numerical results on a hypercubic cell spin glass model the authors study a spin glass hypercubic cell in d dimensions i e hypercubic lattice with l 2 where l is the lattice size for different values of d by means of monte carlo simulation in the limit d to infinity this model converges to the sk model as well as for hypercubic lattices they confirm the usual interpretation of the spin glass phase since this model is more similar to hypercubic lattices than to the infinite range model these results suggest that broken replica symmetry is useful to study finite dimensional hypercubic lattices above a certain critical dimension

essential self adjointness of the graph laplacian we study the operator theory associated with such infinite graphs g as occur in electrical networks in fractals in statistical mechanics and even in internet search engines our emphasis is on the determination of spectral data for a natural laplace operator associated with the graph in question this operator will depend not only on g but also on a prescribed positive real valued function c defined on the edges in g in electrical network models this function c will determine a conductance number for each edge we show that the corresponding laplace operator is automatically essential self adjoint by this we mean that is defined on the dense subspace d of all the real valued functions on the set of vertices g0 with finite support in the hilbert space l2 g0 the conclusion is that the closure of the operator is self adjoint in l2 g0 and so in particular that it has a unique spectral resolution determined by a projection valued measure on the borel subsets of the infinite half line we

symmetric green s functions on complete manifolds on demontre d une facon constructive que l operateur de laplace admet une fonction de green symetrique g x y sur une variete de riemann complete

sharp estimates for capacities and applications to symmetric diffusions we give sharp estimates for the capacity of compact setf x wherex is the state space of a strongly regular dirichlet form of diffusion type these estimates are in terms of the reference measurem and the caratheodory metric onx for instance cap 0 f leqq left mathop smallint limits 0 infty frac dr v r right 1 leqq 2 left mathop smallint limits 0 infty frac r dr v r right 1 wherev r m 0 x f r from these estimates one easily obtains lower estimates for the green function as well as sharp criteria for polarity and for recurrence the latter we apply to the question of hitting the nodal set 0 for the process associated with the operator delta 2 tfrac nabla varphi varphi nabla and to recurrence questions for divergence form operators a ond n witha a ij which degenerate at d

some uniqueness and exact multiplicity results for a predator prey model in this paper we consider positive solutions of a predator prey model with diffusion and under homogeneous dirichlet boundary conditions it turns out that a certain parameter m in this model plays a very important role a good understanding of the existence stability and number of positive solutions is gained when m is large in particular we obtain various results on the exact number of positive solutions our results for large m reveal interesting contrast with that for the well studied casem 0 i e the classical lotka volterra predator prey model

volume growth spectrum and stochastic completeness of infinite graphs we study the connections between volume growth spectral properties and stochastic completeness of locally finite weighted graphs for a class of graphs with a very weak spherical symmetry we give a condition which implies both stochastic incompleteness and discreteness of the spectrum we then use these graphs to give some comparison results for both stochastic completeness and estimates on the bottom of the spectrum for general locally finite weighted graphs

global stability and hopf bifurcations of a beddington deangelis type predator prey system with diffusion and delays the dynamics of a diffusive predator prey system with the beddington deangelis functional response and two delays are considered by choosing the sum of two delays as a bifurcation parameter the stability of the constant equilibria and the existence of hopf bifurcations are investigated by analyzing the characteristic equations and the formulas determining the direction and the stability of the bifurcating periodic solutions are derived by using the center manifold and the normal form theory of partial functional differential equations particularly the sufficient conditions of the global stability of the positive equilibrium are given by the upper lower solutions method

function theory random paths and covering spaces pour des varietes de riemann connexes m on etudie la relation entre la theorie de fonction harmonique sur m les proprietes statistiques des chemins aleatoires sur m et la structure geometrique globale de m

