1Introduction. In this paper, I give a definition of uniform distribution for sequences taking values in certain types of Abelian group—in particular, in groups of prime exponent—and one of independent distribution for sets of such sequences. In the various cases, I find conditions for the properties to hold, which are similar to Weyl's criterion for the uniform distribution of real sequences modulo 1.

Let G be a finite soluble group. In (1) Alperin proves that two system normalizers of G contained in the same Carter subgroup C of G are conjugate in C. In recent unpublished work G.A.Chambers of the University of Wisconsin has proved that, if is a saturated formation, the -normalizers of an A-group are pronormal subgruops; hence, in particular, that two -normalizers contained in an -projector E of an A-group are conjugate in E. In this note we describe an example which shows that in Alperin's theorem the class of nilpotent groups cannot in general be replaced by an arbitary saturated formation without some restriction on the class of soluble groups under consideration. we prove

PROPOSITION. There exists a saturated formationand a group G which has two-normalizers E1and E2contained in an-projector F of G such that E1and E2are not conjugate in F.

1. Introduction. Although Corollaries 4 and 5 to Theorem 1 below appear elsewhere in the literature (1, 2), the proofs given seem to use rather long and involved arguments and refer to other results in the literature for their completeness. The proofs given below are brief and follow quite naturally in sequence with the other corollaries from Theorem 1. The arguments presented are independent of references to the literature except for the reference in the proof of Theorem 1 to Lemma 2·1 of (3), which is well known to topologists. For our purposes this theorem may be stated as: For each open 2-manifold M (non-compact and without boundary), there is a subcomplex L made up of edges of some triangulation of M such that the open simplicial neighbourhood of L is piecewise linearly homeomorphic to M.

1.Introduction. To every closed subalgebra S of a regular algebra R(D) of functions tending to zero at infinity on a discrete space D we define the Rudin spectrum of S to be the partition of D consisting of the equivalence classes of the equivalence relation .

We say a series is summable L if

tends to a finite limit s as x → 1 in the open interval (0, 1) where

1.Introduction. 1.1. Let f(t) be a periodic function with period 2π and integrable in the Lebesgue sense over ( -π,π). We assume as we may without loss of generality, that the Fourier series of f(t) is .

Integral transformations analogous to the Nörlund means have been introduced and investigated by Kuttner, Knopp and Vanderburg(6), (5), (4). It is known that with any regular Nörlund mean (N, p) there is associated a function

regular for |z| < 1, and if we have two Nörlund means (N, p) and (N, r), where (N, pr is regular, while the function is regular for |z| ≤ 1 and different) from zero at z = 1, then q(z) = r(z)p(z) belongs to a regular Nörlund mean (N, q). Concerning Nörlund means Peyerimhoff(7) and Miesner (3) have recently obtained the relation between the convergence fields of the Nörlund means (N, p) and (N, r) on the one hand and the convergence field of the Nörlund mean (N, q) on the other hand.

In 1949, Meyer-König (5) introduced the so-called Sα-method of summability which is one of a family of transformations including the Euler, Borel and Taylor methods. Later in 1959, Jakimowski (3) defined the [F, dn] transformation which includes the Euler method as a special case. If {dn} is given infinite sequence and E denotes the displacement operator (i.e. Eks0 = sk), then the [F, dn] transformation of {sn} is given by {tn} where

Introduction. Continued fractions of the form are called Hurwitzian if b1, …, bh, are positive integers, ƒ1(x), …, ƒk(x) are polynomials with rational coefficients which take positive integral values for x = 0, 1, 2, …, and at least one of the polynomials is not constant. f1(x), …, fk(x) are said to form a quasi-period.

In a recent paper in this journal(1) Baker obtained a result giving effectively computable bounds on the simultaneous approximation of certain numbers of the form (ab-1)nj/n by rational numbers of the formpιq-1 where a, b, n, the nι and q are positive integers while the pι are non-negative integers. In this paper we shall obtain explicit bounds on the simultaneous approximation of certain numbers of the form rβ1, …, rβn where β and each rι, 1 ≤ j ≤ n, are positive rational numbers.

In this paper we have evaluated some integrals involving Fox's Η-function and employed them to establish some expansion formulae for the Η-function involving exponential functions. Some results recently given by MacRobert (7) and Kesarwani (5) have been shown as particular cases.

