whereas in observations of an astrophysical disk we are observing the contributions of many radii . of course , given the simplicity of our model and treatment , the inability to explain the detailed phenomenology of real sources is not surprising . observationally , coherence and time - lag studies must be performed on lightcurves extracted in particular bands . even in the case where the observed emission is thermal , a given waveband picks out emission from a broad range of radii and the detailed lightcurve may be poorly approximated by the dissipation at one radius . in addition , of course , it is possible for one ( or even both ! ) of the energy bands under study to be well above the thermal energy of the disk and , hence , actually originating from an x - ray corona the uncertain relationship between the underlying state of the disk ( @xmath2 or dissipation ) and the emission from the x - ray corona is another barrier in matching this simple model to the observations . future developments of our approach will include more realistic models for relating the disk dynamics to the observables . ultimately , real astrophysical accretion disks are extremely complex systems . even our 1d model may miss key disk behaviors by assuming azimuthal symmetry . for example , the work of dexter & agol ( 2011 ) showed that 2d accretion disks with stochastic temperature fluctuations end up strongly inhomogeneous . however , it is not clear if their stochastic behavior is driven by stochasticity in the accretion flow as captured by our model or is yet another complication in understanding the behavior of the accretion disk . clearly , the full range of interactions present require 3d magnetohydrodynamical models , often with general relativistic effects and radiative transfer , to fully understand . however such models are computationally expensive , and for the foreseeable future it will not be feasible to run a grmhd model for many outer disk viscous timescales . thus , simple models such as that presented in this paper can be viewed as a bridge between the full grmhd simulations and the observations , into which lessons from the simulations ( such as the statistics of the fluctuations or the physics of the emission processes ) can be imported . future development of these models will follow such a direction . we thank phil uttley for helpful discussion about this work . we also thank our anonymous referee for their helpful comments about this work . psc is grateful for support provided by the nsf through the graduate research fellowship program , grant dge1144152 . csr thanks nasa for funding under the astrophysical theory program grant nnx10ae41 g . dexter , j. , & agol , e. 2011 , , 727 , l24 gaskell , c.m . , apj , 2004 , 612 , l21 gilfanov , m. , churazov , e. , & revnivtsev , m. , 2000 , mnras , 316 , 4 ingram a. , & van der klis , m. , 2013 , mnras , 434 , 1476 - 1485 kelly , b. c. , bechtold , j. , & siemiginowska , a. 2009 , apj , 698 , 895 kelly , b. c. , sobolewska , m. , & siemiginowska , a. 2011 , apj , 730 , 52 lyubarskii , y. e. 1997 , mnras , 292 , 679 negoro , h. , & mineshige , s. , 2002 , pasj , 54 , l69 nowak , m. , et al . , 1999 , apj , 510 , 874 poutanen , j. , 2000 aas , head meeting # 5 , # 31.17 ; bulletin of the american astronomical society , vol . 32 , p.1235 pringle , j. , 1981 , ara&a , 19 , 137 scaringi s. et al . , 2013 , mnras , 431 , 2535 shakura , n. i. , & sunyaev , r. a. , 1973 , a&a , 24 , 337 titarchuk , l. , shaposhnikov , n. , & arefiev , v. 2007 , apj , 660 , 556 vaughan s. , fabian a.c . , nandra k. , 2003 , mnras , 339 , 1237 uttley p. , & mchardy i. m. , 2001 , mnras , 323 , l26 uttley , p. , mchardy , i. m. , & vaughan , s. 2005 , mnras , 359 , 345 uttley , p. , wilkinson , t. , cassatella , p. , et al . , 2011 , mnras , 414 , 1 wood , k. s. , titarchuk , l. , ray , p. s. , wolff , m. t. , lovellette , m. n. , & bandyopadhyay , r. m. 2001 , apj , 563 , 246 | we present a non - linear numerical model for a geometrically thin accretion disk with the addition of stochastic non - linear fluctuations in the viscous parameter .
these numerical realizations attempt to study the stochastic effects on the disk angular momentum transport .
we show that this simple model is capable of reproducing several observed phenomenologies of accretion driven systems .
the most notable of these is the observed linear rms - flux relationship in the disk luminosity .
this feature is not formally captured by the linearized disk equations used in previous work . a fourier analysis of the dissipation and mass accretion rates across disk radii show coherence for frequencies below the local viscous frequency .
this is consistent with the coherence behavior observed in astrophysical sources such as cygnus x-1 . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
a widely studied model for growth of porous films is ballistic deposition ( bd ) @xcite , in which the particles incide perperdicularly to the substrate and aggregate at the first contact with the deposit @xcite . bd was originally proposed to describe sedimentary rock formation @xcite and was extended to model thin film growth and related systems by considering other aggregation mechanisms , non - colimated particle flux , or polydispersivity of particle size @xcite . most works on the ballistic - like models address the scaling features of the outer surface of the deposits , particularly for the connections with kardar - parisi - zhang ( kpz ) @xcite roughening . some works also connect the surface growth dynamics and the bulk properties of the porous deposits @xcite . this is an essential step for proposing models of porous materials , which have a large variety of technological applications , frequently in the form of thin films @xcite . some ballistic - like models are in a class of competitive growth models in which uncorrelated deposition ( ud ) is obtained for a certain value of a parameter and , near that value , a crossovers in kinetic roughening is observed @xcite . in the simplest model , particle aggregation follows the bd ( ud ) rule with probability @xmath7 ( @xmath8 ) . it was already studied numerically @xcite and with scaling approaches @xcite . for small @xmath7 , there is an enhancement of characteristic times of the correlated kinetics ( @xmath9 ) by a factor @xmath10 and of the outer surface roughness by a factor @xmath11 ; for a recent discussion of this topic , see ref . these features extend to other ballistic - like models with crossovers to ud @xcite and are related to the lateral aggregation . in case of surface relaxation after aggregation , the exponents in those relations are larger , corresponding to longer crossovers for small @xmath7 @xcite . a renewed interest in these competitive models was recently observed , with a focus on the limitations of scaling relations or with an emphasis on the properties of porous media . @xcite discussed the deviations from the dominant scaling of surface roughness at low @xmath7 , which is essential for a quantitative characterization of surface properties in a variety of growth conditions . the effect of relaxation after collision of incident and aggregated particles was considered in ref . @xcite , also with a focus on surface properties . refs . @xcite considered the effect of a stickness parameter on the aggregation of the incident particles , which may attach to neighboring particles located below the outer surface . simulations in @xmath1 dimensions produced deposits with porosity ranging from very small values to approximately @xmath12 and suggested non - kpz behavior in one of the models @xcite . the first aim of this paper is to study surface and bulk properties of the model proposed in ref . @xcite combining a systematic analysis of simulation data and a scaling approach for small values of the stickness parameter . from the extrapolation of saturation roughness and relaxation times , we show that the model has kpz exponents in @xmath1 dimensions . comparison of roughness distributions confirms kpz scaling in @xmath2 dimensions , thus ruling out the proposal of non - kpz exponents . in the limit of small stickness parameter @xmath0 , the scaling approach shows that the crossover time and the roughness scale as @xmath4 and @xmath5 , respectively , for all substrate dimensions . these results show a shortened crossover when compared to all previously studied models with an ud component @xcite , which is a consequence of subsurface aggregation . the same approach predicts porosity and pore height scaling as @xmath6 and @xmath5 , respectively . these predictions will be confirmed numerically in @xmath1 and @xmath2 dimensions . the approach can be extended to the model introduced in ref . @xcite , with the same crossover exponents due to the similar subsurface aggregation conditions . the rest of this work is organized as follows . in sec . [ model ] , we present the sticky particle deposition model . in sec . [ roughness ] , we analyze the surface roughness scaling of simulated deposits in @xmath1 dimensions . in sec . [ scaling ] , we present a scaling approach that relates surface properties to the stickness parameter , and confirm those predictions with numerical simulations . in sec . [ porosity ] , we extend the scaling approach to relate the porosity and the average pore height with the stickness parameter , again with support from numerical simulations . in sec . [ spd3d ] , we show that the main results extend to @xmath2 dimensions . in sec . [ conclusion ] , we summarize our results and conclusions . in all models discussed in this paper , square particles of size @xmath13 are sequentially released on randomly chosen columns of a one - dimensional discretized substrate of lateral size @xmath14 and move in a direction perpendicular to the substrate . here , @xmath15 is the number of columns . the time interval for deposition of one layer of atoms ( @xmath15 atoms ) is @xmath16 . thus , at time @xmath17 , the number of deposited layers is @xmath18 . the model proposed in ref . @xcite is hereafter called sticky particle deposition ( spd ) . in each site of the trajectory of the incident particle , it interacts with particles in nearest neighbor ( nn ) sites at the same layer ( same height above the deposit ) and particles in next neareast neighbor ( nnn ) sites at the layer immediately below it . this interaction is represented by a probabilistic rule of aggregation at its current position . the probability of aggregation to each neighbor is @xmath19 where @xmath0 is the stickness parameter , @xmath20 is the distance between the centers of the particles ( @xmath21 for nn , @xmath22 for nnn ) , and @xmath23 is an exponent related to the nature of the interaction . in ref . @xcite , the cases @xmath24 and @xmath25 were respectively called coulomb - type and van der walls - type interactions , with most results being presented for the former . here we will restrict the analysis to the case @xmath24 , in which aggregation to nn and nnn have probabilities @xmath0 and @xmath26 , respectively . fig . [ modelspd ] helps to understand the aggregation rules of the spd model and the differences from other ballistic - like models . we first recall the rules of bd and of the next nearest neighbor bd ( bdnnn ) model @xcite . in bd , aggregation occurs at the first contact with a nn occupied site : particle a at position 2 , particle b at position 6 in fig . [ modelspd ] . in bdnnn , aggregation occurs at the first contact with a nnn occupied site : position 1 for particle a and position 5 for particle b in fig . [ modelspd ] . in both cases , the incident particle interacts only with the particles at the top of each column , which are highlighted in [ modelspd ] . the set of top particles is called the outer surface of the deposit . in the spd model , particle a may aggregate at lattice sites marked with circles labeled 1 to 4 , and particle b may aggregate at lattice sites marked with circles labeled 5 to 7 . first , consider the trajectory of particle a. in position 1 , two aggregation trials are executed due to the interaction with two nnn occupied sites ; the probability of aggregation in each trial is @xmath26 . if it does not aggregate there , it moves to the position labeled 2 , in which three aggregation trials are executed : two for interactions with the nn at the same height ( probability @xmath0 for each one ) and one for interaction with the nnn in the layer below , at the left ( probability @xmath26 ) . if the particle does not aggregate at position 2 , then it moves to position 3 and may aggregate there with probability @xmath0 due to the interaction with the nn at the left . if aggregation does not occur in position 3 , the incident particle will permanently aggregate at position 4 , which is the top of the incidence column . relaxation to neighboring columns is not allowed . now we consider the trajectory of particle b. in position 5 , two aggregation trials are executed , each one with probability @xmath26 ( due to interactions with two occupied nnn sites ) . if the particle does not aggregate there , it moves to position 6 , in which three aggregation trials are executed : two for the interactions with the lateral nn ( probability @xmath0 for each trial ) and one for the interaction with the nnn in the layer below ( probability @xmath26 ) . if no aggregation trial is successful at position 6 , the particle moves to position 7 and aggregates there . in contrast to other ballistic - like models ( e. g. bd and bdnnn ) , the spd model allows subsurface aggregation . in fig . [ modelspd ] , position 3 is an example of subsurface position : it is not allowed in bd , nor in bdnnn , nor in any model of solid - on - solid deposition ( which prescribe aggregation at the top of each column ) . in all cases , note that the interaction of an incident particle with an aggregated one is possible in two steps : the first one when they are nnn ( larger distance ) , the second one when they are nn ( smaller distance ) . it represents two possibilities of aggregation in the ingoing part of the trajectory of the incident particle . if the aggregation trials are not accepted , then the incident particle moves to a lower position . in this situation , this particle is in an outgoing trajectory respectively to those aggregated particles . for this reason , no aggregation trial is executed with a nnn aggregated particle in the layer above the current position of the incident particle . for instance , when particle a is at position 3 ( third layer of the deposit ) , we do not execute aggregation trials with the black nnn sites at the fourth layer in fig . [ modelspd ] . the spd model resembles the model introduced in ref . @xcite and the slippery bd model ( sbd ) proposed in ref . @xcite , both studied in three - dimensional deposits ( the latter with line seeds perpendicular to a flat inactive surface ) . most of our simulation work is in @xmath1 dimensions , similarly to ref . @xcite , but in sec . [ spd3d ] we show that the main results are also valid in three - dimensional samples . for simplicity , in the following sections we consider unit values of the lattice constant and of the time of deposition of a layer : @xmath27 , @xmath28 . the outer surface roughness is defined as @xmath29 } ^{1/2 } , \label{defw}\ ] ] where @xmath30 is the height of the top particle of each column , the overbars indicate spatial averages , and the angular brackets indicate configurational averages . in systems with normal ( in opposition to anomalous ) scaling , the roughness follows family - vicsek ( fv ) scaling @xcite as @xmath31 where @xmath32 is the roughness exponent , @xmath33 is a relaxation time , and @xmath34 is a scaling function . in long times ( @xmath35 ) , @xmath36 , so that @xmath37 saturates as @xmath38 where @xmath39 is a model - dependent constant . the saturation time @xmath33 scales as @xmath40 where @xmath41 is the dynamic exponent and @xmath42 is another model - dependent constant . the roughness for @xmath43 scales as @xmath44 with @xmath45 and another model - dependent constant @xmath46 . [ figroughness]a shows the surface roughness evolution of the spd model for three values of @xmath0 in @xmath47 . ( red solid curve ) , @xmath48 ( green dashed curve ) , and @xmath49 ( blue dotted curve ) . the dashed line has slope @xmath50 of kpz scaling . ( b ) saturation roughness as a function of the lattice size for @xmath51 ( red squares ) and @xmath52 ( green triangles ) . the dashed line has slope @xmath53 of kpz scaling . , width=264 ] for short times , there is a crossover from an initial regime of rapid roughness increase to a second regime in which it increases slower . for small @xmath0 , the first regime is mainly of ud and the slope of the @xmath54 plot is near @xmath53 . for @xmath55 , lateral aggregation is frequent , thus the roughness at short times is larger than that of ud ( e. g. @xmath51 in fig . [ figroughness]a ) . it is difficult to find a pure ud regime in this case and to estimate the crossover time with accuracy . after this transient , the growth regime begins , with apparent power law scaling of @xmath37 [ eq . ( [ defbeta ] ) ] . it is difficult to distinguish the different curves for small @xmath0 in fig . [ figroughness]a ; this will be explained by the scaling approach of sec . [ scaling ] . the slope of those curves are near @xmath50 , suggesting kpz scaling . at long times , there is an increase in the saturation roughness as @xmath0 decreases . [ figroughness]b shows the saturation roughness as a function of lattice size @xmath15 for two values of @xmath0 . they seem to be consistent with the kpz exponent @xmath56 . however , linear fits of those plots give slopes slightly smaller than @xmath57 , similarly to what was found in ref . for this reason , a systematic extrapolation of those results is necessary to decide whether the roughness scaling is kpz or not . we proceed by using the same methods of refs . @xcite , in which roughness scaling of various ballistic - like models was studied . effective roughness exponents are defined as @xmath58}{\ln{2 } } . \label{defalphal}\ ] ] assuming that the saturation roughness has scaling corrections as @xmath59 @xcite , where @xmath60 and @xmath61 are constants , we expect @xmath62 , where @xmath63 is another constant . fig . [ alpha ] shows effective exponents as a function of @xmath64 for @xmath51 and @xmath48 , respectively using @xmath65 and @xmath66 . these values of @xmath67 provide the best linear fits of the @xmath68 data for each stickness parameter . the asymptotic ( @xmath69 ) estimates from those fits are @xmath70 and @xmath71 , respectively . for @xmath48 ( red triangles ) with @xmath66 and @xmath51 ( blue squares ) with @xmath72 . , width=264 ] we estimate the dynamical exponent @xmath41 using the method proposed in ref . for each lattice size @xmath15 , a characteristic time @xmath73 is defined as @xmath74 with @xmath75 . the fv relation ( [ fv ] ) shows that @xmath73 is proportional to @xmath33 for fixed @xmath76 , thus @xmath77 . effective dynamical exponents are defined as @xmath78}{\ln{2 } } . \label{defzl}\ ] ] figs . [ zl]a and [ zl]b show @xmath79 for @xmath51 and @xmath48 , respectively , obtained with @xmath80 . in both cases , the exponents oscillate near @xmath81 , suggesting that finite - size corrections are very small . for ( a ) @xmath48 and ( b ) @xmath51.,width=264 ] the estimates of @xmath32 and @xmath41 are in very good agreement with kpz exponents @xmath82 and @xmath83 , which is a strong numerical evidence that this model is in the kpz class in @xmath1 dimensions . a universal scaling is expected in the spd model because there is no change in its symmetries as the stickness parameter changes . in other words , the corresponding hydrodynamic growth equation may have coefficients dependent on the parameter @xmath0 , but the leading spatial derivatives will be the same @xcite . due to the lateral aggregation and consequent excess growth velocity , kpz scaling is expected for any @xmath3 . previous works on ballistic - like models @xcite have already shown that systematic extrapolation of finite - size or finite - time data are necessary to avoid crossover effects . as highlighted in ref . @xcite , this is a consequence of the large fluctuations in height increments , typical of those models . crossovers and finite - size corrections probably are the reasons for the deviations from kpz scaling observed in ref . @xcite . this may also be inferred by comparison with finite - size bd data from ref . @xcite . ref . @xcite suggests @xmath84 for @xmath85 , while ref . @xcite gives effective exponents @xmath86 for bd in the same range of @xmath15 . moreover , the growth exponents @xmath87 in lattice sizes from @xmath88 to @xmath47 for @xmath85 , shown in ref . @xcite , are very near the corresponding estimates for bd in ref . @xcite ( considering minimum linear correlation coefficient @xmath89 in the growth region ) . for small @xmath0 , lateral aggregation is unprobable , thus most particles aggregate at the top of the column of incidence . at short times , the roughness increases as @xcite @xmath90 after a crossover time @xmath91 , kpz scaling appears . our first step is to relate @xmath91 to @xmath0 , for @xmath92 . a typical configuration of two neighboring columns in ud is illustrated in fig . [ columns ] . it has a height difference @xmath93 because their heights increase without correlations . if @xmath94 is large , then a new particle inciding at the right column in fig . [ columns ] may aggregate at a number of positions of order @xmath94 , as indicated by the circles . this means that the number of aggregation trials is of order @xmath94 and the aggregation probability for each trial is of order @xmath0 . thus , the probability of no lateral aggregation after those trials ( i. e. aggregation at the top of the column ) is @xmath95 . the probability that some lateral aggregation occurs is , consequently , @xmath96 the latter approximation requires @xmath97 , which will be confirmed below . the average time for a lateral aggregation event at a given column is @xmath98 . lateral aggregation immediately creates correlations between the heights of neighboring columns , thus the crossover time is @xmath99 . ( [ wrandom ] ) and ( [ plat ] ) at the crossover ( @xmath100 , @xmath101 ) give @xmath102 thus , height fluctuations at the crossover scale as @xmath103 and the crossover time scales as @xmath104 these results confirm that @xmath105 , as the approximation in eq . ( [ plat ] ) requires . the amplitudes of the saturation roughness [ eq . ( [ fv ] ) ] and of the relaxation time [ eq . ( [ scalingttimes ] ) ] scale as @xmath106 and @xmath91 , respectively . following the exponent convention introduced by horowitz and albano @xcite , we have @xmath107 and @xmath108 these results are valid in any spatial dimension because ud properties are not dimension - dependent . the scaling exponents in eqs . ( [ defdelta ] ) and ( [ defy ] ) differ from those obtained in other competitive models with ballistic - like aggregation with probability @xmath7 and ud with probability @xmath8 ; in those systems , @xmath109 and @xmath110 @xcite . in solid - on - solid models with crossovers from ud to correlated growth , the exponents are also different : @xmath111 and @xmath112 . the shorter crossover of the spd model is due to subsurface aggregation , which provides a large number of opportunities ( of order @xmath106 ) for lateral aggregation of the incident particle ( fig . [ columns ] ) . on the other hand , the relation @xmath113 obtained in other competitive models is also obeyed here because it is solely related to ud scaling ( see e. g. the discussion in ref . @xcite ) . we performed simulations of the spd in @xmath47 and small values of @xmath0 , from @xmath52 to @xmath114 , until the steady states ( roughness saturation ) . the saturation roughness @xmath115 and the characteristic times @xmath73 were calculated following the same lines of sec . [ roughness ] . [ wt]a and [ wt]b show @xmath73 and @xmath115 , respectively , as a function of the stickness parameter @xmath0 . since they were measured for constant @xmath15 , they are expected to scale as the amplitudes @xmath42 [ eq . ( [ scalingttimes ] ) ] and @xmath39 [ eq . ( [ wsat ] ) ] , respectively . fits of the data for @xmath116 give exponents @xmath117 [ eq . ( [ defy ] ) and fig . [ wt]a ] and @xmath118 [ eq . ( [ defdelta ] ) and fig . [ wt]b ] . and ( b ) saturation roughness @xmath115 in size @xmath47 as a function of the stickness parameter . solid lines are fits of the data for @xmath119.,width=264 ] the estimate of @xmath120 is in good agreement with the theoretical prediction of eq . ( [ defy ] ) . however , the estimate of @xmath121 shows a discrepancy of @xmath122 from the theoretical prediction of eq . ( [ defdelta ] ) . note that the fits in figs . [ wt]a and [ wt]b considered @xmath123 , which are not very small values of @xmath0 , thus deviations are expected , particularly in the smaller exponent ( @xmath121 ) . unfortunately , it is very difficult to obtain accurate estimates for smaller values of @xmath0 because relaxation times become very large and roughness fluctuations also increase . using smaller system sizes is also inappropriate because it enhances crossover effects @xcite estimated the crossover times for @xmath124 and obtained @xmath125 , which is significanly different from the theoretical prediction in eq . ( [ tc ] ) . however , measuring reliable crossover times is a difficult task , as explained in sec . [ roughness ] . on the other hand , the same work shows that the saturation time for @xmath47 scales as @xmath126 , which is in good agreement with our estimate . the scaling of the amplitude @xmath46 in eq . ( [ defbeta ] ) can be predicted along the same lines of refs . @xcite for other competitive models : @xmath127 for the spd model in @xmath1 dimensions , we obtain @xmath128 . this very small exponent gives a very slow variation of @xmath46 with the stickness parameter . it explains the small distance between the curves for different values of @xmath0 in fig . [ figroughness]a . in ref . @xcite , a model similar to the spd was introduced . the particles incide vertically and , at each site of its trajectory with a nn occupied site , it may aggregate with probability @xmath7 . otherwise , the particle moves down one site . if no lateral aggregation occurs , the particle aggregates at the top of the column of incidence . in fig . [ modelspd ] , particle a may aggregate to positions labeled 2 , 3 , and 4 . in positions 2 and 3 , aggregation trials have probability @xmath7 . if the particle does not aggregate at one of those points , it moves to position 4 and aggregates there . particle b may aggregate at position 6 with probability @xmath7 , otherwise it moves to position 7 and aggregates there . for small @xmath7 , most lateral aggregation trials are rejected , thus ud dominates . large local height fluctuations appear , similarly to fig . [ columns ] . the increase of the local height difference @xmath94 and the probability of lateral aggregation @xmath129 are given by eqs . ( [ deltah ] ) and ( [ plat ] ) , with @xmath0 replaced by @xmath7 . thus , the same reasoning of sec . [ approach ] leads to the same scaling relations of the spd model with @xmath0 replaced by @xmath7 . in the notation of ref . @xcite , exponents @xmath130 and @xmath131 are predicted by our scaling approach . the numerical estimates of that work , @xmath132 and @xmath133 , differ from those predictions , probably because they were obtained by data collapse methods that do not account for scaling corrections . for @xmath85 , the samples have large porosity @xmath134 . when @xmath0 decreases , @xmath135 decreases because ud creates no holes . [ samplesspd ] shows regions of some samples obtained with small stickness parameters . the porosity decrease is accompanied by the formation of longer pores extended in the vertical direction . this is a consequence of the increase of the height fluctuation @xmath106 before a lateral aggregation event [ eq . ( [ deltahc ] ) ] . ( in lattice units ) of samples grown with stickness parameters ( a ) @xmath52 , ( b ) @xmath136 , ( c ) @xmath137 , and ( d ) @xmath138.,width=264 ] the number of deposited layers necessary to attain a steady state value of @xmath135 is relatively small , typically of the order of @xmath91 in eq . ( [ tc ] ) . this is expected because pores are narrow , even in pure bd , thus porosity depends only on short wavelength height fluctuations , which saturate at short times ( in the absence of scaling anomaly @xcite ) . our scaling approach can be used to predict the dependence of the porosity @xmath135 and the average pore height on the parameter @xmath0 , as follows . during the time interval @xmath91 between two lateral aggregation events , the number of particles deposited at the top of a given column is approximately @xmath91 ( note that we are still using unit lattice constant and unit deposition time of a layer , @xmath27 and @xmath28 ) . the size of a long pore produced by the lateral aggregation is @xmath106 [ eq . ( [ deltahc ] ) ] . consequently , for small @xmath0 , the porosity ( pore volume divided by total volume ) is expected to scale as @xmath139 this is valid in the limit of very small @xmath0 , in which @xmath140 . the small exponent in eq . ( [ pscaling ] ) explains why a large decrease of @xmath0 leads only to mild reduction of porosity . this is remarkably illustrated in fig . [ samplesspd ] , in which @xmath0 varies three orders of magnitude , while the porosity decreases from @xmath141 to @xmath142 , i. e. changes by a factor smaller than @xmath143 . the porosity scaling in the spd also differs from other competitive models involving ballistic - type aggregation . examples are the bidisperse ballistic deposition @xcite and the bd - ud competitive model , in which @xmath144 ( @xmath7 is the probability of the ballistic - like component ) . we simulated the spd in size @xmath47 for small values of @xmath0 in order to measure the porosity between times @xmath145 and @xmath146 . in all cases , @xmath147 is much larger than the crossover time and @xmath148 is much smaller than the relaxation time @xmath33 . [ phi]a shows the porosity as a function of the stickness parameter . the linear fit for @xmath149 gives @xmath150 , in excellent agreement with eq . ( [ pscaling ] ) . although these values of @xmath0 are very small , the corresponding values of @xmath6 and of @xmath135 are not very small . thus , scaling corrections are particularly weak in this case . . ( b ) average pore height as a function of the stickness parameter . dashed lines have slopes @xmath151 ( right ) and @xmath152 ( left).,width=264 ] for very small @xmath0 , the pores are long and isolated , as illustrated in fig . [ samplesspd]d . the average pore height is expected to scale as eq . ( [ deltahc ] ) , because a pore is formed only when a lateral aggregation event occurs . however , for @xmath153 or larger , many pores occupy two or more neighboring columns . this can be observed in figs . [ samplesspd]a , [ samplesspd]b , and [ samplesspd]c . here we define pore height as the vertical distance between the aggregation position and the top of the incidence column in any lateral aggregation event . its average value , @xmath154 , is taken over all lateral aggregation events between @xmath147 and @xmath148 in @xmath155 different samples . for small @xmath135 , pores are isolated , thus @xmath154 is a reliable approximation of the average pore height , and is expected to scale as eq . ( [ deltahc ] ) . for @xmath135 not too small , some pores occupy two or more neighboring columns , and all these columns contribute to @xmath154 ( each one had a lateral aggregation event ) . fig . [ phi]b shows @xmath154 as a function of @xmath0 . the slope of that log - log plot evolves from @xmath151 for @xmath156 to @xmath152 for @xmath157 . the latter is @xmath158 smaller than the theoretically predicted value @xmath159 [ eq . ( [ deltahc ] ) ] , which indicates the presence of large scaling corrections . ref . @xcite measured the porosity of samples with @xmath160 , with results in qualitative agreement with ours . however , the low porosity scaling was not addressed there . @xcite suggests that the porosity scales with @xmath7 ( equivalent to @xmath0 ) and with the lattice size @xmath15 . the latter is expected only as vanishing corrections , since porosity does not depend on long wavelength fluctuations . this explains the small ( effective ) exponents @xmath0 and @xmath13 obtained in that work . on the other hand , ref . @xcite estimates the long - time scaling on @xmath7 with exponent @xmath161 , which is to be compared with the theoretical prediction @xmath50 . the discrepancy is probably related to the use of data collapse methods . the aim of this section is to show that the main features of the spd model in @xmath1 dimensions can be extended to @xmath2 dimensions , namely the kpz roughening of the outer surface and the porosity scaling derived by the superuniversal approach of sec . [ scaling ] . the aggregation rules of the spd model have to be extended in this case . first , nn interactions are considered in two substrate directions , with a total of four nn in the same height . secondly , nnn interactions appear with aggregated particles in the same height ( four neighbors ) and with particles at the level immediately below ( four neighbors ) . roughness scaling of ballistic - like models usually show large corrections @xcite . an alternative to search for the universality class of a given model is the comparison of scaled roughness distributions of relatively small systems because the finite - size corrections of those quantities are much smaller @xcite . we simulated the spd model with @xmath48 in substrates of lateral size @xmath88 up to the steady state ( roughness saturation ) . in this regime , the square roughness @xmath162 of several configurations is measured . @xmath163 is the probability density of the square roughness of a given configuration to lie in the range @xmath164 $ ] . this quantity is expected to scale as @xmath165 where @xmath166 is the rms fluctuation of @xmath167 and @xmath168 is a universal function @xcite . fig . [ dist ] shows the scaled roughness distribution of the spd model and the distribution of the restricted solid - on - solid ( rsos ) model @xcite in substrate size @xmath88 . the latter is a well known representative of the kpz class and its roughness distributions have negligible finite - size effects @xcite . the excellent collapse of the curves in fig . [ dist ] is striking evidence that the spd model also belongs to the kpz class in @xmath2 dimensions . ( squares ) and of the rsos model ( solid curve ) in @xmath2 dimensions , with @xmath88 . , width=264 ] we also simulated the spd model in size @xmath47 for small values of @xmath0 and measured the porosity between times @xmath169 and @xmath170 . we observe that the porosity is larger than in the @xmath1-dimensional samples for the same value of @xmath0 . for instance , for @xmath48 , the porosity exceeds @xmath171 . this is a consequence of the larger number of interactions of the incident particle with nn and nnn in @xmath2 dimensions , which facilitates lateral aggregation . [ phi3d ] shows the porosity as a function of @xmath0 , for low values of that parameter . the linear fit for @xmath149 gives @xmath172 , which is also in excellent agreement with eq . ( [ pscaling ] ) . this supports the extension of the scaling approach of sec . [ scaling ] to @xmath2 dimensions . dimensions . the solid line is a linear fit of the data for @xmath173 . , width=264 ] an important consequence of this scaling approach is to facilitate the design of samples with the desired values of porosity and elongate pores . however , one has to take care with the fluctuations in the value of @xmath135 in the first layers of the deposit , typically produced at @xmath174 . we studied surface and bulk properties of porous deposits produced by a model proposed in ref . @xcite , in substrates with one and two dimensions . the model shows a crossover from uncorrelated to correlated growth for small values of the stickness parameter @xmath0 . in @xmath1 dimensions , a systematic analysis of simulation data for saturation roughness and relaxation times shows that the model belongs to the kpz class . finite - size corrections explain the previous claim of deviations from kpz scaling . in @xmath2 dimensions , kpz roughening is confirmed by comparison of roughness distributions . a scaling approach for small values of @xmath0 is proposed to relate the crossover time and the local height fluctuations with that parameter , respectively giving exponents @xmath175 and @xmath159 . these results are consequence of the ud properties , thus they do not depend on the spatial dimension . numerical results confirm these predictions . the crossover exponents are smaller than those of other competitive models that consider aggregation only at the outer surface @xcite . the same approach predicts the porosity scaling as @xmath6 , which is in good agreement with simulation results in @xmath1 and @xmath2 dimensions . this result is important for using the model to produce porous samples representative of real materials . this may also help to model samples with desired porosity and pore height , particularly for the possibility of controlling the scaling properties by changing the kinetics of subsurface aggregation . | we study surface and bulk properties of porous films produced by a model in which particles incide perpendicularly to a substrate , interact with deposited neighbors in its trajectory , and aggregate laterally with probability of order @xmath0 at each position .
the model generalizes ballistic - like models by allowing attachment to particles below the outer surface . for small values of @xmath0 , a crossover from uncorrelated deposition ( ud ) to correlated growth
is observed .
simulations are performed in @xmath1 and @xmath2 dimensions .
extrapolation of effective exponents and comparison of roughness distributions confirm kardar - parisi - zhang roughening of the outer surface for @xmath3 .
a scaling approach for small @xmath0 predicts crossover times as @xmath4 and local height fluctuations as @xmath5 at the crossover , independently of substrate dimension .
these relations are different from all previously studied models with crossovers from ud to correlated growth due to subsurface aggregation , which reduces scaling exponents .
the same approach predicts the porosity and average pore height scaling as @xmath6 and @xmath5 , respectively , in good agreement with simulation results in @xmath1 and @xmath2 dimensions
. these results may be useful to modeling samples with desired porosity and long pores . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
despite more than four decades of research , the emission from pulsars and their surrounding nebulae is still poorly understood . two important problems are the origin of pulsar @xmath0-ray emission and , for pulsar wind nebulae ( pwne ) , the so - called sigma - problem ( see the reviews by * ? ? ? * ; * ? ? ? * ; * ? ? ? the latter problem derives its name from the sigma - parameter , @xmath5 , which is the ratio of the energy density due to poynting flux over the particle energy density , with @xmath6 the local magnetic field strength , @xmath7 the lorentz factor of the pulsar wind , and @xmath8 the particle number density . according to theoretical models , most of the pulsar s rotational energy loss rate ( or spin - down power ) @xmath9 is due to poynting flux , i.e. the pulsar wind should have @xmath10 , but observations of pwne indicate that most of the energy that has been lost is actually contained by the relativistic electrons / positrons . somehow , the high @xmath11 flow is converted into kinetic energy somewhere between the pulsar s magnetosphere and the wind termination shock , which converts the radial pulsar wind into an isotropic , relativistic particle distribution @xcite . a third , and perhaps related , problem is the high wind multiplicity factor . the combination of a high magnetic and rapid rotation results in a strong electric potential in the magnetosphere . this potential will be neutralized by charged particles that are stripped from the surface of the neutron star @xcite . the associated charged particle density is @xmath12 with @xmath13 the pulsar s period and @xmath6 the local magnetic field . a fraction of these particles will escape through open field lines , resulting in a particle flux @xmath14 with @xmath15 the dipole surface magnetic field in units of @xmath16 g. however , x - ray ( e.g. * ? ? ? * ) and tev @xcite observations indicate that the number of relativistic electrons contained by pwne turns out to be orders of magnitude larger than @xmath17 , i.e. @xmath18 , with the multiplicity factor being @xmath19 for a young pulsar like b1509 - 58 @xcite . the origin of the additional plasma is likely electron / positron pair production in the magnetosphere . the pair production occurs in the presence of the high magnetic fields inside the magnetosphere , and requires the presence of high energy photons that are either the result of curvature radiation or inverse compton scattering . the electrons that cause the emission are accelerated due to the extremely large voltage drop across the open field lines @xcite . for the inverse compton scattering seed photons are necessary that are emitted by the hot polar caps of the pulsar , heated due to the bombardment by accelerated particles , or due to the cooling of the young neutron star . despite the many unsolved problems , pulsar research has thrived over the last decade thanks to many advances in observational techniques and numerical simulations . in particular high energy observations have contributed to a wealth of new information on pulsars and pwne ; from high spatial resolution x - ray images with _ chandra _ , revealing torii and jets ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , to a rapid progress in the field of tev astronomy ( * ? ? ? * for a review ) , which have revealed an unexpectedly large population of very extended pwne ( e.g. * ? ? ? * ; * ? ? ? this rapid growth in observational data has recently been augmented by the gev @xmath0-ray observatory _ fermi _ , which has greatly increased the sample of @xmath0-ray - emitting pulsars @xcite . here we present a statistical analysis of two samples of rotation powered pulsars . one is those of x - ray pulsars compiled by @xcite , the other the aforementioned sample of _ fermi_-detected pulsars . our analysis concentrates on what determines the radiative efficiency of pulsars and their pwne . we report a surprisingly strong correlation between the x - ray luminosity of pulsars and their pwne , which inversely correlates with characteristic age , at least for young pulsars . in contrast , the @xmath0-ray emission correlates positively with characteristic pulsar age , as already noted by @xcite . it is well known that the non - thermal x - ray luminosities of pulsars and pwne are strongly correlated with the spin - down luminosity @xmath9 of the pulsar @xcite . the efficiency with which the spin - down luminosity is converted into x - ray emission is usually indicated by the symbol @xmath20 with @xmath21 in the range of @xmath22 . an important question is what determines this efficiency for both @xmath0-ray and x - ray emission . for the pwne @xmath21 may provide information on how well the spin - down luminosity is converted into relativistic particles , i.e. it is related to the sigma - problem . for the statistical analysis described here we use the x - ray properties of pulsars and their nebulae as determined by @xcite . this sample is based on _ chandra _ observations . chandra_-acis instrument that was used has a poor timing resolution , so all pulsar luminosities are a combination of pulsed and unpulsed emission . however , given the high spatial resolution of _ chandra _ , the x - ray flux from the pulsar could be accurately separated from the x - ray flux from the pwn . the x - ray luminosities were derived from the 0.5 - 8 kev fluxes , corrected for interstellar absorption and using the distance estimated listed in the paper . @xcite list only the non - thermal x - ray luminosities , so the contribution of thermal x - ray emission , if present , was ignored . the uncertainties in the distance are the largest source of error in the luminosities . in general , the distance estimates may have errors of order @xmath23 , resulting in luminosity errors of order 4 . of course distance errors affect the pulsar and pwne luminosities in the same way . a source of error for the pwne luminosities may be underestimated flux contributions from low surface brightness emission at large radii . indeed , @xcite recently reported the detection of low x - ray surface brightness structures around several pwne by _ suzaku_. the fraction of the luminosity in these low surface brightness is , however , not more than a factor of 2 in luminosity . so all together , the logarithm of the luminosities may contain errors of the order of one decade . we included in our study a statistical analysis of the @xmath0-ray luminosities , based on the pulsed @xmath0-ray emission of pulsar reported by @xcite . we made sure that the @xmath0-ray luminosities of pulsars that were common to both the x - ray and @xmath0-ray samples were based on the same distance estimates , namely those adopted by @xcite . for both the x - ray and @xmath0-ray samples we excluded millisecond pulsars , and omitted pulsars without good distance estimates . in figure [ fig : eta ] we show the correlation of the x - ray luminosity efficiency @xmath21 of pulsars and pwne versus the characteristic spin - down age @xmath24 . we also included data points for two recently discovered pwne ; those surrounding the high energy pulsar j14003 - 6326 @xcite and axp 1e1547.0- 5408 @xcite . we did not include these in our statistical analysis , although their properties are consistent with the general trends we report below . figure [ fig : eta ] also shows the @xmath0-ray efficiency , as obtained from the pulsed @xmath0-ray luminosities determined by @xcite . this figure reveals the trend that young pulsars appear to have higher x - ray efficiencies than old pulsars . moreover , the behavior is similar for the luminosities of the pulsars and the pwne . figure [ fig : ppdot ] shows the timing properties of the pulsars in the two samples used for our statistical analysis in the @xmath25 diagram . it shows that on average the detected @xmath0-ray pulsars seem to be somewhat older than the x - ray detected pulsars / pwne , consistent with the trend that older pulsars are less efficient in emitting x - rays and more efficient in emitting @xmath0-rays . note , however , that an interpretation of this diagram is far from straightforward , because the detectability of a pulsar also depends on its distance and , for x - rays , on the interstellar absorption column . a trend of decreasing x - ray luminosity with @xmath26 was reported before by @xcite , but figure [ fig : eta ] reveals it to be due to a trend in the efficiency , not in the overall spin - down power . for pulsars with @xmath27 yr the efficiency appears to be more or less constant . in contrast , young pulsars are not so efficient in producing @xmath0-ray emission , as already noted by @xcite . the spread in the data points for a given characteristic is of order of 1 - 2 decades . as discussed above , the uncertainties in the luminosity can explain about 1 decade of this spread , but intrinsic variations , and for the @xmath0-ray emission , beaming properties are expected to contribute to the spread as well . in order to investigate the correlations between @xmath21 and @xmath26 further , one has to avoid @xmath21 as an independent variable , since both @xmath21 and @xmath26 are derived from combinations of the pulsar s period @xmath13 and period derivative @xmath28 ; the spin - down luminosity is given by @xmath29 with @xmath30 g@xmath31 the neutron star s moment of inertia . for that reason we base our regression analysis on the logarithm to denote the 10-based logarithm . ] of the x - ray/@xmath0-ray luminosity , as the quantity to be explained , and the independent variables @xmath32 , @xmath33 as the principle input variables for the model ( c.f . as explained above the errors are dominated by systematic errors , mostly due to difficulties in estimating distances . for that reason we used the unweighted least square method , which means that the errors in the best - fit parameters are based on the variance of the residuals . the disadvantage is that we do not have an intrinsic goodness of fit statistic . however , we can compare two hypotheses using the f - test statistic , defined as : @xmath34 here @xmath35 denotes the observed ( @xmath0-ray / x - ray ) luminosity and we use @xmath36 to indicate the expected value of @xmath35 based on the best - fit parameters , and the subscript @xmath37 and @xmath38 to indicate two different best - fit models , @xmath39 is the number of degrees of freedom and @xmath8 is the sample size . the probability that the improvement in the sum of squared residuals is by pure chance is given by the @xmath40-distribution . from the above it is clear that we fitted the linear relation : @xmath41 we compare this relation , using the f - test , with a functional form @xmath42 . this is essentially assuming that @xmath21 ( equation [ eq : eta ] ) is constant . in addition we fit the relation @xmath43 which corresponds to @xmath44 . this is a functional form that has been used before ( e.g. * ? ? ? * ) , and that seems to work particularly well for the @xmath0-ray emission from pulsars @xcite . since figure [ fig : eta ] suggests that young pulsars are more efficient x - ray emitters than old pulsars , we divided the x - ray sample in an old and a young population . as the precise characteristic age that should be chosen is a bit arbitrary , but lies somewhere between @xmath45 yr and @xmath46 yr , we decided to divide the sample in two more or less equally sized samples . this put the cut at @xmath47 yr . the result of the regression analysis is listed in table [ tab : fits ] , whereas figure [ fig : bestfit ] shows the results of applying the best - fit solutions to the different samples . the best - fit relations for young pulsars and pwne indicate that the x - ray luminosity does not scale with @xmath9 , since in that case we would expect for equation ( [ eq : fullfit ] ) @xmath48 and @xmath49 , whereas we find @xmath50 and @xmath51 and @xmath52 and @xmath53 for pwne and pulsars , respectively . this conclusion is based on both the best - fit parameters and their inferred errors , as on the f - test . the f - test , in fact , indicates that equation ( [ eq : fullfit ] ) provides a better fit than equation ( [ eq : eta ] ) at the 99.99% confidence level . interestingly , the values for @xmath54 and @xmath55 for pulsars and pwne are consistent with each other . for older pulsars the f - values indicate that the x - ray emission efficiency is just as well described by equation ( [ eq : eta ] ) with @xmath56 , as by a linear dependency on @xmath32 and @xmath33 . also this result is true for both the pulsar x - ray luminosity as for the pwn luminosity . the least square fits for the _ fermi_-sample shows that the relation @xmath4 is indeed a good description of the data . for equation ( [ eq : edotfit ] ) we find @xmath57 , close to the expected 1/2 , and for equation ( [ eq : fullfit ] ) we find @xmath58 and @xmath59 , whereas a scaling with @xmath60 implies @xmath61 and @xmath62 , in excellent agreement . the f - tests confirm this , and indicate that a constant value for @xmath63 can be rejected at the 99.9996% confidence level . finally , one may wonder whether the combined @xmath0-ray and x - ray luminosity is more closely correlated with @xmath9 , than the @xmath0-ray and x - ray luminosities individually . unfortunately , this is hard to determine from the present data sets as there are only 15 pulsars that the two samples have in common , with most of them being older pulsars . in fact , the combined x - ray/@xmath0-ray luminosity of all pulsars is dominated by the @xmath0-ray emission , with the exception of the crab pulsar . it is , therefore , not surprising that the best - fit relation between @xmath64 indicates @xmath65 , just like for @xmath0-ray luminosity . however , as the number of @xmath0-ray - detected pulsars will grow in the near future , it may be good to investigate the total radiative output from rotation powered pulsars in more detail , especially around @xmath66 yr , where the radiative output changes from x - ray to @xmath0-ray domination . we presented a statistical study of the x - ray and @xmath0-ray properties of rotation powered pulsars , with the aim of finding what trends underly the efficiency with which spin - down luminosity is converted into high energy radiation . we started our analysis by showing that young pulsars are efficient x - ray emitters , but poor @xmath0-ray emitters , whereas it is the other way around for old pulsars . a trend of low x - ray luminosity as a function of characteristic age was reported by @xcite , but it is shown here that it is related to the efficiency with which spin - down luminosity is converted to x - ray emission , and not just due to an overall decline in spin - down luminosity . in order to be as general as possible in finding trends in x - ray/@xmath0-rayluminosities our main results are based on a regression analysis of x - ray luminosity of the pulsars / pwne @xcite and @xmath0-ray luminosity of pulsars @xcite versus the pulse period and its derivative . this is a very generic method , which includes the possibility that the luminosity depends solely on spin - down luminosity . thus encompassing other possible dependencies , such as on spin - down luminosity . given the trend noted in figure [ fig : eta ] we divided the x - ray sample in two equally sized samples of young ( @xmath1 yr ) and old pulsars . our statistical analysis produced two new findings : 1 ) the x - ray luminosity of pulsars and their surrounding pwne appear closely correlated , with @xmath67 for young pulsars , whereas for old pulsars the best fit gives @xmath68 , the latter being close to , and statistically indistinguishable from , a constant x - ray emission efficiency @xmath21 ; 2 ) young pulsars are more efficient x - ray emitters than old pulsars , and have a different dependency for @xmath69 on @xmath13 and @xmath28 . we also confirm the findings by @xcite that 1 ) the @xmath0-ray luminosity is well described by @xmath4 ; 2 ) younger pulsars are less efficient in producing @xmath0-rays . the latter trend is , therefore , the reverse of the x - ray luminosity ( figure [ fig : eta ] ) . the question is what these findings reveal about pulsar x - ray emission mechanisms . first of all , the connection between the x - ray luminosities of pulsars and their pwne may be surprising , given that the luminosities of the pwne are , in general , affected by both the pulsar wind properties and the environment of the pulsar . in particular for young pwne the luminosity is possibly affected by its interaction with a surrounding supernova remnant ( snr ) . the reverse shock of the snr will during a certain phase of the snr evolution compress the pwn , which naively may be assumed to result in a brightnening of the pwn . however , a recent study by @xcite showed the behavior of the pwn luminosity to be more complex . their study indicates that the compression by the reverse shock leads to a brightening of the radio luminosity , but at the same time to an almost total quenching of the x - ray luminosity for a brief period ( for the specific model they calculated this happened between 18 - 30 kyr ) . apart from this brief phase , the x - ray luminosity traces the spin - down luminosity surprisingly well , with an expected fluctuation of about 0.3 decades in @xmath21 . given that apparently the x - ray luminosities of the pulsars and their pwne seem to trace each other , and have similar dependencies on the period and period derivative , one can not easily use the best - fit functions for young pulsars to derive a correlation of x - ray luminosity with some well known physical pulsar property , such as @xmath70 , @xmath71 , @xmath72 ( the magnetic field at the light cylinder ) , @xmath73 , or @xmath74 . for young pulsars the x - ray emission is poorly fit with a dependency on @xmath75 ( table [ tab : fits ] ) . this suggests that the x - ray luminosity from pulsars and pwne may require a more complex model than the @xmath0-ray luminosity , with its @xmath4 scaling . the best - fit formula does , however , explain the strong dependency on @xmath26 for young pulsars , since @xmath76 . secondly , the similar behavior of the pulsars and pwn x - ray luminosities suggest there is a physical connection between the two . it is usually assumed that the pulsar x - ray emission originates in the magnetosphere , whereas the x - ray emission from the nebulae comes from outside the termination shock . these are two distinct regions , which are separated by a region that encompasses the pulsar light cylinder and the so - called wind - zone @xcite , the region in which the pulsar wind is formed . it is tempting to speculate that the connection between the x - ray emission from the pulsars and pwne may have something to do with the pair multiplicity . models of pair creation in the magnetosphere indicate that the pair multiplicity is a function of the characteristic age ( e.g. * ? ? ? this would mean that both the x - ray emission from the pulsar and from the pwne are somehow proportional to the multiplicity . this is not a completely satisfactory explanation , because the x - ray synchrotron depends on the total energy contained by the pairs , not just by the total number of particles . another explanation for the similar behavior of pulsar and pwn luminosities is offered by the striped wind theory @xcite . according to this theory the alternating magnetic fields generated by obliquely rotating pulsars leads to reconnecting magnetic fields in the wind zone . this transforms magnetic energy into kinetic energy , thereby changing the pulsar wind from a high to a low @xmath11 outflow . according to @xcite the zone in which this heating and acceleration occurs could be the location of pulsed x - ray emission . such a model makes it easier to explain why there is a connection between the pulsar and pwn luminosities , as in both cases the energy is generated in the wind zone . the higher pair multiplicity of young pulsars may be of additional importance as there are simply more particles available to be accelerated . finally , there is one other issue to consider , namely , why young pulsars do not seem to be efficient in generating @xmath0-rays . the fact that it is the other way around for the x - ray emission , perhaps indicates that the pair multiplicity in young pulsars is so high that the electric fields in the magnetosphere are shorted out @xcite . it may also be worthwhile to investigate in more detail what the role of the seed photons for inverse compton upscattering in young pulsars is . the compton scattered photons ( and in some cases curvature radiation generated photons ) are converted in pairs , and form the basis of the pair multiplicity @xcite . the soft seed photons originate from the neutron star surface , and may the result of polar cap heating due to accelerated electron / positron beams , or they are the result of the cooling of the neutron star . the latter would induce a dependency on neutron star age , which could explain why young pulsars behave differently than old pulsars . we have only briefly mentioned the tev detected pwne . the tev luminosities of pwne do not seem to correlate with spin - down luminosity @xcite . the probable reason is that the tev emission from pwne is not so much determined by the current energy production of the pulsar , but reflects the time integrated energy input from the pulsar wind , which in itself scales with the initial spin - period @xmath77 . in x - rays the lifetime of the x - ray synchrotron emitting electrons is short compared to the pulsar lifetime , whereas in tev ( and in the radio ) this is not the case . we can therefore conclude that different parts of the electromagnetic spectrum inform us about different aspects of pulsars . the @xmath0-ray emission and pulsed radio emission is likely related to what happens in the magnetosphere , whereas tev and radio emission from the pwne inform us about the time integrated properties of pulsars . based on our statistical study we have added here the suggestion that the x - ray emission from the pulsar , like the x - ray emission from the pwne , may inform us about what happens in one of the least understood regions surrounding the pulsar : the wind zone . jv is supported by a vidi grant from the netherlands science foundation ( nwo ) . ry is supported by grant - in - aid from the ministry of education , culture , sports , science , and technology ( mext ) of japan , no . 19047004 , no . 21740184 , no . we have made use of the atnf online pulsar catalogue : http://www.atnf.csiro.au/research/pulsar/psrcat . \(2 ) @xmath85 & @xmath79 & @xmath86 & @xmath87 & @xmath88 & @xmath89 & @xmath90 + & @xmath54 & @xmath91 & @xmath92 & @xmath93 & @xmath94 & @xmath95 + & @xmath96-value and associated probability based on a comparison with formula 1 in this table . ] & @xmath97 & @xmath98 & @xmath99 & @xmath100 & @xmath101 + \(3 ) @xmath102 & a & @xmath103 & @xmath104 & @xmath105 & @xmath106 & @xmath107 + & b & @xmath108 & @xmath109 & @xmath110 & @xmath111 & @xmath112 + & c & @xmath113 & @xmath114 & @xmath115 & @xmath116 & @xmath117 + & @xmath96 & @xmath118 & @xmath119 & @xmath120 & @xmath121 & @xmath122 + & @xmath96 & @xmath123 & @xmath124 & @xmath125 & @xmath126 & @xmath127 + sample size & & & & 28 + | we present a statistical analysis of the x - ray luminosity of rotation powered pulsars and their surrounding nebulae using the sample of @xcite and we complement this with an analysis of the @xmath0-ray - emission of _ fermi _ detected pulsars . we report a strong trend in the efficiency with which spin - down power is converted to x - ray and @xmath0-ray emission with characteristic age : young pulsars and their surrounding nebulae are efficient x - ray emitters , whereas in contrast old pulsars are efficient @xmath0-ray emitters .
