navier stokes equations for stochastic particle systems on the lattice we introduce a class of stochastic models of particles on the cubic lattice d with velocities and study the hydrodynamical limit on the diffusive spacetime scale assuming special initial conditions corresponding to the incompressible regime we prove that in dimensiond 3 there is a law of large numbers for the empirical density and the rescaled empirical velocity field moreover the limit fields satisfy the corresponding incompressible navier stokes equations with viscosity matrices characterized by a variational formula formally equivalent to the green kubo formula

semiclassical analysis of low lying eigenvalues ii tunneling we discuss the leading asymptotics of eigenvalue splittings of 2 x2v in the limit as x o and where v is a non negative potential with several zeros for example if eo x e1 x are the two lowest eigenvalues in a situation where v has precisely two zeros a and b related by a symmetry then limx x lin e1 x eo x is given as the distance from a to b in a certain riemann metric

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

the equivalence of certain heat kernel and green function bounds abstract we show that certain upper and lower bounds on the green function and heat kernel of a second order elliptic operator in a bounded region are equivalent and imply a number of apparently stronger bounds we also show that these bounds are equivalent to the harnack inequality except for peculiar regions and are therefore almost always valid

coercive inequalities for gibbs measures abstract we prove the generalized nash and logarithmic nash inequalities for gibbs measures with dirichlet form associated to the kawasaki dynamics

poincare and transportation inequalities for gibbs measures under the dobrushin uniqueness condition in in this paper we establish an explicit and sharp estimate of the spectral gap poincare inequality and the transportation inequality for gibbs measures under the dobrushin uniqueness condition moreover we give a generalization of the liggett s m e theorem for interacting particle systems

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

logarithmic sobolev inequalities for mollified compactly supported measures abstract we show that the convolution of a compactly supported measure on r with a gaussian measure satisfies a logarithmic sobolev inequality lsi we use this result to give a new proof of a classical result in random matrix theory that states that under certain hypotheses the empirical law of eigenvalues of a sequence of random real symmetric matrices converges weakly in probability to its mean we then examine the optimal constants in the lsis for the convolved measures in terms of the variance of the convolving gaussian we conclude with partial results on the extension of our main theorem to higher dimensions

cutoff for the ising model on the lattice introduced in 1963 glauber dynamics is one of the most practiced and extensively studied methods for sampling the ising model on lattices it is well known that at high temperatures the time it takes this chain to mix in l 1 on a system of size n is o logn whether in this regime there is cutoff i e a sharp transition in the l 1 convergence to equilibrium is a fundamental open problem if so as conjectured by peres it would imply that mixing occurs abruptly at c o 1 logn for some fixed c 0 thus providing a rigorous stopping rule for this mcmc sampler however obtaining the precise asymptotics of the mixing and proving cutoff can be extremely challenging even for fairly simple markov chains already for the one dimensional ising model showing cutoff is a longstanding open problem

uniqueness and global markov property for euclidean fields the case of general polynomial interactions we give a general method for proving uniqueness and global markov property for euclidean quantum fields the method is based on uniform continuity of local specifications proved by using potential theoretical tools and exploitation of a suitable fkg order structure we apply this method to give a proof of uniqueness and global markov property for the gibbs states and to study extremality of gibbs states also in the case of non uniqueness in particular we prove extremality for 2 4 also in the case of non uniqueness and global markov property for weak coupling 2 4 which solves a long standing problem uniqueness and extremality holds also at any point of differentiability of the pressure with respect to the external magnetic field

spectral properties of metastable markov semigroups abstract it is shown that if a symmetric markov semigroup e ht on the hilbert space l 2 x is hypercontractive then the approximate degeneracy of the ground state has several consequences concerning other parts of the spectrum of h and concerning the unitary group e iht in particular in the presence of a space inversion symmetry all the eigenvalues occur in pairs with gaps comparable to the gap between the bottom two eigenvalues

