strong solvability of interval max plus systems and applications to optimal control abstract this paper considers the strong solvability of interval max plus systems the solvable entry and solvable interval are introduced to establish the necessary and sufficient condition for strong solvability of interval max plus systems it is proved that such a criterion is equivalent to the criterion proposed by cechlarova and cuninghame green the interval strong solution of interval max plus systems is described and the necessary and sufficient condition for uniquely strong solvability is given the method of testing strong solvability of interval max plus systems is constructive and leads to an effective algorithm to illustrate our results a workshop optimal control problem is addressed

equilibrium fluctuations for a non gradient energy conserving stochastic model in this paper we study the equilibrium energy fluctuation field of a one dimensional reversible non gradient model we prove that the limit fluctuation process is governed by a generalized ornstein uhlenbeck process whose covariances are given in terms of the diffusion coefficient

patterns in eigenvalues the 70th josiah willard gibbs lecture typical large unitary matrices show remarkable patterns in their eigenvalue distribution these same patterns appear in telephone encryption the zeros of riemann s zeta function a variety of physics problems and in the study of toeplitz operators this paper surveys these applications and what is currently known about the patterns

rate of convergence to gaussian measures on n spheres and jacobi hypergroups in this paper we prove central limit theorems of the following kind let s d c r d 1 be the unit sphere of dimension d 2 with uniform distribution d for each k n consider the isotropic random walk x n k n 0 on s d starting at the north pole with jumps of fixed sizes l x n k x k n 1 k for all n 1 then there is some k 0 d such that for all k k 0 d the distributions k of x k k have continuous bounded d densities f k moreover there is a known gaussian measure on s d with d density h such that f k h o 1 k and q k o 1 k for k where o 1 k is sharp we shall derive this rate of convergence in the central limit theorem more generally for a quite general class of isotropic random walks on compact symmetric spaces of rank one as well as for random walks on 0 whose transition probabilities are related to product linearization formulas of jacobi polynomials

kinetic hierarchy and propagation of chaos in biological swarm models we consider two models of biological swarm behavior in these models pairs of particles interact to adjust their velocities one to each other in the first process called bdg they join their average velocity up to some noise in the second process called cl one of the two particles tries to join the other one s velocity this paper establishes the master equations and bbgky hierarchies of these two processes it investigates the infinite particle limit of the hierarchies at large time scales it shows that the resulting kinetic hierarchy for the cl process does not satisfy propagation of chaos numerical simulations indicate that the bdg process has similar behavior to the cl process c 2012 elsevier b v all rights reserved

l2 lower bounds for a special class of random walks we investigate thel 2 speed of convergence to stationarity for a certain class of random walks on a compact connected lie group we give a lower bound on the number of stepsk necessary such that thek fold convolution power of the original step distribution has anl 2 density our method uses work by heckman on the asymptotics of multiplicities along a ray of representations several examples are presented

chaos ergodicity and equilibria in a quantum kac model abstract we introduce quantum versions of the kac master equation and the kac boltzmann equation we study the steady states of each of these equations and prove a propagation of chaos theorem that relates them the quantum kac master equation qkme describes a quantum markov semigroup p n t while the kac boltzmann equation describes a non linear evolution of density matrices on the single particle state space all of the steady states of the n particle quantum system described by the qkme are separable and thus the evolution described by the qkme is entanglement breaking the results set the stage for a quantitative study of approach to equilibrium in quantum kinetic theory and a quantitative study of the rate of destruction of entanglement in a class of quantum markov semigroups describing binary interactions

comparison techniques for random walk on finite groups we develop techniques for bounding the rate of convergence of a symmetric random walk on a finite group to the uniform distribution the techniques gives bounds on the second largest and other eigenvalues in terms of the eigenvalues of a comparison chain with known eigenvalues the techniques yield sharp rates for a host of previously intractable problems on the symmetric group

on the markov sequence problem for jacobi polynomials abstract we give a simple and entirely elementary proof of gasper s theorem on the markov sequence problem for jacobi polynomials it is based on the spectral analysis of an operator that arises in the study of a probabilistic model of colliding molecules introduced by marc kac and the methods developed here yield new estimates relevant to the collision model

