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

eigenvalue gaps for the cauchy process and a poincare inequality abstract a connection between the semigroup of the cauchy process killed upon exiting a domain d and a mixed boundary value problem for the laplacian in one dimension higher known as the mixed steklov problem was established in r banuelos t kulczycki the cauchy process and the steklov problem j funct anal 211 2004 355 423 from this a variational characterization for the eigenvalues n n 1 of the cauchy process in d was obtained in this paper we obtain a variational characterization of the difference between n and 1 we study bounded convex domains which are symmetric with respect to one of the coordinate axis and obtain lower bound estimates for 1 where is the eigenvalue corresponding to the first antisymmetric eigenfunction for d the proof is based on a variational characterization of 1 and on a weighted poincare type inequality the poincare inequality is valid for all symmetric stable processes 0 2 and any other process obtained from brownian motion by subordination we also prove upper bound estimates for the spectral gap 2 1 in bounded convex domains

approaches to collective decision making with fuzzy preference relations abstract recent experimental studies show that the predictive accuracy of many of the solution concepts derived from the collective decision making theory leaves much to be desired in a previous paper the author attempted to explain some of the inaccuracies in terms of the fuzzy indifference regions of the individuals participating in the voting game this paper gives straightforward generalizations of the solutions concepts in terms of the fuzzy social or individual preference relations it turns out that some of these new solution concepts cotain their nonfuzzy counterparts as subsets others in turn are subsets of their nonfuzzy counterparts we also discuss a method of aggregating individual nonfuzzy preferences so as to get a fuzzy social preference relation and furthermore a nonfuzzy social choice set

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

some properties of invariant measures of non symmetric dissipative stochastic systems we consider transition semigroups generated by stochastic partial differential equations with dissipative nonlinear terms we prove an integration by part formula and a logarithmic sobolev inequality for the invariant measure no symmetry or reversibility assumptions are made furthemore we prove some compactness results on the transition semigroup and on the embedding of the sobolev spaces based on the invariant measure we use these results to derive asymptotic properties for a stochastic reaction diffusion equation

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

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

approach to equilibrium of glauber dynamics in the one phase region i the attractive case various finite volume mixing conditions in classical statistical mechanics are reviewed and critically analyzed in particular somefinite size conditions are discussed together with their implications for the gibbs measures and for the approach to equilibrium of glauber dynamics inarbitrarily large volumes it is shown that dobrushin shlosman s theory ofcomplete analyticity and its dynamical counterpart due to stroock and zegarlinski cannot be applied in general to the whole one phase region since it requires mixing properties for regions ofarbitrary shape an alternative approach based on previous ideas of oliveri and picco is developed which allows to establish results on rapid approach to equilibrium deeply inside the one phase region in particular in the ferromagnetic case we considerably improve some previous results by holley and aizenman and holley our results are optimal in the sene that for example they show for the first time fast convergence of the dynamicsfor any temperature above the critical one for thed dimensional ising model with or without an external field in part ii we extensively consider the general case not necessarily attractive and we develop a new method based on renormalizations group ideas and on an assumption of strong mixing in a finite cube to prove hypercontractivity of the markov semigroup of the glauber dynamics

hardy littlewood theory for semigroups soit t t t 0 un semi groupe symetrique sousmarkovien et soit n 2 alors on a les proprietes equivalentes a dim t t n b il existe c 2 0 s t t t f c 2 t n 2 f 1 f l 1 x t 0 en fait a b meme dans le cas n 2

a monte carlo study of polymer adsorption random copolymers and random surfaces we consider a model of random copolymer adsorption in which an n edge self avoiding walk in three dimensions interacts with a plane defining a half space to which the walk is confined each vertex of the walk is randomly labelled a with probability p or b with probability 1 p and only vertices labelled a are attracted to the surface plane the system is quenched i e the labelling is fixed and then the thermodynamic properties are computed we use monte carlo methods to investigate the behaviour of this system we observe self averaging of the energy as n increases and investigate the location of the adsorption transition for various values of p in addition we compare the behaviour of this system with that of a homopolymer adsorbing at a randomly heterogeneous surface consisting of two types of sites only one of which interacts with the monomers of the polymer