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

derivation of the leroux system as the hydrodynamic limit of a two component lattice gas the long time behavior of a couple of interacting asymmetric exclusion pro cesses of opposite velocities is investigated in one space dimension we do not allow two particles at the same site and a collision effect exchange takes place when particles of opposite velocities meet at neighboring sites there are two conserved quantities and the model admits hyperbolic euler scaling the hydrodynamic limit results in the classical leroux system of conservation laws even beyond the appearance of shocks actually we prove convergence to the set of entropy solutions the question of unique ness is left open to control rapid oscillations of lax entropies via logarithmic sobolev inequality estimates the symmetric part of the process is speeded up in a suitable way thus a slowly vanishing viscosity is obtained at the macroscopic level following 4 5 the stochastic version of tartar murat theory of compensated compactness is extended to two component stochastic models

nonuniqueness of the positive dirichlet problem for parabolic equations in cylinders abstract we investigate uniqueness of nonnegative solutions to parabolic equations in cylinders of r n 1 with zero dirichlet boundary condition where bases of cylinders are unbounded domains we give necessary and or sufficient conditions for the uniqueness in terms of geometric properties of the bases

definable sets in ordered structures i this paper introduces and begins the study of a well behaved class of linearly ordered structures the minimal structures the definition of this class and the corresponding class of theories the strongly dlminlmal theories is made in analogy with the notions from stability theory of minimal structures and strongly minimal theories theorems 2 1 and 2 3 respectively provide characterizations of 41 minimal ordered groups and rings several other simple results are collected in 3 the primary tool in the analysis of minimal structures is a strong analogue of ii forking symmetry given by theorem 4 2 this result states that any parametri cally definable unary function in an yminimal structure is piecewise either constant or an ore er preserving or reversing bijection of intervals the results that follow include the existence and uniqueness of prime models over sets theorem 5 1 and a characterization of all bto categorical aminimal structures theorem 6 1 1 introduction the class of linearly ordered structures has long been an im portant subject of concern to model theorists impressive results have been obtained in the study of models of several particular theories that extend the theory of linear order among those theories that have been approached successfully are peano arithmetic the theory of ordered abelian groups that of real closed fields and that of linear order itself yet very little has been done in the way of developing a general model theory for ordered structures in this paper we develop the model theory for a class of linearly ordered structures that we isolate by demanding that a structure in this class satisfy a condition whose effect is that the linear ordering and the algebraic part of the structure behave quite well with respect to one another let l be a finitary first order language and w an l structure a set of n tuples a c x is said to be parametrically definable if there is some l formula

an algorithm for finite element analysis of partly wrinkled membranes a generalization and numerical implementation is presented of the stein hedgepeth 1961 continuum theory for the analysis of partly wrinkled membranes such as the tensioned membrane surfaces which are found in spacecraft structural components the approach has its basis in experimental observations showing that when wrinkles develop in a membrane parallel to the x direction the associated overall contraction in the y direction exceeds that predicted by the poisson s ratio effect

numerical study of the wrinkling of a stretched thin sheet a thin sheet clamped at opposite ends and stretched develops wrinkles parallel to the direction of the applied tensile strain due to the hindered poisson lateral contraction at the clamps to study this phenomenon a variational model recently proposed by puntel deseri and fried is adopted the relevant energy functional includes bending and membranal contributions and is minimized subject to a constraint on the area of the mid surface of the sheet a fourth order partial differential equation is henceforth obtained and numerically implemented using b splines predictions are obtained concerning the number of wrinkles critical applied stretches and scaling relationships for wrinkle amplitude and wavelength both a linearized version of the boundary value problem based on the small slope approximation and a fully nonlinear one are considered their results are found to be in good agreement for the whole range of applied stretches taken into account comparisons with previous analytical results by puntel deseri and fried who used different boundary conditions and an ansatz on the deflection function are also provided the numerical results substantially confirm the validity of the analytical predictions the present work provides then an alternative numerical method for the study of wrinkling in thin sheets and supports the use of analytical and semi analytical solutions as viable options for specific geometries though further investigation particularly experimental is still needed extensive comparisons of the results with other studies available in the literature provide confirmation for the scaling laws and signal that predicted values of the critical stretches may only be accurate for higher length to width aspect ratios

