An extension of the classical summation formula of Plana is obtained. The extension is obtained by using the zeros of a Bessel function of the first kind.

Let D be an integral domain with quotient field K. If α2 − α ∈ D and α3 − α2 ∈ D imply α ∈ D for all elements α of K, then D is called a u-closed domain. A submonoid S of a torsion-free Abelian group is called a grading monoid. We consider the semigroup ring D[S] of a grading monoid S over a domain D. The main aim of this note is to determine conditions for D[S] to be u-closed. We shall show the following Theorem: D[S] is u-closed if and only if D is u-closed.

It is well known that Q is a simple quasigroup if and only if Mlt Q acts primitively on Q. Here we show that Q is a simple quasigroup if and only if Mlt Q acts quasiprimitively on Q, and that Q is a simple right quasigroup if and only if RMlt Q acts quasiprimitively on Q.

In this work we demonstrate that the lower radical class construction on a homomorphically closed class of associative rings generates a radical class for any class of associative rings. We also give a new description of the upper radical class using the construction on an appropriate generating class.

Hoping for a hand-shake between methods from diophantine approximation theory and transcendance theory, we show how zeros estimates from transcendance theory imply Roth's type lemmas (including the product theorem). We also formulate some strong conjectures on lower bounds for linear forms in logarithms of rational numbers with rational coefficients, inspired by the subspace theorem and which would imply, for example, the abc conjecture.

Given two subsets X and Y of ℝ each with at least two points, we describe the surjective linear isometries between the spaces of functions of bounded variation BV(X) and BV(Y): namely, if T : BV(X) → BV(Y) is such an isometry, then there exist α ∈ ℂ, |α| = 1, and a monotonic bijective map h : Y → X such that (Tf)(y) = αf(h(y)) for every f ∈ BV(X) and every y ∈ Y.

Let K be a planar, compact, convex set with circumradius R, diameter d, width w and inradius r, and containing no points of the integer lattice. We generalise inequalities concerning the ‘dual’ quantities (2R − d) and (w − 2r) to rectangular lattices. We then use these results to obtain corresponding inequalities for a planar convex set with two interior lattice points. Finally, we conjecture corresponding results for sets containing one interior lattice point.

Let n be a positive integer, and let (a, b, c) be a primitive Pythagorean triple. In this paper we give certain conditions for the equation (an)x + (bn)y = (cn)z to have positive integer solutions (x, y, z) with (x, y, z) ≠ (2, 2, 2). In particular, we show that x, y and z must be distinct.

We show that for every integer k ≥ 3 and every prime p ≤ 5 there is a group with soluble length k and order .

Let G be a finite connected simple graph. The isomorphism classes of graph bundles and graph coverings over G have been enumerated by Kwak and Lee. Recently, Archdeacon and others characterised bipartite coverings of G and enumerated the isomorphism classes of regular 2p-fold bipartite coverings of G, when G is nonbipartite. In this paper, we characterise bipartite graph bundles over G and derive some enumeration formulas of the isomorphism classes of them when the fibre is a connected bipartite graph. As an application, we compute the exact numbers of the isomorphism classes of bipartite graph bundles over G when the fibre is the path Pn or the cycle Cn.

In this paper we establish a generalised KKM theorem from which many well-known KKM theorems and a fixed point theorem of Tarafdar are extended.

Let K be a planar convex body containing no points of the integer lattice. We give a new inequality relating the inradius and circumradius of K.

In this paper we present an elementary proof of a congruence by subtraction relation. In order to prove congruence by subtraction, we produce a dissection relating equal sub-polytopes. An immediate consequence of this relation is an Euler-type volume identity in ℝ3 which appeared in the Unsolved Problems section of the December 1996 MAA Monthly.

This Euler-type volume identity relates the volumes of subsets of a polytope called wedges that correspond to its faces, edges, and vertices. A wedge consists of the inward normal chords of the polytope emanating from a face, vertex, or edge. This identity is stated in the theorem below.

Euler Volume Theorem. For any three dimensional convex polytope P

This identity follows immediately from

We derive the formula in the title and deduce some consequences. For example we show that the identity map from any compact manifold to itself is always stable as an exponentially harmonic map. This is in sharp contrast to the harmonic or p-harmonic cases where many such identity maps are unstable. We also prove that an isometric and totally geodesic immersion of Sm into Sn is an unstable exponentially harmonic map if m ≠ n and is a stable exponentially harmonic map if m = n.

If a Banach space X contains a complemented subspace isomorphic to ℓ1 and if ε > 0, then there exists a subspace Y of X and a projection P from X onto Y such that Y is (1 + ε)-isometric to ℓ1 and ∥P∥ ≤ 1 + ε. A stronger result for c0 is proved for Banach spaces whose dual unit ball is weak sequentially compact.