pulling polymers adsorbed on a striped surface we consider the adsorption of a linear homopolymer on a striped surface where the monomers have different energies of interaction with each of the two kinds of stripe in addition the polymer is subject to a force that tends to desorb the polymer and the force can be applied at different angles we consider both a partially directed walk model and a self avoiding walk model we solve the problem for the partially directed walk problem essentially completely and give phase diagrams in the force temperature plane for various values of the direction in which the force is applied the phase diagrams show re entrance phenomena and for some force directions degenerate ground states we also present a low temperature treatment that helps to interpret the results from the combinatorial solution for the self avoiding walk model we present monte carlo results and show that the phase diagrams for the partially directed walk and self avoiding walk are qualitatively similar and at low temperatures are in quantitative agreement

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

sharp sobolev inequalities on the sphere and the moser trudinger inequality where de denotes normalized surface measure v is the conformal gradient and q 2n n 2 a modern folklore theorem is that by taking the infinitedimensional limit of this inequality one obtains the gross logarithmic sobolev inequality for gaussian measure which determines nelson s hypercontractive estimates for the hermite semigroup see 8 one observes using conformal invariance that the above inequality is equivalent to the sharp sobolev inequality on rn for which boundedness and extremal functions can be easily calculated using dilation invariance and geometric symmetrization the roots here go back to hardy and littlewood the advantage of casting the problem on the sphere is that the role of the constants is evident and one is led immediately to the conjecture that this inequality should hold whenever possible for example 2 q 0o if n 2 this is in fact true and will be demonstrated in section 2 a clear question at this point is what is the situation in dimension 2 two important arguments 25 26 27 dealt with this issue both motivated by geometric variational problems because q goes to infinity for dimension 2 the appropriate function space is the exponential class responding in part

the log sobolev inequality with quadratic interactions we assume one site measures without a boundary e x dx z that satisfies a log sobolev inequality we prove that if these measures are perturbed with quadratic interactions then the associated infinite dimensional gibbs measure on the lattice always satisfies a log sobolev inequality furthermore we present examples of measures that satisfy the inequality with a phase that goes beyond convexity at infinity

spectral gaps for spin systems some non convex phase examples abstract we prove that a convex phase may be perturbed into a non convex phase preserving the spectral gap properties of the unbounded spin system with nearest neighbour interaction associated to this potential the proof is based on helffer s method hat reduces the spectral properties of the unbounded spin system to some uniform spectral gap of the one dimensional phase we then make use of hardy s criterion for poincare inequalities on the real line to construct our examples

chain polymers near an adsorbing surface we present improved estimates of critical exponents for 3d polymers near a plane adsorbing surface the improvements were possible by simulating rather long chains up to n 2000 by means of a new algorithm apart from minor but significant adjustments in the location of the tricritical point and in the various gamma exponents our main result is that the crossover exponent for critical adsorption is compatible with phi 1 2 this is much lower than previous estimates and suggests that this exponent might be superuniversal

uniformly integrable operators and large deviations for markov processes abstract in this paper we introduce and investigate the notion of uniformly integrable operators on l p e its relations to classical compactness and hypercontractivity are exhibited several consequences of this notion are established such as perron frobenius type theorems independence on p of the spectral radius in l p p continuity of spectral radius and especially the existence of spectral gap in the irreducible case we also present some infinitesimal criteria ensuring the uniform integrability of a positive semigroup or of its resolvent for a essentially irreducible markov process we show that the uniform integrability in l p of some type of resolvent associated with the transition semigroup implies the large deviation principle of level 3 with some rate function given by a modified donsker varadhan entropy functional we also prove that the uniform integrability condition becomes even necessary in the symmetric case finally we present several applications of our results to feynman kac semigroups to the thermodynamical limits of grand ensembles and to non symmetric markov processes given by girsanov s formula

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

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

spectral gap and logarithmic sobolev inequality for kawasaki and glauber dynamics we prove that the spectral gap of the kawasaki dynamics shrink at the rate of 1 l2 for cubes of sizel provided that some mixing conditions are satisfied we also prove that the logarithmic sobolev inequality for the glauber dynamics in standard cubes holds uniformly in the size of the cube if the dobrushin shlosman mixing condition holds for standard cubes

a self avoiding walk model of random copolymer adsorption there is an error in the proof of lemma 3 4 which can be remedied by using 3 22 and 3 13 directly the proof of lemma 3 2 can also be simplified please see pdf for details

