In a previous paper [1], Green's theorem for line integrals in the plane was proved, for Riemann integration, assuming the integrability of Qx−Py, where P(x, y) and Q(x, y) are the functions involved, but not the integrability of the individual partial derivatives Qx and Py. In the present paper, this result is extended to a proof of the Gauss-Green theorem for p-space (p ≥ 2), for Lebesgue integration, under analogous hypotheses. The theorem is proved in the form where Ω is a bounded open set in Rp (p-space), with boundary Ω; g(x) =(g(x1)…, g(xp)) is a p-vector valued function of x = (x1,…,xp), continuous in the closure of Ω; μv,(x) is p-dimensional Lebesgue measure; v(x) = (v1(x),…, vp(x)) and Φ(x) are suitably defined unit exterior normal and surface area on the ‘surface’ ∂Ω and g(x) · v(x) denotes inner product of p-vectors.

To assess the mathematical work of the late M. L. Urquhart is, paradoxically, both easy and extremely difficult. It is easy, in that he never published a mathematical paper in any of the journals. Thus one does not have to read a large volume of published work. On the other hand, Urquhart was far from being mathematically inactive, but he communicated his ideas verbally to his associates. At this point of time it is well nigh impossible to recall all the ideas that he discussed so freely during his lifetime. Conversations have long since been forgotten and ideas are now only vaguely remembered. Consequently any objective assessment of Urquhart's mathematical work is very difficult.

This paper is concerned with the convergence of Rayleigh-Ritz approximations to the solution of an elliptic boundary value problem. Although the work arose in connection with the aerofoil problem (and it is to this problem that the results obtained are immediately applied), the methods here employed are suitable for use on the wider class of problem mentioned above.

Coburn [1] has derived the intrinsic form of the characteristic relations, for the steady, supersonic, three-dimensional motion of a polytropic gas. The purpose of this paper is to obtain a generalized form of these relations and to apply them to obtain two classes of complex-screw motions [2].

The aim of this paper is, briefly, an axiomatization of relativistic kinematics. Before stating the aims in more precise terms, a few words about the origins of the paper will be necessary. The idea of a revision of the axiomatic foundations of relativistic kinematics came up in discussions with the late M. L. Urquhart at the 1963 (3rd) Summer Research Institute of the Australian Mathematical Society, and it was a suggestion by Urquhart which started off the present investigation. Following his suggestion I prepared a preliminary draft containing the outlines of an axiomatic system for Minkowski space-time and passed it on to him. Shortly before his death Urquhart asked Professor D. Elliott to send the manuscript back to me and it was this manuscript which formed the nucleus of the present paper.

The Schwarzschild Interior Solution represents a static sphere the proper density of which has the same value throughout. Though it is sometimes referred to as an “incompressible” sphere it is physically unacceptable since (formally) the speed of sound within it is infinite. Perhaps the most natural analogue of the classical incompressible sphere is therefore a sphere such that the speed of sound is everywhere just equal to the speed of light. This paper investigates spheres of this kind in some detail.

The notion of symmetric (non-linear) mappings has been introduced by Vainberg [3, p. 56]. However, symmetric mappings of this type have not played any important rôle in non-linear functional analysis. Naturally, as in the case of linear mappings, the symmetric mappings should be defined in such a way that they are easy to handle and belong to the most elementary class of non-linear mappings.

We shall consider the following mathematical model of dams of finite capacity. In the time interval (0, ∞) water is flowing into a dam (reservoir). Denote by χ(u) the total quantity of water flowing into the dam in the time interval (0, u). The capacity of the dam is a finite positive number h. If the dam becomes full, the excess water overflows. Denote by δ(u) the total quantity of water demanded in the time interval (0, u). If there is enough water in the reservoir the demand is satisfied, if there is not enough water the difference is supplied from elsewhere Denote by η(t) the content of the dam at time t. η(0) is the initial content.

Although there is no need for a ‘distinguished’ submodule to be given a formal definition in the present paper, we like to indicate the meaning attached to this concept here. Perhaps the shortest way of doing so is to say that a distinguished submodule is a (covariant idempotent) functor from the category of (left) R-modules into itself mapping each R-module into its R-submodule specified by a family of left ideals of R. If is a family left ideals of R, then all elements of an R-module M of orders belonging to , do not, of course, in general form a submodule of M; but, there are certain families such that all the elements of orders from form a submodule in any R-module (distinguished submodules defined by ). Consequently, no particular structural properties of the R-module are involved in the definition of such submodules. In this way we can define radicals (in the sense of Kuroš [4]) of a module. In particular, we feel that an application of this method is an appropriate way in defining the (maximal) torsion submodule of a module.

