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 the Applications of the Linear Recurrence Relationships to Pseudoprimes

We present here some results on the applications of linear recursive sequences of order $2$ to the Fermat pseudoprimes, Fibonacci pseudoprimes, and Dickson pseudoprimes.     

Three-dimensional elliptic solvers for interface problems and applications

Second-order accurate elliptic solvers using Cartesian grids are presented for three-dimensional interface problems in which the coefficients, the source term, the solution and its normal flux may be discontinuous across an interface. One of our methods is designed for general interface problems with variable but discontinuous coefficient. The scheme preserves the discrete maximum principle using constrained optimization techniques. An algebraic multigrid solver is applied to solve the discrete system. The second method is designed for interface problems with piecewise constant coefficient. The method is based on the fast immersed interface method and a fast 3D Poisson solver. The second method has been modified to solve Helmholtz/Poisson equations on irregular domains. An application of our method to an inverse interface problem of shape identification is also presented. In this application, the level set method is applied to find the unknown surface iteratively.     

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.     

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.     

On an optimality property of Ramanujan sums

We evaluate , where the is taken over sequences satisfying . In particular we show that it is attained by taking for all , which reduces the summation over to a Ramanujan sum .

     

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.     

Flat Cyclotomic Polynomials of Order Three

A cyclotomic polynomial _n is said to be of order 3 if n = pqrfor three distinct odd primes p, q, and r. We establish theexistence of an infinite family of such polynomials whose coefficientsdo not exceed 1 in modulus. 2000 Mathematics Subject Classification11B83, 11C08.