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

a fuzzy envelope for hesitant fuzzy linguistic term set and its application to multicriteria decision making decision making is a process common to human beings the uncertainty and fuzziness of problems demand the use of the fuzzy linguistic approach to model qualitative aspects of problems related to decision the recent proposal of hesitant fuzzy linguistic term sets supports the elicitation of comparative linguistic expressions in hesitant situations when experts hesitate among different linguistic terms to provide their assessments the use of linguistic intervals whose results lose their initial fuzzy representation was introduced to facilitate the computing processes in which such expressions are used the aim of this paper is to present a new representation of the hesitant fuzzy linguistic term sets by means of a fuzzy envelope to carry out the computing with words processes this new fuzzy envelope can be directly applied to fuzzy multicriteria decision making models an illustrative example of its application to a supplier selection problem through the use of fuzzy topsis is presented

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

hypercontractivity and spectral gap of symmetric diffusions with applications to the stochastic ising models abstract we describe some results relating the spectral gap and logarithmic sobolev constants we work initially in the context of symmetric diffusions on a finite dimensional manifold and later apply our results to the analysis of certain infinite systems of mildly interacting diffusions

completely analytical interactions constructive description an interactionu is called a completely analytical ca interaction if it satisfies one of 12 given conditions formulated in terms of analyticity properties of the partition functions zv u or correlation decay or truncated correlation bounds or asymptotic behavior of ln zv u v the 12 conditions are presented together with part of the proof of their equivalence the main result of the paper is that each condition is constructive in the following sense instead of checking it in all finite volumesv v it is enough to consider only a finite amount of volumes with restricted size in particular the partition functions z v u for the complex perturbationsu ofu do not vanish for all v v and all with provided this is true only forv with diam v c and but with

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

analytic inequalities isoperimetric inequalities and logarithmic sobolev inequalities abstract there is a simple equivalence between isoperimetric inequalities in riemannian manifolds and certain analytic inequalities on the same manifold more extensive than the familiar equivalence of the classical isoperimetric inequality in euclidean space and the associated sobolev inequality by an isoperimetric inequality in this connection we mean any inequality involving the riemannian volume and riemannian surface measure of a subset and its boundary respectively we exploit the equivalence to give log sobolev inequalities for riemannian manifolds some applications to schrodinger equations are also given

hypercontractivity for the heat semigroup for ultraspherical polynomials and on the n sphere a logarithmic sobolev inequality with respect to the probability measure a 1 x2 12 dx on 1 1 is proved for all 12 from this inequality sharp hypercontractive estimates are derived for the heat semigroups for ultraspherical polynomials and on the n sphere

on an integral criterion for hypercontractivity of diffusion semigroups and extremal functions abstract in the line of investigation of the works by d bakry and m emery lecture notes in math vol 1123 pp 175 206 springer verlag new york berlin 1985 and o s rothaus j funct anal 42 1981 102 109 j funct anal 65 1986 358 367 we study an integral inequality behind the 2 criterion of d bakry and m emery see previous reference and its applications to hypercontractivity of diffusion semigroups with in particular a short proof of the hypercontractivity property of the ornstein uhlenbeck semigroup our exposition unifies in a simple way several previous results interpolating smoothly from the spectral gap inequalities to logarithmic sobolev inequalities and even true sobolev inequalities we examine simultaneously the extremal functions for hypercontractivity and logarithmic sobolev inequalities of the ornstein uhlenbeck semigroup and heat semigroup on spheres

multicriteria decision analysis with fuzzy pairwise comparisons abstract we present a modification of the fuzzy multi criteria method proposed by van laarhoven and pedrycz 1983 first the weights of the decision criteria are calculated by the minimization of a logarithmic regression function next the weights of the decision alternatives are calculated for each criterion separately lastly the fuzzy final scores of the alternatives are determined by an appropriate aggregationm of the calculated weights we will show that our modified method in contrast with the original one yields weights which are optimal with respect to the logarithmic regression function we also present a version of our method with geometric ratio scales rather than saaty s equi distant scale to quantify the gradations of human comparative judgement this approach also enables us to develop an efficient procedure for investigating the scale sensitivity of the calculated weights and the final scores of the alternatives

the logarithmic sobolev inequality for continuous spin systems on a lattice abstract in this article we give cf theorem 1 6 a sufficient condition on a local specification for the associated gibbs state to satisfy a logarithmic sobolev inequality in the final section cf theorem 3 10 we relate our condition to the type of mixing condition introduced by dobrushin and shlosman in connection with their work on complete analyticity

