The main concern of this paper is with the solution of infinite linear systems in which the kernel k is a continuous function of real positive variables m, n which is homogeneous with degree –1, so that If k is a rational algebraic function it is supposed further that the continuity extends up to the axes m = 0, n > 0 and n = 0, m > 0; the possibly additional restriction when k is not rational is discussed in § 1,2.

Let {Pn} be any sequence of real or complex numbers subject to the sole restriction And let If tn → s, whenever sn → s we say that the sequence {sn} is summable Nörlund or summable (N, p) to s.

The Fourier transform F(y) of a function f(t) in L1(Ek) where Ek is the k-dimensional cartesian space will be defined by .

An analogue in a solid of the well known Pascal's theorem (Baker, [1], p. 219) for a conic is established by Baker ([2], pp. 53–54, Ex. 15) after Chasles [6] and by Salmon ([2], p. 142). The same is discussed in detail by Court [8]. The purpose of this paper is to extend it to a projective space of n dimensions or briefly to an n-space Sn. To prove it, we introduce here once again the idea of a set of n+1 associated lines in Sn as indicated in an earlier work (Mandan, [12]) in analogy with a set of 5 associated lines in S4 (Baker, [4], p. 122), and make use of the method of induction.

In this note we discuss the stability at the origin of the solutions of the differential equation where a dot indicates a differentiation with respect to time, and α, β are real-valued functions of any arguments. We tacitly assume that α, β are such that solutions to (1) do in fact exist. Under the transformation equation (1) takes the equivalent familiar form .

The purpose of this paper is to prove that the altitudes of an n-simplex (a simplexin an n-space) S form an associated set of n+1 lines (see Baker, [4] for n = 4) such that any (n–2)-space meeting n of them meets the (n+1)th too. As an immediate consequence 2 quadrics are associated with S, one touching its primes at the respective feet of its altitudes and the other touching n(n+1) primes, n parallel to each of its altitudes and 2 through each of its (n−2)-spaces. Certain special cases are also mentioned.

Consider a Markov process defined in discrete time t = 1, 2, 3, hellip on a state space S. The state of the Process at time time t will be specifies by a random varable Vt, taking values in S. This paper presents some results concerning the behaviour of the saquence V1, V2, V3hellip, considered as a time series. In general, S will be assumed to be a Borel subset of an h-dimensional Euclideam space, where h is finite. The results apply, in particular, to a continuous state space, taking S to be an interval of the realine, or to discrete process having finitely or enumerably many states. Certain results, which are indicated in what follows, apply also to more general (infinite-dimensional) state spaces.

By a linear canonical system we mean a system of linear differential equations of the form where J is an invertible skew-Hermitian matrix and H(t) is a continuous Hermitian matrix valued function. We reserve the name Hami1tonia for real canonical systems with where Ik denotes the k × k unit matrix. In recent years the stability properties of Hamiltonian systems whose coefficient matrix H(t) is periodic have been deeply investigated, mainly by Russian authors ([2], [3], [5], [7]). An excellent survey of the literature is given in [6]. The purpose of the present paper is to extend this theory to canonical systems. The only work which we know of in this direction is a paper by Yakubovič [9].

In [2] we defined an irreducible B(J)-cartesian membrane, and used this to obtain a characterization of an n-sphere by generalizing the definition of simple closed curve given by Theorem 1.2 below. There B(J) is a class A(n) of (n−1)-spheres, but here it is a class of mutually homeomorphic continua. In Theorem 1.1 we give a definition of hereditarily unicoherent continua and generalize this in Section 3 by means of B(J)-cartesian membranes. To do this we paraphrase by a translation some of Wilder's work in [7]. In his Unified Topology [8: p. 674] he gives a principle: “The connectedness of a domain is a special case of the bounding properties of its i-cycles”. We substitute the element J of B(J) for the i-cycle and for “bound” we substitute that “J membrane-bases an irreducible B(J)-cartesian membrane. The very nature of an i-cycle seems to limit the complexity of the point set studied, although the restriction to “nice” manifolds is due partly to the difficulty of the subject matter treated. There are similar difficulties here, but also advantages, in the very general set-theoretic approach by means of B(J)-membranes.