It is well known (Watson, A treatise on Bessel functions) that if Jν(z) is a Bessel function of the first kind, and A is a real constant, then all roots of the equation are either real or purely imaginary.

Lanczos(1, 2) has considered the shifted eigenvalue problem where C is a (p × q) matrix of rank r, CH is its hermitean conjugate and u, v are column vectors of orders p, q respectively. In this note we extend Lanczos' work to cover the more general eigenvalue problem which arises in certain problems in solid state physics (3). In (2), the I's are unit matrices of appropriate orders and the constants a, b are real; the partitioned matrix M is thus hermitean so that its eigenvalues λ are real.

We consider a stochastically continuous process ω(t, α) with independent increments, whose sample functions are bounded in the unit interval 0 ≤ t ≤ 1 for almost all α. If ω(t, α) is a process with independent increments, the characteristic function of ω(t, α) is of the form exp {tψ(u)} where where F is a σ-finite measure with finite mass outside every neighbourhood of o, and a and σ are constants. There is no essential restriction in supposing ω(0, α) = 0.

In this paper certain Bessel type eigenfunction expansions are developed by considering a non-seif-adjoint problem which involves a radiation type condition

The calculus of 4-spinors, covariant with respect to arbitrary transformations in a 6-dimensional Riemann space, was presented earlier in a formulation analogous to the so-called ε-formalism in the 2-spinor calculus. Like the latter it is burdened by the appearance of a multiplicity of fractional spin weights and anti-weights as well as the arbitrariness of the trace of the linear spinor connexion. This paper deals in detail with a variant of the calculus which is more akin to the so-called γ-formalism in the 2-spirior calculus; and which is finally extended to be both gauge-invariant and phase-invariant.

An exact static solution of the system of Einstein–Maxwell field equations in the absence of charge is obtained when both the metric tensor gαβ and the Maxwell field tensor Fαβ exhibits plane symmetry in the sense that both remain invariant under the group of motions that characterize plane symmetry. Our solution generalizes the empty space-time solution with plane symmetry previously obtained by Taub to the situation when static charge-free electromagnetic fields are present. We hope to discuss non-static solutions of Einstein–Maxwell fields with plane symmetry as well as some other considerations in a subsequent paper.

Introduction. The problem of the scattering of high-frequency waves, which emanate from a line source in a homogeneous isotropic dielectric medium and impinge upon a cylindrical obstacle, has been attacked in a variety of ways. In certain cases, where both the shape of the obstacle and the conditions to be satisfied on its boundary are particularly convenient, an exact solution may be found by separation of the wave equation (see, for example, Marcuvitz (l)), but in general some form of approximation is necessary to obtain an explicit answer.

Solutions of the partial differential equation ψzzzz + ψx + 0 of the type ψ = x¼nw(y), where y = zx−¼ and n is an integer, are investigated. The equation occurs as a boundary-layer approximation in certain rotating and stratified fluid flows in which the production of vorticity (due, for example, to changes in the Coriolis parameter with latitude in two-dimensional flows on a beta plane, or by the buoyant generation of vorticity in a Boussinesq fluid) is opposed by a diffusive process. The similarity functions w(y) satisfy the fourth order differential equation

.

These functions have many properties analogous to those of error functions and parabolic cylinder functions. When n is a non-negative integer, there exist polynomial solutions of the latter equation. These have analogies with Hermite polynomials, although they do not form an orthogonal set in any useful sense. The main properties of the similarity functions are listed, including their series expansions, integral representations, asymptotic expansions and recurrence relations. In particular, a pair of independent solutions, Jon(y) and Kon(y), are denned such that Jon(y) and Kon(y) are real for real y and vanish at an exponential rate as y → + ∞. The function Jon(y) also decays exponentially as y → − ∞ if n is negative, and grows algebraically as y → − ∞ if n is positive. Curves of the functions Jon(y) and Kon(y) are given for |n| ≤ 3 and 0 < y < 6, and their more useful properties are listed.

The paper deals with the problem of investigating the effect of magneto-thermo-elastic interactions on the cooling process of an infinite circular cylinder when its boundary is subjected to a time-decaying temperature. The Hooke–Duhamel law of elasticity, the equation of heat conduction together with the electromagnetic equations of Maxwell supplemented by Ohm's law have been used to solve the problem. It is found that the Laplace transformation serves as an important tool for the solution of the problem.