id
string | text
string | len_category
string | source
string |
|---|---|---|---|
math/0212360 | This paper studies the boundary behavior of the Berezin transform on the
C*-algebra generated by the analytic Toeplitz operators on the Bergman space.
| lt256 | arxiv_abstracts |
math/0212361 | In recent works [hep-th/9909147, hep-th/0005259] was found a wonderful
correlation between integrable systems and meromorphic functions. They reduce a
problem of effictivisation of Riemann theorem about conformal maps to
calculation of a string solution of dispersionless limit of the 2D Toda
hierarchy. In [math.CV/0103136] was found a recurrent formulas for coeffciens
of Taylor series of the string solution. This gives, in particular, a method
for calculation of the univalent conformal map from the until disk to an
arbitrary domain, described by its harmonic moments.
In the present paper we investigate some properties of these formulas. In
particular, we find a sufficient condition for convergence of the Taylor series
for the string solution of dispersionless limit of 2D Toda hierarchy.
| lt256 | arxiv_abstracts |
math/0212362 | This note contains a new proof of a theorem of Gang Xiao saying that the
bicanonical map of a surface S of general type is generically finite if and
only if the second plurigenus of S is strictly larger than 2. Such properties
are also studied for adjoint linear systems |K_S+L|, where L is any divisor
with at least 2 linearly independent sections.
| lt256 | arxiv_abstracts |
math/0212363 | A simple proof of (2n)-weak amenability of the triangular Banach algebra T=
[(A A) (0 A)] is given where A is a unital C*-algebra.
| lt256 | arxiv_abstracts |
math/0212364 | We prove a Liouville theorem for the plurisubharmonic functions on complete
Kaelher manifolds. As the applications, we prove a splitting theorem for
complete Kaehler manifolds with nonnegative biscetional curvature in terms of
the linear growth harmonic functions and a optomal gap theorem for such
manifolds.
| lt256 | arxiv_abstracts |
math/0212365 | Let G be a Chevalley group scheme and B<=G a Borel subgroup scheme, both
defined over Z. Let K be a global function field, S be a finite non-empty set
of places over K, and O_S be the corresponding S-arithmetic ring. Then, the
S-arithmetic group B(O_S) is of type F_{|S|-1} but not of type FP_{|S|}.
Moreover one can derive lower and upper bounds for the geometric invariants
\Sigma^m(B(O_S)). These are sharp if G has rank 1. For higher ranks, the
estimates imply that normal subgroups of B(O_S) with abelian quotients,
generically, satisfy strong finiteness conditions.
| lt256 | arxiv_abstracts |
math/0212366 | We give an overview about finiteness properties of soluble S-arithmetic
groups. Both, the number field case and the function field case are covered.
The main result is: If B is a Borel subgroup in a Chevalley group and R is an
S-arithmetic ring, then the group B(R) has finiteness length |S|-1 in the
function field case, and infinite finiteness length in the number field case.
| lt256 | arxiv_abstracts |
math/0212367 | We show that the existence of disintegration for cylindrical measures follows
from a general disintegration theorem for countably additive measures.
| lt256 | arxiv_abstracts |
math/0212368 | In this paper we view some fundamentals of the theory of Hilbert C*-modules
and examine some ways in which Hilbert C*-modules differ from Hilbert spaces.
| lt256 | arxiv_abstracts |
math/0212369 | We study pairs of associative algebras and linear functionals. New results
together with corrected proofs for previously published material are presented.
In particular, we prove the identity ind Mat(n) (#) A=n * ind A$ for
finite-dimensional unital associative algebra A with index 1. [ Symbol (#)
denotes tensor product ]
| lt256 | arxiv_abstracts |
math/0212370 | A general scheme of construction of Drinfeldians and Yangians from quantum
non-twisted affine Kac-Moody algebras is presented. Explicit description of
Drinfeldians and Yangians for all Lie algebras of the classical series A, B, C,
D are given in terms of a Cavalley basis.
| lt256 | arxiv_abstracts |
math/0212371 | We consider a rational-trigonometric deformation in context of rational and
trigonometric deformations. The simplest examples of these deformations are
presented in different fields of mathematics. Rational-trigonometric
differential Knizhnik-Zamolodchikov and dynamical equations are introduced.
| lt256 | arxiv_abstracts |
math/0212372 | We give a review of the systematic construction of hierarchies of soliton
flows and integrable elliptic equations associated to a complex semi-simple Lie
algebra and finite order automorphisms. For example, the non-linear
Schr\"odinger equation, the n-wave equation, and the sigma-model are soliton
flows; and the equation for harmonic maps from the plane to a compact Lie
group, for primitive maps from the plane to a $k$-symmetric space, and constant
mean curvature surfaces and isothermic surfaces in space forms are integrable
elliptic systems. We also give a survey of
(i) construction of solutions using loop group factorizations,
(ii) PDEs in differential geometry that are soliton equations or elliptic
integrable systems,
(iii) similarities and differences of soliton equations and integrable
elliptic systems.
| lt256 | arxiv_abstracts |
math/0212373 | Let $G$ be a graph. For a given positive integer $d$, let $f_G(d)$ denote the
largest integer $t$ such that in every coloring of the edges of $G$ with two
colors there is a monochromatic subgraph with minimum degree at least $d$ and
order at least $t$. For $n > k > d$ let $f(n,k,d)$ denote the minimum of
$f_G(d)$ where $G$ ranges over all graphs with $n$ vertices and minimum degree
at least $k$. In this paper we establish $f(n,k,d)$ whenever $k$ or $n-k$ are
fixed, and $n$ is sufficiently large. We also consider the case where more than
two colors are allowed.
| lt256 | arxiv_abstracts |
math/0212374 | In this paper parent substances with molecules which can be divided into a
skeleton and six univalent substituents, and that have the properties mentioned
in the title, are considered. Two instances are the molecules of benzene and
cyclopropane. The Lunn-Senior's symmetry groups of substitution isomerism of
these compounds are described and upper bounds of the numbers of their
di-substitution and tri-substitution homogeneous derivatives are found. Lists
of the possible simple substitution reactions among di-substitution homogeneous
derivatives, on one hand, and di-substitution heterogeneous, and
tri-substitution homogeneous derivatives, on the other, are given. These
substitution reactions allow for some derivatives to be identified with their
structural formulae.