we divided the x - ray sample in a young ( @xmath1 yr ) and old sample and used linear regression to search for correlations between the logarithm of the x - ray and @xmath0-ray luminosities and the logarithms of the periods and period derivatives . the x - ray emission from young pulsars and their nebulae are both consistent with @xmath2 . for old pulsars and their nebulae
the x - ray luminosity is consistent with a more or less constant efficiency @xmath3 . for the @xmath0-ray luminosity
we confirm that @xmath4 .
we discuss these findings in the context of pair production inside pulsar magnetospheres and the striped wind model .
we suggest that the striped wind model may explain the similarity between the x - ray properties of the pulsar wind nebulae and the pulsars themselves , which according to the striped wind model may both find their origin outside the light cylinder , in the pulsar wind zone . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
since its proposal , the random boolean network model [ 1 ] has successfully described in a qualitative way several important aspects of gene regulation and cell differentiation processes ( for references see [ 2 ] ) . the model is constructed by assigning to each of the genes its regulatory inputs from among the large number of genes present in the network . the model consists of @xmath0 binary variables , corresponding to the two states of gene expression ( off and on ) . in this binary setting , each gene is assigned a logical function on its inputs showing its next activity . while clearly an idealization , much has been learned from this class of large boolean networks , and major features generalize to a class of piecewise linear differential equations [ 3 - 4 ] and a family of polynomial maps [ 5 ] . the research on complex boolean networks , shows that networks behave in three regimes : ordered , critical and chaotic [ 6 ] . it is a very attractive hypothesis that cell types have evolved by natural selection to lie in the ordered regime , close to the critical phase transition , where the most complex coordinated behaviors can occur [ 7 - 8 ] . in this deterministic setting , it is almost an inevitable hypothesis that the distinct cell types of an organism correspond to the distinct attractors of the network . this hypothesis needs to be confirmed by experimental tests . cell differentiation consists in response to a perturbation or signal that places the network in a different basin of attraction from which it flows to a new attractor . obviously , real genetic networks are not boolean nets . in real nets , one has to take into account the molecular dynamics by using stochastic differential equation models . the resulted equations can be solved using monte - carlo methods . here the favored approach is the gillespie algorithm [ 9 ] , which can be used to model discrete molecular events of transcription , translation and gene control in complex reaction networks . boolean networks already impose some formidable computational problems . introducing stochastic models render even smaller networks computationally intractable because the number of reactions one has to consider grows exponentially fast with the number of genes in the network . here , we propose a simplified mean - field model of the genetic regulatory network , where the main idea is to replace all regulatory interactions to any one gene with an average or effective interaction , which takes into account the repression and activation mechanisms . our model attempts to bridge the gap between random boolean networks and more realistic stochastic modeling of regulatory networks . in this model , the regulatory interactions are described by differential equations corresponding to the chemical reactions considered in the genetic network . the same set of chemical reactions can , for example , be used in a stochastic simulation of the network . from this point of view , the proposed model gives a mean - field description of the more accurate stochastic approach . in the mean - field deterministic description , the gene - expression state at a given time and the regulatory interactions among them unambiguously determine the gene - expression state at the next time . in a stochastic system , on the other hand , a given gene - expression state can generate more than one successive gene - expression states , and therefore , different cells of the same population may follow a different gene - expression path . as a result of these considerations , the stochastic model describe the kinetics of gene regulation more accurately than a deterministic model . however , a deterministic mean - field model can be transformed in a stochastic model by incorporating noise . this approach results in a stochastic differential equation or langevin equation . it is well known that the langevin equation is asymptotically equivalent ( under certain conditions ) to the chemical master equation [ 10 ] . therefore , the proposed mean - field model is still relevant for the description of gene regulation and cell differentiation processes . we show that depending on the set of regulatory parameters , the model exhibits differing behaviors corresponding to ordered and chaotic dynamics . this result gives quantitative support to the earlier qualitative results obtained for random boolean networks . also , we show that the system acquires stability by increasing the number of interactions . this conclusion provides a possible explanation of how diversity and stability are created in a biological system , giving rise to a great variety of stable living organisms . for the beginning , let us analyze the gene expression process [ 11 - 14 ] . in a genetic regulatory network , genes can be turned on or off by the binding of proteins to regulatory sites on the dna [ 1 ] . the proteins are known as transcription factors , while the dna binding sites are known as promoters . transcription factors can regulate the production of other transcription factors , or they can regulate their own production . the simplest model of gene expression involves only two steps in which the genetic information is first transcribed into messenger rna ( @xmath1 ) and then translated into proteins ( @xmath2 ) by ribosomes ( @xmath3 ) . the transcription process can be described by a sequence of reactions , in which the rna polymerase ( @xmath4 ) binds to the gene promoter ( @xmath5 ) leading to transcription of a complete @xmath6 molecule : @xmath7 here , @xmath8 corresponds to the complex formed in the intermediate reaction @xmath9 , with constant rate @xmath10 . since the waiting times are independent statistical quantities , the waiting time for the whole sequence of intermediate complex formation is the sum of the waiting times for the individual steps . also , we should note that the central limit theorem [ 10 ] indicates that the lumped reaction of the open complex formation will tend to have a gaussian distribution of waiting times , converging to a @xmath11 function for a very large number of intermediate steps . thus , in terms of reaction rates ( which have units of inverse time ) we have @xmath12 . from the above considerations , it follows that the whole sequence of reactions can be approximated by the following reaction : @xmath13 where @xmath14 let us now analyze the translation process , in which the information initially transcribed into @xmath1 is now translated into proteins @xmath2 . to describe this we consider the following additional reactions : @xmath15 @xmath16 the first reaction idealizes the multistep translation process , under the further idealization that a ribosome ( @xmath3 ) binds the @xmath1 and a protein @xmath2 is produced . the second reaction captures the degradation of @xmath1 . the model described here is intentionally as simple as possible . we believe that such an approach is as important as detailed biological modeling in elucidating the basic physics behind genetic regulatory networks . the main idea of the model is to replace all regulatory interactions to any one gene with an average or effective interaction which takes into account the repression and activation mechanisms . let us now consider a system formed from @xmath0 genes . in this system protein multimers ( which are the transcription factors ) are responsible for gene regulation ( activation and repression ) and are allowed to bind to the promoter site . here , an important problem is to specify which reactions are allowed to take place in the system . we consider that the basic level of expression ( transcrition and translation ) , the @xmath1 and protein degradation reactions always exist , and they represent the minimum set of reactions describing the system . the existence of the other ( activation , repression and multimerization ) reactions is specified by an associated set of binary coefficients ( switches ) @xmath17 , @xmath18 , @xmath19 , @xmath20 , @xmath21 . if the value of such a coefficient is @xmath22 then the reaction exists , if the coefficient is @xmath23 then the reaction does not exist . the possible chemical reactions up to the dimer interaction case are @xmath24 : - transcription : @xmath25 - @xmath1 degradation @xmath26 - translation : @xmath27 - protein degradation : @xmath28 - protein dimerization : @xmath29 where the coefficients @xmath30 are the selection switches . here , a protein dimer is formed by combining two protein monomers . - repression : @xmath31 @xmath32 where the selection is done by setting @xmath33 , @xmath34 . in the first reaction , a protein monomer binds to the promoter and the effect is an inhibition of the basic level of transcription reaction , which actually leads to a gene repression . the existence of the second repression reaction is conditioned by the dimerization reaction between proteins . therefore , the real regulatory coefficient for this reaction is @xmath35 . - activation : @xmath36 @xmath37 in order to have an activation , a protein monomer or a dimer must first bind to the promoter and form another complex . therefore , the existence of an activation reaction is conditioned by the coefficients @xmath33 , @xmath38 and @xmath39 . thus , the activation reactions are in fact regulated by @xmath40 , @xmath41 , where @xmath42 , @xmath43 . we should note that the number of possible reactions is very high . this number depends on the considered length of possible multimers formed by the transcription factors . here we have considered only the reactions up to a possible dimer interaction . however , we will show that these reactions are enough for the purpose of illustrating the interaction mechanism in a more general model , corresponding to higher order multimer interactions . in a steady state the repression and dimerization reactions are in equilibrium and we have : @xmath44[m_{i } ] & = & \widetilde{k}_{i}^{n}[p_{n}m_{i } ] , \\ k_{ij}^{n}[p_{n}][m_{i}m_{j } ] & = & \widetilde{k}_{ij}^{n}[p_{n}m_{i}m_{j } ] , \nonumber \\ k_{ij}[m_{i}][m_{j } ] & = & \widetilde{k}_{ij}[m_{i}m_{j } ] . \nonumber\end{aligned}\ ] ] here by @xmath45 $ ] we understand the concentration of @xmath46 . one can see that each promoter @xmath47 can be in different states corresponding to the transcription factor complex which binds to it : @xmath48 from the above equations one can see that the probabilities associated to these states are in the ratio : @xmath49}{[p_{n } ] } & = & \frac{k_{i}^{n}}{\widetilde{k}_{i}^{n } } [ m_{i}]=a_{i}^{n}x_{i } , \\ \frac{\lbrack p_{n}m_{i}m_{j}]}{[p_{n } ] } & = & \frac{k_{ij}^{n}}{\widetilde{k } _ { ij}^{n}}[m_{i}m_{j}]=\frac{k_{ij}^{n}}{\widetilde{k}_{ij}^{n}}\frac{k_{ij } } { \widetilde{k}_{ij}}x_{i}x_{j } \nonumber \\ & = & ( b_{ij}^{n}+b_{ij}^{n})x_{i}x_{j}=2b_{ij}^{n}x_{i}x_{j}. \nonumber\end{aligned}\ ] ] where @xmath50 are the reduced concentrations variables for the transcription factors . the probability that the promoter @xmath47 is in a free state is given by : @xmath51 where the denominator is the sum over all possible states of the promoter including the free state and the binding states . consequently , the probabilities that the promoter is in a binding state @xmath52 or @xmath53 are : @xmath54 @xmath55 obviously , from the above equations we have : @xmath56 in a steady state , the rate of @xmath57 transcription should be equal to the rate of @xmath57 degradation and therefore we can write the following mean - field equation : @xmath58 where @xmath59 is the reduced concentration of @xmath57 . here , we have covered all the system construction possibilities by using the binary coefficients @xmath39 , @xmath33 , @xmath42 , @xmath60 , @xmath61 . thus , the transcription of gene @xmath62 , can be approximated using the following nonlinear differential equation : @xmath63 where @xmath64 , @xmath65 . also , @xmath66 is a parameter corresponding to the promoter strength ( the ratio between the transcription and the degradation rates ) . also , in a steady state , the rate of translation should be equal to the rate of protein degradation . therefore , for the gene @xmath62 , we can write the second equation : @xmath67=\widetilde{k}_{ii}^{n}[m_{n}],\ ] ] and consequently the second differential equation : @xmath68 where @xmath69 is a coefficient measuring the delay induced by the translation process . let us give some simple examples to the above equations . first , let us consider the case of two genes , where all the monomer interactions are excluded and the interaction is mediated only by dimers ( @xmath70 ) . we consider that the interaction at the dimer level is performed only by homo - dimers ( @xmath71 . also , the interaction is due only to the repression mechanism ( @xmath72 ) , and the repression is performed only by the other gene s homo - dimers ( @xmath73 ) . with these approximations we obtain the well known toggle switch model [ 15 ] : @xmath74 now , let us consider the case in which three genes are repressing each other at a dimer level in a circular way : @xmath75 this means that : @xmath76 ; @xmath77 ( @xmath78 ) ; @xmath72 ; @xmath79 ( @xmath78 and @xmath80 ) . with these assumptions we obtain the well known equations of the repressilator [ 16 ] : @xmath81 the analysis of the repressilator model has shown that in order to obtain sustained oscillations we need strong promoters , which is equivalent to a high rate of expression or a low rate of degradation [ 16 ] . from the above considerations we observe that the multimer interactions can be represented using a tensorial expansion . the coefficients describing the interaction can be condensed in tensors of different ranks , corresponding to the length of considered multimers . thus , in general for a network with @xmath0 genes we can write the following set of equations @xmath82 : @xmath83 one can see that the numerator of the above fraction contains all the activation interactions , while the denominator contains all the repression interactions . therefore , this intrinsic ratio , between activation and repression , defines the dynamics of the gene network . in order to simplify further our description we consider that the transcription and translation processes can be condensed in only one reaction : @xmath84 this means that we are neglecting the delay effects introduced by the translation process , which is equivalent to consider @xmath85 . therefore the dynamics of the system will be described only by the following simplified equations : @xmath86 let us consider the mean - field equations up to the dimer level . obviously , these equations contain an extremely large number of unknown parameters . for example , very few rate constants for the constituent reactions have been measured in cells and one is often ignorant of absolute concentrations of the participating molecular species . meanwhile , the ensemble approach [ 17 ] , which consists in sampling random networks from an ensemble of networks built according to the constraints we know that characterize real genomic systems and then analyzing the typical , or generic , properties of ensemble members , remains one useful approach to making use of the information we are gathering on real systems to understand their large scale dynamical and network connectivity implications . we would like to emphasize that the mean - field model described here has the advantage that all the parameters correspond only to ratios of reaction constant rates , and therefore it does not require the knowledge of absolute values of these constant rates . this characteristic of the model gives us the chance to simplify even more the sampling of the parameters . thus , in the spirit of the ensemble approach let us consider that the coefficients @xmath87 , @xmath88 , @xmath89 , @xmath90 , @xmath91 , @xmath92 are drawn from a gaussian distribution as following : @xmath93 here , @xmath94 is a random variable governed by a gaussian distribution with zero mean and variance equal to one , and @xmath95 . by using this sampling procedure , we assume implicitly that the ratios of reaction constant rates are normally distributed around some average values @xmath96 , @xmath97 , @xmath98 , @xmath99 , @xmath100 . also , high values of the parameters controlling the variance ( @xmath101 ) are useful in creating a large spectrum of values around the average values . for example , because @xmath64 are generated from a gaussian distribution centered at @xmath98 it follows that : @xmath102 or @xmath103 . this means that for @xmath104 the rate of activation reactions can be higher or lower than the rate corresponding to the basic level of expression . in our simulations we have used the following parameters : @xmath105 , @xmath106 , @xmath107 . also , the initial conditions @xmath108 are drawn from an uniform distribution between @xmath23 and @xmath109 . obviously , one can try a different scenario in which for example all these parameters are drawn from an uniform distribution . let us now consider the extreme case when all the dimers formed by the transcription factors are missing from the system : @xmath110 because the number of parameters in the system is still very large one can imagine a huge number of numerical experiments . below we describe a couple of such possible numerical experiments . in the first experiment , the coefficients regulating the monomer interactions @xmath111 are generated randomly such that : @xmath112 the numerical results have shown that for any values of @xmath113 and @xmath114 the system converges only to steady states . in the second experiment , the values of the coefficients regulating the monomer interactions @xmath111 are generated such that the sums : @xmath115 follow a power law distribution : @xmath116 where @xmath117 is the riemann zeta function . recent analysis indicates that a power law distribution of interactions seems to fit better the data observed experimentally for several organisms [ 18 - 20 ] . such a distribution can be obtained from an uniform distribution using the inverse transformation method : @xmath118^ { \frac{1}{1-\omega } } .\ ] ] here , @xmath119 is an uniform distributed random number on the interval @xmath120 $ ] . the exponent of the power law distribution is @xmath121 ( where @xmath122 is the pareto distribution shape parameter ) . the above equation returns a value @xmath123 $ ] , distributed accordingly to a power law with the exponent @xmath124 . in our simulation we have set @xmath125 , @xmath126 and the variable parameter is @xmath124 . the rest of the parameters and the initial states are set as before . the numerical results have shown that for any values @xmath127 the system converges to steady states . these numerical results suggest that protein multimerization might be a necessary condition to generate more complex dynamics in the system . by increasing the number of dimers formed by the transcription factors : @xmath128 one can easily see how the complex behavior emerges in the system . depending on the values of the other regulatory parameters , the model exhibits complex oscillatory and chaotic dynamics . let us now consider the other extreme case when all the dimers formed by the transcription factors are present in the system : @xmath129 the coefficients @xmath111 , @xmath130 , regulating the monomer interactions , are generated randomly as before , with the probabilities @xmath113 and @xmath114 . the other coefficients are determined as following : @xmath131 the rest of the coefficients and the initial conditions are set as before in the numerical experiment for @xmath132 . the numerical results , have shown that for any values of @xmath113 , @xmath114 , @xmath133 , @xmath134 the system mostly converges to steady states similar to those obtained for @xmath132 , however oscillatory solutions have also been observed . in the next numerical experiment we consider that the values of the coefficients regulating the monomer and dimer interactions are generated such that the sums : @xmath135 follows a power law distribution . for @xmath136 and @xmath137 we set @xmath126 , while for @xmath138 and @xmath139 we have @xmath140 . the rest of the coefficients and the initial conditions are set as before . for small values of @xmath141 the system converges mostly to steady states . an example of steady state obtained for @xmath142 and @xmath143 is given in fig . for average values @xmath144 the density of steady states in the solution space decreases , making room for oscillatory solutions . in fig . 1(b ) we give an example of oscillatory solution obtained for @xmath145 . for larger values @xmath146 one can easily obtain chaotic solutions . in fig . 1(c ) we give an example of chaotic solution obtained for @xmath147 . the dynamic behavior of solutions was characterized by performing fourier analysis on long trajectories . the fourier power spectrum is discrete for an oscillatory solution , while in the case of a chaotic solution it shows continuous intervals of frequencies . in fig . 2 we give the typical fourier power spectrum obtained for a single @xmath148 trajectory : ( a ) oscillations ; ( b ) chaos . the above result suggests that for power law distributed interactions ( up to the dimer level ) , there is a transition between order and chaos when the power law exponent @xmath124 increases . in order to characterize this transition one can try to calculate the largest lyapunov exponent for the dynamical system [ 21 ] . a positive value of the largest lyapunov exponent indicates the chaotic behavior of the system , while a negative value indicates a regular dynamics . we have performed several calculations for networks with a modest size ( @xmath149 ) just to confirm the chaotic behavior of the solutions . unfortunately , the computation of the lyapunov exponent is quite intensive and it quickly becomes prohibitive for networks with a very large number of genes , sampled from a huge ensemble of networks ( as described above ) . therefore , in order to analyze this transition we consider a simplified approach , which focuses on steady state solutions , by calculating the average fluctuations : @xmath150 here , @xmath151 is the length of the trajectory ( the number of time steps , after the transient is eliminated ) and @xmath152 is the time average of @xmath148 . the global quantity @xmath153 measures the fluctuations around the average values @xmath154 . obviously , @xmath155 can not distinguish between chaotic and oscillatory trajectories , however it provides a simple and efficient discrimination between ordinary steady state solutions ( fig . 1(a ) ) and the oscillatory and chaotic solutions ( fig . 1(b ) , ( c ) ) . for ordinary steady state solutions @xmath156 , while for oscillatory and chaotic solutions @xmath157 . in fig . we give the results obtained for @xmath158 by averaging over 100 solutions for @xmath159 $ ] . the parameter @xmath153 measures the transition between a phase with high density of ordinary steady state solutions and a phase with low density of steady state solutions . one can see that the transition occurs around @xmath160 . also , one can see that @xmath161 does not seems to depend on the number of genes in the network @xmath162 , @xmath163 , @xmath164 , @xmath165 , @xmath166 . therefore , the average number of interactions per gene at the critical point in a large network ( @xmath167 ) can be easily estimated as : @xmath168 obviously , by increasing @xmath124 the number of interactions per gene decreases . for large values of @xmath124 the number of interactions approaches @xmath169 . these results suggest that the network is more stable for low values of @xmath124 , i.e. when the number of interactions per gene is higher than @xmath170 , and it looses stability when @xmath124 increases , i.e. when the number of interactions is low . we have observed this phenomenon for various values of the system parameters , which suggests that it is an important characteristic of the system . for @xmath172 the numerical simulation becomes difficult because of the higher order combinatorial explosion in the definition of tensor coefficients . the number of coefficients for a tensor with rank @xmath2 grows exponentially fast as @xmath173 , where @xmath0 is the number of genes in the network . therefore , further simplifications are required . here , we consider a simplified system of @xmath0 genes which has a high level of complexity , comparable with the general mean - field system of equations . this simplified system has the advantage that it can make the simulations possible even for very large values of @xmath2 and @xmath0 . we make a further simplification by assuming that @xmath174 this means that the interaction strength at a given multimer level depends slightly on the corresponding tensor rank . with these assumptions we obtain the following set of equations ( @xmath175 ) : @xmath176 now let us assume that all the higher order multimer interactions ( corresponding to a tensor rank @xmath177 ) are generated from the first order monomer interactions in the following recursive way : -repression : @xmath178 -activation : @xmath179 for example , the dimer interaction terms are given by : -repression : @xmath180 -activation : @xmath181 thus , we obtain the following simplified system of differential equations @xmath182 : @xmath183 \left ( \frac{a_{n}\sum\limits_{i}\beta _ { i}^{n}x_{i}-1}{c_{n}a_{n}\sum \limits_{i}\gamma _ { i}^{n}\beta _ { i}^{n}x_{i}-1}\right ) -x_{n}.\ ] ] the advantage of this model consists in the fact that at each iteration step , in the algorithm used to solve the system of differential equations , one has to calculate only the first rank interaction tensors , @xmath184 and @xmath185 , even though the expansion is considered up to the maximum rank @xmath2 . we have performed a numerical experiment in which all the parameters and the initial states are set as before for @xmath186 . also , the coefficients regulating the monomer interactions @xmath187 , @xmath130 are generated using the probabilities @xmath113 , @xmath114 , as in the first numerical experiment , performed for @xmath132 . thus , the variable parameters are @xmath113 , @xmath114 and the maximum rank @xmath2 of tensorial expansion ( which corresponds to the maximum length of multimers mediating the interaction among genes ) . depending on the values of all these parameters , the system exhibits different types of behavior . however , the most important parameter seems to be @xmath2 . for low values @xmath188 the density of steady states in the solution space is very high and therefore the system converges most of the time to a steady state ( fig . 4 ( a ) ) . by increasing @xmath2 to @xmath189 the density of steady states decreases and the typical behavior becomes oscillatory ( fig . 4 ( b ) ) . for larger values of @xmath190 , the dynamics becomes more complex , exhibiting chaotic behavior ( fig . 4 ( c ) ) . the numerical results for this experiment show that there is a smooth transition between order and chaos as the parameter @xmath2 increases . this is a consequence of the fact that by increasing @xmath2 one actually increases the cooperativity among the transcription factors and implicitly the nonlinearity of the system . we have discussed a mean - field model of genetic regulatory networks , in which the regulatory interactions are described by differential equations corresponding to the chemical equations considered in the network . we have shown that , depending on the set of regulatory parameters , the model exhibits differing behaviors corresponding to ordered and chaotic dynamics . this result gives some quantitative support to the earlier qualitative results obtained for random boolean networks [ 2 - 6 ] . however , contrary to boolean networks , by increasing the number of interactions per gene , our model acquires stability . this is an important issue which we would like to address here and in the future . we believe that the stability of the system occurs from the intrinsic construction of the activation / repression ratio . this ratio corresponds to an average interaction , which takes into account all the repression and activation mechanisms acting on any one gene in the network . also , according to the central limit theorem , the variance of the sums , defining the numerator and denominator of this ratio , decreases by considering more terms ( interactions ) . these mechanisms create stability in the system by producing a contraction which keeps the solution bounded . so , by increasing the number of interactions , the system becomes more stable . for specific sets of parameters the solution is not only bounded but it also corresponds to ordered dynamics ( steady states or oscillations ) . by changing the set of parameters , this contraction becomes loose enough that the solution becomes chaotic . in this case the system becomes ergodic and it is able to explore large regions in the solution space . an important role in generating oscillatory and chaotic dynamics is played by the length of transcription factor multimers mediating the interaction among genes . our analysis has shown that protein multimerization is a necessary condition for the discussed mean - field model to generate oscillatory and chaotic dynamics . in summary , we have introduced and provided an initial analysis of a mean - field model for a class of reasonably realistic chemical equations modeling genetic regulatory networks . we presume a critical phase transition occurs as the system goes from order to chaos . this is important because recent evidence tentatively suggests that yeast cells are critical [ 23 ] . indeed , it has been a long standing hypothesis that cells are critical or slightly subcritical to withstand noise [ 24 ] . thus , our results give preliminary support to this hypothesis . it remains for future work to explore in more detail how networks with diverse topologies but similar kinetics behave . if it proves true that biological cells are critical or near critical , our model should be of use in exploring the combinations of network topologies , motifs and kinetic rules that can correspond to such critical behavior . this research was supported by icore under grant no . rt732223 . m. aldana , s. coopersmith , l. p. kadanoff , _ perspectives and problems in nonlinear science . springer applied mathematical sciences series . _ ehud kaplan , jerrold e. marsden , and katepalli r. sreenivasan eds . , 23 - 89 ( 2003 ) . example of typical solutions obtained for m=2 : ( a ) steady state ; ( b ) oscillations ; ( c ) chaos . here , @xmath148 is the reduced concentration of protein monomer ( transcription factor ) @xmath62 as a function of time @xmath191 . the total number of genes in the network is @xmath143.,width=453 ] example of typical solutions obtained for @xmath172 : ( a ) steady states ; ( b ) oscillations ; ( c ) chaos . here , @xmath148 is the reduced concentration of protein monomer ( transcription factor ) @xmath62 as a function of time @xmath191 . the total number of genes in the network is @xmath143 . the parameter @xmath2 is the maximum length of multimers mediating the interaction among genes.,width=453 ] | in this paper , we propose a mean - field model which attempts to bridge the gap between random boolean networks and more realistic stochastic modeling of genetic regulatory networks .
the main idea of the model is to replace all regulatory interactions to any one gene with an average or effective interaction , which takes into account the repression and activation mechanisms .
we find that depending on the set of regulatory parameters , the model exhibits rich nonlinear dynamics .
the model also provides quantitative support to the earlier qualitative results obtained for random boolean networks . *
pacs : * 05.45.-a ; 87.16.yc institute for biocomplexity and informatics university of calgary 2500 university drive nw , calgary alberta , t2n 1n4 , canada |
You are an expert at summarizing long articles. Proceed to summarize the following text:
some time ago monopoles in einstein - yang - mills - higgs(eymh ) model , for @xmath2 gauge group with higgs field in adjoint representation , were studied as a generalization of the t hooft - ployakov monopole to see the effect of gravity on it . in particular , it was found that solutions exist up to some critical value of a dimensionless parameter @xmath1 , characterising the strength of the gravitational interaction , above which there is no regular solution . the existance of these solutions were also proved analytically for the case of infinite higgs mass . also , non abelian magnetically charged black hole solutions were shown to exist in this model for both finite as well as infinite value of the coupling constant for higgs field . the abelian black holes exists for @xmath3 and non abelian black holes exist in a limited region of the @xmath4 plane . recently born - infeld theory has received wide publicity , especially in the context of string theory . bogomolnyi - prasad - sommerfield ( bps ) saturated solutions were obtained in abelian higgs model as well as in @xmath5 sigma model in @xmath6 dimensions in presence of born - infeld term . different models for domain wall , vortex and monopole solutions , containing the born - infeld lagrangian were constructed in such a way that the self - dual equations are identical with the corresponding yang - mills - higgs model . recently non self - dual monopole solutions were found numerically in non abelian born - infeld - higgs theory . in this paper we consider the einstein - born - infeld - higgs(ebih ) model and study the monopole and black hole solutions . the solutions are qualitatively similar to those of eymh model . the black hole configurations have nonzero non abelian field strength and hence they are called non abelian black holes . in sec . ii we consider the model and find the equations of motion for static spherically symmetric fields . in sec iii we find the asymptotic behaviours and discuss the numerical results . finally we conclude the results in sec . we consider the following einstein - born - infeld - higgs action for @xmath2 fields with the higgs field in the adjoint representation s = d^4x with l_g & = & , + l_h & = & - d_^a d^^a -(^a^a - v^2 ) ^2 and the non abelian born - infeld lagrangian , l_bi = ^2 str ( 1 - ) where d_^a = _ ^a + e ^abc a_^b^c , f _ = f_^a t^a = ( _ a_^a - _ a_^a + e ^abca_^ba_^c)t^a and the symmetric trace is defined as str(t_1,t_2 ... ,t_n ) = tr(t_i_1t_i_2 ... t_i_n ) . here the sum is over all permutations on the product of the @xmath7 generators @xmath8 . here we are interested in purely magnetic configurations , hence we have @xmath9 . expanding the square root in powers of @xmath10 and keeping up to order @xmath10 we have the born - infeld lagrangian l_bi = -f_^a f^a + + o ( ) . for static spherical symmetric solutions , the metric can be parametrized as ds^2 = -e ^2(r)dt^2 + e ^2(r)dr^2 + r^2(r)(d^2 + ^2d^2 ) and we consider the following ansatz for the gauge and scalar fields a_t^a(r ) = 0 = a_r^a , a_^a = e_^a , a_^a = -e_^a , and ^a = e_r^a v h(r ) . putting the above ansatz in eq.1 , defining @xmath11 and rescaling @xmath12 and @xmath13 we get the following expression for the lagrangian dr e^+ , where v_1 = ( w)^2 + r^2(h)^2 - ( w)^2 , v_2 = and v_3 = + w^2h^2 + ( h^2 - 1)^2 - . here the prime denotes differentiation with respect to @xmath14 . the dimensionless parameter @xmath1 can be expressed as the mass ratio = with the gauge field mass @xmath15 and the planck mass @xmath16 . note that the higgs mass @xmath17 . in the limit of @xmath18 the above action reduces to that of the einstein - yang - mills - higgs model . for the case of @xmath19 we must have @xmath20 which corresponds to the flat space born - infeld - higgs theory . we now consider the gauge @xmath21 , corresponding to the schwarzschild - like coordinates and rename @xmath22 . we define @xmath23 and @xmath24 . varying the matter field lagrangian with respect to the metric we find the energy - momentum tensor . integrating the @xmath25 component of the energy - momentum we get the mass of the monopole equal to @xmath26 where m = ^2 _ 0^ dr ( nv_1 - n^2v_2 + v_3 ) following t hooft the electromagnetic @xmath27 field strength @xmath28 can be defined as _ = - ^abc^ad_^bd_^c . then using the ansatz(3 ) the magnetic field b^i = ^ijkf_jk is equal to @xmath29 with a total flux @xmath30 and unit magnetic charge . the @xmath25 and @xmath31 components of einstein s equations are & & ( 1 - ( rn) ) = ^2 ( n v_1 - n^2 v_2 + v_3 ) + & & = ( v_1 - 2nv_2 ) . the equations for the matter fields are & & ( anv_4) = a w ( ( w^2 - 1 ) + 2 h^2 - - ( w^2 - 1 ) ) + & & ( anr^2h) = a h ( 2w^2 + g^2r^2(h^2 - 1 ) ) with v_4 = 2w - ( w^2 - 1)^2 - ( w)^3 it is easy to see that @xmath32 can be elliminated from the matter field equations using eq.(12 ) . hence we have to solve three differential equations eqs . ( 11),(13 ) and ( 14 ) for the three fields @xmath33 and @xmath34 . for finite @xmath35 , demanding the solutions to be regular and the monopole mass to be finite gives the following behaviour near the origin & & h = a r + o(r^3 ) , + & & w = 1 - b r^2 + o(r^4 ) , + & & n = 1 - c r^2 + o(r^4 ) , where @xmath36 and @xmath37 are free parameters and @xmath38 is given by c = ^2 ( a^2 + 4b^2 + - ) . in general , with these initial conditions @xmath39 can be zero at some finite @xmath40 where the solutions become singular . in order to avoid this singularity we have to adjust the parameters @xmath36 and @xmath37 suitably . for @xmath41 we require the solutions to be asymptotically flat . hence we impose n = 1 - then for finite mass configuration we have the following expressions for the gauge and the higgs fields & & w = c r^-m e^-r(1 + o ( ) ) + & & h = \ { ll 1 - b r^-gm - 1 e^-gr , & for 0 < g + 1 - r^-2m-2 e^-2r , & for g = 0 and g > . . note that the fields have similar kind of asymptotic behaviour in the eymh model . we have solved the equations of motion numerically with the boundary conditions given by eqs.(16 - 21 ) . for @xmath42 , @xmath43 and @xmath18 they corresponds to the exact prasad - sommerfield solution . for nonzero @xmath44 and finite @xmath45 the qualitative behaviour of the solutions are similar to the corresponding solutions of eymh model . for large @xmath40 these solutions converges to their asymptotic values given as in eqs.(19 - 21 ) . for a fixed value of @xmath35 and @xmath45 we solved the equations increasing the value of @xmath1 . for small value of @xmath1 the solutions are very close to flat space solution . as @xmath1 is increased the minimum of the metric function @xmath39 was found to be decreasing . the solutions cease to exist for @xmath1 greater then certain critical value @xmath46 . for @xmath43 and @xmath47 we find @xmath48 . the profile for the fields for different values of @xmath1 with @xmath49 and @xmath50 are given in figs.1,2 and 3 . the profile for the fields for @xmath51 and @xmath52 are given in fig . we find numerically the mass @xmath53 of the monopole for @xmath54 and @xmath50 . apart from the regular monopoles , magnetically charged black holes can also exist in this model . black hole arises when the field @xmath55 vanishes for some finite @xmath56 . demanding the solutions to be regular near horizon @xmath57 we find the following behaviour of the fields & & n(r_h + ) = n_h+ o(^2 ) , + & & h(r_h + ) = h_h + h_h+ o(^2 ) , + & & w(r_h + ) = w_h + w_h+ o(^2 ) with & & n_h = + & & h_h = \ { 2w_h^2 + g^2 r_h^2 ( h_h^2 - 1 ) } + & & w_h = . here @xmath58 and @xmath59 are arbitrary . for @xmath41 the behaviour of the fields is same as the regular monopole solution as given by eqs.(19 - 21 ) . the black hole has unit magnetic charge with nontrivial gauge field strength . we found numerical solutions to the non abelian black hole for different @xmath57 . for a fixed value of @xmath60 we find the solutions for @xmath61 adjusting the parameters @xmath62 and @xmath63 . for @xmath60 close to zero the solutions approach the regular monopole solutions . the profile for the fields are given in fig.5 . we found the mass of the black hole equals to be @xmath64 for @xmath65 and @xmath50 . in this paper we have investigated the effect of gravity on the born - infeld - higgs monopole . we found that solutions exist only up to some critical value @xmath46 of the parameter @xmath1 . in the limit @xmath18 these solutions reduces to those of eymh monopoles . we also found numerically magnetically charged non abelian black hole solutions in this model . it would be interesting to prove analytically the existence of these solutions for finite value of the parameters . recently dyons and dyonic black holes were found in eymh model numerically and the existence of critical value for @xmath1 was also proved analytically . it may be possible to generalize these solutions to find dyons and dyonic black holes in ebih model . we hope to report on this issue in future . i am indebted to avinash khare for many helpful discussions as well as for a careful manuscript reading . 50 p. breitenlohner , p. forgace and d. maison , nucl . phys . b 383 ( 1992 ) 357 . m. e. ortiz , phys . d 45 ( 1992 ) r2586 . p. breitenlohner , p. forgace and d. maison , nucl . b 442 ( 1995 ) 126 . g. t hooft , nucl . b 79 ( 1974 ) 276 ; a. m. polyakov , jetp lett . 20 ( 1974 ) 194 . k. lee , v. p. nair and e. j. weinberg , phys . d 45 ( 1992 ) 2751 p. c. aichelburg and p. bizon phys . rev . d 48 ( 1993 ) 607 . m. born , proc . london a 143 ( 1934 ) 410 . m. born and m. infeld , proc . london a 144 ( 1934 ) 425 . e. s. fradkin and a. a. tseytlin , phys . b 163 ( 1985 ) 123 ; a. a. tseytlin , nucl . phys . b 276 , ( 1986 ) 391 ; r. g. leigh , mod . phys . lett . a 4 ( 1989 ) 2073 ; a 4 , ( 1989 ) 2767 . p. k. tripathy , phys . d 59 ( 1999 ) 085004 ; s. gonorazky , c. nunez , f. a. schaposnik and g. silva , nucl . phys . b 531 ( 1998 ) 2881 . a. nakamura and k. shiraishi , int phys . a 6 ( 1991 ) 2635 ; hadronic j. 14 ( 1991 ) ; g. gibbons , nucl . b 514 ( 1998 ) 603 . n. grandi , e. f. moreno and f. a. schaposnik , hep - th/9901073 . p. bizon , phys . 64 ( 1990 ) 2844 . s. gonorazky , f. a. schaposnik and g. silva , phys . b 449 ( 1999 ) 187 ; d. brecher , phys . lett . b 442 ( 1998 ) 117 ; a. a. tseytlin , nucl . phys . b 501 ( 1997 ) 41 . p. g. bergmann , m. cahen and a. b. komar , j. math . ( 1965 ) 1 . m. k. prasad and c. m. sommerfield , phys 35 ( 1975 ) 760 . y. brihaye , b. hartmann and j. kunz , phys . b 441 ( 1998 ) 77 . y. brihaye , b. hartmann , j. kunz and n. tell , hep - th/9904065 . | we find static spherically symmetric monopoles in einstein - born - infeld - higgs model in @xmath0 dimensions .
the solutions exist only when a parameter @xmath1 ( related to the strength of gravitational interaction ) does not exceed certain critical value .
we also discuss magnetically charged non abelian black holes in this model .
