on the spectral problem for anyons we consider the spectral problem resulting from the schrodinger equation for a quantum system ofn 2 indistinguishable spinless hard core particles on a domain in two dimensional euclidian space for particles obeying fractional statistics and interacting via a repulsive hard core potential we provide a rigorous framework for analysing the spectral problem with its multi valued wave functions

the dirichlet problem for nonlinear second order elliptic equations iii functions of the eigenvalues of the hessian on etudie le probleme de dirichlet dans un domaine borne de r n a frontiere lisse f d 2 u dans u sur

uniform positivity improving property sobolev inequalities and spectral gaps abstract we introduce a certain kind of strong ergodicity condition to study the existence of spectral gap for markov generator we can estimate the spectral gap using the ergodicity condition and a sobolev type inequality we apply our results to the dirichlet form on wiener spaces riemannian manifolds and loop spaces

fuzzy pairwise extremally disconnected spaces abstract in this paper we generalize the concept of fuzzy extremally disconnected spaces due to ghosh 1992 into a fuzzy bitopological setting and investigate some of its properties using the concepts of i j semi closure i j closure and related notions in a fuzzy setting

estimate of the difference between the kac operator and the schrodinger semigroup an operator norm estimate of the difference between the kac operator and the schrodinger semigroup is proved and used to give a variant of the trotter product formula for schrodinger operators in thelp operator norm this extends helffer s result in thel2 operator norm to the case in thelp operator norm for more general scalar potentials and with vector potentials the method of the proof is probabilistic based on the feynman kac and feynman kac ito formula

the norm convergence of the trotter kato product formula with error bound the norm convergence of the trotter kato product formula with error bound is shown for the semigroup generated by that operator sum of two nonnegative selfadjoint operators a and b which is selfadjoint

conformal invariance and vector operators in the o n model it is widely expected that for a large class of models scale invariance implies conformal invariance a sufficient condition for this to happen is that there exists no integrated vector operator invariant under all internal symmetries of the model with scaling dimension 1 in this article we compute the scaling dimensions of vector operators with lowest dimensions in the o n model we use three different approximation schemes epsilon expansion large n limit and third order of the derivative expansion of non perturbative renormalization group equations we find that the scaling dimensions of all considered integrated vector operators are always much larger than 1 this strongly supports the existence of conformal invariance in this model for the ising model an argument based on correlation functions inequalities was derived which yields a lower bound for the scaling dimension of the vector perturbations we generalize this proof to the case of the o n model with n in left 2 3 4 right

exact boundary behavior of positive large solutions of a nonlinear dirichlet problem abstract in this paper we investigate the exact asymptotic behavior of positive solution to the following singular boundary value problem u b x f u x u 0 in u where is a c 2 bounded domain in r n n 3 f c 1 0 0 is nondecreasing on 0 and b is a function in c l o c 0 1 such that there exist b 1 b 2 0 satisfying for each x 0 b 1 lim d x 0 inf b x h d x lim d x 0 sup b x h d x b 2 where d x d i s t x and h t c t exp t z s s d s d i a m 2 such that z is a continuous function on 0 with z 0 0

infinite dimensional diffusion processes as gibbs measures on c 0 1 z d an infinite lattice system of interacting diffusion processes is characterized as a gibbs distribution on c 0 1 z d with continuous local conditional probabilities using estimates for the vasserstein metric onc 0 1 dobrushin s contraction technique is applied in order to obtain information about macroscopic properties of the entire diffusion process

splitting in large dimension and infrared estimates ii moment inequalities this is the continuation of notes written for the nato asi conference in il ciocco september 96 consisting of the analysis of the links between estimating the splitting between the two first eigenvalues for the schrodinger operator h and the proof of infrared estimates for quantities attached to gaussian type measures these notes were mainly reporting on the old contributions of dyson frohlich glimm jaffe lieb simon and spencer in the 1970s in connection with more recent contributions of pastur khoruzhenko barbulyak and kondrat ev which treat in general more sophisticated models here we concentrate on the simplest model related to field theory and extend the results of barbulyak and kondrat ev by mixing ideas coming from pastur and khozurenko related to the use of bogolyubov s inequality with classical inequalities due to ginibre lebowitz sokal and others or in the case when the temperature t is zero by applying rather elementary estimates on schrodinger operators in order to find