One elementary proof of the spectral theorem for bounded self-adjoint operators depends on an elementary construction for the square root of a bounded positive self-adjoint operator. The purpose of this paper is to give an elementary construction for the unbounded case and to deduce the spectral theorem for unbounded self-adjoint operators. In so far as all our results are more or less immediate consequences of the spectral theorem there is little is entirely new. On the other hand the elementary approach seems to the author to provide a deeper insight into the structure of the problem and also leads directly to the spectral theorem without appealing first to the bounded case. Besides this, our methods for proving uniqueness of the square root and of the spectral family seem to be new even in the bounded case. In particular there is no need to invoke representation theorems for linear functionals on spaces of continuous functions.

Throughout this note X will be a topological space with geometry G of length m–1 with . The terminology will be that of [1].

In [2] the author introduced a self-polar double-N(“CD”): this double-N is associated with a pair of very specially related (“-related”) normal rational curves, in that the spaces Hi, of one row of the double-N are chordal to one of the curves while the spaces Ki of the other row are chordal to the other curve. The double-N might be said to be “associated with” the triple consisting of these two curves and the polarizing quadric.

Let X be a topological space equipped with a binary relation R; that is, R is a subset of the Cartesian square X×X. Following Wallace [5], we write Deviating from [7], we shall follow Wallace [4] to call the relation R continuous if RA*⊂(RA)* for each A⊂X, where * designates the topological closure. Borrowing the language from the Ordered System, though our relation R need not be any kind of order relation, we say that a subset S of X is R-decreasing (R-increasing) if RS ⊂ S(SR ⊂ S), and that S is Rmonotone if S is either R-decreasing or R-increasing. Two R-monotone subsets are of the same type if they are either both R-decresaing or both Rincreasing.

Let Zn be the numer of individuals in the nth generation of a discrete branching process, descended from a single a singel ancestor, for which we put It is well known that the probability generating function of Zn is Fn(s), the n-th functional iterate of F(s), and that if m = EZ1 does not exceed unity, then lim (Harris [1], Chapter 1). In particular, extinction is certain.

If we consider any particular topological semigroup S it may seem reasonable to ask for a characterization of all additions on S which make it a topological semiring. We are interested here in this problem when

(i) S is an (I)-semigroup;

(ii) S is [0, ∞) and the multiplication on S is such that 0 and 1 are Zero and identity respectively.

Let X be a real valued random variable with probability measure P and distribution function F. It will be convenient to take F as the intermediate distribution function defined by . In mathematical analysis it is a little more convenient to use this function rather than , which arise more naturally in probability theory. In all cases we shall consider With this definition, if the distribution function of X is F(x), then the distribution function of −X is 1−F(−x). The distribution of X is symmetrical about 0 if F(x) = 1 − F(−x).

A topological semiring is a system (S, +, ·) where (S, +) and (S, ·) are topological semigroups and · distributes across + as in a ring; that is, for all x, y, z in S, The operations + and · are called addition and multiplication respectively.

In a stationary GI/G/1 queueing system in which the waiting time variance is finite, it can be shown that the serial correlation coefficients {ρn} of a (stationary) sequence of waiting times are non-negative and decrease monotonically to zero. By means of renewal theory we find a representation for Σ∞0 ρn from which necessary and sufficient condition for its finiteness can be found. In M/G/1 rather more can be said: {ρn} is convex sequence, the asymptotic form of ρ n can be given in a nearly saturated queue, and a simple explicit expression for Σ∞0 ρn exists. For the stationary M/M/1 queue we find the ρn's explicitly, illustrate them numerically, and derive a representation which shows that {ρn} is completely monotonic.

A Symplectic lattice L is a free Z-module of finite rank endowed with a non-degenerate alternating bilinear form. Thus we have a bilinear mapping Φ of L × L into the domain of integers Z; we donote Φ (α, β) by α · β (where α, β ∈ L). Then α2 = 0 and α·β = −β·α.