we analyse these solutions numerically . 9.0 in 6.0 in |
You are an expert at summarizing long articles. Proceed to summarize the following text:
the main object of this paper is a stochastic differential equation @xmath0 , \end{gathered}\ ] ] where @xmath1 is a standard wiener process ; @xmath2 is a fractional brownian motion ( fbm ) with the hurst parameter @xmath3 ; @xmath4 is a poisson measure with finite intensity measure . such equation can be used to model processes with jumps on financial markets , where two principal random noises influence the prices . one noise is coming from economical background and has a long - range dependence , which is modeled by the fbm . another noise is intrinsic to the stock exchange , where millions of agents act independently and behave irrationally sometimes ; this is a white noise and it is modeled by a wiener process . although equation were not studied before , many authors considered some particular examples . for a pure fractional stochastic equation without wiener component and jumps @xmath5 existence and uniqueness of a solution to such equation was proved first in @xcite . in @xcite this result was proved under weaker assumptions on coefficients and integrability of the solution was established for @xmath6 . for a homogeneous drift - less equation ( i.e. @xmath7 , @xmath8 ) , the integrability was shown in @xcite for all @xmath9 . mixed stochastic differential equation without jumps @xmath10 was first considered in @xcite , where unique solvability was proved for time - independent coefficients and zero drift , i.e. @xmath11 . later , in @xcite , existence of solution to was proved under less restrictive assumptions , but only locally , i.e. up to a random time . in @xcite global existence and uniqueness of solution to was established under the assumption that @xmath1 and @xmath2 are independent . the latter result was obtained in @xcite and @xcite without the independence assumptions , and it was also shown in @xcite that all moments of the solution are finite for @xmath6 . it is also worth mentioning that article @xcite contains related results , which imply , in particular , that has unique solution for @xmath12 . recently , equation without brownian component ( i.e. @xmath13 ) was considered in @xcite , where the existence of solution is proved under weaker conditions on regularity of @xmath14 ( only hlder continuity in the second variable ) and without the assumptions @xmath15 , @xmath9 . on the other hand , in that paper @xmath16 and @xmath17 is independent of @xmath18 , which are much stronger assumptions than those of the present paper . in this paper we show that has a unique solution . the main result is existence of moments of the solution , which is an important property for applications . the existence of moments is proved for all @xmath9 in inhomogeneous case , which is a novelty in comparison to the results of @xcite . the paper is organized as follows . section [ sec : prelim ] gives basic definitions . in section [ sec : exuniq ] we prove existence and uniqueness of the solution to . section [ sec : exmom ] contains results on integrability of solutions to . let @xmath19 be a complete filtered probability space satisfying the usual assumptions . let also @xmath20 be an @xmath21-wiener process and @xmath22 be an @xmath23-adapted fractional brownian motion ( fbm ) , i.e. a centered gaussian process with the covariance @xmath24}=\frac{1}{2}(s^{2h}+t^{2h}-|t - s|^{2h})$ ] . let also @xmath4 be an @xmath23-adapted poisson measure with intensity measure @xmath25 , i.e. @xmath26}=\pi(dy)dt.\ ] ] we will assume that the intensity measure @xmath25 is finite : @xmath27 it is well known that @xmath2 has a modification with almost surely continuous paths ( even hlder continuous of any order up to @xmath28 ) , and further we will assume that it is continuous itself . now we define how we understand the integrals in . the integral with respect to the wiener process @xmath1 is the standard it integral , and it is well defined as long as @xmath29 almost surely . the integral with respect to @xmath4 is defined as usual . since @xmath15 , the process @xmath30 is well defined and it is a pure jump process , which almost surely has finite number of jumps ; we can assume that it is cadlag . so the integral with respect to @xmath4 is just a finite sum @xmath31 } q(s , x_{s-},\delta l_s),\ ] ] where @xmath32 . as for the integral with respect to the fbm @xmath2 , we use the generalized lebesgue stieltjes integral ( see @xcite ) . its construction uses the fractional derivatives @xmath33 assuming that @xmath34 , \ d_{b-}^{1-\alpha}g_{b-}\in l_\infty[a , b]$ ] , where @xmath35 , the generalized ( fractional ) lebesgue - stieltjes integral @xmath36 is defined as @xmath37 it follows from hlder continuity of @xmath2 that @xmath38 $ ] a.s . then for a function @xmath39 with @xmath40 $ ] we can define integral with respect to @xmath2 through : @xmath41 note that in the case where @xmath39 is piecewise hlder continuous with exponent @xmath42 , this integral is just a limit of forward integral sums @xmath43 . ( this fact is proved in @xcite for hlder continuous functions @xmath39 , but it is easily checked that the proof works for piecewise hlder continuous functions as well . ) hence , for such functions , all usual properties of integral hold : linearity , additivity etc . throughout the paper , the symbol @xmath44 will denote a generic constant , whose value is not significant and can change from one line to another . to emphasize its dependence on some parameters , we will put them into subscripts . in this and the following sections we impose the following assumptions on the coefficients of : 1 . the function @xmath45 is differentiable in @xmath18 and for all @xmath46 , @xmath47 $ ] latexmath:[\[\begin{gathered } 2 . the functions @xmath14 , @xmath49 and @xmath50 are lipschitz continuous in @xmath18 : @xmath51 for all @xmath52 , @xmath47 $ ] . the functions @xmath14 , @xmath49 and @xmath50 are hlder continuous in @xmath53 : for some @xmath54 and for all @xmath55 $ ] , @xmath56 @xmath57 we do not impose any assumptions on the function @xmath58 except joint measurability in all arguments . we will say that a process @xmath59 is a solution to if it is cadlag and has the following properties : * for some @xmath60 @xmath61 for all @xmath62 such that @xmath63,{{\mathbb r } } ) = 0 $ ] ( i.e. @xmath59 is hlder continuous between jumps of the process @xmath64 defined in ) ; * equation holds almost surely for all @xmath47 $ ] . from it follows that , almost surely , @xmath64 has finitely many jumps on @xmath65 $ ] , so a solution @xmath59 is piecewise hlder continuous of order @xmath60 , consequently , the integral @xmath66 is well defined for all @xmath47 $ ] . it also follows that @xmath67 a.s . , thus the integrals @xmath68 and @xmath69 are well defined too . [ thm_ex_sol ] equation has a unique solution . for @xmath70 , consider the following equation : @xmath71 where @xmath72 is a wiener process , @xmath73 is a process with almost surely hlder continuous paths of order @xmath74 . it was proved in @xcite that such equation has a unique solution in the class of hlder continuous processes of order @xmath75 . we denote this solution by @xmath76 . let @xmath77 be the moment of the @xmath78th jump of process @xmath64 . define a sequence of processes @xmath79 recursively as follows . let the process @xmath80 be the solution to . if for @xmath81 the process @xmath82 is constructed , set @xmath83 , @xmath84 , @xmath85 , @xmath86 . on the stochastic basis @xmath87 with @xmath88 , @xmath89\}$ ] , the process @xmath90 is a wiener process , and @xmath91 is hlder continuous of any order @xmath92 , hence we can define @xmath93 , @xmath94 . now we put @xmath95 and show that this process solves . indeed , for @xmath96 , @xmath97 , we have @xmath98 thus , for any @xmath99 we can write @xmath100 i.e. @xmath59 solves . uniqueness follows from a similar reasoning : from uniqueness of solution for and the strong markov property of @xmath1 we get , that for @xmath96 , @xmath97 , the solution of satisfies @xmath101 . on the other hand , @xmath102 , hence any solution of coincides with the one constructed above . we start by making pathwise estimates of the solution to equation without jumps . we fix some @xmath103 and introduce the following notation : @xmath104 } = \sup_{s\le u < v < t } \left(\frac{{\left\vertf(v)-f(u)\right\vert}}{(v - u)^{1-\alpha } } + \int_u^v \frac{{\left\vertf(u)-f(z)\right\vert}}{(z - u)^{2-\alpha}}dz\right).\end{gathered}\ ] ] observe that it follows from that @xmath105}\\\times \int_a^b \left({\left\vertf(s)\right\vert}(s - a)^{-\alpha } + \int_a^s { \left\vertf(s)-f(u)\right\vert}(s - u)^{-1-\alpha}du \right)ds . \end{gathered}\ ] ] [ pathwiseestimatemixed ] for the solution @xmath59 of , the following estimate holds : @xmath106}^{1/(1-\alpha)}\right\}}\left(1+{\left\lverti_b\right\rvert}_{\infty;t}\right),\ ] ] where @xmath107 , @xmath47 $ ] . a similar estimate ( naturally , without @xmath108 ) was obtained for the pure fractional equation in @xcite , but with exponent @xmath109 instead of @xmath110 here . in our proof we will use methods similar to those of @xcite , but we modify them as follows . while in @xcite , the sum @xmath111 is estimated and a version of the gronwall lemma is used , here we will estimate these terms separately and then use a kind of two - dimensional gronwall lemma . we start by estimating @xmath115 : @xmath116 where @xmath117 is as above , @xmath118 , @xmath119 . estimate @xmath120 , @xmath121 by , @xmath122 summing up , we have @xmath123 whence @xmath124 where we have used the estimate @xmath125 further , we estimate @xmath126 : @xmath127 where @xmath128 here to estimate @xmath129 we used the following computation : @xmath130 combining all estimates , we get @xmath131 whence @xmath132 thus , we get the following system of inequalities : @xmath133 with some constant @xmath134 ( which can be assumed to be greater than 1 without loss of generality ) . putting @xmath135 , we get from the first inequality that @xmath136 we remark that @xmath137 and @xmath138 are almost surely finite by the results of @xcite . plugging this to the second inequality and making simple transformations , we arrive at @xmath139 with some constant , which is no longer of interest . substituting this to , we get @xmath140 finally , @xmath141}^{1/(1-\alpha)}\right)\right\}}(1+j_b)\\\le c \exp{\left\{c{\left\lvertb^h\right\rvert}_{0;[0,t]}^{1/(1-\alpha)}\right\}}(1+j_b),\end{gathered}\ ] ] as required . thanks to lemma [ pathwiseestimatemixed ] , it is enough to prove that all moments of @xmath146}\right\}}$ ] and of @xmath108 are finite . the first follows from the fact that @xmath147}$ ] is an almost surely finite supremum of a gaussian family , and @xmath148 , since @xmath149.the first is proved as in ( * ? ? ? * lemma 2.3 ) , but the proof is short and for completeness we repeat it here . denote @xmath150 and write @xmath151}\le c_p(i_b ' + i_b''),\ ] ] where @xmath152}{\left\vert\int_0^{t } b_s dw_s\right\vert}^p \right ] } \le c_p{\mathsf{e}\left[\left(\int_0^t { \left\vertb_s\right\vert}^2ds\right)^{p/2}\right ] } < \infty,\\ i_b '' = { \mathsf{e}\left[\sup_{t\in[0,t]}\left(\int_0^t { \left\vert\int_s^t b_z dw_z\right\vert } ( t - s)^{-1-\alpha}ds\right)^p \right]}.\end{gathered}\ ] ] by the garsia rodemich rumsey inequality ( * ? ? ? * theorem 1.4 ) , for arbitrary @xmath153 , @xmath154 $ ] @xmath155 where _ ( t ) = ( _ 0^t _ 0^t dxdy)^/2 . @xmath156 for @xmath157 by the hlder inequality @xmath158}\le c_{p,\eta}\int_0^t \int_0^t \frac{{\mathsf{e}\left[{\left\vert\int_x^y b_v dw_v\right\vert}^{p}\right ] } } { { \left\vertx - y\right\vert}^{p/2}}dx\,dy\\ \le c_{p,\eta}\int_0^t \frac{{\mathsf{e}\left[\left(\int_x^y ( b_v)^2 dv\right)^{p/2}\right ] } } { { \left\vertx - y\right\vert}^{p/2 } } dx\,dy<\infty,\end{gathered}\ ] ] whence @xmath159}\sup_{t\in[0,t]}\left(\int_0^t ( t - s)^{-1/2-\eta-\alpha}ds\right)^p < \infty,\ ] ] and the statement follows . now turn to equation . to prove existence and uniqueness of its solution , we did not make any assumptions about @xmath4 and @xmath58 , except that @xmath4 has finite activity : @xmath15 . to prove existence of moments , we will make further assumptions on the measure @xmath4 and the coefficient @xmath58 in addition to h1h4 . let @xmath77 be the moment of the @xmath78th jump of process @xmath64 . as in the proof of [ thm_ex_sol ] , consider for @xmath70 the equation @xmath71 where @xmath72 is a wiener process , @xmath73 is an adapted process with almost surely @xmath164-hlder continuous paths for some @xmath74 , and denote the unique solution to this equation by @xmath76 . a reasoning similar to that used in the proof of lemma [ pathwiseestimatemixed ] gives @xmath165}\le k{\left\verty\right\vert } \exp{\left\{k{\left\lvertz\right\rvert}_{0;[0,t]}^{1/(1-\alpha)}\right\}}(1+{\left\lverti_{b , y}\right\rvert}_{\infty;t}),\ ] ] where @xmath166 , the constant @xmath134 depends only on constants in assumptions h1h4 , without loss of generality we assume @xmath167 . hence we get for @xmath168 @xmath169}^{1/(1-\alpha)}\right\}}\left(1 + { \left\lverti_{b}\right\rvert}_{\infty;\tau_1}\right),\end{gathered}\ ] ] where @xmath170 . for convenience denote @xmath171 and assume without loss of generality that @xmath172 , @xmath56 . then @xmath173}^{1/(1-\alpha)}\right\ } } ( 1+j_{b}),\ ] ] where @xmath174 further , @xmath175}^{1/(1-\alpha)}\right\ } } ( 1+j_{b}).\end{gathered}\ ] ] using inequality consequently on @xmath176 , @xmath177 , @xmath178 and estimating jumps of @xmath59 as in , we arrive at @xmath179}{\left\vertx_{t}\right\vert}\le \sup_{t\in[0,t]}{\left\vertx_{t}\right\vert}_1\le ( 3k)^{n(t)}(1+j_{b})^{n(t)+1}{\left\vertx_0\right\vert}_1\exp{\left\{k{\left\lvertb^h\right\rvert}_{0;[0,\tau_1\wedge t]}^{1/(1-\alpha)}\right\}}\\ \times \prod_{n:\tau_n\le t}g(\delta l_{\tau_n})\exp{\left\{k{\left\lvertb^h\right\rvert}_{0;[\tau_n,\tau_{n+1}\wedge t]}^{1/(1-\alpha)}\right\}}\\ \le c(6k)^{n(t)}\left(1+j_{b}^{n(t)+1}\right)\exp{\left\{k{\left\lvertb^h\right\rvert}_{0;[0,\tau_1\wedge t]}^{1/(1-\alpha)}\right\}}\\ \times \prod_{n:\tau_n\le t}g(\delta l_{\tau_n})\exp{\left\{k{\left\lvertb^h\right\rvert}_{0;[\tau_n,\tau_{n+1}\wedge t]}^{1/(1-\alpha)}\right\}}\end{gathered}\ ] ] where @xmath180 denotes the number of jumps of the process @xmath64 on @xmath65 $ ] . hence , by the hlder inequality , @xmath181}{\left\vertx_{t}\right\vert}^{p}\right]}\le c_p\left(\vphantom{{\mathsf{e}\left[\prod_{n=1}^{n(t ) } g(\delta l_{\tau_n})^{4p}\right]}}{\mathsf{e}\left[(6k)^{4pn(t)}\right ] } \left(1+{\mathsf{e}\left[j_{b}^{4p(n(t)+1)}\right]}\right)\right.\\ \times\left.{\mathsf{e}\left[\prod_{n=1}^{n(t ) } g(\delta l_{\tau_n})^{4p}\right]}{\mathsf{e}\left[\prod_{n=0}^{n(t)}\exp{\left\{4kp{\left\lvertb^h\right\rvert}_{0;[\tau_n,\tau_{n+1}\wedge t]}^{1/(1-\alpha)}\right\}}\right]}\right)^{1/4 } , \end{gathered}\ ] ] with @xmath182 . since the jumps of @xmath64 are jointly independent and do not depend on @xmath180 , which has a poisson distribution , we have @xmath183}<\infty$ ] and @xmath184 } = \exp{\left\{({\mathsf{e}\left[g(\delta l_{\tau_1})^{4p}\right]}-1)\pi({{\mathbb r}})t\right\}}\\ = \exp{\left\{\left(\int_{{{\mathbb r}}}g(y)^{4p}\pi(dy)-1\right)\pi({{\mathbb r}})t\right\}}<\infty.\end{gathered}\ ] ] further , write @xmath185 } = { \mathsf{e}\left[{\mathsf{e}\left[j_{b}^{4p(n(t)+1)}\,\middle\vert\ , l\right]}\right]}\end{gathered}\ ] ] from the formula for the solution to , obtained in the proof of theorem [ thm_ex_sol ] , we get that @xmath186 , where @xmath187 is certain non - random measurable function . thus , we can write @xmath188 , where @xmath189 is a non - random bounded function . therefore , since @xmath1 and @xmath2 do not depend on @xmath64 , we obtain @xmath190 } = { \mathsf{e}\left[\sup_{0\le s\le t\le t } { \left\lvert\int_s^\cdot g(u , w , b^h , l ) dw_u\right\rvert}_{\alpha;[s , t]}^{4pk}\right ] } \bigg|_{l = l , k= n(t)+1}.\end{gathered}\ ] ] abbreviate @xmath191 and estimate @xmath192}\\\le { \left\vert\int_0^t g_u(l ) dw_u\right\vert } + { \left\vert\int_0^s g_u(l ) dw_u\right\vert } + \int_s^t { \left\vert\int_u^t g_z(l ) dw_z\right\vert}(t - u)^{-1-\alpha } du.\end{gathered}\ ] ] then @xmath193}^{4pk}\right]}\le 2 ( i_1 + i_2),\end{gathered}\ ] ] where @xmath194 } { \left\vert\int_0^t g_u(l ) dw_u\right\vert}^{4pk}\right ] } \le { \mathsf{e}\left[\left(\int_0^tg_u(l)^2 du\right)^{2pk}\right]}\le k_1^k,\\ i_2 = { \mathsf{e}\left[\sup_{0\le s\le t\le t } \left(\int_s^t { \left\vert\int_u^t g_z(l ) dw_z\right\vert}(t - u)^{-1-\alpha } du\right)^{4pk}\right]},\end{gathered}\ ] ] with some constant @xmath195 independent of @xmath196 . the term @xmath197 is estimated as in the proof of theorem [ moments - mixed ] : from the garsia rodemich rumsey inequality we get for any @xmath153 , @xmath154 $ ] @xmath198 with @xmath199 now for @xmath200 @xmath201}\le c_{\eta}^{4pk}t^{2 - 4pk\eta } \int_0^t \int_0^t \frac{{\mathsf{e}\left[{\left\vert\int_x^y g_v(l ) dw_v\right\vert}^{4pk}\right ] } } { { \left\vertx - y\right\vert}^{2pk}}dx\,dy\\ \le c_{\eta}^{4pk}t^{2 - 4pk\eta } \int_0^t \frac{{\mathsf{e}\left[\left(\int_x^y g_v(l)^2 dv\right)^{2pk}\right ] } } { { \left\vertx - y\right\vert}^{2pk } } dx\,dy \le k_2^{k},\end{gathered}\ ] ] where the constant @xmath202 is independent of @xmath196 . consequently , @xmath203 collecting all estimates , we get @xmath193}^{4pk}\right]}\le k_3^{k}\end{gathered}\ ] ] with @xmath204 independent of @xmath196 . therefore , @xmath185 } \le { \mathsf{e}\left[k_3^{n(t)+1}\right]}<\infty.\end{gathered}\ ] ] consider the last multiple in . denote @xmath205}^{1/(1-\alpha)}$ ] . by the hlder inequality , @xmath206}\le \prod_{n=0}^{n(t ) } { \mathsf{e}\left[e^{q_n z_n}\right]}^{1/q_n}= \prod_{n=0}^{n(t)}{\mathsf{e}\left[{\mathsf{e}\left[e^{q_n z_n}|l\right]}\right]}^{1/q_n},\end{gathered}\ ] ] where @xmath207 using the independence of @xmath2 and @xmath64 , write @xmath208 } = { \mathsf{e}\left[\exp{\left\{4kps ( b - a)^{-\kappa } { \left\lvertb^h\right\rvert}_{0;[a , b]}^{1/(1-\alpha)}\right\}}\right]}\big\vert_{a= \tau_n , b=\tau_{n+1}\wedge t , s=\sum_{n=0}^{n(t ) } \delta_n^{\kappa}}.\ ] ] in view of the self - similarity property of @xmath2 , it is easy to check that @xmath209}^{1/(1-\alpha)}\overset{d}{=}{\left\lvertb^h\right\rvert}_{0;[0,1]}^{1/(1-\alpha)}.\ ] ] consequently , @xmath206}\le { \mathsf{e}\left[{\mathsf{e}\left[\exp{\left\{4kp s { \left\lvertb^h\right\rvert}_{0;[0,1]}^{1/(1-\alpha)}\right\}}\right ] } \big\vert_{s=\sum_{n=0}^{n(t ) } \delta_n^{\kappa}}\right]}\end{gathered}\ ] ] it is easy to check that @xmath210 . therefore , @xmath211 this implies @xmath206}\le { \mathsf{e}\left[\exp{\left\{4kp t^{\kappa } { \left\lvertb^h\right\rvert}_{0;[0,1]}^{1/(1-\alpha)}\right\}}\right]},\end{gathered}\ ] ] which is finite as @xmath212}$ ] is a finite supremum of a gaussian family and @xmath148 . the proof is now complete . bai , l. and ma , j. _ stochastic differential equations driven by fractional brownian motion and poisson point process _ , preprint ( 2012 ) , arxiv : math.pr/1206.2710 , available at http://arxiv.org/pdf/1206.2710v1.pdf . y. hu and d. nualart . _ differential equations driven by hlder continuous functions of order greater than 1/2 _ , in _ stochastic analysis and applications _ , _ abel symp . _ , vol . 2 . springer , berlin , 2007 , pp . 399413 . m. l. kleptsyna , p. e. kloeden , and v. v. anh . _ existence and uniqueness theorems for stochastic differential equations with fractal brownian motion _ , problemy peredachi informatsii 34 ( 1998 ) , no . 4 , pp . 5161 . y. s. mishura and g. m. shevchenko . _ stochastic differential equation involving wiener process and fractional brownian motion with hurst index @xmath214 _ , comm . theory methods 40 ( 2011 ) , no . 1920 , pp . 34923508 . y. s. mishura and g. m. shevchenko . _ mixed stochastic differential equations with long - range dependence : existence , uniqueness and convergence of solutions_. comput . 64 ( 2012 ) , no . 10 , pp . 31173127 | in this paper , we consider a stochastic differential equation driven by a fractional brownian motion ( fbm ) and a wiener process and having jumps . we prove that this equation has a unique solution and
show that all moments of the solution are finite .
fractional brownian motion ; wiener process ; poisson measure ; stochastic differential equation ; moments primary 60g15 ; secondary 60g22 , 60h10 , 60j65 . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
the study of the extended emission in the narrow - line region ( nlr ) around nearby active galactic nuclei ( agn ) allows the investigation of both the agn feeding via gas inflows ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) and feedback via the interaction of the agn radiation and mass outflow with the circumnuclear gas , affecting its kinematics and excitation ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? most studies on the feeding and feedback mechanisms of active galactic nuclei ( agn ) presently available in the literature are based on optical observations , which are affected by dust obscuration , a problem that can be softened by the use of infrared observations . another advantage of infrared spectral region is that , besides observing ionized gas emission , we can also observe emission from molecular gas ( h@xmath10 ) . our group , _ agnifs _ ( for agn integral field spectroscopy ) , has been developing a project to map both the feeding and feedback in nearby agn using near - infrared integral field spectroscopic observations mostly with the instrument nifs at the gemini north telescope . the main findings of our group so far have been that the molecular gas traced by k - band @xmath16 emission , and the ionized gas traced by hi recombination lines and [ feii ] emission , present distinct flux distributions and kinematics . usually the @xmath16 emitting gas is restricted to the plane of the galaxy , while the ionized gas extends also to high latitudes and is associated with the radio emission , when present @xcite . the @xmath16 kinematics is usually dominated by rotation , including in some cases , streaming motions towards the nucleus , while the kinematics of the ionized gas , and in particular of the [ feii ] emitting gas , shows also , in many cases , a strong outflowing component associated with radio jets from the agn . similar results have been found using the spectrograph for integral field observations in the near infrared ( sinfoni ) at the very large telescope ( vlt ) . @xcite found molecular gas inflows towards the nucleus of ngc1097 and @xcite mapped similar h@xmath10 inflows feeding and obscuring the active nucleus of ngc1068 , while @xcite mapped outflows in ionized gas around 7 active galatic nuclei . in this work , we present the gaseous distribution and kinematics of the inner 450pc radius of the narrow - line seyfert 1 galaxy mrk766 ( ngc4253 ) a barred spiral galaxy ( sba ) , located at a distance of 60.6mpc , for which 1@xmath17 corresponds to 294pc at the galaxy . the hst images of this galaxy show some irregular dust filaments around the nucleus @xcite . @xcite show that the radio source appears to be extended to south - east in pa @xmath2 150@xmath18 ( on a scale of @xmath2 1@xmath19 ) . the optical emission is extended beyond the radio structure @xcite . the nir spectrum is well described by @xcite , showing a large number of permited lines of hi , hei , heii and feii , and by forbidden lines of [ sii ] , [ siii ] and [ feii ] . high ionization lines like [ siix ] , [ six ] , [ six ] and [ mgviii ] are also observed . the x - rays observations of this galaxy show that it is a strong variable source , with evidences of the amplitude being larger at @xmath2 2kev . the mass of the supermassive black hole has been accurately measured via reverberation mapping by @xcite , resulting in a mass of 1.76@xmath2010@xmath21m@xmath13 . there is no co observations for this galaxy in the literature . mrk766 was selected for this study because : ( i ) it presents strong near - ir emission lines ( e.g. * ? ? ? * ) , allowing the mapping of the gaseous distribution and kinematics ; and ( ii ) it has radio emission , allowing the investigation of the role of the radio jet @xcite in the gas excitation and kinematics . this paper is organized as follows : in sec . 2 we describe the observations and data reduction procedures . the results are presented in sec . 3 and discussed in sec . 4 . we present our conclusions in sec . the observations of mrk766 were obtained using the gemini near infrared integral field spectrograph ( nifs - @xcite ) operating with the gemini north adaptive optics system altair in june 2010 under the programme gn-2010a - q-42 , following the standard sky - object - object - sky dither sequence . observations were obtained in the j - band using the @xmath22 grating and @xmath23 filter , and in the @xmath24-band using the @xmath25 grating and @xmath26 filter . on - source and sky position observations were both obtained with individual exposure times of 550s . two sets of observations with six on - source individual exposures were obtained : the first , in the j - band , was centred at 1.25@xmath5 m and covered the spectral range 1.14@xmath5 m to 1.36@xmath5 m , and the second , in the @xmath24-band , was centred at 2.3@xmath5 m and covered the spectral range 2.10@xmath5 m to 2.53@xmath5 m . the data reduction procedure included trimming of the images , flat - fielding , sky subtraction , wavelength and spatial distortion calibrations . we also removed the telluric bands and flux calibrated the frames by interpolating a black body function to the spectrum of the telluric standard star . these procedures were executed using tasks contained in the nifs software package which is part of gemini iraf package , as well as generic iraf tasks . in order to check our flux calibration , we extracted a nuclear spectrum with the same aperture of a previous spectrum of the galaxy by @xcite . the two spectra are very similar to each other ( considering the difference in spectral resolution ) , with the largest difference in flux being about 5% at 2.2@xmath5 m . the final ifu data cube for each band contains @xmath274200 spectra , with each spectrum corresponding to an angular coverage of 005@xmath28005 , which translates into @xmath215@xmath2815pc@xmath29 at the galaxy and covering the inner 3@xmath283(@xmath2 900@xmath28900pc@xmath29 ) of the galaxy . the full width at half maximum ( fwhm ) of the arc lamp lines in the j - band is 1.65 , corresponding in velocity space to 40kms@xmath1 , while in the @xmath24-band the fwhm of the arc lamp lines is 3.45 , corresponding to 45kms@xmath1 . the angular resolution obtained from the fwhm of the spatial profile of the flux distribution of the broad component of the pa@xmath11 and br@xmath9 lines and is 021@xmath30003 for the j - band and 019@xmath30003 for the @xmath24-band , corresponding to 60pc and 55pc at the galaxy , respectively . in the top - left panel of fig.[galaxia ] we present an optical image of mrk766 obtained with the lick observatory nickel telescope @xcite . in the top - right panel we present an optical image of mrk766 obtained with the hubble space telescope ( hst ) wide field planetary camera 2 ( wfpc2 ) through the filter f606w @xcite . in the bottom panels we present , to the left , a zoom of the hst image within the field - of - view ( fov ) covered by the nifs observations and to the right an image obtained from the nifs data cube within a continuum window centred at @xmath31 m . in fig.[teste ] we present two ifu spectra integrated within a 025@xmath28025 aperture : one at the nucleus and the other at 05 east of it ( position a ) , chosen randomly with the purpose of just presenting a characteristic extranuclear spectrum . the nucleus was defined to be the location of the peak flux in the continuum . we list in table 1 the emission line fluxes we could measure from these two spectra , which comprise 20 emission lines from the species [ pii ] , [ feii ] , heii , hi , @xmath32 , [ six ] and [ caviii ] . they were measured with the _ splot _ task in iraf and the uncertainties were estimated as the standard deviation of the average of 6 measurements . in order to map the flux distributions as well as the centroid velocity and velocity dispersion fields , we used the profit routine @xcite to fit the profiles of [ pii]@xmath33 m , [ six]@xmath3 m , [ feii]@xmath6 m , pa@xmath34 m , @xmath35 m and br@xmath36 m emission lines at each pixel over the whole fov . these emission lines were chosen because they have the highest signal - to - noise ( s / n ) ratios among their species ( coronal , ionized and molecular gas ) . the flux values ( as well as those of the central wavelength and width of the profile , see next sections ) were obtained by the fit of the profiles using both gaussian and gauss - hermite ( gh ) series . we found out that the latter gave better fits to most lines , except for the [ six ] line , for which the gh fits introduced extra wings in some regions where the line was weak . we decided then to adopt the paramenters of the fit obtained from the gh for all lines except for [ s ix ] , for which we adopted the fit with gaussians . in the case of pa@xmath11 and br@xmath9 we have fitted also a broad component to the line . this was done via a modification of the profit routine to fit the broad component and subtract its contribution from the profiles in order to generate a datacube only with the narrow component . the steps in this procedure were : i ) fit only one gaussian to the broad component ; ii ) subtract it from the spectra where it is present , and iii ) fit the narrow component . in fig.[flux - distributions ] we present the resulting flux distribution maps , where we have masked out the bad fits by using the chi - square map , which is an output from the profit routine . all maps have their peak fluxes at the same position , which also coincides with that of the peak of the continuum : the nucleus . the [ pii ] and [ six ] flux distributions are the most compact , reaching about 05 from the nucleus in all directions in the case of the former , and being more extended to the south - west in the case of the latter . another coronal line ( not shown in the figure ) , [ caviii]@xmath37 m , also shows a similarly compact flux distribution , indicating that the coronal line region is compact but resolved , extending up to 150 pc from the nucleus , what is a typical radius for this region ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the highest levels of the [ pii ] flux distribution are more elongated towards the south - east . this elongation is also observed in the [ feii ] emission , which reaches 08 ( 240 pc ) from the nucleus in that direction . the pa@xmath11 flux distribution is the most extended in all directions , reaching up to 1@xmath17 from the nucleus . the br@xmath9 flux distribution is very similar to that of pa@xmath11 , although noisier due to its lower flux . the h@xmath10 flux distribution is somewhat distinct , being elongated from north - east to south - west , thus approximately perpendicular to the elongation of the [ feii ] flux distribution , reaching 15 ( 440 pc ) from the nucleus towards the south - west . . [ flux - distributions ] in fig . [ razoes ] , we present line - ratio maps obtained from the flux maps , where regions with bad fits were masked out . the average uncertainties in the line - ratio values are @xmath2 10% . in the left panel we present the [ feii]@xmath38m/@xmath7 ratio map , which can be used to investigate the excitation mechanism of [ feii ] ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the values of [ feii]/pa@xmath11 for mrk766 range from 0.2 ( most locations ) to 1.0 with the highest values between 0.6 and 1 being observed between 02 and 06 to the south - east of the nucleus . another line ratio that can be used to investigate the [ feii ] excitation mechanism is [ feii]@xmath39m/[pii]@xmath40 m . values larger than 2 indicate that shocks have passed through the gas destroying the dust grains , releasing the fe and enhancing its abundance and emission ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? we present this ratio map in the central panel of fig . [ razoes ] . the lowest values ( @xmath412 ) , are observed to the north and north - west of the nucleus while the highest values of @xmath2 7.0 are observed in a narrow strip at @xmath206 to the south - east of the nucleus , approximately at the border of the region with the highest values of [ feii]/pa@xmath11 . in the right panel of fig . [ razoes ] we present the @xmath35m / br@xmath9 ratio map , which is useful to investigate the excitation of the @xmath32 emission line ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . in mrk766 , the values of this ratio range from 0.2 to 2.0 . the lowest values are observed at the nucleus and in most regions to the south and south - east , except for the region between 02 and 06 to the south - east where the values increase to @xmath20.8 ( where both [ feii]/pa@xmath11 and [ feii]/[pii ] show larger values ) . such increased values are also observed to the north where they reach @xmath2 1.25 . at approximately 1 arcsec to the south - west , the values reach up to @xmath22 , although the fit of the lines is not so good and the uncertainty is high there . [ cols="<,<,^,^",options="header " , ] the profit routine ( riffel 2010 ) that we have used to obtain the flux of the emission lines , provide also the centroid velocity ( v ) , velocity dispersion ( @xmath42 ) and higher order gauss - hermite moments ( @xmath43 and @xmath44 ) , which have been used to map the gas kinematics . in fig . [ velocity ] we present the centroid velocity fields after subtraction of the heliocentric systemic velocity of 3853@xmath3017 km s@xmath1 , which was obtained through a model fitted to the pa@xmath11 velocity field , as discussed in sec . the uncertainties in the velocity maps range from 5 to 20 km s@xmath1 depending on the s / n ratio of the spectra ( which decrease from the center towards the border of the mapped region ) . the white regions in the figures represent locations where the s / n was not high enough to allow the fitting of the line profiles . all velocity fields show blueshifts to the east ( left in the figures ) and redshifts to the west , with the line of nodes oriented at a position angle of approximately @xmath45 ( see sec . 4.3 ) , with the isovelocity lines showing an approximate `` spider diagram '' characteristic of rotation . 6 shows the velocity dispersion maps corresponding to the centroid velocity maps of fig . [ velocity ] . as in the case of the centroid velocities , the uncertainties in the velocity dispersion maps range from 5 to 20 km s@xmath1 depending on the s / n ratio of the spectra . the white regions in the figures represent locations where the s / n was not high enough to allow the fitting of the line profiles . the [ feii ] @xmath42 map shows the highest values of up to 150 kms@xmath1 to the south - east of the nucleus and lowest values , down to 75 kms@xmath1 , to the north - west . the [ pii ] @xmath42 map has medium values with soft deviations . the pa@xmath11 @xmath42 map shows high values at the nucleus and also 0@xmath194 to the south and 0@xmath196 to the north and lower values to the east , south and west of this central region . the higher values at the nuclear region may be due to residual contamination from the broad component of the line . the @xmath16 emitting gas presents the lowest @xmath42 values , which are smaller than 70 kms@xmath1 at most locations . we do not show the @xmath43 and @xmath44 maps because their values are low and do not present any systematic behavior . channel maps along the emission line profiles are shown in figs . 7 , 8 , 9 and 10 for the [ six ] , [ feii ] , pa@xmath11 and @xmath16 emission lines , respectively . each panel presents the flux distribution in logarithmic units integrated in velocity bins centred at the velocity shown in the top - left corner of each panel ( relative to the systemic velocity of the galaxy ) . the central cross marks the position of the nucleus . we do not show channel maps for [ pii ] and br@xmath9 because the [ pii ] maps are similar to those of [ feii ] and those for br@xmath9 are similar to those of pa@xmath11 but noisier . in fig . 7 , the channel maps along the [ six ] emission line profile show the flux distributions integrated within velocity bins of 25 kms@xmath1 ( corresponding to one spectral pixel ) . at the highest velocities the emission is extended 05 to the south / south - west , and at the lowest velocities , the [ six ] is concentrated in the nucleus . in fig . 8 , the channel maps along the [ feii ] emission - line profile show the flux distributions integrated within velocity bins of 105 kms@xmath1 ( corresponding to three spectral pixels ) for the highest velocities and 50kms@xmath1 for the central panels ( corresponding to two spectral pixels ) . all [ feii ] channel maps present flux distributions which are elongated towards the south - east , up to@xmath209 ( 270pc ) from the nucleus . both the highest blueshifts and highest redshifts , which reach @xmath46kms@xmath1 , are also observed to the south - east of the nucleus . fig . 9 shows the channel maps for the pa@xmath11 emitting gas for the same velocity bins as for [ feii ] . the highest blueshifts and redshifts are observed mostly at the nucleus , but are probably due to residuals of a broad component to the line which was fitted and subtracted . the flux distributions are more extended and more symmetrically distributed around the nucleus than those of the [ feii ] channel maps . 10 shows the channel maps for the @xmath16 emitting gas , for velocity bins of 30kms@xmath1 . the highest blueshifts and redshifts , reaching @xmath2130kms@xmath1 , are observed to the north - east and south - west of the nucleus respectively , following the line of nodes of the galaxy , as seen in fig . [ velocity ] . for zero and positive velocities there is a structure extending from the nucleus to the south - west . in order to further map the excitation of the circum - nuclear line - emitting region we constructed a spectral diagnostic diagram with the ratios [ feii]@xmath6m@xmath47 vs. @xmath35m / br@xmath9 @xcite , shown in fig[diagnostico ] . typical values for the nuclei of seyfert galaxies range between 0.6 and 2.0 for both ratios @xcite , while for starbursts the values are smaller than 0.6 and for liners the values are larger than 2 , as shown in the top panel of fig.[diagnostico ] . in this figure , black filled circles represent seyfert ratios , blue open circles , starbursts ratios and red crosses represent ratios of low - ionization nuclear emission - line regions ( liners ) . most ratios present starburst and seyfert values , with a few liner values . the locations from where the distinct line ratios originate are shown in the bottom panel of fig.[diagnostico ] . seyfert ratios are found at the nucleus , in most regions to the north and between 02 and 06 to the south - east . in a few locations to the north of the nucleus , between the nucleus and the region of enhanced ratios ( at 0206 ) and beyond this region , starburst ratios are found . liner ratios are only found at the south - west border of the mapped region . @xmath41.25@xmath5m / pa@xmath11 versus h@[email protected]@xmath5m / br@xmath9 line - ratio diagnostic diagram . the dashed lines delimit regions with ratios typical of starbursts ( blue open circles ) , seyferts ( black filled circles ) and liners ( red crosses ) . bottom panel : spatial position of each point from the diagnostic diagram . ] the excitation of warm h@xmath10 has been the subject of many previous studies ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . summarizing , these studies have shown that the @xmath16 emission lines can be excited by two mechanisms : ( i ) fluorescent excitation through absorption of soft - uv photons ( 912 - 1108 ) in the lyman and werner bands @xcite and ( ii ) collisional excitation due to heating of the gas by shocks , in the interaction of a radio jet with the interstellar medium @xcite or heating by x - rays from the central agn @xcite . the second mechanism is usually referred to as thermal process since it involves the local heating of the emitting gas , while the first is usually called a non - thermal process . previous studies have verified that non - thermal processes are not important for most galaxies studied so far ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . in the case of mrk766 , the h@xmath10/br@xmath9 ratio ( fig . [ razoes ] ) is larger than 0.6 to the north - east of the nucleus and in the arc - shaped region between @xmath48 and @xmath49 to the south - east supporting seyfert excitation there . the origin of the h@xmath10 excitation could be fluorescence or thermal excitation . one possible evidence for fluorescence ( a non - thermal process ) is a ratio between the h@xmath10 lines 2.24@xmath5m/2.12@xmath5 m higher than 0.6 @xcite . we could measure this ratio at position a , where the value is @xmath270.2 , favoring thermal excitation . at a number of other positions , the 2.24@xmath5 m line is fainter , but wherever it could be measured , the line ratio is smaller than 0.2 . this line ratio thus seem to favor thermal excitation due to heating by x - rays or shocks from a radio jet ( which seems to be present to the se ) in the region with seyfert excitation " , although there is no clear signature of shocks such as an increase in velocity dispersion as observed in the [ feii ] emission . in the remaining regions , the h@xmath10/br@xmath9 ratio is smaller than 0.6 , supporting starburst excitation via heating from shocks in sne winds and/or by uv radiation from young stars . the presence of starbursts in the nuclear region is in agreement with the results of @xcite , who reported the observation of the pah 3.3@xmath5 m feature in the infrared spectrum of mrk766 within the inner 150 pc , suggesting the presence of recent star formation there . the mixed seyfert and starburst excitation is also seen in the diagnostic diagram of fig . [ diagnostico ] . using the [ feii]@xmath6m@xmath47 and [ feii]@xmath6m@xmath50[pii]@xmath40 m line - ratio maps shown in fig.[razoes ] , we can investigate the excitation mechanism of [ feii ] . the first ratio [ feii]@xmath6m@xmath47 is controlled by the ratio between the volumes of partially to fully ionized gas regions , as the [ feii ] emission is excited in partially ionized gas regions . in agns , such regions can be created by x - ray ( e.g. * ? ? ? * ) and/or shock ( e.g. * ? ? ? * ) heating of the gas . for starburst galaxies , [ feii]/@xmath51 and for supernovae for which shocks are the main excitation mechanism , this ratio is larger than 2 @xcite . the values of [ feii]/@xmath7 range from @xmath20.2 to the north - west to @xmath21.0 in the arc - shaped region between 02 and 06 to the south - east of the nucleus . @xcite have obtained a 3.6 cm radio image of mrk766 and found an extended emission to the se , at the location of the arc - shaped region where there is an enhancement of the [ feii]/pa@xmath11 ratio . the variation of this line ratio , and its correlation with the radio structure suggest that excitation by shocks from the radio jet is indeed important at this location . on the other hand , we can not rule out the possible contribution from supernovae as well . the above conclusion is also supported by the [ feii]@xmath6m/[pii]@xmath33 m line - ratio map ( central panel of fig.[razoes ] ) . these two lines have similar ionization temperatures , and their parent ions have similar ionization potentials and relative recombination coefficients . values larger than 2 indicate that the shocks have passed through the gas destroying the dust grains , releasing the fe and enhancing its abundance and thus emission @xcite . for supernovae remnants , where shocks are the dominant excitation mechanism , [ feii]/[pii ] is typically higher than 20 @xcite . for mrk766 , to the south - east of the nucleus , where there is the radio structure , [ feii]/[pii ] values reach @xmath210 ( fig.[razoes ] ) , suggesting that shocks are indeed important in agreement with the highest values obtained for the [ feii]/@xmath7 at the same locations . in other regions , typical values are [ feii]/[pii]@xmath41 2 , indicating almost no contribution from shocks . finally , the diagnostic diagram of fig.[diagnostico ] confirms seyfert excitation in the nucleus and in the south - east arc and regions surrounding the nucleus and non - seyfert values in the other regions . and the low [ feii]/[pii ] ratios in these other regions suggest also that sne winds should not be important , favoring ionization by young stars instead . the mass of ionized gas in the inner 900@xmath28900pc@xmath29 of the galaxy can be estimated using ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) : @xmath52 , \ ] ] where @xmath53 is the integrated flux for the @xmath54 emission line and @xmath55 is the distance to mrk766 . we have assumed an electron temperature @xmath56 and electron density @xmath57@xmath58 @xcite . the mass of warm molecular gas can be obtained using @xcite : @xmath59,\ ] ] where @xmath60 is the integrated flux for the @xmath61 m emission line and we have used the vibrational temperature @xmath62 @xcite . we used the br@xmath9/pa@xmath11 line ratio in order to estimate the effect of the reddening in the observed fluxes for these lines . we constructed a reddening map using the pa@xmath11/br@xmath9 line ratio . the resulting map is very noisy with an additional uncertainty relative to other line ratios because the lines are in different spectral bands ( k and j ) . the e(b - v ) values are also mostly very small . thus , instead of using this map to correct the whole br@xmath9 flux distribution for reddening as we would introduce too much noise we have estimated an average value for @xmath63 using the the integrated fluxes for br@xmath9 and pa@xmath11 emission lines over the whole field of view , following @xcite and adopting the extinction law of @xcite . adopting this @xmath64 value , the fluxes for the emission lines in the k band increase by about 10% . the effect of the reddening is negligible for the line ratios of fig . [ razoes ] , since the lines are from the same band and the reddening has no effect on the discussion of the gas excitation presented above . on the other hand , its effect is not negligible for the estimate of the ionized and molecular gas masses , which have thus been corrected . integrating over the whole ifu field , we obtain the following reddening - corrected values : @xmath65 6.82@xmath66 erg@xmath67@xmath68 and @xmath697.3@xmath70erg@xmath67@xmath68 . the resulting masses are @xmath71 7.6@xmath72 and @xmath73 1.32@xmath74 @xmath75 . the above values are similar to those we have obtained in previous studies , which are in the range @xmath76 m@xmath77 [email protected]@xmath79 m@xmath80 and 66m@xmath77 m@xmath813300 m@xmath80 , respectively . the mass of molecular gas is thus 10@xmath82 times smaller than that of the ionized gas but , as discussed in @xcite , this h@xmath10 mass represents only that of warm gas emitting in the near - ir . the total mass of molecular gas is dominated by the cold gas , and the usual proxy to estimate the cold h@xmath10 mass has been the co emission . a number of studies have derived the ratio between the cold and warm h@xmath10 gas masses by comparing the masses obtained using the co and near - ir emission . @xcite obtained ratios in the range 10@xmath8310@xmath84 ; using a larger sample of 16 luminous and ultraluminous infrared galaxies , @xcite derived a ratio m@xmath85/m@xmath86 = 1 - 5@xmath87 . more recently , @xcite compiled from the literature values of mcold derived from co observations and [email protected]@xmath5 m luminosities for a larger number of galaxies , covering a wider range of luminosities , morphological and nuclear activity types . from that , an estimate of the cold h2 gas mass can be obtained from @xmath88 where @xmath89 is the luminosity of the [email protected]@xmath5 m line . the resulting mass value is m@xmath90 @xmath2 9.8@xmath2810@xmath91 . all the velocity fields shown in fig . [ velocity ] suggest rotation in the inner 450pc of mrk766 . in order to obtain the systemic velocity , orientation of the line of nodes and an estimate for the enclosed mass , we fitted a model of circular orbits in a plane to the @xmath7 and h@xmath92 m velocity fields . the expression for the circular velocity is given by @xcite : @xmath93 where @xmath94 is the projected distance from the nucleus in the plane of the sky , @xmath95 is the corresponding position angle , @xmath96 is the mass inside @xmath94 , @xmath97 is the newtow s gravitational constant , @xmath98 is the systemic velocity , @xmath99 is the inclination of the disc ( @xmath100 for a face - on disc ) , @xmath101 is the position angle of the line of nodes and @xmath102 is a scale length projected in the plane of the sky . the location of the kinematical center was not allowed to vary , being fixed to the position of the peak of the continuum . the equation above contains five free parameters , which can be determined by fitting the model to the observations . this was done using the levenberg - marquardt least - squares fitting algorithm , in which initial guesses are given for the free parameters . the best fit model for pa@xmath11 is shown in fig.[residuals1 ] ( top - left panel ) and the best fit model for h@xmath92 m in fig.[residuals2 ] . in both figures we show the residual maps ( observed velocity field - model ) for [ feii ] ( bottom - left panel ) , @xmath16 ( bottom - right panel ) and @xmath7 ( top - right panel ) . the parameters derived from the fit of the pa@xmath11 are : the systemic velocity corrected to the heliocentric reference frame @xmath103kms@xmath1 , @xmath104 , @xmath105 , @xmath106 and @xmath107 pc . we can compare the near - ir line - emitting gas kinematics with results obtained in the optical at larger scales . @xcite present long - slit spectroscopy of mrk 766 at kpc scales with the slit oriented along pa@xmath108 . they found that the kinematic of the high - excitation gas ( traced by the [ oiii]@xmath109 emission ) is more perturbed than that of the low - ionization gas ( traced by h@xmath110 and h@xmath11 ) , showing radial motions consistent with gas outflows from the nucleus . the low - ionization gas seems to be dominated by rotation in the plane of the galaxy with a velocity amplitude of @xmath27130 kms@xmath1 . at distances smaller than 15 from the nucleus , the velocity amplitude is @xmath11150kms@xmath1 , which is somewhat smaller than the amplitude that we have derived . this is expected , since the slit used by @xcite was not oriented along the major axis of the galaxy . @xcite quote a photometric major axis orientation of 105@xmath112 , based on an large scale continuum image at @xmath45960 . the pa of the line of nodes @xmath101 that we have found is 25@xmath112 smaller than this value . on the other hand , our @xmath101 is in reasonable agreement with the value listed at the hyperleda ( @xmath113 * ? ? ? [ galaxia ] shows that mrk766 presents a bar with size of 4.5 kpc . the orientation of the bar is similar to that of the photometric major axis considered by @xcite . as the bar is broad an luminous and the outer parts of the galaxy are faint , we believe they have mistakenly concluded that the direction of the bar was that of the major axis . the systemic velocity and @xmath99 are in reasonable agreement with the values listed at the hyperleda @xcite and ned databases ( @xmath114kms@xmath1 and @xmath115 and @xmath96 and @xmath102 are similar to values found for other seyfert galaxies using the same model ( e.g. * ? ? ? * ) . for the h@xmath92 m fit we found a much more compact velocity field than that of pa@xmath11 , with a scale lenght @xmath116 pc . the other parameters were practically the same . this signature of a more compact rotating disk in h@xmath10 than in pa@xmath11 , is similar to that we have found for mrk1066 @xcite , indicating that h@xmath10 presents a `` colder '' and more ordered kinematics . an exception is the region to the south - west , which seems to show a detached kinematics . this region is probably a molecular cloud that is not in the galaxy disc . the residuals shown in fig.[residuals1 ] show blueshifts in the borders of the measured field in pa@xmath11 to the north north - east of the nucleus which we attribute to poor fits of the lines in this region . more significant are the redshift residuals to the south - south - east , a region where the largest residuals in the [ feii ] velocity field are also observed . there is where the enhanced [ feii ] velocity dispersion and the radio structure are also located . in addition , in this same region , the [ feii ] flux distributions in the channel maps of fig . 8 show both blueshifts and redshifts , with velocities of up to 250 kms1 . we interpret these results as being due to emission of gas in a one - sided outflow oriented along the position angle @xmath2 135@xmath18 . the observation of both blueshifts and redshifts in the channel maps supports that its axis lies approximately in the plane of the sky . the main residuals in the h@xmath10 velocity field that are not in the borders of the field ( where the line fits are poorer ) are the redshifts observed to the north northwest . as this is the near side of the galaxy , we speculate that these residuals could be due to inflows in the plane of the galaxy . these residuals are seen along the direction of the bar at pa@xmath117 . we speculate that they may be associated to inflows along the bar , as predicted by theoretical models ( e.g * ? ? ? * ) and as measured in a few cases ( e.g. * ? ? ? * ) . in previous studies , we have found inflows along nuclear dusty spirals @xcite . indeed , numerical simulations by @xcite have shown that if a central smbh is present , shocks can extend all the way to the vicinity of the smbh and generate gas inflows consistent with the accretion rates inferred in local agn . similar residuals are also seen in the pa@xmath11 and [ feii ] residual maps : redshifts to north northwest , also suggesting inflows . we rule out the possibility of these redshifts being due to a counterpart of the south - east outflow once it is observed in redshift over the far side of the galaxy . a possible counterpart should be in blueshift and behind the near side of the galaxy plane . we do not see such a component ; one possibility is that it is hidden by the galaxy plane . the residuals shown in fig.[residuals2 ] , after the subtraction of the circular velocity model fitted to the h@xmath10 velocity field shows similar residuals to the south south - east for pa@xmath11 and [ feii ] , but show additional residuals in the northern part of the field . in the case of the h@xmath10 residuals , redshifts are observed also in the region to the south - west , that we have interpreted as due to a detached cloud , that is probably not in the galaxy plane . with the goal of quantifying the feedback from the agn in mrk766 , we estimate the ionized - gas mass outflow rate through a circular cross section with radius @xmath118pc located at a distance of @xmath119 from the nucleus to the south - east . this geometry corresponds to a conical outflow with an opening angle of @xmath2 64@xmath112 , estimated from fig.[sliceferro ] . the mass outflow rate can be obtained using : @xmath120 and the filling factor ( @xmath121 ) can be obtained from @xmath122 where @xmath123 is the proton mass , @xmath124 the electron density , @xmath125 is velocity of the outflowing gas and @xmath126 and @xmath127 are the luminosity and the emission coefficient of pa@xmath11 @xcite . we have assumed that @xmath128@xmath58 , @xmath1291.43@xmath130 erg s@xmath1 , @xmath131 erg @xmath132s@xmath1 and @xmath133kms@xmath134kms@xmath1 where @xmath135 is the angle between the wall of the cone ( from where we observe the line - of - sight velocity component of 147kms@xmath1 ) and the plane of sky . the latter velocity value was obtained directly from the channel maps considering that the structure seen to south - east is due to the emission of the walls of the cone . as described above , the axis of the cone seems to lie close to the plane of the sky . from the estimated aperture of the cone , we adopt a maximum angle between the cone and the plane of the sky of @xmath136 . under these assumptions we obtain @xmath137 and then @xmath138 yr@xmath1 . the value found here for @xmath139 is in good agreement with those found in @xcite , which range from 0.1 to 10@xmath140 yr@xmath1 , it is of the same order of that obtained by @xcite , of 8@xmath141 yr@xmath1 , and is also within the range of the values found by @xcite , which range from 2.5 to 120 @xmath141 yr@xmath1 . following @xcite , we can use the above mass outflow rate to estimate the kinetic power of the outflow using : @xmath142 where @xmath143 is the velocity of the outflowing gas and @xmath42 is its velocity dispersion . using @xmath144kms@xmath1 ( from fig.