an operator theoretic proof of an estimate on the transfer operator let be a nonnegative number denote 1 as the smallest integer which is larger than 1 let k max 2 1 4 and let v be a positive potential on rn of class ck such that for all multi index satisfying k we have v x c 1 v x 1 we prove that e tve2t e tv e2t v l l2 o t1 2inf 1 2 as t decreases to zero our techniques rely on estimates on commutators copyright 1999 academic press soit un reel strictement positif notons 1 le plus petit entier superieur ou egal a 1 soit v un potentiel positif sur rn de classe ck tel que pour tout multiindice satisfaisant k on ait v x c 1 v x 1 nous prouvons que e tve2t e tv e2t v l l2 o t1 2inf 1 2 quand t decroit vers zero nos techniques utilisent des estimations sur des commutateurs

remarks on decay of correlations and witten laplacians brascamp lieb inequalities and semiclassical limit as it appears in recent articles by helffer or sjostrand and naddaf spencer the analysis in the context of the statistical mechanics of measures of the type exp x dxis connected with the analysis of suitable witten laplacians on 1 forms for illustrating this point of view we present here remarks about the brascamp lieb inequalities and its extensions and prove the decay of the correlation in some cases when is weakly non convex

the dirichlet problem for nonlinear second order elliptic equations i monge ampegre equation on considere le probleme de dirichlet pour des equations elliptiques non lineaires pour une fonction reelle u definie dans la fermeture d un domaine borne dans r n avec une frontiere c

spectral gap estimates for interacting particle systems via a bochner type identity abstract we develop a general technique based on a bochner type identity to estimate spectral gaps of a class of markov operator we apply this technique to various interacting particle systems in particular we give a simple and short proof of the diffusive scaling of the spectral gap of the kawasaki model at high temperature similar results are derived for kawasaki type dynamics in the lattice without exclusion and in the continuum new estimates for glauber type dynamics are also obtained

on the correlation for kac like models in the convex case the aim of this paper is to stu the behavior asm tends to of a family of measures exp m x dx m on m where m is a potential on m which is a perturbation in a suitable sense of the harmonic potential j x j 2

estimates of logarithmic sobolev constant for finite volume continuous spin systems letm be a compact connected riemannian manifold with or without boundary we study the logarithmic sobolev constant for stochastic ising models on m z d let be a sequence of cubes inz d we show that the logarithmic sobolev constant for the finite systems onm a shrinks at most exponentially fast in d 1 d d 2 which is sharp in order for the classical ising models withm 1 1 moreover a geometrical lemma proved by l e thomas is also improved

boundary behavior of large solutions to the monge ampere equations with weights abstract this paper is concerned with the monge ampere equation det d 2 u x b x f u x x where is a strictly convex bounded smooth domain in r n with n 2 and b c which is positive in but may be vanishing on the boundary we find a new structure condition on f which plays a crucial role in the boundary behavior of strictly convex large solutions our results are obtained in a more general setting than those in cirstea and trombetti 2008 12 where f is regularly varying at infinity with index p n

low temperature expansions for the gibbs states of weakly interacting quantum ising lattice systems low temperature expansions for the gibbs states of weakly interacting transverse ising like models are developed by conditioning the states on a sub algebra of observables the conditioned states have effective classical hamiltonians which are estimated by the solution to a simple implicit equation provided the interaction is sufficiently weak but fixed independent of temperature and the temperature is sufficiently low exponential clustering of the correlation functions holds

on singular boundary value problems for the monge ampere operator abstract we consider different types of singular boundary value problems for the monge ampere operator the approach is based on existing regularity theory and a subsolution supersolution method nonexistence and uniqueness results are also given

existence and estimates of solutions to a singular dirichlet problem for the monge ampere equation given a strictly convex smooth and bounded domain in rn we establish the existence of a negative convex solution in c c with zero boundary value to the singular monge ampere equation det d2u p x g u an associated dirichlet problem will be employed to provide a necessary and sufficient condition for the solvability of the singular boundary value problem estimates of solutions will also be given and regularity of solutions will be deduced from the estimates

