An axiomatic approach to relativity theory is proposed which is based entirely on a single reflexive relation called signalling. This generalises and simplifies earlier schemes of Kronheimer and Penrose and of Carter which are based on causality relations. Many of the features of space-times, such as photon paths and topology, have their counterparts in general signal spaces.

Suppose F: (Cn+1 × Cp, 0) → (C, 0) is a germ of a holomorphic function, and (S, 0) ⊂ (Cn+1, 0) is a germ of some hypersurface in (Cn+1, 0). The quasicaustic Q(F) of F is defined as Q(F) = {a ∈ Cp; F(•, a) has a critical point on S}. We investigate the structure of quasicaustics corresponding to boundary singularities. The procedure for calculating the modules of logarithmic vector fields is given. The minimal set of generators for the Whitney's cross-cap singular variety is explicitly calculated.

Modifying the concept underlying Daneš' drop theorem, Rolewicz introduced the notion of the drop property of a norm which was later generalised to the weak drop property of a norm. Kutzarova extended the discussion to consider the drop property for closed bounded convex sets. Here we characterise the drop and weak drop properties for such sets by upper semi-continuous and compact valued subdifferential mappings.

Our main result is that if G(x1, …, xn) = 0 is a system of homogeneous equations such that for every partition of the positive integers into finitely many classes there are distinct y1,…, yn in one class such that G(y1, …, yn) = 0, then, for every partition of the positive integers into finitely many classes there are distinct Z1, …, Zn in one class such that

In particular, we show that if the positive integers are split into r classes, then for every n ≥ 2 there are distinct positive integers x1, x1, …, xn in one class such that

We also show that if [1, n6 − (n2 − n)2] is partitioned into two classes, then some class contains x0, x1, …, xn such that

(Here, x0, x2, …, xn are not necessarily distinct.)

We give a general result for the lower bound of the volume of a compact convex set K in Ed in terms of the volumes of orthogonal sections of K.

We give holomorphic differentials of some algebraic function field K of complex dimension one which is a generalisation of a hyperelliptic field.

An estimate is obtained on the Hausdorff and fractal dimensions of global attractors of semilinear partial differential equations with delay: ẋ(t) = Ax(t) + f(xt). The method employed is to associate such an equation with a nonlinear semigroup on a product space and then appeal to the upper estimate due to Constantin, Foias and Teman on topological dimensions of global attractors for general nonlinear dynamical systems.

Let X be a complete regular space. We denote by Cb(X) the Banach space of all real-valued bounded continuous functions on X endowed with the supremumnorm.

In this paper we give a characterisation of weakly compact operators defined from Cb(X) into a Banach space E which are β∞-continuous, where β∞ is a locally convex topology on Cb(X) introduced by Wheeler. We also prove that (Cb(X), β∞) has the strict Dunford-Pettis property and, if X is a σ-compact space, (Cb(X), β∞), has the Dunford-Pettis property.

At the Second Oklahoma Conference on Convexity and Related Combinatorial Geometry (1980) G.Sierksma posed several problems dealing with the generalised Helly and Radon numbers of a convexity space. The aim of this note is to give answers to and comment on some of Sierksma's questions.

I shall consider the following problem: given a stable, conservative, single-exit q-matrix, Q, over an irreducible state-space S and a μ-subinvariant measure, m, for Q, determine all Q-processes for which m is a μ-invariant measure. I shall provide necessary and sufficient conditions for the existence and uniqueness of such a process.

We prove several commutativity theorems for unital rings with polynomial constraints on certain subsets, which improve and generalise the recent results of Grosen, and Ashraf and Quadri.

Shortly connected and shortly linked inverse semigroups arise in the study of inverse semigroups determined by the lattices of inverse subsemigroups and by the partial automorphism semigroups. It has been shown that every shortly linked inverse semigroup is shortly connected, but the question of whether the converse is true has not been addressed. Here we construct two examples of combinatorial shortly connected inverse semigroups which are not shortly linked. One of them is completely semisimple while the other is not.

In this paper we give some new types of inequalities which can be regarded as discrete forms of the Wirtinger inequality and several new isoperimetric-type inequalities.

Some general theorems concerning the relative behaviour of moduli of smoothness are established. In particular an open problem raised by Dickmeis, Nessel and van Wickeren is negated.

In the present paper a simple proof of a theorem of Sargent on CλP-integral of Burkill is given.

Let R be a commutative ring with unit, T an R-module, and n a positive integer. It is proved that T is n-flat over R if B⊗RT is B-torsionfree for each n–generated commutative R-algebra B. The converse holds if T is n–generated, in which case T is actually flat over R. Several other instances of the converse are established, but it is shown that the converse fails in general, even for R an integral domain, T an ideal of R, and n = 1.

We give a description of the structure of the semigroups for which each principal ideal is a retract. The globally idempotent case is solved quickly using a—suitably modified—construction which has been developed by Tully for the study of semigroups in which each ideal is a retract. The general case can be treated by a naturally obtained semigroup of subsets of the semigroup constructed for the globally idempotent case.

The purpose of this paper is to study the chain recurrent sets under persistent dynamical systems, and give a necessary condition for a persistent dynamical system to be topologically stable. Moreover we show that the various recurrent sets depend continuously on persistent dynamical systems.

The resolvent operator and the moment generating function of a reflected Ornstein-Uhlenbeck process are obtained. These results are then applied to determine the long-run average cost and the total expected discounted cost of operating a finite storage system with content-dependent release rate.