Let us denote by α the set of n real numbers α1, …, αn, and by ck(α) and hk(α) the elementary and complete symmetric functions of degree k in α1, …, αn, and by ck(α) and hk(α) the elementary and complete symmetric functions of degree k in αl, …, αn, i.e. ck(α) is the sum of all possible products of k different αi and hk(α) is the sum of all possible products of k αi, where now in any product one or more αi may be repeated any number of times.

We introduce a class of regular extensions of regular semigroups, called enlargements; a regular semigroup T is said to be an enlargement of a regular subsemigroup S if S = STS and T = TST. We show that S and T have many properties in common, and that enlargements may be used to analyse a number of questions in regular semigroup theory.

By a theorem of Perron, a non-negative irreducible (n × n) matrix A = (aμν) has a positive fundamental root σ, the “ maximal root of A ”, such that the moduli of all other eigenvalues of A do not exceed σ. If we put

σ lies between R and r. Since σ is not changed if A is transformed by a positive diagonal matrix D(p1,…pn, σ lies also between the expressions

By a theorem of Frobenius, to σ as an eigenvalue of A belongs a positive eigenvector ξ, = (x1 …, xn), satisfying

Tate cohomology of finite groups [5] is very good at emphasising periodic cohomological behaviour and hence at the study of free actions on spheres [8]. Tate cohomology of spaces was introduced by Swan [10] for finite dimensional spaces to systematically ignore free actions, and hence to simplify various arguments in fixed point theory.

In this paper we determine the group of endotrivial modules for certain symmetric and alternating groups in characteristic p. If p = 2, then the group is generated by the class of Ωn(k) except in a few low degrees. If p > 2, then the group is only determined for degrees less than p2. In these cases we show that there are several Young modules which are endotrivial.

We give upper and lower bounds on the count of positive integers n ≤ x dividing the nth term of a non-degenerate linearly recurrent sequence with simple roots.

An element $a$ in a ring $R$ with identity is called strongly clean if it is the sum of an idempotent and a unit that commute. And $a\in R$ is called strongly $\pi$-regular if both chains $aR\supseteq a^2R\supseteq\cdots$ and $Ra\supseteq Ra^2\supseteq\cdots$ terminate. A ring $R$ is called strongly clean (respectively, strongly $\pi$-regular) if every element of $R$ is strongly clean (respectively, strongly $\pi$-regular). Strongly $\pi$-regular elements of a ring are all strongly clean. Let $\sigma$ be an endomorphism of $R$. It is proved that for $\varSigma r_ix^i\in R[[x,\sigma]]$, if $r_0$ or $1-r_0$ is strongly $\pi$-regular in $R$, then $\varSigma r_ix^i$ is strongly clean in $R[[x,\sigma]]$. In particular, if $R$ is strongly $\pi$-regular, then $R[[x,\sigma]]$ is strongly clean. It is also proved that if $R$ is a strongly $\pi$-regular ring, then $R[x,\sigma]/(x^n)$ is strongly clean for all $n\ge1$ and that the group ring of a locally finite group over a strongly regular or commutative strongly $\pi$-regular ring is strongly clean.

I have begun to study the laws of the struggle for life by a group of species living in the same environment in such a way that some devour others. I have used for this purpose the so called principle of encounter, considering the encounters of the individuals of the various species, and what follows by reason of the actions that the individuals exercise on one another.

Let X be a topological space, E a real or complex topological vector space, and C(X, E) the vector space of all bounded continuous E-valued functions on X. The notion of the strict topology on C(X, E) was first introduced by Buck (1) in 1958 in the case of X locally compact and E a locally convex space. In recent years a large number of papers have appeared in the literature concerned with extending the results contained in Buck's paper (1); see, for example, (14), (15), (3), (4), (12), (2), and (6). Most of these investigations have been concerned with generalising the space X and taking E to be the scalar field or a locally convex space.