almost sure weak convergence for the generalized orthogonal ensemble the generalized orthogonal ensemble satisfies isoperimetric inequalities analogous to the gaussian isoperimetric inequality and an analogue of wigner s law let v be a continuous and even real function such that v x tracev x n defines a uniformly p convex function on the real symmetric n n matrices x for some p 2 then dx e v x dx z satisfies deviation and transportation inequalities analogous to those satisfied by gaussian measure 6 27 but for the schatten cp norm the map that associates to each x msn its ordered eigenvalue sequence induces from a measure which satisfies similar inequalities it follows from such concentration inequalities that the empirical distribution of eigenvalues converges weakly almost surely to some non random compactly supported probability distribution as n

mean field bounds and correlation inequalities i prove a new correlation inequality for a class ofn component classical ferromagnets 1 n 4 this inequality implies that the correlation functions decay exponentially and the spontaneous magnetization is zero above the mean field critical temperature

convergence speed for simple symmetric exclusion an explicit calculation for the infinite volume simple symmetric nearest neighbor exclusion process in one dimension we investigate the speed of convergence to equilibrium from a particular initial distribution we use duality to reduce the analysis to that of the two particle process which we further reduce to a random walk reflecting rightward at zero whose generator is self adjoint on l 2 z we obtain the spectral representation of the generator and use asymptotic analysis to show that convergence is slow

boundary behavior of k convex solutions for singular k hessian equations abstract we discuss the existence and boundary behavior of k convex solution to the singular k hessian problem s k d 2 u x b x f u x x u x 0 x where s k d 2 u k 1 2 n is the k hessian operator r n n 2 is a smooth bounded strictly convex domain here the weight function b x is not necessarily bounded on another interest is that f u as u 0 our approach mainly relies on karamata s regular variation theory and the construction of suitable sub and super solutions

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

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

boundary blow up solutions to the k hessian equation with a weakly superlinear nonlinearity abstract by constructing new sub and super solutions we are concerned with determining values of for which there exist k convex solutions to the boundary blow up k hessian problem s k d 2 u x h x u x k ln u x 0 for x u x as dist x 0 here k 1 2 n s k d 2 u is the k hessian operator 0 and is a smooth bounded strictly convex domain in r n n 2 we suppose that the nonlinearity behaves like u k ln u as u which is more complex and difficult to deal with than the nonlinearity grows like u p with p k or faster at infinity further several new results of nonexistence global estimates and estimates near the boundary for the solutions are also given

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

correlation at low temperature i exponential decay abstract the present paper generalizes the analysis in ann h poincare 1 2000 59 math j ams 8 1997 123 of the correlations for a lattice system of real valued spins at low temperature the gibbs measure is assumed to be generated by a fairly general hamiltonian function with pair interaction the novelty as compared to 2 20 is that the single site self energies of the spins are not required to have only a single local minimum and no other extrema our derivation of exponential decay of correlations goes through the spectral analysis of a deformed laplacian closely related to the witten laplacian studied in 2 20 we prove that this laplacian has a spectral gap above zero and argue that this implies exponential decay of the correlations

gaussian bounds for correlations in lattice spin systems in quite generaln component ferromagnetic spin systems it is proved that an arbitrary correlation function is bounded by the corresponding correlation function of a gaussian model the bound is useful for the analysis of high temperature behavior of the system similar bounds for truncated correlation functions are also obtained for a class of single component spin systems

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

mean field bound and ghs inequality a new proof of the mean field bounds for magnetizations is presented it applies to any single component spin system which allows ghs inequality and to ann vector model forn 3 and to ann solid sphere model for all values ofn provided that the interactions are ferromagnetic and translation invariant

fuzzy sets and functions on fuzzy spaces abstract we generally follow the terminology of azad j math anal appl 82 1981 14 32 and ming j math anal appl 76 1980 571 599 in addition to the fundamental concepts for fuzzy sets we emphasize the usefulness of the concepts of fuzzy point fuzzy elementhood for fuzzy sets a and b a new characterization is given for relations a b and a b this knowledge permits us to combine the two definitions of fuzzy point fuzzy elementhood in the third section some results are given concerning various special types of fuzzy sets in fuzzy topological spaces and the fuzzy semi regular fuzzy regular spaces defined by azad in the last section the definitions of h almost continuous w almost open functions urysohn space which are defined by hussain wilansky and noiri respectively are extended to fuzzy sets furthermore some results are obtained in the functions of the fuzzy topological spaces defined by azad and those are defined here

