a note on self adjoint extensions of the laplacian on weighted graphs abstract we study the uniqueness of self adjoint and markovian extensions of the laplacian on weighted graphs we first show that for locally finite graphs and a certain family of metrics completeness of the graph implies uniqueness of these extensions moreover in the case when the graph is not metrically complete and the cauchy boundary has finite capacity we characterize the uniqueness of the markovian extensions

on the heat kernel of the bergmann metric on algebraic varieties the restriction of the standard fubini study metric of ycpn l to m m is an incomplete kahler metric g called the bergmann metric let us consider the operators d and 3 defined on c1 functions and c1 1 forms on m m respectively we define the domain 9y d of d to be the set of c1 functions 2 f defined on m m such that both f and df are in l similarly we define the domain 92 6 of to be the set of c1 1 forms co such that both co and 2 6 o are in l we then define the laplacian a with respect to the bergmann metric by a d with domain 9 a given by the set of c2 functions f such that f e 9 d and df e 6 the main purpose of this paper is to show that

adjoint based optimal control using meshfree discretizations the paper at hand presents a combination of optimal control approaches for pdes with meshless discretizations applying a classical lagrangian type particle method to optimization problems with hyperbolic constraints several adjoint based strategies differing in the sequential order of optimization and discretization of the lagrangian or eulerian problem formulation are proposed and compared the numerical results confirm the theoretically predicted independence principle of the optimization approaches and show the expected convergence behavior moreover they exemplify the superiority of meshless methods over the conventional mesh based approaches for the numerical handling and optimization of problems with time dependent geometries and freely moving boundaries

heat equations on minimal submanifolds and their applications soit m n une sous variete a n dimensions immergee de facon minimale de m n l l 1 soit d m un domaine compact on demontre des theoremes de comparaisons pour les noyaux de la chaleur h x y t et k x y t pour les conditions aux limites de dirichet et de neumann

a review of the decomposition method in applied mathematics abstract the decomposition method can be an effective procedure for analytical solution of a wide class of dynamical systems without linearization or weak nonlinearity assumptions closure approximations perturbation theory or restrictive assumptions on stochasticitiy

on the upper estimate of the heat kernel of a complete riemannian manifold let m be a complete non compact riemannian manifold whose sectional curvature is bounded between two constants k and k then one expects that the heat diffusion in such a manifold behaves like the heat diffusion in euclidean space the purpose of this paper is to give a justification of such a statement in 5 j cheeger and the third author found a lower estimate of the heat kernel by comparing it with the heat kernel of the space form whose curvature is the lower bound of the curvature of the manifold this lower estimate is sharp if we insist the dependence should be on the lower bound of the ricci curvature alone it remains to give an upper estimate of the heat kernel one does not however expect to have a comparison theorem for the upper bound because it is more sensitive to the geometry of the manifold in fact the heat kernel of the upper hialf space and the heat kernel of the complete hyperbolic manifold with finite volume have quite different behavior this is reflected by the fact that the laplacian has no discrete spectrum in the first case while infinite number of discrete eigenvalues may exist in the latter case what we will prove here is that in any case the heat kernel has to decay in a manner similar to the euclidean heat kernel thus we will prove that for any constant c 4 there exists c1 depending on c t the bound of the curvature of m and x so that for all t e 0 t

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 kernel and curvature bounds in ricci flows with bounded scalar curvature abstract in this paper we analyze ricci flows on which the scalar curvature is globally or locally bounded from above by a uniform or time dependent constant on such ricci flows we establish a new time derivative bound for solutions to the heat equation based on this bound we solve several open problems 1 distance distortion estimates 2 the existence of a cutoff function 3 gaussian bounds for heat kernels and 4 a backward pseudolocality theorem which states that a curvature bound at a later time implies a curvature bound at a slightly earlier time using the backward pseudolocality theorem we next establish a uniform l 2 curvature bound in dimension 4 and we show that the flow in dimension 4 converges to an orbifold at a singularity we also obtain a stronger e regularity theorem for ricci flows this result is particularly useful in the study of kahler ricci flows on fano manifolds where it can be used to derive certain convergence results