proof of a mckean conjecture on the rate of convergence of boltzmann equation solutions the present work provides a definitive answer to the problem of quantifying relaxation to equilibrium of the solution to the spatially homogeneous boltzmann equation for maxwellian molecules under really mild conditions on the initial datum and a weak physically consistent angular cutoff hypothesis our main result theorem 1 contains the first precise statement that the total variation distance between the solution and the limiting maxwellian distribution admits an upper bound of the form c e varlambda b t varlambda b being the least negative eigenvalue of the linearized collision operator and c a constant depending only on the initial datum the validity of this quantification was conjectured about fifty years ago by henry p mckean as to the proof of our results we have taken as point of reference an analogy between the problem of convergence to equilibrium and the central limit theorem of probability theory highlighted by mckean

on the solvability of interval max min matrix equations abstract max min algebra is an algebraic structure in which classical addition and multiplication are replaced by maximum and minimum respectively the notation a x c b where a b and c are given interval matrices represents an interval max min matrix equation the paper deals with the solvability of interval matrix equations in max min algebra we define three types of solvability of interval max min matrix equations namely the strongly universal universal and weakly universal solvability we provide the equivalent conditions for each type of solvability that can be verified in polynomial times

a counter example to cercignani s conjecture for the d dimensional kac model kac s d dimensional model gives a linear many particle binary collision model from which under suitable conditions the celebrated boltzmann equation in its spatially homogeneous form arise as a mean field limit the ergodicity of the evolution equation leads to questions about the relaxation rate in hope that such a rate would pass on the boltzmann equation as the number of particles goes to infinity this program starting with kac and his one dimensional spectral gap conjecture at 1956 finally reached its conclusion in the maxwellian case in a series of papers by authors such as janvresse maslen carlen carvalho loss and geronimo but the hope to get a limiting relaxation rate for the boltzmann equation with this linear method was already shown to be unrealistic although the problem is still important and interesting due to its connection with the linearized boltzmann operator a less linear approach via a many particle version of cercignani s conjecture is the grounds for this paper in our paper we extend recent results by the author from the one dimensional kac model to the d dimensional one showing that the entropy entropy production ratio n still yields a very strong dependency in the number of particles of the problem when we consider the general case

interval systems of max separable linear equations for a given matrix interval a a a and a given vector interval b b b the notation a x b if max represents an interval system of linear max separable equations we define and characterize several types of solutions and solvability of interval systems of max separable linear equations

rates of convergence for gibbs sampling for variance component models this paper analyzes the gibbs sampler applied to a standard variance component model and considers the question of how many iterations are required for convergence it is proved that for k location parameters with j observations each the number of iterations required for convergence for large k and j is a constant times 1 log k log j this is one of the first rigorous a priori results about time to convergence for the gibbs sampler a quantitative version of the theory of harris recurrence for markov chains is developed and applied

the linear fokker planck equation for the ornstein uhlenbeck process as an almost nonlinear kinetic equation for an isolated n particle system it is long known that the fokker planck equation with prescribed constant coefficients of diffusion and linear friction describes the ensemble average of the stochastic evolutions in velocity space of a brownian test particle immersed in a heat bath of fixed temperature apparently it is not so well known that the same partial differential equation but now with constant coefficients which are functionals of the solution itself rather than being prescribed describes the kinetic evolution in the n limit of an isolated n particle system with certain stochastic interactions here we discuss in detail this recently discovered interpretation

cercignani s conjecture is sometimes true and always almost true we establish several new functional inequalities comparing boltzmann s entropy production functional with the relative h functional first we prove a longstanding conjecture by cercignani under the nonphysical assumption that the boltzmann collision kernel is superquadratic at infinity the proof rests on the method introduced in 39 combined with a novel use of the blachman stam inequality if the superquadraticity assumption is not satisfied then it is known that cercignani s conjecture is not true however we establish a slightly weakened version of it for all physically relevant collision kernels thus extending previous results from 39 finally we consider the entropy entropy production version of kac s spectral gap problem and obtain estimates about the dependence of the constants with respect to the dimension the first two results are sharp in some sense and the third one is likely to be too they contain all previously known entropy estimates as particular cases this gives a first coherent picture of the study of entropy production according to a program started by carlen and carvalho 12 ten years ago these entropy inequalities are one step in our study of the trend to equilibrium for the boltzmann equation in both its spatially homogeneous and spatially inhomogeneous versions

