gibb s functionals on subshifts we consider functionals on one dimensional subshifts which have prescribed randon nikodym derivative under transportation by conjugating homeomorphisms and investigate their relation to ruelle s transfer operator in particular we show that two sided functionals essentially are products of a functional which are supported on stable and unstable leaves we also prove the meromorphicity of the fourier transform of correlation functions for axioma follows in a more general setting

on the durfee size of kronecker products of characters of the symmetric group and its double covers abstract we investigate kronecker products of characters for s n and spin products for the double covers s n we give bounds on the durfee size of these products and provide information on the existence of special constituents also all products of durfee size at most 2 are explicitly described

convergence of dynamical zeta functions i study poles and zeros of zeta functions in one dimensional maps numerical and analytical arguments are given to show that the first pole of one such zeta function is given by the first zero ofanother zeta function this describes convergence of the calculations of the first zero which is generally the physically interesting quantity some remarks on how these results should generalize to zeta functions of dynamical systems with pruned symbolic dynamics and in higher dimensions follow

spectral decomposition of the renyi map the authors construct a generalized spectral decomposition of the frobenius perron operator of the general beta adic renyi map using a general iterative operator method applicable in principle to any mixing dynamical system they also explicitly define appropriate rigged hilbert spaces which provide mathematical meaning to the formally obtained spectral decomposition the explicit construction of the eigenvalues and eigenvectors allows one to show that the essential spectral radius of the frobenius perron operator decreases as the smoothness of the domain increases the reason for the change of the spectrum from the unit disk to isolated eigenvalues is the existence of coherent states of the frobenius perron operator which are not infinitely differentiable

the character covering numbers of the alternating groups we prove that the character covering number of an n 6 is n n where x is the smallest integer not less than x

multiple hooks in kronecker products of sn characters and a conjecture of berele and imbo abstract in this paper we prove a conjecture due to a berele and t imbo concerning the appearance of multiple hook characters as components in kronecker products of complex irreducible characters of the symmetric group our proof is constructive and the required multiple hooks are computed combinatorially

the quest for the ultimate anisotropic banach space we present a new scale u t s p with s t 0 and 1 le p infty of anisotropic banach spaces defined via paley littlewood on which the transfer operator associated to a hyperbolic dynamical system has good spectral properties when p 1 and t is an integer the spaces are analogous to the geometric spaces considered by gou ezel and liverani when p 1 and 1 1 p s t 0 t 1 p the spaces are somewhat analogous to the geometric spaces considered by demers and liverani in addition just like for the microlocal spaces defined by baladi tsujii the spaces u t s p are amenable to the kneading approach of milnor thurson to study dynamical determinants and zeta functions

a formula for the kronecker products of schur functions of hook shapes abstract the kronecker product of two homogeneous symmetric polynomials p 1 and p 2 is defined by means of the frobenius map by the formula p 1 p 2 f f 1 p 1 f 1 p 2 when p 1 and p 2 are schur functions s i and s j respectively then the resulting product s i s j is the frobenius characteristic of the tensor product of the irreducible representations of the symmetric group corresponding to the diagrams i and j taking the scaler product of s i s j with a third schur function s k gives the so called kronecker coefficient c ijk s i s j s k which gives the multiplicity of the representation corresponding to k in the tensor product in this paper we give an explicit formula for c ijk where i and j are hooks and k is arbitrary

a markov partition that reflects the geometry of a hyperbolic toral automorphism we show how to construct a markov partition that reflects the geometrical action of a hyperbolic automorphism of the n torus the transition matrix is the transpose of the matrix induced by the automorphism in u dimensional homology provided this is non negative here u denotes the expanding dimension that condition is satisfied at least for some power of the original automorphism under a certain non degeneracy condition on the galois group of the characteristic polynomial the nu rectangles are constructed by an iterated function system and they resemble the product of the projection of a u dimensional face of the unit cube onto the unstable subspace and the projection of minus the orthogonal n u dimensional face onto the stable subspace

the structure of random partitions of large integers random partitions of integers are treated in the case where all partitions of an integer are assumed to have the same probability the focus is on limit theorems as the number being partitioned approaches the limiting probability distribution of the appropriately normalized number of parts of some small size is exponential the large parts are described by a particular markov chain a central limit theorem and a law of large numbers holds for the numbers of intermediate parts of certain sizes the major tool is a simple construction of random partitions that treats the number being partitioned as a random variable the same technique is useful when some restriction is placed on partitions such as the requirement that all parts must be distinct

