toeplitz matrices and random walks with memory abstract we use a technique based on toeplitz matrices to calculate the probability distribution for certain random walks on a lattice in continuous time where the walker can take steps of various sizes in each direction and where the probability of a step depends on the nature of a finite set of previous steps if k ij is the rate constant for a step of j units given a history of type i then we can solve the random walk problem for the special case when the sum over k ij is independent of j

on trace identities and universal eigenvalue estimates for some partial differential operators in this article we prove and exploit a trace identity for the spectra of schrodinger operators and similar operators this identity leads to universal bounds on the spectra which apply to low lying eigenvalues eigenvalue asymptotics and to partition functions traces of heat operators in many cases they are sharp in the sense that there are specific examples for which the inequalities are saturated special cases corresponding to known inequalities include those of hile and protter and of yang introduction in this article we prove and exploit an identity for the spectra of self adjoint operators h modeled on the dirichlet laplacian or more generally on schrodinger operators of the form p a x 2 v x 1 where p 1i is the usual momentum operator in convenient units and a x is the magnetic vector potential we recover and extend several known inequalities involving sums differences and ratios of eigenvalues let j j 1 denote the ordered eigenvalues of the dirichlet laplacian on a bounded d dimensional domain with zero dirichlet boundary conditions and recall that hile and protter hipr80 proved that d 4 1 n n j 1 j n 1 j 2 thereby extending an earlier inequality of payne polya and weinberger papowe56 in the last few years it has become clear that these and many similar relationships can be realized as special cases of abstract variational bounds involving the interplay among commutators of 2 a cartesian coordinate xj and the corresponding derivative xj for this analysis and various extensions of 2 see ha88 ho90 ha93 hami95 while inequalities of this type have been fairly sharp for low lying eigenvalues they have been mostly disappointing for higher eigenvalues yang ya91 however received by the editors september 28 1995 1991 mathematics subject classification primary 35j10 35j25 58g25

estimates on eigenvalues of laplacian in this paper we study eigenvalues of laplacian on either a bounded connected domain in an n dimensional unit sphere sn 1 or a compact homogeneous riemannian manifold or an n dimensional compact minimal submanifold in an n dimensional unit sphere sn 1 we estimate the k 1 th eigenvalue by the first k eigenvalues as a corollary we obtain an estimate of difference between consecutive eigenvlaues our results are sharper than ones of p c yang and yau 25 leung 19 li 20 and harrel ii and stubbe 12 respectively from weyl s asymptotical formula we know that our estimates are optimal in the sense of the order of k for eigenvalues of laplacian on a bounded connected domain in an n dimensional unit sphere sn 1

trace identities for commutators with applications to the distribution of eigenvalues we prove trace identities for commutators of operators which are used to derive sum rules and sharp universal bounds for the eigenvalues of periodic schrodinger operators and schrodinger operators on immersed manifolds in particular we prove bounds on the eigenvalue lambda n 1 in terms of the lower spectrum bounds on ratios of means of eigenvalues and universal monotonicity properties of eigenvalue moments which imply sharp versions of lieb thirring inequalities in the case of a schrodinger operator on an immersed manifold of dimension d we derive a version of reilly s inequality bounding the eigenvalue lambda n 1 of the laplace beltrami operator by a universal constant times vertical bar vertical bar h vertical bar n 2 infinity 2 d

on the cauchy problem for the fast diffusion equation abstract for u 0 1 u 0 l loc 1 r n the author studies the existence of a kind of weak solution to the cauchy problem u t div u m 1 d u in r n 0 t u x 0 u 0 x 0 in r n where m 0 is a constant the uniqueness and regularity of solutions are also discussed

eigenvalue inequalities for klein gordon operators we consider the pseudodifferential operators hm associated by the prescriptions of quantum mechanics to the klein gordon hamiltonian p 2 m2 when restricted to a bounded open domain rd when the mass m is 0 the operator h0 coincides with the generator of the cauchy stochastic process with a killing condition on the operator h0 is sometimes called the fractional laplacian with power 12 cf r banuelos t kulczycki eigenvalue gaps for the cauchy process and a poincare inequality j funct anal 211 2 2004 355 423 r banuelos t kulczycki the cauchy process and the steklov problem j funct anal 234 2006 199 225 e giere the fractional laplacian in applications http www eckhard giere de math publications review pdf we prove several universal inequalities for the eigenvalues 0 1 2 of hm and their means k 1k l 1k l among the inequalities proved are k cst k 1 d for an explicit optimal semiclassical constant depending only on the dimension d for any dimension d 2 and any k k 1 d 1d 1 k furthermore when d 2 and k 2j k j d21 d d 1 kj 1d finally we present some analogous estimates allowing for an operator including an external potential energy field i e hm v x for v x in certain function classes

on the application of maximum entropy to the moments problem the maximum entropy approach to the solution of classical inverse problem of moments in which one seeks to reconstruct a function p x where x 0 from the values of a finite set n 1 of its moments is studied it is shown that for n 4 such a function always exists while for n 2 and n 3 the acceptable values of the moments are singled out analytically the paper extends to the general case where the results were previously bounded to the case n 2