| lt256 | arxiv_abstracts |
math/0212375 | An optimum solution free from degeneration is found to the system of linear
algebraic equations with empirical coefficients and right-hand sides. The
quadratic risk of estimators of the unknown solution vector is minimized over a
class of linear systems with given square norm of the coefficient matrix and
length of the right-hand side vector. Empirical coefficients and right-hand
sides are assumed to be independent and normal with known variance. It is found
that the optimal estimator has the form of a regularized minimum square
solution with an extension multiple. A simple formula is derived showing
explicitly the dependence of the minimal risk on parameters.
| lt256 | arxiv_abstracts |
math/0212376 | We establish a Liouville type theorem for some conformally invariant fully
nonlinear equations
| lt256 | arxiv_abstracts |
math/0212377 | In many everyday categories (sets, spaces, modules, ...) objects can be both
added and multiplied. The arithmetic of such objects is a challenge because
there is usually no subtraction. We prove a family of cases of the following
principle: if an arithmetic statement about the objects can be proved by
pretending that they are complex numbers, then there also exists an honest
proof.
| lt256 | arxiv_abstracts |
math/0212378 | We show that in characteristic 2, the Steinberg representation of the
symplectic group Sp(2n,q), q a power of an odd prime p, has two irreducible
constituents lying just above the socle that are isomorphic to the two Weil
modules of degree (q^n-1)/2.
| lt256 | arxiv_abstracts |
math/0212379 | A clone on a set X is a set of finitary operations on X which contains all
the projections and is closed under composition.
The set of all clones forms a complete lattice Cl(X) with greatest element O,
the set of all finitary operations. For finite sets X the lattice is "dually
atomic": every clone other than O is below a coatom of Cl(X).
It was open whether Cl(X) is also dually atomic for infinite X. Assuming the
continuum hypothesis, we show that there is a clone C on a countable set such
that the interval of clones above C is linearly ordered, uncountable, and has
no coatoms.
| lt256 | arxiv_abstracts |
math/0212380 | Let $F\subseteq H\subseteq G$ be closed subgroups of a locally compact group.
In response to a 1972 question by Eymard, we construct an example where the
homogeneous factor-space $G/F$ is amenable in the sense of Eymard-Greenleaf,
while $H/F$ is not. (In our example, $G$ is discrete.) As a corollary which
answers a 1990 question by Bekka, the induced representation $\ind_H^G(\rho)$
can be amenable in the sense of Bekka even if $\rho$ is not amenable. The
second example, answering another question by Bekka, shows that
$\ind_H^G(\rho)$ need not be amenable even if both the representation $\rho$
and the coset space $G/H$ are amenable.
| lt256 | arxiv_abstracts |
math/0212381 | A group is coherent if all its finitely generated subgroups are finitely
presented. In this article we provide a criterion for positively determining
the coherence of a group. This criterion is based upon the notion of the
perimeter of a map between two finite 2-complexes which is introduced here. In
the groups to which this theory applies, a presentation for a finitely
generated subgroup can be computed in quadratic time relative to the sum of the
lengths of the generators. For many of these groups we can show in addition
that they are locally quasiconvex.
As an application of these results we prove that one-relator groups with
sufficient torsion are coherent and locally quasiconvex and we give an
alternative proof of the coherence and local quasiconvexity of certain
3-manifold groups. The main application is to establish the coherence and local
quasiconvexity of many small cancellation groups.
| lt256 | arxiv_abstracts |
math/0212382 | The goal of this technical note is to show that the geometry of generalized
parabolic towers cannot be essentially bounded. It fills a gap in author's
paper "Combinatorics, geomerty and attractors of quasi-quadratic maps", Annals
of Math., 1992.
| lt256 | arxiv_abstracts |
math/0212383 | This paper gives a short summary of the central role played by Ed Brown's
"twisting cochains" in higher Franz-Reidemeister (FR) torsion and higher
analytic torsion. Briefly, any fiber bundle gives a twisting cochain which is
unique up to fiberwise homotopy equivalence. However, when they are based, the
difference between two of them is a higher algebraic K-theory class measured by
higher FR torsion. Flat superconnections are also equivalent to twisting
cochains.
| lt256 | arxiv_abstracts |
math/0212384 | The rational cohomology of a coadjoint orbit ${\cal O}$ is expressed as
tensor product of the cohomology of other coadjoint orbits ${\cal O}_k$, with $
\hbox{dim} {\cal O}_k< \hbox{dim} {\cal O}$.
| lt256 | arxiv_abstracts |
math/0212385 | In this note we prove a decomposition related to the affine fundamental group
and the projective fundamental group of a line arrangement and a reducible
curve with a line component. We give some applications to this result.
| lt256 | arxiv_abstracts |
math/0212386 | We give sufficient conditions for a group of homeomorphisms of a Peano
continuum X without cut-points to be a convergence group. The condition is that
there is a collection of convergence subgroups whose limit sets `cut up' X in
the correct fashion. This is closely related to the result in [E Swenson, Axial
pairs and convergence groups on S^1, Topology 39 (2000) 229-237].
| lt256 | arxiv_abstracts |
math/0212387 | This is an expository introduction to fusion rules for affine Kac-Moody
algebras, with major focus on the algorithmic aspects of their computation and
the relationship with tensor product decompositions. Many explicit examples are
included with figures illustrating the rank 2 cases. New results relating
fusion coefficients to tensor product coefficients are proved, and a conjecture
is given which shows that the Frenkel-Zhu affine fusion rule theorem can be
seen as a beautiful generalization of the Parasarathy-Ranga Rao-Varadarajan
tensor product theorem. Previous work of the author and collaborators on a
different approach to fusion rules from elementary group theory is also
explained.
| lt256 | arxiv_abstracts |
math/0212388 | If we assume the Thesis that any classical Turing machine T, which halts on
every n-ary sequence of natural numbers as input, determines a PA-provable
formula, whose standard interpretation is an n-ary arithmetical relation f(x1,
>..., xn) that holds if, and only if, T halts, then standard PA can model the
state of a deterministic universe that is consistent with a probabilistic
Quantum Theory. Another significant consequence of this Thesis is that every
partial recursive function can be effectively defined as total.
| lt256 | arxiv_abstracts |
math/0212389 | This article describes various moduli spaces of pseudoholomorphic curves on
the symplectization of a particular overtwisted contact structure on S^1 x S^2.
