ABSTRACT Using an eigenfunction expansion to solve the transport equation, complemented by Monte Carlo simulations, we show that ultrarelativistic shocks can be effective particle accelerators even when they fail produce large amplitude turbulence in downstream plasma. This finding contradicts widely held belief a uniform magnetic field perpendicular shock normal inhibits acceleration first-ord...

‎Radial basis functions (RBFs) are a powerful tool for approximating the solution of high-dimensional problems‎. ‎They are often referred to as a meshfree method and can be spectrally accurate‎. ‎In this paper, we analyze a new stable method for evaluating Gaussian radial basis function interpolants based on the eigenfunction expansion‎. ‎We develop our approach in two-dimensional spaces for so...

In this paper we study the asymptotic behavior of u -capacities small sets and its application to analysis eigenvalues Dirichlet-Laplacian on a bounded planar domain with hole. More precisely, consider two (sufficiently regular) open connected Ω ω ℝ 2 , containing origin. First, if ε is close 0 function defined Ω, compute an expansion -capacity [see formula in PDF] as → 0. As byproduct, for N t...

In doing p-adic analysis on spaces of classical modular functions and forms, it is convenient and traditional to broaden the notion of “modular form” to a class called “overconvergent p-adic modular forms.” Critical for the analysis of the p-adic Banach spaces composed of this wider class of forms is the “Atkin U -operator”, which is completely continuous and whose spectral theory (still not ve...

In this article we describe a numerical method to solve a nonhomogeneous diffusion equation with arbitrary geometry by combining the method of fundamental solutions (MFS), the method of particular solutions (MPS), and the eigenfunction expansion method (EEM). This forms a meshless numerical scheme of the MFS-MPS-EEM model to solve nonhomogeneous diffusion equations with time-independent source ...

The numerical evaluation of a class of Hilbert transforms using an eigenfunction approach is considered. Judicious selection of convergence accelerators allows for the efficient evaluation of the resulting series. The approach is particularly accurate for functions having a Gaussian-like asymptotic behavior. For more slowly decreasing functions, the accuracy of the evaluation decreases.

This paper provides explicit analytical characterizations for first hitting time densities for Cox–Ingersoll–Ross (CIR) and Ornstein–Uhlenbeck (OU) diffusions in terms of relevant Sturm–Liouville eigenfunction expansions. Starting with Vasicek (1977) and Cox, Ingersoll and Ross (1985), the Gaussian Ornstein– Uhlenbeck and Feller’s (1951) square-root diffusions are among the most commonly used s...