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

hypercontractivity and the bakry emery criterion for compact lie groups on etudie l ingredient essentiel du critere de bakry emery on utilise ce critere pour obtenir des estimations ameliorees de la constante hypercontractive

stochastic analysis on product manifolds dirichlet operators on differential forms abstract we define a de rham complex over a product manifold infinite product of compact manifolds and dirichlet operators on differential forms associated with differentiable measures in particular gibbs measures which generalize the notions of bochner and de rham laplacians we give probabilistic representations for corresponding semigroups and study properties of the corresponding stochastic dynamics

fuzzy credibility relation method for multiple criteria decision making problems combining the concepts of fuzzy set theory entropy ideal and grey relation analysis a fuzzy grey relation method for multiple criteria decision making problems is proposed first triangular fuzzy numbers and linguistic values characterized by triangular fuzzy numbers are used to evaluate the importance weights of all criteria and the superiority of all alternatives versus various criteria above the alternative level then the concept of entropy is utilized to solve the adjusted integration weight of all objective criteria above the alternative level furthermore using the concepts of ideal the grey ration grades of various alternatives versus ideal solution are ranked to determine the best alternative finally a numerical example of selecting most appropriate company to build a new highway is used to demonstrate the applicability of proposed method the study results show that this method is an effective means for tackling mcdm problems in fuzzy and grey environments

coercive inequalities for hormander type generators in infinite dimensions abstract we prove log sobolev inequalities for hormander type generators in infinite dimensions and prove the strong exponential decay to equilibrium for corresponding semigroups

a fuzzy multi criteria decision making model by associating technique for order preference by similarity to ideal solution with relative preference relation generally classical multi criteria decision making mcdm methods were extended to encompass uncertainty and vagueness of messages under fuzzy environment for solving decision making problems especially for technique for order preference by similarity to ideal solution topsis in the fuzzy extension of topsis fuzzy numbers comparison and aggregation based on fuzzy preference relation are important issues to compute distance values between alternatives and ideal or anti ideal solution or rank feasible alternatives because lots of messages are reserved by fuzzy preference relation however fuzzy preference relation on pair wise comparison is commonly too complex to calculate to avoid the drawback we use a relative preference relation improved from fuzzy preference relation in the fuzzy extension of topsis for computing distance values between alternatives and ideal or anti ideal solution or obtaining relative closeness coefficients of alternatives thus the relative preference relation on fuzzy numbers will be associated with topsis under fuzzy environment to develop a fuzzy multi criteria decision making fmcdm model through the association above fmcdm problems can be easily solved by the model further we compare the proposed model with other methods to demonstrate the model s feasibility and rationality

relaxed criteria of the dobrushin shlosman mixing condition an interacting particle system glauber dynamics which evolves on a finite subset in thed dimensional integer lattice is considered it is known that a mixing property of the gibbs state in the sense of dobrushin and shlosman is equivalent to several very strong estimates in terms of the glauber dynamics we show that similar but seemingly much milder estimates are again equivalent to the dobrushin shlosman mixing condition hence to the original ones found by stroock and zegarlinski this may be understood as the absence of intermediate speed of convergence to equilibrium

the equivalence of the logarithmic sobolev inequality and the dobrushin shlosman mixing condition given a finite range lattice gas with a compact continuous spin space it is shown cf theorem 1 2 that a uniform logarithmic sobolev inequality cf 1 4 holds if and only if the dobrushin shlosman mixing condition cf 1 5 holds as a consequence of our considerations we also show cf theorems 3 2 and 3 6 that these conditions are equivalent to a statement about the uniform rate at which the associated glauber dynamics tends to equilibrium in this same direction we show cf theorem 3 19 that these ideas lead to a surprisingly strong large deviation principle for the occupation time distribution of the glauber dynamics