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

discontinuity of the magnetization in one dimensional 1 x y 2 ising and potts models results from percolation theory are used to study phase transitions in one dimensional ising and q state potts models with couplings of the asymptotic form j sub x y approx const absolute value x y sup 2 for translation invariant systems with well defined lim sub x infinity x sup 2 j sub x j sup possibly o or infinity we establish 1 there is no long range order at inverse temperatures beta with beta j sup less than or equal to1 2 if beta j sup q then by sufficiently increasing j sub 1 the spontaneous magnetization m is made positive 3 in models with 0 j sup infinity the magnetization is discontinuous at the transition point as originally predicted by thouless and obeys m beta sub c less than or equal to 1 beta sub c j sup sup 1 2 4 for ising q 2 models with j sup infinity it is noted that the correlation function decays as beta approx c beta absolute value x y sup 2 whenever beta beta sub c points 1 3 are deduced from previous percolation results by utilizing the fortuin kasteleyn representation which also yields other results of independent interest relating potts models with different values of q

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

instability of the liouville property for quasi isometric riemannian manifolds and reversible markov chains on montre par un exemple que la propriete de liouville n est pas un invariant quasi isometrique des varietes de riemann et des chaines de markov reversibles

bifurcation of steady state solutions in predator prey and competition systems we discuss steady state solutions of systems of semilinear reaction diffusion equations which model situations in which two interacting species u and v inhabit the same bounded region it is easy to find solutions to the systems such that either u or v is identically zero such solutions correspond to the case where one of the species is extinct by using decoupling and global bifurcation theory techniques we prove the existence of solutions which are positive in both u and v corresponding to the case where the populations can co exist

uniqueness of l 1 solutions for the laplace equation and the heat equation on riemannian manifolds on deduit une hypothese geometrique optimale sur une variete de riemann complete qui assure des proprietes d unicite pour des solutions l 1 de l equation de laplace et de la chaleur

ollivier ricci curvature for general graph laplacians heat equation laplacian comparison non explosion and diameter bounds abstract discrete time random walks on a finite set naturally translate via a one to one correspondence to discrete laplace operators typically ollivier curvature has been investigated via random walks we first extend the definition of ollivier curvature to general weighted graphs and then give a strikingly simple representation of ollivier curvature using the graph laplacian using the laplacian as a generator of a continuous time markov chain we connect ollivier curvature with the heat equation which is strongly related to continuous time random walks in particular we prove that a lower bound on the ollivier curvature is equivalent to a certain lipschitz decay of solutions to the heat equation this is a discrete analogue to a celebrated ricci curvature lower bound characterization by renesse and sturm our representation of ollivier curvature via the laplacian allows us to deduce a laplacian comparison principle by which we prove non explosion and improved diameter bounds

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

heat semigroup on a complete riemannian manifold let m be a complete riemannian manifold and p t x y the minimal heat kernel on m let p t be the associated semigroup we say that m is stochastically complete if int m p t x y dy 1 for all t 0 x in m we say that m has the c 0 diffusion property or the feller property if p tf vanishes at infinity for all t 0 whenever f is so let x 0 in m and let kappa r 2 geq inf ric x rho x x 0 leq r rho is the riemannian distance we prove that m is stochastically complete and has the c 0 diffusion property if int infty c kappa r 1 dr infty by studying the radial part of the riemannian brownian motion on m

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

multiplicity and uniqueness of positive solutions for a predator prey model with b d functional response abstract in this paper we consider positive solutions of a predator prey model with beddington deangelis functional response first we establish the existence of positive solutions and present some sufficient and necessary conditions mainly using the standard fixed point index theory in cone second the multiplicity uniqueness and stability of positive solutions are obtained by means of bifurcation theory perturbation theory of eigenvalues and the fixed point index theory the parameters k and m play very important roles in deciding the number of positive solutions which are showed explicitly in this paper

on the conservativeness and the recurrence of symmetric jump diffusions abstract sufficient conditions for a symmetric jump diffusion process to be conservative and recurrent are given in terms of the volume of the state space and the jump kernel of the process a number of examples are presented to illustrate the optimality of these conditions in particular the situation is allowed to be that the state space is topologically disconnected but the particles can jump from a connected component to the other components

