In a 1975 paper [8], Heinig established the following three inequalities:

where A = p/(p + s − λ) with p, s, λ real numbers satisfying p + s > λ,p > 0;

where B = p/(2p + sp − λ −1) with p, s, λ real numbers satisfying 2p +sp > λ, + 1, p > 0;

where is a sequence of nonnegative real numbers,

and C = p[l + l/(p + s−λ)] with p, s, λ real numbers satisfying s > 0, p ≥ 1, and p +s > λ 0.

While expanding upon the work of H. M. Srivastava [6] on generalizations of an interesting identity of Carlson, R. G. Buschman and H. M. Srivastava [2] proved a number of double-series identities and listed various cases of reducibility of certain hypergeometric series in two variables (cf. [1], p. 150, equation (29)). The object of the present paper is to derive three new classes of combinatorial series identities (contained in Theorems 1, 2 and 3 below) which unify and extend the results of these earlier papers ([2], [6]). A multiple-series analogue of one of the combinatorial series identities presented here is also recorded.

In two previous papers the author studied the Jacobson and nil redicals of the algebra of a commutative semigroup over a field [8] and over a commutative ring with unity [9]. This work is continued here.

In this paper we list the elliptic curves defined over Q √ − 1, Q√ −2 or Q√ − 3 which have good reduction away from 2. The possible invariants of such curves are given in Table 1, and their minimal equations in Tables 2, 3 and 4. These extend (and agree with) results of Ogg[4] and Stroeker [10], by a different method.

Let K be a field with char K ≡ 2,3. We consider the problem of finding rational points over K on the family of elliptic curves Fλ, given in homogeneous coordinates (over ) by

In a long sequence of notes in the Comptes Rendus and elsewhere, and in the papers [1], [2], [3], [6], [7], Lichnerowicz and his collaborators have studied the ‘classical infinite-dimensional Lie algebras’, their derivations, automorphisms, co-homology, and other properties. The most familiar of these algebras is the Lie algebra of C∞ vector fields on a C∞ manifold. Another is the Lie algebra of ‘Poisson brackets’, that is, of C∞ functions on a C∞ symplectic manifold, with the Poisson bracket as composition; some questions concerning this algebra are of considerable interest in the theory of quantization – see, for instance, [2] and [3].

It has been persuasively argued (e.g. by Lawvere[8]) that the mathematical world picture needed to develop the physics of continuous bodies and fields should involve a cartesian closed category of smooth morphisms between smooth spaces. As far as the foundations of the calculus of variations are concerned, the need for such a category was recognized by K. T. Chen(cf. [2]).

We show that there is an adjoint dual equivalence between realcompact Alexandroff spaces and the Alexandroff σ-frames. This gives a corresponding adjoint duality for realcompact Tychonoff spaces. Consequently we characterize lattice theoretically the cozero-sets of a topological space.

The purpose of this note is to count almost complex structures on products of some lens spaces.

There are spectra P−k constructed from stunted real projective spaces as in [1] such that H*(P−k) is the span in ℤ/2[x, x−1] of those xi with i ≥ −k. (All cohomology groups have ℤ/2-coefficients unless specified otherwise.) Using collapsing maps, these form an inverse system

which is similar to those of Lin ([15], p. 451). It is a corollary of Lin's work that there is an equivalence of spectra

where holim is the homotopy inverse limit ([3], ch. 5) and Ŝ–1 the 2-adic completion of a sphere spectrum. One may denote by this holim (P–κ), although one must constantly keep in mind that , but rather

The object of this paper is to describe certain families of unstable elements in the homotopy groups of spheres at an odd prime. In so doing we completely account for the image of J as possible Hopf invariants of unstable elements. The analogous result for p = 2 was obtained in [13]. In addition we will discuss other periodic phenomena. Our main results have been independently obtained by Bendersky[5] using BP*. Our methods, however, are entirely geometric, and we actually construct the elements, rather than detect them. Our basic tool is the map .All our constructions are made in BΣp and transferred over.

1. In this paper, the notation and terminology of [2] will be used throughout. As in [2], we consider the space of bounded measurable functions of period 2π, with the sup norm. We let K0ρ denote the subspace consisting of those functions in the space considered which have the further property that the conjugate function belongs to Lrρ 1. Again as in [2], we consider regular summability methods given by a sequence-to-sequence matrix transformation D = (dnk), where

and where, for fixed n,

A sequence of Gamma-type approximation operators n(f) (n = 1,2,…) (which reduce to the Gamma operators, Gn (n = 1,2,…), defined by Müller in a particular case) has been defined. Uniform convergence of n(f) towards f on fixed intervals [x1, x2] (0 ≦ x1 ≦ x2 < ∞) has been proved and a simple example has been given in support of the proof.

The characters (extremal positive definite central functions) of discrete nilpotent groups are studied. The relationship between the set of characters of G and the primitive ideals of the group C*-algebra C*(G) is investigated. It is shown that for a large class of nilpotent groups these objects are in 1–1 correspondence. One proof of this exploits the fact that faithful characters of certain nilpotent groups vanish off the finite conjugacy class subgroup. An example is given where the latter property fails.

Existence is established on some groups for Characteristic functions which are prime in the sence of being indecomposable for product factorization.

A problem of fundamental importance in Spectral Theory consists of finding criteria for an operator to be of scalar-type in the sense of N. Dunford [1]. One relatively general approach in determining such criteria is based on the method of integral transforms (see for example [4], [5], [6], [11], [12]). For example, if X is a Banach space and T is a continuous linear operator on X, then the group {eitT; t real} exists. As noted by several authors (e.g. [4], [6]), this group can then be effectively used for analysing the operator T.

Let p = (p(i): i ≥ 0) be a sequence of numbers satisfying 0 ≤ p(i) < 1 for i = 0,1,2,…, and let G be a random graph with vertex set ℤ = {…, — 1, 0, 1,…} and with edge set defined as follows: for each pair i, j of vertices, where i ≤ j, there is an edge joining i and j with probability p(j — i), independently of the presence or absence of all other edges. We explore the connectedness of G, showing that G is almost surely connected if and only if Σip(i) = ∞ and the (positive) greatest common divisor of the set {i ≥ 1: p(i) < 0} equals 1; if one of these two conditions fails to hold then G is almost surely disconnected. Corresponding results hold in higher dimensions, for random graphs defined on the vertex sets ℤd where d ≥ 2.

Processes related to the empirical distribution are shown to converge to a time changed Brownian motion. In particular, the time change of the Brownian motion differs from the time change obtained by using the distribution function: this gives rise to two different representations of a related Brownian bridge.

A modified kernel is devised which avoids the irregular values which plague the natural integral equations for the exterior problems of harmonic elastic waves in a homogeneous isotropic medium.

Spatially homogeneous cosmological models of the Bianchi type-V are considered in the Brans-Dicke scalar-tensor theory of gravitation. Exact solutions are given in the vacuum case as well as for models filled with dust, radiation or stiff matter.