If f(t) belongs to L(0, R) for every positive R and is such that the integral

converges for x > 0, then F(s) exists for complex s(s ╪ 0) not lying on the negative real axis and

for any positive ξ at which f(ξ+) and f(ξ−) both exist.

We define an operator Lk, t[F(x)]by

Under the above conditions on f(t), it is known that for all points t of the Lebesgue set for the function f(t),

Let Ln, x denote the differentiation operator

Suppose that

converges for some x¬ 0; then, if f(t) belongs to L(R−1, R) for every R>1,

In (6) we studied a topology T on the product set X × Y of two topological spaces X and Y which was defined by the requirement that each mapping from X × Y which was continuous in each variable separately was also continuous in T; we called (X × Y, T) the tensor product of X and Y, and denoted it by X ⊗ Y. Theorem (3·2) of (6) indicated that X ⊗ Y was rarely completely regular; as complete regularity is of importance in analytic problems, we consider here a ‘completely regular tensor product’ . Roughly speaking, gives a tensor product in the category of completely regular topological spaces. The categorical properties of are discussed in section 5.

This paper describes new detailed Monte Carlo investigations into bond and site percolation problems on the set of eleven regular and semi-regular (Archimedean) lattices in two dimensions.

The propagation of electromagnetic waves through isotropic media containing a continuous energy-gain mechanism is considered and the existence of perfect resonance properties is investigated for both homogeneous and inhomogeneous layers.

Let (E, C) be a partially ordered vector space with positive cone C. The order-bound topology Pb(6) (order topology in the terminology of Schaefer(9)) on E is the finest locally convex topology for which every order-bounded subset of E is topologically bounded.

We let G denote an infinite compact group and G its dual. We use the notation of our book ((l), Chapters 7 and 8). Recall A(G) denotes the Fourier algebra of G (an algebra of continuous functions on G), and ℒ∞(G) denotes its dual space under the pairing 〈ƒ,φ〉 (ƒ ∈ A(G), φ ∈ ℒ∞(G)). Further, note ℒ∞(G) is identified with the C*-algebra of bounded operators on L2(G) commuting with left translation. The module action of A(G) of ℒ∞(G) is defined by the following: for ƒ ∈ A(G), φ ℒ∞(G), ƒ. φ ∈ ℒ∞(G) by 〈g, ƒ . φ〉 = 〈 ƒg, φ〉, g ∈ A(G) Also ‖ƒ . φ‖∞ ≥ ‖ƒ‖A ‖φ‖∞.

A many-body variational calculation is carried out using a trial wave function which is a product of pair functions. An exact integro-differential equation for the pair function is derived, valid for any system of more than three Bose particles.

1. It often happens that the greater parts of two variables x, y may be connected linearly but there may be other disturbances of both. A treatment given by one of us ((1), section 4·42, pp. 216–17) assumes that the other disturbances are independent, as for observational errors. This is not altogether satisfactory because unless the variations of xr, yr are much greater than the standard errors there is an asymmetry in the treatment corresponding to the well-known distinction between the two regression lines.

It is shown that for the Cesàro means (C, α) with α > - 1, and for a certain class of more general Nörlund means, summability of the series σan implies uniform summability of the series σan zn in a Stolz angle at z = 1.

If B is a normal matrix and (B) denotes the series summability field with the usual Banach space topology, then the vectors {ek} (ek = {0,0,..., 1,0,...}) are said to form a Toplitz basis for (B) relative to a method H if H — Σakek = a for each a = {ak}ε(B). It is shown for example that the above relation holds for B = (C,α), α> − 1 , and H = Abel method; also for B = (C,α) and H = (C,β) with 0 ≤ α ≤ β.

Applications are made to theorems on summability factors.

If f is meromorphic in D = {,|z| < 1}, we say that f has the asymptotic value a at ζ, ζ ∈ C(={|z| = 1}), if there exists an arc Γ such that Γ ends at ζ, (Γ − {ζ}) ⊂ D, and f has the limit a as |z| → 1 on Γ. The class (M) originally defined by G. R. MacLane ((8), p. 8), consists of those functions that are non-constant and holomorphic (meromorphic) in D and that have asymptotic values at a dense subset of C. In (8), p. 36, MacLane has shown the following three conditions are sufficient for a non-constant holomorphic function f to be in :

1. Let g(n) = g(n; m) denote the number of partitions of a natural number n into powers of a given natural number m (> 2). Then the generating function of g(n) is

(1) where we take g(0) = 1.

In the following a distribution or a generalized function f is denned, as by Gelfand and Shilov(2) or by Temple (3), as a continuous linear functional on the space K of infinitely differentiable test functions ø having compact support. The value of f at a test function ø will be denoted by (f, ø).

A sequence of test functions {øn} is said to be a null sequence if

(1) the support of each øn is contained in some fixed domain D independent of n,

(2) the sequence converges uniformly to zero in D, as n tends to infinity, for all m.

The algebraic invariants of the Riemann tensor for a static axially symmetric gravitational field are examined. Contrary to common expectation they are shown to be well defined in a region of the field known to be physically singular.

Introduction: Let X1, X2, …, Xn be n (n ≤ 2) independent observations on a random variable X with distribution function F. Also let L = L (X1, X2, …, Xn) be a linear statistic and Q = Q (X1, X2, …, Xn) be a homogeneous quadratic statistic. In this paper, we consider the problem of characterizing a class of probability distributions by the linear regression of the statistic Q on the other statistic L. In section 2, we obtain a characterization of a class of probability distributions, which includes the normal and the Poisson distributions. In section 3, a class of distributions including the gamma, the binomial and the negative binomial distributions is characterized.

The success of approximate methods in the calculation of transmission coefficients of waves penetrating potential barriers rests, not so much on any satisfactory approximation theory, but on a comparison with a particular barrier, namely the parabolic barrier. The paucity of exact analytical solutions of barrier problems has prevented an explicit detailed analysis of the errors involved, but two recently devised barriers with analytical solutions are investigated here, in order to compare the exact values of the transmission coefficients (both as to their mathematical structure and numerically) with their approximate forms. An estimate is also made of the effect of neglecting poles existing in the complex propagation plane.

Two extensions of the Rikitake model are examined; the first includes viscous friction and the second includes time delay. By use of Liapounov's direct method the viscous system is shown to have an unstable limit cycle associated with two of the three singular points. By numerical integration true ‘reversals of field’ are shown to occur for both the viscous and time delay cases.

A recent search for analytical profiles in plane-stratified isotropic gyrational media has revealed particular Epstein models for permittivity and magneto-electric properties of the media when subjected to normally incident electromagnetic waves. The reflexion and transmission coefficients of such a model are derived and possible transparencies are discussed.

1. There has been considerable debate on the conditions under which a given function f(r) can be expanded as a series of the orthogonal functions where υ1, υ2, … are the zeros of the Hankel function regarded as a function of its order. Here k, a are positive constants and 0 < a ≤ r < ∞. The formal expansion can be written as

where