gibbs measures and power spectra for type i intermittent maps abstract gibbs measures of dynamical systems are investigated using the corresponding correlation function and power spectrum the method is shown to be equivalent to the concept of order q power spectra which has been developed previously a technique for the computation of these spectra is presented as an application we specially focus on the treatment of type i intermittent model maps and investigate their phase transition in the vicinity of the phase transition point the order q power spectra are shown to obey some typical scaling relation which is captured neither by the topological pressure nor by the ordinary power spectrum

periodic orbit expansions for classical smooth flows the authors derive a generalized selberg type zeta function for a smooth deterministic flow which relates the spectrum of an evolution operator to the periodic orbits of the flow this relation is a classical analogue of the quantum trace formulae and selberg type zeta functions

rectangular kronecker coefficients and plethysms in geometric complexity theory abstract we prove that in the geometric complexity theory program the vanishing of rectangular kronecker coefficients cannot be used to prove superpolynomial determinantal complexity lower bounds for the permanent polynomial moreover we prove the positivity of rectangular kronecker coefficients for a large class of partitions where the side lengths of the rectangle are at least quadratic in the length of the partition we also compare rectangular kronecker coefficients with their corresponding plethysm coefficients which leads to a new lower bound for rectangular kronecker coefficients moreover we prove that the saturation of the rectangular kronecker semigroup is trivial we show that the rectangular kronecker positivity stretching factor is 2 for a long first row and we completely classify the positivity of rectangular limit kronecker coefficients that were introduced by manivel in 2011

noncommutative algebras for hyperbolic diffeomorphisms the gibbs states of classical equilibrium statistical mechanics can be extended to states on non commutative algebras satisfying the kubo martin schwinger boundary condition this way of looking at gibbs states is applied here to the study of differentiable dynamical systems when some strong or weak hyperbolicity conditions are satisfied

lyapunov exponents and anomalous diffusion of a lorentz gas with infinite horizon using approximate zeta functions we compute the lyapunov exponent the generalized lyapunov exponents and the diffusion constant for a lorentz gas on a square lattice thus having infinite horizon approximate zeta functions written in terms of probabilities rather than periodic orbits are used in order to avoid the convergence problems of cycle expansions the emphasis is on the relation between the analytic structure of the zeta function where a branch cut plays an important role and the asymptotic dynamics of the system the lyapunov exponent for the corresponding map agrees with the conjectured limit map 2 log r c o r and we derive an approximate value for the constantc in good agreement with numerical simulations we also find a diverging diffusion constantd t logt and a phase transition for the generalized lyapunov exponents

smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics this paper reviews various applications of the theory of smooth dynamical systems to conceptual problems of nonequilibrium statistical mecanics we adopt a new point of view which has emerged progressively in recent years and which takes seriously into account the chaotic character of the microscopic time evolution the emphasis is on nonequilibrium steady states rather than the traditional approach to equilibrium point of view of boltzmann the nonequilibrium steady states in presence of a gaussian thermostat are described by srb measures in terms of these one can prove the gallavotti cohen fluctuation theorem one can also prove a general linear response formula and study its consequences which are not restricted to near equilibrium situations at equilibrium one recovers in particular the onsager reciprocity relations under suitable conditions the nonequilibrium steady states satisfy the pairing theorem of dettmann and morriss the results just mentioned hold so far only for classical systems they do not involve large size i e they hold without a thermodynamic limit

compact locally maximal hyperbolic sets for smooth maps fine statistical properties compact locally maximal hyperbolic sets are studied via geometrically defined functional spaces that take advantage of the smoothness of the map in a neighborhood of the hyperbolic set this provides a self contained theory that not only reproduces all the known classical results but also gives new insights on the statistical properties of these systems

galerkin differences for acoustic and elastic wave equations in two space dimensions abstract the galerkin difference method a finite element method built using standard galerkin projection techniques but employing nonstandard basis functions was originally developed for one space dimension in 1 here the method is extended to two space dimensions using a tensor product construction theoretical and computational evidence shows the method behaves as expected for the acoustic wave equation for the elastic wave equation the approximations are found to be at least as accurate as predicted but with a free surface the scheme may exhibit an unexpected superconvergence extension to curvilinear mapped grids is also considered for acoustics in all cases the use of a tensor product construction allows for efficient solution of the linear system involving the mass matrix which implies optimal linear time solutions with respect to the number of degrees of freedom