differential inequalities for riesz means and weyl type bounds for eigenvalues we derive differential inequalities and difference inequalities for riesz means of eigenvalues of the dirichlet laplacian r z x k z k

universal inequalities for the eigenvalues of schrodinger operators on submanifolds we establish inequalities for the eigenvalues of schr o dinger operators on compact submanifolds possibly with nonempty boundary of euclidean spaces of spheres and of real complex and quaternionic projective spaces which are related to inequalities for the laplacian on euclidean domains due to payne polya and weinberger and to yang but which depend in an explicit way on the mean curvature in later sections we prove similar results for schr o dinger operators on homogeneous riemannian spaces and more generally on any riemannian manifold that admits an eigenmap into a sphere as well as for the kohn laplacian on subdomains of the heisenberg group among the consequences of this analysis are an extension of reilly s inequality bounding any eigenvalue of the laplacian in terms of the mean curvature and spectral criteria for the immersibility of manifolds in homogeneous spaces

on eigenvalues of a system of elliptic equations and of the biharmonic operator abstract let be a bounded domain in an n dimensional euclidean space r n we study eigenvalues of an eigenvalue problem of a system of elliptic equations u grad div u u in u 0 estimates for eigenvalues of the above eigenvalue problem are obtained furthermore we obtain an upper bound on the k 1 th eigenvalue k 1 we also obtain sharp lower bound for the first eigenvalue of two kinds of eigenvalue problems of the biharmonic operator on compact manifolds with boundary and positive ricci curvature

universal bounds for eigenvalues of the polyharmonic operators we study eigenvalues of polyharmonic operators on compact riemannian manifolds with boundary possibly empty in particular we prove a universal inequality for the eigenvalues of the polyharmonic operators on compact domains in a euclidean space this inequality controls the kth eigenvalue by the lower eigenvalues independently of the particular geometry of the domain our inequality is sharper than the known payne polya weinberg type inequality and also covers the important yang inequality on eigenvalues of the dirichlet laplacian we also prove universal inequalities for the lower order eigenvalues of the polyharmonic operator on compact domains in a euclidean space which in the case of the biharmonic operator and the buckling problem strengthen the estimates obtained by ashbaugh finally we prove universal inequalities for eigenvalues of polyharmonic operators of any order on compact domains in the sphere

universal bounds for eigenvalues of the biharmonic operator abstract in this paper we study the eigenvalues of the clamped plate problem 2 u u in d u d u d 0 where d is a bounded connected domain in an n dimensional complete minimal submanifold of a unit m sphere s m 1 or of an m dimensional euclidean space r m let 0 1 2 k be the eigenvalues of the above problem we obtain universal bounds on k 1 in terms the first k eigenvalues independent of the domains for example when d is contained in an n dimensional complete minimal submanifold of s m 1 we show that k 1 1 k i 1 k i 1 k n i 1 k k 1 i 1 2 2 n 4 i 1 2 n 2 1 2 i 1 k k 1 i 1 2 4 i 1 2 n 2 1 2 from which one can obtain a more explicit upper bound on k 1 in terms of 1 k see corollary 1 when d is contained in a complete n dimensional minimal submanifold of r m we prove the inequality k 1 1 k i 1 k k 8 n 2 n 2 1 2 1 k i 1 k i k 1 i 1 2 which generalizes the main theorem in cheng and yang 2006 10 that states that the same estimate holds when d is a connected and bounded domain in r n

the cauchy problem for ut δu m when 0 m 1 this paper deals with the cauchy problem for the nonlinear diffusion equation u t u u m 1 0 on 0 x rn u 0 u0 when 0 m 1 fast diffusion case we prove that there exists a global time solution for any locally integrable function u0 hence no growth condition at infinity for u0 is required moreover the solution is shown to be unique in that class behavior at infinity of the solution and l loc regularizing effects are also examined when m max n 2 n 0 1

commutators spectral trace identities and universal estimates for eigenvalues using simple commutator relations we obtain several trace identities involving eigenvalues and eigenfunctions of an abstract self adjoint operator acting in a hilbert space applications involve abstract universal estimates for the eigenvalue gaps as particular examples we present simple proofs of the classical universal estimates for eigenvalues of the dirichlet laplacian as well as of some known and new results for other differential operators and systems we also suggest an extension of the methods to the case of non self adjoint operators

domain independent upper bounds for eigenvalues of elliptic operators let q c ir m be a bounded open set aq its boundary and a the laplacian on rm consider the elliptic differential equation 1 au au in q u 0 on aq it is known that the eigenvalues i of 1 satisfy n m n 2 i l n 1l 1 provided that n l an in this paper we abstract the method used by hile and protter 2 to establish 2 and apply the method to a variety of second order elliptic problems in particular to all constant coefficient problems we then consider a variety of higher order problems and establish an extension of 2 for problem 1 where the laplacian is replaced by a more general operator in a hilbert space