[disper ] ) and @xmath145kms@xmath1 we obtain @xmath146ergs@xmath1 which is in good agreement with the values obtained for seyfert galaxies and compact radio sources @xcite . this value is also similar to that obtained for mrk1157 @xcite , of @xmath147ergs@xmath1 , it is within the range of those found by @xcite , between 0.6 to 50@xmath148ergs@xmath1 . in order to compare the above value of @xmath149 with the bolometric luminosity , we estimate the latter as 10 times the x - ray luminosity , of 3.5@xmath148 erg s@xmath1 @xcite , resulting in @xmath150 0.08 l@xmath151 . finally , we can calculate the mass acretion rate to feed the active nucleus from@xcite @xmath152 where , @xmath153 is the nuclear bolometric luminosity , @xmath154 is the efficiency of conversion of the rest mass energy of the accreted material into radiation and @xmath155 is the light speed . the bolometric luminosity was already estimated as 3.5@xmath156 erg s@xmath1 . assuming @xmath1570.1 which is a typical value for a geometrially thin , optically thick accretion disc @xcite , we obtain an accretion rate of @xmath1581.4@xmath2810@xmath159 m@xmath13yr@xmath1 , which is about three orders of magnitude smaller than the mass outflow rate , a ratio compared with those found in our previous studies . we have mapped the gas flux distribution , excitation and kinematics from the inner @xmath2450pc radius of the seyfert1 galaxy mrk766 using near - ir j - and k - band integral - field spectroscopy at a spatial resolution of @xmath260pc ( 020 ) . the main conclusions of this work are : * the emission - line flux distributions of molecular hydrogen h@xmath10 and low - ionization gas are extended to at least @xmath2300pc from the nucleus ; * the h@xmath10 line emission is most extended along pa=70@xmath112 , which is close to the position angle of the line of nodes of the gas kinematics ; * the [ feii ] emission is most extended approximately along the perpendicular direction to the line of nodes of the gas kinematics ; * the coronal line [ six ] emission is resolved and extends up to @xmath2150pc from the nucleus ; * the emission - line ratios [ feii]/@xmath7 and @xmath32/br@xmath9 show a mixture of starburst and seyfert type excitation ; the seyfert values dominates at the nucleus , to the north - west and in an arc - shaped region between 02 and 06 to the south - east where a radio jet has been observed , while starburst values are present at the nucleus and other regions ; * the enhancement of the [ feii]/[pii ] line ratio at the location of the radio jet , as well as the corresponding increase in the [ fe ii ] flux and velocity dispersion support a contribution from shocks to the gas excitation in the arc - shaped region to the south - east ; in the remaining regions , the favoured excitation mechanism is uv radiation from young stars ; * the @xmath16 gas kinematics is dominated by rotation in a compact disc with a velocity amplitude of 140 km s@xmath1 and low velocity dispersion ( 40 - 60 km s@xmath1 , consistent with orbital motion in the plane of the galaxy ) ; * the kinematics of the ionized gas is also dominated by rotation , but channel maps in [ fe ii ] show in addition an outflowing component to the south - east , with an axis lying close to the plane of the sky reaching velocities of @xmath27 300 km s@xmath1 , probably associated with the radio jet . * the mass outflow rate in ionized gas is estimated to be @xmath2 10.7 m@xmath80 yr@xmath1 and the power of the outflow estimated to be @xmath160 ; * the mass of ionized gas is @xmath161 while the mass of the hot molecular gas is @xmath162 and the estimated cold molecular gas mass is m@xmath90 @xmath2 9.8@xmath2810@xmath91 . + the distinct flux distributions and kinematics of the h@xmath10 and [ feii ] emitting gas , with the first more restricted to the plane of the galaxy and in compact rotation and the second related with the radio jet and in outflow are common characteristics of 8 seyfert galaxies ( eso428-g14 , ngc4051 , ngc7582 , ngc4151 , mrk1066 , mrk1157 , mrk79 and mrk766 ) we have studied so far using similar integral - field observations and 2 others ( circinus and ngc2110 ) using long - slit observations . these results again suggest as those found in previous studies that the h@xmath10 emission is tracer of the agn feeding , while the [ feii ] is a tracer of its feedback . this work is based on observations obtained at the gemini observatory , which is operated by the association of universities for research in astronomy , inc . , under a cooperative agreement with the nsf on behalf of the gemini partnership : the national science foundation ( united states ) , the science and technology facilities council ( united kingdom ) , the national research council ( canada ) , conicyt ( chile ) , the australian research council ( australia ) , ministrio da cincia e tecnologia ( brazil ) and south - east cyt ( argentina ) . 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 . we acknowledge the usage of the hyperleda database ( http://leda.univ-lyon1.fr ) . this work has been partially supported by the brazilian institutions cnpq , capes and fapergs . barbosa f. k. b. , storchi - bergmann t. , cid fernandes r. , winge c. , schmitt h. , 2006 , mnras , 371 , 170 . bentz m. c. , walsh j. l. , barth a. j. , baliber n. , bennert v. n. , canalizo g. , filippenko a. v. , ganeshalingam m. , gates e. l. , greene j. e. , hidas m. g. , hiner k. d. , lee n. , li w. , malkan m. a. , minezaki t. , sakata y. , serduke f. j. d. , silverman j. m. , steele t. n. , stern d. , street r. a. , thornton c. e. , treu t. , wang x. , woo j. h. , yoshii y. , 2009 , apj , 705 , 199b . black j. h. , van dishoeck e. f. , 1987 , apj , 322 , 412 . boller , t. , keil , r. , trmper , j. , obrien , p. t. , reeves , j. & page , m. 2001 , a&a , 365 , 146 . cardelli , j. a. , clayton , g. c. & mathis , j. s. , 1989 , apj , 345 , 245 . crenshaw , d. m. , & kraemer , s. b. 2007 , apj , 659,250 . crenshaw , d. m. , kraemer , s. b. , schmitt h. r. , kaastra , j. s. , arav , n. , gabel , j. r. & korista , k. t. 2009 , apj , 698 , 281 . crenshaw , d. m. , kraemer , s. b. , schmitt , h. r. , jaff , y. l. , deo , r. p. , collins , n. r. & fischer , t. c. 2010a , apj , 139 , 871 . crenshaw , d. m. , schmitt , h. r. , kraemer , s. b. , mushotsky , r. f. & dunn , j. p. 2010b , apj , 708 , 419 . dale d. a. , sheth k. , helou g. , regan m. w. , httemeister s. , 2005 , aj , 129 , 2197d . davies r. i. i. , sternberg a. , lehnert m. d. , tacconi - garman l. e. , 2005 , apj , 633 , 105 davies r. i. , maciejewski w. , hicks e. k. s. , tacconi l. j. , genzel r. , engel h. , apj , 702 , 114d . dors o. l. jr . , riffel , r. a. , cardaci m. v. , hgele g. f. , krabbe a. c. , prez - montero , e.,rodrigues , i. , 2012 , mnras , 252 , 260 . fathi k. , storchi - bergmann t. , riffel r. a. , winge c. , axon d. j. , robinson a. , capetti a. , marconi a. , 2006 , apj , 641 , l25 ferruit , p. , wilson , a. s. , & mulchaey , j. 2000 , apjs , 128 , 139 . fischer , t. c. , crenshaw , d.m . , kraemer , s. b. , schmitt , h. r. & trippe , m. l. 2010 , aj , 140 , 577 . fischer , t. c. , crenshaw , d.m . , kraemer , s. b. , schmitt , h. r. , mushotsky , r. f. & ward , m. j. 1993 , apj , 727 , 71 . frank j. , king a. r. , raine d. j. , 2002 , accretion power in astrophysics , 3rd edn . cambridge univ . press , cambridge . forbes d. a. , ward m. j. , 1993 , apj , 416 , 150 . gonzlez delgado r. m. , & prez e. , 1996 , mnras , 278 , 737 . guillard p. , boulanger f. , cluver m. e. , appleton p. n. , pineau des forts g. , ogle p. , 2010 , a&a , 518 , 59 . holt j. , tadhunter c. , morganti r. , bellamy m. , gonzlez delgado r. m.,tzioumis a. , inskip k. j. , 2006 , mnras , 370 , 1633 . hollenbach d. , mckee c. f. , 1989 , apj , 342 , 306 . hunt l. k. , malkan m. a. , rush b. , bicay m. d. , nelson b. o. , stanga r. m. , webb w. , 1999 , apjs , 125 , 349 . kukula m. j. , pedlar a.,baum s. a. , odea c. p. , 1995 , mnras , 276 , 1262 . larkin j. e. , armus l. , knop r. a. , soifer b. t. , matthews k. , 1998 , apjs , 114 , 59 . malkan m. a. , gorjian v. , tam r. , 1998 , apjs , 117 , 25 . maloney p. r. , hollenbach d. j. , tielens a. g. g. m. , 1996 , apj , 466 , 561 . mazzalay x. , rodrguez - ardila a. , komossa s. , 2010 , mnras , 405 , 1315 . mazzalay x. et al . , 2013 , mnras , 428 , 2389 . mcgregor p. j. et al . , 2003,proc . spie , 4841 , 1581 . morganti r. , tadhunter c. n. , oosterloo t. a. , 2005 , a&a , l9 , l13 . mller - snchez f. , davies r. i. , eisenhauer f. , tacconi l. j. , genzel r. , sternberg a. , 2006 , a&a , 454 , 492 . mller - snchez f. , prieto m. a. , hicks e. k. s. , vives - arias h. , davies r. i. , malkan m. , tacconi l. j. , genzel r. , 2011 , apj . , 739 , 69 . mller snchez f. , davies r. i. , genzel r. , tacconi l. j. , eisenhauer f. , hicks e. k. s. , friedrich s. , sternberg a. , 2009 , apj , 691 , 749 . mundell c. g. , shone d. l. , 1999 , mnras , 304 , 475 . oliva e. et al . , 2001 , a&a , 369 , l5 . riffel r. , rodrguez - ardila a. , pastoriza m. g. , 2006b , a&a , 457 , 61 . riffel r. , rodrguez - ardila a. , aleman i. , brotherton m. s. , pastoriza m. g. , bonatto c. j. & dors o. l. 2013a , mnras , 430 , 2017 . riffel r. a. , 2010 , ap&ss , 327 , 239 . riffel r. a. , sorchi - bergmann t. , winge c. , barbosa f. k. b. , 2006a , mnras , 373 , 2 . riffel r. a. , storchi - bergmann t. , winge c. , mcgregor p. j. , beck t. , schmitt h. , 2008 , mnras , 385 , 1129 . riffel r. a. , storchi - bergmann t. , dors o. l. , winge c. , 2009 , mnras , 393 , 783 . riffel r. a. , storchi - bergmann t. , 2010 , mnras , 411 , 469 . riffel r. a. , storchi - bergmann t. , nagar n. m. , 2010 , mnras , 404 , 166 . riffel r. , riffel r. a. , ferrari f. , storchi - bergmann t. , 2011 , mnras , 416 , 493 . riffel r. a. et al . , 2013b , mnras , 429 , 2587 . riffel r. a. , storchi - bergmann t. , winge , c. 2013 , mnras , 430 , 2249 . rodrguez - ardila a. , pastoriza m. g. , viegas s. , sigut t. a. a. , pradhan a. k. , 2004 , a&a , 425 , 457 . rodrguez - ardila a. , contini m. , viegas s. m. , 2005 , mnras , 357 , 220 . rodrguez - ardila a. , prieto m. a. , viegas s. , gruenwald r. , 2006 , apj , 653 , 1098 . rodruez - ardila a. , riffel r. , pastoriza m. g. , 2005 , mnras , 364 , 1041 . riffel r. , rodrguez - ardila a. , pastoriza m. g. , 2006b , a&a , 457 , 61 . scoville , n. z. , hall , d. n. b. , kleinmann , s. g. , & ridgway , s. t. 1982 , 253 , 136 .. schmitt h. r. , kinney a. l. , 1996 , apj , 463 , 498 . schnorr mller , a. , storchi - bergmann , t. , riffel , r. a. , ferrari , f. , steiner , j. e. , axon , d. j. , robinson , a. 2011 , mnras , 413 , 149 simpson c. , forbes d. a. , baker a. c. , ward m. j. , 1996 , mnras , 283 , 777 . storchi - bergmann t. , winge c. , ward m. , wilson a. s. , 1999 , mnras , 304 , 35 . storchi - bergmann t. , dors o. , jr , riffel r. a. , fathi k. , axon d. j. , robinson a. , 2007 , apj , 670 , 25 . storchi - bergmann t. , mcgregor p. , riffel r. , rogemar a. , simes lopes r. , beck t. , dopita m. , 2009 , mnras , 394 , 1148 . storchi - bergmann t. , simes lopes r. , mcgregor p. riffel , rogemar a. , beck t. , martini p. , 2010 , mnras , 402 , 819 . veilleux s. , cecil g. , bland - hawthorn j. , 2005 , ara&a , 43 , 769 . veilleux s. a. , goodrich r. w. , hill g. j. , 1997 , apj , 477 , 631 . wilson a. s. , braatz j. a. , heckman t. m. , krolic j. h. , miley g. k. , 1993 , apj , 419 , l61 . | we have mapped the emission - line flux distributions and ratios as well as the gaseous kinematics of the inner 450 pc radius of the seyfert 1 galaxy mrk766 using integral field near - ir j- and k@xmath0-band spectra obtained with the gemini nifs at a spatial resolution of 60 pc and velocity resolution of 40 kms@xmath1 .
emission - line flux distributions in ionized and molecular gas extend up to @xmath2 300 pc from the nucleus .
coronal [ six]@xmath3 m line emission is resolved , being extended up to 150 pc from the nucleus . at the highest flux levels ,
the [ feii]@xmath41.257@xmath5 m line emission is most extended to the south - east , where a radio jet has been observed.the emission - line ratios [ feii]@xmath6m/@xmath7 and @xmath8m / br@xmath9 show a mixture of starburst and seyfert excitation ; the seyfert excitation dominates at the nucleus , to the north - west and in an arc - shaped region between 02 and 06 to the south - east at the location of the radio jet . a contribution from shocks at this location
is supported by enhanced [ feii]/[pii ] line ratios and increased [ feii ] velocity dispersion .
the gas velocity field is dominated by rotation that is more compact for h@xmath10 than for pa@xmath11 , indicating that the molecular gas has a colder kinematics and is located in the galaxy plane .
there is about 10@xmath12 m@xmath13 of hot h@xmath10 , implying @xmath2 10@xmath14 m@xmath13 of cold molecular gas . at the location of the radio jet
, we observe an increase in the [ feii ] velocity dispersion ( 150 km s@xmath1 ) , as well as both blueshift and redshifts in the channel maps , supporting the presence of an outflow there .
the ionized gas mass outflow rate is estimated to be @xmath2 10 m@xmath13 yr@xmath1 , and the power of the outflow @xmath2 0.08 l@xmath15 .
[ firstpage ] galaxies : individual ( mrk766 ) galaxies : active galaxies : seyfert galaxies : nuclei galaxies : kinematics and dynamics |
You are an expert at summarizing long articles. Proceed to summarize the following text:
the existence of inactive supermassive black holes in the nuclei of massive nearby galaxies ( magorrian et al . 1998 ) is a sure sign that most galaxies underwent substantial active phases . furthermore , the correlation between the amount of material accreted ( black hole mass ) and the galaxy mass indicates important links between galaxy formation and agn activity . one implication is that the most massive black holes live in the most massive galaxies which reside in the most massive dark matter halos . the @xmath0 quasars being discovered in the sloan digital sky survey ( see fan , this volume ) have exceptionally high luminosities ( @xmath1 ) . the available evidence suggests that these luminosities are not amplified by gravitational lensing ( fan et al . 2003 ) or beaming ( pentericci et al . 2002 ; willott et al . 2003 ) . assuming that these quasars are radiating at the eddington limit gives a lower limit on their black hole masses of several billion solar masses . these are comparable to the black holes inside the largest dominant cluster ellipticals at the present time . applying local calibrations from this black hole mass to galaxy and halo masses suggests that these quasars reside in halos with mass @xmath2 ( e.g. fan et al . lcdm simulations show that structure grows ` bottom - up ' with the largest halos typically collapsing at the latest epochs . halos with mass @xmath3 at @xmath4 are therefore extremely rare ( the fraction of mass in such halos is of order @xmath5 ; sheth & tormen 1999 ) there is evidence that at least some of these quasars are located in massive galaxies . sub - millimetre photometry of several of them show very high far - infrared luminosities , implying extreme starbursts of @xmath6 ( bertoldi et al . 2003 ; priddey et al . the most distant quasar , sdssj1148 + 5251 at @xmath7 , contains @xmath8 of molecular gas , again indicative of a massive , primeval galaxy ( walter et al . these distant quasars , which are thought to be residing in massive galaxies , are therefore ideal places to search for the first clusters of galaxies . the most straightforward method for identifying galaxies at such high redshifts is the lyman - break technique a large discontinuity in the spectral slope due to absorption by neutral hydrogen ( e.g. steidel et al . 1996 ) . the increase in the optical depth of hydrogen absorption at such high redshifts ( e.g. songaila & cowie 2002 ) makes the lyman - break technique even more effective at @xmath9 than at lower redshifts . to search for galaxies in these putative mass overdensities , we carried out a search for lyman - break galaxies in fields of three @xmath10 sdss quasars . we have imaged three quasar fields with the gmos - north imaging spectrograph on the gemini - north telescope : sdssj1030 + 0524 at @xmath11 , sdssj1048 + 4637 at @xmath12 and sdssj1148 + 5251 at @xmath7 . the imaging field - of - view is 5.5 arcmin on a side , equivalent to a co - moving size of 13 mpc at @xmath13 . observations were carried out in queue mode during november and december 2003 . the typical seeing fwhm of the observations is in the range 0.5 to @xmath14 . typical exposure times are @xmath15 hours in the @xmath16-band and @xmath17 hours in the @xmath18-band . the relative exposure times were designed to give similar sensitivity in the two bands for very red objects with colours of @xmath19 . full details of these observations and their reduction will be published elsewhere . -band magnitude derived from recovery of simulated galaxies . the curves are quite similar for the three different quasar fields . dotted lines indicate the location of a completeness ratio of 0.8 and the adopted complete magnitude limit of @xmath20 . ] detection of objects in the reduced images was performed using the sextractor software ( bertin & arnouts 1996 ) . the @xmath16-band was selected as the primary detection waveband since @xmath0 galaxies are expected to have @xmath21 and may therefore be undetected at @xmath18 . sextractor was run in `` double - image '' mode to determine @xmath18-band measurements for objects detected in @xmath16 . magnitudes on the ab system were measured in circular apertures of diameter @xmath22 . aperture corrections were applied statistically to the @xmath16-band magnitudes by fitting a linear function to the difference between the total magnitude and aperture magnitude as a function of aperture magnitude . typical magnitude limits ( @xmath23 limits in @xmath24 apertures ) are @xmath25 and @xmath26 . to assess the completeness of the @xmath16-band catalogues we consider both the observed number counts and the recovery of simulated objects . the number counts in the three fields do not differ significantly from each other . they agree well with the @xmath16-band counts determined from a much larger area survey ( 0.2 square degrees or thirty times the gmos - north field - of - view ) by capak et al . the number counts in the quasar fields begin to change slope at @xmath27 and turn over at @xmath28 , indicating this is where the sample becomes incomplete . the source recovery as a function of magnitude was determined by populating the images with artificial galaxies and then using sextractor to attempt to detect these objects . about 10000 artificial galaxies with magnitudes in the range @xmath29 were placed into copies of the @xmath16 images of each quasar . regions of the images already occupied by objects were masked out of the process to eliminate incompleteness due to blending . the resulting completeness ratio is plotted as a function of magnitude for the three quasar fields in fig.[fig : comp ] . the completeness in all the fields is fairly flat at close to 1 up to @xmath30 and then begins to decline . the rapid decline occurs at @xmath27 and the completeness drops to 0.5 by @xmath31 . all the fields have completeness @xmath32 at @xmath20 and we adopt this as the magnitude at which completeness begins to become an issue . this analysis with simulated objects agrees well with the results for the number counts discussed previously . the @xmath0 sdss quasars have @xmath33 and colours of @xmath34 respectively for j1030 , j1048 and j1148 . quasars and star - forming galaxies have broadly similar spectra over the rest - frame wavelength range @xmath35 nm probed by the @xmath18 and @xmath16 filters . their spectra are dominated by a large break due to absorption by neutral hydrogen . therefore one would expect companion galaxies to have comparable colours to the quasars . a colour - magnitude diagram for objects in the field of sdssj1030 + 0524 is shown in fig.[fig : colmags ] . most objects have colours in the range @xmath36 as is well known from previous surveys ( dickinson et al . 2004 ; capak et al . 2004 ) . also plotted are curves showing the colour - magnitude relation for two different types of galaxy . model galaxy spectra were generated from the bruzual & charlot ( 2003 ) spectral synthesis code with lyman forest absorption evolution matching the observations of songaila & cowie ( 2002 ) . the upper curve is a present - day l@xmath37 elliptical which formed all of its stars in a starburst starting at @xmath38 with a characteristic timescale of 1gyr and evolved since without merging . note that there is no dust extinction assumed for this model , but in reality the dust extinction would increase with redshift ( due to evolution and @xmath39-correction ) making the galaxy fainter than plotted at higher redshifts . the lower curve is a l@xmath37 lyman - break galaxy model where the galaxy is observed 0.5gyr into a constant star formation rate starburst . a search was made for objects which could plausibly be high - redshift galaxies . the @xmath18-band dropout selection criteria adopted were snr in the @xmath16-band @xmath40 and a colour of @xmath41 . possible candidates were inspected and magnitudes checked to ensure their unusual colours are not spurious . a total of four objects satisfying these criteria were found ; two in the field of sdssj1030 + 0524 , shown in fig.[fig : colmags ] with filled symbols , and one in each of the other two fields . only one of these four galaxies has a magnitude brighter than the completeness limit of @xmath20 . this object has @xmath42 which is quite close to the colour selection value and the size of the uncertainty means it is quite plausible that photometric errors have scattered the colour into the dropout range . the measured colour is about @xmath43 away from the colours of the sdss quasars , which suggests that if it is a high - redshift galaxy , then it is most likely foreground to the quasars with a redshift in the range @xmath44 . the other three objects have @xmath45 and only lower limits on their @xmath46 colours which lie in the range @xmath47 . -band in the field of the @xmath11 quasar sdssj1030 + 0524 . the completeness limit of @xmath20 is shown with a dot - dashed line . the high - redshift galaxy selection criterion of @xmath48 is marked with the dashed line . the dotted line represents the colour of an object which is just detected at the @xmath23 level in the @xmath18-band as a function of the @xmath16-band magnitude . most objects with measured zero or negative flux in the @xmath18-band are plotted at @xmath49 . the exceptions to this are objects which pass the @xmath18-band dropout selection criteria discussed in the text . these are shown with star symbols and lower limits on the @xmath18-band @xmath23 line . the dots at @xmath48 indicate sources which are detected at less than @xmath50 at @xmath16-band and hence have very large uncertainties on their magnitudes and colours and in some cases may be spurious . the labelled curves show the colour and magnitude as a function of redshift for an evolving l@xmath37 elliptical ( upper curve ) and a non - evolving l@xmath37 lyman - break galaxy ( lower curve ) see text for more details . ] our @xmath16-band images are complete at the greater than 80% level to @xmath20 . at @xmath51 , every single @xmath16-band object detected on our images has a counterpart at the @xmath52 level in the @xmath18-band image . at fainter magnitudes , this is no longer true and our constraints on the @xmath46 colours of objects at these magnitudes becomes very weak due to uncertainty in both the @xmath16 and @xmath18 magnitudes . we now consider the number of @xmath18-band dropouts we could have expected to find under the assumption that the quasar fields are `` random '' and show no enhancement due to the existence of the quasars . the best comparison dataset which goes deep enough over a wide area is the _ hubble space telescope _ acs imaging of the goods regions ( giavalisco et al . these observations give a surface density of objects with @xmath53 and @xmath48 of 0.02 arcmin@xmath54 ( dickinson et al 2004 ; bouwens et al . the total sky area we have surveyed with gmos - north is 82 arcmin@xmath55 . therefore on the basis of the goods observations we would expect 1.6 @xmath18-band dropouts in our total area . our finding of one dropout is entirely consistent with the expectations for blank fields . our observations show that these quasar fields do not exhibit an excess of luminous companion galaxies . the magnitude limit of @xmath20 corresponds to a uv luminosity of @xmath56 at a redshift of @xmath13 . this is equivalent to an unobscured star formation rate of @xmath57 , assuming the conversion given in madau , pozzetti & dickinson ( 1998 ) . the few known galaxies at redshifts @xmath58 discovered in narrow - band surveys have star formation rates derived from their uv luminosities comparable to this limit ( hu et al . 2002 ; kodaira et al . for comparison , the millimeter detections of dust in j1048 and j1148 imply total star formation rates @xmath59 in the host galaxies of the quasars ( bertoldi et al . 2003 ) . how do we explain the lack of companion galaxies with high star formation rates ? one possibility is that galaxies undergoing intense star formation do exist but they are heavily extinguished by dust . such galaxies may be detectable via their far - infrared emission and we are conducting a program of sub - mm imaging around these quasars to identify such galaxies . deep observations with the irac camera on the _ spitzer space telescope _ would also be able to reveal these obscured galaxies . another possibility is that massive companion galaxies exist but they are not observed during a period of intense star formation . this possibility seems unlikely since the youth of the universe at this redshift means galaxies would likely be gas - rich and forming stars , especially if located in a dense environment which is not yet virialized . finally , we raise a question mark over the notion that these rare massive black holes actually reside in the rarest , densest peaks in the dark matter distribution . perhaps the correlation between black hole mass and dark matter halo mass ( or halo circular velocity see wyithe & loeb 2004 ) does not apply to these quasars and they are actually located inside less massive dark matter halos . bertin , e. , arnouts , s. 1996 , a&as , 117 , 393 bertoldi , f. , et al . 2003 , a&a , 406 , l55 bouwens , r. j. , et al . 2004 , apj , 606l , 25 bruzual , g. , charlot , s. 2003 , mnras , 344 , 1000 capak , p. , et al . 2004 , aj , 127 , 180 dickinson , m. , et al . 2004 , apj , 600 , l99 fan , x. , et al . 2001 , aj , 122 , 2833 fan , x. , et al . 2003 , aj , 125 , 1649 giavalisco , m. , et al . 2004 , apj , 600 , l93 hu , e. m. , et al . 2002 , apj , 568 , l75 kodaira , k. , et al . 2003 , pasj , 55l , 17 madau , p. , pozzetti , l. , dickinson , m. 1998 , apj , 498 , 106 magorrian j. , et al . 1998 , aj , 115 , 2285 pentericci , l. , et al . 2002 , aj , 123 , 2151 priddey , r. s. , et al . 2003 , mnras , 344l , 74 sheth , r. k. , tormen , g. 1999 , mnras , 308 , 119 songaila , a. , cowie , l. l. , 2002 , aj , 123 , 2183 steidel , c. c. , et al . 1996 , apj , 462l , 17 walter , f. , et al . 2003 , nature , 424 , 406 willott , c. j. , mclure , r. j. , jarvis , m. j. 2003 , apj , 587 , l15 wyithe , j. s. b. , loeb , a. 2004 , apj , submitted , astro - ph/0403714 | we have obtained deep , multi - band imaging observations around three of the most distant known quasars at redshifts @xmath0 .
standard accretion theory predicts that the supermassive black holes present in these quasars were formed at a very early epoch .
if a correlation between black hole mass and dark matter halo mass is present at these early times , then these rare supermassive black holes will be located inside the most massive dark matter halos .
these are therefore ideal locations to search for the first clusters of galaxies .
we use the lyman - break technique to identify star - forming galaxies at high redshifts .
our observations show no overdensity of star - forming galaxies in the fields of these quasars .
the lack of ( dust - free ) luminous starburst companions indicates that the quasars may be the only massive galaxies in their vicinity undergoing a period of intense activity . |
You are an expert at summarizing long articles. Proceed to summarize the following text:
natural processes proceed simultaneously on many different scales interacting with each other . this is especially important in astrophysics , where an astronomical observation or a hypothesized system can only be understood by modeling processes on scales spanning many orders of magnitude . thus , also nuclear physics and astrophysics are closely entwined . nuclear reactions power quiescent burning of stars and cause the most powerful explosions known . they not only release or transform energy but also change the composition of the matter in which they occur and thus are responsible for the range of chemical elements found on our planet and throughout the universe . finally , to understand the properties of matter at extreme density and/or density , the nuclear equation of state has to be known . it determines the properties of neutron stars and the latest stages of the life of stars with more than @xmath1 which end their lives in a supernova explosion . it is evident that nuclear physics input is essential for many astrophysical models and this fact is represented in the field of nuclear astrophysics . there are different perceptions on how to define this field . some limit it to the application of nuclear physics to reactions of astrophysical interest . a more comprehensive , and perhaps more adequate , definition includes the more astrophysical aspects in the investigation of nuclear processes and nucleosynthesis in astrophysical sites through reaction networks . in any case , the interests of astrophysics emphasize different aspects than those of basic nuclear physics and this makes nuclear astrophysics a distinct research area . from the nuclear point of view , astrophysics involves low - energy reactions with light projectiles on light , intermediate , and heavy nuclei . although nuclear physics has moved to higher energies in the last decades , low - energy reactions are not well enough explored and still offer considerable challenges to both experiment and theory , even for stable target nuclei . explosive conditions in astrophysics favor the production of nuclei far off stability and reaction rates for these have to be predicted across the nuclear chart . this proves very difficult , especially for first - principle methods , due to the complexity of the nuclear many - body problem . the calculation of astrophysical reaction rates also includes special requirements and processes not studied in nuclear physics so far . among these are effects appearing in low energy , subcoulomb reactions and reactions on excited states of target nuclei . depending on the conditions , plasma effects also have to be considered because these alter the reaction rates . this includes , e.g. , the shielding of coulomb barriers through electrons , pycnonuclear burning in the lattice of a high - density plasma , and the modification of nuclear partition functions at very high plasma temperatures . this paper focuses on nuclear reactions in astrophysics and , more specifically , on reactions with light ions ( nucleons and @xmath0 particles ) proceeding via the strong interaction . it is intended to present the basic definitions essential for studies in nuclear astrophysics , to point out the differences between nuclear reactions taking place in stars and in a terrestrial laboratory , and to illustrate some of the challenges to be faced in theoretical and experimental studies of those reactions . the sensitivity of the reaction rates to the uncertainties in the prediction of various nuclear properties is explored and some guidelines for experimentalists are also provided . the discussion revolves around the relevant quantities for astrophysics , which are the astrophysical reaction rates . the impact of using different models or data is always presented with respect to the possible modification of the reaction rates , not the cross sections . at first , the basic equations through which nuclear processes enter astrophysical models are introduced in sec . [ sec : netrate ] . the astrophysically relevant energy ranges are defined in sec . [ sec : energies ] . the thorough discussion of the special stellar effects affecting reactions in sec . [ sec : stellareffects ] is the heart of this review . stellar and effective cross sections are derived in sec . [ sec : stellar ] , then the relation between laboratory reactions and those occurring in a stellar plasma is investigated in sec . [ sec : stellarexp ] . as it is important to realize when individual reactions are important and when they are not , reaction equlibria are introduced in sec . [ sec : equilibria ] . cross sections in a plasma are affected by the free electrons present which shield the nuclear charge . this is explained in sec . [ sec : electro ] . finally , astrophysically relevant reaction mechanisms are reviewed in sec . [ sec : mech ] and reactions through isolated resonances ( sec . [ sec : reso ] ) , in systems with high nuclear level density ( sec . [ sec : statmod ] ) , and direct reactions ( sec . [ sec : direct ] ) are discussed separately . this includes a detailed discussion of the sensitivities of the rates on the required input in secs . [ sec : relevance ] and [ sec : directsensi ] . section [ sec : conclusion ] presents a brief conclusion . nuclear reactions are the engine of stellar evolution and determine the overall production of the known chemical elements and their isotopes in a variety of nucleosynthesis patterns . a detailed understanding of the characteristic production and depletion rates of nuclei within the framework of a nucleosynthesis process is crucial for reliable model predictions and the interpretation of the observed abundances . instead of the number @xmath2 of nuclei of a given species per volume @xmath3 ( the number density @xmath4 ) , it is advantageous to use a quantity independent of density changes : the abundance @xmath5 , where @xmath6 is the plasma density and @xmath7 denotes avogadro s number . the change of abundances @xmath8 with time due to nuclear processes is traced by coupled differential equations . due to the nature of the involved reactions and the vastly different timescales appearing , the equation system is non - linear and stiff . in addition , for complete solubility of the coupled equations the number of equations @xmath9 has to equal the number of involved nuclei acting as reaction partners and thus an equation matrix of size @xmath10 has to be solved . nucleosynthesis processes include thousands of nuclides and tens of thousands reactions . this still makes it impossible to fully couple such a reaction network to a full set of hydrodynamic equations as would be required for a complete modeling of nucleosynthesis in a given astrophysical site . a reaction network generally can be written as @xcite @xmath11 where @xmath12 numbers the nucleus , @xmath13 is the @xmath14th rate for destruction or creation of the @xmath15th nucleus without a nuclear projectile involved ( this includes spontaneous decay , lepton capture , photodisintegration ) , and @xmath16 is the rate of the @xmath14th reaction involving a nuclear projectile and creating or destroying nucleus @xmath15 . similarly , we have three - body reactions where nucleus @xmath15 is produced or destroyed together with two other ( or similar ) nuclei . reactions with more participants ( denoted by @xmath17 above ) are unlikely to occur at astrophysical conditions and are usually neglected . the quantities @xmath18 , @xmath19 , and @xmath20 are positive or negative integer numbers specifying the amount of nuclei @xmath15 produced or destroyed , respectively , in the given process . as shown below , the rates @xmath21 , @xmath22 , and @xmath23 contain the abundances of the interacting nuclei . rates of type @xmath21 depend on one abundance ( or number density ) , rates @xmath22 depend on the abundances of two species , and rates @xmath23 on three . the rates @xmath13 appearing in the first term of eq . ( [ eq : network ] ) are reactions per time and volume , and only contain the abundance @xmath24 . for example , @xmath25 is simply @xmath26 for @xmath27-decays . the factor @xmath28 is the usual decay constant ( with the unit 1/time ) and is related to the half - life @xmath29 of the decaying nucleus @xmath14 . it has to be noted that some decays depend on the plasma temperature and thus @xmath30 is not always constant , even for decays ( see eq . ( [ eq : stelldecay ] ) in sec . [ sec : stellar ] ) . two - body rates @xmath22 include the abundances of two interacting particles or nuclei . in general , target @xmath31 and projectile @xmath32 follow specific thermal momentum distributions @xmath33 and @xmath34 in an astrophysical plasma . with the resulting relative velocities @xmath35 , the number of reactions per volume and time is given by @xmath36 and involves the reaction cross section @xmath37 as a function of velocity , the relative velocity @xmath35 and the thermodynamic distributions of target and projectile @xmath33 and @xmath34 . the evaluation of this integral depends on the type of particles ( fermions , bosons ) and distributions which are involved . however , many two - body reactions can be simplified and effectively expressed similarly to one - body reactions , only depending on one abundance ( or number density ) . if reaction partner @xmath32 is a photon , the relative velocity is always @xmath38 and the quantities in the integral do not depend on @xmath34 . this simplifies the rate expression to @xmath39 where @xmath40 stems from an integration over a planck distribution for photons of temperature @xmath41 . this is similar to the decay rates introduced earlier and therefore we replaced @xmath22 by @xmath21 in our notation and can include this type of reaction in the first term of eq . ( [ eq : network ] ) . a similar procedure is used for electron captures by protons and nuclei . because the electron is about 2000 times less massive than a nucleon , the velocity of the nucleus is negligible in the center - of - mass system in comparison to the electron velocity ( @xmath42 ) . the electron capture cross section has to be integrated over a fermi distribution of electrons . the electron capture rates are a function of the plasma temperature @xmath41 and the electron number density @xmath43 . in a neutral , completely ionized plasma , the electron abundance @xmath44 is equal to the total proton abundance @xmath45 and thus @xmath46 again , we have effectively a rate per target @xmath47 ( with unit 1/time ) similar to the treatment of decays earlier and a rate per volume including the number density of only one nucleus . we denote the latter by @xmath21 and use it in the first term of eq . ( [ eq : network ] ) . this treatment can be applied also to the capture of positrons , being in thermal equilibrium with photons , electrons , and nuclei . furthermore , at high densities ( @xmath48g@xmath49 ) the size of the neutrino scattering cross section on nucleons , nuclei , and electrons ensures that enough scattering events occur to lead to a continuous neutrino energy distribution . then also the inverse process to electron capture ( neutrino capture ) can occur as well as other processes like , e.g. , inelastic scattering , leaving a nucleus in an excited state which can emit nucleons and @xmath0 particles . such reactions can be expressed similarly to photon and electron captures , integrating over the corresponding neutrino distribution . in the following , we focus on the case of two interacting nuclei or nucleons as these reactions will be extensively discussed in the following sections . ( we mention in passing that eq . ( [ eq:2body ] ) can be generalized to three and more interacting nuclear species by integrating over the appropriate number of distributions , leading to rates @xmath23 and higher order terms in eq . ( [ eq : network ] ) . ) the velocity distributions in the rate definition in eq . ( [ eq:2body ] ) can be replaced by energy distributions . furthermore , it can be shown that the two distributions can be replaced by a single one in the center - of - mass system.@xcite then the two - body rate @xmath22 is defined as an interaction of two reaction partners with an energy distribution @xmath50 according to the plasma temperature @xmath41 and a reaction cross section @xmath51 specifying the probability of the reaction in the plasma : @xmath52 the factor @xmath53 with the kronecker symbol @xmath54 is introduced to avoid double counting . the nuclear cross section @xmath55 is defined as in standard scattering theory by @xmath56 however , in an astrophysical plasma reactions not only proceed on the ground state of a nucleus but also from excited states . this is implied in the notation for the modified cross section @xmath57 , contrasting the usual laboratory cross section ( denoted by @xmath55 or @xmath58 ) for reactions acting only on the ground state of the target nucleus . the implications of using such a cross section modified in the stellar plasma , instead of the usual laboratory one , will be discussed in sec . [ sec : stellar ] . it should be noted that @xmath59 may be a function not only of energy but also of plasma temperature . the distribution of kinetic energies of nuclei in an astrophysical plasma with temperature @xmath41 follows a maxwell - boltzmann distribution ( mbd ) @xmath60 and we obtain finally : @xmath61 here , @xmath62 denotes the reduced mass of the two - particle system and @xmath63 is the reaction rate per particle pair or _ reactivity _ under stellar conditions . the angle brackets stand for the appropriate averaging , i.e. integration , over the energy distribution . for the remainder of the paper we will be concerned with the determination of this reactivity and the involved cross sections , respectively . before we proceed to the details of the determination of the reaction cross sections , it is instructive to further investigate the rate equation and to derive the relevant energies at which the nuclear reaction cross sections have to be known . although the integral in eq . ( [ eq : rate ] ) runs to infinity , the mbd folded with the cross section selects a comparatively narrow energy range with non - negligible contributions to the total value of the integral . historically , this energy range is called the gamow window because gamow realized early on the astrophysical relevance of the fact that if the energy dependence of the cross section is dominated by the coulomb barrier between the projectile and the target the integrand in eq . ( [ eq : rate ] ) can be factorized as@xcite @xmath64 where @xmath65 is the astrophysical @xmath65-factor @xmath66 which is assumed to be only weakly dependent on the energy @xmath67 for non - resonant reactions . the second exponential in eq . ( [ eq : penet ] ) is called the gamow factor and contains an approximation of the coulomb penetration with the sommerfeld parameter @xmath68 where @xmath69 , @xmath70 are the charges of projectile @xmath32 and target @xmath31 , respectively , and @xmath62 is their reduced mass . while the first exponential ( the tail of the mbd ) decreases with increasing energy , the gamow factor increases , leading to a confined peak of the integrand , the so - called gamow peak . the location of the peak @xmath71 is shifted to higher energies with respect to the maximum of the mbd at @xmath72 ( @xmath73 mev when @xmath74 is the plasma temperature in gk ) . the width of the peak gives the astrophysically relevant energy range in which most of the reactions will take place . in absence of a coulomb barrier the energy dependence of the non - resonant cross section is roughly given by the one of the wave number of the particle ( @xmath75 ) folded with the angular momentum barrier . this does not , however , lead to a relevant shift of the peak of the integrand compared to the peak of the mbd . thus , the effective energy window for neutrons is simply the peak of the mbd . the above considerations concerning the location and size of the energy window have given rise to simple approximation formulae extensively used by experimentalists to estimate the energies of interest . for instance , with a charged projectile the location @xmath76 and width @xmath77 of the gamow window is often computed from @xmath78 which is derived from eq . ( [ eq : penet ] ) assuming a gaussian shape of the peak and using appropriate numerical constants , yielding @xmath76 and @xmath77 in mev.@xcite this approximation of the gamow window @xmath79 is valid for some but not all cases because it is oversimplified . the above factorization with the given sommerfeld parameter including the charges of projectile @xmath32 and target @xmath31 implicitly assumes that the energy dependence of the cross section is given by the coulomb penetration in the entrance channel @xmath80 of the reaction @xmath81 . it has been realized , however , that sometimes resonances below the gamow window derived with the above approximation significantly contribute to the reaction rate for certain capture reactions.@xcite in those considered cases the energy dependence of the cross section is dominated by the energy dependence of the @xmath82 width in the exit channel instead of the charged particle width in the entrance channel . this can be generalized@xcite and leads to the important realization that the energy dependence of the integrand @xmath83 has to be numerically examined in order to derive reliable energy windows . this has been generally and extensively studied in ref . . here , only a few examples are shown . figure [ fig : sn112pa ] shows a comparison between the actual @xmath83 and the integrand assumed with the standard approximation . in this case , the relative shift of the energy window is to higher energy because of the higher coulomb barrier in the exit channel . in other cases , the shifts can also be to much lower energy than predicted by the standard approximation . for a detailed understanding of these differences one has to realize that both resonant and hauser - feshbach cross sections ( see secs . [ sec : reso ] , [ sec : statmod ] ) can be expressed as@xcite @xmath84 with @xmath85 being either breit - wigner widths or averaged hauser - feshbach widths , depending on the context . the width of the entrance channel is given by @xmath86 , the one of the exit channel by @xmath87 , and the total width including all possible emission channels from a given resonance or compound state with spin @xmath88 by @xmath89 it has become common knowledge that a cross section of the form shown in eq . ( [ eq : cs ] ) is determined by the properties of the smaller width in the numerator if no other channels than the entrance and exit channel contribute significantly to @xmath90 ( see also sec . [ sec : sensigeneral ] ) . then @xmath90 cancels with the larger width in the numerator and the smaller width remains . ( the effect is less pronounced and requires a more detailed investigation when other channels are non - negligible in @xmath90 . ) in consequence , the energy - dependence of the cross section will then be governed by the energy dependence of this smallest @xmath91 . only if this happens to be the charged - particle ( averaged ) width in the entrance channel , the use of the standard formula for the gamow window ( eqs . [ eq : e0approx ] , [ eq : deltaapprox ] ) will be justified . since @xmath92 and @xmath93 have different energy dependences , it will depend on the specific energy ( weighted by the mbd ) which of the widths is smaller . therefore , the above approximation should not be applied blindly but rather the actual gamow windows have to be determined from the true energy dependence by inspection of the integrand in eq . ( [ eq : rate ] ) . extensive tables of revised effective energy ranges for astrophysics from such a numerical inspection are given in ref . . these ranges can be shifted by several mev to higher or lower energy compared to the ones obtained with the standard formula . furthermore it is found that the assumption of a gaussian shape of @xmath83 is untenable for the majority of cases with intermediate and heavy target nuclei . rather , the integrand @xmath83 may show a pronounced asymmetry around its maximum value . therefore , the energy of the maximum alone is not sufficient to determine the astrophysically relevant energy range . although derived from cross sections of a specific model prediction , the energy windows given in ref . are supposed to be robust . this can be understood by realizing that they mainly depend on the relative energy dependence of the acting reaction channels and not the absolute value of the cross sections . this dependence is governed by the relative energy and the coulomb barrier seen in each reaction channel . therefore , the limits of the energy windows are set by the knowledge of the charges of the nuclei involved in the different reaction channels , and the reaction @xmath94-values . only the latter may be unknown for nuclei far off stability and mass measurements may have an impact . another important consequence of using the correct energy dependence is that different reactions may not have necessarily the same effective energy window , even when projectile and target nucleus are the same . this is immediately seen when considering the case of a reaction with a positive @xmath94-value for one channel but a negative one for another reaction channel . the astrophysically relevant energy window of the exothermic reaction may lie below the threshold of the other reaction channel . obviously , the relevant energy window for the endothermic channel can not open below the threshold energy and thus has to lie at higher energies than the one for the capture . a randomly chosen example for such a reaction pair would be @xmath95pd(p,@xmath82)@xmath96ag and @xmath95pd(p , n)@xmath95ag , with the energy windows at @xmath97 mev and @xmath98 mev , respectively , for a plasma temperature of 2 gk.@xcite below the ( p , n ) threshold at 5.06 mev , the ( p , n ) cross section is zero and does not give a contribution to the integral in eq . ( [ eq : rate ] ) . this also further illustrates the limitation of the standard approximation which yields identical energy windows for reactions with identical entrance channels but different exit channels . it has to be noted , though , that the effective energy windows only point out the energy range contributing mostly to the rate integral at given stellar temperature but do not make a statement on the size of the rate or its astrophysical relevance . it has already been mentioned that eq . ( [ eq : cs ] ) applies to reactions either exhibiting isolated resonances treatable by a breit - wigner resonance formula or smooth cross sections stemming from an averaging over a large number of narrowly spaced and unresolved resonances . therefore the derived energy windows are also applicable to obtain the relevant energy ranges in which narrow resonances have to be considered . they do not , however , specify the relative strengths of the resonances within a given window . definition ( [ eq : sfactor ] ) only makes sense when using the laboratory cross section @xmath99 ( see eq . ( [ eq : labcs ] ) in the following section ) . strictly speaking , with the laboratory cross section the energy windows apply to laboratory measurements only , i.e. to the determination of the ground state component of the actual stellar cross section . for low stellar temperatures and positive @xmath94-values ( see the discussion in sec . [ sec : stellar ] ) , however , this will dominate the stellar cross section . relevant energy windows can also be derived numerically for stellar cross sections ( see next section ) , of course , in the same manner . since these can not be measured ( yet ) , they may be of limited use , though . a general scrutiny of the astrophysically relevant energy windows up to a plasma temperature of 5 gk ( above this temperature , reaction equilibria are established which do not require the knowledge of individual rates ; see , e.g. , refs . ) shows that the appearing interaction energies are small by nuclear physics standards . for neutron - induced reactions , the encountered maximum energies are a few hundred kev , depending on the examined nucleosynthesis process ( e.g. , in the @xmath100 process they are more like 8 - 60 kev).@xcite the relevant energy windows are shifted to higher energies for charged reactants , with a few mev for reactions with protons and several mev up to about 10 mev for reactions involving @xmath0 particles . the formulae given in eqs . ( [ eq : e0approx ] ) , ( [ eq : deltaapprox ] ) are inadequate for the determination of these energies and should not be used anymore . in an astrophysical plasma , nuclei quickly ( on the timescale of nuclear reactions and scattering ) reach thermal equilibrium with all plasma components . this allows thermal excitation of nuclei which follows a boltzmann law and gives rise to the _ stellar _ cross section @xmath101 where the sum runs over all excited states @xmath102 of the target nucleus @xmath31 ( for simplicity , here we assume the projectile @xmath32 , i.e. the second reaction partner , does not have excited states ) with spin factor @xmath103 and excitation energy @xmath104 . thus , the stellar cross section is the sum of cross sections @xmath105 ( evaluated at their respective center - of - mass energies @xmath106 with @xmath107 when @xmath108 ) for reactions on a nucleus in excited state @xmath102 , weighted by the population coefficient @xmath109 . the above relation can be derived from a saha equation@xcite . it is to be noted that the stellar cross section depends on energy _ and _ temperature , contrary to the usual cross section which only is a function of energy . the quantity @xmath110 is the partition function of the nucleus . often , the partition function normalized to the ground state @xmath111 \end{split } \label{eq : partfuncint}\ ] ] is used ( the ground state is labeled as @xmath112 , the first excited state as @xmath113 , ) . equation ( [ eq : partfuncint ] ) shows how the computation can be extended beyond the energy of the highest known discrete energy level @xmath114 by using an integration of a nuclear level density @xmath115 over a range of excitation energies @xmath116 . likewise , the sums appearing in ( [ eq : csstar ] ) can be amended with a supplemental integration over the level density above the last discrete level used . although the product of the boltzmann factor @xmath117 and @xmath115 does not have a trivial energy dependence , it has been shown that for the application of ( [ eq : partfuncint ] ) at temperatures @xmath118 gk it is sufficient to integrate only up to @xmath119 mev.@xcite temperatures above 10 gk are encountered in some explosive astrophysical events , in accretion disks , and in the formation of neutron stars and black holes . nuclear transformations in such environments are described in reaction equilibria between several or all possible reactions ( with the exception of reactions mediated via the weak interaction because they are too slow in most cases ) , replacing full reaction networks by simplified abundance equations ( see , e.g. , refs . for details ) but still containing the partition functions . at such high temperatures , a straightforward application of ( [ eq : partfuncint ] ) would overestimate the partition function because continuum effects have to be taken into account . these can be treated by approximated correction factors to @xmath115 and extending the integration to @xmath120 mev.