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Diffusion determines the compact manifold. (arXiv:1104.1012 [math.AP])
We provide a short proof for the theorem that two compact Riemannian manifolds are isomorphic if and only there exists an order isomorphism which intertwines between the heat semigroups on the manifolds.
arXiv:1104.1012 [math.AP]
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From forms to semigroups. (arXiv:1104.1013 [math.AP])
Form methods give a very efficient tool to solve evolutionary problems on Hilbert space. They were developed by T. Kato [Kat] and, in slightly different language by J.L. Lions. In this expository article we give an introduction based on [AE2]. The main point in our approach is that the notion of closability is not needed anymore. The new setting is particularly efficient for the Dirichlet-to-Neumann operator and degenerate equations. Besides this we give several other examples. This presentation starts by an introduction to holomorphic semigroups. Instead of the contour argument found in the literature, we give a more direct argument based on the Hille--Yosida theorem.
arXiv:1104.1013 [math.AP]
> "We develop a method for evaluation of A. Einstein's strength of systems of partial differential and difference equations based on the computation of Hilbert-type dimension polynomials of the associated differential and difference field extensions. Also we present algorithms for such computations, which are based on the Gr\"obner basis method adjusted for the modules over rings of differential, difference and inversive difference operators. The developed technique is applied to some fundamental systems of PDEs of mathematical physics such as the diffusion equation, Maxwell equations and equations for an electromagnetic field given by its potential. In each of these cases we determine the strength of the original system of PDEs and the strength of the corresponding systems of partial difference equations obtained by forward and symmetric difference schemes. In particular, we obtain a method for comparing two difference schemes from the point of view of their strength."
> "We develop a method for evaluation of A. Einstein's strength of systems of partial differential and difference equations based on the computation of Hilbert-type dimension polynomials of the associated differential and difference field extensions. Also we present algorithms for such computations, which are based on the Gr\"obner basis method adjusted for the modules over rings of differential, difference and inversive difference operators. The developed technique is applied to some fundamental systems of PDEs of mathematical physics such as the diffusion equation, Maxwell equations and equations for an electromagnetic field given by its potential. In each of these cases we determine the strength of the original system of PDEs and the strength of the corresponding systems of partial difference equations obtained by forward and symmetric difference schemes. In particular, we obtain a method for comparing two difference schemes from the point of view of their strength."
> "Partial differential equations are fundamental tools in mathematics,sciences and engineering. This book is mainly an exposition of the various algebraic techniques of solving partial differential equations for exact solutions developed by the author in recent years, with emphasis on physical equations such as: the Calogero-Sutherland model of quantum many-body system in one-dimension, the Maxwell equations, the free Dirac equations, the generalized acoustic system, the Kortweg and de Vries (KdV) equation, the Kadomtsev and Petviashvili (KP) equation, the equation of transonic gas flows, the short-wave equation, the Khokhlov and Zabolotskaya equation in nonlinear acoustics, the equation of geopotential forecast, the nonlinear Schrodinger equation and coupled nonlinear Schrodinger equations in optics, the Davey and Stewartson equations of three-dimensional packets of surface waves, the equation of the dynamic convection in a sea, the Boussinesq equations in geophysics, the incompressible Navier-Stokes equations and the classical boundary layer equations. In linear partial differential equations, we focus on finding all the polynomial solutions and solving the initial-value problems. Intuitive derivations of Lie symmetry of nonlinear partial differential equations are given. These symmetry transformations generate sophisticated solutions with more parameters from relatively simple ones. They are also used to simplify our process of finding exact solutions. We have extensively used moving frames, asymmetric conditions, stable ranges of nonlinear terms, special functions and linearizations in our approaches to nonlinear partial differential equations. The exact solutions we obtained usually contain multiple parameter functions and most of them are not of traveling-wave type."
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"This mini-course of 20 lectures aims at highlights of spectral theory for self-adjoint partial differential operators, with a heavy emphasis on problems with discrete spectrum. Part I: Discrete Spectrum (ODE preview, Laplacian - computable spectra, Schroedinger - computable spectra, Discrete spectral theorem via sesquilinear forms, Laplace eigenfunctions, Natural boundary conditions, Magnetic Laplacian, Schroedinger in confining well, Variational characterizations, Monotonicity of eigenvalues, Weyl's asymptotic, Polya's conjecture, Reaction-diffusion stability, Thin fluid film stability) Part II: Continuous Spectrum (Laplacian on whole space, Schroedinger with $-2sech^2$ potential, Selfadjoint operators, Spectra: discrete and continuous, Discrete spectrum revisited)"
"Kashiwara-Schapira style sheaf theory is used to justify analytic continuability of solutions of a Laplace transformed Schroedinger equation with a small parameter. This partially proves the description of the Stokes phenomenon for WKB asymptotics predicted by Voros in 1983."