We prove that factor groups of cartesian powers of finite non-abelian simple groups cannot be countably infinite. Thisis not our main result, but it had been our original aim. The proof is based on a similar fact concerning σ-complete Boolean algebras, and on a representation of certain subcartesian powers of a group in its group ring over a Boolean ring. This representation, to which we give the name “Boolean power”, will be our central theme, and we begin with it.

Let be a space of points x, a σ-field of subsets of a σ-finite measure on . The elements of will be called measurable sets and all the sets considered in this paper are measurable sets. A real-valued point function t(x) on will be said to be measurabl if, for each real α, the set {x: t(x)≦ α} is measurable. Let (S), S C denote the σ-field of all measurable subsets of S. A real-valued function f(·) on will be called a set function.

Let α(n) be a multiplicative arithmetic function. H. Delange [1] has proved that if |α(n)| ≦ 1 for all n and for a certain constant ρ, , where if ρ = 1 then then . He applied this result to several problems such as uniform distribution (mod 1) of certain types of sequences.

Although the harmonic series diverges, there is a sense in which it “nearly converges”. Let N denote the set of all positive integers, and S a subset of N. Then there are various sequences S for which converges, but for which the “omitted sequence” N–S is, in intuitive sense, sparse, compared with N. For example, Apostol [1] (page 384) quotes, without proof the case where S is the set of all Positive integers whose decimal representation does not invlove the digit zero (e.g. 7∈S but 101 ∉ S); then (1) converges, with T < 90.

Nevile [2] has shown that if Rt is a certain measure of the rate of growth of the national income in Harrod's growth model of an economy, then Rt satisfies the non-linear recurrence relation , where 0 < k < and −1 < c < 1. The definition of Rt ([2] p. 369) is such that Rt > 0 for all t. Nevile has pointed out features of the model that indicate that it may be unstable. In this paper I propose to show that the model is, in general. unstable, but that proper choice of the initial values R0R1 apparently leads to stability. In order to do this, we require the conditions (if any) under which Rt converges.

The following paper is a sequel to the author's earlier paper [2]. In that paper some general results were obtained which described the motion of a fluid with a free surface subsequent to a given initial state and prescribed boundary conditions of a certain type. The analysis was based on a linearized theory but gravity effects were included. Viscosity, compressibility and surface tension effects were neglected. Among the problems treated was that of the normal symmetric entry of a thin wedge into water at rest. This Water entry problem has attracted a considerable amount of attention since the pioneer paper by Wagner [5]. Both linear and non-linear approximations have been used but all papers apart from [2] neglect gravity on the assumption that in the early stages of the penetration this is unimportant. One of the objects of [2] was to determine the solution with the gravity terms retained. A formal solution was obtained but no attempt was made to analyse this quantitatively. In the present paper we examine the extent of this effect in some detail. It will be of help to the reader to have some familiarity with the first three or four sections of [2] but in order to make the present paper self-contained we shall first reintroduce the notation used there and quote the necessary results from that paper without proof.

Let E be a topological linear space over the real number field. Throughout of this paper, we denote by G an open subset of E, by ∂G the boundary of G and by the closure of G. The totality of all circled open neighbourhoods of the zero element denoted by U.

Soit f(x) continue strictement croissante pour x ∈ [0, a0] et telle que 0 < f(x) < x pour x ∈ 0, a0]. Il est connu que l'équation fonctionnelle d'Abel ainsi que l'équation de Schröder possèdent une infinité de solutions continues strictement croissantes.

In this paper we shall use the integral operator method of Bergman, B[1–6], to investigate solutions of the partial differential equation where s > −1. In particular, information concerning the growth, and location of singularities, of solutions of (1.1) will be obtained. Equations of the form (1.1) with s = 1, 2, arise from the (n+k+1)-dimensional Laplace equation Δn+k+1u = 0 in the “axially symmetric” coordinates x1, …xn, p where the relationship between cartesian and “axially symmetric” coordinates is given by

In a recent paper by P. D. Finch and myself [1], the solution for the limiting distribution of a moving average queueing system was obtained. In this paper the system is generalised to the case of batch arrivals in batches of size ρ > 1.

Throughout this paper, G denotes a Hausdorff locally compact Abelian group, X its character group, and Lp(G) (1 ≦ p ≦ ∞) the usual Lebesgue space formed relative to the Haar measure on G. If f ∈ Lp(G), we denote by Tp[f] the closure (or weak closure, if p = ∞) in Lp(G) of the set linear combinations of translates of f.