s shaped global bifurcation curve and hopf bifurcation of positive solutions to a predator prey model where d is a bounded domain in r n 1 with smooth boundary d and a b c d m are positive constants problem 1 0 is known as the predator prey model with holling tanner type interactions where u and v represent the densities of the prey and predator respectively hence we are only interested in positive solutions by which we mean that both u and v are positive in d it is easy to see that a 1 is necessary for the existence of positive solutions to 1 0 and we shall also assume that d 1 the case d 1 is rather different and will not be discussed here we are mainly interested in studying the number of positive solutions of 1 0 and the stability of these solutions in particular we shall show that when b d c and m fall into certain range the solution set u v a of 1 0 forms a s shaped smooth curve and that hopf bifurcation occurs along this curve this not only confirms rigorously similar numerical observations on 1 0 made in 4 but also shows that the corresponding parabolic system

stochastic completeness for graphs with curvature dimension conditions abstract we prove pointwise gradient bounds for heat semigroups associated to general possibly unbounded laplacians on infinite graphs satisfying the curvature dimension condition cd k using gradient bounds we show stochastic completeness for graphs satisfying the curvature dimension condition

minimal graphs over riemannian surfaces and harmonic diffeomorphisms we construct a parabolic entire minimal graph s over a finite topology complete riemannian surface sigma of curvature 1 and infinite area thus of non parabolic conformal type the vertical projection of this graph yields a harmonic diffeomorphism from s onto sigma the proof uses the theory of divergence lines to construct minimal graphs we also generalize a theorem of r schoen let g 1 and g 2 be two complete metrics on a orientable surface s with compact boundary and suppose int s r 2 k g 2 d sigma g 2 le c ln 2 r for some c 0 and all r 0 if there is a harmonic diffeomorphism from s g 1 to s g 2 then s g 1 is parabolic

a sears type self adjointness result for discrete magnetic schrodinger operators abstract in the context of a weighted graph with vertex set v and bounded vertex degree we give a sufficient condition for the essential self adjointness of the operator w where is the magnetic laplacian and w v r is a function satisfying w x q x for all x v with q v 1 the condition is expressed in terms of completeness of a metric that depends on q and the weights of the graph the main result is a discrete analogue of the results of i oleinik and m a shubin in the setting of non compact riemannian manifolds

on the indices of fixed points of mappings in cones and applications abstract assume that k is a cone in a banach space and a k k is completely continuous we obtain a formula for the index in k of a fixed point of a under the assumption that a linearization exists and satisfies an invertibility condition we then use this formula to generalize some results of amann on the number of fixed points of a to the case where k has empty interior

on stochastic completeness of jump processes we prove the following sufficient condition for stochastic completeness of symmetric jump processes on metric measure spaces if the volume of the metric balls grows at most exponentially with radius and if the distance function is adapted in a certain sense to the jump kernel then the process is stochastically complete we use this theorem to prove the following criterion for stochastic completeness of a continuous time random walk on a graph with a counting measure if the volume growth with respect to the graph distance is at most cubic then the random walk is stochastically complete where the cubic volume growth is sharp

the classical theory of minimal surfaces we present here a survey of recent spectacular successes in classical minimal surface theory we highlight this article with the theorem that the plane the helicoid the catenoid and the one parameter family frtgt2 0 1 of riemann minimal examples are the only complete properly embedded minimal planar domains in r 3 the proof of this result depends primarily on work of colding

uniqueness and stability of a predator prey model with c m functional response in this paper we are concerned with positive solutions of a predator prey model with crowley martin functional response under dirichlet boundary conditions first we establish the existence of positive solutions when a c is close to 1 1 next the stability properties of nonnegative solutions are discussed by spectral analysis in addition we obtain a complete understanding of the uniqueness and non uniqueness of positive solutions by the lyapunov schmidt procedure and the perturbation technique at last some numerical simulations are presented to supplement the analytic results in one dimension

range of fluctuation of brownian motion on a complete riemannian manifold we investigate the escape rate of the brownian motion w x t on a complete noncompact riemannian manifold assuming that the manifold has at most polynomial volume growth and that its ricci curvature is bounded below we prove that dist w x t x ct log t for all large t with probability 1 on the other hand if the ricci curvature is nonnegative and the volume growth is at least polynomial of the order n 2 then formula math again for all large t with probability 1 where e 0