upper bounds for eigenvalues of the discrete and continuous laplace operators in this paper we are concerned with upper bounds of eigenvalues of laplace operator on compact riemannian manifolds and finite graphs while on the former the laplace operator is generated by the riemannian metric on the latter it reflects combinatorial structure of a graph respectively eigenvalues have many applications in geometry as well as in combinatorics and in other fields of mathematics we develop a universal approach to upper bounds on both continuous and discrete structures based upon certain properties of the corresponding heat kernel this approach is perhaps much more general than its realization here basically we start with the following entries 1 an underlying space m with a finite measure 2 a well defined laplace operator on functions on m so that is a self adjoint operator in l m with a discrete spectrum 3 if m has a boundary then the boundary condition should be chosen so that it does not disrupt self adjointness of and is of dissipative nature 4 a distance function dist x y on m so that dist 1 for an appropriate notion of gradient

a sharp bound for the ratio of the first two eigenvalues of dirichlet laplacians and extensions the first author s work was partially supported by fondecyt chile project 0132 88 and by a summer research fellowship provided by the research council of the university of missouri columbia he would like to thank the physics department and others at the universidad de chile for their hospitality during his visit in april 1990 when much of this research was completed the second author was supported in part by fondecyt chile projects 0132 88 and 1238 90 both authors also thank fritz gesztesy for general comments and encouragement

universal inequalities for eigenvalues of the buckling problem on spherical domains in this paper we study the eigenvalues of the buckling problem on domains in a unit sphere we obtain universal bounds on the k 1 th eigenvalue in terms of the first k eigenvalues independent of the domains

aronson benilan estimates for the fast diffusion equation under the ricci flow abstract we study the fast diffusion equation with a linear forcing term 0 1 u t div u p 1 u r u under the ricci flow on a complete manifold m such that m r 2 has bounded curvature and nonnegative isotropic curvature where 0 p 1 and r r x t is the evolving scalar curvature of m at time t we prove aronson benilan and li yau hamilton type differential harnack estimates for positive solutions of 0 1 in addition we use similar method to prove certain li yau hamilton estimates for the heat equation and conjugate heat equation which extend those obtained in cao and hamilton 2009 cao 2008 and kuang and zhang 2008 to the noncompact setting

multiple step random walks distribution functions from toeplitz matrices abstract toeplitz matrices are used to calculate the distribution function for a random walk on a lattice in continuous time where multiple hops covering an arbitrary number of lattice sites are allowed the method employed is similar to that used to calculate the complete end to end distance distribution for polymers

universal bounds for eigenvalues of a buckling problem in this paper we investigate an eigenvalue problem for a biharmonic operator on a bounded domain in an n dimensional euclidean space which is also called a buckling problem we introduce a new method to construct nice trial functions and we derive a universal inequality for higher eigenvalues of the buckling problem by making use of the trial functions thus we give an affirmative answer for the problem on universal bounds for eigenvalues of the buckling problem which was proposed by payne polya and weinberger in 14 and this problem has been mentioned again by ashbaugh in 1

asymptotic behavior of the nonlinear diffusion equation nt n 1nx x the asymptotic behavior of the equation nt ln n xx is studied on the finite interval 0 x 1 with the boundary conditions n 0 t n 1 t n0 and initial data n x 0 n0 we prove that asymptotically ln n x t n0 a exp 2t n0 21 2 sin x and also provide rigorous upper and lower bounds on the asymptotic amplitude a in terms of integrals of nonlinear functions of the initial data the rigorous bounds are compared to values of a obtained from computer experiments the lower bound l 23 2 exp li 1 q where li is the logarithmic integral is euler s constant and q 2 f n x 0 n0 1 sin x dx is found to be the best known estimate of a

existence of solutions to initial value problem for a parabolic monge ampere equation and application abstract for the initial value problem of the parabolic monge ampere equation v t v xx rxv x v xx v x 2 0 x t r 0 t arising from the optimal investment of mathematical finance theory we establish the existence of solutions whose application is also given here the initial function is unbounded and a special property is required for the solution to satisfy

inequalities for eigenvalues of selfadjoint operators we establish several inequalities for eigenvalues of selfadjoint operators in hilbert space the results are quite general in particular let be a region in r n its boundary and the laplace operator in r n

universal inequalities for the eigenvalues of laplace and schrodinger operators on submanifolds we establish inequalities for the eigenvalues of schrodinger oper ators on compact submanifolds possibly with nonempty boundary of euclidean spaces of spheres and of real complex and quaternionic projective spaces which are related to inequalities for the laplacian on euclidean domains due to payne polya and weinberger and to yang but which depend in an explicit way on the mean curvature in later sections we prove similar results for schrodinger operators on homogeneous riemannian spaces and more generally on any rie mannian manifold that admits an eigenmap into a sphere as well as for the kohn laplacian on subdomains of the heisenberg group among the consequences of this analysis are an extension of reilly s inequality bounding any eigenvalue of the laplacian in terms of the mean curvature and spectral criteria for the immersibility of manifolds in homogeneous spaces