on a constrained 2 d navier stokes equation the planar navier stokes equation exhibits in absence of external forces a trivial asymptotics in time nevertheless the appearence of coherent structures suggests non trivial intermediate asymptotics which should be explained in terms of the equation itself motivated by the separation of the different time scales observed in the dynamics of the navier stokes equation we study the well posedness and asymptotic behaviour of a constrained equation which neglects the variation of the energy and moment of inertia

ranking alternatives with fuzzy weights using maximizing set and minimizing set ranking alternatives both qualitative as well as quantitative in a multicriterion environment employing experts opinion preference structure using fuzzy numbers and linguistic variables are presented in this paper fuzzy weights wj i of alternatives a i are computed using standard fuzzy arithmetic concept of maximizing set and minimizing set is introduced to decide total utility or ordering value of each of the alternatives a numerical example is provided at the end to illustrate the method 1999 published by elsevier science b v all rights reserved

uniqueness of gibbs states for quantum lattice systems we prove uniqueness of euclidean gibbs states for certain quantum lattice systems with unbounded spins we use dobrushin s uniqueness criterion the necessary estimates for the vasershtein distance between the corresponding one point conditional distributions with boundary conditions differing only at one side are obtained by proving a log sobolev inequality on the infinite dimensional single spin loop spaces some important classes of concrete examples to which all this applies are discussed

poincare inequality for weighted first order sobolev spaces on loop spaces abstract let e be the loop space over a compact connected riemannian manifold with a torsion skew symmetric connection let l d be the ornstein uhlenbeck operator on a nonempty connected component d of the loop space e and let v d r be the restriction on d of the potential in the logarithmic sobolev inequality found by l gross on the loop group by s aida and by f z gong and z m ma on the loop space respectively we prove that the schrodinger operator l v l d v always has a spectral gap at the bottom 0 v of its spectrum and thus has its ground state transformed operator 1 l v 0 v where is the unique ground state of l v in particular our result proves l gross s conjecture about the existence of a spectral gap for the ground state transform of the schrodinger operator studied by him on the loop group in addition in all the above cases we identify the domain of the dirichlet forms associated with the ground state transforms as weighted first order sobolev spaces with weight given by 2 thus establishing a poincare inequality for them all these results are consequences from some new results in this paper on dirichlet forms characterizing certain classes with spectral gaps and from results by s aida and m hino

gaussian kernels have only gaussian maximizers a gaussian integral kernel g x y on rn x rn is the exponential of a quadratic form in x and y the fourier transform kernel is an example the problem addressed here is to find the sharp bound of g as an operator from l p rp to l q rn and to prove that the l p rn functions that saturate the bound are necessarily gaussians this is accomplished generally for 1 q in some special cases besides greatly extending previous results in this area the proof technique is also essentially different from earlier ones a corollary of these results is a fully multidimensional multilinear generalization of young s inequality

generalizing topsis for fuzzy multiple criteria group decision making in this paper we generalize topsis to fuzzy multiple criteria group decision making fmcgdm in a fuzzy environment topsis is one of the well known methods for multiple criteria decision making mcdm most of the steps of topsis can be easily generalized to a fuzzy environment except max and min operations in finding the ideal solution and negative ideal solution thus we propose two operators up and lo which satisfy the partial ordering relation on fuzzy numbers to the generalization of topsis in generalized topsis these two operations up and lo are employed to find ideal and negative ideal solutions under a fuzzy environment then the fmcgdm problem can be solved effectively and efficiently

convex sobolev inequalities derived from entropy dissipation we study families of convex sobolev inequalities which arise as entropy dissipation relations for certain linear fokker planck equations extending the ideas recently developed by the first two authors a refinement of the bakry emery method is established which allows us to prove non trivial inequalities even in situations where the classical bakry emery criterion fails the main application of our theory concerns the linearized fast diffusion equation in dimensions d 1 which admits a poincare but no logarithmic sobolev inequality we calculate bounds on the constants in the interpolating convex sobolev inequalities and prove that these bounds are sharp on a specified range in dimension d 1 our estimates improve the corresponding results that can be obtained by the measure theoretic techniques of barthe and roberto as a by product we give a short and elementary alternative proof of the sharp spectral gap inequality first obtained by denzler and mccann in further applications of our method we prove convex sobolev inequalities for a mean field model for the redistribution of wealth in a simple market economy and the lasota model for blood cell production

