Kevin A. Broughan

Adic Topologies for the Rational Integers

A topology on Z, which gives a nice proof that the set of prime integers is infinite, is char- acterised and examined. It is found to be homeomorphic to Q, with a compact completion homeo- morphic to the Cantor set. It has a natural place in a family of topologies on Z, which includes the p-adics, and one in which the set of rational primes P is dense. Examples from number theory are given, including the primes and squares, Fermat numbers, Fibonacci numbers and k-free numbers.

Fast floating-point processing in Common Lisp

Lisp, one of the oldest higher-level programming languages, has rarely been used for fast numerical (floating-point) computation. We explore the benefits of Common Lisp, an emerging new language standard with some excellent implementations, for numerical computation. We compare it to Fortran in terms of the speed of efficiency of generated code, as well as the structure and convenience of the language. There are a surprising number of advantages to Lisp, especially in cases where a mixture of symbolic and numeric processing is needed.

The holomorphic flow of the riemann zeta function

The flow of the Riemann zeta function, s = ζ(s), is considered, and phase portraits are presented. Attention is given to the characterization of the flow lines in the neighborhood of the first 500 zeros on the critical line. All of these zeros are foci. The majority are sources, but in a small proportion of exceptional cases the zero is a sink. To produce these portraits, the zeta function was evaluated numerically to 12 decimal places, in the region of interest, using the Chebyshev method and using Mathematica. The phase diagrams suggest new analytic properties of zeta, of which some are proved and others are given in the form of conjectures.

On dominated terms in the general knapsack problem

A necessary and sufficient condition for the identification of dominated terms in a general knapsack problem is derived. By general, we mean a knapsack problem which is unbounded, equality constrained and which has a parametric right-hand side. The given condition yields recently published results in the literature. A report on computational experiments for large-scale knapsack problems, demonstrating the effectiveness of this approach, is included.

Vanishing of the integral of the Hurwitz zeta function

A number of authors have considered mean values of powers of the modulus of the Hurwitz zeta function ζ(s, a), see [3, 4, 5, 6, 7]. In this paper, the mean of the function itself is considered. First a functional equation relating the Riemann zeta function to sums of the values of the Hurwitz zeta function at rational values of a is derived. This functional equation underlies the vanishing of the integral of the Hurwitz zeta function. Consider the values of the function at negative integers:

Holomorphic flows on simply connected regions have no limit cycles

The dynamical system or flow ż = f(z), where f is holomorphic on C, is considered. The behavior of the flow at critical points coincides with the behavior of the linearization when the critical points are non-degenerate: there is no center-focus dichotomy. Periodic orbits about a center have the same period and form an open subset. The flow has no limit cycles in simply connected regions. The advance mapping is holomorphic where the flow is complete. The structure of the separatrices bounding the orbits surrounding a center is determined. Some examples are given including the following: if a quartic polynomial system has four distinct centers, then they are collinear.

A Note on Reducing the Number of Variables in IntegerProgramming Problems

A necessary and sufficient condition for identification of dominatedcolumns, which correspond to one type of redundant integer variables,in the matrix of a general Integer Programming problem, isderived. The given condition extends our recent work on eliminatingdominated integer variables in Knapsack problems, and revises arecently published procedure for reducing the number of variables ingeneral Integer Programming problems given in the literature. Areport on computational experiments for one class of large scaleKnapsack problems, illustrating the function of this approach, isincluded.

Fortran to Lisp translation using f2cl

The need for both algebraic and numerical capabilities within mathematical computation systems has highlighted the need to translate numerical software written in Fortran77 to Common Lisp, a language favoured by the algebraic computation community. This paper reports the current state of f2cl, a translator written to achieve that end. We describe the translation process giving details of Lisp equivalents of Fortran expressions, as well as discussing features of Fortran that have no straightforward equivalent in Lisp.

SENAC: a high-level interface for the NAG library

and the iteration is to begin at the point with coordinates [2, – 1, 2, 2]. Suppose the user selects the routine e041bf from the NAG Library to perform the minimization. There is a daunting amount of work involved writing code that can call this routine. Initialization of variables, writing subroutines for function values and derivatives, and formatting of the output all require detailed work. This should be unnecessary for a problem with

Metrization of spaces having Cech dimension zero

In this paper we prove that a necessary and sufficient condition for a topological space to have a compatible metric taking values in a closed zero dimensional subset of the real numbers is that the space be metrizable and have Cech dimension zero. We also prove that the two topological properties Cech dimension zero and metrizable, when taken together, are equivalent to a single topological property namely, having a development consisting of families of open sets which partition the space. These theorems are an extension of the theorem given in [I]: a metrizable space has Cech dimension zero if and only if there exists a metric for the space, compatible with the topology, taking values in some subset of the real numbers with zero as its only cluster point. The proofs of the theorems depend on this characterization.