Siegel defined an E-function to be a function which admits a power series expansion

with complex coefficients βn, belonging to some algebraic number field K (of finite degree over the rationals Q), satisfying the following conditions:

(i) Given ε > 0, the maximum of the absolute values of the conjugates of βn, satisfies

In this paper we shall discuss the boundary value problem consisting of the nonlinear ordinary differential equation of the second order, and the boundary conditions.

Let I be a homogeneous ideal in the polynomial ring R = Λ[X1, X2, …, XN], where Λ is a field or, more generally, an Artin ring†. Then R|I has an induced structure as a graded R-module and its homogeneous elements of degree n form a Λ-module of finite length. If this length is denoted by H(n, R|I), then H(n, R|I), considered as a function of n, is often known as the Hilbert function of the ideal I although, in other contexts, it is called the Hilbert function of the graded module R|I. We shall adopt the latter terminology.

In continuation of the two previous papers (10; 11), this paper was originally written at the Indian Institute of Technology, Kharagpur and revised at the University of Sydney under the advice of Prof. T. G. Room. Although the altitudes of a general simplex S(A) in n-space (n > 2) do not concur as they do for a triangle (n = 2), yet we observe that its Monge point, M (1; 5), is an appropriate analogue of the orthocentre of a triangle such that M coincides with its orthocentre when it is orthogonal (or orthocentric). In consistency with the previous papers (10; 11; 13; 15) we shall call M as the S-point of S(A) and denote it as S as explained in § 1.2. The altitudes of S(A) are all met by the (n − 2)-spaces normal to its plane faces at their orthocentres, each parallel to of them, thus indicating the associated character of the altitudes as discussed separately in 2 other papers (12; 16). Before we introduce an orthogonal simplex and develop its properties in regard to its γ-altitudes and associated hyperspheres, we come across a number of intermediate ones of special interest. Two special types are treated here and the other two are developed in 2 other papers (13; 15).

Suppose we have a set of n points P1, …, Pn in a unit square. Let

and let Sn be the upper bound of d1+…dn taken over all positions of P1, …, Pn in the square. We prove that

Over a field of characteristic p the group algebra of a finite group has a non-trivial radical if and only if the order of the group is divisible by the prime p. It would be of interest to determine the powers of the radical in the non-semi-simple case [2, p. 61]. In the particular case of p-groups the solution to the problem is known through the work of Jennings [6]. We here consider the special case of group algebras whose radicals have square zero and we relate this condition to the structure of the group itself.

1. Suppose throughout that a, k are positive numbers and that p is the integer such that k—l≦p<k. Suppose also that φ(w), ψ(w) are functions with absolutely continuous (p+1)th derivatives in every interval [a, W] and that φ(w) is positive and unboundedly increasing. Let λ ={λn} be an unboundedly increasing sequence with λ1 > 0.

Given a series and a number m ≧0, we write

otherwise,

and A(w) = AO(w).

In two previous papers [1], [2] the confluent form

of the δ-algorithm [3]

was established, and various properties which the confluent form of the algorithm possesses were discussed. It was shown, among other things, that if in (1)

and the notation

is used, then (1) is satisfied by

and further that under certain conditions, and for a certain n,

identically. It is the purpose of this note to derive another confluent form of the Ɛ-algorithm and to discuss its properties.

In a very recent paper [1], Basil Gordon discusses generalizations of Jacobi's identity

where x and z are complex numbers and |x| <1. He notes that some of its consequences, inter alia Euler's formula

are of interest in number theory and combinatory analysis. He proves the apparently new and striking result

where |s|<1, and also considers the possibility of generalizations. His methods are algebraic and quite simple, but perhaps do not make obvious what underlies such formulae. It may be worth while to do so, especially since the details become simpler and the presentation more perspicuous. The method given here assumes no more knowledge than his does, although the new proof is expressed in terms of theta-functions, in simple properties of which, formulae such as (3) have their origin. Further, (3) appears in a slightly more symmetrical form.