gaussian upper bounds for the heat kernels of some second order operators on riemannian manifolds abstract we describe a method of obtaining gaussian upper bounds on heat kernels which unifies and improves recent results for hypoelliptic operators in divergence form and for laplace beltrami operators on complete riemannian manifolds

ultracontractivity and the heat kernel for schrodinger operators and dirichlet laplacians abstract abstract connections between integral kernels of positivity preserving semigroups and suitable lp contractivity properties are established then these questions are studied for the semigroups generated by v and h the dirichlet laplacian for an open connected region as an application under a suitable hypothesis sobolev estimates are proved valid up to of the form x c 0 x h k 2 where 0 is the unique positive l2 eigenfunction of h

outlier eigenvalues for deformed i i d random matrices we consider a square random matrix of size n of the form a y where a is deterministic and y has iid entries with variance 1 n under mild assumptions as n grows the empirical distribution of the eigenvalues of a y converges weakly to a limit probability measure beta on the complex plane this work is devoted to the study of the outlier eigenvalues i e eigenvalues in the complement of the support of beta even in the simplest cases a variety of interesting phenomena can occur as in earlier works we give a sufficient condition to guarantee that outliers are stable and provide examples where their fluctuations vary with the particular distribution of the entries of y or the jordan decomposition of a we also exhibit concrete examples where the outlier eigenvalues converge in distribution to the zeros of a gaussian analytic function

definable sets in ordered structures on introduit la notion de theorie o minimale des structures ordonnees une telle theorie etant telle que les sous ensembles definissables de ses modeles soient particulierement simples

traps for reflected brownian motion consider an open set d r d d 2 and a closed ball b d let e x tb denote the expectation of the hitting time of b for reflected brownian motion in d starting from x d we say that d is a trap domain if supx e x tb a domain d is not a trap domain if and only if the reflecting brownian motion in d is uniformly ergodic we fully characterize the simply connected planar trap domains using a geometric condition and give a number of less complete results for d 2

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

on equivalence of spin and field pictures of lattice systems we investigate the spin and field systems on a lattice connected by the kac siegert transform it is shown that the structures of corresponding theories are equivalent in the sense of isomorphy of space of gibbs states and order parameters using the idea of equivalence of spin and field pictures we exhibit a class of lattice systems possessing infinitely uncountably many ground states the systems of this type with infinite range slow decaying interactions are expected to have a spin glass phase transition

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

isoperimetry for gibbs measures we show that a strong mixing condition implies a bakry bobkov ledoux inequality for a probability measure on infinite dimensional space

spectral gap for zero range dynamics we give a lower bound on the spectral gap for symmetric zero range processes under some conditions on the rate function we show that the gap shrinks as n 2 independent of the density for the dynamics localized on a cube of size n d we follow the method outlined by lu and yau where a similar spectral gap is proved for kawasaki dynamics

hypercontractivity in noncommutative lpspaces we introduce and study a setup for hypercontractivity in the interpolating family of noncommutative lpspaces associated to a locally normal state on an inductive limitc algebra

mixing of the symmetric exclusion processes in terms of the corresponding single particle random walk we prove an upper bound for the e mixing time of the symmetric exclusion process on any graph g with any feasible number of particles our estimate is proportional to trw g ln v e where v is the number of vertices in g and trw g is the 1 4 mixing time of the corresponding single particle random walk this bound implies new results for symmetric exclusion on expanders percolation clusters the giant component of the erdos renyi random graph and poisson point processes in rd our technical tools include a variant of morris s chameleon process

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

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

the logarithmic sobolev constant of the lamplighter we give estimates on the logarithmic sobolev constant of some finite lamplighter graphs in terms of the spectral gap of the underlying base also we give examples of application

perturbation of singular equilibria of hyperbolic two component systems a universal hydrodynamic limit we consider one dimensional locally finite interacting particle systems with two conservation laws which under the eulerian hydrodynamic limit lead to two by two systems of conservation laws t x u 0 t u x u 0

