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.

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.

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 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.

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 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.

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 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.

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.

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 x1, x2,…, xn, be n consecutive observations generated by a stationary time series {x}, t = 0, ±1, ±2,…, with E(xt2) < ∈. The periodogram of the set of observations, which may be defined as a function In of angular frequency with range [0, π] that is proportional to , plays an important part in methods of making inferences about the structure of {xt}, particularly its spectral distribution function or spectral density.