difference equations isoperimetric inequality and transience of certain random walks the difference laplacian on a square lattice in rn has been stud ied by many authors in this paper an analogous difference operator is studied for an arbitrary graph it is shown that many properties of the laplacian in the continuous setting e g the maximum principle the harnack inequality and cheeger s bound for the lowest eigenvalue hold for this difference oper ator the difference laplacian governs the random walk on a graph just as the laplace operator governs the brownian motion as an application of the theory of the difference laplacian it is shown that the random walk on a class of graphs is transient the random walks we consider are defined as follows let k be a connected graph i e a one dimensional simplicial complex for a vertex x e k let m x denote the number of edges emanating from x the probability that a particle moves from x to another vertex y e k is l m x if x and y are connected by an edge and it is zero otherwise as observed by courant friedrichs and lewy cfl for the case of a square lattice in the plane this random walk is intimately related to the difference analog of the laplacian

stochastic incompleteness for graphs and weak omori yau maximum principle abstract we prove an analogue of the weak omori yau maximum principle and khas minskii s criterion for graphs in the general setting of keller and lenz our approach naturally gives the stability of stochastic incompleteness under certain surgeries of graphs it allows to develop a unified approach to all known criteria of stochastic completeness incompleteness as well as to obtain new criteria

non glassy ground state in a long range antiferromagnetic frustrated model in the hypercubic cell abstract we analyzed the statistical mechanics of a long range antiferromagnetic model defined on a d dimensional hypercube both at zero and finite temperatures the associated hamiltonian is derived from a recently proposed complexity measure of boolean functions in the context of neural networks learning processes we show that depending on the value of d the system either presents a low temperature antiferromagnetic stable phase or the global antiferromagnetic order disappears at any temperature in the last case the ground state is an infinitely degenerated non glassy one composed by two equal size anti aligned antiferromagnetic domains we also present some results for the ferromagnetic version of the model

on uniqueness class for a heat equation on graphs abstract we study the uniqueness class problem for a class of heat equations defined on weighted graphs as an application we recover the volume growth criterion for stochastic completeness of grigor yan huang and masamune in the case of weighted graphs

dynamics of a diffusive predator prey system with a nonlinear growth rate for the predator abstract in this paper we investigate a diffusive predator prey system with holling type ii functional response and a nonlinear growth rate for the predator our results include the global attractivity of constant equilibria and non existence of non constant positive steady states these results give some ranges for the model parameters within which spatiotemporal pattern formation is impossible

a probabilistic approach to martin boundaries for manifolds with ends we use martingale methods and coupling arguments to prove results of li and tam 1987 and donnelly 1986 characterizing positive and bounded harmonic functions respectively on certain manifolds with ends

diffusion determines the recurrent graph abstract we consider diffusion on discrete measure spaces as encoded by markovian semigroups arising from weighted graphs we study whether the graph is uniquely determined if the diffusion is given up to order isomorphism if the graph is recurrent then the complete graph structure and the measure space are determined up to an overall scaling as shown by counterexamples this result is optimal without the recurrence assumption the graph still turns out to be determined in the case of normalized diffusion on graphs with standard weights and in the case of arbitrary graphs over spaces in which each point has the same mass these investigations provide discrete counterparts to studies of diffusion on euclidean domains and manifolds initiated by arendt and continued by arendt biegert ter elst and arendt ter elst a crucial step in our considerations shows that order isomorphisms are actually unitary maps up to a scaling in our context

a liouville property for schrodinger operators 6 examples 27 6 1 thick and thin sets in r 27 6 2 big and small measures in r 29 6 3 a scattered measure in r which is big 31 6 4 counter example to theorem 1 2 32 6 5 counter example to theorem 5 2 34 6 6 harmonic functions with the third boundary condition 35

qualitative analysis on a diffusive predator prey model with toxins abstract this paper is concerned with the diffusive predator prey model with toxins subject to dirichlet boundary conditions the uniform persistence of positive solution is given under certain conditions in addition by the liapunov schmidt method the existence and stability of the bifurcation solution from a double eigenvalues are investigated moreover by the fixed point index theory and perturbation theory of eigenvalues the uniqueness stability and multiplicity of coexistence states are analyzed when some key parameter changes finally some numerical simulations are presented to verify the theoretical conclusions and further to reflect the importance of parameters to the number of coexistence states