This contact structure appears when one considers a closed self dual form on a
4-manifold as a symplectic form on the complement of its zero locus. The
article is focussed mainly on disks, cylinders and three-holed spheres, but it
also supplies groundwork for a description of moduli spaces of curves with more
punctures and non-zero genus.
| lt256 | arxiv_abstracts |
math/0212390 | Combinatorics is a fundamental mathematical discipline as well as an
essential component of many mathematical areas, and its study has experienced
an impressive growth in recent years. One of the main reasons for this growth
is the tight connection between Discrete Mathematics and Theoretical Computer
Science, and the rapid development of the latter. While in the past many of the
basic combinatorial results were obtained mainly by ingenuity and detailed
reasoning, the modern theory has grown out of this early stage, and often
relies on deep, well developed tools. This is a survey of two of the main
general techniques that played a crucial role in the development of modern
combinatorics; algebraic methods and probabilistic methods. Both will be
illustrated by examples, focusing on the basic ideas and the connection to
other areas.
| lt256 | arxiv_abstracts |
math/0212391 | Differential complexes such as the de Rham complex have recently come to play
an important role in the design and analysis of numerical methods for partial
differential equations. The design of stable discretizations of systems of
partial differential equations often hinges on capturing subtle aspects of the
structure of the system in the discretization. In many cases the differential
geometric structure captured by a differential complex has proven to be a key
element, and a discrete differential complex which is appropriately related to
the original complex is essential. This new geometric viewpoint has provided a
unifying understanding of a variety of innovative numerical methods developed
over recent decades and pointed the way to stable discretizations of problems
for which none were previously known, and it appears likely to play an
important role in attacking some currently intractable problems in numerical
PDE.
| lt256 | arxiv_abstracts |
math/0212392 | Aim of this paper is to review some basic ideas and recent developments in
the theory of strictly hyperbolic systems of conservation laws in one space
dimension. The main focus will be on the uniqueness and stability of entropy
weak solutions and on the convergence of vanishing viscosity approximations.
| lt256 | arxiv_abstracts |
math/0212393 | The Monge-Ampere equation, plays a central role in the theory of fully non
linear equations. In fact we will like to show how the Monge-Ampere equation,
links in some way the ideas comming from the calculus of variations and those
of the theory of fully non linear equations.
| lt256 | arxiv_abstracts |
math/0212394 | In the study of conformal geometry, the method of elliptic partial
differential equations is playing an increasingly significant role. Since the
solution of the Yamabe problem, a family of conformally covariant operators
(for definition, see section 2) generalizing the conformal Laplacian, and their
associated conformal invariants have been introduced. The conformally covariant
powers of the Laplacian form a family $P_{2k}$ with $k \in \mathbb N$ and $k
\leq \frac{n}{2}$ if the dimension $n$ is even. Each $P_{2k}$ has leading order
term $(- \Delta)^k$ and is equal to $ (- \Delta) ^k$ if the metric is flat.
| lt256 | arxiv_abstracts |
math/0212395 | Classical multiscale analysis based on wavelets has a number of successful
applications, e.g. in data compression, fast algorithms, and noise removal.
Wavelets, however, are adapted to point singularities, and many phenomena in
several variables exhibit intermediate-dimensional singularities, such as
edges, filaments, and sheets. This suggests that in higher dimensions, wavelets
ought to be replaced in certain applications by multiscale analysis adapted to
intermediate-dimensional singularities.
My lecture described various initial attempts in this direction. In
particular, I discussed two approaches to geometric multiscale analysis
originally arising in the work of Harmonic Analysts Hart Smith and Peter Jones
(and others): (a) a directional wavelet transform based on parabolic dilations;
and (b) analysis via anistropic strips. Perhaps surprisingly, these tools have
potential applications in data compression, inverse problems, noise removal,
and signal detection; applied mathematicians, statisticians, and engineers are
eagerly pursuing these leads.
| 256 | arxiv_abstracts |
math/0212396 | Random matrices have their roots in multivariate analysis in statistics, and
since Wigner's pioneering work in 1955, they have been a very important tool in
mathematical physics. In functional analysis, random matrices and random
structures have in the last two decades been used to construct Banach spaces
with surprising properties. After Voiculescu in 1990--1991 used random matrices
to classification problems for von Neumann algebras, they have played a key
role in von Neumann algebra theory. In this lecture we will discuss some new
applications of random matrices to operator algebra theory, namely applications
to classification problems for $C^*$-algebras and to the invariant subspace
problem relative to a von Neumann algebra.
| lt256 | arxiv_abstracts |
math/0212397 | Modular forms appear in many facets of mathematics, and have played important
roles in geometry, mathematical physics, number theory, representation theory,
topology, and other areas. Around 1994, motivated by technical issues in
homotopy theory, Mark Mahowald, Haynes Miller and I constructed a topological
refinement of modular forms, which we call {\em topological modular forms}. At
the Zurich ICM I sketched a program designed to relate topological modular
forms to invariants of manifolds, homotopy groups of spheres, and ordinary
modular forms. This program has recently been completed and new directions have
emerged. In this talk I will describe this recent work and how it informs our
understanding of both algebraic topology and modular forms.
| lt256 | arxiv_abstracts |
math/0212398 | We describe the percolation model and some of the principal results and open
problems in percolation theory. We also discuss briefly the spectacular recent
progress by Lawler, Schramm, Smirnov and Werner towards understanding the phase
transition of percolation (on the triangular lattice).
| lt256 | arxiv_abstracts |
math/0212399 | Cet expos\'e est consacr\'e \`a la preuve de la correspondance de Langlands
pour les groupes $\GL_r$ sur les corps de fonctions.
-----
This article is devoted to the proof of the Langlands correspondence for the
groups $GL_r$ over function fields.
| lt256 | arxiv_abstracts |
math/0212400 | Is there a mathematical theory underlying intelligence? Control theory
addresses the output side, motor control, but the work of the last 30 years has
made clear that perception is a matter of Bayesian statistical inference, based
on stochastic models of the signals delivered by our senses and the structures
in the world producing them. We will start by sketching the simplest such
model, the hidden Markov model for speech, and then go on illustrate the
complications, mathematical issues and challenges that this has led to.
| lt256 | arxiv_abstracts |
math/0212401 | Let $\Gamma$ be a finite subgroup of $\SL_2(\C)$. We consider $\Gamma$-fixed
point sets in Hilbert schemes of points on the affine plane $\C^2$. The direct
sum of homology groups of components has a structure of a representation of the
affine Lie algebra $\ag$ corresponding to $\Gamma$. If we replace homology
groups by equivariant $K$-homology groups, we get a representation of the
quantum toroidal algebra $\Ut$. We also discuss a higher rank generalization
and character formulas in terms of intersection homology groups.
| lt256 | arxiv_abstracts |
math/0212402 | This article discusses the recent transcendental techniques used in the
proofs of the following three conjectures. (1)~The plurigenera of a compact
projective algebraic manifold are invariant under holomorphic deformation.
