1. Good (4) has proved the formula

where Pn(x) is the Legendre polynomial of degree n and t is any integer greater than n. The form of the right-hand side suggests that (1) may be of use in deriving arithmetic properties of Pn(x).

Elsewhere (1) the writer indicated a connexion between divisibility properties of Pm(a) for special values of a and the complex multiplication of elliptic functions. If p = 2m + 1 is an odd prime, put

In a recent note (1) I sharpened existing theorems, which state, roughly speaking, that unless a symmetric convex body K has an excessively small homogeneous minimum, it must have a reasonably small inhomogeneous minimum; in other words, if the translates of K centred at points of the integer lattice form an efficient packing, then by expanding them we may derive an efficient covering. In this note I will prove the converse, that a bad packing leads to a bad covering; the result is a much less useful one, as the worst case occurs when we foolishly try to pack rather spiky bodies point to point. A result of this kind has previously been proved by Mahler (4), but his result is less precise from our present point of view; a full account is given by Cassels(3).

Asymptotic theory of partitions of unipartite numbers has been extensively investigated after the classic work of Hardy and Ramanujan on the subject. The more complicated problem of bipartite partitions has been only recently broached. Auluck(1) has deduced asymptotic formulae for p(m, n), the number of partitions of the bipartite number (m, n), both for m fixed and n → ∞, and for both m and n tending to infinity in such a way that m/n = 0(1) and n/m = 0(1). It will be shown here that the formula obtained for the first case is actually valid for m = 0(n¼). This proves the conjecture made earlier by the author (3).

1. Discussion of results

1·1. Introduction: The classical theorem of Lagrange states that the order of a subgroup of a finite group G divides the order, (G), of G. More generally, if H and K are subgroups of G, and H ≥ K, then (G:K) = (G:H)(H:K), where (G:K) denotes the index of K in G, etc. We call a number a possible order of a subgroup of G if it is a divisor of (G), and a possible order of a subgroup of G containing a subgroup H if it is a divisor of (G) and a multiple of (H). In this paper we discuss conditions on G for the existence of subgroups of every possible order, the existence of subgroups of every possible order containing arbitrary subgroups, and similar properties.

By an iteration of a regular summation matrix B = (bmn) with a regular matrix A = (amn), or B.A, we shall mean the summation method defined by the matrix B operating on the sequence {tn} of A transforms of a sequence {sn}. Thus the sequence {τn} of B. A is given by

The unitary irreducible representations on a Hilbert space of the universal covering group G of the 3 + 2 deSitter group are determined, using Harish-Chandra's theory of semi-simple Lie groups. These irreducible representations of G are reduced out with respect to representations of the universal covering group K of the direct product R2 × R3 of a two-dimensional and a three-dimensional rotation group. It is convenient to distinguish between two types of irreducible representations of G, called ‘irreducible case’ and ‘reducible case’ representations. The former are more numerous in a certain sense, while the latter may be regarded as special cases.

Let (j, µ) be an irreducible unitary representation of K, where j denotes the (ray) representation of R3 involved (2j is a non-negative integer) and µ that of R2 (µ is a real number). Then in an ‘irreducible case’ representation of G, each (j, µ) that occurs at all occurs with multiplicity j + ½ if 2j is odd, or j + ½ ± ½ if 2j is even. In a ‘reducible case’ representation of G, the multiplicity of the (j, µ)'s is smaller, and may even be 1 for all (j, µ)'s which occur. (The analogous problem for the 4 + 1 deSitter group has been solved by L. H. Thomas by reducing out the irreducible representations of the whole group with respect to those of the four-dimensional rotation group R4. However, each irreducible representation of the 4 + 1 deSitter group contains any irreducible representation of R4 at most once, so that the more general Harish-Chandra theory is not then needed to carry out the classification.)