complete mathematical analysis of predator prey models with linear prey growth and beddington deangelis functional response the dynamics of a predator prey model with a beddington deangelis functional response and linear intrinsic growth rate of the prey population is fully analyzed conditions on local and global stability of the interior equilibrium are established the equilibria type are determined all possible global asymptotic behaviors of the system are considered including the determination of the extinction conditions and existence of periodic orbits it is shown that mutual interference between predators can alone stabilize predator prey interactions even when only a linear intrinsic growth rate of the prey population is considered in the mathematical model additional biological implications and a set of numerical simulations supporting the analysis are also presented

the influence of oppositely classified examples on the generalization complexity of boolean functions in this paper we analyze boolean functions using a recently proposed measure of their complexity this complexity measure motivated by the aim of relating the complexity of the functions with the generalization ability that can be obtained when the functions are implemented in feed forward neural networks is the sum of a number of components we concentrate on the case in which we use the first two of these components the first is related to the average sensitivity of the function and the second is in a sense a measure of the randomness or lack of structure of the function in this paper we investigate the importance of using the second term in the complexity measure and we consider to what extent these two terms suffice as an indicator of how difficult it is to learn a boolean function we also explore the existence of very complex boolean functions considering in particular the symmetric boolean functions

the effect of mutual interference between predators on a predator prey model with diffusion abstract we consider a diffusive predator prey model with beddington deangelis functional response under homogeneous dirichlet boundary conditions the effect of large k which represents the extent of mutual interference between predators is extensively studied by making use of the fixed point index theory we obtain a complete understanding of the existence uniqueness and stability of positive steady states when k is sufficiently large moreover we present some numerical simulations that supplement the analytic results in one dimension

coexistence theorems of steady states for predator prey interacting systems on donne des conditions necessaires et suffisantes pour l existence des solutions positives des etats stationnaires pour des systemes predateur proie sous des conditions aux limites de dirichlet sur r n

rich dynamics of a stochastic michaelis menten type ratio dependent predator prey system abstract stochastic predator prey systems with different functional responses have been studied but the dynamics of a michaelis menten type ratio dependent predator prey system with stochastic perturbation is not investigated systematically in this paper we give the asymptotic behavior of this system sufficient criteria for the existence of a stationary distribution and ergodicity are obtained which means the species are permanent besides we show the situations in which the species are non persistence finally examples and simulations are carried on to verify these results

analysis of the physical laplacian and the heat flow on a locally finite graph we study the physical laplacian and the corresponding heat flow on an infinite locally finite graph with possibly unbounded valence

expansion in generalized eigenfunctions for laplacians on graphs and metric measure spaces we consider an arbitrary selfadjoint operator on a separable hilbert space to this operator we construct an expansion in generalized eigenfunctions in which the original hilbert space is decomposed as a direct integral of hilbert spaces consisting of general eigenfunctions this automatically gives a plancherel type formula for suitable operators on metric measure spaces we discuss some growth restrictions on the generalized eigenfunctions for laplacians on locally finite graphs the generalized eigenfunctions are exactly the solutions of the cor responding difference equation

large time behavior of the heat equation on complete manifolds with non negative ricci curvature sur une variete de riemann complete m on considere le noyau de la chaleur h x y t qui resout l equation de la chaleur t f x t 0 sur mx 0 avec la donnee initiale f x 0 f x en posant f x t m h x y t f y dy

symmetric markov chains in ℤ d how fast can they move consider a reversible markov chain x n which takes values in a subset of d if the steps of the chain are uniformly bounded and the invariant measure satisfies a mild regularity condition varopoulos carne and kesten have obtained estimates on p x n x text 0 lambda n 1 2 which exhibit a gaussian tail in but blow up as n following kesten s approach we derive bounds which are uniform in n in some special cases our main result however is an example which shows that in general the estimates of varopoulos carne and kesten are essentially best possible