A function φ(p) is operationally related to h(t) when they satisfy the integral equation

provided that the integral is convergent and R(p)> 0.

As usual, we shall denote (1) by the symbolic expression

φ(p) ≑ h(t).

The object of this paper is to evaluate some infinite integrals involving E-functions by the methods of the operational calculus. Most of the results obtained are believed to be new.

In a previous note [3], Mennicke and I showed that the relations

(8, 7|2, 3): A8=B7=(AB)2=(AB)2=(A-1B)2=(A-1B)3=E

define a group of order 10752. As we remarked, the results of §§ 2, 3 of that note are not restricted in their application to this group; they apply to the group

[3, 7]+: B7=(AB)2=(A-1B)3=E

and to any factor group of this group which in turn has Klein's simple group of order 168, defined by

(4,7|2, 3): A4=B7=(AB)2=(A-1B)3=E,

as a factor group. In this note I use these results to establish alternative “weaker” definitions for Klein's group and for two groups discussed by Sinkov [4], namely (8, 7|2, 3) defined above and a factor group of this group of order 1344. These latter groups are eloquently discussed by Coxeter [1].

A cardinal number which is too large to be reached by some process is generally said to be inaccessible by that process. Many kinds of inaccessible cardinals have been discussed and for a general survey the book of H. Bachmann [1, Chapter 7] may be consulted. We consider here two inaccessibility properties. We shall denote the cardinal of a set X by |X|. The first inaccessibility property will be called regularity: the cardinal| X| will be said to be regular if there does not exist a disjoint cover {X1: i ε I} of X such that

(i)|X1|<|X|, for each i in I, and

(ii)|I|<|X|.

The torsion of beams of L-cross-section was studied for the first time, from a mathematical standpoint, by Kotter [1]. He solved the problem in the case of an L-section both arms of which are infinite. Some time later, Trefftz [2], in his work on the torsion of beams of polygonal cross-section, applied his method also to an infinite L-section. In 1934, Seth [3] solved the case of a beam of an L-section with only one infinite arm. In 1949, Arutyanyan [4] solved the torsion problem of an L-section that has both arms finite, but of equal length, reducing the problem to that of solving an infinite system of equations.

If f(z) is a function of the complex variable z, regular in a neighbourhood of z = 0, with f (0) = 0 and f' (0) ≠ 0, then the equation w = f(z) admits a unique solution, regular in some neighbourhood of w = 0, given by

where C is an appropriate contour encircling z = 0. These formulae are well-known, being stages in the proof of the classical reversion † formula

due to Lagrange.

A formal solution of the coupled linear thermoelastic equations representing oscillatory motion of an elastic body is obtained, and the decay of the normal modes of vibration examined in detail. The main results of Zener's theory of the thermal damping of a vibrating body are then derived, and somewhat amplified, and consistency with the foregoing normal mode analysis established.

This paper is concerned with infinitesimal transverse displacements of homogeneous isotropic elastic plates. The method uses moments of the fundamental equations of orders 0, 1, 2, 3. Assuming a form for the shear stresses tα3, these equations enable one to determine the mean values of the transverse displacements instead of the weighted mean values associated with plate theories of all but the classical type. The relevant moments of the stresses and displacements are expressed in terms of three functions satisfying three differential equations of the fourth order, the solutions of which may be expressed in terms of six independent functions. Thus six boundary conditions may be satisfied. Equating two, three and four of the above functions to zero in turn gives plate theories involving four, three and two boundary conditions respectively. The method is illustrated by assuming that the shear stresses are quadratic functions of the distance from the mid-plane of the material.

The purpose of the study of Diophantine equations is to find out as much as one can about the rational solutions of a given indeterminate set of equations. In geometrical language, this is the investigation of the rational points on a given variety. The simplest type of problem for which no satisfactory theory is known is that of the cubic surface—the homogeneous cubic equation in four variables. Throughout this note we shall exclude the case of a cubic cone or cylinder, whose theory follows trivially from that of a cubic curve; however, we do not assume that the cubic surface is non-singular.