In an irreducible representation of G, only those irreducible representations (j, µ) of K may occur for which the fractional part of µ is fixed throughout the representation of G. This fractional part of µ helps to characterize the representations of G, as do the two polynomials in the infinitesimal elements, one of degree 2 and the other of degree 4, which generate all those polynomials which have the property of commuting with all members of the Lie algebra of G.

Representations of the inhomogeneous Lorentz group may be obtained from representations of G by the process of contraction, as defined by Inonu and Wigner, just as representations of the former group may in turn be contracted to obtain representations of the Galilei group.

Introduction. Let be the complex separable Hilbert space. We say that the closed linear operator T, with domain dense in. , is represented by the infinite matrix H if T is the operator T˜1(H) defined† by H (with respect to some complete orthonormal set). We define an (nJ)-matrix as a Hermitian matrix H = [hij]i, j ≥ 1 for which hij = 0 when i − j > n and hij ╪ 0 when i − j = n. (Thus a Jacobi matrix is a (1J)-matrix.) If, in addition, hij = 0 when 0 < i − j < n, we call H an (nJ ┴)-matrix.

1. Introduction. Let 0 < λ < 1 and remove from the closed unit interval the open interval of length λ concentric with the unit interval. From each of the two remaining closed intervals of length ½(1 − λ) remove the concentric open interval of length ½λ(1 − λ). From each of the four remaining closed intervals of length ¼λ(1 − λ)2 remove the concentric open interval of length ¼λ(l − λ)2, etc. The remaining set is a perfect non-dense set of Lebesgue measure zero and is the Cantor set for λ = ⅓. Let Tλr be the Cartesian product of this set with the set similar to it obtained by magnifying it by a factor r > 0. Letting L be Carathéodory linear measure (1) and letting G be Gillespie linear square(2), Randolph(3) has established the following relations:

1. Introduction. In this note Abel-type summability methods (Aλ) are defined and some of their properties investigated.

Let

and let {sn} be any sequence of numbers. If

is convergent for all x in the open interval (0,1) and tends to a finite limit s as x → 1 in (0,1), we shall say that the sequence is Aλ-convergent to s and write sn → s (Aλ;). The A0 method is the ordinary Abel method.

1. Introduction. The definition of the order of an integral function of one complex variable extends immediately to integral functions of several complex variables to give us their total order. In the case of integral functions of two complex variables, Lelong(5) has shown that the order on any analytic plane through the origin is equal to the total order except for an exceptional set of planes of exterior capacity 0 on which the order is less than the total order.

It is often necessary to obtain expansions in spherical harmonics that belong to a given irreducible representation of a symmetry group. This is the case, for instance, when the cellular method is applied to investigate the band structure of a metal, where expansions are required that reproduce the symmetry of the group of the k vector (see Bouckaert, Smoluchowski and Wigner(4); von der Lage and Bethe (9)). Another instance where such expansions are necessary appears when hybrid orbitals are obtained for a central atom in a molecule of given symmetry (Kimball (8)). In this case lower order spherical harmonics are considered and tables for them up to l = 2 (functions s, p and d in real form) are given in the literature (cf. for example Eyring, Walter and Kimball (5)). However, interest has recently arisen in hybrids that include f functions (Shirmazan and Dyatkina (12)) and an extension of these tables appears to be desirable.

A simple solution of a general problem involving a bifurcated wave guide is presented. The purpose of the work is to explain a new and simple method of solving such problems and to exhibit an organic connexion between Sturm–Liouville theory and the theory of two-part boundary-value problems.

Asymptotic solutions of the differential equation

for large positive values of u, are examined; z is a complex variable in a domain Dz in which P1(z) and z2q(z) are regular and p1(z) does not vanish. In this paper it is shown that there exist Airy-type expansions of the solutions of this equation which are valid uniformly with respect to z in a domain in which z = 0 and z = z0 are interior points. If Dz is unbounded and the equation has a regular singularity at infinity, Airy-type expansions exist which are valid at z = 0, z = z0 and z = δ. If p(z) = constant + O (│z│-1) as │ z │ → ∞ in Dz, similar expansions also exist. The results given here are new.