@xcite a more rigorous treatment of the correction would be desireable but the sheer number of involved nuclei proves prohibitive for fully microscopic approaches . transitions ) in a compound reaction involving the nuclei a and f , and proceeding via a compound state ( horizontal dashed line ) with spin @xmath121 and parity @xmath122 in the compound nucleus c. the reaction @xmath94 values for the capture reaction ( q@xmath123 ) and the reaction a@xmath124f ( q@xmath125=q@xmath126 ) are given by the mass differences of the involved nuclei . above the last state , transitions can be computed by integrating over nuclear level densities ( shaded areas ) . [ fig : cnscheme],scaledwidth=80.0% ] the use of stellar cross sections in the calculation of reaction rates assures an important property , the reciprocity of forward and reverse rate . a scheme of the energetics and the transitions between nuclear levels in the involved nuclei is shown in fig . [ fig : cnscheme ] . for a reaction @xmath127 involving only one initial level @xmath102 in nucleus @xmath31 and one final level @xmath128 in nucleus @xmath129 ( this can also be the compound nucleus @xmath130 if the ejectile @xmath131 is a photon ) the well - known reciprocity relation between forward ( @xmath132 ) and reverse ( @xmath133 ) reaction cross section is@xcite @xmath134 where @xmath135 , @xmath136 are the reduced masses , @xmath137 , @xmath138 the center - of - mass energies relative to the levels @xmath102 , @xmath128 , respectively , and @xmath139 the spin factors as before . this relation connects one initial with one final state and therefore is not applicable to the regular laboratory reaction cross sections which connect one initial state ( usually the ground state @xmath112 ) of the target nucleus with a number of possible final states @xmath140 in order to obtain a cross section obeying reciprocity one has to construct a theoretical quantity called _ effective cross section _ the effective cross section is a sum over all energetically possible transitions of initial levels to final levels ( capture : in nuclei @xmath31 and @xmath130 , otherwise : in nuclei @xmath31 and @xmath129 ; as indicated in fig . [ fig : cnscheme ] ) , applying to compound reactions as well as to direct reactions ( see sec . [ sec : mech ] ) . it includes all the transitions shown by arrows in fig . [ fig : cnscheme ] and therefore sums over all final levels @xmath128 _ and _ initial levels @xmath102 @xmath142 as before , the relative center - of - mass energy of a transition proceeding from level @xmath102 is denoted by @xmath143 , and @xmath144 for @xmath108 . the first summand ( @xmath112 ) in the sum over @xmath102 is just the laboratory cross section @xmath145 . note that the effective cross section is only a function of energy , like the usually defined cross section , and does not depend on temperature , contrary to the stellar cross section . when interchanging the labels @xmath102 and @xmath128 , a similar quantity @xmath146 is obtained for the reverse direction commencing on levels @xmath128 . it is straightforward to show that the two effective cross sections obey the reciprocity relations @xmath147 which are identical to the one in ( [ eq : reci_single ] ) for a single transition between two states in two nuclei . the relative energies of the transitions proceeding on the ground states of the two target nuclei for forward and reverse reaction are denoted by @xmath148 and @xmath149 , respectively . the effective cross section is , of course , unmeasureable . its usefulness becomes apparent when we combine definition ( [ eq : csstar ] ) of the stellar cross section with the definition ( [ eq : rate ] ) for the astrophysical reaction rate . since excited states of target nuclei are populated in a stellar plasma according to ( [ eq : csstar ] ) , we have to sum over the rates for reactions from each level and weight each summand with the population factor @xmath150 this means that projectiles with mb distributed energies are acting on each level @xmath102 separately . insertion of definition ( [ eq : csstar ] ) for the population factor @xmath109 leads to @xmath151 where @xmath152 is the normalized partition function as defined in ( [ eq : partfuncint ] ) . in order to obtain an expression similar to the original single mbd , the integral can be transformed by replacing @xmath153 , with @xmath154 and this yields @xmath155 } e^{-\frac{e_0^a}{kt}}\,de_0^a } = \nonumber \\ & = \frac{1}{g^a_0 } \sum_\mu { \int \limits_{0}^\infty \frac{g_\mu^a}{g_0^a } \sigma^\mu_{aa}(e_\mu^a ) \left[e_0^a - e_\mu^\mathrm{x}\right ] e^{-\frac{e_0^a}{kt}}\,de_0^a } \quad . \label{eq : stellrate3}\end{aligned}\ ] ] in the last line above , the lower limit of the integration was reset to zero . this is allowed because cross sections at negative energies do not give any contribution to the integral . it has been pointed out in ref . that it is mathematically equivalent when sum and integral are exchanged , leading to@xcite @xmath156 e^{-\frac{e_0^a}{kt}}\,de_0^a } = \nonumber \\ & = \left(\frac{8}{m_{aa } \pi}\right)^{1/2 } ( kt)^{-3/2}\frac{1}{g^a_0 } \int \limits_{0}^\infty { \left\ { \sum_\mu { \frac{g_\mu^a}{g_0^a } \sigma^\mu_{aa}(e_\mu^a ) e_\mu^a } \right\ } e^{-\frac{e_0^a}{kt}}\,de_0^a } = \nonumber \\ & = \left(\frac{8}{m_{aa } \pi}\right)^{1/2 } ( kt)^{-3/2}\frac{1}{g^a_0 } \int \limits_{0}^\infty { \sigma^\mathrm{eff}_{aa } e_0^a e^{-\frac{e_0^a}{kt}}\,de_0^a } = \frac{\langle \sigma^\mathrm{eff } v\rangle_{aa}}{g^a_0 } \quad . \label{eq : effrate}\end{aligned}\ ] ] the last line was obtained by realizing that the expression in the curly brackets is identical to @xmath157 , with the effective cross section from ( [ eq : effcs ] ) . thus , the weighted sum over many mbds acting on the thermally populated excited states is reduced to a single mbd acting on an effective cross section and divided by the normalized partition function . in terms of relevant physics , this means that the boltzmann factor in the population probability is offset by shifting down each mbd to the same relative energy.@xcite now @xmath59 in ( [ eq : rate ] ) can be identified as @xmath158 , introducing a temperature dependence while @xmath141 is conveniently independent of @xmath41 . it may be confusing that the stellar reactivity frequently is written as @xmath159 instead of @xmath160 . this is not meant to imply that the stellar cross section as defined in ( [ eq : csstar ] ) is inserted in a single integral over a mbd as shown in ( [ eq : rate ] ) . rather , the angle brackets imply a separate integration for each populated state as performed in ( [ eq : stellrate1 ] ) and ( [ eq : stellrate2 ] ) in this case . equation ( [ eq : effrate ] ) not only simplifies the numerical calculation of the stellar rate but also allows to better understand certain details . for instance , it can immediately be seen that stellar rates obey a reciprocity relation because the effective cross sections do . a similar expression has to hold for the reverse reactivity @xmath161 as for the forward reactivity @xmath162 , being @xmath163 this expression is derived in the same manner as ( [ eq : effrate ] ) but by starting from thermally populated excited states in the final nucleus . with the help of ( [ eq : reci_eff2 ] ) we can express the reverse reactivity of ( [ eq : revreac ] ) in terms of the forward reactivity:@xcite @xmath164 where @xmath165 is the reaction @xmath94-value of the forward reaction . the reciprocity relation ( [ eq : reci_eff2 ] ) applies to photodisintegration and captures as well . in relating the photodisintegration rate @xmath21 to the capture rate , however , it has to be assumed that the denominator @xmath166 of the planck distribution for photons appearing in the photodisintegration reactivity @xmath167 can be replaced by @xmath168 , similar to the one of a mbd with the same temperature @xmath41 . with this approximation and realizing that @xmath169 , one obtains@xcite @xmath170 in the same manner as ( [ eq : revrate ] ) . using the approximation of the denominator resulting in ( [ eq : revphoto ] ) is very important for the application in reaction networks . employing the expressions ( [ eq : revreac ] ) and ( [ eq : revphoto ] ) avoids numerical inconsistencies in network calculations which may arise when forward and reverse rates are calculated separately ( or even from different sources ) . the proper balance between the two reaction directions can only be achieved in such a treatment . furthermore , simplified equations for reaction equilibria ( see sec . [ sec : equilibria ] ) can be derived which prove important in the modeling and understanding of nucleosynthesis at high temperature . how large is the error stemming from the approximation involved in the derivation of ( [ eq : revphoto ] ) ? although mathematically unsound , it turns out that setting @xmath171 is a good approximation for the calculation of the rate integrals and introduces an error of less than a few percent for astrophysically relevant temperatures and rate values.@xcite in other words , the contributions to the integral in ( [ eq : lambda ] ) are negligible at the low energies where planck and mbds differ considerably ( see also fig . 3.5 in ref . , and ref . ) . this is assured by either a sufficiently large and positive @xmath172 , which causes the integration over the planck distribution to start not at zero energy but rather at a sufficiently large threshold energy , or by vanishing effective cross sections at low energy due to , e.g. , a coulomb barrier . the assumption may not be valid for s - wave neutron captures with very small ( of the order of @xmath173 ) or negative @xmath94-values , but the required correction still is only a few % as can be shown in numerical comparisons between photodisintegration rates calculated with the two versions of the denominator . such a comparison was performed with the code smaragd ( version 0.8s ; see sec . [ sec : codes ] ) and the results are shown in figs . [ fig : mbcomp_proton ] , [ fig : mbcomp_alpha ] , and [ fig : mbcomp_neut ] . generally , larger errors appear at lower temperature . this results in astrophysical irrelevance of the errors in many cases because either the rates are too slow ( especially for rates involving charged projectiles ) or the target nuclei in question are so short - lived that they will never be produced at low plasma temperature . the largest error found was between 50 and 100% for a few heavy nuclei at the driplines for proton- or @xmath0-capture at @xmath174 gk . for neutron captures , the errors when applying the standard approximation for the reverse rate were never larger than 10% at any investigated temperature , even at the driplines . in the figures , errors of 5% and smaller are not presented in detail because they are assumed to be negligible , especially given the remaining uncertainties in the prediction of the rates far from stability . figure [ fig : mbcomp_proton ] shows target nuclei for proton capture where errors reach 510% at @xmath175 gk when computing the photodisintegration rate from ( [ eq : revphoto ] ) . as expected , this occurs close to the dripline where the reaction @xmath94-value is small or negative . at @xmath176 gk , however , the errors for all nuclei with mass number @xmath177 are below 5% already and thus negligible . figure [ fig : mbcomp_alpha ] shows target nuclei for @xmath0 capture where errors maximally reach 550% at @xmath175 and 510% at @xmath178 gk , respectively , when computing the photodisintegration rate from ( [ eq : revphoto ] ) . again , this occurs for some ( but not all , depending on the energy - dependence of the effective cross section ) @xmath0 captures with small or negative @xmath94-values . at @xmath178 gk even fewer rates are affected and the maximal error is below 10% . the shown rates have little or no astrophysical impact because those nuclei can only be reached at higher temperatures . for instance , the @xmath82-process ( p - process ) significantly photodisintegrates nuclei close to stability at @xmath179 gk and the rp - process also requires such high temperatures and probably does not proceed beyond @xmath180.@xcite figure [ fig : mbcomp_alpha ] shows target nuclei for neutron capture where errors reach 510% at any plasma temperature . as expected , captures with low @xmath94-value , either at the dripline or in the vicinity of closed shells , exhibit the largest errors but those do not exceed 10% . nuclei so far from stability are expected to be synthesized only at higher temperature . for @xmath181 gk a smaller number of nuclei shows errors above 4% . in fact , for most reactions across the chart the error introduced in the reverse rate due to the approximation of the planck distribution is less than 1% . having assured that the approximation of the planck distribution in the derivation of ( [ eq : revphoto ] ) does not introduce a considerable error in the obtained rates , it is important to realize that further conditions have to be fulfilled to allow the application of the relations discussed above . we started from introducing a thermal population @xmath109 in ( [ eq : csstar ] ) above and consequently all derivations up to here depend on the assumption that the excited states in all participating nuclei are occupied according to this population factor . this is a valid assumption for most astrophysical plasmas and most nuclei reach thermal equilibrium very rapidly through collisions and interactions with photons and other plasma components . however , there are some nuclei which exhibit long - lived isomeric states ( well - known examples are @xmath182al , @xmath183lu , and @xmath184ta)@xcite with such spins that they can not be easily excited or de - excited through electromagnetic transitions . at sufficiently high temperature they may still get into equilibrium , sometimes through couplings to intermediate states , but the relevant transitions have to be carefully studied . this can be achieved through an internal reaction network , not connecting different nuclei but rather including the different levels within one nucleus.@xcite levels not being in thermal equilibrium can be included in regular networks in such a manner as if they were a different nucleus . also in this case , however , the populating and depopulating reactions have to be known explicitly . this also applies when ensembles of excited states are in equilibrium but the different ensembles within a nucleus are not . then each ensemble can be treated as a separate species in a reaction network and the reactions connecting the ensembles have to be included explicitly . although not discussed in further detail here , it is worth mentioning that also weak interactions are affected by the thermal population of excited states . for example , the @xmath27-decay half - life @xmath185 of a nucleus will be changed relative to its ground state half - life when the ground state becomes depopulated and excited states with different decay half - lives are populated . thus , the decay `` constant '' @xmath186 is actually temperature - dependent @xmath187 where @xmath188 and @xmath189 is the decay lifetime of the excited state . similar considerations also apply to other processes , such as electron capture and neutrino - induced reactions . using the reciprocity relations derived above allows to simplify the full reaction networks as defined in ( [ eq : network ] ) . such simplifications are instructive because they enable us to study nucleosynthesis properties which are independent of details in the hydrodynamic evolution of the system or even , as shown below , independent of individual reaction rates . they usually go along with a restriction to only types of reactions in the network which are actually necessary instead of blindly evolving a large system of differential equations . such an approach is not always feasible but considerable understanding of nucleosynthesis has been gained in the past through such means by circumventing the necessity of computationally intensive calculations . such simplifications remain important today because it is still impossible to couple multi - dimensional hydrodynamic simulations to full reaction networks . furthermore , restriction to the essential often provides a much better insight into the physical processes than a brute - force full network calculation . here we are concerned with the simplifications because it has to be understood when it is necessary to know astrophysical reaction rates and when not . setting the abundance change @xmath190 on the left - hand side of ( [ eq : network ] ) implies that the sum of all rates destroying the nuclear species @xmath15 is exactly balanced by all production rates and the net change in abundance is therefore zero , leaving the abundance constant . this is called _ steady flow equilibrium_. it is especially useful with reaction chains where most rates ( perhaps except one ) are in steady flow equilibrium . then the slowest reaction sets the timescale of the reaction flow and all other reactions adjust . as long as steady flow is upheld , no full reaction network has to be solved . rather , the ratios of the steady flow abundances of the involved nuclei are related by the ratios of their net destruction rates ( or , equivalently , production rates as these have to be the same ) . for illustration , let us assume a chain of reactions @xmath191 connecting nuclei through reactions with the same projectile , where all net reactions are in steady flow and therefore the same , except for the one starting at nucleus @xmath31 . then @xmath192 and @xmath193 the slowest rate sets the abundance of @xmath194 through @xmath195 . the use of a complete set of coupled differential equations is not required anymore but the important rates still have to be known . steady state considerations are helpful when investigating hydrostatic hydrogen burning of stars through the pp - chains and the cno cycles.@xcite in the past they have also been used for sequences of neutron captures in the s - process on nuclei in between magic numbers.@xcite the fact that separate steady flows can be assigned to each mass region between closed shells has been termed _ local approximation _ in s - process studies . a slightly different concept is to assume equilibrium between a forward and its reverse rate . this is the case when the two rates are equal or , in practice , very close . since stellar rates obey the simple reciprocity relations ( [ eq : revrate ] ) and ( [ eq : revphoto ] ) , respectively , it is trivial to show that @xmath196 for a reaction @xmath197 and @xmath198 for a reaction @xmath199 . the individual rates do not appear anymore in the relation between the abundances . note that this does not imply that the abundances remain constant , they still depend on @xmath41 which may vary with time as well as on @xmath200 and @xmath201 . depending on the plasma density reaction @xmath202c is very sensitive to the density and will not get into equilibrium for @xmath203 g/@xmath204 . ] , above @xmath205 gk all reactions ( with the exception of the weak interaction ) achieve equilibrium . it can be shown that the equilibrium abundance of a nucleus @xmath31 can be calculated from a set of three equations@xcite @xmath206 where @xmath207 is the mass number , @xmath208 , @xmath209 are the abundances of the free neutrons and protons , respectively , and @xmath210 the nuclear mass unit . the binding energy of the nucleus with neutron number @xmath211 and proton number @xmath212 is denoted by @xmath213 . the sums run over all species of nuclei in the plasma , including neutrons and protons . equation ( [ eq : masscons ] ) expresses mass conservation and ( [ eq : chargeconv ] ) is the charge conservation . the unknown abundances @xmath214 , @xmath208 , and @xmath209 are obtained with the above equation set . note that reactions mediated by the weak interaction are not included in the equilibrium and @xmath44 may be time - dependent . again , individual rates are not required to determine the abundances . when all abundances in the network obey the above relations , full _ nuclear statistical equilibrium _ ( nse ) is achieved . in this case , no reaction rates have to be known . in realistic cases , more or less extended groups of nuclei are in statistical equilibrium and the relative abundances within a group can be described by equations similar to ( [ eq : nse ] ) . the different groups are connected by comparatively slow reactions not being in equilibrium , which determine the abundance level of one group with respect to another group similar to what was shown above for the steady state equilibrium . the rates of these slow , connecting reactions have to be known explicitly . this is called _ quasi - statistical equilibrium _ ( qse ) . it appears in various kinds of high - temperature burning , such as hydrostatic oxygen and silicon burning in massive stars and different explosive scenarios . a special kind of qse is the ( n,@xmath82)@xmath215(@xmath82,n ) equilibrium or _ waiting point approximation _ , often used in r - process calculations.@xcite this is nothing else than a qse within an isotopic chain , where neutron captures and ( @xmath82,n ) reactions are in equilibrium under very neutron - rich conditions ( @xmath216 @xmath49 ) and @xmath217 gk . for the r - process the network is reduced to neutron captures and their inverse reactions , and to @xmath218 decays ( with possible subsequent neutron emission ) . the decays are not in equilibrium and determine the timescale with which matter is processed from small @xmath212 to the heaviest nuclei . when ( n,@xmath82)@xmath215(@xmath82,n ) equilibrium is achieved , the abundances of nuclei within an isotopic chain are connected by @xmath219 which connects the abundance of nucleus @xmath31 ( mass number @xmath220 ) with the one of nucleus @xmath221 ( mass number @xmath222 ) . there is an exponential dependence on the neutron separation energy @xmath223 of @xmath221 . also in this type of equilibrium there is no dependence on the individual capture or photodisintegration rates . the r - process flow to higher elements , however , depends on the @xmath218-decay rates which connect the isotopic chains and are not in equilibrium . they are very slow compared to the rates in equilibrium and that is why `` waiting points '' are established , which are just the nuclei ( usually only one or two within a chain ) with the highest abundances according to ( [ eq : nggn ] ) . the r - process can not proceed until they decay and their decay rates have to be known . a similar equilibrium , but between proton captures and ( @xmath82,p ) , is reached in the late phase of the rp - process on the surface of mass accreting neutron stars.@xcite there , the waiting points are established close to the proton dripline . nucleosynthesis under extreme conditions , such as encountered in some explosive scenarios , involves exotic nuclei far from stability . according to the above , reaction rates are not needed for all of them because reaction equilibria are established at such extreme conditions . required are nuclear masses ( to determine @xmath94-values or binding energies ) as well as spectroscopic information and nuclear level densities ( entering the calculation of the partition functions @xmath110 ) . but although nse , qse , and the waiting point approximation do not contain the rates explicitly , they implicitly depend on them because they determine whether nuclei are participating in the equilibrium or not . the higher the rates , the lower the temperature at which equilibrium is reached . with a time - dependent @xmath41 evolution , this means that the rates determine whether equilibrium is reached earlier with increasing @xmath41 or the freeze - out happens later with decreasing @xmath41 . in principle , stellar cross sections @xmath224 as defined in ( [ eq : csstar ] ) correspond to physically measurable quantities , contrary to the purely theoretical effective cross sections introduced in ( [ eq : effcs ] ) . in practice , cross sections @xmath225 measured in terrestrial laboratories do not include thermal effects . therefore the rates derived from them do not obey reciprocity relations . to derive astrophysical rates appropriate for the utilization in reaction networks , laboratory cross sections almost always have to be supplemented by theory to account for the additional transitions not included in the measurement . the energy- and temperature - dependent stellar enhancement factor for the cross sections ( again obtained via theory ) @xmath226 compares the stellar cross section including reactions from thermally populated excited states to the cross section obtained with reactions proceeding from the ground state of the target nucleus only . another definition of a stellar enhancement factor involves the rates or reactivities , @xmath227 in fact , @xmath228 is the astrophysically interesting quantity because it shows the deviation introduced in the _ rate _ at a chosen stellar temperature @xmath41 when using the laboratory cross section instead of the stellar cross section . then the fraction of the ground state contribution to the stellar rate is @xmath229 . on the other hand , @xmath230 is supposed to provide information for the experimentalist on how much off the measured @xmath231 is in comparison to @xmath224 at each energy for a given stellar temperature . to this end it was quoted in literature occasionally . however , a much more useful definition is @xmath232 which weights the excited states appropriately for a straightforward comparison . this is the same as @xmath228 only if @xmath233 is independent of @xmath67 . this is not necessarily fulfilled because the energy - dependence of laboratory and effective cross section may be different and this would lead to a different evaluation of the reaction rate integral . ( it turns out that in practice @xmath233 is often more slowly varying with energy than the cross section across the relevant energy window ( see sec . [ sec : energies ] ) and that it can be approximated by an energy - independent factor in this case . ) it is recommended to use only either @xmath233 or @xmath228 , depending on whether an energy - dependent measure is desired or one independent of interaction energy . as has become clear from the derivation of the stellar rate in sec . [ sec : stellar ] there is no simple relation between @xmath228 and @xmath230 . it should be noted that while the stellar population factors @xmath109 are normalized to unity and the normalized partition functions @xmath234 can not become smaller than unity , the three types of stellar enhancement factors defined above can assume any positive value , _ larger or smaller _ than unity . in which case do we have to expect large deviations @xmath235 from the laboratory value ? ( the above definition assures @xmath236 ; without thermal effects @xmath237 . ) again , a scrutiny of the effective cross section appearing in ( [ eq : sefrate ] ) helps to understand the various dependences . let us start with the dependence on stellar temperature . naively , one would assume that the higher the temperature , the larger the stellar enhancement will become . indeed , this is often the case but there are further intricacies . there are two dependences on @xmath41 appearing in ( [ eq : sefrate ] ) , one in the rate in the numerator and the other in the normalized partition function in the denominator . the latter is monotonically increasing with increasing @xmath41 . the integration of the product of the effective cross section and the mbd may show a different @xmath41-dependence , however , which even may not prove to be monotonic . this can be understood by combining the knowledge of the relevant energy windows from sec . [ sec : energies ] with definition ( [ eq : effcs ] ) of the effective cross section . larger @xmath41 shifts the energy window to higher relative energy , both in the entrance channel @xmath238 with the relevant @xmath148 becoming larger and in the exit channel @xmath239 ( or @xmath240 ) with the relevant @xmath241 becoming larger ( see fig . [ fig : cnscheme ] ) . since the effective cross section sums over transitions with relative energies @xmath242 and @xmath243 , it becomes obvious that the higher the relevant energy window , the more transitions are included . this does not necessarily result in an increased rate although it often will . it is also conceivable that the additionally included transitions at large @xmath41 have small cross sections and do not provide a considerable increase in the rate . depending on the type of reaction , cross sections for transitions already included at small @xmath41 may also decrease with increasing relative energy . in these cases , @xmath228 will be reduced at larger @xmath41 because @xmath234 is always increasing . the stellar `` enhancement '' may even become smaller than unity and not live up to its name anymore . this behavior should not be viewed as monotonic , either , because as additional transitions become accessible at even larger @xmath41 the stellar rate again may increase faster than @xmath234 . it is to be expected , however , that @xmath228 is decreasing to very small values for very large @xmath41 after having reached a maximum value , i.e. , for very high ( but not necessarily astrophysically important ) temperatures @xmath244 will always be achieved . this is because with increasing relevant energy @xmath245 other reaction channels become increasingly important , reducing the cross sections of the individual transitions . additionally , more unbound levels will be included in the effective cross section . these may lose particles to other reaction channels ( e.g. , through pre - equilibrium emission ; see also sec . [ sec : statmod ] ) and also not contribute to the effective cross section anymore . it was suggested in ref . to include only bound states in the definition of the effective cross section . although this may be a good approximation , it may neglect some transitions which still can contribute to the effective cross section at high temperature . note , however , that unbound states may not be in thermal equilibrium although the timescale for reaching equilibrium under stellar burning conditions is short compared to the one of a nuclear reaction.@xcite a more suitable cutoff , if required , would be the energy @xmath246 appearing as cutoff in the calculation of the partition function in ( [ eq : partfuncint ] ) , although this may include already too many transitions with negligible cross sections . examples for the above considerations are shown in figs . [ fig : ossef1 ] , [ fig : ossef2 ] for neutron - induced reactions on os isotopes . neutron capture on @xmath247os is important to understand the re - os cosmochronometer which can be used for age determinations in our galaxy.@xcite modern measurements of neutron capture in the astrophysically relevant energy range have reached a precision that requires the inclusion of the @xmath228 correction , even when it is only a few tens of percent.@xcite due to low - lying levels ( @xmath248 at 9.75 and 74.3 kev , @xmath249 at 75 kev , and @xmath250 at 100 kev ) in @xmath247os , @xmath228 is higher for @xmath247os(n,@xmath82)@xmath251os at low temperature than for neutron captures on neighboring isotopes . the enhancement factors of the reactions @xmath247os(n,@xmath82 ) and @xmath251os(n,@xmath82 ) both rise to a maximum and decline from there , soon reaching values below unity and thus not being `` enhancements '' anymore . while the @xmath228 for both capture reactions stay below a factor of two ( and reach only a few tens of percent at the s - process temperature relevant for the cosmochronometer ) , the @xmath228 of @xmath252os(n,@xmath0)@xmath253w is also shown here to give an example for larger values ( see below for the even larger factors encountered for photodisintegration reactions ) . having realized the importance of constraining @xmath148 by the relevant energy window at a given temperature and the definition of the range of energies @xmath254 of relevant excited states , we can arrive at a more general understanding of when @xmath255 will considerably differ from unity . in the assessment of which excited states in the target are contributing to the stellar rate , it is incorrect to directly use the boltzmann population factors @xmath109 from eq . ( [ eq : csstar ] ) ! these are only appropriate when each state is bombarded by its own mbd of projectiles . they can not be used when we calculate the rate as usual , integrating over just one mbd with the energy scale being relative to the ground state . for levels at excitation energy @xmath256 and zero spin . the weights are independent of stellar temperature but their slopes depend on the maximal transition energy @xmath76 as indicated by the labels.[fig : excweights1],scaledwidth=64.0% ] as shown in eqs . ( [ eq : stellrate2])([eq : effrate ] ) the transformation to a single mbd is possible because the boltzmann weight @xmath257 in the population factor offsets the exponentials in the mbd for each state . this offset is not a complete cancellation but results in a transformation of the population @xmath109 . the individual weights are transformed to linearly declining functions of the excitation energy @xmath258 this is also readily seen in the weighting of the reactions from excited states in the definition of the effective cross section by eq . ( [ eq : effcs ] ) . these relative _ effective weights _ @xmath259 are the ones to be employed when using the standard definition of the rate as in ( [ eq : rate ] ) , instead of the relative weights @xmath260 . contrary to the @xmath260 , the relative effective weights @xmath259 are not explicitly temperature dependent anymore . however , they depend on the energy @xmath76 of the transitions to the ground state which are the transitions with the highest possible relative energy . in the application to astrophysical reaction rates the range of @xmath76 is given by the relevant energy windows as discussed in sec . [ sec : energies ] . it is a relatively narrow range , depending on stellar temperature and the type of reaction ( and thus introducing an implicit temperature dependence ) . due to the linearity in excitation energy @xmath104 contrary to the @xmath260 which show an exponential decrease almost all states up to @xmath76 contribute . neglecting the spin weights @xmath261 , one has to consider levels up to @xmath262 to include 90% of the levels with non - negligible weight . depending on the nucleus and considered reactions , some levels with large spin values and/or large reaction cross sections may require the inclusion of all levels up to very close to @xmath263 . the relative effective weights @xmath259 as the product of the population factors with their respective mbds are shown in fig . [ fig : excweights1 ] . understanding the @xmath259 we are ready to make some general statements on the magnitude of @xmath255 in various reactions in a range of astrophysically relevant temperatures . for example , neutron captures have their relevant energies around the maximum of the mbd , @xmath264 . this is between a few kev up to several tens of kev for the s - process@xcite ( see also sec . [ sec : energies ] ) . therefore only ( exothermic ) captures on nuclei with low - lying excited states below several tens of kev will exhibit @xmath265 . for light and intermediate mass nuclei , the average level spacings typically are larger than @xmath266 kev and thus @xmath255 remains close to unity , with a few notable exceptions . the r - process@xcite involves neutron captures at temperatures of @xmath267 gk which translates to a location of the mbd peak at @xmath268 kev and we will expect slightly larger @xmath255 than for the s - process on average . and @xmath269 in the initial and final nucleus , respectively , under the assumption of a positive reaction @xmath94-value @xmath270 . in both nuclei a few low - lying states are explicitly numbered , above them nuclear level densities are indicated ( shaded areas ) . [ fig : excweights2],scaledwidth=80.0% ] reactions having their relevant energy window determined by charged particle widths have considerably larger @xmath76 , in the range of about @xmath271 mev , depending on the stellar temperature and the charges of the interacting particles . accordingly , @xmath255 can already become large at rather low temperature because transitions from many excited states have to be considered . only in light , strongly bound nuclei with level spacings of several mev above the ground state , @xmath255 may still remain close to unity . the relative effective weights @xmath272 derived in ( [ eq : effweights ] ) also allow to understand a rule - of - thumb which has been known , but never quantified , for quite some time ( see , e.g. , ref . ) . the rule states that exothermic reactions ( @xmath273 ) usually have smaller @xmath255 than their endothermic inverses . defining the forward reactions by the exothermic reaction and the reverse reaction by its endothermic counterpart , this means that @xmath274 and frequently @xmath275 . this is not immediately comprehensible upon inspection of the boltzmann weights @xmath260 as these seem to act similarly in the initial and final nuclei . as stated above , however , it is a mistake to straightforwardly use the @xmath260 together with a single mbd . the rule becomes obvious when using the appropriate @xmath259 . the situation is sketched in fig . [ fig : excweights2 ] where the behavior of the weights is shown for two nuclei @xmath31 and @xmath129 being the target and the final nucleus of a reaction , respectively . for the relevant interaction energy set to @xmath76 the maximum relative transition energy is @xmath148 in the target nucleus and @xmath276 in the final nucleus . provided that @xmath277 the energy range of possible transitions @xmath278 in the final nucleus is larger than in the target nucleus . in consequence , the relative effective weights @xmath279 in nucleus @xmath129 decline slower with increasing excitation energy and thus a larger range of levels contributes . this assumes similar spin structure in the two nuclei , of course , and the rule may not work at small @xmath280 when @xmath259 , @xmath279 are similar and the spin factors @xmath261 , @xmath281 dominate . the largest reaction @xmath94-values are encountered in capture reactions . for instance , neutron captures close to stability exhibit @xmath94-values of the order of @xmath282 mev , for highly proton - rich nuclei they can reach @xmath283 mev . similar values are found for proton captures around stability and on the neutron - rich side of the nuclear chart . in the light of the above it is not surprising that endothermic photodisintegration reactions exhibit very large @xmath228 of the order of several hundreds to thousands as illustrated in table [ tab : photosefs ] . an exception to the above rule @xmath274 , however , has recently been discovered.@xcite although the effective weight @xmath284 may be slowly decreasing with increasing excitation energy @xmath104 of a level , the corresponding cross section @xmath105 may decrease much faster , even exponentially . this is because with increasing @xmath104 the relative interaction energy in that channel @xmath143 is reduced . if the cross section @xmath105 is strongly decreasing with decreasing energy as it is the case in the presence of a coulomb barrier or at high relative angular momentum transitions on excited states will cease to importantly contribute to the effective cross section even when being strongly weighted . this is the reason why charged particle reactions show only moderate values of @xmath255 at low @xmath41 even though their relevant energy window may be high above the reaction threshold . this , of course , acts in the entrance channel of a reaction as well as in its exit channel . the point , however , is to realize that the barriers may be different in the two channels , leading to a different suppression of contributions . for instance , a ( n , p ) or ( n,@xmath0 ) reaction has a coulomb barrier only in the exit channel , ( p,@xmath0 ) reactions have different barriers in entrance and exit channel . this has the consequence that low - energy transitions in the exit channel of an exothermic reaction may be suppressed in such a manner as to yield @xmath285 . whether this is the case will strongly depend on the @xmath94-value because it determines the range of transition energies to be efficiently suppressed . the higher the barrier , the larger @xmath94-value is allowed while still permitting to suppress most transitions from excited states . the stellar temperature also plays a role but its impact is smaller . a global search across the chart of nuclides for reactions exhibiting @xmath285 was performed in refs . . the study focused on identifying cases which are interesting for experiments by requiring @xmath286 and the stellar enhancement factors for forward and reverse reaction to differ by more than 10% . at stellar temperatures @xmath287 gk about 1200 reactions were found but not all of them are astrophysically interesting.@xcite figure [ fig : qsupp ] shows the dependence of the @xmath94-values on the charge of the target nucleus of two types of reactions selected from the total set . the envelope from the maximal @xmath280 appearing for each type at each charge @xmath288 illustrates the action of the increasing coulomb barrier . it can be clearly seen that larger maximal @xmath289 is allowed with increasing charge @xmath288 . the increase with increasing charge is steeper for ( @xmath0,n ) reactions than for ( p , n ) reactions due to the higher coulomb barrier for @xmath0 particles . the scatter at a fixed charge number @xmath288 is due to the range of @xmath94-values found within an isotopic chain . summarizing the above , comparing the position of the relevant energy window with the average level spacing in a nucleus already gives an estimate of the magnitude of the stellar enhancement to be expected . when measuring reaction cross sections for astrophysics it is desireable to perform the experiment in the reaction direction showing smallest @xmath255 in order to stay as close as possible to the actually required stellar cross section . this favored direction turns out to be the exothermic one in the vast majority of cases , i.e. , a reaction with positive @xmath94-value , with comparatively few exceptions ( compared to the total number of conceivable reactions ) as discussed above . it should be remembered that the reverse rate can always be calculated using the reciprocity relations ( [ eq : revrate ] ) and ( [ eq : revphoto ] ) . , appearing in ( [ eq : revrate ] ) , ( [ eq : revphoto ] ) , when @xmath290 . ] finally , it is worth mentioning that similar considerations for the transformation of effective weights and deviations @xmath255 from the ground state values also apply to other types of reactions , not only the ones discussed here . for instance , also for electron captures and other reactions mediated by the weak interaction an effective cross section , effective weights , and relevant energy windows can be derived , applying the appropriate energy distributions . they may become more complicated , however , than the ones obtained for the mbd because an explicit dependence on the chemical potential is required in the general case . moreover , neutrinos are not in thermal equilibrium with nuclei and thus their temperature will enter as an additional parameter . the plasma temperature in the astrophysical sites where nuclear reactions occur is so high that the nuclei are fully ionized and embedded in a cloud of free electrons . this situation affects reactions and decays and has to be considered when preparing a rate to be used in astrophysical reaction networks . similar to the treatment of stellar enhancement factors , these corrections also have to be modeled theoretically and experimental rates have to be corrected for them . for completeness , some of the effects are discussed briefly without going into detail here . decays ( and electron captures ) are affected not only by the thermal excitation of the nucei as shown in ( [ eq : stelldecay ] ) but also by the electrons surrounding the nuclei . a nucleus in an atom can be converted by , e.g. , capture of an electron from the atomic k - shell . in the plasma , electrons are captured from the electron cloud and this may alter the half - life of a nucleus considerably . a well - known example for this is the decay of @xmath291be . its lifetime under central solar conditions ( @xmath292 d ) is almost double the one under terrestrial , non - ionized conditions ( @xmath293 d ) . also the @xmath218 decay lifetimes are modified by a change in the electron emission probability . the plasma electrons reduce the phasespace available for emission to the continuum and thus increase @xmath294 . on the other hand , bound - state decay , i.e. the placement of the emitted electron into a low - lying atomic shell , becomes possible even when it were forbidden in an atom because of the occupation of available electron shells . similar considerations apply to charged - current reactions with electron neutrinos . these effects act in addition to the alteration of the lifetimes through thermal excitation of the nucleus and depend on the plasma temperature , density , and composition , which may all affect the distribution of the electrons throughout the plasma . nuclear reactions with charged nuclei are affected by _ electron screening_. electrons in the vicinity of the nucleus shield part of its charge and thus effectively lower its coulomb barrier . theoretical predictions of cross sections and rates always assume bare nuclei without any electrons and therefore have to be corrected for screening . the magnitude of the screening is strongly dependent on the temperature @xmath41 , density @xmath6 , and composition @xmath295 of the plasma . at high density , electron screening can increase the rate by several orders of magnitude and lead to _ pycnonuclear _ burning at much lower temperatures than otherwise needed to ignite nuclear burning . moreover , dynamical effects due to the fast movement of electrons may also play a role , although there is discussion on whether this is important when all reactants are in thermodynamic equilibrium.@xcite it is apparent that the theoretical treatment of screening is complicated and we are far from complete microscopic descriptions.@xcite the proper inclusion of screening effects remains a challenging problem in plasma physics . under most conditions , the screened reactivity can be decomposed into the regular stellar reactivity and a screening factor@xcite @xmath296 this implies that the coulomb potential @xmath297 seen by the reaction partners can be described by the bare coulomb potential and an effective screening potential @xmath298 @xmath299 then the screening factor acquires the form @xmath300 . the challenge is in the determination of @xmath301 depending on the plasma conditions . an often used static approximation , being appropriate for early burning stages of stars , is _ weak screening _ in the debye - hckel model.@xcite weak screening assumes that the average coulomb energy of each nucleus is much smaller than the thermal energy , i.e. @xmath302 . then a nucleus will be surrounded by a polarized sphere of charges , with a radius @xmath303 with @xmath304 where the sum runs over all charged plasma components . the screening factor @xmath305 is transformed to @xmath306 with @xmath307 . for _ strong screening _ in high density plasmas it is more appropriate to use the ion - sphere model instead of the debye - hckel approximation.@xcite the ion - sphere model is equivalent to the wigner - seitz model used in condensed matter theory . another type of screening is observed in nuclear experiments in the laboratory . there , nuclei are present in atoms , molecules , or metals , each with specific electron charge distributions around the nucleus . although completely different from plasma screening , this type of screening has to be understood because it is especially important at the low interaction energies of astrophysical relevance . the _ measured _ reaction cross sections have to be corrected to obtain the bare cross section which can be compared to theory or used to determine the rate . atomic screening can be treated in the adiabatic approximation , leading to @xmath308 with the sommerfeld parameter @xmath309 from ( [ eq : sommerfeld ] ) . the screening potential @xmath310 in this approximation is given by the difference in the electron binding energy of the target atom and the atom made from target atom plus projectile @xmath311 . in light systems , the velocity of the atomic electrons is comparable to the relative motion between the nuclei . therefore a dynamical model is more appropriate.@xcite however , the adiabatic approximation provides an upper limit on the expected screening effect on the cross section . there seem to be discrepancies between theory and laboratory determinations of @xmath310 , the latter often yielding much larger values of @xmath310 . some of them have been resolved through improved stopping powers used in the determination of the experimental cross sections,@xcite while others remain puzzling , especially regarding cross sections of nuclei implanted in metals.@xcite thus , the laboratory screening seems to be less understood than the stellar screening . having discussed the special requirements of astrophysics for the determination of the stellar rates in the preceding sections , we turn to the question of how to obtain the cross sections @xmath312 required in eqs . ( [ eq : rate ] ) , ( [ eq : csstar ] ) and ( [ eq : effcs ] ) at the energies of astrophysical relevance . it has become apparent that astrophysical rates include more transitions than usually obtained in straightforward laboratory reaction cross section measurements . cleverly designed experiments may study some of them but especially at larger stellar temperatures , as they are typical for explosive burning , theory will be indispensable for providing the appropriate _ stellar _ reaction rates . even at stability , the small cross sections of charged - particle reactions at astrophysically relevant energies ( see sec . [ sec : energies ] ) pose a considerable challenge for current measurements and future experiments have to employ novel techniques or new facilities to address this problem . moreover , hot astrophysical environments produce highly unstable nuclei which can not be studied in the laboratory , yet . reaction cross sections of nuclei far from stability at astrophysical energies probably will never be experimentally determined . therefore reaction networks for explosive nucleosynthesis have to include the majority of their reaction rates from theoretical predictions although experiments may help to determine nuclear properties and some of the transitions required for the calculation of effective cross sections . although reaction theory dates back as far as the 1950s , the special requirements of astrophysics and the need for cross section predictions of nuclei far from stability provide an interesting and stimulating environment for the application and further developments of different approaches . the challenges are manyfold . on one hand , astrophysical energies are very low and , as we will see below , different reaction mechanisms may contribute or interfere . on the other hand , even if the reaction mechanism is unique and well understood , nuclear properties entering the reaction model have to be predicted for nuclei far off stability . this proves challenging even for modern nuclear structure calculations . although a fully microscopic treatment is preferrable , good parameterizations and averaged quantities are still necessary in many cases due to the sheer number of reactions and involved nuclei , especially for intermediate and heavy nuclei consisting of more than @xmath313 nucleons and thus not allowing the application of few - body models . finally , the interpretation of experiments has to be supported by theory . this latter case may involve different methods than the one dealing with the prediction of astrophysical rates because experiments may be conducted at higher energies and theory is needed to extract the information to be included in the rates.@xcite for example , the properties of excited states and their spectroscopic factors can be studied by ( d , p ) reactions at comparatively high energy which are not directly relevant in astrophysics . in the following i focus on theory for the prediction of astrophysical reaction rates . theoretical models can be roughly classified in three categories:@xcite 1 . models involving adjustable parameters , such as the @xmath314-matrix@xcite or the @xmath315-matrix@xcite methods ; parameters are fitted to the available experimental data and the cross sections are extrapolated down to astrophysical energies . these fitting procedures , of course , require the knowledge of data , which are sometimes too scarce for a reliable extrapolation . ab initio " models , where the cross sections are determined from the wave functions of the system . the potential model@xcite , the distorted wave born approximation ( dwba)@xcite , and microscopic models@xcite are , in principle , independent of experimental data . more realistically , these models depend on some physical parameters , such as a nucleus - nucleus or a nucleon - nucleon interaction which can be reasonably determined from experiment only . the microscopic generator coordinate method ( gcm ) provides a basic " description of a nucleonic system , since the whole information is obtained from a nucleon - nucleon interaction . since this interaction is nearly the same for all light nuclei , the predictive power of the gcm is high for such nuclei . the above models can be used for low level - density nuclei only . this condition is fulfilled in most of the reactions involving light nuclei ( @xmath316 ) . however when the level density near threshold is large ( i.e. more than a few levels per mev ) , statistical models , using averaged optical transmission coefficients , are more suitable ( see sec . [ sec : statmod ] ) . the nuclear level density ( nld ) at the compound nucleus excitation energy corresponding to the astrophysical energy window determines which reaction mechanism is applicable and which model to choose . the compound formation energy @xmath317 is given by the astrophysical energy @xmath76 relative to the ground state of the target nucleus and the separation energy of the projectile @xmath318 . around stability , @xmath318 usually is high and dominates @xmath319 . statistical models will then be applicable for intermediate and heavy nuclei with sufficiently high nld . even at stability , however , the nld may not be high enough at nuclei with closed shells ( see sec . [ sec : statmod ] ) . approaching the driplines , the neutron- or proton - separation energies strongly decrease , resulting in low @xmath319 for neutron- or proton - induced reactions , respectively . this leads to low nld at @xmath319 even for intermediate and heavy nuclei.@xcite isolated resonances but also direct reactions will become important.@xcite in the following , reactions at intermediate ( sec . [ sec : reso ] ) , high ( sec . [ sec : highnld ] ) , and low ( sec . [ sec : direct ] ) compound nld are discussed separately although there may be contributions from several reaction mechanisms simultaneously , especially in systems with low and intermediate nld . the discussion will focus on models more or less applicable for large - scale predictions across the nuclear chart . further models are presented in , e.g. , refs . . resonances in reaction cross sections are important for the majority of nuclei . depending on the number of nucleons in the target nucleus resonances appear in the reaction cross section at lower or higher energy and their average spacing also depends on the structure of the nucleus . astrophysical energy windows cover regions of widely spaced , isolated resonances to regions of a large number of overlapping , unresolved resonances . accordingly , different approaches have to be combined . the latter region is more important for reactions between charged reactants because the coulomb barrier shifts the relevant energy window to higher energy compared to reactions where neutrons determining the location of the window ( see sec . [ sec : energies ] ) . isolated resonances in the low and intermediate nld regimes can be treated in the @xmath314-matrix@xcite or the @xmath315-matrix@xcite approaches or by applying simple single - level or multi - level breit - wigner formulae . in all these methods , the resonance properties ( resonance energy , spin , partial and total widths ) have to be known . often , an inverse approach is used and the resonance properties are derived from experimental data by , e.g. , @xmath314-matrix fits . where this is impossible , nuclear theory has to be invoked to predict the required quantities . this remains problematic , however , because the reaction cross sections are very sensitive to the resonance properties . the astrophysical rates are also sensitive but since their calculation involves an integration over an energy range , only strong resonances truly contribute and others may be averaged out . nevertheless , there are large uncertainties in reaction rates off stability due to the unknown resonance contributions ( see also sec . [ sec : sensi ] ) . cluster models ( see sec . [ sec : optmod ] and ref . ) have been successful in describing resonant cross sections in light nuclei but can not be easily applied to nuclei at intermediate and heavy mass . although resonances with the same spin @xmath320 interfere and single resonances may also show interference with a direct reaction ( sec . [ sec : direct ] ) , the single - level breit - wigner formula ( bwf ) is often used:@xcite @xmath321 it is quoted here despite its restrictions because it allows to demonstrate some important principles important for the relation between stellar and laboratory rates . equation ( [ eq : breit ] ) gives the bwf for @xmath322 non - interfering resonances . the wave number is denoted by @xmath323 . the total width @xmath324 of a resonant state @xmath15 in the compound nucleus is the sum over the widths of the individual decay channels @xmath325 , also including transitions to other reaction channels beyond the exit channel @xmath239 . the widths of the individual decay channels are summed over transitions to all possible final states in the channel . thus , @xmath326 . figure [ fig : cnscheme ] shows the energy scheme and the contributing transitions in each channel . when the resonance energy @xmath327 is known , the widths @xmath328 and @xmath329 can be calculated from the transmission coefficients obtained by solution of a schrdinger equation in the optical model ( see sec . [ sec : optmod ] ) and a spectroscopic factor ( see sec . [ sec : direct ] ) . both resonance energies and spectroscopic factors should , in principle , be predictable in the shell model ( see , e.g. , refs . ) or other microscopic theories but this is currently not feasible for all nuclei across the nuclear chart . different approaches yield results which differ more than it is tolerable in the calculations of astrophysical reaction rates.