(2)~There exists no smooth Leviflat hypersurface in the complex projective
plane. (3)~A generic hypersurface of sufficiently high degree in the complex
projective space is hyperbolic in the sense that there is no nonconstant
holomorphic map from the complex Euclidean line to it.
| lt256 | arxiv_abstracts |
math/0212403 | In the first part of this paper we try to explain to a general mathematical
audience some of the remarkable web of conjectures linking representations of
Galois groups with algebraic geometry, complex analysis and discrete subgroups
of Lie groups. In the second part we briefly review some limited recent
progress on these conjectures.
| lt256 | arxiv_abstracts |
math/0212404 | Nonlinear analysis has played a prominent role in the recent developments in
geometry and topology. The study of the Yang-Mills equation and its cousins
gave rise to the Donaldson invariants and more recently, the Seiberg-Witten
invariants. Those invariants have enabled us to prove a number of striking
results for low dimensional manifolds, particularly, 4-manifolds. The theory of
Gromov-Witten invariants was established by using solutions of the
Cauchy-Riemann equation. These solutions are often refered as
pseudo-holomorphic maps which are special minimal surfaces studied long in
geometry. It is certainly not the end of applications of nonlinear partial
differential equations to geometry. In this talk, we will discuss some recent
progress on nonlinear partial differential equations in geometry. We will be
selective, partly because of my own interest and partly because of recent
applications of nonlinear equations. There are also talks in this ICM to cover
some other topics of geometric analysis by R. Bartnik, B. Andrew, P. Li and
X.X. Chen, etc.
| 256 | arxiv_abstracts |
math/0212405 | We survey old and recent results on the problem of finding a complete set of
rules describing the behavior of the power function, i.e. the function which
takes a cardinal $\kappa$ to the cardinality of its power $2^\kappa$.
| lt256 | arxiv_abstracts |
math/0212406 | There have been many generalizations of Shoenfield's Theorem on the
absoluteness of $\Sigma^1_2$ sentences between uncountable transitive models of
$\mathrm{ZFC}$. One of the strongest versions currently known deals with
$\Sigma^2_1$ absoluteness conditioned on $\mathrm{CH}$. For a variety of
reasons, from the study of inner models and from simply combinatorial set
theory, the question of whether conditional $\Sigma^2_2$ absoluteness is
possible at all, and if so, what large cardinal assumptions are involved and
what sentence(s) might play the role of $\mathrm{CH}$, are fundamental
questions. This article investigates the possiblities for $\Sigma^2_2$
absoluteness by extending the connections between determinacy hypotheses and
absoluteness hypotheses.
| lt256 | arxiv_abstracts |
math/0212407 | This article describes the mean curvature flow, some of the discoveries that
have been made about it, and some unresolved questions.
| lt256 | arxiv_abstracts |
math/0212408 | The Manin-Mumford conjecture in characteristic zero was first proved by
Raynaud. Later, Hrushovski gave a different proof using model theory. His main
result from model theory, when applied to abelian varieties, can be rephrased
in terms of algebraic geometry. In this paper we prove that intervening result
using classical algebraic geometry alone. Altogether, this yields a new proof
of the Manin-Mumford conjecture using only classical algebraic geometry.
| lt256 | arxiv_abstracts |
math/0212409 | We consider the algebro-geometric consequences of integration by parts.
| lt256 | arxiv_abstracts |
math/0212410 | State space models have long played an important role in signal processing.
The Gaussian case can be treated algorithmically using the famous Kalman
filter. Similarly since the 1970s there has been extensive application of
Hidden Markov models in speech recognition with prediction being the most
important goal. The basic theoretical work here, in the case $X$ and $Y$ finite
(small) providing both algorithms and asymptotic analysis for inference is that
of Baum and colleagues. During the last 30-40 years these general models have
proved of great value in applications ranging from genomics to finance.
Unless the $X,Y$ are jointly Gaussian or $X$ is finite and small the problem
of calculating the distributions discussed and the likelihood exactly are
numerically intractable and if $Y$ is not finite asymptotic analysis becomes
much more difficult. Some new developments have been the construction of
so-called ``particle filters'' (Monte Carlo type) methods for approximate
calculation of these distributions (see Doucet et al. [4]) for instance and
general asymptotic methods for analysis of statistical methods in HMM [2] and
other authors.
We will discuss these methods and results in the light of exponential mixing
properties of the conditional (posterior) distribution of $(X_1,X_2,...)$ given
$(Y_1,Y_2,...)$ already noted by Baum and Petrie and recent work of the authors
Bickel, Ritov and Ryden, Del Moral and Jacod, Douc and Matias.
| 256 | arxiv_abstracts |
math/0212411 | A classical limit theorem of stochastic process theory concerns the sample
cumulative distribution function (CDF) from independent random variables. If
the variables are uniformly distributed then these centered CDFs converge in a
suitable sense to the sample paths of a Brownian Bridge. The so-called
Hungarian construction of Komlos, Major and Tusnady provides a strong form of
this result. In this construction the CDFs and the Brownian Bridge sample paths
are coupled through an appropriate representation of each on the same
measurable space, and the convergence is uniform at a suitable rate.
Within the last decade several asymptotic statistical-equivalence theorems
for nonparametric problems have been proven, beginning with Brown and Low
(1996) and Nussbaum (1996). The approach here to statistical-equivalence is
firmly rooted within the asymptotic statistical theory created by L. Le Cam but
in some respects goes beyond earlier results.