semi open sets semi continuity and semi open mappings in fuzzy bitopological spaces abstract in this paper the concepts of fuzzy semi open sets fuzzy semi continuity and fuzzy semi open mappings due to azad j math anal appl 82 1981 14 32 have been generalized to the fuzzy bitopological setting and some of its basic properties are discussed several characterizing theorems of fuzzy semi continuity and its relationship with fuzzy continuity are investigated

fuzzy extremally disconnected spaces abstract an investigation of extremally disconnected fuzzy topological spaces is done in this paper such a class of spaces is characterized in various ways by using the concepts of q neighborhoods semi q neighbourhoods closure closure semi closure and related notions in fuzzy setting

convergence of a splitting method of high order for reaction diffusion systems in this article we prove the convergence of a splitting scheme of high order for a reaction diffusion system of the form u t m u f u 0 where m is an m m matrix whose spectrum is included in rz 0 this scheme is obtained by applying a richardson extrapolation to a strang formula

the random walk representation of classical spin systems and correlation inequalities ii the skeleton inequalities on utilise la representation marche aleatoire pour prouver les premieres d une nouvelle famille d inegalites de correlation pour des modeles reticules ferromagnetiques 4 ces inegalites etablissent que les sommes partielles finies du developpement en perturbation resomme du propagateur pour la fonction a 4 points forment un ensemble alterne de limites superieures et inferieures rigoureuses pour la fonction a 4 points exactes

boundary blow up solutions to the k hessian equation with singular weights abstract in this paper we study the k convex solutions to the boundary blow up k hessian problem s k d 2 u h x u p for x u x as dist x 0 here k 1 2 n s k d 2 u is the k hessian operator and is a smooth bounded strictly convex domain in r n n 2 we show the existence nonexistence uniqueness results global estimates and estimates near the boundary for the solutions our approach is largely based on the construction of suitable sub and super solutions

on fuzzy pairwise a continuity and fuzzy pairwise pre continuity in this paper the concepts of fuzzy open closed sets fuzzy pre open pre closed sets fuzzy continuous pre continuous mappings due to bin shahna fuzzy sets and systems 44 1991 303 308 have been generalized to the fuzzy bitopological setting and some of its basic properties are studied characterizations of these mappings along with their relationship with certain other types of mappings are investigated

dobrushin uniqueness theorem and logarithmic sobolev inequalities we formulate a condition on a local specification e on a countable product space m m being a riemannian manifold or a discrete set 1 1 assuring that the corresponding set of gibbs measures consists of a unique measure satisfying a logarithmic sobolev inequality

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

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

the random walk representation of classical spin systems and correlation inequalities ferromagnetic lattice spin systems can be expressed as gases of random walks interacting via a soft core repulsion by using a mixed spinrandom walk representation we present a unified approach to many recently established correlation inequalities as an application of these inequalities we obtain a simple proof of the mass gap for the 4 2 quantum field model we also establish new upper bounds on critical temperatures

absolute continuity of heat kernel measure with pinned wiener measure on loop groups let t 0 k be a connected compact lie group equipped with an ad k invariant inner product on the lie algebra of k associated to this data are two measures 0 t and t 0 on l k the space of continuous loops based at e e k the measure 0 t is pinned wiener measure with variance t while the measure 0 t is a heat kernel measure on l k the measure 0 t is constructed using a k valued brownian motion while the measure 0 t is constructed using a l k valued brownian motion in this paper we show that 0 t is absolutely continuous with respect to 0 t and the radon nikodym derivative d 0 t d 0 t is bounded

harnack inequalities for log sobolev functions and estimates of log sobolev constants by using the maximum principle and analysis of heat semigroups harnack inequalities are studied for log soboley functions from this some lower bound estimates of the log soboley constant are presented by using the spectral gap inequality and the coupling method the resulting inequalities either recover or improve the corresponding ones proved by chung and yau especially harnack inequalities and estimates of log sobolev constants can be dimension free

