In this paper, we use the results of (3) to show that a condition which is formally much weaker than that given by Sherman in (11) is necessary and sufficient for the commutativity of a C*-algebra. The basic idea behind the proof is then used again to obtain a characterization in order-theoretic terms of the self-adjoint elements in the centre of a C*-algebra. We shall use the notation and terminology of (3).

It is found that the correlation functions of quantum stochastic processes are very similar to the corresponding functions for classical stochastic processes. It is shown also that if the state for the process satisfies the KMS boundary condition then this is reflected in the analyticity of the continuation of the correlation.

In the structure theory of Banach spaces as developed in (1), an important role is played by subspaces which are the ranges of projections having norm properties akin to those of the classical Banach spaces. A linear projection e on a Banach space V is called an M-projection if

and an L-projection if, instead

A closed subspace J of V is called an M-Summand if it is the range of an M-projection and an M-Ideal if J0 is the range of an L-projection in V′. Every M-Summand is an M-Ideal but the reverse is false.

Approximating sequences have been extensively studied in many branches of mathematics, for example, in number theory (approximating real numbers by rationals) and in numerical analysis (approximations to functions by polynomials). In (1), A. Hyllengren introduced a type of approximating sequence ‘majorizing sequences’ which he used in solving a problem in complex analysis. In this note we study a very similar concept, which is general enough to be applicable to any separable metric space, and which turns out to have strong connexions with the theory of Hausdorff measures (as did Hyllengren's majorizing sequences).

A typical Riesz product on the circle is the weak* limit

where – 1 ≤ rk ≤ 1, øk ∈ R, λT is Haar measure, and the positive integers nk satisfy nk+1/nk ≥ 3. A classical result of Zygmund (11) implies that either µ is absolutely continuous with respect to λT (when ) or µ is purely singular (when ).

For t > 0, let

where xn+r = xr ≥ 0 and xr+1 + xr+2 > 0 for each r.

Let f(x, y) be a bounded real function on the unit square. Consider the Riemann integrals (if they exist)

Certain infinite products occur in potential problems involving plane boundaries and elements, such as line charges, which produce logarithmic potentials. In this paper, the author derives a theorem which is used to evaluate a particular product which arose in the calculation of the electric potential distribution in a flat, square sheet of conducting material when an electric current enters and leaves respectively at line-contact electrodes attached to adjacent corners of the square.

1. Given a series

we will throughout write {sn} for the sequence of its partial sums; that is to say,

We will write p, q to denote the sequences {pn}, {qn}; we will write throughout

and will take P−1 = p−1 = 0.

In the contour integral

f and g are regular functions of z in a neighbourhood of the contour C and of the complex parameters (α1, α2, …, αp) = α in a domain of . N is a positive parameter and asymptotic expansions are considered as N → + ∞. The method of steepest descents provides an asymptotic expansion for each fixed value of α, but this is non-uniform with respect to α near critical values α0 at which certain saddle points tend to coincidence. A more complicated expansion, involving Airy functions or generalizations thereof, is valid and uniform for α near α0. This expansion is known to be valid in a neighbourhood of α0 which does not depend on N, in contrast to certain other expansions having regions of validity which contract to a point or surface as N → ∞. It is thereby suggested that the actual region of validity has a definite geometrical shape determined by the functions f and g and the contour C, just as steepest descents expansions are valid in regions bounded by Stokes' lines. In this paper, procedures are derived for the determination of such regions, subject to assumptions concerning f, g and C. The validity of the expansion is established in regions so determined.

Let denote the group of transformations of the upper-half complex plane U onto itself of the form

If, on U, we introduce the Riemannian metric ds = |dz| y−1 (z = x + iy), then U becomes a model of the hyperbolic plane and its group of isometries. The set of elements of type I, the orientation-preserving isometries form a subgroup of index two in , which we denote by .

Let , where pk are complex numbers, have 0 < ρ ≤ ∞ for radius of convergence and assume that P(x) ≠ 0 for α ≤ x < ρ (α < ρ is some real constant). Assuming that is convergent for all (x ∈ [0, ρ), we define the P-limit of the sequence s = {sk} by

This, so called, power method of limitation (see (3), Definition 9 and (1) Definition 6) will be denoted by P. The best known power methods are Abel's (P(x) = 1/(1 – x), α = 0, ρ = 1) and Borel's (P(x) = ex, α = 0, ρ = ∞). By Cp we denote the set of all sequences, P-limitable to a finite limit and by the set of all sequences, P-limitable to zero.

In this paper the classical Hardy transformation (5) has been extended to distributions. The celebrated Hankel transformation, Y-transformation and H-transformation are particular cases of the Hardy transformation. The Hankel transformation has been extended to distributions by Zemanian(12). The distributional theory has been explored by Zemanian(13), Koh and Zemanian(6), and Dubey and Pandey(2). The famous Y- and its reciprocal H-transformations have not been extended to distributions. The inversion and some of the properties of these distributional transformations can be deduced as particular cases of the results contained in this paper.

Consider a differential game of survival governed by the differential equation

in , with pay-off

where tF is the entry time of the trajectory (t, x(t)) into a given terminal set F. Under suitable conditions on f, g, h and the terminal set F, it was shown in (3) that the question of existence of value of such a game can be approached by considering a certain pair of partial differential equations called the Isaacs-Bellman equations.

In this paper, we improve on the results of two previous papers (8, 9) by establishing a general existence theorem (section 1·3, below) for a class of periodic oscillations of a wide class of non-linear differential equations of the second order in the real domain which are perturbations of the autonomous differential equation

where g(x) is strictly non-linear. We then, by way of illustrating the power of the theorem, apply it to the problems which Morris (section 2·2 below), Shimuzu (section 2·3 below) and Loud (section 2·5 below) set themselves on the existence of periodic oscillations of certain differential equations which are perturbations of equations of the form (1·1·1).

An investigation of the random or stochastic integral equations of the form

and

is presented, where ω ∈ Ω, the supporting set of the probability measure space (Ω, A, P). The existence and uniqueness of a random solution, a second-order stochastic process, of the equations is considered. Several theorems utilizing fixed point theorems and successive stochastic approximations give sufficient conditions for the existence of a random solution.

Let Zn = n−½(X1 + X2 + … + Xn), where {Xn} is a sequence of independent and identically distributed random variables with EX1 = 0, and a common distribution function F and characteristic function ω. Suppose |ω|r is integrable for some integer r ≥ 1. For all n ≥ r, then Zn has a probability density function fn obtained by using the inversion formula.

In electrostatic, elastostatic and hydrodynamic problems involving a plane annulus in space, two mixed boundary value potential problems are frequently encountered. For reasons given below, we consider here the axisymmetric cases of them; and we describe them using cylindrical polar coordinates (r, θ, z) so chosen that the annulus is a < r < b, z = 0.