This talk demonstrates the analogy between these results and those from the
coupling method for proving stochastic process limit theorems. These two
classes of theorems possess a strong inter-relationship, and technical methods
from each domain can profitably be employed in the other. Results in a recent
paper by Carter, Low, Zhang and myself will be described from this perspective.
| 256 | arxiv_abstracts |
math/0212412 | Recently, a number of authors have investigated the conditions under which a
stochastic perturbation acting on an infinite dimensional dynamical system,
e.g. a partial differential equation, makes the system ergodic and mixing. In
particular, one is interested in finding minimal and physically natural
conditions on the nature of the stochastic perturbation. I shall review recent
results on this question; in particular, I shall discuss the Navier-Stokes
equation on a two dimensional torus with a random force which is white noise in
time, and excites only a finite number of modes. The number of excited modes
depends on the viscosity $\nu$, and grows like $\nu^{-3}$ when $\nu$ goes to
zero. This Markov process has a unique invariant measure and is exponentially
mixing in time.
| lt256 | arxiv_abstracts |
math/0212413 | Spielman and Teng introduced the smoothed analysis of algorithms to provide a
framework in which one could explain the success in practice of algorithms and
heuristics that could not be understood through the traditional worst-case and
average-case analyses. In this talk, we survey some of the smoothed analyses
that have been performed.
| lt256 | arxiv_abstracts |
math/0212414 | Adaptive mesh refinement techniques are nowadays an established and powerful
tool for the numerical discretization of PDE's. In recent years, wavelet bases
have been proposed as an alternative to these techniques. The main motivation
for the use of such bases in this context is their good performances in data
compression and the approximation theoretic foundations which allow to analyze
and optimize these performances. We shall discuss these theoretical
foundations, as well as one of the approaches which has been followed in
developing efficient adaptive wavelet solvers. We shall also discuss the
similarities and differences between wavelet methods and adaptive mesh
refinement.
| lt256 | arxiv_abstracts |
math/0212415 | Many problems in physics, material sciences, chemistry and biology can be
abstractly formulated as a system that navigates over a complex energy
landscape of high or infinite dimensions. Well-known examples include phase
transitions of condensed matter, conformational changes of biopolymers, and
chemical reactions. The energy landscape typically exhibits multiscale
features, giving rise to the multiscale nature of the dynamics. This is one of
the main challenges that we face in computational science. In this report, we
will review the recent work done by scientists from several disciplines on
probing such energy landscapes. Of particular interest is the analysis and
computation of transition pathways and transition rates between metastable
states. We will then present the string method that has proven to be very
effective for some truly complex systems in material science and chemistry.
| lt256 | arxiv_abstracts |
math/0212416 | The paper opens with an overview of the discussion of international
comparisons (including goals) in mathematics education. Afterwards, the two
most important recent international studies, the PISA Study and TIMSS-Repeat,
are described. After a short description of the qualitative-quantitative
debate, a qualitatively oriented small-scale study is described. The paper
closes with reflection on the possibilities and limitations of such studies.
| lt256 | arxiv_abstracts |
math/0212417 | Laurent Lafforgue has been awarded the Fields Medal for his proof of the
Langlands correspondence for the full linear groups $\mathop{\rm
GL}\nolimits_{r}$ ($r\geq 1$) over function fields.
This article is a brief introduction to the Langlands correspondence and to
Lafforgue's theorem.
| lt256 | arxiv_abstracts |
math/0212418 | Vladimir Voevodsky was born in 1966. He studied at Moscow State University
and Harvard university. He is now Professor at the Institute for Advanced Study
in Princeton.
Among his main achievements are the following: he defined and developed
motivic cohomology and the ${\mathbf A}^1$-homotopy theory of algebraic
varieties; he proved the Milnor conjectures on the $K$-theory of fields. This
article is a brief introduction to this work, for which Voevodsky was awarded
the Fields Medal.
| lt256 | arxiv_abstracts |
math/0212419 | This paper is an updated version of ANT-0372 (2002 dec 4) with the same
title. Several errors are corrected in this version.
An example of the kind of results obtained is: Let K/\Q be an abelian
extension with N = [K:\Q] > 1, N odd. Let h(K) be the class number of K.
Suppose that h(K) > 1. Let p be a prime dividing h(K). Let r_p be the rank of
the p-class group of K. Then p \times (p^{r_p}-1) and N are not coprime.
The paper is at elementary level and contains a lot of numerical examples.
| lt256 | arxiv_abstracts |
math/0212420 | This is a revised version of ANT-0332: "A support problem for the
intermediate Jacobians of l-adic representations", by G. Banaszak, W. Gajda &
P. Krason, which was placed on these archives on the 29th of January 2002.
Following a suggestion of the referee we have subdivided the paper into two
separate parts: "Support problem for the intermediate Jacobians of l-adic
representations", and "On Galois representations for abelian varieties with
complex and real multiplications".
Our results on the image of Galois and the Mumford-Tate conjecture for some
RM abelian varieties are contained in the second paper. Both papers were
accepted for publication.
| lt256 | arxiv_abstracts |
math/0301001 | Every real algebraic variety is isomorphic to the set of totally mixed Nash
equilibria of some three-person game, and also to the set of totally mixed Nash
equilibria of an $N$-person game in which each player has two pure strategies.
From the Nash-Tognoli Theorem it follows that every compact differentiable
manifold can be encoded as the set of totally mixed Nash equilibria of some
game. Moreover, there exist isolated Nash equilibria of arbitrary topological
degree.
| lt256 | arxiv_abstracts |
math/0301002 | For unbounded operators A,B and C in general, with C closure of [A,B] does
not lead to the uncertainty relation ||Au|| ||Bu|| >= |<C u,u> |/2. If A,B and
C are part of the generators of a unitary representation of a Lie group then
the uncertainty principle above holds.
| lt256 | arxiv_abstracts |
math/0301003 | This paper is a sequel to [LoMa] where moduli spaces of painted stable curves
were introduced and studied. We define the extended modular operad of genus
zero, algebras over this operad, and study the formal differential geometric
structures related to these algebras: pencils of flat connections and Frobenius
manifolds without metric. We focus here on the combinatorial aspects of the
picture. Algebraic geometric aspects are treated in [Ma2].
| lt256 | arxiv_abstracts |
math/0301004 | We study the extraordinary dimension function dim_{L} introduced by
\v{S}\v{c}epin. An axiomatic characterization of this dimension function is
obtained. We also introduce inductive dimensions ind_{L} and Ind_{L} and prove
that for separable metrizable spaces all three coincide. Several results such
as characterization of dim_{L} in terms of partitions and in terms of mappings
into $n$-dimensional cubes are presented. We also prove the converse of the
Dranishnikov-Uspenskij theorem on dimension-raising maps.
