a greedy randomized adaptive search procedure for maximum independent set an efficient randomized heuristic for a maximum independent set is presented the procedure is tested on randomly generated graphs having from 400 to 3 500 vertices and edge probabilities from 0 2 to 0 9 the heuristic can be implemented trivially in parallel and is tested on an mimd computer with 1 2 4 and 8 processors computational results indicate that the heuristic frequently finds the optimal or expected optimal solution in a fraction of the time required by implementations of simulated annealing tabu search and an exact partial enumeration method

the logarithmic sobolev inequality for discrete spin systems on a lattice for finite range lattice gases with a finite spin space it is shown that the dobrushin shlosman mixing condition is equivalent to the existence of a logarithmic sobolev inequality for the associated unique gibbs state in addition implications of these considerations for the ergodic properties of the corresponding glauber dynamics are examined

regularity of invariant measures on finite and infinite dimensional spaces and applications abstract in this paper we prove new results on the regularity i e smoothness of measures solving the equation l 0 for operators of type l b on finite and infinite dimensional state spaces e in particular we settle a conjecture of i shigekawa in the situation where h is the gross laplacian e h is an abstract wiener space and b id e v where v takes values in the cameron martin space h using gross logarithmic sobolev inequality in an essential way we show that is always absolutely continuous w r t the gaussian measure and that the square root of the density is in the malliavin test function space of order 1 in l 2 furthermore we discuss applications to infinite dimensional stochastic differential equations and prove some new existence results for l 0 these include results on the inverse problem i e we give conditions ensuring that b is the vector logarithmic derivative of a measure we also prove necessary and sufficient conditions for to be symmetrizing i e l is symmetric on l 2 finally a substantial part of this work is devoted to the uniqueness of symmetrizing measures for l we characterize the cases where we have uniqueness by the irreducibility of the associated classical dirichlet forms

analytic and geometric background of recurrence and non explosion of the brownian motion on riemannian manifolds we provide an overview of such properties of the brownian motion on complete non compact riemannian manifolds as recurrence and non explosion it is shown that both properties have various analytic characterizations in terms of the heat kernel green function liouville properties etc on the other hand we consider a number of geometric conditions such as the volume growth isoperimetric inequalities curvature bounds etc which are related to recurrence and non explosion

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

ergodicity ofl2 semigroups and extremality of gibbs states we extend the classical results of holley stroock on the characterization of extreme gibbs states for the ising model in terms of the irreducibility resp ergodicity of the corresponding glauber dynamics to the case of lattice systems with unbounded linear spin spaces we first develop a general framework to discuss questions of this type using classical dirichlet forms on infinite dimensional state spaces and their associated diffusions we then describe concrete applications to lattice models with polynomial interactions i e the discretep d models of euclidean quantum field theory in addition we prove the equivalence of extremality and shift ergodicity for tempered gibbs states of these models and also discuss this question in the general framework

least energy solutions for fractional kirchhoff problems with logarithmic nonlinearity abstract in this paper we study the existence of least energy solutions to the following fractional kirchhoff problem with logarithmic nonlinearity m u s p p p s u h x u p 2 u ln u u q 2 u x u 0 x r n where s 0 1 1 p n s r n is a bounded domain with lipschitz boundary m u s p p u s p 1 p with 1 and u s p is the gagliardo seminorm of u h c may change sign 0 is a parameter q 1 p s and p s is the fractional p laplacian when p q p s and h is a positive function on the existence of least energy solutions is obtained by restricting the discussion on nehari manifold when 1 q p and h is a sign changing function on two local least energy solutions are obtained by using the nehari manifold approach

a greedy randomized adaptive search procedure for the two partition problem we present a greedy randomized adaptive search procedure grasp for the network 2 partition problem the heuristic is empirically compared with the kernighan lin k l method on a wide range of instances the grasp approach dominates k l with respect to solution value on a large percentage of the instances tested the ability of grasp to find optimal solutions is assessed by comparing its performance with a general purpose mixed integer programming package