on the exponential fitting of composite multiderivative linear multistep methods exponentially fitted composite multiderivative linear multistep methods of orders up to 5 are derived a numerical investigation of these formulae indicates that they may be a stable for all choices of the fitting parameters an important aspect of the class of integration formulae considered is that each formula contains a built in local error estimate a facility lacking in most other exponentially fitted formulae some preliminary numerical results are given and these show that for problems for which exponential fitting is appropriate the new formulae provide a large saving in computational effort compared with conventional ones

deconvolution density estimation on so n this paper develops nonparametric deconvolution density estimation over so n the group of n x n orthogonal matrices of determinant 1 the methodology is to use the group and manifold structures to adapt the euclidean deconvolution techniques to this lie group environment this is achieved by employing the theory of group representations explicit to so n general consistency results are obtained with specific rates of convergence achieved under sufficient smoothness conditions application to empirical bayes prior estimation and inference is also discussed

analyzing glauber dynamics by comparison of markov chains a popular technique for studying random properties of a combinatorial set is to design a markov chain monte carlo algorithm for many problems there are natural markov chains connecting the set of allowable configurations which are based on local moves or glauber dynamics typically these single site update algorithms are difficult to analyze so often the markov chain is modified to update several sites simultaneously recently there has been progress in analyzing these more complicated algorithms for several important combinatorial problems in this work we use the comparison technique of diaconis and saloff coste to show that several of the natural single point update algorithms are efficient the strategy is to relate the mixing rate of these algorithms to the corresponding nonlocal algorithms which have already been analyzed this allows us to give polynomial time bounds for single point update algorithms for problems such as generating planar tilings and random triangulations of convex polygons we also survey several other comparison techniques along with specific applications which have been used in the context of estimating mixing rates of markov chains

control solvability of interval systems of max separable linear equations abstract for a given matrix interval a a a and a given vector interval b b b the notation a x b if max represents an interval system of linear max separable equations several types of solvability of interval systems are known we define control solvability weak control solvability and weak universal solvability and give necessary and sufficient condition for an interval system to be solvable according to each definition

spectral gap for the kac model with hard sphere collisions abstract we prove the analog of the kac conjecture for hard sphere collisions giving a computable close estimate on the spectral gap that is independent of the number of particles previous work has focused on the case in which the collision rates are independent of the particle velocities the case of so called maxwellian molecules the new methods introduced here allow us to deal with collision rates that are not bounded from below we also obtain information on the structure of the gap eigenfunction

cut off phenomenon in the uniform plane kac walk we consider an analogue of the kac random walk on the special orthogonal group so n so n in which at each step a random rotation is performed in a randomly chosen 2 plane of rnrn we obtain sharp asymptotics for the rate of convergence in total variance distance establishing a cut off phenomenon in the large nn limit in the special case where the angle of rotation is deterministic this confirms a conjecture of rosenthal ann probab 22 1994 398 423 under mild conditions we also establish a cut off for convergence of the walk to stationarity under the l2l2 norm depending on the distribution of the randomly chosen angle of rotation several surprising features emerge for instance it is sometimes the case that the mixing times differ in the total variation and l2l2 norms our estimates use an integral representation of the characters of the special orthogonal group together with saddle point analysis

sampling based approaches to calculating marginal densities abstract stochastic substitution the gibbs sampler and the sampling importance resampling algorithm can be viewed as three alternative sampling or monte carlo based approaches to the calculation of numerical estimates of marginal probability distributions the three approaches will be reviewed compared and contrasted in relation to various joint probability structures frequently encountered in applications in particular the relevance of the approaches to calculating bayesian posterior densities for a variety of structured models will be discussed and illustrated

on the trend to global equilibrium for spatially inhomogeneous kinetic systems the boltzmann equation as part of our study of convergence to equilibrium for spatially inhomogeneous kinetic equations started in 21 we derive estimates on the rate of convergence to equilibrium for solutions of the boltzmann equation like o t our results hold conditionally to some strong but natural estimates of smoothness decay at large velocities and strict positivity which at the moment have only been established in certain particular cases among the most important steps in our proof are 1 quantitative variants of boltzmann s h theorem as proven in 52 60 based on symmetry features hypercontractivity and information theoretical tools 2 a new quantitative version of the instability of the hydrodynamic description for non small knudsen number 3 some functional inequalities with geometrical content in particular the korn type inequality which we established in 22 and 4 the study of a system of coupled differential inequalities of second order by a treatment inspired from 21 we also briefly point out the particular role of conformal velocity fields when they are allowed by the geometry of the problem