gradient flow and entropy inequalities for quantum markov semigroups with detailed balance abstract we study a class of ergodic quantum markov semigroups on finite dimensional unital c algebras these semigroups have a unique stationary state and we are concerned with those that satisfy a quantum detailed balance condition with respect to we show that the evolution on the set of states that is given by such a quantum markov semigroup is gradient flow for the relative entropy with respect to in a particular riemannian metric on the set of states this metric is a non commutative analog of the 2 wasserstein metric and in several interesting cases we are able to show in analogy with work of otto on gradient flows with respect to the classical 2 wasserstein metric that the relative entropy is strictly and uniformly convex with respect to the riemannian metric introduced here as a consequence we obtain a number of new inequalities for the decay of relative entropy for ergodic quantum markov semigroups with detailed balance

a very simple proof of the lsi for high temperature spin systems abstract we present a very simple proof that the o n model satisfies a uniform logarithmic sobolev inequality lsi if the difference between the largest and the smallest eigenvalue of the coupling matrix is less than n this condition applies in particular to the sk spin glass model at inverse temperature 1 4 it is the first result of rapid relaxation for the sk model and requires significant cancellations between the ferromagnetic and anti ferromagnetic spin couplings that cannot be obtained by existing methods to prove log sobolev inequalities the proof also applies to more general bounded and unbounded spin systems it uses a single step of zero range renormalisation and bakry emery theory for the renormalised measure

a monte carlo study of adsorption of random copolymers on random surfaces we study the adsorption problem of a random copolymer on a random surface in which a self avoiding walk in three dimensions interacts with a plane defining a half space to which the walk is confined each vertex of the walk is randomly labelled a with probability pp or b with probability 1 pp and only vertices labelled a are attracted to the surface plane each lattice site on the plane is also labelled either a with probability ps or b with probability 1 ps and only lattice sites labelled a interact with the walk we study two variations of this model in the first case the a vertices of the walk interact only with the a sites on the surface in the second case the constraint of selective binding is removed that is any contact between the walk and the surface that involves an a labelling either from the surface or from the walk is counted as a visit to the surface the system is quenched in both cases i e the labellings of the walk and of the surface are fixed as thermodynamic properties are computed we present monte carlo simulation results which provide evidence for second order transitions in both cases we observe that in both cases the adsorption location depends on both ps and pp the dilution of the interactive sites on the surface and the walk we compare critical properties for the two cases and find that the adsorption location varies for different models while the crossover exponent is independent of the details of the models

dirichlet forms and dirichlet operators for gibbs measures of quantum unbounded spin systems essential self adjointness and log sobolev inequality for each 0 1 we consider the dirichlet form mu gamma and the associated dirichlet operator h mu gamma for the gibbs measure of quantum unbounded spin systems interacting via superstable and regular potential the gibbs measure is related to the gibbs state of the system via a functional euclidean integral procedure the configuration space for the spin systems is given by omega e mathbb z v e left omega in c left left 0 1 right mathbb r d right omega left 0 right omega left 1 right right we formulate dirichlet forms in the framework of rigged hilbert spaces which are related to the space under appropriate conditions on the potential we show that the dirichlet operator h mu left gamma right is essentially self adjoint on the domain of smooth cylinder functions we give sufficient conditions on the potential so that the corresponding gibbs measure is uniformly log concave ulc this property gives the spectral gap of the dirichlet operator h mu left gamma right at the lower end of the spectrum furthermore we prove that under the conditions of ulc the unique gibbs measure satisfies the log sobolev inequality ls we use an approximate argument used in the study of the same subjects for loop spaces which in turn is a modification of the method originally developed by s albeverio yu g kondratiev and m rockner

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

the log sobolev inequality for unbounded spin systems abstract in this note we would like to give a proof of log sobolev inequality for un bounded spin systems with weaker assumptions on the potentials than previously obtained