surfaces generated by moving least squares methods an analysis of moving least squares m l s methods for smoothing and interpolating scattered data is presented in particular theorems are proved concerning the smoothness of interpolants and the description of m l s processes as projection methods some properties of compositions of the m l s projector with projectors associated with finiteelement schemes are also considered the analysis is accompanied by examples of univariate and bivariate problems

on the heat operators of normal singular algebraic surfaces soit x une surface algebrique singuliere normale sur c plongee dans l espace projectif p n c l ensemble singularite s de x est un ensemble fini de points isoles en restreignant la metrique de fubini study de p n c a 7b h x s on obtient une variete de riemann incomplete 7b h g on considere le laplacien d agissant sur les fonctions de carre integrable sur 7b h g on demontre que 1 le laplacien est auto adjoint 2 l operateur de la chaleur exp t est de classe trace et il existe une constante k 0 telle que tr exp t kt 2 0 t t 0

bounds for laplace eigenvalues of kahler metrics abstract we prove inequalities for laplace eigenvalues of kahler manifolds generalising to higher eigenvalues the classical inequality for the first laplace eigenvalue due to bourguignon li and yau in 1994 we also obtain similar eigenvalue inequalities for analytic varieties in kahler manifolds

convergence of calabi yau manifolds abstract in this paper we study the convergence of calabi yau manifolds under kahler degeneration to orbifold singularities and complex degeneration to canonical singularities including the conifold singularities and the collapsing of a family of calabi yau manifolds

degenerating hermitian metrics and spectral geometry of the canonical bundle abstract let x h be a compact and irreducible hermitian complex space of complex dimension m in this paper we are interested in the dolbeault operator acting on the space of l 2 sections of the canonical bundle of reg x the regular part of x more precisely let d m 0 l 2 m 0 reg x h l 2 m 1 reg x h be an arbitrarily fixed closed extension of m 0 l 2 m 0 reg x h l 2 m 1 reg x h where the domain of the latter operator is c m 0 reg x we establish various properties such as closed range of d m 0 compactness of the inclusion d d m 0 l 2 m 0 reg x h where d d m 0 the domain of d m 0 is endowed with the corresponding graph norm and discreteness of the spectrum of the associated hodge kodaira laplacian d m 0 d m 0 with an estimate for the growth of its eigenvalues several corollaries such as trace class property for the heat operator associated to d m 0 d m 0 with an estimate for its trace are derived finally in the last part we provide several applications to the hodge kodaira laplacian in the setting of both compact irreducible hermitian complex spaces with isolated singularities and complex projective surfaces

algebraic equations with exponential terms abstract the decomposition method is applied to algebraic equations containing exponential terms the n term approximation n is rapidly damped as n increases yielding an oscillating convergence of superior accuracy n 5 and n 9 for k 2

sobolev spaces laplacian and heat kernel on alexandrov spaces we prove the compactness of the imbedding of the sobolev space w 1 2 0 omega into l 2 omega for any relatively compact open subset omega of an alexandrov space as a corollary the generator induced from the dirichlet energy form has discrete spectrum the generator can be approximated by the laplacian induced from the dc structure on the alexandrov space we also prove the existence of the locally holder continuous heat kernel

extension of the piecewise parabolic method to multidimensional ideal magnetohydrodynamics an extension of the piecewise parabolic method to treat multidimensional ideal magnetohydrodynamical equations is presented in this paper the multidimensional scheme is constructed from a one dimensional functioning code based on the dimensional splitting method originally suggested by strang the functioning code is built upon a nonlinear riemann solver for ideal mhd equations recently developed by the authors the correctness of the scheme is tested in the steepening of waves in both one and two dimensional situations and in various mhd shock tube problems which involve all the discontinuities in ideal mhd the robust character of the scheme is demonstrated in the shock tube problems and in the interaction between mhd shocks and a cloud the results of these problems show that the scheme keeps the principal advantages of a high order godunov scheme robust operation in the presence of very strong waves thin shock fronts with little attendant noise generation and thin contact discontinuity

