Let $\Omega \in \mathbb C$ be a domain such that $K:= \mathbb C \setminus \Omega $ is compact and non-polar. Let $(q_k)_{k>0}$ be a sequence of polynomials with $n_k$, the degree of $q_k$ satisfying $n_k \to \infty $, and let $(q_k^{(m)})_k$ denote the sequence of mth derivatives. We provide conditions which ensure that the preimages $(q_k^{(m)})^{-1}(\{a\})$ uniformly equidistribute on $\partial \Omega $, as $k \to \infty $, for every $a \in \mathbb C$ and every $m = 0, 1, \ldots .$

A hybrid asymptotic-numerical method is developed to approximate the mean first passage time (MFPT) and the splitting probability for a Brownian particle in a bounded two-dimensional (2D) domain that contains absorbing disks, referred to as “traps”, of asymptotically small radii. In contrast to previous studies that required traps to be spatially well separated, we show how to readily incorporate the effect of a cluster of closely spaced traps by adapting a recently formulated least-squares approach in order to numerically solve certain local problems for the Laplacian near the cluster. We also provide new asymptotic formulae for the MFPT in 2D spatially periodic domains where a trap cluster is centred at the lattice points of an oblique Bravais lattice. Over all such lattices with fixed area for the primitive cell, and for each specific trap set, the average MFPT is smallest for a hexagonal lattice of traps.

In this paper, two formulations in explicit form to derive the fundamental solutions for two and three dimensional unsteady unbounded Stokes flows due to a mass source and a point force are presented, based on the vector calculus method and also the Hörmander’s method. The mathematical derivation process for the fundamental solutions is detailed. The steady fundamental solutions of Stokes equations can be obtained from the unsteady fundamental solutions by the integral process. As an application, we adopt fundamental solutions: an unsteady Stokeslet and an unsteady potential dipole to validate a simple case that a sphere translates in Stokes or low-Reynolds-number flow by using the singularity method instead by the traditional method which in general limits to the assumption of oscillating flow. It is concluded that this study is able to extend the unsteady Stokes flow theory to more general transient motions by making use of the fundamental solutions of the linearly unsteady Stokes equations.

In this paper, a detailed description of the resolvent of the Laplace–Beltrami operator in $n$-dimensional hyperbolic space is given. The resolvent is an integral operator with the kernel (Green’s function) being a solution of a hypergeometric differential equation. Asymptotic analysis of the solution of this equation is carried out.

The modified embedded atom method (MEAM) with the universal form of embedding function and a modified energy term along with the pair potential has been employed to determine the potentials for alkali metals: Na, K, by fitting to the Cauchy pressure (C12 − C44)/2, shear constants Gν = (C11 − C12 + 3C44)/5 and C44, the cohesive energy and the vacancy formation energy. The obtained potentials are used to calculate the phonon dispersions of these metals. Using these calculated phonons we evaluate the local density of states of neighbours of vacancy using Green’s function method. The local density of states of neighbours of vacancy has been used to calculate mean square displacements of these atoms and formation entropy of vacancy. The calculated mean square displacements of both 1st and 2nd neighbours of vacancy are found to be lower than that of host atom. The calculated phonon dispersions agree well with the experimental phonon dispersion curves and the calculated results of vacancy formation entropy compare well with the other available results.

The aim of this paper is to find a concrete bound for the error involved when approximating the nth Hermite function (in the oscillating range) by an asymptotic formula due to D. Dominici. This bound is then used to study the accuracy of certain approximations to Hermite expansions and to Fourier transforms. A way of estimating an unknown probability density is proposed.

In this work, we consider the periodic boundary value problem where a,c∈L1(0,T) and f is a Carathéodory function. An existence theorem for positive periodic solutions is proved in the case where the associated Green function is nonnegative. Our result is valid for systems with strong singularities, and answers partially the open problem raised in Torres [‘Weak singularities may help periodic solutions to exist’, J. Differential Equations232 (2007), 277–284].