| lt256 | arxiv_abstracts |
math/0301005 | A Kaehler-Nijenhuis manifold is a Kaehler manifold M, with metric g, complex
structure J and Kaehler form F, endowed with a Nijenhuis tensor field A that is
compatible with the Poisson stucture defined by F in the sense of the theory of
Poisson-Nijenhuis structures. If this happens, and if either AJ=JA or AJ=-JA, M
is foliated by im A into non degenerate Kaehler-Nijenhuis submanifolds. If A is
a non degenerate (1,1)-tensor field on M, (M,g,J,A) is a Kaehler-Nijenhuis
manifold iff one of the following two properties holds: 1) A is associated with
a symplectic structure of M that defines a Poisson structure compatible with
the Poisson structure defined by F; 2) A and its inverse are associated with
closed 2-forms. On a Kaehler-Nijenhuis manifold, if A is non degenerate and
AJ=-JA, A must be a parallel tensor field.
| lt256 | arxiv_abstracts |
math/0301006 | For isolated complex hypersurface singularities with real defining equation
we show the existence of a monodromy vector field such that complex conjugation
intertwines the local monodromy diffeomorphism with its inverse. In particular,
it follows that the geometric monodromy is the composition of the involution
induced by complex conjugation and another involution. This topological
property holds for all isolated complex plane curve singularities. Using real
morsifications, we compute the action of complex conjugation and of the other
involution on the Milnor fiber of real plane curve singularities. These
involutions have nice descriptions in terms of divides for the singularity.
| lt256 | arxiv_abstracts |
math/0301007 | We determine the class of the (closure of the) Schottky locus in genus 4 in
the Igusa- and the Voronoi compactification of A_4 and comment on the weight 8
modular form which vanishes on it.
| lt256 | arxiv_abstracts |
math/0301008 | We define stacks of uniform cyclic covers of Brauer-Severi schemes, proving
that they can be realized as quotient stacks of open subsets of
representations, and compute the Picard group for the open substacks
parametrizing smooth uniform cyclic covers. Moreover, we give an analogous
description for stacks parametrizing triple cyclic covers of Brauer-Severi
schemes of rank 1, which are not necessarily uniform, and give a presentation
of the Picard group for substacks corresponding to smooth triple cyclic covers.
| lt256 | arxiv_abstracts |
math/0301009 | It is demonstrated that in the (projective plane over) Galois fields GF(q)
with q=2^n and n>2 (n being a positive integer) we can define, in addition to
the temporal dimensions generated by pencils of conics, also time coordinates
represented by aggregates of (q+1)-arcs that are not conics. The case is
illustrated by a (self-dual) pencil of conics endowed with two singular conics
of which one represents a double real line and the other is a real line pair.
Although this pencil does not generate the ordinary (i.e., featuring the past,
present and future) arrow of time over GF(2^n), there does exist a
pencil-related family of (q+1)-arcs, not conics, that closely resembles such an
arrow. Some psycho(patho)logical justifications of this finding are presented,
based on the "peculiar/anomalous" experiences of time by a couple of
schizophrenic patients.
| lt256 | arxiv_abstracts |
math/0301010 | Let $S$ be a surface of nonpositive curvature of genus bigger than 1 (i.e.
not the torus). We prove that any flat strip in the surface is in fact a flat
cylinder. Moreover we prove that the number of homotopy classes of such flat
cylinders is bounded.
| lt256 | arxiv_abstracts |
math/0301011 | This is the first issue of a semi-formal bulletin dealing with Selection
Principles in Mathematics (SPM) -- this field deals with all sorts of studies
of diagonalization arguments, especially in topology (covering properties,
sequences of covers, etc.) and infinite combinatorics (cardinal characteristics
of the continuum), and their applications to other areas of mathematics
(function spaces, game theory, group theory, etc.)
| lt256 | arxiv_abstracts |
math/0301012 | Let $\bar{P}$ be a sequence of length $2n$ in which each element of
$\{1,2,...,n\}$ occurs twice. Let $P'$ be a closed curve in a closed surface
$S$ having $n$ points of simple auto-intersections, inducing a 4-regular graph
embedded in $S$ which is 2-face colorable. If the sequence of
auto-intersections along $P'$ is given by $\bar{P}$, we say that is a {\em $P'$
2-face colorable solution for the Gauss Code $\bar{P}$ on surface $S$} or a
{\em lacet for $\bar{P}$ on $S$}. In this paper we present a necessary and
sufficient condition yielding these solutions when $S$ is Klein bottle. The
condition take the form of a system of $m$ linear equations in $2n$ variables
over $\Z_2$, where $m \le n(n-1)/2$. Our solution generalize solutions for the
projective plane and on the sphere. In a strong way, the Klein bottle is an
extremal case admitting an affine linear solution: we show that the similar
problem on the torus and on surfaces of higher connectivity are modelled by a
quadratic system of equations.
| lt256 | arxiv_abstracts |
math/0301013 | In this paper we prove over fields of characteristic zero that the zero slice
of the motivic sphere spectrum is the motivic Eilenberg-Maclane spectrum. As a
corollary one concludes that the slices of any spectrum are modules over the
motivic Eilenberg-MacLane spectrum. To prove our result we analyze the unstable
homotopy type of the symmetric powers of the T-sphere.
| lt256 | arxiv_abstracts |
math/0301014 | For all $m > 0$ we build a two-dimensional family of smooth manifolds of real
dimension $3m + 2$ and use it to interpolate between the anticanonical family
in the complex projective space of dimension $m + 1$ and its mirror dual. The
main tool is the notion of self-dual manifold.
| lt256 | arxiv_abstracts |
math/0301015 | A new construction of the real number system, that is built directly upon the
additive group of integers and has its roots in the definition due to Henri
Poincar\'e of the rotation number of an orientation preserving homeomorphism of
the circle.
| lt256 | arxiv_abstracts |
math/0301016 | We summarize the known methods of producing a non-supercompact strongly
compact cardinal and describe some new variants. Our Main Theorem shows how to
apply these methods to many cardinals simultaneously and exactly control which
cardinals are supercompact and which are only strongly compact in a forcing
extension. Depending upon the method, the surviving non-supercompact strongly
compact cardinals can be strong cardinals, have trivial Mitchell rank or even
contain a club disjoint from the set of measurable cardinals. These results
improve and unify previous results of the first author.