stein s method and plancherel measure of the symmetric group we initiate a stein s method approach to the study of the plancherel measure of the symmetric group a new proof of kerov s central limit theorem for character ratios of random representations of the symmetric group on transpositions is obtained the proof gives an error term the construction of an exchangeable pair needed for applying stein s method arises from the theory of harmonic functions on bratelli diagrams we also find the spectrum of the markov chain on partitions underlying the construction of the exchangeable pair this yields an intriguing method for studying the asymptotic decomposition of tensor powers of some representations of the symmetric group

convergence of random walks on the circle generated by an irrational rotation fix 0 1 consider the random walk on the circle s1 which proceeds by repeatedly rotating points forward or backward with probability 1 2 by an angle 2 this paper analyzes the rate of convergence of this walk to the uniform distribution under discrepancy distance the rate depends on the continued fraction properties of the number 2 we obtain bounds for rates when is any irrational and a sharp rate when is a quadratic irrational in that case the discrepancy falls as k 1 2 up to constant factors where k is the number of steps in the walk this is the first example of a sharp rate for a discrete walk on a continuous state space it is obtained by establishing an interesting recurrence relation for the distribution of multiples of which allows for tighter bounds on terms which appear in the erd s turan inequality

entropy dissipation estimates in a zero range dynamics we study the exponential decay of relative entropy functionals for zero range processes on the complete graph for the standard model with rates increasing at infinity we prove entropy dissipation estimates uniformly over the number of particles and the number of vertices

interval max plus matrix equations abstract this paper deals with the solvability of interval matrix equations in max plus algebra max plus algebra is the algebraic structure in which classical addition and multiplication are replaced by and where a b max a b and a b a b the notation a x c b represents an interval max plus matrix equation where a b and c are given interval matrices we define four types of solvability of interval max plus matrix equations i e the tolerance weak tolerance left weak tolerance and right weak tolerance solvability we derive the necessary and sufficient conditions for checking each of them whereby all can be verified in polynomial time

spectral gap for stochastic energy exchange model with nonuniformly positive rate function we give a lower bound on the spectral gap for a class of stochastic energy exchange models in 2011 grigo et al introduced the model and showed that for a class of stochastic energy exchange models with a uniformly positive rate function the spectral gap of an n component system is bounded from below by a function of order n 2 in this paper we consider the case where the rate function is not uniformly positive for this case the spectral gap depends not only on n but also on the averaged energy e which is the conserved quantity under the dynamics under some assumption we obtain a lower bound of the spectral gap which is of order c e n 2 where c e is a positive constant depending on e as a corollary of the result a lower bound of the spectral gap for the mesoscopic energy exchange process of billiard lattice studied by gaspard and gilbert j stat mech theory exp 2008 2008 p11021 j stat mech theory exp 2009 2009 p08020 and the stick process studied by feng et al stochastic process appl 66 1997 147 182 are obtained

localized spherical deconvolution we provide a new algorithm for the treatment of the deconvolution problem on the sphere which combines the traditional svd inversion with an appropriate thresholding technique in a well chosen new basis we establish upper bounds for the behavior of our procedure for any l p loss it is important to emphasize the adaptation properties of our procedures with respect to the regularity sparsity of the object to recover as well as to inhomogeneous smoothness we also perform a numerical study which proves that the procedure shows very promising properties in practice as well

interval strong solutions of interval systems of max plus linear equations abstract this paper considers the existence and uniqueness of interval strong solutions of interval systems of max plus linear equations a necessary and sufficient condition for the existence of interval strong solutions is presented the proof of the existence of interval strong solutions is constructive and results in a formula for computing such solutions a necessary and sufficient condition for the uniqueness of interval strong solutions is established by testing the unique solvability of a finite number of subsystems rather than all subsystems on this basis a polynomial algorithm is developed to verify the uniqueness of interval strong solutions

the cut off phenomenon for random reflections ii complex and quaternionic cases in this paper we generalize the random reflections problem ono n considered in an earlier paper to the complex and quaternionic cases we give precise estimates on the speed of convergence to stationarity for specific examples of random walks onu n andsp n for which the one step distribution is a certain probability measure concentrated on reflections our results show that in both cases the so called cut off phenomenon occurs atk0 1 2n logn