a low storage curvilinear discontinuous galerkin method for wave problems the low storage curvilinear discontinuous galerkin lsc dg method reduces the storage requirements for solving symmetric and linear linear symmetric conservation laws in curvilinear domains when compared to the standard discontinuous galerkin method we perform a semidiscrete a priori convergence analysis of lsc dg and determine sufficient conditions on sequences of meshes to guarantee the same rate of convergence as the original upwind dg method on curvilinear domains computational results confirm the optimal order convergence on sample mesh sequences that satisfy the sufficiency conditions additional results show that the sufficient conditions for optimal order convergence of lsc dg may also be necessary conditions

determination of resonance spectra for bound chaotic systems we consider the computation of the eigenvalues of the evolution operator the resonance spectrum by means of the zeros of a zeta function in particular we address the problems of applying this formalism to bound chaotic systems caused by e g intermittency and non completeness of the symbolic dynamics for bound intermittent systems we derive an approximation of the zeta function with the aid of this zeta function it is argued that bound systems with long time tails have branch cuts in the zeta function and traces of the evolution operator approaching unity as a power law we also show that the dominant time scale can be much longer than the period of the shortest periodic orbit as is for example the case for the hyperbola billiard isolated zeros of the zeta function for the hyperbola billiard are evaluated by means of a cycle expansion crucial for the success of this approach is the identification of a sequence of periodic orbit responsible for a logarithmic branch cut in the zeta function semiclassical implications are briefly discussed

zeta functions and transfer operators for piecewise monotone transformations given a piecewise monotone transformationt of the interval and a piecewise continuous complex weight functiong of bounded variation we prove that the ruelle zeta function z of t g extends meromorphically to z 1 where lim g tn 1 g tg 1 n and thatz is a pole of if and only ifz 1 is an eigenvalue of the corresponding transfer operator l we do not assume that l leaves a reference measure invariant

on the thermodynamic approach towards time series analysis a scheme is presented which allows for the computation of the topological pressure and power spectra associated with equilibrium measures from a given time series its convergence properties are demonstrated on numerical data sets obtained from the lozi and the henon map numerical evidence for sharp peaks in the power spectrum of the non hyperbolic phase is given and supported by estimates using periodic orbit expansions

kronecker products characters partitions and the tensor square conjectures abstract we study the remarkable saxl conjecture which states that tensor squares of certain irreducible representations of the symmetric groups s n contain all irreducibles as their constituents our main result is that for sufficiently large n they contain representations corresponding to young diagrams of hooks two row and diagrams obtained from hooks and two rows by adding a finite number of squares for that we develop a new sufficient condition for the positivity of kronecker coefficients in terms of characters and apply this tool using combinatorics of rim hook tableaux in combination with known results on unimodality of certain partition functions we also present connections and speculations on random characters of s n

resonances in chaotic dynamics we present a discussion and some numerical results on the actual possibility of making accessible by numerical techniques the complex singularities of the power spectrum resonances for a chaotic signal henon s transformation is investigated in detail showing that the position of the leading resonance in the complex frequency plane determines the kind of mixing rate in the time evolution

on transfer operators for continued fractions with restricted digits for any non empty subset i of the natural numbers let lambda i denote those numbers in the unit interval whose continued fraction digits all lie in i define the corresponding transfer operator formula for formula where re rs theta i is the abscissa of convergence of the series formula when acting on a certain hilbert space hi rs we show that the operator li rs is conjugate to an integral operator ki rs if furthermore rs is real then ki rs is selfadjoint so that li rs hi rs hi rs has purely real spectrum it is proved that li rs also has purely real spectrum when acting on various hilbert or banach spaces of holomorphic functions on the nuclear space c omega 0 1 and on the frechet space c infty 0 1 the analytic properties of the map rs m li rs are investigated for certain alphabets i of an arithmetic nature for example i primes i squares i an arithmetic progression i the set of sums of two squares it is shown that rs m li rs admits an analytic continuation beyond the half plane re rs theta i

boundary estimates for the elastic wave equation in almost incompressible materials we study the half plane problem for the elastic wave equation subject to a free surface boundary condition with particular emphasis on almost incompressible materials a normal mode analysis is developed to estimate the solution in terms of the boundary data showing that the problem is boundary stable the dependence on the material properties which is difficult to analyze by the energy method is made transparent by our estimates the normal mode technique is used to analyze the influence of truncation errors in a finite difference approximation our analysis explains why the number of grid points per wave length must be increased when the shear modulus mu becomes small compared to the first lame parameter lambda that is for almost incompressible materials when the surface waves are scaled to have unit wave length our analysis predicts that the grid size must be proportional to mu lambda 1 2 for a second order method for a fourth order method the grid size can be proportional t