a matrix li yau hamilton estimate for kahler ricci flow in this paper we prove a new matrix li yau hamilton lyh estimate for kahler ricci flow on manifolds with nonnegative bisectional curvature the form of this new lyh estimate is obtained by the interpolation consideration originated in ch by chow this new inequality is shown to be connected with perelman s entropy formula through a family of differential equalities in the rest of the paper we show several applications of this new estimate and its corresponding estimate for linear heat equation these include a sharp heat kernel comparison theorem generalizing the earlier result of li and tian lt a manifold version of stoll s theorem st on the characterization of algebraic divisors and a localized monotonicity formula for analytic subvarieties which sharpens the bishop volume comparison theorem motivated by the connection between the heat kernel estimate and the reduced volume monotonicity of perelman p we prove a sharp lower bound heat kernel estimate for the time dependent heat equation which is in a certain sense dual to perelman s monotonicity of the reduced volume as an application of this new monotonicity formula we show that the blow down limit of a certain type long time solution is a gradient expanding soliton we also illustrate the connection between the new lyh estimate and the hessian comparison theorem of fin on the forward reduced distance local monotonicity formulae on entropy and forward reduced volume are also derived

modeling of two phase flows with surface tension by finite pointset method fpm a meshfree method for two phase immiscible incompressible flows including surface tension is presented the continuum surface force csf model is used to include the surface tension force the incompressible navier stokes equation is considered as the mathematical model application of implicit projection method results in linear second order partial differential equations for velocities and pressure these equations are then solved by the finite pointset method fpm which is a meshfree and lagrangian method the fluid is represented as finite number of particles and the immiscible fluids are distinguished by the color of each particle the interface is tracked automatically by advecting the color functions for each particle two test cases laplace s law and the rayleigh taylor instability in 2d have been presented the results are found to be consistent with the theoretical results

a review of the decomposition method and some recent results for nonlinear equations abstract the decomposition method can be an effective procedure for solution of nonlinear and or stochastic continous time dynamical systems without usual restrictive assumptions this paper is intended as a convenient tutorial review of the method 1

ends of locally symmetric spaces with maximal bottom spectrum let x be a symmetric space of non compact type and gnx a locally sym metric space then the bottom spectrum l1 gnx satisfies the inequality l1 gnx e l1 x we show that if equality l1 gnx l1 x holds then gnx has either one end which is necessarily of infinite volume or two ends one of infinite volume and another of finite vol ume in the latter case gnx is isometric to r 1 n endowed with a multi warped metric where n is compact

a convenient computational form for the adomian polynomials abstract recent important generalizations by g adomian stochastic systems academic press 1983 have extended the scope of his decomposition method for nonlinear stochastic operator equations see also iterative method inverse operator method symmetrized method or stochastic green s function method very considerably so that they are now applicable to differential partial differential delay and coupled equations which may be strongly nonlinear and or strongly stochastic or linear or deterministic as subcases thus for equations modeling physical problems solutions are obtained rapidly easily and accurately the methodology involves an analytic parametrization in which certain polynomials an dependent on the nonlinearity are derived this paper establishes simple symmetry rules which yield adomian s polynomials quickly to high orders

an extension of the decomposition method for solving nonlinear equations and its convergence in this paper we propose an extension of the decomposition method for solving nonlinear equations after a summary of adomian s decomposition method adm a new kind of decomposition strategy for nonlinear functions in nonlinear equations is proposed and a generalized decomposition method gdm for solving nonlinear equations is presented under a rather mild condition the convergence of the gdm is proved also some concrete examples are studied to illustrate with numerical results how powerful the gdm is finally some general remarks conclude this study