Results 1 
4 of
4
Integrated Density Of States For Random Metrics On Manifolds
 Proc. London Math. Soc
, 2002
"... We study ergodic random Schrödinger operators on a covering manifold, where the randomness enters both via the potential and the metric. We prove measurability of the random operators, almost sure constancy of their spectral properties, the existence of a selfaveraging integrated density of states a ..."
Abstract

Cited by 8 (6 self)
 Add to MetaCart
We study ergodic random Schrödinger operators on a covering manifold, where the randomness enters both via the potential and the metric. We prove measurability of the random operators, almost sure constancy of their spectral properties, the existence of a selfaveraging integrated density of states and a Subin type trace formula.
Data Analysis and Representation on a General Domain using Eigenfunctions of Laplacian
, 2007
"... We propose a new method to analyze and represent data recorded on a domain of general shape in R d by computing the eigenfunctions of Laplacian defined over there and expanding the data into these eigenfunctions. Instead of directly solving the eigenvalue problem on such a domain via the Helmholtz ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
(Show Context)
We propose a new method to analyze and represent data recorded on a domain of general shape in R d by computing the eigenfunctions of Laplacian defined over there and expanding the data into these eigenfunctions. Instead of directly solving the eigenvalue problem on such a domain via the Helmholtz equation (which can be quite complicated and costly), we find the integral operator commuting with the Laplacian and diagonalize that operator. Although our eigenfunctions satisfy neither the Dirichlet nor the Neumann boundary condition, computing our eigenfunctions via the integral operator is simple and has a potential to utilize modern fast algorithms to accelerate the computation. We also show that our method is better suited for small sample data than the KarhunenLoève Transform/Principal Component Analysis. In fact, our eigenfunctions depend only on the shape of the domain, not the statistics of the data. As a further application, we demonstrate the use of our Laplacian eigenfunctions for solving the heat equation on a complicated domain.
Weyls Law: Spectral Properties of the Laplacian in Mathematics and Physics
 Mathematical Analysis of Evolution, Information and Complexity. WileyVerlag, 2009. 27 Rajendra Bhatia, Linear Algebra to Quantum Cohomology: The Story of Alfred Horn’s Inequalities, Amer. Math. Monthly 108(2001
"... Weyl’s law is in its simplest version a statement on the asymptotic growth of the eigenvalues of the Laplacian on bounded domains with Dirichlet and Neumann boundary conditions. In the typical applications in physics one deals either with the Helmholtz wave equation describing the vibrations of a st ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Weyl’s law is in its simplest version a statement on the asymptotic growth of the eigenvalues of the Laplacian on bounded domains with Dirichlet and Neumann boundary conditions. In the typical applications in physics one deals either with the Helmholtz wave equation describing the vibrations of a string, a membrane
Approximating L²invariants of amenable covering spaces: A heat kernel approach
, 1996
"... In this paper, we prove that the L² Betti numbers of an amenable covering space can be approximated by the average Betti numbers of a regular exhaustion, under some hypotheses. We also prove that some L² spectral invariants can be approximated by the corresponding average spectral invariants of a re ..."
Abstract
 Add to MetaCart
In this paper, we prove that the L² Betti numbers of an amenable covering space can be approximated by the average Betti numbers of a regular exhaustion, under some hypotheses. We also prove that some L² spectral invariants can be approximated by the corresponding average spectral invariants of a regular exhaustion. The main tool which is used is a generalisation of the ”principle of not feeling the boundary” (due to M. Kac), for heat kernels associated to boundary value problems.