a priori estimates for symmetrizing measures and their applications to gibbs states abstract we prove existence and uniform a priori estimates for tempered gibbs states of certain classical lattice systems with unbounded spins nonharmonic pair potentials and infinite radius of interaction we use an alternative characterization of gibbs measures in terms of their radon nikodym derivatives w r t local shifts of the configuration space and corresponding integration by parts formulas

ito type measure valued stochastic differential equations a class of ito type measure valued stochastic differential equations is studied on a locally compact polish space the sdes are driven by countably many brownian motions with interactions caused by the diffusion and the drift coefficients through countably many continuous functions explicit conditions are presented for the existence uniqueness and ergodicity of the solution

the brezis nirenberg result for the fractional laplacian the aim of this paper is to deal with the non local fractional counterpart of the laplace equation involving critical non linearities studied in the famous paper of brezis and nirenberg 1983 namely our model is the equation su u u 2 2u in u 0 in rn where s is the fractional laplace operator s 0 1 is an open bounded set of rn n 2s with lipschitz boundary 0 is a real parameter and 2 2n n 2s is a fractional critical sobolev exponent in this paper we first study the problem in a general framework indeed we consider the equation lku u u 2 2u f x u 0 in u 0 in rn where lk is a general non local integrodifferential operator of order s and f is a lower order perturbation of the critical power u 2 2u in this setting we prove an existence result through variational techniques then as a concrete example we derive a brezis nirenberg type result for our model equation that is we show that if 1 s is the first eigenvalue of the non local operator s with homogeneous dirichlet boundary datum then for any 0 1 s there exists a non trivial solution of the above model equation provided n 4s in this sense the present work may be seen as the extension of the classical brezis nirenberg result to the case of non local fractional operators

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

the nehari manifold for fractional p laplacian equation with logarithmic nonlinearity on whole space abstract we study a fractional p laplacian equation in the whole space with the sign changing logarithmic nonlinearity by using fibrering maps and nehari manifold we obtain the existence of at least two nontrivial solutions our result extends a recent result by tian 2017 the main difficulty is the lack of logarithmic sobolev inequality concerning to fractional p laplacian

relative entropy and mixing properties of infinite dimensional diffusions let be a diffusion process taking values on the infinite dimensional space t z where t is the circle and with components satisfying the equations d i i dw i b i dt for some coefficients i and b i i z suppose we have an initial distribution and a sequence of times t n such that lim n s tn exists where s t is the semi group of the process we prove that if i and b i are bounded of finite range have uniformly bounded second order partial derivatives and inf i i 0 then is invariant

on the ergodic properties of glauber dynamics we show that if there is an infinite volume gibbs measure which satisfies a logarithmic sobolev inequality with local coefficients of moderate growth then the corresponding stochastic dynamics decays to equilibrium exponentially fast in the uniform norm

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

elliptic equations for measures on infinite dimensional spaces and applications we introduce and study a new concept of a weak elliptic equation for measures on infinite dimensional spaces this concept allows one to consider equations whose coefficients are not globally integrable by using a suitably extended lyapunov function technique we derive a priori estimates for the solutions of such equations and prove new existence results as an application we consider stochastic burgers reaction diffusion and navier stokes equations and investigate the elliptic equations for the corresponding invariant measures our general theorems yield a priori estimates and existence results for such elliptic equations we also obtain moment estimates for gibbs distributions and prove an existence result applicable to a wide class of models

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

multiplicity results for variable order fractional laplacian equations with variable growth abstract in this paper we study the multiplicity of solutions for an elliptic type problem driven by the variable order fractional laplace operator involving variable exponents more precisely we consider s u v x u u p x 2 u u q x 2 u in u 0 in r n where n 1 s r n r n 0 1 is a continuous function is a bounded domain in r n with n 2 s x y for all x y s is the variable order fractional laplace operator 0 is a parameter v 0 is a continuous function 0 are two parameters and p q c under some suitable assumptions we show that the above problem admits at least two distinct solutions by applying the mountain pass theorem and ekeland s variational principle then we prove that these two solutions converge to two solutions of a limit problem as moreover we obtain the existence of infinitely many solutions for the limit problem

