It is known that there is a connexion between the functional equation of the theta-function

and the reciprocity formula for Gauss sums. The latter formula is usually written as

where p and q are positive integers, mutually prime, and not both odd.

If x1, x2, …., xk, …. are independent random variables each of which is subjected to a distribution law σ = σ(x) independent of k and having a finite positive dispersion, then x1 + x2 + …. + xn is known to obey the Gauss law as n→ + ∞, no matter how σ (x) be chosen. There arises, however, the question whether it is nevertheless possible to determine the elementary law σ (x) from the asymptotic behaviour of the distribution law of x1 + x2 + …. + xn for very large but finite values of n. It will be shown that the answer is affirmative under very general conditions.

The idea of the following proof was communicated to me some years ago by Mr Edward Carpenter of Millthorpe, Derbyshire, formerly Fellow of Trinity Half, Cambridge; who remarked that it seemed to afford a demonstration of Taylor's Theorem which came very naturally and directly from the definition of a differential coefficient. The chief difficulty seemed to arise in dealing with the negligible small quantities which are produced in great numbers. However, I found it not difficult to complete the proof for the case when all the successive differential coefficients of f(x) are finite and continuous.

1. In his Miscellanea Analytica of the year 1762, Waring stated the identities

where sp stands for the sum of the pth powers of n quantities α, β, γ,…. If the identities be written in the uncontracted form

it is readily seen that they hold not merely for certain powers of certain quantities, but for any quantities whatever; and that, indeed, for the elements of the array

we may legitimately substitute in order the elements of the array

thus obtaining the identities

These latter were given by Binet in 1812, and are usually, but with little real justification, spoken of as “Binet's Identities.”

Let be a polynomial with distinct real zeros whose separation is defined by δ(f) = min i≠j(ai-aj ). We establish upper estimates for δ(f′-kf) in terms of n, k, and δ(f). The results give sufficient conditions for the inverse operator (D – kl)−1 to preserve the reality of the zeros of a polynomial.

Let Bnf; x) denote the Bernstein polynomial of degree n on [0,1] for a function f(x) defined on this interval. Among the many properties of Bernstein polynomials, we recall in particular that if f(x) is convex in [0,1] then (i) Bn(f;x) is convex in [0,1] and (ii) Bn(f;x)≧Bn+1(f;x), (n = l,2,…). Recently these properties have been the subject of study for Bernstein polynomials over triangles [1].

The authors have recently treated (2) the problem of finding subsets E of the real line , of type Fσ, such that E–E contains an interval and the k-fold vector sum (k)E is of measure zero. Positive results can be obtained, for all k, on the basis of a recent theorem of J. A. Haight (3), following earlier partial results (1), (4) for k ≦ 7; and indeed in these cases the problem has a solution with E a perfect set. An analogous problem, apparently in most respects subtler than the first, is the following. Do there exist finite regular Borel measures μ on such that is absolutely continuous (where is the adjoint of μ) and the kth convolution power μk is singular? Both problems are of interest in the general context of elucidating the properties of the measure algebra or, more generally, M(G) for locally compact abelian G. The second problem may be regarded as an attempt to provide (at least one aspectof) a multiplicity theory for the first.

It is sometimes the case that geometrical theorems, which are usually enunciated as properties of the triangle or the quadrilateral, may be stated more succinctly, and in a form that better suggests generalisation, as properties of the complete quadrilateral. Thus the theorem that is usually stated in the following form, “The three straight lines that join the middle points of opposite sides and the middle points of diagonals of a quadrilateral are concurrent and bisect one another at the point of concurrence,” may be enunciated more simply as a property of the straight lines that join the middle points of the three pairs of opposite sides of a complete four-point; and the latter form of enunciation suggests how the theorem may be generalised by projection. So too the figure that consists of four Points, any one of which is the orthocentre of the triangle formed by the other three, is better described as a complete four-point in which opposite sides intersect at right angles, (which may be called an orthic four-point).

Let G be a group and K a field. We denote by (KG) the group of units of the group ring of G over K and for a group X we denote by T(X) the setof torsion elements of G i.e., the set of all elements of finite order.