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

critical ising on the square lattice mixes in polynomial time the ising model is widely regarded as the most studied model of spin systems in statistical physics the focus of this paper is its dynamic stochastic version the glauber dynamics introduced in 1963 and by now the most popular means of sampling the ising measure intensive study throughout the last three decades has yielded a rigorous understanding of the spectral gap of the dynamics on mathbb z 2 everywhere except at criticality while the critical behavior of the ising model has long been the focus for physicists mathematicians have only recently developed an understanding of its critical geometry with the advent of sle cle and new tools to study conformally invariant systems

on the structure of almost einstein manifolds in this paper we study the structure of the limit space of a sequence of almost einstein manifolds which are generalizations of einstein manifolds roughly speaking such manifolds are the initial manifolds of some normalized ricci flows whose scalar curvatures are almost constants over space time in the l1 sense ricci curvatures are bounded from below at the initial time under the non collapsed condition we show that the limit space of a sequence of almost einstein manifolds has most properties which is known for the limit space of einstein manifolds as applications we can apply our structure results to study the properties of kahler manifolds

approach to equilibrium of glauber dynamics in the one phase region ii the general case we develop a new method based on renormalization group ideas block decimation procedure to prove under an assumption of strong mixing in a finite cube 0 a logarithmic sobolev inequality for the gibbs state of a discrete spin system as a consequence we derive the hypercontractivity of the markov semigroup of the associated glauber dynamics and the exponential convergence to equilibrium in the uniform norm in all volumes multiples of the cube 0

uniform and l 2 convergence in one dimensional stochastic ising models we study the rate of convergence to equilibrium of one dimensional stochastic ising models with finite range interactions we donot assume that the interactions are ferromagnetic or that the flip rates are attractive the infinitesimal generators of these processes all have gaps between zero and the rest of their spectra we prove that if one of these processes is observed by means of local observables then the convergence is seen to be exponentially fast with an exponent that is any number less than the spectral gap moreover this exponential convergence is uniform in the initial configuration

absence of second order phase transitions in the dobrushin uniqueness region the dobrushin uniqueness theorem assures that in a very large class of high temperature classical statistical mechanical lattice models with short or long range many body interactions and arbitrary compact spin space there are no first order phase transitions it will be shown that for the same class of interactions there are also no second order phase transitions

dobrushin s uniqueness for quantum lattice systems with nonlocal interaction based on dobrushin s fundamental criterion we prove uniqueness of euclidean gibbs states for a certain class of quantum lattice systems with unbounded spins nonharmonic pair potentials and infinite radius of interaction the necessary estimates on dobrushin s coefficients are obtained from the log sobolev inequality which holds for the one point conditional distributions on the infinite dimensional single spin loop spaces

asymptotics of the fast diffusion equation via entropy estimates we consider non negative solutions of the fast diffusion equation ut um with m 0 1 in the euclidean space mathbb r d d 3 and study the asymptotic behavior of a natural class of solutions in the limit corresponding to t for m mc d 2 d or as t approaches the extinction time when m mc for a class of initial data we prove that the solution converges with a polynomial rate to a self similar solution for t large enough if m mc or close enough to the extinction time if m mc such results are new in the range m mc where previous approaches fail in the range mc m 1 we improve on known results

extensions of the topsis for group decision making under fuzzy environment the aim of this paper is to extend the topsis to the fuzzy environment owing to vague concepts frequently represented in decision data the crisp value are inadequate to model real life situations in this paper the rating of each alternative and the weight of each criterion are described by linguistic terms which can be expressed in triangular fuzzy numbers then a vertex method is proposed to calculate the distance between two triangular fuzzy numbers according to the concept of the topsis a closeness coefficient is defined to determine the ranking order of all alternatives by calculating the distances to both the fuzzy positive ideal solution fpis and fuzzy negative ideal solution fnis simultaneously finally an example is shown to highlight the procedure of the proposed method at the end of this paper