random rotations characters and random walks on so n we analyze a random walk on the orthogonal group so n given by repeatedly rotating by a fixed angle through randomly chosen planes of r we derive estimates of the rate at which this random walk will converge to haar measure on so n using character theory and the upper bound lemma of diaconis and shashahani in some cases we are able to establish the existence of a cut off phenomenon for the random walk this is the first interesting such result on a non finite group

entropy production estimates for boltzmann equations with physically realistic collision kernels we establish strict entropy production bounds for the boltzmann equation with the hard sphere collision kernel using these entropy production bounds we prove results asserting that the rate at which strongl 1 convergence to equilibrium occurs is uniform in wide classes of initial data this extends our previous results in this direction which applied only to a very special collision kernel moreover the present results provide computable lower bounds compactness arguments are entirely avoided the uniformity is an important ingredient in our study of scaling limits of solutions of the non spatially homogeneous boltzmann equation and is the main focus of this paper however the results obtained here provide the only framework known to us in which one can obtain computable estimates on the time it takes a solution of the spatially homogeneous boltzmann equation with initial data far from equilibrium to reach any given small strongl 1 neighborhood of equilibrium

hydrodynamic limit of a disordered lattice gas we consider a model of lattice gas dynamics in d in the presence of disorder if the particle interaction is only mutual exclusion and if the disorder field is given by i i d bounded random variables we prove the almost sure existence of the hydrodynamical limit in dimension d 3 the limit equation is a non linear diffusion equation with diffusion matrix characterized by a variational principle

the random k cycle walk on the symmetric group we study the random walk on the symmetric group s n generated by the conjugacy class of cycles of length k we show that the convergence to uniform measure of this walk has a cut off in total variation distance after frac n k log n steps uniformly in k o n as n rightarrow infty the analysis follows from a new asymptotic estimation of the characters of the symmetric group evaluated at cycles

interval systems of max separable linear equations an interval system of equations is weakly solvable if at least one of its subsystems is solvable and it is strongly solvable if all its subsystems are solvable we give necessary and sufficient conditions enabling efficient testing of weak and strong solvability over max plus and max min algebras

cut off phenomenon for random walks on free orthogonal quantum groups we give bounds in total variation distance for random walks associated to pure central states on free orthogonal quantum groups as a consequence we prove that the analogue of the uniform plane kac walk on this quantum group has a cut off at n ln n 2 1 cos theta this is the first result of this type for genuine compact quantum groups we also obtain similar results for mixtures of rotations and quantum permutations

on the optimal choice of coefficients in a truncated wild sum and approximate solutions for the kac equation we study an approximate solution of the boltzmann problem for kac s caricature of a maxwellian gas by using a truncated and modified expansion of wild type we choose the coefficients in the wild sum approximation using a criterion based on exactly reproducing the behavior of the leading modes

kac s program in kinetic theory this paper is devoted to the study of propagation of chaos and mean field limits for systems of indistinguishable particles undergoing collision processes the prime examples we will consider are the many particle jump processes of kac and mckean kac in proceedings of the third berkeley symposium on mathematical statistics and probability 1954 1955 vol iii pp 171 197 1956 mckean in j comb theory 2 358 382 1967 giving rise to the boltzmann equation we solve the conjecture raised by kac proceedings of the third berkeley symposium on mathematical statistics and probability 1954 1955 vol iii pp 171 197 1956 motivating his program on the rigorous connection between the long time behavior of a collisional many particle system and the one of its mean field limit for bounded as well as unbounded collision rates

hyperbolic versus parabolic asymptotics in kinetic theory toward fluid dynamic models in this work we are interested in the hyperbolic limits in kinetic theory we propose a nonstandard scaling to be understood as a sort of intermediate hyperbolic limit which connects the macroscopic hyperbolic limiting behavior of the physical system with the microscopic properties usually obtained under parabolic scalings we present our main result by means of a general kinetic frame for the intermediate hyperbolic limit which covers some well known examples in kinetic theory vlasov poisson fokker planck systems and linear relaxation for boltzmann type equations in semiconductor theory among others we will also apply our methods to deal with the kac approach to boltzmann operators