In this paper we consider finite generation and finite presentability of Rees matrix semigroups (with or without zero) over arbitrary semigroups. The main result states that a Rees matrix semigroup M[S; I, J; P] is finitely generated (respectively, finitely presented) if and only if S is finitely generated (respectively, finitely presented), and the sets I, J and S\U are finite, where U is the ideal of S generated by the entries of P.

We study the profile near quenching time for the solutions of the first and second initial boundary value problems (IBVP) for a semilinear heat equation. Under certain conditions, one-point quenching occurs for both first and second IBVPs. Furthermore, we derive the asymptotic self-similar quenching rate for both problems.

A short proof of Chow's generalization of Widder's inequality and generalizations of some related Hardy's results are given.

It is well known that any map of n regions on a sphere may be coloured in five or fewer colours. The purpose of the present note is to prove the following

Theorem. If Pn(λ)denotes the number of ways of colouring any ma: of n regions on the sphere in λ (or fewer) colours, then

(1)

This inequality obviously holds for λ = 1, 2, 3 so that we may confine attention to the case λ > 4. Furthermore it holds for n = 3, 4 since the first region may be coloured in λ ways, the second in at least λ — 1 ways, the third in at least λ — 2 ways, and the fourth, if there be one, in at least λ — 3 ways.

If R is a ring with identity and M is a left R-module then it is well known that the following statements are equivalent:

(1) M is flat.

(2) The functor −⊗M preserves embeddings of right ideals into R.

This paper investigates situations in which the analogous statements are equivalent in the context of S-sets over a monoid S.

We study the differential geometry of hypersurfaces in hyperbolic space. As an application of the theory of Lagrangian singularities, we investigate the contact of hypersurfaces with families of hyperspheres or equidistant hyperplanes.

AMS 2000 Mathematics subject classification: Primary 53A35. Secondary 58C27

The abstract theory of positive compact operators (acting in a partially ordered Banach space) has proved to be particularly useful in the theory of integral equations. In a recent paper (2) it was shown that many of the now classical theorems for positive compact operators can be extended to certain classes of non-compact operators. One result, proved in (2, Theorem 5), was a fixed point theorem for compressive k-set contractions (k<l). The main result of this paper (Theorem 3.3) shows that some of the hypotheses of (2, Theorem 5) are unnecessary. We use techniques based on those used by M. A. Krasnoselskii in the proof of Theorem 4.12 in (4), which is the classical fixed point theorem for compressive compact operators, to obtain a complete generalisation of this classical result to the k-set contractions (k < 1). It should be remarked that J. D. Hamilton has extended the same result to A-proper mappings (3, Theorem 1). However apparently it is not known, even in the case when we are dealing with a Π1-space, whether k-set contractions are A-proper or not.

We consider a class of Jacobi matrices with periodically modulated diagonal in a critical hyperbolic (‘double root’) situation. For the model with ‘non-smooth’ matrix entries we obtain the asymptotics of generalized eigenvectors and analyse the spectrum. In addition, we reformulate a very helpful theorem from a paper by Janas and Moszynski in its full generality in order to serve the needs of our method.

Let the Laurent expansion of the Riemann zeta function ξ(s) about s=1 be written in the form

It has been discovered independently by many authors that, in terms of this notation, the coefficient

In the general theory of relativity, as in many other branches of theoretical physics, the material and energetical content of spacetime is considered, in the first instance, as an extended field, which is specified by means of field quantities (energy-momentum-stress tensor, charge-current density, electromagnetic field strength). From this point of view corpuscles (material particles, photons) are constructs obtained by first considering the field quantities as nonvanishing only within certain world tubes, and then passing by limiting processes to the idealisation in which these world tubes are shrunk into world lines. More precisely, this passage to the corpuscular description may be thought of as accomplished by replacing the original field by successive members of a sequence of field distributions, satisfying the same field laws, which cluster more and more in the neighbourhood of the world lines, and for which in some significant sense the total measure approaches a finite limit. Each such world line, together with the limiting measures of those portions of the field quantities associated therewith, is then a corpuscle; the form of the world line determines the motion of the corpuscle, and the associated “corpuscular quantities” its physical attributes (mass or energy, momentum, charge).