optimal concentration for su 1 1 coherent state transforms and an analogue of the lieb wehrl conjecture for su 1 1 we derive a lower bound for the wehrl entropy in the setting of su 1 1 for asymptotically high values of the quantum number k this bound coincides with the analogue of the lieb wehrl conjecture for su 1 1 coherent states the bound on the entropy is proved via a sharp norm bound the norm bound is deduced by using an interesting identity for fisher information of su 1 1 coherent state transforms on the hyperbolic plane mathbb h 2 and a new family of sharp sobolev inequalities on mathbb h 2 to prove the sharpness of our sobolev inequality we need to first prove a uniqueness theorem for solutions of a semi linear poisson equation which is actually the euler lagrange equation for the variational problem associated with our sharp sobolev inequality on mathbb h 2 uniqueness theorems proved for similar semi linear equations in the past do not apply here and the new features of our proof are of independent interest as are some of the consequences we derive from the new family of sobolev inequalities

fuzzy mcdm based on ideal and anti ideal concepts this paper presents a novel fuzzy multiple criteria decision making mcdm based on the concepts of ideal and anti ideal points the concepts of fuzzy set theory and hierarchical structure analysis are used to develop a weighted suitability decision matrix to evaluate the weighted suitability of different alternatives versus various criteria the distance of different alternatives versus positive ideal solution and negative ideal solution are then obtained by using the proposed ranking method finally the relative approximation values of various alternatives versus positive ideal solution are ranked to determine the best alternative

symmetric ornstein uhlenbeck semigroups and their generators we provide necessary and sufficient conditions for a hilbert space valued ornstein uhlenbeck process to be reversible with respect to its invariant measure for a reversible process the domain of its generator in l p is characterized in terms of appropriate sobolev spaces thus extending the meyer equivalence of norms to any symmetric ornstein uhlenbeck operator we provide also a formula for the size of the spectral gap of the generator those results are applied to study the ornstein uhlenbeck process in a chaotic environment necessary and sufficient conditions for a transition semigroup r t to be compact hilbert schmidt and strong feller are given in terms of the coefficients of the ornstein uhlenbeck operator we show also that the existence of spectral gap implies a smoothing property of r t and provide an estimate for the appropriately defined gradient of r t finally in the hilbert schmidt case we show tha t for any the function r t is an almost classical solution of a version of the kolmogorov equation

the statistical mechanics of random copolymers random copolymers are polymers with two or more types of monomer where the monomer sequence is determined by some random process once determined the sequence is fixed so random copolymers are an example of a system with quenched randomness we review the statistical mechanics of random copolymers focusing on self avoiding walk models where there are two types of monomers a and b which are randomly distributed along the polymer chain theoretical approximate and numerical results are reviewed for models of the random copolymer adsorption localization and collapse phase transitions we concentrate on what is known about the existence of phase transitions the morita approximation and results about self averaging we also discuss in less detail the replica trick and numerical methods including monte carlo methods exact enumeration and transfer matrix methods important open problems are identified throughout and highlighted in the conclusions

some dichotomy results for central gaussian convolution semigroups on compact groups consider a central gaussian convolution semigroup t t 0 on a connected compact semisimple group then either the measure t is singular with respect to haar measure for all t 0 or there exists a time t such that t is absolutely continuous with respect to haar measure and admits a continuous density

ranking fuzzy numbers with maximizing set and minimizing set abstract up to now these are five methods of ranking n fuzzy numbers in order but these methods contain some confusions and occasionally conflict with intuition this paper introduces the concept of maximizing set and minimizing set to decide the ordering value of each fuzzy number and uses these values to determine the order of the n fuzzy numbers in addition we give a method for calculating the ordering value of each fuzzy number with triangular trapezoidal and two sided drum like shaped membership functions

stability of the bakry emery theorem on rn abstract we establish quantitative stability estimates for the bakry emery bound on logarithmic sobolev and poincare constants of uniformly log concave measures more specifically we show that if a 1 uniformly log concave measure has almost the same logarithmic sobolev or poincare constant as the standard gaussian measure then it almost splits off a gaussian factor our results are dimension free leading to dimension free stability estimates for gaussian concentration of lipschitz functions the proofs are based on stein s method optimal transport and an approximate integration by parts identity relating measures and near extremals in the associated functional inequality

refined convex sobolev inequalities this paper is devoted to refinements of convex sobolev inequalities in the case of power law relative entropies a nonlinear entropy entropy production relation improves the known inequalities of this type the corresponding generalized poincare type inequalities with weights are derived optimal constants are compared to the usual poincare constant

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