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

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

how the unbounded drift shapes the dirichlet semigroups behaviour of non gaussian gibbs measures abstract we investigate the dirichlet operator semigroups of non gaussian gibbs measures on linear spin lattice systems we prove that dirichlet semigroups preserve spaces of l 2 differentiable functions we show how the non linearity and unboundedness of drift term in dirichlet operator affects the structure of the spaces involved

sharp bounds for ground state eigenfunctions on domains with horns and cusps abstract we prove a lower bound on the ground state eigenfunction of the dirichlet laplacian for horn shaped and cusped domains the novelty of the method is that for horn shaped domains it implies a new result for dimension two and more importantly is valid for higher dimensions moreover the same technique applies just as easily to cusps finally we construct an example to show that the estimates are sharp

a c0 discontinuous galerkin method for the stationary quasi geostrophic equations of the ocean abstract this work concerns the development of a finite element algorithm for the stationary quasi geostrophic equations to treat the large scale wind driven ocean circulation the algorithm is developed based on the streamfunction formulation involving fourth order gradients of the streamfunction here we examine the adaptation of a relatively inexpensive nonconforming method based on c 0 basis functions we develop the variational form of the method and establish consistency the method weakly enforces continuity of the normal flux across interelement boundaries and stabilization is achieved via nitsche s method explicit expression for the choice of the stabilization parameter is derived moreover the theoretical error estimate for the linear stommel munk model is performed numerical results from several benchmark problems on rectangular and curved domains are provided to demonstrate the accuracy and robustness of the method we also provide the mediterranean sea example to demonstrate the capability of the approach to the wind driven ocean circulation simulation

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

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

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

log sobolev inequalities for infinite one dimensional lattice systems it is shown that a unique gibbs measure of infinite spin system with short range interaction on one dimensional lattice satisfies log sobolev inequality

large time behavior of the lp norm of schrodinger semigroups abstract letv 0 v c0 rv with v 3 be such that h 1 2 v 0 but for any e 0 1 2 1 e v is not positive we determine the exact rate of divergence of the norm of e th as a map from l to l a number of related problems are discussed

an invariance principle for branching diffusions in bounded domains we study branching diffusions in a bounded domain d of mathbb r d in which particles are killed upon hitting the boundary partial d it is known that any such process undergoes a phase transition when the branching rate beta exceeds a critical value a multiple of the first eigenvalue of the generator of the diffusion we investigate the system at criticality and show that the associated genealogical tree when the process is conditioned to survive for a long time converges to aldous continuum random tree under appropriate rescaling the result holds under only a mild assumption on the domain and is valid for all branching mechanisms with finite variance and a general class of diffusions

explicit constants for gaussian upper bounds on heat kernels on utilise des inegalites logarithmiques de sobolev pour obtenir des bornes superieures gaussiennes sur les noyaux de la chaleur associes a des operateurs d ordre 2 elliptiques et hypoelliptiques

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

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

spline based finite element method for the stationary quasi geostrophic equations on arbitrary shaped coastal boundaries abstract this work concerns a b spline based finite element algorithm for the stationary quasi geostrophic equations to treat the large scale wind driven ocean circulation on arbitrary shaped domains the algorithm models arbitrary shaped coastal boundaries on intra element or embedded boundaries dirichlet boundary conditions on the embedded boundaries are weakly imposed and stabilization is achieved via nitsche s method we employ a hierarchical local refinement approach to improve the geometrical representation of curved boundaries results from several benchmark problems on rectangular and curved domains are provided to demonstrate the accuracy and robustness of the method we also provide the mediterranean sea example that illustrates the effectiveness of the approach in the wind driven ocean circulation simulation

uniqueness and the global markov property for euclidean fields the case of general exponential interaction the uniqueness and the global markov property for the regular gibbs measure corresponding to the interaction u lambda varphi lambda int limits lambda d 2 x int d varrho alpha e alpha varphi 0 x for 0 d a probability measure with support in 2 sqrt pi 2 sqrt pi is proved

