Commutation rules between the Laplace operator and other basic operators (include the divergence, the curl, and the gradient operator) are established. These rules were seldom noticed in the past. However, it is shown that they lead to an alternative derivation of the electromagnetic fields due to an arbitrary current distribution in a homogeneous environment without referring to the vector pot...

Approximations of Laplace-Beltrami operators on manifolds through graph Laplacians have become popular tools in data analysis and machine learning. These discretized operators usually depend on bandwidth parameters whose tuning remains a theoretical and practical problem. In this paper, we address this problem for the unnormalized graph Laplacian by establishing an oracle inequality that opens ...

We investigate the Dirichlet solution for a semianalytic continuous function on the boundary of a semianalytic bounded domain in the plane. We show that the germ of the Dirichlet solution at a boundary point with angle greater than 0 lies in a certain quasianalytic class used by Ilyashenko in his work on Hilbert’s 16 problem. With this result we can prove that the Dirichlet solution is definabl...

We consider the M/M/1-PS queue with processor sharing. We study the conditional sojourn time distribution of an arriving customer, conditioned on the number of other customers present. A new formula is obtained for the conditional sojourn time distribution, using a discrete Green’s function. This is shown to be equivalent to some classic results of Pollaczeck and Vaulot from 1946. Then various ...

The purpose of the present work is to establish decorrelation estimates for some random discrete Schrödinger operator in dimension one. We prove that the Minami estimates are consequences of the Wegner estimates and Localization. We also prove decorrelation estimates at distinct energies for the random hopping model and Schrödinger operators with alloy-type potentials. These results are used to...

Abstract Based on the recent work [4] for compact potentials, we develop the spectral theory for the one-dimensional discrete Schrödinger operator Hφ = (−∆+ V )φ = −(φn+1 + φn−1 − 2φn) + Vnφn. We show that under appropriate decay conditions on the general potential (and a nonresonance condition at the spectral edges), the spectrum of H consists of finitely many eigenvalues of finite multiplicit...

This paper presents a smoothing method which preserves features for a triangular mesh even when large-scale noise are included because of measurement errors. First, scale-dependent discrete Laplacian is introduced along with boundary Laplacian to deal with an open mesh. Then, a method for feature detection which uses the values by these Laplacians is constructed. Furthermore, anisotropic diffus...

We study the ‘asymmetric’ Hubbard model, where hoppings of electrons depend on their spin. For strong interactions and sufficiently asymmetric hoppings, it is proved that the ground state displays phase separation away from half-filling. This extends a recent result obtained with Freericks and Lieb for the Falicov-Kimball model. It is based on estimates for the sum of lowest eigenvalues of the ...

In this paper we study intermittency for the parabolic Anderson equation ∂u/∂t = κ∆u + ξu, where u : Z d × [0, ∞) → R, κ is the diffusion constant, ∆ is the discrete Laplacian, and ξ : Z d × [0, ∞) → R is a space-time random medium. We focus on the case where ξ is γ times the random medium that is obtained by running independent simple random walks with diffusion constant ρ starting from a Pois...

This paper provides equivalent characterization of the discrete maximum principle for Galerkin solutions of general linear elliptic problems. The characterization is formulated in terms of the discrete Green’s function and the elliptic projection of the boundary data. This general concept is applied to the analysis of the discrete maximum principle for the higher-order finite elements in one-di...