geometric analysis of φ4 fields and ising models parts i and ii we provide here the details of the proof announced in 1 that ind 4 dimensions the even 4 euclidean field theory with a lattice cut off is inevitably free in the continuum limit in the single phase regime the analysis is nonperturbative and is based on a representation of the field variables or spins in ising systems as source sink creation operators in a system of random currents which may be viewed as the mediators of correlations in this dual representation the onset of long range order is attributed to percolation in an ensemble of sourceless currents and the physical interaction in the 4 field and other aspects of the critical behavior in ising models are directly related to the intersection properties of long current clusters an insight into the criticality of the dimensiond 4 is derived from an analogy foreseen by k symanzik with the intersection properties of paths of brownian motion other results include the proof that in certain respect the critical behavior in ising models is in exact agreement with the mean field approximation in high dimensionsd 4 but not in the low dimensiond 2 for which we establish the universality of hyperscaling

the existence and asymptotic behavior of boundary blow up solutions to the k hessian equation abstract in this paper we consider the existence and asymptotic behavior of k convex solution to the boundary blow up k hessian problem s k d 2 u h x f u for x u x as dist x 0 where k 1 2 n s k d 2 u is the k hessian operator is a smooth bounded strictly convex domain in r n n 2 h c is positive in but is not necessarily bounded on and f is a smooth positive function that satisfies the so called keller osserman condition further results are obtained for the special case that is a ball our approach to show the existence and asymptotic behavior exploits the method of sub and super solutions and karamata regular variation theory

decay of correlations and uniqueness of gibbs lattice systems with nonquadratic interaction the aim of this paper is to develop the classical lattice models with unbounded spin to the case of nonquadratic polynomial interaction we demonstrate that the distinct relation between the growths of potentials leads to the uniqueness and the fast decay of correlations for gibbs measure

on a k hessian equation with a weakly superlinear nonlinearity and singular weights abstract consider the existence nonexistence and global estimates of k convex solutions to the boundary blow up k hessian problem s k d 2 u x h x u x k ln u x 0 for x u x as dist x 0 here k 1 2 n s k d 2 u is the k hessian operator 0 is a smooth bounded strictly convex domain in r n n 2 and h x is a positive weight function which is singular near the boundary we first give the existence and nonexistence results of k convex solution to the above boundary blow up problem on a larger range of h and then we show that there is a k convex solution provided that h x grows fast near and u k ln u grows slow at it turns out that this case is more difficult to handle than the case in which h x grows slow near and u k ln u grows fast at this needs some new ingredients in the arguments

invariance implies gibbsian some new results we investigate stationary distributions of stochastic gradient systems in riemannian manifolds and prove that under certain assumptions such distributions are symmetric these results are extended to countable products of finite dimensional manifolds and applied to gibbs distributions in the case where the single spin spaces are riemannian manifolds in particular we obtain a new result concerning the question whether all invariant measures are gibbsian actually we consider a more general object weak elliptic equations for measures which on the one hand yields the results obtained stronger than the above mentioned statements and on the other hand enables us to give simpler proofs of more general than previously known facts applications to concrete models of lattice systems over d with not necessarily compact spin space are presented also in the case d 3 under certain assumptions of decay of interaction

estimates of logarithmic sobolev constant an improvement of bakry emery criterion this paper is mainly devoted to estimate the logarithmic sobolev abbrev l s constant for diffusion operators on manifold or in rd in most cases we study the lower bounds but a generalization to a korzeniowski j funct anal 71 1987 363 370 theorem 1 for the upper bound is also presented theorem 1 5 based on a simple observation due to j d deuschel and d w stroock j funct anal 92 1990 30 48 of the comparison between the l s constants for different potentials the powerful bakry emery criterion for the l s inequality is improved considerably in the paper especially for the manifolds with non positive sectional curvatures theorem 1 3 1 in terms of our notation r inf x p r infx tx m x 1 ricc hessv x x where x p is the distance betweenxand an arbitrary fixed pointp m the improvement can be roughly stated as follows the condition infr 0 r 0 for which the criterion is available is now replaced by supr 0 r 0