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

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

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

poincare s inequalities and talagrand s concentration phenomenon for the exponential distribution we present a simple proof based on modified logarithmic sobolev inequalities of talagrand s concentration inequality for the exponential distribution we actually observe that every measure satisfying a poincare inequality shares the same concentration phenomenon we also discuss exponential integrability under poincare inequalities and its consequence to sharp diameter upper bounds on spectral gaps

how to avoid a compact set abstract a first order expansion of the r vector space structure on r does not define every compact subset of every r n if and only if topological and hausdorff dimension coincide on all closed definable sets equivalently if a r k is closed and the hausdorff dimension of a exceeds the topological dimension of a then every compact subset of every r n can be constructed from a using finitely many boolean operations cartesian products and linear operations the same statement fails when hausdorff dimension is replaced by packing dimension

differential equation for the amplitude of wrinkles introduction t he renewed interest in the phenomenonof wrinkling of membranes fueled by applications to the aerospace industry has spurred considerable activity on both the theoretical and computational aspectsof the subject althoughthe wrinklingphenomenonis known to be a case of shell buckling it seems unwise to utilize this knowledge in actual computations because treating the membrane as a shell with a vanishinglysmall thickness leads to the frustrating effects of numerical instability there exist on the other hand techniques based on themembrane theory alone that permit us to obtain with relative ease and numerical robustness not only the extent of thewrinkledzones but also the eldof directionsof thewrinkles 1i4 all that is needed therefore is a means for calculating the amplitude and wavelength of the wrinkles in an already known wrinkled domain in this note we derive an ordinary differential equation whose solution delivers such information for clarity of the presentation the equation is obtained under a number of simplifying assumptions mostly involving the smallness of the deformations however the method is straightforwardlyextendable to the general case our guiding principle has been to keep only those aspects of the model that are clearly determinant of the result while leaving all other aspects for future re nement we believe that this philosophy will serve to highlight the essential features rather than the technicalities of the proposedequation note that even under a geometrically andmaterially linear regime the resultingdifferentialequation is highly nonlinear the physical reason for this nonlinearityis to be foundin that becausewrinklingentailsno stress in thedirectionperpendicular to the wrinkles there exists a severe constraint between the wavelength and amplitude of the wrinkles this can be phrased intuitively as follows for a given amount of wrinkling strain the larger the number of wrinkles the smaller their amplitude to the best of our knowledge the only attempt in a direction similar to the one proposed herein is due to wong and pellegrino in their work these authors present a model basedmainly on the particular case of shearing of a rectangular panel their results are very valuable and will be used here as a benchmark for validation purposes

error estimates of b spline based finite element methods for the stationary quasi geostrophic equations of the ocean abstract this paper presents theoretical error estimates of b spline based finite element methods for the streamfunction formulation of the stationary quasi geostrophic equations which describe the large scale wind driven ocean circulation we introduce variational formulations of the streamfunction formulation inspired by the interior penalty discontinuous galerkin method dirichlet boundary conditions are weakly enforced in the formulations and stabilizations are achieved via nitsche s method existence and uniqueness of the approximation are proved and optimal error estimates in the energy norm are demonstrated under a small data assumption numerical experiments are performed to verify the theoretical error estimates on rectangular and l shape geometries

an inequality for relative entropy and logarithmic sobolev inequalities in euclidean spaces for a class of density functions q x on rn we prove an inequality between relative entropy and the weighted sum of conditional relative entropies of the following form d p q const i 1n i d pi y1 yi 1 yi 1 yn qi y1 yi 1 yi 1 yn for any density function p x on rn where pi y1 yi 1 yi 1 yn and qi x1 xi 1 xi 1 xn denote the local specifications of p respectively q and i is the logarithmic sobolev constant of qi x1 xi 1 xi 1 xn thereby we derive a logarithmic sobolev inequality for a weighted gibbs sampler governed by the local specifications of q moreover the above inequality implies a classical logarithmic sobolev inequality for q as defined for gaussian distribution by gross this strengthens a result by otto and reznikoff the proof is based on ideas developed by otto and villani in their paper on the connection between talagrand s transportation cost inequality and logarithmic sobolev inequality

