A note on circulant transition matrices in Markov chains

Let g and n be positive integers and gcd(g,n)=1. Let C=(c_ij) be a g-circulant transition matrix of order n of Markov chain. We are interested in studying and lim_k→∞C^k.     

On a Problem of Dobrowolski and Williams and the Pólya-Vinogradov Inequality

In [2] E. Dobrowolski and K.S. Williams considered a problem of obtaining estimates for the sum _n=a+1 ^a+N f(n),for a certain class of functions f. One specific application of their result is a new method for estimating character sums. In particular, they obtain a form of the Pólya-Vinogradov inequality with the constant 1/(2 log 2). In this note we improve their estimates and obtain, in particular, a form of the Pólya-Vinogradov inequality with the constant 1/(3 log 3). A nice feature of our estimate is that it is obtained by a very simple argument.     

Analysis and algorithms for the computation of the excited states of helium

In this paper, we study a dimensionally scaled helium atom model for excited states of helium. The mathematical analysis of the corresponding effective energy potential is presented. Two simple numerical algorithms are developed for the computation of the excited states of helium. Comparison between our numerical results and those in the existing literature is given to indicate the accuracy and efficiency of the proposed algorithms.     

Developing a time-domain finite-element method for modeling of electromagnetic cylindrical cloaks

In this paper we propose a time-domain finite element method for modeling of electromagnetic cloaks. The permittivity and permeability of the cloak model are described by the Drude dispersion model. The model to be solved is quite challenging in that we have to solve a coupled problem with different partial differential equations given in different regions. Our method is based on a mixed finite element method using edge elements with different types of meshes in different regions. Numerical results demonstrate that our algorithm is quite effective for simulating cloaks in time-domain. To our knowledge, this is the first cloak simulation carried out by the time-domain finite element method.     

A symbolic operator approach to several summation formulas for power series II

Here expounded is a kind of symbolic operator method that can be used to construct many transformation formulas and summation formulas for various types of power series including some old ones and more new ones.     

On divisibility properties of certain multinomial coefficients—II

A recent conjecture of Myerson and Sander concerns divisibility properties of certain multinomial coefficients. We obtain results in this direction by further pursuing a line of attack developed earlier by the first author.     

K3 surfaces, rational curves, and rational points

We prove that for any of a wide class of elliptic surfaces X defined over a number field k, if there is an algebraic point on X that lies on only finitely many rational curves, then there is an algebraic point on X that lies on no rational curves. In particular, our theorem applies to a large class of elliptic K3 surfaces, which relates to a question posed by Bogomolov in 1981.