logarithmic sobolev inequality for lattice gases with mixing conditions let mu lambda l lambda gc denote the grand canonical gibbs measure of a lattice gas in a cube of sizel with the chemical potential and a fixed boundary condition let mu lambda l n c be the corresponding canonical measure defined by conditioning mu lambda l lambda gc on sigma x in lambda eta x n consider the lattice gas dynamics for which each particle performs random walk with rates depending on near by particles the rates are chosen such that for everyn andl fixed mu lambda l n c is a reversible measure suppose that the dobrushin shlosman mixing conditions holds for mu l lambda gc forall chemical potentials we prove that smallint f log fd mu lambda l n c leqq const l 2 d sqrt f for any probability densityf with respect to mu lambda l n c here the constant is independent ofn orl andd denotes the dirichlet form of the dynamics the dependence onl is optimal

logarithmic sobolev inequality for generalized simple exclusion processes let be a probability measure on the set 0 1 r for some r and l a cube of width l in d denote by gc l the grand canonical product measure on the configuration space on l with as the marginal measure here the superscript indicates the grand canonical ensemble the canonical ensemble denoted by c l n is defined by conditioning gc l given the total number of particles to be n consider the exclusion dynamics where each particle performs random walk with rates depending only on the number of particles at the same site the rates are chosen such that for every n and l fixed the measure c l n is reversible we prove the logarithmic sobolev inequality in the sense that flogfd c l n for any probability density f with respect to c l n here the constant is independent of n or l and d denotes the dirichlet form of the dynamics the dependence on l is optimal

a probabilistic interpretation of the macdonald polynomials the two parameter macdonald polynomials are a central object of algebraic combinatorics and representation theory we give a markov chain on partitions of k with eigenfunctions the coefficients of the macdonald polynomials when expanded in the power sum polynomials the markov chain has stationary distribution a new two parameter family of measures on partitions the inverse of the macdonald weight rescaled the uniform distribution on cycles of permutations and the ewens sampling formula are special cases the markov chain is a version of the auxiliary variables algorithm of statistical physics properties of the macdonald polynomials allow a sharp analysis of the running time in natural cases a bounded number of steps suffice for arbitrarily large k

spectral gap for kac s model of boltzmann equation we consider a random walk on s n 1 the standard sphere of dimension n 1 generated by random rotations on randomly selected coordinate planes i j with 1 i j n this dynamic was used by marc kac as a model for the spatially homogeneous boltzmann equation we prove that the spectral gap on s n 1 is n 1 up to a constant independent of n

dynamic properties of the local linearization method for initial value problems some dynamic properties of the local linearization ll scheme for the numerical integration of initial value problems in ordinary differential equations odes are investigated specifically the general conditions under which this scheme preserves the stationary points and periodic orbits of the odes and the local stability at these steady states are studied these dynamic properties are also examined by means of numerical experiments and the results are compared with those achieved by other numerical schemes in addition a brief review of the computational implementations of the ll scheme is also presented

the fast johnson lindenstrauss transform and approximate nearest neighbors we introduce a new low distortion embedding of ell 2 d into ell p o log n p 1 2 called the fast johnson lindenstrauss transform fjlt the fjlt is faster than standard random projections and just as easy to implement it is based upon the preconditioning of a sparse projection matrix with a randomized fourier transform sparse random projections are unsuitable for low distortion embeddings we overcome this handicap by exploiting the heisenberg principle of the fourier transform i e its local global duality the fjlt can be used to speed up search algorithms based on low distortion embeddings in ell 1 and ell 2 we consider the case of approximate nearest neighbors in ell 2 d we provide a faster algorithm using classical projections which we then speed up further by plugging in the fjlt we also give a faster algorithm for searching over the hypercube

ornstein uhlenbeck limit for the velocity process of an n particle system interacting stochastically an n particle system with stochastic interactions is considered interactions are driven by a brownian noise term and total energy conservation is imposed the evolution of the system in velocity space is a diffusion on a 3n 1 dimensional sphere with radius fixed by the total energy in the n limit a finite number of velocity components are shown to evolve independently and according to an ornstein uhlenbeck process

transport inequalities gradient estimates entropy and ricci curvature we present various characterizations of uniform lower bounds for the ricci curvature of a smooth riemannian manifold m in terms of convexity properties of the entropy considered as a function on the space of probability measures on m as well as in terms of transportation inequalities for volume measures heat kernels and brownian motions and in terms of gradient estimates for the heat semigroup 2004 wiley periodicals inc