b spline based finite element method for the stationary quasi geostrophic equations of the ocean abstract we present a b spline based conforming finite element method of the streamfunction formulation of the stationary one layer quasi geostrophic equations for the study of the large scale wind driven ocean circulation the method encompasses standard simplifications of these equations in particular the linear stommel and stommel munk models a variational form of the method is developed and its consistency is established in this formulation dirichlet boundary conditions are enforced only weakly and stabilization is achieved via nitsche s method stability parameters are evaluated by solving a local generalized eigenvalue problem on the dirichlet boundary and by monitoring the condition number of the resulting linear systems comparisons of the results from our simulations with previously published results and convergence studies lead us to conclude that our finite element discretization is accurate

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

tension field theory a general theory of the tension field is developed for application to the analysis of wrinkling in isotropic elastic membranes undergoing finite deformations the principal contribution is a partial differential equation describing a geometrical property of tension trajectories this is one of a system of two equations which describes the state of stress independently of the deformation this system is strongly elliptic at any stable solution whereas the deformation is described by a system of parabolic type controllable solutions i e those states that can be maintained in any isotropic elastic material by application of edge tractions and lateral pressure alone are obtained the general axisymmetric problem is solved implicitly and the theory is applied to the solution of two representative examples existing small strain theories are shown to correspond to a singular limit of the general theory at which the underlying system changes from elliptic to parabolic type

on modified logarithmic sobolev inequalities for bernoulli and poisson measures we show that for any positive functionfon the discrete cube 0 1 n ent np f pqe np1f df 2where npis the product measure of the bernoulli measure with probability of successp as well as related inequalities which may be shown to imply in the limit the classical gaussian logarithmic sobolev inequality as well as a logarithmic sobolev inequality for poisson measure we further investigate modified logarithmic sobolev inequalities to analyze integrability properties of lipschitz functions on discrete spaces in particular we obtain under modified logarithmic sobolev inequalities some concentration results for product measures that extend the classical exponential inequalities for sums of independent random variables

decay of correlations under dobrushin s uniqueness condition and its applications an estimate on the correlation of functionals of gibbs fields satisfying dobrushin s uniqueness condition is given as a consequence a result of gross saying that the truncated pair correlation function decays in the same weighted summability sense as the potential can be extended to the whole dobrushin uniqueness region applications to the central limit theorem and the second derivative of the pressure are also given

imposing dirichlet boundary conditions with nitsche s method and spline based finite elements a key challenge while employing non interpolatory basis functions in finite element methods is the robust imposition of dirichlet boundary conditions the current work studies the weak enforcement of such conditions for b spline basis functions with application to both second and fourth order problems this is achieved using concepts borrowed from nitsche s method which is a stabilized method for imposing constraints on surfaces conditions for the stability of the system of equations are derived for each class of problem stability parameters in the nitsche weak form are then evaluated by solving a local generalized eigenvalue problem at the dirichlet boundary the approach is designed to work equally well when the grid used to build the splines conforms to the physical boundary of interest as well as to the more general case when it does not through several numerical examples the approach is shown to yield optimal rates of convergence copyright 2010 john wiley sons ltd

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

structures having o minimal open core the open core of an expansion of a dense linear order is its reduct in the sense of definability generated by the collection of all of its open definable sets in this paper expansions of dense linear orders that have o minimal open core are investigated with emphasis on expansions of densely ordered groups the first main result establishes conditions under which an expansion of a densely ordered group has an o minimal open core specifically the following is proved let r be an expansion of a densely ordered group r that is definably complete and satisfies the uniform finiteness property then the open core of r is o minimal two examples of classes of structures that are not o minimal yet have o minimal open core are discussed dense pairs of o minimal expansions of ordered groups and expansions of o minimal structures by generic predicates in particular such structures have open core interdefinable with the original o minimal structure these examples are differentiated by the existence of definable unary functions whose graphs are dense in the plane a phenomenon that can occur in dense pairs but not in expansions by generic predicates the property of having no dense graphs is examined and related to uniform finiteness definable completeness and having o minimal open core