explicit eigenvalue estimates for transfer operators acting on spaces of holomorphic functions abstract we consider transfer operators acting on spaces of holomorphic functions and provide explicit bounds for their eigenvalues more precisely if is any open set in c d and l is a suitable transfer operator acting on bergman space a 2 its eigenvalue sequence n l is bounded by n l a exp a n 1 d where a a 0 are explicitly given

correlation spectrum of quenched and annealed equilibrium states for random expanding maps we show that the integrated transfer operators for positively weighted independent identically distributed smooth expanding systems give rise to annealed equilibrium states for a new variational principle the unique annealed equilibrium state coincides with the unique annealed gibbs state using work of ruelle 1990 and fried 1995 on generalised fredholm determinants for transfer operators we prove that the discrete spectrum of the transfer operators coincides with the correlation spectrum of these invariant measures yielding exponential decay of correlations and with the poles of an annealed zeta function defined also for complex weights a modified integrated transfer operator is introduced which describes the relativised quenched states studied e g by kifer 1992 and conditions including srb ensuring coincidence of quenched and annealed states are given for small random perturbations we obtain stability results on the quenched and annealed measures and spectra by applying perturbative results of young and the author 1993

one dimensional gibbs states and axiom a diffeomorphisms on etudie la mecanique statistique en equilibre de systemes reticules classiques a une dimension a interactions a decroissance exponentielle pour de tels systemes les transformees de fourier des fonctions de correlation de paires sont meromorphes dans une bande et les residus des poles peuvent s exprimer en termes de distributions de gibbs on applique ces resultats aux diffeomorphismes de l axiome a de smale

the thermodynamic formalism for expanding maps letf x x be an expanding map of a compact space small distances are increased by a factor 1 a generating function z is defined which countsf periodic points with a weight one can express in terms of nonstandard fredholm determinants of certain transfer operators which can be studied by methods borrowed from statistical mechanics in this paper we review the spectral properties of the transfer operators and the corresponding analytic properties of z gibbs distributions and applications to julia sets are also discussed some new results are proved and some natural conjectures are proposed

on the kronecker product of sn characters abstract let be irreducible s n characters and assume appears in the kronecker product with maximal first part 1 then 1 min i i a similar result holds for the maximal first column we also give a recursive formula for as an application we show that if n 1 1 1 then s n 2 3 2 3 2 3 s n 1 where denotes the outer tensor product these results are applied to study the character where runs through the partitions with no more then k parts this character is closely related to the polynomial identities of the algebra of k k matrices

geometric complexity theory ii towards explicit obstructions for embeddings among class varieties in k d mulmuley and m sohoni siam j comput 31 2001 pp 496 526 henceforth referred to as part i we suggested an approach to the p vs np and related lower bound problems in complexity theory through geometric invariant theory in particular it reduces the arithmetic characteristic zero version of the np not subseteq p conjecture to the problem of showing that a variety associated with the complexity class np cannot be embedded in a variety associated with the complexity class p we shall call these class varieties associated with the complexity classes p and np this paper develops this approach further reducing these lower bound problems which are all nonexistence problems to some existence problems specifically to proving the existence of obstructions to such embeddings among class varieties it gives two results towards explicit construction of such obstructions the first result is a generalization of the borel weil theorem to a class of orbit closures which include class varieties the second result is a weaker form of a conjectured analogue of the second fundamental theorem of invariant theory for the class variety associated with the complexity class nc these results indicate that the fundamental lower bound problems in complexity theory are in turn intimately linked with explicit construction problems in algebraic geometry and representation theory the results here were announced in k d mulmuley and m sohoni in advances in algebra and geometry hyderabad 2001 hindustan book agency new delhi india 2003 pp 239 261

bounds on kronecker coefficients via contingency tables abstract we present both upper and lower bounds for the kronecker coefficients and the reduced kronecker coefficients in term of the number of certain contingency tables various asymptotic applications and generalizations are also presented

time compact difference methods for wave propagation in discontinuous media in b gustafsson and e mossberg siam j sci comput 26 2004 pp 259 271 a fourth order one step method was constructed for the solution of wave propagation problems the method is based on the first order system form of the pde and uses a staggered grid both in space and time the method was also applied with good results to a problem with discontinuous coefficients without using any special procedure across the discontinuity in this paper we will analyze a model problem from acoustics and demonstrate the theoretical foundation for this behavior furthermore we shall present more detailed numerical experiments which confirm the theoretical results

hook young diagrams with applications to combinatorics and to representations of lie superalgebras definition et etude des tableaux k l semi standards applications a la combinatoire et aux representations des superalgebres de lie des diagrammes de young a crochets