painting a graph with competing random walks let x1 x2 be independent random walks on zdn d 3 each starting from the uniform distribution initially each site of zdn is unmarked and whenever xi visits such a site it is set irreversibly to i the mean of ai the cardinality of the set ai of sites painted by i once all of zdn has been visited is 12nd by symmetry we prove the following conjecture due to pemantle and peres for each d 3 there exists a constant d such that limn var ai hd n 14 d where h3 n n4 h4 n n4 logn and hd n nd for d 5 we will also identify d explicitly and show that d 1 as d this is a special case of a more general theorem which gives the asymptotics of var ai for a large class of transient vertex transitive graphs other examples include the hypercube and the caley graph of the symmetric group generated by transpositions

semi classical limit for random walks let g u be a discrete symmetric random walk on a compact lie group g with step distribution and let t be the associated transition operator on l2 g the irreducibles vp of the left regular representation of g on l2 g are finite dimensional invariant subspaces for t and the spectrum of t is the union of the sub spectra a t 1vp on the irreducibles which consist of real eigenvalues apl ap dim vp our main result is an asymptotic expansion for the spectral measures dim vp m dim vp j along rays of representations in a positive weyl chamber t i e for sequences of representations kp k c n with k oo as a corollary we obtain some estimates on the spectral radius of the random walk we also analyse the fine structure of the spectrum for certain random walks on u n for which tm is essentially a direct sum of harper operators

interval max plus systems of linear equations abstract in this paper we shall deal with solvability of interval systems of linear equations in max plus algebra max plus algebra is an algebraic structure in which classical addition and multiplication are replaced by and where a b max a b a b a b the notation a x b represents an interval system of linear equations where a a a b b b are given interval matrix and interval vector respectively and a solution is from a given interval vector x x x we can define several types of solvability of interval systems in this paper we define six types of solvability of interval max plus systems of linear equations and give necessary and sufficient conditions for them

a spectral radius formula for the fourier transform on compact groups and applications to random walks abstract let g be a compact group not necessarily abelian let be its unitary dual and for f l1 g let f n f f denote n fold convolution of f with itself and f the fourier transform of f in this paper we derive the following spectral radius formula f n 1 1 n max r g f r where f r is the spectral radius of f r thereby extending the well known beurling gelfand spectral radius formula for the fourier transform on a locally compact abelian group we also establish a partial result in this direction for arbitrary regular borel measures on g as applications we give conceptual and rather short proofs of existing results concerning convergence in the total variation norm of random walks on compact groups and uniform convergence of the corresponding densities we also describe how these results may be transferred to homogeneous spaces

counting semiregular permutations which are products of a full cycle and an involution character theoretic methods and the group algebra of the symmetric group are used to derive properties of the number of permutations with only p cycles for an arbitrary but fixed p which are expressible as the product of a full cycle and a fixed point free involution this problem has application to single face embeddings of p regular graphs on surfaces of given genus

the phonon boltzmann equation properties and link to weakly anharmonic lattice dynamics for low density gases the validity of the boltzmann transport equation is well established the central object is the one particle distribution function f which in the boltzmann grad limit satisfies the boltzmann equation grad and much refined cercignani argue for the existence of this limit on the basis of the bbgky hierarchy for hard spheres at least for a short kinetic time span the argument can be made mathematically precise following the seminal work of lanford in this article a corresponding program is undertaken for weakly nonlinear both discrete and continuum wave equations our working example is the harmonic lattice with a weakly nonquadratic on site potential we argue that the role of the boltzmann f function is taken over by the wigner function which is a very convenient device to filter the slow degrees of freedom the wigner function so to speak labels locally the covariances of dynamically almost stationary measures one route to the phonon boltzmann equation is a gaussian decoupling which is based on the fact that the purely harmonic dynamics has very good mixing properties as a further approach the expansion in terms of feynman diagrams is outlined both methods are extended to the quantized version of the weakly nonlinear wave equation the resulting phonon boltzmann equation has been hardly studied on a rigorous level as one novel contribution we establish that the spatially homogeneous stationary solutions are precisely the thermal wigner functions for three phonon processes such a result requires extra conditions on the dispersion law we also outline the reasoning leading to fourier s law for heat conduction

propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous maxwellian gas we prove an inequality for the gain term in the boltzmann equation for maxwellian molecules that implies a uniform bound on sobolev norms of the solution provided the initial data has a finite norm in the corresponding sobolev space we then prove a sharp bound on the rate of exponential convergence to equilibrium in a weak norm these results are then combined using interpolation inequalities to obtain the optimal rate of exponential convergence in the strong l 1 norm as well as various sobolev norms these results are the first showing that the spectral gap in the linearized collision operator actually does govern the rate of approach to equilibrium for the full non linear boltzmann equation even for initial data that is far from equilibrium

a fifth order exponentially fitted formula a fifth order second derivative formula is developed for stiff systems of ordinary differential equations the exponentially fitted formulas possess a large region of absolute stability and numerical results demonstrate increased accuracy with the same computational effort when compared with similar fourth order formulas

exponentially fitted third derivative three step methods for numerical integration of stiff initial value problems in this paper we consider the construction of exponentially fitted third derivative three step method of order six for the numerical integration of stiff initial value problems the numerical analysis of the stability of the new method was discussed and some numerical experiments confirming theoretical expectation are provided finally the numerical results show that the new method is a stable and compete favorably with the existing methods in terms of efficiency and accuracy

the eigenvalues of kac s master equation in this paper we derive formulae for the eigenvalues and spectral gap of the master equation for general collision kernels we prove a conjecture of mark kac s on the existence of a spectral gap independent of the number of particles we relate the eigenvalues to the nonlinear eigenvalues that occur in the exact solutions of model boltzmann equations due to m ernst

poincare and logarithmic sobolev inequality for ginzburg landau processes in random environment we consider reversible conservative ginzburg landau processes in a random environment whose potential are bounded perturbations of the gaussian potential evolving on a d dimensional cube of length l we prove in all dimensions that the spectral gap of the generator and the logarithmic sobolev constant are of order l 2 almost surely with respect to the environment

dimension reduction and feedback stabilization for max plus linear systems and applications in vlsi array processors this paper investigates the dimension reduction and feedback stabilization of max plus linear systems and applies them to control and optimize the very large scale integration vlsi array processors we introduce the weakly similar relation between max plus matrices and the pseudoequivalent relation between autonomous max plus linear systems and point out that two systems are pseudoequivalent if and only if their state matrices are weakly similar the reduced system is defined by using the pseudoequivalence whose dimension is determined by the row rank of the original state matrix we focus on obtaining a reduced system which maintains the stability and retains the steady state period an algorithm of polynomial complexity is developed to find such a reduced system the reduced system is then used to design a state feedback controller to stabilize a max plus linear system finally we use the vlsi array processors as an example to demonstrate how the presented methods work in practical applications

global spectral gap for dirichlet kac random motions we prove that the global spectral gap for any dirichlet kac random motion is equal to the convergence rate of the limit motion

uniform contractivity in wasserstein metric for the original 1d kac s model we study here a very popular 1d jump model introduced by kac it consists of n velocities encountering random binary collisions at which they randomly exchange energy we show the uniform in n exponential contractivity of the dynamics in a non standard monge kantorovich wasserstein precisely the mkw metric of order 2 on the energy the result is optimal in the sense that for each n the contractivity constant is equal to the l 2 spectral gap of the generator associated to kac s dynamic as a corollary we get an uniform but non optimal contractivity in the mkw metric of order 4 we use a simple coupling that works better that the parallel one the estimates are simple and new to the best of our knowledge

cutoff for general spin systems with arbitrary boundary conditions the cutoff phenomenon describes a sharp transition in the convergence of a markov chain to equilibrium in recent work the authors established cutoff and its location for the stochastic ising model on the d dimensional torus n d for any d 1 the proof used the symmetric structure of the torus and monotonicity in an essential way here we enhance the framework and extend it to general geometries boundary conditions and external fields to derive a cutoff criterion that involves the growth rate of balls and the log sobolev constant of the glauber dynamics in particular we show there is cutoff for the stochastic ising model on any sequence of bounded degree graphs with subexponential growth under arbitrary external fields provided the inverse log sobolev constant is bounded for lattices with homogenous boundary such as all plus we identify the cutoff location explicitly in terms of spectral gaps of infinite volume dynamics on half plane intersections analogous results establishing cutoff are obtained for nonmonotone spin systems at high temperatures including the gas hard core model the potts model the antiferromagnetic potts model and the coloring model 2014 wiley periodicals inc