@xcite according to ( [ eq : labcs ] ) , we take @xmath112 for the usual laboratory cross section . it is interesting to note that also resonances located below the reaction threshold may contribute due to their finite width reaching above the threshold . these are called _ sub - threshold resonances_.@xcite in reactions with a large , positive @xmath94-value the energy dependence of the partial width in the exit channel @xmath330 can be neglected . this is not true when the @xmath94-value is small or negative . as has become obvious in the discussion of the stellar cross section in sec . [ sec : stellar ] , for the astrophysical reaction rate , the effective cross section has to be employed in the integration for the reaction rate and thus a weighted sum over excited target states @xmath102 has to be performed and we obtain @xmath331 with the effective weights @xmath332 taken from ( [ eq : effweights ] ) and @xmath143 . this can be simply achieved by replacing @xmath333 in ( [ eq : breit ] ) by @xmath334 ( summing over all possible transitions in the entrance channel ) and dividing the resulting cross section by the normalized partition function of the target nucleus @xmath335 . this can easily be shown when combining definitions ( [ eq : effcs ] ) and ( [ eq : breit ] ) for a single resonance with spin @xmath320 , @xmath336 this results in @xmath337 for the stellar rate . some simplifications can be made depending on the resonance widths . for simplicity , the derivations are given for a single resonance with spin @xmath320 , for @xmath338 the contributions can be added . a frequently used simplification is the one for narrow resonances , assuming that the widths @xmath329 in the numerator in the integral and the boltzmann factor @xmath339 do not change across the width of the resonance.@xcite then their values can be taken at the resonance energy and the integration can be performed analytically ( see also eq . ( [ eq : hfaverage ] ) ) , yielding @xmath340 where the resonance energy @xmath341 is given relative to the ground state of the target nucleus . how does this compare to the ground state rate usually measured in the laboratory ? the stellar rate can be recast as @xmath342 with @xmath343 the second line in ( [ eq : narrowcontrib ] ) was obtained by neglecting the energy dependence of @xmath344 and @xmath345 . this is a valid assumption provided the reaction has a sizeable , positive @xmath94-value . as can be seen from ( [ eq : narrowcontrib ] ) , the contributions from excited states vanish quickly because often @xmath328 is strongly energy dependent and vanishes fast with decreasing energy ( remember that @xmath143 ) . this is certainly true for reactions between charged particles and low resonance energies . nevertheless , the resonant transitions from excited states may dominate a resonant stellar rate.@xcite for broader resonances the above approximation can not be used and the rate has to be determined by numerical integration of eq . ( [ eq : rate ] ) . the wings of broad resonances can also contribute significantly to the rate even when the resonance energy is outside the relevant energy window for the rate . sometimes the values for @xmath346 , @xmath344 , and @xmath347 are known experimentally at the resonance energy . then a frequently used approach in experimental nuclear physics is to assume that @xmath344 and @xmath347 are approximated by energy - independent values and the energy - dependence of @xmath346 is only due to a barrier penetration factor derived from an optical model . even if the energy dependence of @xmath344 , @xmath347 is accounted for explicitly , this type of extrapolation does not include the stellar enhancement and is only valid for laboratory cross sections . the stellar rate must be calculated from a weighted sum of resonant contributions , as shown in ( [ eq : stellbw ] ) , both for the value at the resonance energy and in the extrapolation . it follows from ( [ eq : sumgamma ] ) that the only difference in the energy dependence , however , stems from the width in the entrance channel where @xmath346 has to be replaced by @xmath348 . the additional transitions to excited states in the target nucleus can be measured in principle . if they are not available , @xmath348 at the resonance energy has to be predicted from theory . also the extrapolation is more involved because @xmath348 will have a different energy dependence than @xmath346 . the same methods can be used as in the extrapolation of @xmath346 but they have to be applied to all contributing transitions separately . the discussion of electron screening in the stellar plasma in sec . [ sec : electro ] applied to nonresonant rates . it can be shown that the same screening corrections can be applied for resonant rates when @xmath349.@xcite a more complicated form arises for @xmath350 , see refs . and for further details . microscopic models are based on basic principles of quantum mechanics , such as the treatment of all nucleons , with exact antisymmetrization of the wave functions . the hamiltonian of an @xmath220-nucleon system is @xmath351 where @xmath352 is the kinetic energy and @xmath353 a nucleon - nucleon interaction.@xcite the schrdinger equation associated with this hamiltonian can not be solved exactly when @xmath354 . the quantum monte carlo method represents a significant breakthrough in this direction , but is currently limited to @xmath355.@xcite in addition its application to continuum states is not feasible for the moment ( it has been applied to the d(@xmath0,@xmath82)@xmath356li reaction but the @xmath0+d relative motion is described by a nucleus - nucleus potential).@xcite in cluster models , it is assumed that the nucleons are grouped in clusters and internal wave functions describing the relative cluster motions are generated.@xcite the main advantage of cluster models with respect to other microscopic theories is its ability to deal with reactions , as well as with nuclear spectroscopy . over the past years , much work has been devoted to the improvement of the internal wave functions : multicluster descriptions@xcite , large - basis shell model extensions@xcite , or monopolar distortion@xcite . the main limitation arises from the number of channels included in the wave function , which reduces the validity of the model at low energies . also large nlds require many channels in the wave functions . therefore the application of cluster models is limited to light nuclei . due to the complexity of the nucleon - nucleon ( nn ) interaction , one often resorts to working with effective interactions instead of solving microscopic models based on nn potentials . widely used in calculating different reaction mechanisms is the optical model . in that model , the complicated many - body problem posed by the interaction of two nuclei is replaced by the much simpler problem of two particles interacting through an effective potential , the so - called optical potential.@xcite such an approach is usually feasible only with few contributing channels . always included is elastic scattering . that is why optical potentials can be derived from elastic scattering data . the time - independent radial schrdinger equation is numerically solved with an optical potential which provides a mean interaction potential , averaging over individual nn interactions . in consequence , single - particle resonances can not be described in such a model . however , resonances stemming from potential scattering can still be found . elementary scattering theory yields expressions for the elastic cross section and the reaction cross section . the latter includes all reactions and inelastic processes which cause loss of flux from the elastic channel . with the diagonal element @xmath357 of the s - matrix ( sometimes also called scattering matrix or collision matrix ) , the reaction cross section for spinless particles is then given by@xcite @xmath358 this can be generalized to other outgoing channels @xmath27 , not just the elastic one . the elements of the s - matrix are complex , in general , and related to the scattering amplitude @xmath359 of the outgoing wave function @xmath360 which , in turn , is nothing else than the transition amplitude @xmath361 connecting entrance channel @xmath0 and exit channel @xmath27 , with @xmath362 being the reduced mass in the entrance channel and @xmath363 a legendre polynomial . the imaginary part of the optical potential gives rise to an absorption term in the solution of the schrdinger equation , thus removing flux from the considered channels . therefore , the matrix element @xmath357 is also related to the transmission coefficient @xmath364 which describes the absorption of the projectile by the nucleus . important for practical application is that the phase shifts @xmath365 can be derived from elastic data . the optical model is well suited for describing transitions between states of intermediate and heavy nuclei . it has been and is still used also to treat reactions with light nuclei although other methods exist for these . the optical model can be used to compute the widths ( @xmath328 , @xmath329 , @xmath345 ) appearing in the bwf for resonances , see ( [ eq : breit ] ) above . as mentioned before , the relevant energy windows for astrophysics also include compound nucleus excitation energies with such high nld that individual resonances can not be separated because the average resonance width @xmath366 becomes larger than the average level spacing @xmath367 . in fact , this is the case for the majority of reactions included in astrophysical reaction networks . instead of explicitly dealing with a large number of unknown resonances , one moves to averaged resonance properties . starting with the bwf given in ( [ eq : breit ] ) , the sum of individual resonances can be replaced by an average over an energy interval @xmath368 using the mathematical relation@xcite @xmath369 here , the angle brackets denote the average as defined by the above equation . note that the approximation for narrow resonances , as also used in ( [ eq : narrowres ] ) and ( [ eq : narrowgs ] ) , was applied to arrive at the last line of ( [ eq : hfaverage ] ) . this is obviously allowed because of the assumption of a large number of narrowly spaced resonances . with this we rewrite the sum over resonances in the bwf as @xmath370 the number of resonances @xmath371 within an energy interval @xmath368 was replaced by the nld @xmath115 in the last line . the averaged widths , the nld , and the @xmath372 are energy - dependent , of course . the width fluctuation coefficients @xmath372 account for the different averaging in the last line @xmath373 in terms of physics , they describe non - statistical correlations between the widths in the channels @xmath238 and @xmath239 . in practice , they differ from unity only close to channel openings.@xcite making use of the relation between transmission coefficients obtained from the solution of the schrdinger equation with an optical potential and the averaged widths @xmath374 the cross section for the statistical model of compound reactions can be written as @xmath375 the summation in the denominator runs over all channels @xmath38 leading to the same compound nucleus , not only ( but including ) @xmath238 and @xmath239 . thus , this sum is equivalent to @xmath376 . also , the sums over channel spins @xmath14 and partial waves @xmath377 are explicitly written to emphasize that the transmission coefficients must include these quantum numbers . each transmission coefficient includes transitions from states at the compound energy @xmath378 ( @xmath379 being the separation energy of the projectile in the compound nucleus ) . while @xmath380 only includes those to the state @xmath102 in the target nucleus , the @xmath381 include all transitions allowed by energetics and quantum selection rules . comparing ( [ eq : hf ] ) with ( [ eq : breit ] ) and ( [ eq : narrowgs ] ) it is readily seen that the statistical model cross section is an averaged breit - wigner cross section for narrow resonances , when @xmath382 . completely equivalently to the breit - wigner case , it can be shown that for the calculation of the stellar rate it is sufficient to replace @xmath380 by @xmath383 in ( [ eq : hf ] ) and divide the resulting integral by the normalized partition function @xmath335 , giving@xcite @xmath384 the total transmission coefficients @xmath385 in each channel @xmath386@xmath387 , @xmath388 , include a sum over final states @xmath389 in that channel . similar to the treatment of the partition functions in ( [ eq : partfuncint ] ) , the sum over discrete states can be extended by an integration over a level density above the energy @xmath390 of the last discrete state included , @xmath391 the integration is over the nld @xmath392 in the channel @xmath393 , i.e. in the target nucleus @xmath31 for channel @xmath387 , in the final nucleus @xmath129 for channel @xmath388 , and so on . the transmission @xmath394 is the same as @xmath395 , only that it is a transition to an artifical state with given @xmath396 and weighted by the nld @xmath397 . the relative transition energy in channel @xmath393 is @xmath398 , where @xmath399 is the channel separation energy . the reader is advised to consult fig . [ fig : cnscheme ] to get an overview of the included transitions and their relative energies . particle transmission coefficients have to obey spin selection rules and thus @xmath400 here the angular momentum @xmath401 and the channel spin @xmath402 are connected by @xmath403 including the particle spin @xmath100 . each @xmath404 can be directly obtained from the solution of the ( time - independent , radial ) schrdinger equation at the energy @xmath405 with an appropriate optical potential . the calculation of radiative transmission coefficients proceeds equivalently to ( [ eq : parttrans ] ) but electromagnetic selection rules ( see , e.g. , appendix b of ref . ) have to be obeyed . the parities @xmath406 , @xmath407 and the angular momentum @xmath377 select the type of allowed electromagnetic transition ( e1 , e2 , m1 , m2 , etc . ) and accordingly the appropriate description has to be invoked to calculate the transition strength @xmath404 . to phenomenologically account for pre - equilibrium particle emission at higher compound excitation energy ( see sec . [ sec : statmodmod ] ) , sometimes the integration in ( [ eq : tottrans ] ) is only carried out to a cut - off energy @xmath408 , with an appropriately chosen @xmath409 ( e.g. , the energy at which the @xmath82 transmission exceeds a certain fraction of the total transmission ) for the @xmath82 transmission coefficient appearing in the numerator of ( [ eq : hfrate ] ) . the total transmission coefficient @xmath410 in the denominator , however , always has to include the full integration up to @xmath319 . the statistical model of compound reactions was initially developed by bohr , who conceived the independence hypothesis.@xcite it states that the projectile forms a compound system with the target , shares its energy among all of the nucleons , and finally the compound nucleus decays by emitting photons or particles independently of the formation process . this implicitly requires long reaction timescales as the compound nucleus has to live long enough to establish complete statistical equilibrium among the nucleons . compared to the direct mechanism ( sec.[sec : direct ] ) the timescale is about @xmath411 orders of magnitude longer and includes many degrees of freedom . in the independence hypothesis , the ( laboratory ) cross section can be factorized into two terms @xmath412 the formation cross section @xmath413 and a branching ratio describing the probability for decay to the observed channel @xmath388 . an early implementation of this was the weisskopf - ewing theory.@xcite since then , the hauser - feshbach approach has been widely used , which also incorporates conservation of angular momentum partially lifting the independence assumption but thus being more realistic.@xcite equation ( [ eq : hf ] ) is the cross section from the full hauser - feshbach formalism . nevertheless , although too simplified , eq . ( [ eq : independence ] ) is sometimes useful when estimating the relative feeding of different reaction channels . although it might seem tempting to conclude that the cross section of a reaction proceeding through the compound mechanism should be smooth because it is formed from the superposition of amplitudes from a very large number of states with random phases , this is a wrong assumption.@xcite it was first shown in ref . that the cross sections can continue to show large fluctuations . the usual hauser - feshbach equations do not account for these fluctuations . therefore , a meaningful comparison to experimental data is only possible after averaging the data over a sufficiently wide energy range , comparable to the average resonance widths . when using the statistical model to compute astrophysical reaction rates ( or when deriving rates experimentally directly ) this is taken care of automatically . however , when using beams with a very narrow energy spread it should be noted that the results can not be directly compared to calculations.@xcite it is worthwhile to point out that the reaction rate is rather `` forgiving '' to deviations around a `` true '' cross section value , provided the deviations go both ways and can cancel within the integration in ( [ eq : rate ] ) . therefore , the statistical model approach may even be applicable in the presence of small but isolated resonances as long as their average contribution is correctly accounted for . this is closely connected to the question of the applicability of the statistical model . this does not necessarily mean the actual validity of the model but rather its suitability to calculate astrophysical reaction rates . as pointed out above , the model cross sections may deviate from the actual ones with large consequences for the rate . the point is that the cross section _ averaged _ within the relevant energy window is described correctly so that the evaluation of the integral yields the `` true '' value . this is a different and less stringent criterion than one asking for a close reproduction of experimental cross sections . the rule - of - thumb@xcite of 10 levels within the relevant energy window has been found quite sound on average in simple numerical tests.@xcite it has to be emphasized that this is an average value . sometimes even one or a few resonances , if broad enough to almost fill the energy window , may be sufficient . on the other hand , it is always desireable to get as much experimental information as possible for any nucleus but especially for those with low nld in the relevant energy window . is in gk ) : @xmath414 ( 0 ) , @xmath415 ( 1 ) , @xmath416 ( 2 ) , @xmath417 ( 3 ) , @xmath418 ( 4 ) , @xmath419 ( 5 ) , @xmath420 ( 6 ) , @xmath421 ( 7 ) , @xmath422 ( 8) , @xmath423 ( 9 ) . [ fig : pgapplic],scaledwidth=80.0% ] using the criterion of 10 levels per energy window , applicability maps for neutron- , proton- , and @xmath0-induced reactions were shown in ref . , showing the lowest stellar temperature at which the statistical model is applicable to predict the rate . it is important to note that those maps were derived with the approximations ( [ eq : e0approx ] ) and ( [ eq : deltaapprox ] ) for the gamow windows and not with the correct energy windows as explained in sec . [ sec : energies ] . within the accuracy with which the plots can be read , this may not impact capture reactions too much , however , especially the energy windows for ( n,@xmath82 ) reactions have not been changed.@xcite for comparison , fig . [ fig : pgapplic ] shows the minimal temperatures required for ( p,@xmath82 ) ( although on a coarser temperature grid ) for a region of the nuclear chart when applying the correct energy windows . it has to be kept in mind , though , that reactions with different exit channels do not exhibit the same relevant energy window , as explained in sec . [ sec : energies ] . the general picture arising is the same as described in ref . . the statistical model can be applied to the majority of neutron capture reactions , with exceptions close to magic neutron numbers or low neutron separation energy , both leading to a low nld at the compound formation energy @xmath319 . since the relevant energy windows for charged - particle capture are shifted to higher energy with respect to the ones for ( n,@xmath82 ) , the applicability is even broader . this may also apply to reactions with a charged particle in entrance or exit channel but no general statement can be made because it depends on which width determines the location of the energy window . finally , endothermic reactions always require higher temperature to have appreciable rates but that does not necessarily mean that the compound nucleus is formed at high excitation energy . therefore , they may require even higher temperature until the statistical model becomes applicable . in the preceding section , the applicability of the statistical model ( hauser - feshbach model , hfm ) due to the required average nld has been discussed . modern reaction theory knows a multitude of models , each suited to a particular reaction type and mechanism . these all imply certain approximations . of course , nature is continuous , many types of reactions occur simultaneously and we have to choose an approximation suited to describe the dominant effects . astrophysical energy windows prefer low projectile energies but still may include transitional regions between several reaction mechanisms . discussed below are two types of modifications to the rate calculations : 1 ) accounting for additional reaction mechanisms , and 2 ) modifying the hfm itself to provide a smooth transition to low nld regimes . the inclusion of isospin conservation is a further modification which is discussed in a separate subsection of sec . [ sec : relevance ] . direct reactions are known to be relevant at high projectile energies but were also found to significantly contribute at low energies in nuclei with low nld . they are further discussed in sec . [ sec : direct ] . semi - direct@xcite and multistep reactions@xcite also occur on a faster timescale than equilibrated compound reactions . they become important at several to several tens of mev projectile energy and are thus outside the astrophysically relevant energy range . the capture of @xmath0-particles on intermediate and heavy nuclei may barely reach such energies but only at high plasma temperature . it was shown that semi - direct capture is negligible even for very neutron - rich nuclei at r - process conditions.@xcite when assessing the importance of additional mechanisms it is essential to consider the change they bring about for the astrophysical rates . even if the ground state transitions ( as usually explored in theoretical and experimental nuclear physics ) may barely reach the required high energy for additional processes , such as semi - direct reactions , this may not affect the reaction rates much . this is because they also include transitions from excited states according to ( [ eq : effrate ] ) and ( [ eq : effweights ] ) , which proceed at lower relative energy and thus remain unaffected by modifications of the cross sections at higher energy . the hfm assumes that the compound nucleus has sufficient time to distribute the energy gained through the interaction with the projectile among all nucleons of the compound system before it decays . at a high compound formation energy @xmath319 of several tens of mev transitions occurring before this energy - equilibration lead to multistep reactions and pre - equilibrium emission of particles and photons.@xcite the required compound excitation energy is too high to be of astrophysical relevance . the second type of modifications introduced above is discussed in the following , focussing on two ideas . the first is that subsequent emission of several particles may also occur when the final nucleus is produced at an energy above a further particle emission threshold . astrophysically this can be relevant in nuclei close to the driplines with low separation energies . it can be treated approximately by applying an iterative application of ( [ eq : hf ] ) , ( [ eq : tottrans ] ) , and ( [ eq : parttrans ] ) to transitions to states in each system formed in each emission process . this has been used , for example , for calculating neutrino - induced particle emission which is relevant to the construction of neutrino detectors.@xcite in such calculations , the formation transmission coefficients @xmath424 in ( [ eq : hf ] ) are replaced by neutrino transmission coefficients @xmath425 describing the population of compound states by neutrino interactions ( using , e.g. , the random phase approximation ) . neutrino reactions select high excitation energies and thus multi - particle emission will also be important at stability . the same approach can be used to determine @xmath27-delayed particle emission when @xmath27-decay produces a daughter nucleus in an excited state above a particle - separation energy . it can also be used to study @xmath27-delayed fission . both processes are important in the r - process.@xcite the second idea revolves around the fact that regular hfm calculations assume a compound formation probability independent of the compound nld at the compound formation energy . therefore the sum in ( [ eq : cs ] ) runs over all @xmath426 pairs ( a high - spin cutoff is introduced in practical application of the model because spin values far removed from the spins appearing in the initial and final nuclei do not contribute significantly to the transmission coefficients ) . the availability of compound states and doorway states defines the applicability of the hfm.@xcite relying on an average over resonances , the hfm is not applicable with a low nld at compound formation . furthermore , not all spins and parities will be available with equal probability at each @xmath319 , especially at low nld . on average the hfm will then overpredict the resonant cross section ( unless single resonances dominate ) because it will overestimate the compound formation probability . this can be treated by introducing a modification of the formation cross section which includes the compound nld dependence . the summands of ( [ eq : cs ] ) will then be weighted according to the available number of states with the given @xmath426 . ( formally this is the same as assuming @xmath426 dependent potentials for particle channels . ) in the most general case this will require a dependence on a @xmath426-dependent compound nucleus nld at @xmath319 . note that the standard hfm only includes the nld of the compound nucleus in the photon transmission coefficients ( see sec . [ sec : relevance ] ) to determine the endpoints of the @xmath82-transitions . the assumption that all spins and parities are available can be lifted in several steps . a parity - dependent , global nld @xmath392 was used in the calculation of the transmission coefficients for nuclei without experimentally determined excited states ( see sec . [ sec : relevance]).@xcite shortly thereafter , the parity - dependence of the compound formation was implemented in a modified hfm in version 4.0w of the code non - smoker@xmath427 ( see also sec . [ sec : codes]).@xcite a discussion of the implications of a parity - dependent compound formation for astrophysical neutron capture is given in ref . . although the parities are not equidistributed up to sizeable excitation energies , the impact on _ stellar _ rates remains small ( in comparison to other uncertainties ) because the effective cross section ( [ eq : effcs ] ) also includes transitions from excited states which washes out the selectivity on parity . this gives rise to factors of two modifications close to the neutron dripline ( but see sec . [ sec : equilibria ] ) . additionally , non - smoker@xmath427 offered the option of weighting the hfm cross section by a function depending on the total nld since version 4.0w.@xcite an improved implementation with @xmath426 dependent weighting of the summands , thus implicitly accounting for a low nld at the compound formation energy , is introduced in the smaragd code and will be used for a future update of large - scale reaction rate predictions.@xcite preliminary results with this modification are shown in fig . [ fig : avdc ] in sec . [ sec : avdc ] . obviously , these modifications of the hfm depend on the nld treatment to obtain the spin- and parity distribution . see the paragraphs on the nld in sec . [ sec : relevance ] for further details . there are different ingredients required to calculate the hfm cross section with the formulas given in the preceding sections . which ingredients impact different parts of the calculation in what manner is discussed below , but how a change in the transmission coefficients ( of certain or all included transitions ) affects the resulting cross section can only be understood with the help of ( [ eq : cs ] ) . similar to the determination of the energy - dependence of the cross section ( which is crucial in the derivation of the relevant energy windows in sec . [ sec : energies ] ) , the sensitivity of the cross section and rate to a change in the nuclear properties of the participating nuclei depends on which transmission coefficient ( or averaged width ) actually affects the cross section while the others cancel from ( [ eq : cs ] ) . as already pointed out in sec . [ sec : energies ] , the discussion applies to both the bwf and the hfm . thus , the sensitivities are rather well known in certain parts of the experimental community studying resonances and in the field of nuclear data evaluation . since they do not seem to be so well known in nuclear astrophysics and their implications also have to be interpreted in terms of stellar cross sections and astrophysical reaction rates , it is helpful to outline briefly the main points here . by comparison to ( [ eq : hf ] ) and ( [ eq : hfrate ] ) we find that ( [ eq : cs ] ) transforms to @xmath428 for each @xmath429-dependent summand . in laboratory cross sections ( usually with @xmath112 ) only few @xmath429-summands contribute due to the spin selection rules , and so the application of @xmath430 to determine the sensitivities to changes in the widths is straightforward . the situation is different for stellar rates and ratios @xmath431 because transitions from excited target states additionally contribute and more terms in the sum may be relevant , depending on the nucleus and the plasma temperature . it is interesting to note that @xmath410 includes @xmath432 in both cases . depending on whether the average entrance transmission @xmath433 significantly defines the size of @xmath432 , a variation of the entrance transmission will affect the total transmission more or less this can lead to a different sensitivity of laboratory cross sections to the entrance channel than to the transmission in the exit channel , even if everything else is comparable . the interpretation of experimental results concerning the impact on stellar rates has also to proceed carefully . for example , if a strong dependence is found on the entrance transmission coefficient and some deficiencies when comparing the model to experimental data , this does not necessarily mean that this is of relevance to the astrophysical rate . or strength functions @xmath434 can be determined experimentally , it is more useful to obtain the latter because they are directly proportional to the transmission coefficients @xmath381 without an additional dependence on the compound nld.@xcite ] since the astrophysical rate involves @xmath432 also in the numerator , it may more often cancel with the denominator , even if it would not for @xmath433 in the numerator . only for rates with low sef @xmath435 . this shows again that stellar rates and laboratory cross sections do not have a one - to - one correspondence and additional , mostly theoretical considerations have to be included . several special cases can appear in ( [ eq : fraction ] ) : ( i ) the larger of the two transmissions in the denominator also dominates the numerator ; then the cross section or rate will change similarly to a change of the smaller transmission coefficient in the numerator and be oblivious to any others ; ( ii ) three channels will be important when neither of the transmission coefficients in the numerator dominates the total transmission ; then any change in the two transmissions in the numerator will be translated into an equal change in the cross section but a change in the one determining the denominator will result in an inverse proportional change in the cross section . in the general case when @xmath410 is not dominated by a single channel the situation is more complicated as any change in a transmission coefficient will not fully affect the cross section at a similar level . a helpful visual aid to estimate the relative importance of the different channels is a sensitivity plot . i define the sensitivity @xmath436 as a measure of a change in the cross section @xmath437 as the result of a change in a transmission coefficient ( or width ) by the factor @xmath438 , with @xmath439 when no change occurs and @xmath440 when the cross section changes by the same factor as the transmission coefficient ( or width ) : @xmath441 plotting @xmath436 as a function of the c.m . energy yields a plot like the example shown in fig . [ fig : sensi ] for the reaction @xmath442ru(p,@xmath82)@xmath443rh . its astrophysically relevant energy window is @xmath444 mev for the typical p - process temperature @xmath445 gk.@xcite it can clearly be seen in fig . [ fig : sensi ] that the sensitivities are very different at lower and higher energies . for example , a measurement closely below the neutron threshold would be in a region where @xmath436 is largest for the @xmath82 transmission coefficient ( or width ) but smallest for the proton transmission , just the opposite of what is found in the astrophysically relevant energy region . above the neutron threshold the situation is even more complicated because there is additional sensitivity to the neutron channel ( dominating @xmath446 ) , although not as large as to @xmath447 . if any discrepancy between measured and predicted cross sections was found , it would be hard to disentangle the different contributions . in this example , no information on the astrophysically important proton transmission coefficient can be extracted from a measurement at higher energies . the above example also shows that some of the standard assumptions usually used in nuclear physics experiments do not apply . for instance , it is usually assumed that the @xmath448width is always smaller than particle widths and therefore a capture reaction will always only be sensitive to the @xmath448width . many , if not most , astrophysical reactions with charged particles , however , proceed close to or below the coulomb barrier . this leads to very narrow charged particle widths and they may well become smaller than the @xmath448widths . even neutron widths close to a neutron threshold become smaller than photon widths ( which are relatively independent of energy compared to the particle widths ) . therefore it is always important to closely inspect the widths and to perform a thorough sensitivity study when investigating the astrophysical impacts of changes in the hfm inputs . from the general considerations above it also follows that it is advantageous to use reactions with neutrons in one channel for investigating the sensitivity to the charged - particle optical potential , i.e. , using ( @xmath0,n ) , ( n,@xmath0 ) , ( p , n ) , or ( n , p ) reactions . except within a few kev above the neutron threshold , the neutron width will always be much larger than the charged - particle width , even at higher than astrophysical energies . therefore it will cancel with the denominator in eq . and leave the pure energy dependence of the charged - particle width . on the other hand , the neutron channel may not be open at the energies required to study the astrophysically important charged particle width and extrapolations have to be performed . furthermore , this shows that it is difficult to obtain information on the neutron potential from reactions . but this also implies that the sensitivity of astrophysical rates to the neutron optical potential is not high . nuclear properties and how they affect the calculation of transmission coefficients ( or averaged widths ) are briefly discussed in the following . this is by no means meant to be an exhaustive listing and discussion but rather some exemplary points are taken to explain the special requirements in the calculation of astrophysical rates and to point out the challenges . further information on the nuclear input used in statistical model calculations for nuclear physics and nuclear astrophysics can be found , e.g. , in refs . and references therein . while the hfm has been used extensively to study reaction data , its application in nuclear astrophysics has a slightly different focus . standard nuclear physics investigations use the hfm by including measured or known properties of nuclei . then reaction cross sections can be reproduced with high accuracy . stellar cross sections not only require the modification of the hfm shown above but also the inclusion of mostly unknown further input because the reactions proceed at much lower energy and/or involve unstable nuclei . the required properties have to be predicted _ globally _ for a large number of nuclei and it has to be realized which properties are important in which astrophysical process . this is where the challenge lies and where basic research is necessary , going beyond a mere application of a seasoned model . although intellectually the most satisfying , modern microscopic models still are not applicable for large - scale predictions and do not produce all the necessary data with sufficient reliability . this calls for clever combinations of microscopic models and parameterizations ( which can also use dependences derived from microscopic calculations for a limited range of nuclei ) . furthermore , additional information from experiments is required to constrain such models . [ [ masses ] ] masses : + + + + + + + nuclear masses or , rather , mass differences determine separation energies and reaction @xmath94-values . in this way , they also determine the range of transition energies to be considered in each reaction channel and through this the relative importance of a channel . except in the nse equations ( [ eq : nse ] ) , masses always appear in mass differences . this poses a potential difficulty when a certain mass region is not fully explored experimentally yet . care has to be taken to avoid artifical breaks and structures in the mass surface when calculating mass differences from a mix of experimental and theoretical masses . this is usually considered in codes especially written for astrophysical applications . on the other hand , it is expected that mass differences can be measured as well as predicted with higher accuracy than single masses . this seems reassuring for the calculation of astrophysical rates far from stability . a change in the mass of a nucleus impacts the rates in two different manners . first , the separation energies are altered in the reaction channels including the nucleus . this leads to a change in the transition energies in these channels ( see fig . [ fig : cnscheme ] ) . if the change is large , also more or fewer transitions may become possible . although charged - particle transitions sensitively depend on the interaction energy , it has to be realized that even in this case the change in @xmath94-value _ plus _ ejectile energy has to be considerable to have a sizeable impact on the rate . the entrance channel is not affected . secondly , a change in the @xmath94-value changes the relation between forward and reverse rate as shown in ( [ eq : revrate ] ) and ( [ eq : revphoto ] ) . it is highly sensitive to a small change in @xmath94-value due to the exponential dependence . this impacts the temperature at which the two rates become comparable and at which equilibria are reached . on the other hand , if forward and reverse rates are different by many orders of magnitude ( i.e. for large @xmath280 ) , there may not be much astrophysical impact , nevertheless . [ [ properties - of - ground - and - excited - states ] ] properties of ground and excited states : + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + it is standard procedure to include spin , parity , and excitation energy of low - lying discrete levels when calculating the transmission coefficients . information about discrete states comes from experiments or from nuclear theory ( single particle states and shell model states ) . close to stability , a large number of excited states are well known . however , often excited states of nuclides not produced in reactions on stable target nuclei are only partially known experimentally , even if the nuclide itself is stable . obviously , the situation worsens further off stability . therefore , nuclear spectroscopy is important to provide the database for reaction modelling . discrete states are not only important in the hfm but also for resonant reactions treated in an @xmath314-matrix or bwf approach ( unbound states ) and even more for the treatment of direct reactions ( see sec . [ sec : direct ] ) ( bound states ) . it is very important to have a complete set of excited states because a large number of missing levels at and below a given excitation energy would lead to an incorrect prediction . therefore it is important to set a cut - off in the excitation energy above which no states are included ( even if there are some data ) and below which the assumption of a complete level set holds . above the cut - off , a nld is employed ( see below ) . it is difficult to define a useful cut - off energy but blindly including all existing data leads to worse results than neglecting too many , provided a reliable nld description is used . usually , an educated guess has to be made by comparing the level data to nld predictions . [ [ nuclear - level - density ] ] nuclear level density : + + + + + + + + + + + + + + + + + + + + + + it should be noted that the level density @xmath115 used throughout this paper follows the `` experimental '' definition of number of observed levels per energy interval @xmath449 , where @xmath450 is the observed number of levels with spin @xmath320 and parity @xmath406 in a small energy interval around an excitation energy @xmath67 . it is not to be confused with a state density @xmath451 appearing in microscopic nuclear theory , such as the shell model . the terms `` level density '' and `` state density '' are used inconsistently in the literature . for the relation of the two types of density , see , e.g. , refs . . in the regular hfm , the nuclear level density only enters in the calculation of the transmission coefficients ( eq . [ eq : tottrans ] ) when there are no or not enough discrete states known and therefore its importance depends on how many low - lying , discrete states were included ( see above ) . because of the energetics connected to particle emission ( see fig . [ fig : cnscheme ] ) , usually only a small fraction of the particle width is due to transitions calculated with a level density at astrophysically relevant projectile energies as long as discrete excited states are known . this is especially true for neutron - capture reactions due to their low interaction energies . reactions with charged particles in entrance or exit channel prefer somewhat higher projectile energies due to the location of the relevant energy window ( sec . [ sec : energies ] and thus may show slightly larger sensitivity to the level density in the target and final nucleus . ( these sensitivities are different from those encountered in usual nuclear reaction studies where projectile energies are much larger . ) on the other hand , there is a larger range of energies available for @xmath82-transitions ( see fig . [ fig : cnscheme ] ; grey areas signify transitions calculated by integration over a level density , as shown in the second term on the right hand side of eq . [ eq : tottrans ] ) and thus the impact of the level density in the compound nucleus will be largest . in most cases , it is accurate to assume that a variation of the level density will mostly ( or only ) affect the @xmath82-widths . generally , the impact of a change in the nld will be larger for nuclides with fewer known discrete states , i.e. far from stability . on the other hand , the @xmath94-values of the astrophysically relevant reactions become lower and thus ground state transitions to the final nucleus may again dominate . it is instructive to remember which excitation energies are the most important ones . for particle transmission coefficients the transitions with highest relative energy are most important , i.e. to the ground state and the lowest excitation energies in both target and final nucleus ( similar considerations apply to direct reactions ) . discrete levels or a nld have to be known there . for electromagnetic transitions , the relevant excitation energy is around @xmath452 in the compound nucleus ( see the discussion of electromagnetic transmission coefficients below and ref . for a detailed explanation ) . due to the low excitation energies involved it is important to a ) include the correct ground state spin and parity , and b ) to account for a possible parity dependence of the nld at low excitation . both is automatically ensured when experimental information is available up to sufficiently high excitation energies . otherwise , ground state properties and nlds have to be predicted . this introduces additional uncertainties in rates far from stability . until recently , astrophysical rate predictions made use of equally distributed parities @xmath453.@xcite modern rate predictions include parity - dependent nlds in various manners . either , microscopic nlds are directly utilized in calculations@xcite or a parity - dependence is applied to a total nld.@xcite the advantage of the latter approach is that it can conveniently be applied to total nlds from different sources and for a large number of nuclei . the total nld @xmath115 is not changed but the parities are redistributed according to excitation energy . this approach was used to study the impact of a parity - dependence across the nuclear chart . since capture reactions mainly populate higher lying states , for which an equipartition of parities already is a good assumption , the impact of a parity - dependent level density is small , unless very low @xmath319 is encountered due to low projectile separation energies and low plasma temperatures.@xcite according to the discussion above , however , the impact is larger in particle emission channels and for direct reactions . a modified hfm is introduced in sec . [ sec : statmodmod ] above , which accounts for the relative level distribution at the compound formation energy @xmath319 . again , the impact of a parity - distribution is small for sufficiently large @xmath319 and only becomes important far from stability at low separation energies ( in the r - process).@xcite however , it remains doubtful whether this actually is of astrophysical relevance as it is not clear whether the involved nuclei can be produced outside of equilibria ( see sec . [ sec : equilibria ] ) and whether the uncertainty introduced by using the statistical model for nuclei with such low nld at @xmath319 is not much larger than the impact of the parity - dependence . on the other hand , a dependence on the total nld and on the @xmath320 distribution in the further modified hfm may be more important at a larger range of excitation energies and thus also for nuclei closer to stability ( see sec . [ sec : statmodmod ] ) . in any case , the uncertainties introduced by the predicted nlds in rates far from stability are overall much smaller than the uncertainties stemming from other input to the hfm calculations , such as optical potentials and photon transmission coefficients . [ [ optical - potentials ] ] optical potentials : + + + + + + + + + + + + + + + + + + + optical potentials are required in the solution of the radial schrdinger equations to determine the particle transmission coefficients as shown in ( [ eq : parttrans ] ) . together with the electromagnetic transitions strengths ( see below ) , the unknown optical potentials at low energy give rise to the largest uncertainties in astrophysical reaction rate predictions . there is a combination of two problems involved in the determination of appropriate optical potentials for astrophysical applications : the prediction of optical potentials for highly unstable nuclei and the extension to the astrophysically relevant energies . there is a large amount of reaction and scattering data along the line of stability and many parameterizations ( usually of the saxon - woods shape ) , partly mass- and/or energy - dependent , are available . however , most scattering experiments to derive optical potential parameters have been performed at several tens of mev , far above the astrophysically relevant energy window ( see sec . [ sec : energies ] ) . even at stability , there are almost no data ( not even reaction data ) for charged - particle reactions at astrophysical energies . measuring low - energy cross sections for charged - particle reactions is especially problematic due to the coulomb barrier causing the astrophysically relevant cross sections to be tiny . even more problematic is the standard way to obtain information on optical potentials through elastic scattering experiments . the scattering cross section at low energy becomes indistinguishable from rutherford scattering . in the optical model , an imaginary part of the potential appears whenever there is loss of flux from the elastic channel due to any kind of inelastic process . the reaction cross section is sensitive to both real and imaginary part of the optical potential because they determine the relation of real and imaginary part of the nuclear wavefunction and thus the transmission coefficient.@xcite microscopic approaches to derive the optical potential are preferrable over parameterizations when predicting rates far from stability , especially because the available parameterizations were derived at far too high projectile energies . especially the imaginary part of the potential may vary strongly with energy , due to the energy - dependence in the available reaction channels included in the imaginary part . nevertheless , any more sophisticated theoretical approach also includes some parameters which have to be constrained by comparison with experiment and so even in this case there may be uncertainties at low energy ( see below and the further discussion of how additional reaction mechanisms impact the optical potential in sec . [ sec : directintro ] ) . an optical potential widely used for interactions of nuclei with neutrons and protons uses the brckner - hartree - fock approximation with reid s hard core nucleon - nucleon interaction and adopts a local density approximation.@xcite a low - energy modification of this potential was provided specially for astrophysical applications.@xcite the latter has become the standard potential in predictions of astrophysical rates and is generally very successful when compared to the scarce experimental data at low energy . as stated in sec . [ sec : sensigeneral ] , astrophysical rates are rather insensitive to the neutron potential . regarding rates involving charged particles in entrance or exit channel , an important difference to cross sections at higher energies is in the fact that astrophysical cross sections are mostly sensitive to the charged particle widths instead of the @xmath82- or neutron widths . a series of ( p,@xmath82 ) and ( p , n ) reactions was measured close to astrophysical energies recently ( see , e.g. , refs . , and references therein ) . the latter reactions are especially useful for testing the proton potential because the neutron width will ( almost always ) be larger than the proton width at all energies . despite of the overall good agreement when using the standard potential , certain systematic deviations at low energy were found recently ( see , e.g. , figs . [ fig : pg1 ] , [ fig : pg2 ] , more examples are shown in ref . ) . it was found that an increase in the strength of the imaginary part at low energies considerably improves the reproduction of the data ( denoted by `` mod jlm '' in figs . [ fig : pg1 ] , [ fig : pg2 ] , [ fig : dens1]).@xcite an increased absorption is permitted within the previous parameterization because the isoscalar and especially the isovector component of the imaginary part is not well constrained at low energies.@xcite therefore , the change has to be energy - dependent ( i.e. acting only at low energy ) . more low - energy data is required to obtain a better picture . figures [ fig : pg1 ] , [ fig : pg2 ] also shows results obtained with another recent , lane - consistent new parameterization of the jlm potential.@xcite although it showed improved performance at higher energy , it yields worse agreement at astrophysically low energy . this is understandable as neither does it include the additional modifications of ref . , nor can it constrain well the low energy part because it was fitted to data at higher energy.@xcite similar considerations apply to another recent reevaluation of the standard potential.@xcite required input to the calculation of this type of optical potentials is the nuclear density distribution @xmath454 . figure [ fig : dens1 ] shows results when employing a droplet model density@xcite and one from a hartree - fock - bogolyubov model ( hfb-2).@xcite for the reactions considered here , the droplet description yields better agreement to the data in both absolute scale and energy dependence of the theoretical s - factor . for comparison , [ fig : dens2 ] also shows the results when employing the optical potentials of refs . with both densities . in the original work , hfb densities were employed.@xcite for further information on how nuclear density distributions affect the astrophysical rates see the subsection on density distributions below , including figs . [ fig : pgdens1]@xmath215[fig : ngdens2 ] . the standard potential for neutrons and protons seems to work very well compared to the situation encountered when exploring the adequacy of @xmath0+nucleus optical potentials for astrophysics . global parameterizations describing scattering , reaction , and decay data have been notoriously hard to find for @xmath0-particle potentials . somewhat surprisingly , a mass- and energy - independent potential of saxon - woods type has been quite successful and is widely used to evaluate reaction data and also for astrophysical applications.@xcite the potential was fitted to scattering data at 26.7 mev @xmath0-energy for a wide range of nuclei . however , it became obvious early on that it may be impossible to find a global potential with a predictive power comparable to those for nucleonic projectiles , especially at low energy.