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.

Consider a positive regular Markov chain X0, X1, X2,… with s(s finite) number of states E1, E2,… E8, and a transition probability matrix P = (pij) where = , and an initial probability distribution given by the vector p0. Let {Zr} be a sequence of random variables such that and consider the sum SN = Z1+Z2+ … ZN. It can easily be shown that (cf. Bartlett [1] p. 37), where λ1(t), λ2(t)…λ1(t) are the latent roots of P(t) ≡ (pijethij) and si(t) and t′i(t) are the column and row vectors corresponding to λi(t), and so constructed as to give t′i(t)Si(t) = 1 and t′i(t), si(o) = si where t′i(t) and si are the corresponding column and row vectors, considering the matrix .

Our present view of the universe suggest that the set of mutually receding galaxies may provide a natural substratum for the propagation of light. It is shown that this assumption leads to a consistent derivation and interpretation of special relativity, along the lines evvisaged by Lorentz but requiring also the employment of Einstein's measurement definitions. The time-dilatation and Fitzgerald contraction effects emerge as intelligible consequences of this approach, and their interaction with an associated anisotropy effect produces the relativity of simultaneity, the reciprocity phenomenon and the results described by Einstein's principles; the approach provides a definitive resolution of the “clock paradox” within the framework of Special Relativity.

One of the elementary applications of the Rankine-Hugoniot shock relations which relate conditions on the two sides of a plane shock wave is that of determining the flow when a piston is pushed with constant velocity ū into a tube containing gas at rest. A shock wave races into the undisturbed gas at a constant speed Ū whose value can easily be found in terms of ū and the constants which specify the uniform condition of the gas at rest. If, however, the piston is suddenly brought to rest after a finite time the subsequent behaviour of the shock wave is very difficult to determine. A rarefaction wave is generated at the piston, and, as the velocity of the shock is subsonic relative to the gas behind it, this eventually overtakes the shock wave causing it to weaken. Since the energy supplied is finite the ultimate speed of the shock will tend to that of a sound wave. The analytical treatment of the flow behind the shock is made difficult by the entropy gradients which arise because of the variation in shock strength. It is further complicated by the disturbances which are reflected off the piston and give rise to a secondary interaction with the shock. Indeed, it seems safe to say that a complete description of the motion would certainly depend on some form of numerical integration.

The purpose of this note is to establish the following characterisation of the radical: Theorem. Let R be a ring with the minimum condition for left ideals. Then the radical of R is the intersection of the maximal nilpotent subrings of R.

The principal theorem to be proved in this part is: Theorem II. If in IIn a normal rational curve, ρ, and a quadric primal S are such that there is a proper simplex inscribed in ρ and self-polar with regard to S, then there exist sets of N, = (2n+1/2), chords of р every two of which are conjugate with regard to S. A set can be constructed to contain any pair of chords of р which are conjugate with regard to S.

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.

We consider a single-server queueing system with first-come first-served queue discipline in which (i) customers arrive at the instants 0 = A0 < A1 < A2 < …, with time interval between the mth and (m+1)th arrivals

Let Λ be the set of inequivalent representations of a finite group over a field . Λ is made the basis of an algebra over the complex numbers , called the representation algebra, in which multiplication corresponds to the tensor product of representations and addition to direct sum. Green [5] has shown that if char (the non-modular case) or if is cyclic, then is semi-simple, i.e. is a direct sum of copies of . Here we consider two modular, non-cyclic cases, viz, where is or 4 (alternating group) and is of characteristic 2.

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 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 .

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.

This note is concerned with arithmetic properties of power series with integral coefficients that are lacunary in the following sense. There are two infinite sequences of integers {rn} and {sn}, satisfying such that It is also assumed that f(z) has a positive radius of convergence, Rf say, where naturally . A power series with these properties will be called admissible.

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.

Let ρ and σ be two congruences on a completely 0-simple semigroup. Suppose that there is a maximal chain of congruences from ρ to σ which is of finite length. Then, as we shall show, any maximal chain of congruences from ρ to σ finite and of the same length.

In this we will study analytic solutions to the linear functional equation where f and h are given functions, x is a given complex number and the function g is to be found. This is a generalization of Schröder's functional equation. The results obtained are global in nature and the solutions holomorphic. The equation will be viewed from the standpoint of linear operator theory. When studied in this manner one arrives at a general operator inversion formula.

Let be a p×q matrix of linear forms in the n+1 coordinates in a projective space Πn. Then points which satisfy the q equations in general span a space Πn-q, but will span a space Πn-q+1 if a set μ,={μβ of multipliers can be found such that Such a set μ can be found if and only if the equations have solutions.