We prove an approximation formula for the Riemann zeta function. We show that a classical theorem:

uniformly in the domain ½ ≤ σ < 1, is an immediate consequence of our approximation formula. Our method is real and free from complex analysis.

In a recent paper (Bull. Austral. Math. Soc. 13 (1975), 241–245), Tarafdar has considered nonexpansive self mappings on a subset X of a locally convex vector space E and proved an extension to E of a theorem of Göhde. The purpose of this paper is to show that the condition f: X → X, in Göhde-Tarafdar's Theorem in the above paper, may be weakened to f: X → E with f(∂X) ⊆ X. As a consequence, it is further shown that an extension to E of a well-known common fixed point theorem of Belluce and Kirk due to Tarafdar remains true on domains that are not necessarily bounded or quasi-complete.

This study is a continuation of an earlier paper on the regularity series of a convergence space. The notions of a R-Hausdorff series and the T3-modification of a convergence space are introduced, and their relationship with the regularity series is studied. The concept of a symmetric space is shown to be useful in studying T3-compactifications. Several examples are given; one being a Hausdorff convergence space with an arbitrarily large regularity series.

Let A = {ai} be a finite set of integers and let p and m denote the cardinalities of A + A = {ai+aj} and A - A {ai–aj}, respectively. In the paper relations are established between p and m; in particular, if max {ai–ai-1} = 2 those sets are characterized for which p = m holds.

The author and Ira J. Papick have termed an integral domain R a going-down ring if R ⊂ T satisfies going-down for each domain T containing R. The present paper investigates conditions which, for an integral extension A ⊂ B of domains, imply that A (respectively B ) is going-down whenever B (respectively A ) is going-down. This explains the “descent” (respectively “ascent”) in the title. Two typical results (the first about descent, the second about ascent) are given next.

We call a topological space completely regular if points and closed sets can be separated by continuous real-valued functions, and we call a topological space Tychonoff or T3α if it is T1 and completely regular. It is well known that the Tychonoff spaces are precisely the gauge spaces, that is the topological spaces whose topologies are induced by separating families of pseudometrics. V.G. Boltjanskiĭ has given analogous characterizations for T0, T1, T2, regular, and regular-T1 spaces in terms of families of “metric-like” functions. The purpose of this note is to fill in a case missing in Boltjanskiĭ's paper, the completely regular spaces, and to relate Boltjanskiĭ's work to results on quasi-uniformization by William J. Pervin and results about semi-gauge spaces by J.V. Michalowicz.

A new brief proof of the fact stated in the title is presented.

In previous papers we have proved that if G is a ω*-closed subspace of the conjugate space B* of a normed linear space B, then every b ∈ B can be extended within B, from G to the whole B*, with an arbitrarily small increase of the norm. Here we give some extensions of this result to the case when B* is replaced by a normed linear space E and B by any linear subspace V of E*, and some applications to separation and Helly type theorems.

An elementary proof is given of a sufficient optimality condition recently proven by B.D. Craven. This proof avoids the use of a transposition theorem and this allows for a strengthening of Craven's result.

The Fibonacci group F(2, 7) has been known to be cyclic of order 29 for about five years. This was first established by computer coset enumerations which exhibit only the result, without supporting proofs. The working in a coset enumeration actually contains proofs of many relations that hold in the group. A hand proof that F(2, 7) is cyclic of order 29, based on the working in computer coset enumerations, is presented here.