1. The simple group of order 6048 has a permutation representation of degree 28 and two of degree 63. These are conspicuous in SU(3,32), the representation as a group of 3-rowed matrices, unitary and of determinant + 1, whose elements all belong to GF(32), or to J as we name this field. When these matrices as linear transformations in the projective plane ϖ over J they leave invariant the unit Hermitian form

and the single capital U will henceforth signify the group so acting. There are 28 isotropic points n in ϖ satisfying H3 = 0; they furnish the representation of degree 28. The other 63 points p in ϖ are non-isotropic and furnish one of the representations of degree 63. The other is furnished by 63 triangles T; the vertices of a T are all p and every pair of them is linked in the antipolarity Π set up by H3; if their coordinate vectors are y and z then y′z = 0. Each p is a vertex of 3 T. These matters are fully explained in (7). Those points linked by II to a given point A are collinear, on the polar of A; no p lies on its polar, every n does. The polars t of points n are called tangents of H3 = 0, and the only isotropic point on t is its pole. The polars s of points p are secants of H3 = 0: each consists of 4 n with 3 pairs of linked p and, given s, it is these 3 pairs that complete those T having a vertex at the pole of s.

Let G be a locally compact Abelian group, written adoptively, with Haar measure m, L1(G) the group algebra of G, and M(G) the Banach algebra of all bounded, complex-valued, regular, countably additive measures on G. For a general account of L1(G) and M(G) see Rudin (7).

Of particular importance in the structure theory of finitely generated groups are those groups G having the property ℳd that for any two sets of generators g1, …, gd; h1, …, hd of G there is an automorphism θ of G such that (i = 1, …, d). Gaschütz(1) proved that all groups G having† ℳd also possess a further manifestation of symmetry, the property that if M and N are finite normal subgroups of G, any isomorphism between G/M and G/N can be induced by an automorphism of G. The case in which G is a finite p-group had previously been considered by Taunt (6) and Liebeck(2), who proved that for such groups the above property on the quotient groups of G is equivalent to ℳd.

Sierpiński ((6), (7)) proved that the continuum hypothesis implies the following proposition:

Euclidean space E3 can be decomposed into three sets Si (i = 1, 2, 3) such that, for some three straight lines Di in E3, the intersection of each line parallel to Di with the corresponding set Si is finite.

Let be a plane closed convex curve, and regard as the boundary of a billiard table. If a perfectly elastic billiard ball is set moving on the table, its path will be a polygon inscribed in ; and this will be closed if and only if the motion is periodic. We seek periodic paths which are triangles.

In (6), Kiang gives an example of a simply connected (closed) region D in 3-space R3, bounded by a smooth 2-sphere ∂D, whose (harmonic) Green's function Γ has a critical point for any choice of pole p ∈ Int D—contrary to the analogous 2-dimensional situation. It is not known whether any position of p exists, for which Γ is non-degenerate in the sense of Morse theory; however, we show here the following result.

1. The purpose of this note is to prove a result which includes certain classical theorems generally thought of as being unconnected; in explicit terms, a result about the Fourier series of a periodic Lebesgue-integrable function showing that the series is summable at a point by a Nörlund method (N, pn) defined as usual ((2), p. 64) if pn ↓ 0, Σpn = ∞ and the point is in a certain subset of the Lebesgue set. More precisely, the purpose is to prove Theorem I on the Nörlund summability of Fourier series and to derive from it the well-known Theorems A, B which follow and the recent extension of Theorem A in Theorem A' which appears later and is due to Sahney (8).

An alternative procedure is developed for examining the solutions of the Low equation. The ambiguities of those solutions which have a finite number of zeros are shown to satisfy the conditions derived earlier by Castillejo, Dalitz and Dyson. The method presented here has the advantage that it makes no special use of the positiveness of the cut-off function, and so can be used when this function changes sign. It is shown that the arbitrariness of the solution increases with the number of these sign changes, but that a completely negative cut-off function still satisfies the CDD conditions.

The theory of Fourier transforms

can be developed from the functional equation K(s) K(1 – s) = 1, where K(s) is the Mellin transform of the kernel k(x).