We give arithmetic characterizations which allow us to determine algorithmically when the semigroup ring associated to a simplicial affine semigroup is Cohen-Macaulay and/or Gorenstein. These characterizations are then used to provide information about presentations of this kind of semigroup and, in particular, to obtain bounds for the cardinality of their minimal presentations. Finally, we show that these bounds are reached for semigroups with maximal codimension.

In an earlier paper1 the author investigated the relation existing between the induced matrices of a group of permutation matrices and the table of group characters of the irreducible representations of the corresponding symmetric group. It was found that the traces of a particular set of induced matrices sufficed to give, by a relatively simple transformation, the complete table of characters.It was remarked also that for n > 4 the set of compound matrices of permutation matrices, on the other hand, could at most provide only part of the table; for in fact the number of compounds, n + 1. is then less than P (n), the numbe'r of partitions of n. For this reason the subject was not pursued into further detail.

The paper deals with superlinear elliptic boundary value problems depending on a parameter. Given appropriate hypotheses concerning the asymptotic behaviour of the nonlinearity, we prove lower bounds on the number of solutions. The results generalize a theorem due to Lazer and McKenna within the framework of quasi-Banach spaces of Besov and Triebel-Lizorkin spaces.

In [7, Section 5], Glimm showed that if φ and ψ are inequivalent pure states of a liminal C*-algebra A such that the Gelfand-Naimark-Segal (GNS) representations πφ and πψ cannot be separated by disjoint open subsets of the spectrum  then ½ (φ+ψ) is a weak*-limit of pure states. We extend this to arbitrary C*-algebras (and more general convex combinations) by means of what we hope will be regarded as a transparent proof based on the notion of transition probabilities. As an application, we show that if J is a proper primal ideal of a separable C*-algebra A then there exists a state φ in (the pure state space) such that J=ker πφ (Theorem 3). The significance of this is discussed below after the introduction of further notation and terminology.

There are various methods in existence for the practical solution of a set of simultaneous equations

Some of these methods are appropriate to special systems, as for example to the axisymmetric “normal equations” of Least Squares. In many applications, however, as in problems of statistical correlation of many variables, it may be desired not merely to solve a given set of equations but to obtain as much knowledge as possible about the system or matrix of coefficients; perhaps to evaluate its determinant and various minors, such as principal minors, possibly also to determine the elements of the adjugate matrix, or the reciprocal matrix. The examination of the sign of successive principal minors of an axisymmetric determinant, in order to find the signature of the corresponding quadratic form, is a case in point; and there are many such applications.

In a former paper published in these Proceedings it was shown that an integral function of order less than 1 cannot have any asymptotic periods, and it was suggested that a function of order 1 can have at most a set Kω(k=±1, ±2, …). This was subsequently found to be the case. Meromorphic functions for which K, the exponent of convergence of the poles, is less than ρ, the order, behave in many ways like integral functions, so we should expect that (i) if 0≦κ<ρ1 there should be no asymptotic periods, (ii) 0≦κ<ρ1 either none or else a single sequence kω(k = ±1, ±2, …). It will be shown that this is so.

Whittaker's studies of the relations between physics and philosophy are contained in about forty books and papers, nearly all of which were published during the last fifteen years of his life. His approach is mainly historical and is a natural sequence to his History of the Theories of Aether and Electricity. Many of the publications were based on lectures delivered to various groups and there is naturally a certain amount of repetition.

Let R be a ring. An element p of R is a prime element if pR = Rp is a prime ideal of R. A prime ring R is said to be a Unique Factorisation Ring if every non-zero prime ideal contains a prime element. This paper develops the basic theory of U.F.R.s. We show that every polynomial extension in central indeterminates of a U.F.R. is a U.F.R. We consider in more detail the case when a U.F.R. is either Noetherian or satisfies a polynomial identity. In particular we show that such a ring R is a maximal order, that every height-1 prime ideal of R has a classical localisation in which every two-sided ideal is principal, and that R is the intersection of a left and right Noetherian ring and a simple ring.