@xcite the number of optical potential parameters can be reduced by using folding potentials @xmath455 for the real part,@xcite @xmath456 in this expression @xmath22 is the separation of the centers of mass of the two interacting nuclei , @xmath457 and @xmath458 are their respective nucleon densities and @xmath21 is an adjustable strength factor . the factor @xmath21 may differ slightly from unity because it accounts for the effects of antisymmetrization and the pauli principle . the effective nucleon - nucleon interaction @xmath459 for the folding procedure is usually of the ddm3y type.@xcite density distributions have to be taken from experiment or theory ( see the section on matter density distributions below ) . a global parameterization of the real part with such folding potentials was found based on extensive scattering data.@xcite unfortunately , there is no simple description for the imaginary part , for which shape and strength have to be energy - dependent . correlations with the compound nld , different parametrizations for the energy - dependence of the strength ( fermi - function , brown - rho dependence ) , and an energy - dependence in the relative strength of volume and surface imaginary parts have been suggested.@xcite again , those extrapolations to low energy are only loosely constrained due to the lack of scattering data . based on apparently different potentials required for the description of @xmath0-particles in entrance and exit channel and the fact that a potential fitted to reaction data is able to describe a number of reactions but does not reproduce scattering data , it was suggested that there may be some dependence on nuclear temperature and that absorption and emission potentials may be different from scattering potentials.@xcite moreover , at energies close to or below the coulomb barrier , so - called `` threshold anomalies '' have been observed , a rapid variation of optical potential parameters with energy.@xcite it has been shown that the dispersion relation connecting real and imaginary part of the optical potential is essential to describe these.@xcite there is a large literature on different local and global parameterizations of @xmath0+nucleus potentials , mostly at high energy , underlining the lack of a coherent nuclear physics treatment . a complete review can not be provided here but see , e.g. , refs . and references therein for further details . for astrophysical applications , early reaction rate predictions made use of an equivalent square well potential.@xcite later on , the potential of ref . was used . with accumulating evidence that low - energy data deviates from the predicted cross sections more complicated parameterizations were tried , based on various combinations of scattering and reaction data , but these did not lead to a consistent picture . on the other hand , the potential by ref . ( see also refs . ) was fitted to simultaneously reproduce low - energy reaction cross sections of @xmath460nd(n,@xmath0)@xmath461ce , @xmath462sm(n,@xmath0)@xmath463nd , and @xmath463sm(@xmath0,@xmath82)@xmath464gd . although the potential does not describe scattering data ( at higher energy ) it was found to work surprisingly well for low - energy cross sections for target nuclei across a large mass range @xmath465.@xcite sharing the same imaginary part with the potential of ref . , it predicts systematically lower cross sections at low energy due to a shallower real part . although not fully satisfactory yet , this shows that the main change required by a global @xmath0-potential is to reduce the predicted low - energy cross sections by factors of about @xmath466 . the only and remarkable exception to these factors known so far is in the comparison to experimental data for @xmath463sm(@xmath0,@xmath82)@xmath464gd.@xcite this reaction is important in the astrophysical context not just generally in p - process calculations@xcite but specifically in deriving astronomical timescales from the nd / sm abundance ratios measured in meteoritic inclusions or , vice versa , to determine production ratios of these elements in supernovae.@xcite the astrophysical s - factors were measured from @xmath467 mev , the astrophysically relevant energy window extends from 9 mev downwards . above 11.5 mev the s - factors obtained with the potential of ref . are too high by a factor of three but the data is well described with the potential of ref . and a potential derived from scattering data at 20 mev.@xcite however , below 11.5 mev the energy - dependence changes and requires further modifications of the potentials . at the lowest measured energy of 10.2 mev , the measured s - factor differs by a factor of 10 from the one predicted with the potential of ref . . in ref . a potential with energy - dependent imaginary part was fitted to describe the data . it predicts s - factors at and below 9 mev which are orders of magnitude lower than those predicted with the global potentials . although the original problem was a too low calculated nd / sm ratio as compared to what is found in meteorites , such low s - factors yield much too high ratios and additional astrophysical dilution effects have to be invoked in an ad hoc manner.@xcite the s - factors are shown in fig . [ fig : sm144a ] , where also new results obtained with the recent potential of ref . are included . it should be noted that this reaction may be a special case because it is strongly endothermic . nevertheless , it was shown that the stellar enhancement factor is lower in the capture direction ( see also sec . [ sec : stellarexp]).@xcite since the extrapolation to astrophysical energies strongly hinges on the data points at the lowest energies , an independent remeasurement of this reaction at comparable or lower energies is highly desireable . further issues in the determination of s - factors below the coulomb barrier are illustrated in figs . [ fig : sm144a]@xmath215 [ fig : sm144rcoul2 ] . figure [ fig : sm144a ] shows the results obtained when using the routine of ref . ( as also used in refs . ) for the solution of the schrdinger equation and the determination of the @xmath0-particle transmission coefficients . it does not work properly when applied at energies at or below the coulomb barrier as a comparison to results obtained with a modern routine shows ( fig . [ fig : sm144b ] ) . the discrepancy between predictions and measurement is even enhanced when using an appropriate method . since the s - factors are very sensitive also to the wavefunction far outside the nuclear center , they show a strong dependence to the shape and width of the effective barrier ( determined by the sum of nuclear potential and coulomb potential ) . results obtained with the optical potentials of refs . and with fine - tuned energy - dependences of the real and/or imaginary part also have a strong sensitivity to the coulomb radius parameter , as shown in figs . [ fig : sm144rcoul1 ] and [ fig : sm144rcoul2 ] . this is often overlooked because the sensitivity is much lower at higher energies . also , when using the potentials of refs . there is no sensitivity to the coulomb radius in the investigated energy range . these additional complications show that any extrapolation to subcoulomb energies has to be performed very carefully and that it is difficult to construct a global potential . further experimental data ( on scattering and reactions ) are especially in demand for improving the optical potentials ( see also the discussion of additional reaction mechanisms contributing to the absorptive part of the potential in sec . [ sec : directintro ] ) . currently , progress is hampered by the lack of systematic reaction ( and scattering ) data at astrophysically relevant energies , even at stability . [ [ electromagnetic - transitions ] ] electromagnetic transitions : + + + + + + + + + + + + + + + + + + + + + + + + + + + + first , a few words on the energies of the emitted @xmath82-rays and their significance for changes in @xmath82-ray strength functions ( drawn from experiment or from theory ) are in order . as shown in fig . [ fig : cnscheme ] , the energies of emitted @xmath82-rays are in the range @xmath468 . therefore , the behavior of @xmath448strength functions at low energy have to be known . since the strength of the @xmath82-transition scales with some power of @xmath469 , @xmath82-transitions with higher energies are favored . on the other hand , the number of available endpoints of the transitions increases with increasing excitation energy of the nucleus because the nld increases rapidly . this competition between transition strength and nld gives rise to a peak in the @xmath82-emission energies as shown in fig . [ fig : gammapeak ] . this peak is fragmented when certain transitions to discrete excited states are dominating . this is mainly the case far off stability for captures with low @xmath94-values , forming a compound nucleus with low nld.@xcite figures [ fig : gammaenergies1 ] , [ fig : gammaenergies2 ] show examples for the @xmath82-energies which maximally contribute to the reaction rate integral . interestingly , it can be seen that for astrophysically relevant projectile energies , the @xmath82-energies with the strongest impact are between @xmath470 mev unless the level density is so low that only few transitions are allowed ( usually for highly unstable nuclei).@xcite then the relevant @xmath471 is @xmath472 ( in the figures this is the ground state with @xmath473 ) instead of the almost constant value below @xmath319 . the @xmath82-emission peak defines the range of @xmath82-energies at which changes in the strength function have largest impact as well as the excitation energies at which the nld is most important . this also holds for the reverse reaction ( photodisintegration ) under stellar conditions because the additional , linear weight @xmath474 from ( [ eq : effweights ] ) has a much weaker @xmath471 dependence then both the nld and the @xmath82-strength . in consequence , changes in the strength function around this energy have the largest impact . testing strength function models outside the energy range will not be relevant to astrophysics . unfortunately , such low energies can not be probed by photodisintegration experiments because they are below the particle separation energies , at least close to stability . such experiments would allow to study strength functions in the most direct way ( but they can not test the astrophysically relevant rates , either , see sec . [ sec : stellarexp ] ) . other types of experiments are complicated by the fact that the observables are generated by a convolution of different nuclear properties ( such as the dependence on the nld or different spin selectivities of transitions ) which have to be known and disentangled . at least the dominant @xmath82-transitions ( e1 and m1 ) have to be included in the calculation of the total photon width for astrophysics . some codes offer the possibility of including higher order transitions . there are two issues involved : 1 ) obtaining , understanding , and modeling photon strength functions ( psf ) at stability ; 2 ) predicting strength functions far from stability by using parameterized or microscopic models , predicting the nuclear properties ( e.g. , deformation ) entering the descriptions . despite of decades of experience in studying nuclear reaction data , the understanding of the electromagnetic transitions between nuclear states is limited and predictions are subject to considerable uncertainties , even at stability.@xcite the paradigm in the field of studying electromagnetic transitions is the validity of the reciprocity theorem ( [ eq : reci_single ] ) also when applied to photon emission and absorption , and the independence of the psfs from the nuclear structure of the initial and final states ( except for spin and parity , selecting the allowed multipolarity of the radiation).@xcite this is called the brink hypothesis.@xcite this is also the basis for the construction of the effective cross section ( [ eq : effcs ] ) , the expression for the reverse rate ( [ eq : revphoto ] ) , and the introduction of equilibrium abundances ( sec . [ sec : equilibria ] ) . the brink hypothesis has been studied extensively in experiments and its violation would have grave consequences not only in nuclear reaction theory but also for astrophysical reaction rates and network calculations . among the collective modes of nuclei the electric dipole ( e1 ) excitation has the special property that most of its strength is concentrated in the isovector giant dipole resonance ( gdr ) . macroscopically , this strong resonance is described as a vibration of the charged ( proton ) matter in the nucleus against the neutral matter ( neutrons ) . the transmission coefficient in a nucleus with charge number @xmath212 , neutron number @xmath211 , and mass number @xmath475 can be parameterized as@xcite @xmath476 here , @xmath477 is the proton mass , @xmath478 accounts for the neutron - proton exchange contribution,@xcite and the summation over @xmath15 includes two terms which correspond to the split of the gdr in statically deformed nuclei , with oscillations along ( @xmath479 ) and perpendicular ( @xmath480 ) to the axis of rotational symmetry . in this deformed case , the two resonance energies are related to the mean value calculated by the relations@xcite @xmath481 the deformation parameter @xmath309 is the ratio of the diameter along the nuclear symmetry axis to the diameter perpendicular to it , and is obtained from the experimentally known deformation or from mass model predictions . many microscopic and macroscopic models have been devoted to the calculation of the gdr energies @xmath482 and widths @xmath483 . of special interest here is the low - energy tail of the gdr . it is a long - standing question of nuclear physics to specify how much of the e1 strength is still present at energies far below the gdr maximum , which also encompasses the astrophysically relevant energy region . theoretically it has been shown that it is justified to describe the gdr by a lorentzian also below the particle emission thresholds.@xcite various experimental attempts to determine the low - energy extension of the gdr for heavier nuclei have led to conflicting results . neutron - capture experiments often have indicated an overshoot of the lorentzian over the observed e1 strength at the low - energy tail of the gdr.@xcite on the basis of these data theoretical explanations have been proposed to explain the differences.@xcite photon - scattering experiments , however , are in many cases in good agreement with the lorentzian extrapolation.@xcite unfortunately , they do not access the astrophysically relevant energies . also other experiments are used to extract psfs , e.g. , @xmath484he - induced reactions.@xcite recent investigations have shown that the e1 strength can be described by lorentzians for a large range of nuclei.@xcite it was also shown that indirect determinations of pdfs are prone to large uncertainties due to the experimental difficulties and some claims of enhancement@xcite at low @xmath471 have been premature.@xcite however , in some nuclei extra strength at low energy with respect to the smooth lorentzian was found consistently and denoted as `` pygmy dipole resonance '' ( pdr).@xcite the pdr has experimentally been studied so far in spherical nuclides around @xmath485 , @xmath486 , and in the doubly magic @xmath487pb.@xcite theoretical approaches describe the pdr as caused by an oscillation of excessive neutrons against the symmetric proton - neutron system ( see , e.g. , refs . ) . other oscillation modes were also proposed ( e.g. , scissor modes ) which may also add strength beyond the lorentzian tail of the gdr . in any case , extra strength in the low - energy tail only has an astrophysical impact if it is within the relevant energy range defined above . although a pdr may lead to an increase by several orders of magnitude in the astrophysical capture rate,@xcite this depends sensitively on its location and width . different models give varying predictions of these crucial properties . depending on the microscopic model used , the pygmy resonance is sometimes predicted at too high or too low an energy as to have any astrophysical consequence.@xcite further investigation of this issue is required . it is important to note that the uncertainties in predicting the pdr enter additionally to the general uncertainties still present in the prediction of gdr energies and widths . together with the predictions of optical potentials for the particle transmission coefficients , these are among the largest uncertainties in the determination of astrophysical rates . psfs of higher multipole order are even less studied due to their small contributions to cross sections . there are several descriptions available for m1 transitions , starting from psfs from the simple single particle approach to more sophisticated ( but less thoroughly tested ) models.@xcite unless energy - independent psfs are employed , a relevant energy window similar to the one for e1 transitions will arise . [ [ isospin ] ] isospin : + + + + + + + + isospin conservation restricts transitions to certain final states with the same isospin as the initial and compound states , i.e. @xmath488 . isospin conservation is not absolute and cross section measurements of isospin - forbidden reactions give an estimate of the size of the isospin breaking ( or isospin mixing ) . internal isospin mixing due to the coulomb interaction and external mixing via the other reaction channels have to be distinguished . the hfm equation as shown in ( [ eq : hf ] ) with the transmission coefficients ( [ eq : tottrans ] ) does not account for isospin conservation unless it is included in the transmission coefficients . in other words , complete isospin mixing is assumed . the calculation of the transmission coefficients can be generalized to explicitly treat the contributions of the dense background states with isospin @xmath489 and the isobaric analog states with @xmath490.@xcite in reality , compound nucleus states do not have unique isospin and for that reason an isospin mixing parameter @xmath491 was introduced which is the fraction of the width of @xmath492 states leading to @xmath493 transitions.@xcite for complete isospin mixing @xmath494 , for pure @xmath493 states @xmath495 . in the case of overlapping resonances for each involved isospin , @xmath496 is directly related to the level densities @xmath497 and @xmath498 , respectively . isolated resonances can also be included via their internal spreading width @xmath499 and a bridging formula was derived to cover both regimes.@xcite in order to determine the mixing parameter @xmath500 , experimental information for excitation energies of @xmath492 levels can be used where available.@xcite experimental values for spreading widths are also tabulated.@xcite inspection of the tables shows that internal mixing dominates and that the associated spreading width is nearly independent of mass number and excitation energy , facilitating the extrapolation to unstable nuclei.@xcite similarly to the standard treatment for the @xmath493 states ( the regular transmission coefficients as shown above ) , a nld can be invoked above the last experimentally known @xmath492 level . since the @xmath492 states in a nucleus ( @xmath212,@xmath211 ) are part of multiplet , they can be approximated by the levels ( and nld ) of the nucleus ( @xmath212@xmath2151,@xmath211 + 1 ) , only shifted by a certain energy @xmath501 . this displacement energy can be calculated and it is dominated by the coulomb displacement energy @xmath502.@xcite thus , the uncertainties involved are the same as in the prediction of the nld and discrete excited states . the inclusion of the explicit treatment of isospin has two major effects on statistical cross section calculations in astrophysics which will be discussed below:@xcite the suppression of @xmath82-widths for reactions involving self - conjugate nuclei and the suppression of the neutron emission in proton - induced reactions . non - statistical effects , i.e. the appearance of isobaric analog resonances , can be included in the treatment of the mixing parameter @xmath496 but will not be further discussed here.@xcite the isospin selection rule for e1 transitions is @xmath503 with transitions @xmath504 being forbidden.@xcite an approximate suppression rule for @xmath488 transitions in self - conjugate nuclei can also be derived for m1 transitions.@xcite in the case of ( @xmath0,@xmath82 ) reactions on targets with @xmath505 , the cross sections will be heavily suppressed because @xmath506 states can not be populated due to isospin conservation . a suppression will also be found for capture reactions leading into self - conjugate nuclei , although somewhat less pronounced because @xmath506 states can be populated according to the isospin coupling coefficients . this cross section suppression can be implemented as a suppression of the photon transmission coefficient . some older reaction rate calculations treated this suppression of the @xmath82-widths completely phenomenologically by dividing by rather arbitrary factors of 5 and 2 , for ( @xmath0,@xmath82 ) reactions and nucleon capture reactions , respectively.@xcite this can be improved by explicitly accounting for population and decay of @xmath507 and @xmath508 states , and considering isospin mixing by the parameter @xmath496.@xcite an astrophysically important reaction with pronounced isospin suppression effect is the reaction @xmath509ca(@xmath0,@xmath82)@xmath510ti which is responsible for the production of the long - lived @xmath510ti in supernovae.@xcite decay @xmath82-emission of @xmath510ti is observed in supernova remnants and can be used to test supernova models.@xcite furthermore , assuming incomplete isospin mixing , the strength of the neutron channel will be suppressed in comparison to the proton channel in proton - induced reactions.@xcite this leads to a smaller cross section for ( p , n ) reactions and an increase in the cross section of ( p,@xmath82 ) reactions above the neutron threshold , as compared to calculations neglecting isospin ( i.e. implicitly assuming complete isospin mixing with @xmath494 ) . the isospin mixing parameter was varied in the theoretical investigation of a @xmath511v(p,@xmath82)@xmath512cr experiment.@xcite it was found that complete isospin mixing closely reproduced the measured cross sections when width fluctuation corrections were considered . width fluctuation corrections affect the ( p,@xmath82 ) cross sections above as well as below the neutron threshold , whereas incomplete isospin mixing only reduces the cross sections above the threshold . thus , the two corrections can be discriminated . mainly from this result it was concluded that contrary to width fluctuation corrections isospin can be neglected . however , a closer investigation of the @xmath492 levels in @xmath512cr ( using results from refs . ) shows that isospin mixing should be rather complete already at the neutron threshold ( since the first @xmath492 state is almost 1 mev below the threshold).@xcite this is also true for lighter targets . for reactions on heavier nuclei ( @xmath513 ) , however , the neutron and proton threshold , respectively , will still be in a region of incomplete isospin mixing and therefore isospin effects should be detectable there . on the other hand , this effect is not as important in the calculation of astrophysical reaction rates as the suppression of the @xmath82-width because of the averaging over an energy range in the calculation of the rate , washing out the cusp effect . [ [ nuclear - matter - density - distribution ] ] nuclear matter density distribution : + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + the density distribution @xmath454 of neutrons and protons inside a nucleus is needed to calculate the optical potentials for some choices of potentials ( see above ) . there are charge density distributions available from electron scattering experiments@xcite on stable nuclei but the majority of density distributions for application across the nuclear chart comes from theoretical predictions . rates are mostly sensitive to optical potentials , and thus densities , at large radii because of the low astrophysical energies . density distributions are required in the determination of optical potentials and the potentials used in direct capture calculations . some optical potentials rely on nuclear density distributions from a certain model which were used when fitting the remaining open parameters to experimental data.@xcite although modern microscopic models have considerably improved in predicting nuclear masses and radii , the differences between different models still are large . because of the sensitivity of the transmission coefficients to the optical potentials , one would expect that small differences in the nuclear density distributions can give rise to large differences in the rates , especially at low plasma temperatures . as an example , figs . [ fig : pgdens1 ] , [ fig : pgdens2 ] show comparisons of ( p,@xmath82 ) rates obtained when using densities from hartree - fock - bogolyubov ( hfb ) models and from the droplet model in the jlm potentials while leaving all other input unchanged.@xcite densities from hfb-2 were included in the recommended input parameter library for hauser - feshbach calculations ripl-2 ( ref . ) and the hfb-14 densities are in its successor ripl-3 ( ref . ) . the left panel shows the comparison of rates obtained within the `` family '' of hfb models . the right panel shows a comparison of rates obtained with droplet model and hfb-14 densities . while the ratios stay well within a factor of two even with the seasoned droplet densities , it is interesting to note that they reach unity when approaching the proton dripline . figures [ fig : ngdens1 ] , [ fig : ngdens2 ] are the same as above but for ( n,@xmath82 ) rates at @xmath175 gk with stable and neutron - rich target nuclei . here , the largest ratios appear for very neutron - rich nuclei but again recede to unity when approaching the neutron dripline . overall , the maximal ratios are higher than for proton capture but stay within a factor of ten . this is partly due to the fact that nuclei further from stability are involved where the disagreement between different models becomes larger . the main reason , however , is the lower relevant temperature because the differences in the rates due to the density distributions decrease with higher @xmath41 , as the average transmission coefficients become less sensitive to the nuclear surface region . a further example of the impact of nuclear density distributions on calculated reaction @xmath65-factors is shown in figs . [ fig : dens1 ] , [ fig : dens2 ] in the section on optical potentials above . [ [ deformation ] ] deformation : + + + + + + + + + + + + deformations are implicitly present when taking excited states from experiments or theory , or nuclear density distributions from theoretical models . there the problem lies in the fact that most results of microscopic models available for large - scale calculations assume sphericity . the hfm using transmission coefficients as described in ( [ eq : tottrans ] and ( [ eq : parttrans ] ) can not describe reactions on deformed nuclei because it assumes @xmath377 to be a good quantum number . the coupled - channel model has to be invoked for a rigorous treatment.@xcite it is computationally very expensive and thus not suited for large - scale calculations . furthermore , the effective cross sections ( [ eq : effcs ] ) require to include even more transitions than in standard , laboratory nuclear reactions . fortunately , it has been shown that experimental data can be well described in a spherical hfm using an effective optical potential which is obtained by averaging over nuclear orientation.@xcite this leads to a spherical potential with larger diffuseness . to compute this modified potential , the explicit inclusion of the nuclear deformation is required and is usually taken from microscopic or macroscopic - microscopic models . a deformation parameter may also enter the description of the nld employed in the hfm calculation . but certain nld descriptions include the deformation implicitly , like the one of ref . where the deformation is contained in the microscopic correction . the splitting of the gdr in deformed nuclei can also be accounted for phenomenologically by a dependence on a deformation parameter . this has the largest impact on the rates among the possibilities of the inclusion of deformation discussed here . finally , fission transmission coefficients ( see below ) are also very sensitive to deformation parameters . usually , the deformation is already included in an effective fission barrier , leading to double - humped fission barriers.@xcite [ [ width - fluctuation - corrections ] ] width fluctuation corrections : + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + the width fluctuation coefficients ( wfc ) defined in ( [ eq : wfc ] ) impact the reaction cross section only closely around channel openings , with a few kev to tens of kev . contrary to isospin competition cusps ( see above ) , the modify the cross section above and below the channel threshold.@xcite generally , they enhance the elastic channel and reduce the other open channels accordingly to obey flux conservation . above the neutron threshold the behavior may be not so obvious because of the dominance of the neutron channel with respect to other channels . the transfer of strength from the dominant neutron channel to the elastic channel results in a marked reduction in competition with other exit channels and actually increases the strengths in those other channels . there are different ways to implement the calculation of the wfc , depending on the assumptions taken.@xcite an explicit form for the @xmath372 , requiring the knowledge of the level width distribution in the compound nucleus , is obtained by assuming that all transmission coefficients are small.@xcite this approach has been widely used , even when the transmission coefficients were not small.@xcite an alternative approach is to recognize that the main effect of the correlations is in the elastic channel , with smaller effects on the other channels . this leads to a modification of the transmission coefficients and an additional factor applied in the elastic channel only . this is the more general hrtw method which has been used in most astrophysical applications.@xcite another implementation of the hrtw method gives a general formula without restricting assumptions for the additional factor but is complicated to apply.@xcite overall , in astrophysical rates the impact of the wfc is small because of the low energies encountered . even when the relevant energy window covers the neutron threshold , the difference when choosing one or the other description is barely noticeable because of the energy averaging taking place in the rate integration . the wfc are important , however , when trying to compare theoretical and experimental cross sections close to channel openings . [ [ fission ] ] fission : + + + + + + + + neutron - induced ( and @xmath218-delayed ) fission of extremely neutron - rich nuclides is important in determining the endpoint of the r - process and the amount of intermediate nuclei produced by the fission process.@xcite it can be included in the hfm by using an additional exit channel in ( [ eq : tottrans ] ) describing the fission process . the fission transmission coefficient is calculated from the penetration probability through a fission barrier . since most of the astrophysical fission occurs at energies below or close above the barrier , the resulting rates are very sensitive to the height and width of the fission barrier . barrier predictions from various models show large differences and thus there are considerable uncertainties ( reaching several orders of magnitude ) in the resulting fission rates . these uncertainties have been explored in ref . and recommendations for comparative rate sets were given . the prediction of fission barriers remains a challenge to current microscopic models . another issue concerns the fission fragment distribution . earlier studies of the r - process have used simple fission barriers and assumed symmetric fission.@xcite recent years have seen the advent of improved predictions using more sophisticated statistical models and their results are being included in rate calculations.@xcite as for the barriers , considerable uncertainties may exist for extremely neutron - rich nuclei , though . on one hand , fission determines how far the r - process can synthesize elements and whether it could reach the region of long - lived superheavy nuclei . on the other hand , the fission rate together with the fragment distribution impacts the abundances of intermediate and heavy r - process nuclei . intermediate r - process nuclei ( including the rare earth peak ) have a contribution from fission fragments.@xcite if the fission timescale is short compared to the r - process timescale , fission cycling can occur whereby the fission fragments capture neutrons and follow an r - process evolution until they fission again.@xcite this exponentially enhances the final r - abundances . the number of possible fission cycles depends on the fission rates and thus is very sensitive to the fission barriers . the final abundance level is less sensitive to the fragment distribution but the distribution will determine the details of which nuclei receive contributions from fission . in principle , one would think that any code implementing the hfm should give the same result . this is obviously true when using the same implementation of the hfm _ and _ the same descriptions of the required input . however , both may differ among different codes . when quoting results it is therefore essential to not only always specify the exact version of the used program but also what selections regarding the properties described above have been made . there is a variety of codes which have been and are used in data evaluation . these usually focus on higher energies than astrophysically relevant . they also may include further reaction mechanisms which may not be relevant for astrophysical application ( see sec . [ sec : statmodmod ] ) . finally , they use experimental knowledge ( either directly or by renormalizing theoretical results ) or local parameterizations of nuclear properties . this way , high accuracy may be achieved locally for one or a few nuclei but no global prediction , essential for astrophysics , can be made . astrophysical codes are especially written for global predictions of reaction rates and thus focus on low - energy cross sections . this includes using different internal numerics ( see also figs . [ fig : sm144a]@xmath215[fig : sm144rcoul2 ] for issues regarding the calculation of subcoulomb s - factors ) but also different choices of the used input . global treatments by global parameterizations or ( semi-)microscopic models are preferred . experimental information may be used where available to test those global approaches and , of course , to locally improve astrophysical reaction rates . however , a fair comparison of the global predictions of different astrophysical and other codes is only possible when a similar philosophy is used in determining their input ( and by using the same experimental input where unavoidable ) . most importantly , however , astrophysical codes directly account for the additional transitions required for the calculation of the true astrophysical reaction rate with thermally excited nuclei in the stellar plasma . they properly include the effective cross section defined in ( [ eq : effcs ] ) and thus implicitly use the correct weighting factors of transitions from excited states as derived in ( [ eq : effweights ] ) . early large - scale reaction rate and cross section predictions made use of a code developed at caltech.@xcite the original code@xcite was developed further@xcite but no further tables of reaction rates were published , ready for use in astrophysical reaction networks ( see also ref . for a comparison of these early codes ) . the work of refs . was not only important for astrophysical modeling but also in nuclear physics . prior to these calculations , all hitherto experimentally studied reactions had featureless excitation functions and tests of the statistical theory were hindered by this . the large - scale calculations allowed to identify the cases suited to study competition between different channels.@xcite another influential development is the one of the smoker code.@xcite it went beyond the previous codes by including a more sophisticated calculation of the transmission coefficients in all channels by explicitly solving the schrdinger equation with optical potentials and including several new global parameterizations of the further inputs . a set of neutron - capture rates for r - process nucleosynthesis was published in ref . and complete sets of neutron - induced as well as charged - particle induced reactions calculated with this code became included in the first version of reaclib , a library of theoretical and experimental cross sections which can directly be used in astrophysical reaction networks.@xcite the reaclib format , using fits of reaction rates to a function of 7 temperature - dependent terms ( see , e.g. , ref . ) , has become a standard widely used in the astrophysics community.@xcite a series of codes some closely related , other only loosely based on smoker has appeared since then . the code non - smoker extended the functionality by not only updating the input data but also including an improved , global nld description and isospin effects.@xcite although it already included many possibilities for descriptions of nuclear properties to use , often the term `` non - smoker calculations '' is used synonymously for the extended tables in refs . calculated with a chosen input set . in a parallel development , the code most also updated the input physics and provided a different selection of treatments of nuclear properties.@xcite tables of reaction rates ( but no fits in reaclib format ) were provided online for several versions of the code.@xcite the newly written , but still closely related , non - smoker@xmath427 code included several changes.@xcite apart from several updates of included nuclear data , also the internal numerical calculations were modified , the isospin suppression was improved , additional choices for microscopic and macroscopic predictions of nuclear properties were offered . the innovative web interface allows access to the code from anywhere through a simple web browser . additional switches can be set and different nuclear properties provided in an optional input file which is uploaded to the server running the code . the resulting cross sections and reaction rates are immediately displayed . certain nuclear properties , e.g. optical potentials , can not only be uploaded as data but also as formulae because the code includes a simple equation parser . non - smoker@xmath427 has been used for the astrophysical analysis of a large number of experimental results , especially for p - process nucleosynthesis ( see , e.g. , refs . and references therein ) . its development has been frozen at version v5.8.1w but it is still available and used for calculations . most recently , the code smaragd ( statistical model for astrophysical reactions and global direct reactions ) continues and extends the development initiated with non - smoker@xmath427 of a user - friendly , easily extensible code tailored for astrophysical reaction rates.@xcite the code is written completely in fortran90/95 ( with exception of the routines handling the web interface and the function parser , which are written in c ) and has a modular structure , making changes easy . the latest nuclear data can also swiftly be included through files or web downloads . internally , the numerics and solvers for the schrdinger equation have been improved to be more accurate at low , subcoulomb energies and to consistently also calculate direct processes as required for astrophysics ( see sec . [ sec : directintro ] ) . recent developments in parameterized or microscopic predictions of global nuclear properties ( masses , nlds , density distributions , optical potentials , photon strength functions ) have been included . it also uses the modified hfm discussed in sec . [ sec : statmodmod ] . reaction rates are provided in tabular form as well as in the reaclib format through an automated fit routine . future versions will include multiple particle emission , follow @xmath82-cascades explicitly , and allow the calculation of fission rates . code versions below v1.0s are not public , those below v2.0s do not include direct reactions , yet . direct reactions are included in a number of ways , as discussed in sec . [ sec : direct ] below . at later stages of the code development it is planned that users may upload modules providing nuclear data , numerical methods to compute required properties , or even altering the functionality of the program . however , before these improvements in versatility are made accessible , a new large - scale calculation will provide a new set of published reaction rates between the driplines , intended to improve on and supersede the non - smoker rates@xcite which currently are used by the majority of astrophysical modelers worldwide.@xcite in its most general definition , the term `` direct reaction '' includes all processes directly connecting the initial and final states of a nuclear reaction without formation of an intermediate compound system . this includes elastic scattering as described in the optical model , and inelastic scattering which predominantly excites collective states.@xcite the latter includes coulomb excitation which has been found to be important in heavy ion collisions due to the high coulomb barriers involved.@xcite in astrophysically relevant reactions , especially with @xmath0-particles , energies may also be close to or below the coulomb barrier and coulomb excitation may also become important , depending on the structure of the target nucleus.@xcite here , we focus on direct reactions when some ( if it is a stripping reaction ) or all ( if it is a capture or charge - exchange reaction ) nucleons of the projectile are incorporated in the target nucleus . in a pick - up reaction , one or more nucleons from the target nucleus are added to the projectile to form the ejectile , again in a direct manner . pick - up and stripping reactions are subsumed under the term `` transfer reactions '' . in contrast to the hfm model , direct reactions excite only few degrees of freedom because most of the nucleons included in the system of target nucleus plus projectile remain spectators . a nucleon of the projectile reaches its final state without sharing any energy with any of the other nucleons present and the excess energy is emitted as a discrete photon carrying the energy difference between initial and final state . direct reactions can be identified experimentally because of their angular dependence of the differential cross sections , being peaked in forward direction . direct processes are also faster by at least 5 orders of magnitude than compound reactions , with reaction timescales of the order of @xmath514 s. this is comparable to the time the projectile requires to cover a distance of the size of a nucleus . therefore , direct reactions are important at high projectile energies when compound formation is disfavored . although the notion of direct processes was inspired originally by angular distributions of low - energy reactions , it was assumed for a long time that higher energies are the domain where they are dominating.@xcite in resonant reactions at lower energy , it is sometimes necessary to include a non - resonant background ( which may show interference with resonances ) but experimentally it is often difficult to distinguish between a direct component and contributions from tails of resonances . however , in systems with low nld , and thus widely spaced resonances , direct reactions become important even at astrophysically low interaction energies because compound formation is suppressed.@xcite this even applies to intermediate and heavy nuclei far off stability , e.g. , for neutron capture in nuclei with low neutron separation energy.@xcite the direct capture cross section can become considerably larger than the compound cross section . figure [ fig : dchf ] compares ( n,@xmath82 ) cross sections at 30 kev from direct capture and from hfm for a number of isotopes . with decreasing neutron separation energy , the direct component plays an increasingly important role because the compound nucleus is formed at lower excitation energy and thus also at lower nld . elements with inherently low nld , such as sn , show a larger direct contribution for all isotopes . also for nuclei at shell closures the nld is low and the importance of direct reactions enhanced relative to the hfm.@xcite similar considerations may apply to proton - induced reactions on proton - rich nuclei . direct processes are not only important to be included in reaction rate predictions . all possible processes ( elastic and other direct ones , compound - elastic and compound nucleus reactions ) have to be taken into account in the analysis of scattering and reaction data when extracting optical potentials . the elastic scattering cross section @xmath515 ( which is a direct process ) , the one for reactions @xmath516 , and the total @xmath517 are related by @xmath518 where we distinguished between elastic scattering at the optical potential @xmath519 and compound - elastic scattering @xmath520 . the cross section @xmath521 includes all inelastic processes , i.e. reactions . often , one reaction mechanism is dominating by far and then @xmath521 can be identified with the cross section for that mechanism , e.g. the compound cross section @xmath520 for resonant processes ( including the hfm ) as discussed in secs . [ sec : reso ] and [ sec : statmod ] . but this is not always the case as depending on the nucleus and the projectile energy there may be additional mechanisms contributing to @xmath521 in some cases . therefore , using an optical potential derived from scattering implies that the absorption term is due to some reaction(s ) but does not define the reaction mechanism(s ) . using such a potential in a pure hfm implicitly assumes that the missing flux from the elastic channel is due to the compound mechanism only . this may not be appropriate when direct processes are non - negligible ( this comprises direct reactions as discussed below but also coulomb excitation at low energy ) and will require a modification of the optical potential depending on which mechanism is to be studied . this is also implicitly contained in the idea of the modified hfm briefly discussed in sec . [ sec : statmodmod ] . earlier calculations of direct neutron capture have made use of a simple hard sphere capture model ( see sec . [ sec : dc ] ) in a combination of direct and hfm capture , not just to simplify calculations but also because it also allows the assumption that the contribution to direct capture potential absorption due to the tail of distant resonances is already included in the statistical model averaging.@xcite astrophysical rates can be calculated from cross sections by applying ( [ eq : distrirate ] ) , ( [ eq : rate ] ) , and ( [ eq : effrate ] ) , regardless of the reaction mechanism . since each discrete transition appearing in a direct process obeys the reciprocity relation ( [ eq : reci_single ] ) , a similar effective cross section ( [ eq : effcs ] ) can be derived as for the compound case , resulting in the same weighting factors ( [ eq : effweights ] ) of transitions from the excited states . also the same reciprocity relations ( [ eq : revrate ] ) and ( [ eq : revphoto ] ) apply , provided that thermal population of the states in all participating nuclei is valid . in the following only a few methods are outlined which have been used to calculate direct reactions for astrophysics . only the basic equations for the reaction from one initial state to one final state are given but the actual rate equations can straighforwardly be obtained by using the methods described in secs . [ sec : stellar ] and [ sec : stellarexp ] . it should be noted that here not only light targets are implied but also intermediate and heavy ones ( see fig . [ fig : dchf ] ) , for which microscopic models are not feasible ( see sec . [ sec : mechintro ] ) . furthermore , low projectile energies are implied as these are required for calculating astrophysical reaction rates . nevertheless , direct reactions at higher energies can be used to extract certain properties , such as spin assignments and spectroscopic factors ( see , e.g. , ref . ) , of stable and unstable nuclei which are required for the calculation of the cross sections and the rates . direct transfer reactions can be treated by solving the time - independent schrdinger equation with optical potentials in the entrance and exit channels . a simple implementation of this is the distorted wave born approximation ( dwba ) . the differential cross section for the one - nucleon or cluster transfer @xmath522 with @xmath523 , @xmath524 for @xmath525 and @xmath526 or 3 is given in zero - range dwba by@xcite @xmath527 with the zero - range normalization constant @xmath528 . the reduced cross section without spin - orbit coupling is given by @xmath529 with the reduced transition amplitude @xmath530^ * \chi_{aa}^{(+)}\left(k_{aa},r \right)\ , dr \quad.\ ] ] as before , the quantities @xmath62 , @xmath531 and @xmath532 , @xmath533 are the reduced masses and wave numbers in the entrance and exit channel , respectively . the orbital angular momentum quantum number @xmath377 , the spin quantum number @xmath100 , and the total angular momentum quantum number @xmath14 refer to the nucleon or cluster @xmath534 bound in the residual nucleus @xmath194 . the spectroscopic factor and the isospin clebsch - gordan coefficient for the partition @xmath535 are given by @xmath130 and @xmath536 , respectively . the optical wavefunctions in the entrance and exit channels are given by @xmath537 and the time - reversed solution @xmath538 . the bound state wave function is denoted by @xmath539 and the @xmath540 are the usual spherical harmonics . expressions similar to the above are obtained when the finite range of the interaction potential is taken into account.@xcite important for the successful application is to keep the number of open parameters as small as possible . for this reason , folding potentials ( see the paragraphs on optical potentials in sec . [ sec : relevance ] ) were used in many astrophysical applications of the model , with @xmath21 either determined from scattering data or from global dependences.@xcite this leaves the spectroscopic factor @xmath541 which is usually determined by simply comparing the calculated magnitude of the differential cross section to measurements . this is mainly done with ( d , p ) or ( d , n ) reactions at energies above the astrophysically relevant ones.@xcite the partial width @xmath542 appearing in ( [ eq : breit ] ) can be related to spectroscopic factors for a particle @xmath393 in a state @xmath102 by@xcite @xmath543 the single particle width @xmath544 can be derived from scattering phase shifts and this offers a different experimental access to spectroscopic factors ( see sec . [ sec : avdc ] for further methods to determine spectroscopic factors).@xcite further required input includes , of course , masses or separation energies and nuclear spectroscopic information which have a similar importance as for the hfm , discussed in sec . [ sec : relevance ] . in the absence of experimental data , the spectroscopic factor can be calculated microscopically from the overlap between initial and final state wave functions , e.g. , in the shell model.@xcite however , there is some ambiguity because this overlap is not well defined in different microscopic approaches.@xcite it has been shown that the spectroscopic factors for depositing or picking up a single nucleon are related to the occupation factors of the participating quasi - particle states ( see also sec . [ sec : avdc]).@xcite as in the case of the hfm , many codes have been used for dwba calculations over the years , especially for the analysis of data at intermediate and high energies . the code tetra has been written especially for application at astrophysically relevant low energies.@xcite it has been applied successfully to astrophysically relevant reactions with light and intermediate target nuclei ( see , e.g. , refs . , and references therein ) . is it necessary to go beyond the dwba ? there are three fundamental assumptions contained in the dwba treatment:@xcite 1 . the reaction proceeds directly from initial to final state and all particles except the transferred one(s ) remain unaffected spectators . the wave function for the relative motion between the reactands is assumed to be correctly described by the optical potential . the reaction is assumed to be sufficiently weak to be treated in lowest order . to relax the first two assumptions , the coupled - channel born approximation ( ccba ) was introduced.@xcite under rare circumstances the transfer amplitudes may be large and the third assumption has to be relaxed . this leads to a full coupled - channels treatment for the reaction.@xcite one has to be aware of the fact that the relevant energies remain low for astrophysical reaction rates , also because transitions from excited states contribute considerably . this is contrary to what one is used to in the investigation of reactions proceeding at several tens of mev . due to the low energies involved , the reaction channel is weak ( compared to , e.g. , elastic scattering ) and the third assumption is valid . the usual concern with the second assumption is that the optical potential also has to describe well the wave function even deep in the nuclear interior . the deep region , however , is crucial for reactions at higher energy whereas at the low astrophysical energies most contributions to the overlap integrals stem from regions close to the surface of the nucleus or even from outside of the nuclear radius . as long as these regions are described well by the optical potentials , the dwba should work . the first assumption implies that either no indirect processes exist or that they can be treated separately ( incoherently ) as was suggested by the above , separate discussion of compound reactions and other mechanisms . in the hfm it is assumed that interference terms cancel and thus also interference with direct reactions should cancel on average . interference with isolated resonances can be treated explicitly by adding an interference term , e.g. , between the s - factor of the direct reaction @xmath545 and the one of a breit - wigner resonance @xmath546 ( see sec . [ sec : reso ] ) @xmath547 where @xmath548 is the energy - dependent , relative phase shift.@xcite a potential model can also be used to calculate direct capture ( dc ) . although microscopic models are an alternative for light systems ( see sec . [ sec : mechintro ] ) , a dc potential model has the advantage that it can be applied also to heavier nuclei . the dc cross section for a particular transition is determined by the overlap of the scattering wave function in the entrance channel , the bound - state wave function in the exit channel , and the electromagnetic multipole transition operator . the dc cross section is then given by@xcite @xmath549 the polarisation @xmath37 of the electromagnetic radiation can be @xmath550 . the wave number in the entrance channel and for the emitted radiation is given by @xmath532 and @xmath551 , respectively . the multipole expansion of the transition matrices @xmath552 including electric dipole ( e1 ) and quadrupole ( e2 ) transitions as well as magnetic dipole ( m1 ) transitions is given by @xmath553 the rotation matrices depend on the angle between @xmath554 and @xmath555 which is denoted by @xmath556 , where @xmath557 . defining @xmath558 we can write for the transition matrices for the electric dipole ( e@xmath559 = e1 ) or quadrupole ( e@xmath559 = e2 ) transition @xmath560 in the above expressions the quantum numbers for the channel spin in the entrance channel and for the transferred angular momentum are denoted by @xmath561 and @xmath562 , respectively . the quantities @xmath563 , @xmath564 and @xmath565 ( @xmath566 , @xmath567 and @xmath568 ) are the spins ( magnetic quantum numbers ) of the target nucleus @xmath31 , residual nucleus @xmath194 and projectile @xmath32 , respectively . for magnetic dipole transitions ( m@xmath559 = m1 ) we obtain @xmath569 \nonumber\\ & - ( l_a\ , 0\ , s_a\ , m_a \mid j_a\ , m_a ) ( j_a\ , m_a\ , i_a\ , m_b\ ! -\ ! m_a \mid i_b\ , m_b ) \nonumber\\ & \times ( i_a\ , m_b\ ! -\ ! m_a\ , 1\ , \delta\ , \mid i_a\ , m_a ) \nonumber \\ & \times \mu_a \ , \delta_{j_a j_b } \sqrt{\ , ( i_a+1)/i_a } \biggr\ } \biggl\ { \frac{\hbar c}{2 m_{\rm p } c^2 } \biggr\}\ , \delta_{l_a l_b } \ : \hat{l}_a\ : i_{l_b j_b i_b ; l_a j_a}^{\rm m1 } \quad,\label{eq : kim4}\end{aligned}\ ] ] where @xmath570 is the racah coefficient , the @xmath571 are the magnetic moments and @xmath572 is the mass of the proton . the overlap integrals in ( [ eq : kim3 ] ) and ( [ eq : kim4 ] ) are given as @xmath573 for the electric dipole ( e@xmath559 = e1 ) or quadrupole ( e@xmath559 = e2 ) transition , and by @xmath574 for the magnetic dipole transition ( m@xmath559 = m1 ) . the radial part of the bound state wave function in the exit channel and the scattering wave function in the entrance channel is given by @xmath575 and @xmath576 , respectively . the radial parts of the electromagnetic multipole operators are@xcite @xmath577 \quad , \nonumber\\ { \cal o}^{\rm e1}(r)&= \frac{3}{\hat{\rho}^3}\left [ ( \hat{\rho}^2 - 2 ) \sin \hat{\rho } + 2 \hat{\rho } \cos \hat{\rho } \right ] r \quad , \nonumber\\ { \cal o}^{\rm e2}(r)&= \frac{15}{\hat{\rho}^5}\left [ ( 5 \hat{\rho}^2 - 12 ) \sin \hat{\rho } + ( 12 - \hat{\rho}^2 ) \hat{\rho } \cos \hat{\rho } \right ] r^2 \quad .