| lt256 | arxiv_abstracts |
math/0301017 | A mathematical relation between elements of one- and multi-dimensional
discrete Fourier transforms (DFT) is found. A method of analysing the
multi-dimensional data by their single one-dimensional (1-D) DFT is offered. An
experiment of filtering a two-dimensional image using the single 1-D DFT is
carried out.
| lt256 | arxiv_abstracts |
math/0301018 | We examine spaces of connected tri-/univalent graphs subject to local
relations which are motivated by the theory of Vassiliev invariants. It is
shown that the behaviour of ladder-like subgraphs is strongly related to the
parity of the number of rungs: there are similar relations for ladders of even
and odd lengths, respectively. Moreover, we prove that - under certain
conditions - an even number of rungs may be transferred from one ladder to
another.
| lt256 | arxiv_abstracts |
math/0301019 | The graded algebra Lambda defined by Pierre Vogel is of general interest in
the theory of finite-type invariants of knots and of 3-manifolds because it
acts on the corresponding spaces of connected graphs subject to relations
called IHX and AS. We examine a subalgebra Lambda_0 that is generated by
certain elements called t and x_n with n >= 3. Two families of relations in
Lambda_0 are derived and it is shown that the dimension of Lambda_0 grows at
most quadratically with respect to degree. Under the assumption that t is not a
zero divisor in Lambda_0, a basis of Lambda_0 and an isomorphism from Lambda_0
to a sub-ring of Z[t,u,v] is given.
| lt256 | arxiv_abstracts |
math/0301020 | A new filtration of the spaces of tri-/univalent graphs B_m^u that occur in
the theory of finite-type invariants of knots and 3-manifolds is introduced.
Combining the results of the two preceding articles, the quotients of this
filtration are modeled by spaces of graphs with two types of edges and four
types of vertices, and an upper bound for dim B_m^u in terms of the dimensions
of the filtration quotients is given. The degree m up to which B_m^u is known
is raised by two for u=0 and u=2.
| lt256 | arxiv_abstracts |
math/0301021 | The first step in investigating colour symmetries for periodic and aperiodic
systems is the determination of all colouring schemes that are compatible with
the symmetry group of the underlying structure, or with a subgroup of it. For
an important class of colourings of planar structures, this mainly
combinatorial question can be addressed with methods of algebraic number
theory. We present the corresponding results for all planar modules with N-fold
symmetry that emerge as the rings of integers in cyclotomic fields with class
number one. The counting functions are multiplicative and can be encapsulated
in Dirichlet series generating functions, which turn out to be the Dedekind
zeta functions of the corresponding cyclotomic fields.
| lt256 | arxiv_abstracts |
math/0301022 | A triangular quantum deformation of $ osp(2/1) $ from the classical
$r$-matrix including an odd generator is presented with its full Hopf algebra
structure. The deformation maps, twisting element and tensor operators are
considered for the deformed $ osp(2/1)$. It is also shown that its subalgebra
generated by the Borel subalgebra is self-dual.
| lt256 | arxiv_abstracts |
math/0301023 | A semialgebraic bijection from the field of p-adic numbers to itself minus
one point is constructed. Semialgebraic p-adic sets are classified up to
semialgebraic bijection. A cell decomposition theorem for restricted analytic
p-adic maps is proven, in analogy with the cell decomposition theorem for
polynomial maps by Denef. This cell decomposition is used to show that a
certain algebra (built up with analytic and subanalytic p-adic functions) is
closed under p-adic integration. This solves a conjecture of Denef on
parametrized analytic p-adic integrals. Local (analytic) singular series are
shown to be in this algebra. Subanalytic p-adic sets are classified up to
subanalytic bijection. Multivariate Kloosterman sums are studied modulo powers
of p. A qualitative decay rate is obtained when this power goes to infinity.
This is a multivariate analogue of a result of Igusa's. Also Presburger groups
are studied. A dimension for Presburger sets is defined, Presburger sets are
classified up to definable bijection, and elimination of imaginaries is proven.
Grothendieck rings of several classes of valued fields are calculated.
| 256 | arxiv_abstracts |
math/0301024 | We consider the eigenvalue equation for the largest eigenvalue of certain
kinds of non-compact linear operators given as the sum of a multiplication and
a kernel operator. It is shown that, under moderate conditions, such operators
can be approximated arbitrarily well by operators of finite rank, which
constitutes a discretization procedure. For this purpose, two standard methods
of approximation theory, the Nystr\"om and the Galerkin method, are
generalized. The operators considered describe models for mutation and
selection of an infinitely large population of individuals that are labeled by
real numbers, commonly called continuum-of-alleles (COA) models.
| lt256 | arxiv_abstracts |
math/0301025 | In this paper we define a new class of the quantum integrable systems
associated with the quantization of the cotangent bundle $T^*(GL(N))$ to the
Lie algebra $\frak{gl}_N$. The construction is based on the Gelfand-Zetlin
maximal commuting subalgebra in $U(\frak{gl}_N)$. We discuss the connection
with the other known integrable systems based on $T^*GL(N)$. The construction
of the spectral tower associated with the proposed integrable theory is given.
This spectral tower appears as a generalization of the standard spectral curve
for integrable system.
| lt256 | arxiv_abstracts |
math/0301026 | We explore certain restrictions on knots in the three-sphere which admit
non-trivial Seifert fibered surgeries. These restrictions stem from the
Heegaard Floer homology for Seifert fibered spaces, and hence they have
consequences for both the Alexander polynomial of such knots, and also their
knot Floer homology. In particular, we show that certain polynomials are never
the Alexander polynomials of knots which admit homology three-sphere Seifert
fibered surgeries. The knot Floer homology restrictions, on the other hand,
apply also in cases where the Alexander polynomial gives no information, such
as the Kinoshita-Terasaka knots.
| lt256 | arxiv_abstracts |
math/0301027 | We start the general structure theory of not necessarily semisimple finite
tensor categories, generalizing the results in the semisimple case (i.e. for
fusion categories), obtained recently in our joint work with D.Nikshych. In
particular, we generalize to the categorical setting the Hopf and quasi-Hopf
algebra freeness theorems due to Nichols-Zoeller and Schauenburg, respectively.