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

compactness and the maximal gibbs state for random gibbs fields on a lattice we prove for a general class of gibbsian random field on that the set of tempered gibbs states is compact this class contains the euclidean random fields moreover if the interaction is attractive there is a unique minimal and maximal gibbs state and are unique translation invariant ant and have the global markov property we also prove that uniqueness of the tempered gibbs state is equivalent to the magnetizationsm qx being equal which is true if the pressure is differentiable

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

the nehari manifold for a semilinear elliptic equation with a sign changing weight function abstract the nehari manifold for the equation u x a x u x b x u x 1u x for x together with dirichlet boundary conditions is investigated exploiting the relationship between the nehari manifold and fibrering maps i e maps of the form t j tu where j is the euler functional associated with the equation we discuss how the nehari manifold changes as changes and show how existence and non existence results for positive solutions of the equation are linked to properties of the manifold

a critical kirchhoff type problem involving a nonlocal operator abstract in this paper we show the existence of non negative solutions for a kirchhoff type problem driven by a nonlocal integrodifferential operator that is m u z 2 l k u f x u u 2 2 u in u 0 in r n where l k is an integrodifferential operator with kernel k is a bounded subset of r n m and f are continuous functions z is a functional norm and 2 is a fractional sobolev exponent

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

existence and stability of standing waves for nonlinear fractional schrodinger equation with logarithmic nonlinearity abstract in this paper we consider the nonlinear fractional logarithmic schrodinger equation by using a compactness method we construct a unique global solution of the associated cauchy problem in a suitable functional framework we also prove the existence of ground states as minimizers of the action on the nehari manifold finally we prove that the set of minimizers is a stable set for the initial value problem that is a solution whose initial data is near the set will remain near it for all time

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

regularity of invariant measures the case of non constant diffusion part we prove regularity i e smoothness of measures on rdsatisfying the equationl 0 wherelis an operator of typelu tr au b u hereais a lipschitz continuous uniformly elliptic matrix valued map andbis merely square integrable we also treat a class of corresponding infinite dimensional cases where rdis replaced by a locally convex topological vector spacex in this cases is proved to be absolutely continuous w r t a gaussian measure onxand the square root of the radon nikodym density belongs to the malliavin test function space d2 1

multiple solutions for the semilinear elliptic equations with the sign changing logarithmic nonlinearity abstract in this paper we study the semilinear elliptic equations with the sign changing logarithmic nonlinearity under some conditions we obtain at least two nontrivial solutions by nehari manifold and logarithmic sobolev inequality here the results are quite different from these in the polynomial case

reversibility in infinite hamiltonian systems with conservative noise the set of stationary measures of an infinite hamiltonian system with noise is investigated the model consists of particles moving in r 3 with bounded velocities and subject to a noise that does not violate the classical laws of conservation see ovy following lo we assume that the noise has also a finite radius of interaction and prove that translation invariant stationary states of finite specific entropy are reversible with respect to the stochastic component of the evolution therefore the results of lo imply that such invariant measures are superpositions of gibbs states

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

existence of solutions for kirchhoff type problem involving the non local fractional p laplacian abstract the purpose of this paper is to investigate the existence of weak solutions for a kirchhoff type problem driven by a non local integro differential operator of elliptic type with homogeneous dirichlet boundary conditions as follows m r 2 n u x u y p k x y d x d y l k p u f x u in u 0 in r n where l k p is a non local operator with singular kernel k is an open bounded subset of r n with lipshcitz boundary m is a continuous function and f is a caratheodory function satisfying the ambrosetti rabinowitz type condition we discuss the above mentioned problem in two cases when f satisfies sublinear growth condition the existence of nontrivial weak solutions is obtained by applying the direct method in variational methods when f satisfies suplinear growth condition the existence of two nontrivial weak solutions is obtained by using the mountain pass theorem

one half approximation algorithms for the k partition problem the k partition problem seeks to cluster the vertices of an edge weighted graph g v e into k sets of v k vertices each such that the total weight of the edges having both endpoints in the same cluster is maximized bottom up type heuristics based on matchings are presented for this problem these heuristics are shown to yield solutions that are at least one half the weight of the optimal solution for k equal to v 3 and v 4