\label{11}\end{aligned}\ ] ] in the long wave - length approximation applicable as long as @xmath578 these quantities reduce to @xmath579 usually , only the dominant e1 transitions have to be taken into account . possible exceptions are captures far from stability with very low reaction @xmath94-values because for these cases no final states may be energetically accessible through e1 transitions . however , because the astrophysical reaction rate involves summing over transitions originating from excited states , a larger spin range may be available and e1 ( from excited target states ) may again dominate . for e1 transitions , the above expressions reduce to@xcite @xmath580 the coefficients @xmath581 are calculated in ls coupling to @xmath582 in the above expressions , the energy of the emitted photon is @xmath469 . the orbital and total angular momentum quantum numbers of the nuclei in the entrance and exit channels are @xmath583 , @xmath584 , @xmath585 and @xmath586 , respectively . the spin quantum number , the orbital and total angular momentum quantum numbers are characterized by @xmath65 , @xmath47 and @xmath587 , respectively , with indices @xmath32 , @xmath31 and @xmath194 corresponding to the projectile , target and residual nucleus , respectively . the notation @xmath588 stands for the @xmath589 symbol . the radial wave functions in the entrance and exit channels are given by @xmath590 and @xmath591 , respectively . the spectroscopic factor and the isospin clebsch - gordan coefficient for the partition @xmath592 are given by @xmath130 and @xmath593 , respectively . the dc potential model has been successfully applied to many reactions with light target nuclei at astrophysical energies , e.g. , for @xmath291be(p,@xmath82)@xmath594b and @xmath291li(n,@xmath82)@xmath594li.@xcite for further calculations see , e.g. , refs . and references therein . many calculations were performed with the dc potential model code tedca , which is tailored to treat low - energy reactions of astrophysical interest.@xcite again , folding potentials ( see sec . [ sec : relevance ] ) were the key to reduce the number of parameters . spectroscopic factors were taken from experiment or shell model calculations . it was also used to extend calculations to intermediate and heavy target nuclei.@xcite other astrophysical calculations have not made use of the full dc equations but used simplifying assumptions . resonant and dc rates on the proton - rich side based on hard sphere scattering wavefunctions in the entrance channel with resonance properties , final states , and spectroscopic factors taken from shell model calculations were provided for target nuclei with mass number @xmath595.@xcite in these calculations , however , the resonant part exceeds the direct part of the cross section by several orders of magnitude . astrophysical neutron capture on stable and neutron - rich nuclei was calculated in the hard - sphere model for e1 capture by refs . . in this model , the e1 neutron capture cross section can be written as@xcite @xmath596 where @xmath597 is the hard sphere radius and the multiplicity @xmath389 is the number of incident channel spins which can lead to the same final state with spin @xmath598 . it is @xmath599 for @xmath600 , or for @xmath601 and @xmath602 . the value @xmath603 applies for @xmath601 and @xmath604 . the quantity @xmath605 is the orbital angular momentum of the final bound state @xmath128 and @xmath606 is the spectroscopic factor of this state . the dimensionless parameter @xmath607 is given by @xmath608 the advantage of this approach is that no explicit wave functions for scattering and bound states are required . on the other hand , the correct overlap of the wave functions may yield more accurate cross sections , especially for low projectile energies when considerable contributions to ( [ eq : overlape ] ) are coming from far outside the nuclear radius . ref . showed the importance of direct neutron capture in the r - process based on the hard - sphere model . for the validity of the low - order potential model approach for dc while neglecting higher - order processes , similar arguments can be made as were presented for the dwba towards the end of sec . [ sec : dwba ] . for astrophysics , a large number of rates for highly unstable nuclei have to predicted . similar to the uncertainties of the hfm discussed in sec . [ sec : relevance ] the dwba and dc predictions are sensitive to certain inputs , such as masses ( @xmath94-values ) , excited and bound state properties , spectroscopic factors , optical and bound state potentials . when folding potentials are used , a ( weak ) dependence on nuclear matter density distributions appears additionally . the discussion of these quantities in sec . [ sec : relevance ] also applies here . however , some of the quantities appearing in the hfm are sums over individual transitions and averaged quantities whereas individual transitions are determining the direct reaction cross sections . thus , direct reactions are more sensitive to nuclear properties impacting those individual transitions , including spins , parity , energy of bound and excited states , @xmath94-values , and spectroscopic factors . due to the angular momentum barrier and the low projectile energies , low partial waves contribute most to the e1 cross section , i.e. s - waves when initial and final states have differing parity , p - waves when they have the same parity . assuming equal spins and parities , low - lying states are contributing more than higher ones . the strong sensitivity to individual transitions , however , may be reduced in the astrophysical rate , when using the weighted sums ( [ eq : rate ] ) , ( [ eq : effrate ] ) , ( [ eq : effweights ] ) over transitions from excited target states to available final states . for neutron - rich sn and pb isotopes , predictions for 30 kev neutron capture employing input from different microscopic or semi - microscopic approaches were compared in detail.@xcite the dc calculation was tested in comparison to experimental data for @xmath487pb(n,@xmath82)@xmath609pb . in this case , a discrepancy between the data obtained in an activation measurement and the data from a high - resolution resonance counting experiment was resolved by showing that the difference is due to the dc contribution to the cross section which is only included in the cross section from the activation measurement . for very neutron - rich isotopes of sn and pb it was found that the resulting cross sections differ by orders of magnitude with the different inputs ( see figs . [ fig : dccompsn ] , [ fig : dccomppb ] ) . this is mainly due to the sensitivity of the cross section to the predicted location of the low - spin bound states with respect to the neutron separation energy.@xcite the dc calculation can be tested for @xmath610sn(n,@xmath82 ) because there is experimental information on the bound states in @xmath611sn.@xcite this nucleus is predicted to be close to or directly in the r - process path.@xcite an independent calculation confirmed the original work.@xcite fortunately for astrophysics , very similar cross sections ( within a factor of 3 ) are computed for this reaction with input from the different microscopic approaches . unfortunately for nuclear physics , the reaction is not a good case to select a preferred microscopic model for the same reason . spectroscopic data for neutron - richer isotopes would be necessary . furthermore , spectroscopic factors were set to unity in these calculations . this is a good assumption for the states in @xmath611sn ( and was recently confirmed by ref . ) but is not valid in mid - shell . however , the uncertainty introduced by the predicted bound state energies exceeds by far the one introduced by the spectroscopic factors when only particle states are considered and hole states are neglected.@xcite the latter have very small spectroscopic factors and are negligible because they involve a reordering process in the final nucleus ( see also ref . for another example of a reordering process in @xmath612s and its dependence on deformation ) . regular dc cross sections are obtained by summing over all allowed transitions to energetically accessible final states , @xmath613 where each summand contains the appropriate spectroscopic factor . following the derivations in sec . [ sec : stellar ] , the astrophysical rate will contain an effective cross section and a partition function . similarly to the hfm , it was suggested ( refs . ) that the sum over final states may be ( partially ) replaced by an integration over the nld in the final nucleus @xmath614 , @xmath615 + \nonumber \\ & + \int_{e^\mathrm{x}_{\nu^f}}^{s_\mathrm{n } } \sum_{j^f\pi^f } \rho_f(e^f , j^f,\pi^f ) \sigma_\mathrm{dc}^{\mu\rightarrow f}(e , e^f , j^f,\pi^f,\bar{\mathcal{s}}_{e^fj^f\pi^f } ) \ , de^f \quad . \label{eq : avdcmu}\end{aligned}\ ] ] similarly , the summation over initial states @xmath102 in ( [ eq : effcs ] ) can be ( partially or fully ) replaced by an integration over the nld in the target nucleus @xmath616 , @xmath617 + \nonumber \\ & + \int_{e^\mathrm{x}_{\mu^f}}^{e_\mathrm{proj}(t ) } \sum_{j^i\pi^i } \rho_i(e^i , j^i,\pi^i ) \bar{\sigma}_\mathrm{dc}^{i\rightarrow ( \nu , f)}(e , e^i , j^i,\pi^i,\bar{\mathcal{s}}_{e^ij^i\pi^ie^fj^f\pi^f } ) \ , de^i \quad . \label{eq : avdceff}\end{aligned}\ ] ] through the nld , the cross sections @xmath618 and @xmath619 not only include transitions to discrete final states but also `` average '' transitions to states described by the nld . therefore i call this _ averaged direct capture _ ( adc ) . as in the hfm , the adc cross section may only include the ground state transitions and nlds above the ground state when all excited state properties are unknown . the advantage of this is that the sensitivity to the location of discrete states relative to the projectile separation energy ( as seen in ref . ) is washed out and the change in cross section from one isotope to the next is smoother . of course , this may not properly describe nuclear structure details in the cross sections when only few states are available but it can give a more appropriate estimate of the magnitude of the rate for astrophysics ( which involves an averaging over a relevant energy range and includes more transitions than the laboratory rate ) than relying on just a single ( semi-)microscopic approach . obviously , the adc cross section will be sensitive to the nld . contrary to the hfm , where the nld mostly impacts the @xmath82-widths and has its strongest effect at several mev excitation energy , captures to low - lying states ( assuming spins and parities are favorable ) are dominating and thus the nld at low excitation energy will be relevant . this is why it is important to use a proper spin- _ and _ parity - dependent nld description . it was found recently , however , that thermal excitation of target nuclei reduces the sensitivity to the parity dependence in the nld for astrophysical rates in the hfm.@xcite we can expect a similar effect for direct capture , although the number of possible transitions is more limited.@xcite appropriate spectroscopic factors are a further important ingredient in adc calculations . in the integral of ( [ eq : avdcmu ] ) these are _ averaged _ spectroscopic factors @xmath620 , describing the average overlap between the initial state @xmath102 and the final bound states with given spin and parity at an excitation energy @xmath621 . the doubly averaged spectroscopic factors @xmath622 appearing in the integrand of ( [ eq : avdceff ] ) are even more complicated , as they involve the average overlap between all initial states with given spin and parity at excitation energy @xmath623 and all final states . currently , the only spectroscopic factors to be found in literature ( from experiment or theory ) are for transitions connecting the ground state @xmath112 of the target nucleus with the final states . spectroscopic factors for transitions from excited states have yet to be calculated . they are needed not just for the averaged dc model employing nlds but also for the regular dc model when applied to compute astrophysical reaction rates ( see also the discussion of the stellar enhancement and the effective weights in sec . [ sec : stellarexp ] ) . spectroscopic factors for one - nucleon capture ( or transfer ) on a target in the ground state can easily be computed from the occupation numbers @xmath624 as calculated , e.g. , from bcs or lipkin - nogami pairing.@xcite then the spectroscopic factor for putting a nucleon in state @xmath14 with spin @xmath625 is just @xmath626 for a target nucleus with an even number of nucleons of the same type as the projectile . the occupation probability is the one of the target nucleus . in a chain of linked reactions , the total processing efficiency is given by the slowest reaction(s ) . therefore , when considering sequences of capture reactions , e.g. , in the s- , r- , rp - processes , the rates on such target nuclei with even nucleon number will be the slowest ( and their reverse photodisintegrations the fastest ) and thus they will have the largest astrophysical impact . when the target nucleus has an odd number of nucleons of the projectile type , the spectroscopic factor is @xmath627 the occupation probability @xmath628 is always taken in the nucleus with even number of nucleons . in ( [ eq : occodd ] ) this is the final nucleus of the reaction . the expressions for the extraction of a nucleon from a given state follow from the fact that the time - reversed reactions have to show the same spectroscopic factor . the occupation probabilities can be calculated from microscopic theory , e.g. , using bcs or lipkin - nogami pairing on a single - particle basis.@xcite in the absence of calculated spectroscopic factors several different approaches have been used in the past . often , spectroscopic factors were set constant to 1.0 ( e.g. , refs . ) or to 0.1 ( e.g. , refs . ) . these values can already be seen as averaged spectroscopic factors @xmath620 for low - lying particle - states . useful for the application to the adc approach is the construction of an excitation energy - dependent , averaged spectroscopic factor to be employed along with the nld . as an example for this , in studying neutron capture on the astrophysically important nucleus @xmath510ti the dc component was estimated implementing a distributed spectroscopic strength.@xcite in this case , transitions to @xmath629 and @xmath630 states in @xmath612ti are dominating . a distribution @xmath631 of the @xmath629 and @xmath630 strengths was assumed , reaching from the ( experimentally known ) location of the lowest @xmath629 and @xmath630 state , respectively , to the neutron separation energy . with this `` smearing out '' of the states and of the strength and due to the @xmath632 dependence of the e1 transition probability , then the calculation is reduced to computing a transition to an effective bound state with full spectroscopic strength at an energy of @xmath633{\widetilde{e^3_\gamma}}\quad,\ ] ] where the average transition energy is given by @xmath634 such an approach accounts for the uncertainties in spectroscopic strength and location of excited states and can be viewed as a zeroth approximation to the adc . comparison between experimental and averaged ( eq . [ eq : holzerspectfact ] ) spectroscopic factors as function of excitation energy for @xmath635xe+n.@xcite ] another suggestion for the functional form of the energy - dependence of the average spectroscopic factors was made in refs . . the spectroscopic factors describe the overlap between the antisymmetrized wave functions of target+nucleon and the final state . the number of final state configurations increases with increasing excitation energy @xmath636 and the overlap of initial and final state wavefunctions decreases . thus , also the spectroscopic factor decreases . in a simple approximation , the energy dependence of the spectroscopic factor for single nucleon transfer can be parameterized by a fermi function with @xmath637 and the parameters @xmath638 . this is motivated by the excitation - energy dependence of the occupation probabilities . figure [ fig : avdcspect ] shows how well averaged spectroscopic factors with the functional dependence ( [ eq : holzerspectfact ] ) compare to experimental ones for @xmath635xe+n.@xcite there is also a connection between single - particle spectroscopic factors @xmath639 and the partial resonance widths @xmath640 appearing in the bwf ( see sec . [ sec : reso ] ) @xmath641 where @xmath130 is the isospin clebsch - gordan coefficient , @xmath362 the reduced mass of the system nucleus+particle , and @xmath642 is the dimensionless single - particle reduced width.@xcite the penetrability @xmath363 for the relative angular momentum @xmath377 can be expressed in terms of the regular and irregular coloumb wavefunctions @xmath643 and @xmath644 @xmath645 in literature , a value of @xmath646 is often assumed for an average single - particle reduced width . comparing ( [ eq : spwidth ] ) and ( [ eq : reduwidth ] ) , it can be seen that an average @xmath642 can be calculated from solutions of the radial schrdinger equation ( giving @xmath328 ) with an optical potential.@xcite since @xmath647 , there is a direct connection to the average transmission coefficients @xmath648 appearing in the hfm ( see sec . [ sec : statmod ] ) which can be used to consistently derive the average combined value @xmath649 . the averaged spectroscopic factors for transitions from excited target states in ( [ eq : avdceff ] ) can be estimated in this way . sums of the contributions to the total direct capture cross sections with the averaged dc model and the standard dc for @xmath635xe(n,@xmath82 ) as function of excitation energy of the final nucleus.@xcite ] the integrals appearing in ( [ eq : avdcmu ] ) and ( [ eq : avdceff ] ) always contain products @xmath650 of nld and averaged spectroscopic factors . with the relation between strength function @xmath651 and reduced width ( see , e.g. , ref . ) we can derive @xmath652 and thus ( partially ) eliminate the nld within the integrals as it is implicitly contained in the @xmath653 or @xmath654 . the strength function @xmath653 is only defined for resonance states above the projectile separation energy . for estimating transitions to bound states it was suggested to construct an `` internal strength function '' @xmath655 , extending the regular strength function below the separation energy.@xcite this is motivated by the relation ( [ eq : avspect ] ) and the fact that the spectroscopic factors describe the structure of bound states as well . the adc was investigated in ref . for the reactions @xmath635xe(n,@xmath82)@xmath656xe and @xmath463sm(n,@xmath82)@xmath657sm , using ( [ eq : lanelynn ] ) with an internal strength function . the adc cross section ( for the target in the ground state ) was compared to a calculation summing over transitions to bound states in the standard potential model ( as described in sec . [ sec : dc ] ) and with experimental data . the energy - dependence of @xmath655 was chosen very similar to the one found for the average spectroscopic factors @xmath658 , with independent parameters @xmath659 , @xmath660 , @xmath661 . the parameters were determined by requiring @xmath662 at the neutron separation energy . simultaneously , it was required that the adc cross section integrated up to the excitation energy of the last included state yields the same cross section as obtained with the standard potential model . although this is not suited for a prediction , it can be used to assess the validity of the assumptions . including only s - wave neutrons , the experimental cross sections were reproduced within 25% for @xmath635xe(n,@xmath82 ) and 2% for @xmath463sm(n,@xmath82 ) . figure [ fig : holzeravdc ] displays a comparison between the results from the averaged direct neutron capture and standard direct neutron capture for @xmath635xe . for predictions across the nuclear chart , the spectroscopic factors and/or internal strength functions can be obtained from optical model single - particle states and/or occupation numbers of quasi - particle states as shown in ( [ eq : occeven ] ) and ( [ eq : occodd ] ) . for simpler application and to increase computational speeds , these can be parameterized according to ( [ eq : holzerspectfact ] ) . spectroscopic factors for transitions from excited target states or the doubly averaged factors required in ( [ eq : avdceff ] ) remain an open problem . the code smaragd ( see sec . [ sec : codes ] ) will also include a global dc treatment using an adc model and energy - dependent spectroscopic factors , making use of a combination of the above approaches intended to also yield consistency with the hfm.@xcite such an adc approach aims at providing robust predictions despite of considerable differences between microscopic predictions.@xcite preliminary results for this adc are shown in fig . [ fig : avdc ] , along with results from the hfm ( sec . [ sec : statmod ] ) and a modified hfm ( sec . [ sec : statmodmod ] ) . the final rate ( or cross section ) is the sum of the modified hfm value and the adc one . interestingly , for the isotopes shown here ( except for @xmath663 ) this sum is approximated by the unmodified hfm result within a factor of 10 . this is in accordance with ref . ( see figure 3 therein ) , where the dc contribution also almost replaces the standard hfm values . this shows that it seems justified to use unmodified hfm rates as crude estimate of the total rates for exotic nuclei . there is no fast highway to improvements in predictions of reaction rates , not even a wide road . historically , the fields of standard nuclear physics and nuclear data for applications have taken another direction . therefore instead of following a beaten track rather a new , narrow path has to be driven step by step through a jungle of complications and details . this is only possible in a concerted effort of theory and experiment , the latter involving both large - scale rare - isotope production sites and smaller facilities . there is not one `` most important '' nucleus or `` most important '' reaction in nuclear astrophysics . which nuclei , nuclear properties , and reactions are at the center of attention depends on the astrophysical process studied . therefore , systematical studies are needed as well as information on specific nuclei and reactions . on the theory side large - scale studies of general trends and dependences are required as well as detailed predictions of individual nuclear properties . as shown above , the calculation of astrophysical rates , even when experimental information is present , involves a number of specialities not encountered in usual nuclear physics investigations . thus , nuclear astrophysics is heavily relying on advances in nuclear theory and experiment but also requires its own special developments in theory and experiment which justify its existence as a separate field . nuclear physicists working at the boundary to astrophysics have to be aware of these special requirements and it is one of the aims of this work to having outlined a number of them . a further noteable fact is that it is necessary to not only point out special effects or important possible improvements but to actually apply them across the nuclear chart and produce large - scale sets of reaction rates which can be readily used by astrophysicists . this implies that they are made accessible in a form suitable to be implemented in astrophysical models . it should not be forgotten that the other essential aspect of nuclear astrophysics is the astrophysical modeling using reaction networks . only in conjunction with this part of the field progress can be made . the models set the stage and define the ranges of conditions within which nuclear processes occur . nevertheless , it can be treacherous to rely too strongly on a certain model . reliable nuclear models and astrophysical reaction rates should cover a large range of possibilities and provide a sound base for pinpointing the sites of nucleosynthesis processes or even for discovering new types of nucleosynthesis in different astrophysical models . moreover , the modifications of the cross sections and rates in a stellar plasma are an interesting topic in itself and warrant an independent study even without connection to a specific astrophysical site . the prediction of astrophysical reaction rates takes nuclear physics in a new direction and tests nuclear theory at the limits . due to the finite number of nuclei , however , this is a finite task . this is also why parameterizations or phenomenological models may still have their justification , if designed in an appropriate manner . we do not have to extrapolate to infinity but within rather limited ranges of nucleon numbers . it is reassuring that the overall abundance distributions obtained by combining several postulated nucleosynthesis processes are already closely resembling what we find in nature , even when explosive events and highly unstable nuclei are involved . this tells us that the most important properties seem to be described acceptably well . there are remarkable exceptions , however , both in explaining abundance distributions ( e.g. , of the p - nuclei , the light s - process elements , isotopic anomalies in meteorites , the heaviest nuclei at the endpoint of the r - process , and many more ) and in assigning astrophysical sites ( e.g. , to the r - process , probably also to parts of the p - process ) . a detailed understanding and reliable reaction rates are also essential for using nuclear cosmochronometry to determine astronomical timescales . to go beyond previous estimates and reach a new level of detail , however , requires a large , dedicated effort of both experiment and theory . this work was supported by the swiss national science foundation , grant 200020 - 105328 , and by the european commission within the ensar / thexo project . k. wildermuth and y. c. tang , in _ a unified theory of the nucleus , clustering phenomena in nuclei _ , vol . 1 , k. wildermuth and p. kramer , a unified theory of the nucleus , clustering phenomena in nuclei , vol . 1 , ed . k. wildermuth and p. kramer ( vieweg , braunschweig , germany , 1977 ) . a. j. koning , s. hilaire and m. c. duijvestijn , _ proceedings of the international conference on nuclear data for science and technology , nice , 2227 april 2007 _ , eds . o. bersillon , f. gunsing , e. bauge , r. jacqmin , s. leray ( edp sciences , les ulis , 2008 ) pp . 211214 ; online available at http://www.talys.eu f .- k . thielemann , m. arnould and j. w. truran , in _ advances in nuclear astrophysics _ , e. vangioni - flam , j. audouze , m. cass , j .- chize , j. tran thanh van ( editions frontires , gif - sur - yvette , 1986 ) , p. 525 . t. rauscher , w. bhmer , k .- kratz , w. balogh and h. oberhummer , _ proc . nuclei and atomic masses _ , eds . m. de st . simon , o. sorlin ( editions frontires , gif - sur - yvette , 1995 ) , p. 683 . w. balogh , r. bieber , h. oberhummer , t. rauscher , k .- kratz , p. mohr and g. staudt , _ proc . workshop on heavy element nucleosynthesis _ , e. somorjai , zs . flp ( atomki , debrecen , 1994 ) , p. 67 ; arxiv : nucl - th/9404010 . h. herndl , r. hofinger and h. oberhummer , _ proc . origin of matter and evolution of galaxies _ , eds . s. kubono , t. kajino , k. i. nomoto , i. tanihata ( world scientific , singapore , 1998 ) , p. 233 ; arxiv : nucl - th/9803012 . j. dobaczewski , h. flocard and j. treiner , _ nucl . phys . _ * a422 * ( 1984 ) 103 . j. dobaczewski , i. hamamoto , w. nazarewicz and j.a . sheikh , _ phys . _ * 72 * ( 1994 ) 981 . m. m. sharma , m. a. nagarajan and p. ring , _ phys . b _ * 312 * ( 1993 ) 377 . m. m. sharma , g. a. lalazissis and p. ring , _ phys . b _ * 317 * ( 1993 ) 9 . p. mller and j. r. nix , _ nucl . phys . _ * a361 * ( 1981 ) 117 . p. mller , j. r. nix , w. d. myers and w. j. swiatecki , _ at . data nucl . data tables _ * 59 * ( 1995 ) 185 . p. mller , j. r. nix and k .- l . kratz , _ at . data nucl . data tables _ * 66 * ( 1997 ) 131 . | this review focuses on nuclear reactions in astrophysics and , more specifically , on reactions with light ions ( nucleons and @xmath0 particles ) proceeding via the strong interaction .
it is intended to present the basic definitions essential for studies in nuclear astrophysics , to point out the differences between nuclear reactions taking place in stars and in a terrestrial laboratory , and to illustrate some of the challenges to be faced in theoretical and experimental studies of those reactions .
the discussion revolves around the relevant quantities for astrophysics , which are the astrophysical reaction rates .
the sensitivity of the reaction rates to the uncertainties in the prediction of various nuclear properties is explored and some guidelines for experimentalists are also provided .
_ caminante , son tus huellas el camino , y nada ms ; + caminante , no hay camino , se hace camino al andar .
+ al andar se hace camino y al volver la vista atrs + se ve la senda que nunca se ha de volver de pisar .
+ caminante , no hay camino sino estelas en la mar . + antonio machado _ |
You are an expert at summarizing long articles. Proceed to summarize the following text:
understanding the string - black hole transition is an important problem in string theory . the radius of a schwarzschild blackhole becomes smaller with rising temperature and at a certain temperature the blackhole transits to a gas of strings @xcite . this is a difficult problem to address as it needs a proper understanding of non - perturbative effects in string theory . the finite temperature of a schwarzschild blackhole breaks supersymmetry and string loop corrections are uncontrolled . in @xcite it was demonstrated that the problem of the string - blackhole transition can be formulated in a space - time with @xmath7 boundary conditions . this enabled one to use the ads / cft correspondence @xcite to map the string - blackhole transition phenomenon to a gross - witten - wadia ( gww ) type phase transition @xcite in the boundary gauge theory defined on @xmath8 . the compactness and positive curvature of the space @xmath2 permits one to integrate out all other modes to get an effective multi - trace unitary matrix model for the zero mode of the polyakov line . based on the works @xcite , this type of effective unitary matrix model was analyzed in @xcite to show the existence of the gww type transition . the @xmath9 part of the gauge theory effective action was also calculated in a double scaled region near the transition temperature . the @xmath9 part is universally given in terms of @xmath10 , where @xmath11 satisfies the following differential equation @xmath12 and @xmath13 is the painleve ii function , and @xmath14 is a scaled variable proportional to @xmath15 . the derivation of the effective unitary matrix model from the gauge theory is a subtle one . in a weakly coupled gauge theory one may demonstrate this explicitly in perturbation theory at large @xmath0 @xcite . however the situation is less clear in the strong coupling regime . one difficulty comes from the gregory - laflamme transition for a small @xmath7 blackhole . this transition breaks the @xmath4 symmetry of @xmath16 and the question arises whether there is a new zero mode associated with this transition , and whether the unitary matrix model is a good description after this transition . in @xcite it has been shown , using a supergravity analysis within the ads / cft correspondence , that even at strong coupling , the unitary matrix model serves as an effective description . given the physical relevance of the gww transition , it is important to see if this phenomenon occurs when one is not dealing with the dynamics ( however complicated ) of a single unitary matrix or the quantum mechanics of a single unitary matrix . it is not at all obvious that this large-@xmath0 transition occurs in more complicated models of non - commuting matrices and gauge theories . in the past this question has been explored by douglas and kazakov@xcite in their study of two - dimensional yang - mills theory on @xmath17 . however this problem too , gets recast into a problem of a single unitary matrix because the partition function turns out to be the heat kernel on the unitary group . in order to answer these questions , there seems to be no analytic tools as is usually the case with complicated dynamical problems . hence we use numerical monte carlo methods to explore and exhibit the large-@xmath0 gww transition and also study the question of r - symmetry breaking at large @xmath0 . since the full @xmath1 sym theory on @xmath8 is too difficult , in the first run we study the gauge theory restricted to the zero modes of the bosonic sector . it is likely that this reduction captures the essential features of the dynamics . it is motivated by the fact that the metric of the small schwarzschild black hole is uniform on @xmath2 . regarding fluctuations in the bulk , the zero mode gauge theory has correspondence with fluctuations in the bulk which are independent of @xmath2 and only depend on the radial @xmath18 coordinate and time . the importance of exhibiting this transition lies in the fact that in the vicinity of the critical temperature @xmath3 , the system goes critical and the fluctuations give rise to universal formulas ( [ painleve ] ) which solely depend on the multi - critical point which is characterized by the exponent @xmath19 in the scaling law @xmath20 . hence the formulas for the black hole cross over which were derived using the effective unitary matrix model in @xcite are also valid while working directly with the zero mode sector of the gauge theory . this paper is organized as follows . in section 2 , we introduce the zero mode action of the bosonic part of the @xmath1 sym theory on @xmath8 . in section 3 , we discuss the numerical studies of the gww - type phase transition . in section 4 , we study the @xmath4 r - symmetry using monte carlo simulation . section 5 is devoted to conclusions and future outlook . we study the @xmath1 sym theory , when all the bosonic fields are restricted to their zero modes on @xmath2 . @xmath21^{2 } + m^{2 } \sum_{\mu=1}^{d } { { \rm tr\,}}m^{2}_{\mu } ( t ) \right ) , \nonumber \\ \label{action}\end{aligned}\ ] ] and @xmath22 is a covariant derivative defined by @xmath23 . \label{covariant}\end{aligned}\ ] ] @xmath24 is the dimensionality of the model , and the dynamical variables @xmath25 and @xmath26 ( @xmath27 ) are @xmath28 hermitian matrices , which can be regarded as the gauge field and the @xmath29 adjoint scalars , respectively . this model has a @xmath30 gauge symmetry @xmath31 the euclidean time @xmath14 in the action ( [ action ] ) has a finite extent @xmath32 , which is the inverse temperature @xmath33 . both the gauge and the scalar fields obey the periodic boundary conditions @xmath34 while this model has three parameters , @xmath32 , @xmath35 and @xmath36 , these are not independent of each other , as @xmath36 can always be set to unity by the following redefinitions @xmath37 and rescaling of the fields @xmath38 the periodic boundary condition ( [ periodic_bc ] ) prevents us from fixing the @xmath39 gauge . however we can fix a gauge where the gauge field is static and diagonal : @xmath40 where @xmath41 $ ] . the indices @xmath42 run over @xmath43 . this gives rise to the fadeev - popov term @xmath44 whose derivation is given in full detail in @xcite . in the following , we study the action @xmath45 we study the model ( [ action_gf ] ) numerically by monte carlo simulation . the details of the algorithm are given in @xcite . we simulate the model with the time direction discretized . we apply the heat bath algorithm to the scalar fields and metropolis algorithm to the gauge field , respectively . it turns out that taking 10 lattice points of the time direction is enough and that increasing the lattice points further does not affect the result . in this section , we study the gww type phase transition of the simplest unitary matrix model , for which an analytical solution is available @xcite . this model is useful to test the accuracy of the numerical method . we start with the numerical simulation of the unitary matrix model consisting only of @xmath47 without adjoint scalar fields , where @xmath48 @xmath49 denotes the path - ordered product . we consider the partition function @xmath50 and define @xmath51 for an integer @xmath52 . in the static and diagonal gauge ( [ gauge_fixing ] ) , @xmath53 these are the moments of the density of eigenvalues : @xmath54 . the first two moments are given by @xmath55 the third - order transition at the point @xmath56 is the gww transition . this is a transition between the gapped and ungapped phases of the eigenvalue distribution of the unitary matrix model . for a generic unitary matrix model , all @xmath57 s show a similar non - analytical behavior like @xmath58 , because near the gap opening point , the relevant operator is given by a linear combination of @xmath57@xcite . we first verify this result numerically using monte carlo simulation . to this end , we take static and diagonal gauge ( [ gauge_fixing ] ) and add the fadeev - popov term ( [ fp_ghost ] ) . namely , we apply the metropolis algorithm to the action @xmath59 we plot the vev s @xmath60 against @xmath61 in figure [ gw-0502227 ] for @xmath62 , and find that they actually agree with the result ( [ tru_analytic ] ) . we next study the saddle point of the gauge field by adding the chemical potential @xmath63 to the action ( [ action_gf ] ) . namely , we study the matrix model @xmath64 where the terms @xmath65 and @xmath66 are defined in ( [ action ] ) and ( [ fp_ghost ] ) , respectively , and @xmath67 is the polyakov line defined in @xmath68 . we first study the @xmath69 case , in which the numerical simulation of large @xmath0 is reachable at a reasonable cpu time . the phase transition of the one - dimensional matrix quantum mechanics with respect to the temperature has been studied in @xcite in the absence of the chemical potential . the polyakov line @xmath70 is small in the low - temperature region , while it is large in the high - temperature region . we focus on the low - temperature region @xmath71 , in which the polyakov line @xmath70 is small for @xmath72 . we plot the result of the @xmath69 , @xmath73 and @xmath74 case in figure [ d2-case ] . the graph above indicates a signature of the phase transition ( possibly third or higher order ) near the critical point @xmath75 at which the polyakov line is @xmath76 . this is expected from the fact that @xmath77 near the transition point . to understand the nature of the transition we first numerically plot the derivative @xmath78 in figure [ d2-case ] ( left ) . the derivative seems to be continuous , and hence the possible transition should at least be of third order . numerical errors prevent us from going further and calculating the higher derivatives directly from our data . instead , in figure [ d2-case ] ( right ) we try to fit our data with analytic functions in the regime @xmath79 and @xmath80 and extrapolate the information about derivatives from the fitted functions . it should be noted that the fitted functions do not necessarily represent the correct analytic form of the exact answer , but they can be viewed as a close approximation . we fit the vev @xmath70 with the function @xmath81 we exploit the fact that in the large-@xmath0 limit , @xmath82 is 0 at @xmath83 and @xmath84 as @xmath85 . and from the fact that @xmath86 and its first derivative @xmath87 are continuous at the critical point @xmath88 , we obtain the condition for @xmath89 and @xmath90 . @xmath91 the parameters @xmath92 are fitted as @xmath93 in this case , the coefficients @xmath94 are @xmath95 and @xmath96 , respectively . we find that the contribution of the terms @xmath97 for @xmath98 and @xmath99 for @xmath80 is small compared to the rest of the terms in ( [ u1_fit ] ) . since its second derivative @xmath100 is discontinuous at the critical point @xmath101 , this system undergoes the gww type third - order phase transition . the vev @xmath102 is small when @xmath98 , and in this region @xmath102 is closer to zero at larger @xmath0 . when @xmath80 , it is fitted with the following function similarly to the unitary matrix model . @xmath103 next , we turn our attention to the high - temperature case @xmath104 , in which the vev s of the polyakov line @xmath105 are large even in the absence of the chemical potential . we plot @xmath106 against @xmath107 for @xmath69 , @xmath108 , @xmath109 case for @xmath74 in figure [ d2-high ] . in contrast to the low - temperature case , we find that there is no gww type third - order phase transition in this case and that the vev s of the polyakov line increase monotonically . we next study a different dimensionality , @xmath110 . similarly , we plot the vev s @xmath60 against @xmath107 in the @xmath110 , @xmath111 , @xmath73 case for @xmath112 in figure [ d6-case ] . we read off the critical point as @xmath113 then , we fit them with the functions ( [ u1_fit ] ) and ( [ u2_fit ] ) . in this case , the parameters are @xmath114 the coefficients @xmath94 are @xmath115 and @xmath116 , which suggests that the contribution of the @xmath94 terms is smaller than that of the rest of the terms in ( [ u1_fit ] ) . we find that the result is similar to the @xmath69 case . in this section , we study the spontaneous breaking of the @xmath4 r - symmetry of the model ( [ action_gf ] ) by monte carlo simulation . throughout this section , we focus on the @xmath110 dimensional case . in analogy with the ikkt - type matrix model @xcite , we consider the following observable @xcite , @xmath117 in our case , we integrate the operator @xmath118 with respect to the time direction and obtain the `` integrated moment of inertia tensor '' @xmath119 we define the eigenvalues of this @xmath120 matrix @xmath121 , which are all real positive , as @xmath122 with the specific order @xmath123 we consider the following @xmath4 invariant quantity @xcite , @xmath124 this quantity measures the variance of the eigenvalue distribution of @xmath121 . using large-@xmath0 factorization we get @xmath125 using the fact that the vev of any @xmath4 two - tensor is proportional to @xmath126 , i.e. @xmath127 , we get @xmath128 . this relationship is not true in general and we expect @xmath129 at finite @xmath0 . in the case when @xmath130 is non - zero , the width of the eigenvalue distribution of @xmath121 is non - zero . the above scenario implies that the dominant contribution of the path integral comes from the configurations for which the eigenvalues of @xmath131 are not equal and consequently the @xmath4 symmetry is broken . hence by plotting the vev s of the eigenvalues of @xmath131 and measuring the width of the distribution , we can figure out the possibility of @xmath4 symmetry breaking at large @xmath0 . this leads us to evaluate the vev s of these eigenvalues @xmath132 in the large-@xmath0 limit . after diagonalization , the residual @xmath4 transformations permute the eigenvalues @xmath122 . hence an unbroken @xmath4 symmetry implies , @xmath133 whereas a broken @xmath4 symmetry implies that for some @xmath134 . to this end , we extrapolate the large-@xmath0 limit from the simulation of finite @xmath0 . if the eigenvalues @xmath132 are all equal in the large-@xmath0 limit , this suggests that the @xmath4 symmetry is unbroken . we first study the @xmath4 r - symmetry breaking when gauge field @xmath135 is integrated . to this end , we update the gauge field @xmath135 , as well as the scalar fields @xmath26 via the usual algorithm . we extrapolate the large-@xmath0 limit from the finite-@xmath0 results of @xmath136 for the high - temperature @xmath137 and the middle - temperature @xmath138 cases . we plot the eigenvalues @xmath132 against @xmath139 in figure [ teig1-vacuum ] . it turns out that the eigenvalues @xmath132 converge to the same value in the large-@xmath0 limit . this behavior is qualitatively the same for other parameter regions of the action ( [ action_gf ] ) . this indicates that the @xmath4 r - symmetry is unbroken in the matrix model ( [ action_gf ] ) . we next study the @xmath4 r - symmetry breaking in the specific configurations of the gauge field , which correspond to @xmath7 and a black hole . to this end , we put the constraints on the gauge fields . in the following , we focus on the high - temperature @xmath140 and massive @xmath141 case , and take @xmath142 . [ [ uniform - distribution ] ] 1 . uniform distribution + + + + + + + + + + + + + + + + + + + + + + + + we take the diagonal part of the gauge field ( [ gauge_fixing ] ) as @xmath143 in this case , the polyakov line @xmath67 satisfies @xmath144 for any nonzero integer @xmath52 . in the ads / cft correspondence uniform distribution , which is depicted in figure [ gauge - distribution ] ( 1 ) , corresponds to the @xmath145 spacetime @xcite . to realize this configuration , we skip the metropolis algorithm to update the gauge field @xmath135 and fix the configuration of the gauge field to be ( [ gauge - uni ] ) . we update only the scalar fields @xmath146 via heat bath algorithm . [ [ clumped - distribution ] ] 2 . clumped distribution + + + + + + + + + + + + + + + + + + + + + + + + in the clumped distribution , we constrain the gauge fields in a small region @xmath147 $ ] , which is opposite to the gapped distribution . this distribution is depicted in figure [ gauge - distribution ] ( 2 ) . similarly to the gapped distribution , we take @xmath148 . if @xmath149 goes out of the region @xmath150 $ ] , we automatically reject that configuration . this configuration coming from a gapped distribution of eigenvalues corresponds to the blackhole state as can be indicated by an analysis of large-@xmath0 perturbation theory around the gapped phase @xcite . similar to the case when we updated the configuration of the gauge field , we make a large-@xmath0 extrapolation of the eigenvalues @xmath132 . we plot in figure [ teig1-specific ] the eigenvalues @xmath132 against @xmath139 for the high - temperature @xmath140 and @xmath73 case . in these cases , too , the eigenvalues @xmath132 converge to the same value at large @xmath0 . we find that the @xmath4 r - symmetry of the scalar field is unbroken for these configurations of the gauge field . in this paper , we have exhibited the gww large-@xmath0 phase transition using monte carlo simulation in the zero mode reduction of the bosonic part of the @xmath1 sym theory on the @xmath8 space . we have studied the saddle point by adding a chemical potential to the reduced action , and observed a third - order phase transition in the large-@xmath0 limit . its significance is that the large-@xmath0 transition signals critical behavior and the the properties of the model in the vicinity of the critical point are universal . hence we expect that the @xmath9 free energy is given by ( [ painleve ] ) . we have also numerically found that the @xmath4 r - symmetry is not spontaneously broken , in the large-@xmath0 limit . in the @xmath151 and @xmath152 unitary matrix models the physical mechanism for the gww transition is well understood . in the @xmath151 models the repulsion between eigenvalues , from the measure , and their attraction in the potential well , are competing effects which lead to this transition @xcite . limit , in terms of the saddle point in the space of the young tableaux density . this density and the eigenvalue density provide a very interesting phase space picture of the large-@xmath0 transition . ] in the @xmath152 models the phase transition is signaled when the fermi level reaches the hump ( maximum ) of the potential @xcite . in the more complicated models we have explored , there are typically non - commuting matrices and explanation seems to be less obvious . in the future it would be instructive to go beyond the zero mode approximation and develop numerical methods to include the variation of the fields on @xmath2 . also , it would be interesting to be able to include the adjoint fermions of the gauge theory in the numerical calculation . + the authors would like to thank rajesh gopakumar , shiraz minwalla , shingo takeuchi and toby wiseman for valuable discussions . thanks sourendu gupta for help in the use of the computer system . part of the simulations were performed on the computer cluster of the physics theory group of the national technical university of athens . srw would like to thank the j. c. bose fellowship of the govt . of india . l. alvarez - gaume , p. basu , m. marino and s. r. wadia , `` blackhole / string transition for the small schwarzschild blackhole of @xmath153 and critical unitary matrix models , '' eur . phys . j. c * 48 * , 647 ( 2006 ) [ hep - th/0605041 ] . l. susskind , `` some speculations about blackhole entropy in string theory , '' hep - th/9309145 . g. t. horowitz and j. polchinski , `` a correspondence principle for blackholes and strings , '' phys . d * 55 * , 6189 ( 1997 ) [ hep - th/9612146 ] . a. sen , `` extremal blackholes and elementary string states , '' mod . lett . a * 10 * , 2081 ( 1995 ) [ hep - th/9504147 ] . m. j. bowick , l. smolin and l. c. r. wijewardhana , `` does string theory solve the puzzles of blackhole evaporation ? , '' gen . * 19 * , 113 ( 1987 ) . m. j. bowick , l. smolin and l. c. r. wijewardhana , `` role of string excitations in the last stages of blackhole evaporation , '' phys . lett . * 56 * , 424 ( 1986 ) . j. m. maldacena , `` the large n limit of superconformal field theories and supergravity , '' adv . * 2 * , 231 ( 1998 ) [ int . j. theor . phys . * 38 * , 1113 ( 1999 ) ] [ hep - th/9711200 ] . d. j. gross and e. witten , `` possible third order phase transition in the large n lattice gauge theory , '' phys . d * 21 * , 446 ( 1980 ) . s. wadia , `` a study of u(n ) lattice gauge theory in two - dimensions , '' efi-79/44-chicago s. r. wadia , `` n = infinity phase transition in a class of exactly soluble model lattice gauge theories , '' phys . b * 93 * , 403 ( 1980 ) . l. alvarez - gaume , c. gomez , h. liu and s. wadia , `` finite temperature effective action , ads(5 ) blackholes , and 1/n expansion , '' phys . d * 71 * , 124023 ( 2005 ) [ hep - th/0502227 ] . p. basu and s. r. wadia , `` r - charged ads(5 ) black holes and large n unitary matrix models , '' phys . d * 73 * , 045022 ( 2006 ) [ hep - th/0506203 ] . b. sundborg , `` the hagedorn transition , deconfinement and n = 4 sym theory , '' nucl . b * 573 * , 349 ( 2000 ) [ hep - th/9908001 ] . a. m. polyakov , `` gauge fields and space - time , '' int . j. mod . a * 17s1 * , 119 ( 2002 ) [ hep - th/0110196 ] . o. aharony , j. marsano , s. minwalla , k. papadodimas and m. van raamsdonk , `` the hagedorn / deconfinement phase transition in weakly coupled large n gauge theories , '' adv . * 8 * , 603 ( 2004 ) [ hep - th/0310285 ] . h. liu , `` fine structure of hagedorn transitions , '' hep - th/0408001 . j. hallin and d. persson , `` thermal phase transition in weakly interacting , large n(c ) qcd , '' phys . b * 429 * , 232 ( 1998 ) [ hep - ph/9803234 ] . o. aharony , j. marsano , s. minwalla , k. papadodimas and m. van raamsdonk , `` a first order deconfinement transition in large n yang - mills theory on a small @xmath2 , '' phys . d * 71 * , 125018 ( 2005 ) [ hep - th/0502149 ] . m. r. douglas and v. a. kazakov , phys . b * 319 * , 219 ( 1993 ) [ hep - th/9305047 ] . k. furuuchi , e. schreiber and g. w. semenoff , `` five - brane thermodynamics from the matrix model , '' hep - th/0310286 . n. kawahara , j. nishimura and k. yoshida , `` dynamical aspects of the plane - wave matrix model at finite temperature , '' jhep * 0606 * , 052 ( 2006 ) [ hep - th/0601170 ] . n. kawahara , j. nishimura and s. takeuchi , `` phase structure of matrix quantum mechanics at finite temperature , '' jhep * 0710 * , 097 ( 2007 ) [ arxiv:0706.3517 [ hep - th ] ] . m. hanada , j. nishimura and s. takeuchi , `` non - lattice simulation for supersymmetric gauge theories in one dimension , '' phys . lett . * 99 * , 161602 ( 2007 ) [ arxiv:0706.1647 [ hep - lat ] ] . k. n. anagnostopoulos , m. hanada , j. nishimura and s. takeuchi , `` monte carlo studies of supersymmetric matrix quantum mechanics with sixteen supercharges at finite temperature , '' arxiv:0707.4454 [ hep - th ] . n. kawahara , j. nishimura and s. takeuchi , `` exact fuzzy sphere thermodynamics in matrix quantum mechanics , '' jhep * 0705 * , 091 ( 2007 ) [ arxiv:0704.3183 [ hep - th ] ] . o. aharony , j. marsano , s. minwalla and t. wiseman , `` black hole - black string phase transitions in thermal 1 + 1 dimensional supersymmetric yang - mills theory on a circle , '' class . grav . * 21 * , 5169 ( 2004 ) [ hep - th/0406210 ] . o. aharony , j. marsano , s. minwalla , k. papadodimas , m. van raamsdonk and t. wiseman , `` the phase structure of low dimensional large n gauge theories on tori , '' jhep * 0601 * , 140 ( 2006 ) [ hep - th/0508077 ] . n. ishibashi , h. kawai , y. kitazawa and a. tsuchiya , `` a large - n reduced model as superstring , '' nucl . b * 498 * , 467 ( 1997 ) [ hep - th/9612115 ] . t. hotta , j. nishimura and a. tsuchiya , `` dynamical aspects of large n reduced models , '' nucl . b * 545 * , 543 ( 1999 ) [ hep - th/9811220 ] . j. ambjorn , k. n. anagnostopoulos , w. bietenholz , t. hotta and j. nishimura , `` large n dynamics of dimensionally reduced 4d su(n ) super yang - mills theory , '' jhep * 0007 * , 013 ( 2000 ) [ hep - th/0003208 ] . j. ambjorn , k. n. anagnostopoulos , w. bietenholz , t. hotta and j. nishimura , `` monte carlo studies of the iib matrix model at large n , '' jhep * 0007 * , 011 ( 2000 ) [ hep - th/0005147 ] . j. ambjorn , k. n. anagnostopoulos , w. bietenholz , f. hofheinz and j. nishimura , `` on the spontaneous breakdown of lorentz symmetry in matrix models of superstrings , '' phys . d * 65 * , 086001 ( 2002 ) [ hep - th/0104260 ] . k. n. anagnostopoulos and j. nishimura , `` new approach to the complex - action problem and its application to a nonperturbative study of superstring theory , '' phys . d * 66 * , 106008 ( 2002 ) [ hep - th/0108041 ] . j. nishimura , `` exactly solvable matrix models for the dynamical generation of space - time in superstring theory , '' phys . d * 65 * , 105012 ( 2002 ) [ hep - th/0108070 ] . j. nishimura , t. okubo and f. sugino , `` gaussian expansion analysis of a matrix model with the spontaneous breakdown of rotational symmetry , '' prog . * 114 * , 487 ( 2005 ) [ hep - th/0412194 ] . e. witten , `` anti - de sitter space , thermal phase transition , and confinement in gauge theories , '' adv . theor . math . * 2 * , 505 ( 1998 ) [ hep - th/9803131 ] . s. dutta and r. gopakumar , `` free fermions and thermal ads / cft , '' arxiv:0711.0133 [ hep - th ] . | in the study of the small ten - dimensional schwarzschild blackhole , the blackhole to string transition is an important problem . in @xcite
, a possible identification is made between the gross - witten - wadia ( gww ) type third - order large-@xmath0 phase transition in the boundary gauge theory and the string - black hole transition in the bulk . in this paper
, we exhibit the existence of the gww transition by monte carlo simulation in the zero mode bosonic action of the finite - temperature @xmath1 sym theory on @xmath2 . exhibiting
this transition in the truncated but highly non - trivial gauge theory implies that in the vicinity of the critical temperature @xmath3 , the system goes critical , and the fluctuations give rise to universal formulas derived in @xcite .
we also discuss the issue of @xmath4 r - symmetry breaking .
@xmath5 [ cols= " < " , ] + + + monte carlo studies of the gww phase transition + in large-@xmath0 gauge theories takehiro azuma , pallab basu@xmath6 and spenta r. wadia address after dec . 1 , 2007 : university of british columbia , vancouver , canada . ] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.