We also give categorical versions of the theory of distinguished group-like
elements in a finite dimensional Hopf algebra, of Lorenz's result on degeneracy
of the Cartan matrix, and of the absence of primitive elements in a finite
dimensional Hopf algebra in zero characteristic. We also develop the theory of
module categories and dual categories for not necessarily semisimple finite
tensor categories; the crucial new notion here is that of an exact module
category. Finally, we classify indecomposable exact module categories over the
simplest finite tensor categories, such as representations of a finite group in
positive characteristic, representations of a finite supergroup, and
representations of the Taft Hopf algebra.
| 256 | arxiv_abstracts |
math/0301028 | We characterize the connection between closed and $\sigma$-finite measures on
orthogonal projections of von Neumann algebras.
| lt256 | arxiv_abstracts |
math/0301029 | We introduce the p-adic analogue of Arakelov intersection theory on
arithmetic surfaces. The intersection pairing in an extension of the p-adic
height pairing for divisors of degree 0 in the form described by Coleman and
Gross. It also uses Coleman integration and is related to work of Colmez on
p-adic Green functions. We introduce the p-adic version of a metrized line
bundle and define the metric on the determinant of its cohomology in the style
of Faltings. It is possible to prove in this theory analogues of the Adjunction
formula and the Riemann-Roch formula.
| lt256 | arxiv_abstracts |
math/0301030 | Recently, I defined a squarefree module over a polynomial ring $S = k[x_1,
>..., x_n]$ generalizing the Stanley-Reisner ring $k[\Delta] = S/I_\Delta$ of a
simplicial complex $\Delta \subset 2^{1, ..., n}$. In this paper, from a
squarefree module $M$, we construct the $k$-sheaf $M^+$ on an $(n-1)$ simplex
$B$ which is the geometric realization of $2^{1, ..., n}$. For example,
$k[\Delta]^+$ is (the direct image to $B$ of) the constant sheaf on the
geometric realization $|\Delta| \subset B$.
We have $H^i(B, M^+) = [H^{i+1}_m(M)]_0$ for all $i > 0$. The
Poincare-Verdier duality for sheaves $M^+$ on $B$ corresponds to the local
duality for squarefree modules over $S$. For example, if $|\Delta|$ is a
manifold, then $k[\Delta]$ is a Buchsbaum ring whose canonical module is a
squarefree module giving the orientation sheaf of $|\Delta|$ with the
coefficients in $k$.
| lt256 | arxiv_abstracts |
math/0301031 | Recently N. Nitsure showed that for a coherent sheaf F on a noetherian scheme
the automorphism functor Aut_F is representable if and only if F is locally
free. Here we remove the noetherian hypothesis and show that the same result
holds for the endomorphism functor End_F even if one asks for representability
by an algebraic space.
| lt256 | arxiv_abstracts |
math/0301032 | We extend the Jacquet-Langlands'correspondence between the Hecke-modules of
usual and quaternionic modular forms, to overconvergent p-adic forms of finite
slope. We show that this correspondence respects p-adic families and is induced
by an isomorphism between some associated eigencurves.
| lt256 | arxiv_abstracts |
math/0301033 | We consider the two permutation statistics which count the distinct pairs
obtained from the last two terms of occurrences of patterns t_1...t_{m-2}m(m-1)
and t_1...t_{m-2}(m-1)m in a permutation, respectively. By a simple involution
in terms of permutation diagrams we will prove their equidistribution over the
symmetric group. As special case we derive a one-to-one correspondence between
permutations which avoid each of the patterns t_1...t_{m-2}m(m-1) in S_m and
such ones which avoid each of the patterns t_1...t_{m-2}(m-1)m. For m=3, this
correspondence coincides with the bijection given by Simion and Schmidt in
their famous paper on restricted permutations.
| lt256 | arxiv_abstracts |
math/0301034 | This work investigates the existence and properties of a certain class of
solutions of the Hill's equation truncated in the interval [tau, tau + L] -
where L = N a, a is the period of the coefficients in Hill's equation, N is a
positive integer and tau is a real number. It is found that the truncated
Hill's equation has two different types of solutions which vanish at the
truncation boundaries tau and tau + L: There are always N-1 solutions in each
stability interval of Hill's equation, whose eigen value is dependent on the
truncation length L but not on the truncation boundary tau; There is always one
and only one solution in each finite conditional instability interval of Hill's
equation, whose eigen value might be dependent on the boundary location tau but
not on the truncation length L.
The results obtained are applied to the physics problem on the electronic
states in one dimensional crystals of finite length. It significantly improves
many known results and also provides more new understandings on the physics in
low dimensional systems.
| 256 | arxiv_abstracts |
math/0301035 | In this short survey we look at a few basic features of p-adic numbers,
somewhat with the point of view of a classical analyst. In particular, with
p-adic numbers one has arithmetic operations and a norm, just as for real or
complex numbers.
| lt256 | arxiv_abstracts |
math/0301036 | Using noncocommutative coproduct properties of the quantum algebras, we
introduce and obtain, in a bipartite composite system, the Barut-Girardello
coherent state for the q-deformed $su_{q}(1,1)$ algebra. The quantum coproduct
structure ensures this normalizable coherent state to be entangled. The
entanglement disappears in the classical $q \to 1$ limit, giving rise to a
factorizable state.
| lt256 | arxiv_abstracts |
math/0301037 | In this paper we study the orthogonality conditions satisfied by Jacobi
polynomials $P_n^{(\alpha,\beta)}$ when the parameters $\alpha$ and $\beta$ are
not necessarily $>-1$. We establish orthogonality on a generic closed contour
on a Riemann surface. Depending on the parameters, this leads to either full
orthogonality conditions on a single contour in the plane, or to multiple
orthogonality conditions on a number of contours in the plane. In all cases we
show that the orthogonality conditions characterize the Jacobi polynomial
$P_n^{(\alpha, \beta)}$ of degree $n$ up to a constant factor.
| lt256 | arxiv_abstracts |
math/0301038 | We study the boundary of the nonnegative trigonometric polynomials from the
algebraic point of view. In particularly, we show that it is a subset of an
irreducible algebraic hypersurface and established its explicit form in terms
of resultants.
| lt256 | arxiv_abstracts |
math/0301039 | In this paper we study the possibility to define irreducible representations
of the symmetric groups with the help of finitely many relations. The existence
of finite bases is established for the classes of representations corresponding
to two-part partitions and to partitions from the fundamental alcove.
| lt256 | arxiv_abstracts |
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