In this paper I show that reciprocities can be obtained which are analogous to the Fourier transforms above but which develop from the much more general functional equation

The reciprocities are obtained by using fractional integration. In addition to the reciprocities we have analogues of the Parseval theorem and of the discontinuous integrals usually associated with Fourier transforms.

In order to simplify the analysis I confine myself to the case n = 1 and to L2 space.

Asymptotic expansions of certain generalized Bernoulli polynomials are obtained, some in terms of elementary functions and others in terms of gamma functions and their derivatives. The latter results can also be written in terms of elementary functions, by using the known asymptotic expansions of the gamma functions, and the leading terms are obtained in this way. The results obtained give the asymptotic form of the coefficients occurring in all the usual central-difference formulae of the calculus of finite differences.

The purpose of this paper is to establish a few approximation formulae with error estimates for the integration of rapidly oscillating functions, and to present an expression for the remainder term of an expansion formula obtained previously (see (3)). Moreover, we shall give a certain sharpest possible estimation for the error term involved in the numerical integration of periodic functions. Some results contained in earlier papers ((2), (4), (5)) will be employed and sharpened.

The irregular radial Coulomb wave function in repulsive fields is expressed, for any angular momentum value, in series of Bessel functions of arbitrary order. Previous expansions, obtained for angular momenta zero and one, can now be obtained as special cases. The expansion can be most conveniently used for high-energy particles.

Exact analytical solutions of certain second-order linear differential equations are often employed as approximate solutions of other second-order differential equations when the solutions of this latter equation cannot be expressed in terms of the standard transcendental functions. The classical exposition of this method has been given by Jeffreys (6); approximate solutions of the equation (using Jeffreys's notation)

are given in terms of solutions either of the equation

or of the equation

where h is a large parameter. A complete history of this technique is given in the author's recent text An introduction to phase-integral methods (Heading (5)).

1. The equation considered here is of the form

where a, b are constants, h(x) is differentiable and h′(x), p(t) are continuous in x, t respectively. The primary object of the paper is to prove the following

Theorem 1. Suppose that

(i) a > 0,b > 0;

(ii) h(0) = 0, h(x)/x ≥ δ > 0 (x ≠ 0);

(iii) h′(x) ≤ c for all x where ab > c > 0.

The time-dependent solutions of an infinite set of differential-difference equations arising from queueing theory and models of ‘living’ polymer are expressed in terms of modified Bessel functions. Explicit solutions are available for constant values of a parameter describing the arrival rate or monomer concentration; for time-dependent parameter a formal solution is obtained in terms of a function which satisfies a Volterra type integral equation of the second kind. These results are used as the basis of a numerical method of solving the infinite set of differential equations when the time-dependent parameter itself satisfies a differential equation.

In the investigation of preference orders which are explanations of expenditure data which associates a quantity vector xr with a price vector pr (r = 1, …, k), in respect to some n goods, there is considered the class of functions φ, with gradient g, which are increasing and convex in some convex region containing the points xr, such that gr = g(xr) has the direction of pr. † Let ur = pr/er, where so that

1. A type of problem which frequently occurs in probability theory and statistics can be formulated in the following way. We are given real-valued functions f(x), gi(x) (i = 1, 2, …, k) on a space (typically finite-dimensional Euclidean space). Then the problem is to set bounds for Ef(X), where X is a random variable taking values in , about which all we know is the values of Egi(X). For example, we might wish to set bounds for P(X > a), where X is a real random variable with some of its moments given.

The Feynman method of quantization by summation over q-histories is extended to a summation over q, p-histories, corresponding to the classical variational method of treating q and p as independent variables. The equivalence of this q-p summation to the usual quantization procedure is shown in the Heisenberg picture by choosing convenient canonical coordinates and in the Schrödinger picture by obtaining the usual representations of the q, p operators.

A proof by means of graphical techniques is given of an inequality conjectured by Nakanishi concerning the Feynman integrand for any term in the perturbation expansion of the scattering amplitude.

1. This paper is a continuation of a previous one(1), in which dispersion relations satisfied by the scattering amplitudes for a non-relativistic two-channel scattering process were derived. It was shown that for the process described by the integral equations

with

the scattering amplitudes Mi(E, τ) (where E is the incident energy and τ is the momentum transfer), satisfy the relations