1. Introduction. The object of this investigation is to obtain by means of contour integrals exact solutions of certain nth-order differential equations, together with their n independent power-series solutions, their asymptotic solutions and the relationship between these two types of solution. The Stokes phenomenon associated with the changing constants in these asymptotic solutions will then be investigated by various methods. Paper I is concerned with the properties of the various contour-integral solutions of the equations under consideration. The manner in which the arbitrary constants in the general asymptotic solution must change as the argument of the independent variable z varies is dealt with in paper II. This phenomenon is considered in a way that has proved profitable for the approximate solution of more general linear differential equations of the nth order that are approximately identical with those considered here hi certain regions of the complex z–plane. In later publications, these approximations will be described together with their application to explicit physical problems.

1. Introduction. In paper I(2), solutions have been found for the nth-order differential equations (I, 3), (I, 9), (I, 18), namely,

where ϑ = wd/dw, m is any rational fraction and a any constant. Power-series solutions have been found for these equations, together with integral representations and their asymptotic expressions valid for restricted ranges of arg z. The object of this second paper is to consider these asymptotic solutions in more detail, and to extend these expressions to all values of arg z. The Stokes phenomenon will be manifest throughout, and this will be treated in a manner suitable for further application.

A direct method of integrating the equation y″ + g(x) y = h(x), with the two-point linear boundary conditions y′(a) + αy(a) = A, y′(b) + βy(b) = B, is based on the factorization of the equation into two first-order linear equations v′ − sv = h and y′ + sy = v, where s is a solution of the Riccati equation s′ − s2 = g. The first-order equations for v and y are integrated in succession, one in the direction of x increasing, and one in the direction of x decreasing, one boundary condition being used in each of these integrations. The appropriate solution of the Riccati equation is determined by the boundary condition at the end of the range from which the integration of the equation for v is started. The process is compared with the matrix factorization method of Thomas and Fox, and its stability discussed.

Proof of convergence of the Crank-Nicolson procedure, an ‘implicit’ numerical method for solving parabolic partial differential equations, is given for the case of the classical ‘problem of limits’ for one-dimensional diffusion with zero boundary conditions. Orders of convergence are also given for different classes of initial functions. Results do not support the validity of so-called h2-extrapolation in some cases.

It is shown that a combination of the methods developed in the two previous papers in the series allows the reformulation of general relativity as a technique for extrapolating physical theory to distant regions, whose accessibility to experiment is limited. General relativity is thus shown to be one of a number of extrapolation theories allowed by the spread theory of Paper II.

1. Introduction. Among the more notable attempts to derive a generalization of Einstein's gravitational theory is the recent one of Einstein and Schrodinger ((1)–(8)). This was formulated by dropping the symmetry of the fundamental tensor gμν and the components of the affine connexion. The most serious defect of these non-symmetric theories is that the field equations, in their original form, do not determine the motion of electrically charged particles in an electromagnetic field, as has been proved by Infeld(9), Callaway (10) and Bonnor (n). Together with the lack of an energy-momentum tensor and a geometric description of the paths of charged particles, this seems to indicate that the concept of motion is missing in this type of theory. It is clear that one of the most important results which should follow from a generalization of Einstein's gravitational theory is the correct equations of motion of charged particles in an electromagnetic field.

A brief account is given of the fundamental properties of a new generalization ((1), (2)) of Einstein's gravitational theory. The field equations are then solved exactly for the case of a static spherically symmetric gravitational and electric field due to a charged particle at rest at the origin of the space-time coordinates. This solution provides information about the gravitational field produced by the electric energy surrounding a charged particle and yields the Coulomb potential field. The solution satisfies the required boundary conditions at infinity, and it reduces to the Schwarzschild solution of general relativity when the charge is zero.