Every algebraic equation can be uniformized by automorphic functions belonging to a certain group of bilinear transformations. In certain cases, such as for hyperelliptic equations, this group is a subgroup of the monodromic group of a differential equation of the form

where R(z) is a rational function which, in general, contains unknown parameters as coefficients. A conjecture of E. T. Whittaker regarding the values of these parameters for the hyperelliptic case is proved for a wide variety of algebraic equations whose branch points possess certain symmetric properties, and is extended to equations of higher type. In several cases, the uniformizing functions belong to subgroups of the groups of the Riemann-Schwarz triangle functions.

The bivariate distribution corresponding to the univariate negative binomial, and the corresponding distribution when sampling is without replacement, are investigated. Formulas are derived for the factorial moment generating functions and for the regression equations, which are linear.

Canonical forms of the four-dimensional complex Lie algebras are obtained by considering the roots of certain well-defined vectors of the algebras. A complete set of characters of the algebras is also given, enabling any given four-dimensional complex Lie algebra to be identified with one of the canonical forms.

A result proved elsewhere concerning the structure of linear algebras of genus one is used (i) to classify Jordan algebras of genus one, and (ii) to obtain necessary and sufficient conditions for an algebra of genus one to be simple. The class of simple Jordan algebras of genus one and dimension not less than 4 over an algebraically closed field is shown to coincide with the class of simple Jordan algebras of degree two.

In this paper are considered certain numbers called moduli associated with a rectangular matrix with complex elements. These moduli have to satisfy a set of conditions analogous to those satisfied by the modulus of a complex number. For a complex rectangular matrix A = (aij) it is shown that R(A), C(A) and | A |° are moduli of A where:

where cmin(H) and cmax(H) denote, respectively, the minimum and maximum characteristic values of the hermitian matrix H, A* being the transpose conjugate of A. Using the various properties of a modulus of a matrix and taking R(A), C(A) and | A |° as the moduli, a number of known results about the characteristic values of a matrix are obtained and extended. Relations between |A|°, |A |° R(A), p(A) and C(A), ɣ(A) are also studied. These relations provide a number of results about estimates of bounds of characteristic values of sums and products of matrices.

Metrisable Lie algebras have been defined by Tsou and Walker (1957). Their definition is adopted below in § I.

The object of the present paper is to display something of the geometrical background of such algebras, particularly for those taken over the field of real numbers.

§ I is introductory. In § 2 appears a statement, made as brief as possible because it is wholly classical, of the relationship between the vector-space of the Lie algebra L and the associated affine and projective spaces A and P. Some properties of metrisable Lie algebras are then examined in terms of the geometry of P, which provides an (n –I)-dimensional map of the n-dimensional algebra L. It is assumed throughout the paper that the Lie algebras under discussion are non-abelian, since the projective map of an abelian algebra presents nothing of interest.

As the present work is intended as no more than a preliminary, it is confined, so far as its applications are concerned, to a discussion of metrisable algebras of dimensions 3 and 4 and to one example of an algebra of dimension 6.

I have been privileged in preparing the paper to have access to the typescript of the paper by Tsou and Walker referred to above, and also to the doctoral thesis (1955) of the former. I am greatly indebted to them both, and also to Dr Paul Cohn and to a referee for suggestions regarding certain details of presentation.

The E-functions were defined by MacRobert [3] in 1937; they are denoted by E (p; αr: q; ps. z).

In § 3 of this paper, I prove a new expansion for E(p; αr: q; ps: z) which is similar to an expansion due to MacRobert [2], viz.,

where

The Mathieu functions of integral order [1] are the solutions with period π or 2π of the equation

The eigenvalues associated with the functions ceN and seN, where N is a positive integer, denoted by aN and bN respectively, reduce to

aN = bN = N2

when q is zero. The quantities aN and bN can be expanded in powers of q, but the explicit construction of high order coefficients is very tedious. In some applications the quantity of most interest is aN – bN, which may be called the “width of the unstable zone“. It is the object of this note to derive a general formula for the leading term in the expansion of this quantity, namely

Suppose first that N is an odd integer. Then there is an expansion

where

These functions π satisfy

and

On Substituting (3) in (1), one obtains the algebraic equation

where

Explicitly,

{11} = q

{lm} = 0 otherwise.

Let K be a field. We denote by K[t] the integral domain of all polynomials in an indeterminate t with coefficients in K, and by K(t) the quotient field of K[t], i.e. the field of all formal rational functions of t over K. A valuation |f| of the elements f of K(t) can be defined by

for f ≠ 0, and |0| = 0, where e > 1. This valuation is multiplicative, and has the properties

In the solution of many problems in applied mathematics it is often convenient to have expansions for functions, F(r), which satisfy the boundary conditions

In some recent work on hydrodynamic and hydromagnetic stability the author has found certain types of expansions for such functions which have proved very useful and which appear to be novel in this connection. In this note two types of such expansions will be considered and the principles underlying them will be described.

The problem of the isotropic elastic sphere under the application of equal and opposite point couples ±N at its poles has been treated by M. Sadowsky [1] and by A. Huber [2]. The object of this note is to obtain their results by elementary means.

The problem is an example of the torsion of a shaft of varying circular section, the surface of the shaft being obtained by rotating a curve ξ = α about the z-axis.

It was proved by Roth in a recent paper that if α is any real algebraic number, and if K > 2, then the inequality

has only a finite number of solutions in relatively prime integers p, q (q > 0) The object of the present paper is to prove that the lower bound for κ can be reduced if conditions are imposed on p and q. The result obtained is as follows.

It is our purpose here to show that, using results already in the literature, it is easy to prove the following and similar theorems.

For every positive integer d, there exists an integer Ψ (d) such that if K is an algebraic number field of degree d over the field of rational numbers then every cubic form f(x1 x2, …, xn) over K, with n ≥ Ψ(d), has a non-trivial zero in K.

In the present paper a simple technique will be developed for the arithmetical determination of certain class group components and class number factors in finite number fields. This technique is based on classical theories (Hilbert's work on inertia groups, the theory of absolutely Abelian fields as class fields of congruence groups, absolute class fields of number fields). In keeping with the traditional approach to the subject we shall use here the language of ideal theory. The only non-classical concepts to be used (which, however, are of fundamental importance) are those of the inertia groups and the congruence groups associated to p-adic fields. We shall also give some illustrations of the use of our technique in some special cases. Further applications will follow in subsequent papers.

The indentation produced by an axially symmetrical punch bearing on the plane surface of an elastic half-space has been considered by Harding and Sneddon [1], who used Hankel transforms and a well-known pair of dual integral equations, and for the case of a spherical punch they took the indenting surface to be part of the approximating paraboloid of revolution. Chong [2], also using these dual integral equations has treated the case of a symmetrical punch of polynomial form and considers a two-termed expansion for a spherical punch. More recently, Payne [3] has given the exact solution for a spherical punch using either oblate spheroidal coordinates or toroidal coordinates.