Let G be a locally compact topological group, let H be a closed subgroup and let G/H be the space of left cosets = xH with the natural topology. We denote by μ a non-negative measure in G/Hdefined on the ring of Baire sets. G acts by left multiplication as a transitive group of homeomorphisms on G/H: Every t ∈ G defines the homeomorphism We write, for E ⊂ G/H, tE = . The measure μ is called stable (cf. [3], [4]) if from t ∈ G, E ⊂ G/H and μ(E) = 0 follows μ(tE) = 0. We say that μ is locally finite [3], [5] if every set of positive measure contains a subset of positive finite measure.

This note extends the concept of the inner automorphism, but here applies only to those finite groups G for which some member of the lower central series is Abelian. In general (e.g. when G is metabelian) the construction yields an endomorphism semigroup, but in the special case where Gis nilpotent (and may therefore, for our present purposes, be considered as a p-group) a group of automorphisms results.

Potential problems in which different conditions hold over two different parts of the same boundary can often be conveniently reduced to the solution of a pair of dual integral equations. In some problems, however, the boundary condition is such that different conditions hold over three different parts of the boundary and, in such cases, the integral equations involved are frequently of the form

where f(r), g(r) are specified functions of r, p = ± ½ and ø(u) is to be found. Such equations might well be called triple integral equations and, in this note, I point out certain special cases which I have found to be capable of solution in closed form.

We will consider a reflexive Banach space 𝔅, with real or complex scalars, and a bounded operator in 𝔅 with a real spectrum.

There are a number of well known theorems on the mutual independence of forms, either linear or quadratic, in normal variables. Some of these theorems can only hold when the system of variables is normal or degenerate and so the possibility of certain forms being independent characterises the normal distribution. The theorems on characterisation have usually been proved by consideration of the necessary properties of the characteristic function. Here we shall be considering the characteristic function of the variables but we shall make more use of cumulant theory than previous authors. To do so we have first to prove that the existence of cumulants of all orders is implied by the independence conditions. The basic theorems we use are those from general cumulant theory and the special theorems of Cramer and of Marcinkiewicz. An advantage of the methods of this paper is that it is possible to show that some of the characterisation theorems require neither of these special theorems. For example, spherical symmetry is a very strong condition and so neither theorem is required whereas both theorems are needed for the most general theorems on the independence of two linear forms. Throughout we take the class of normal distributions to include the degenerate normal, that is, the distribution of a sure variable.

By using certain fractional integrals and derivatives it is possible to construct a continuum of Hilbert spaces within the space L2 (0, ∞); these are the spaces gλ of functions f(x) for which 1xλf(λ)(x) є L2(0, ∞), and they exhibit invariance properties under generalized Fourier transformations. They are described in (6) and (7).

Polynomial solutions of a few binomial congruences have been known for a long time. For instance Legendre showed that the congruence has a solution this being the expansion of as far as the term of degree m — 3. [1] It seems that only restricted types, e.g. (1), have been investigated.

We deal with questions about the possible embeddings of two given groups A and B in a group P such that the intersection of A and B is a given subgroup H. The data, consisting of the “constituents” A and B with the “amalgamated” subgroup H, form an amalgam.1 According to a classical theorem of Otto Schreier [5], every amalgam of two groups can be embedded in a group F, the “free product of A and B with amalgamated subgroup H” or the “generalized free product” of the amalgam. This has the property that every group P in which the amalgam is embedded and which is generated by the amalgam, is a homomorphic image of it. Hence theorems on the existence of certain embedding groups P can be interpreted also as theorems on the existence of certain normal subgroups of F.

This is a sequel to a recent paper [1] on the construction by the hodograph method of trans-sonic nozzle-flows of a perfect gas. At the end of that paper it was shown how we can obtain regular flows that are ultimately uniform (as required in the test section of a supersonic wind tunnel), and the object now is to give some quantitative examples of such flows. The gas is supposed to have the polytropic equation of state Pρ−γ = constant, and the calculations have been made for the case γ = 1.4, with the Mach number M = 2.25 at the test section. The results, which are exhibited graphically, are indicative of what may be expected for other supersonic values of M, and it is hoped that they may be significant for the design of wind tunnels.

This note is concerned with a question arising from some work of D. R. Smart (2). In the introduction we indicate the nature of the problem. Notations will be explained only in as far as they differ from those of (2).

An investigation has been made into the numerical solution of non-singular linear integral equations by the direct expansion of the unknown function f(x) into a series of Chebyshev polynomials of the first kind. The use of polynomial expansions is not new, and was first described by Crout [1]. He writes f(x) as a Lagrangian-type polynomial over the range in x, and determines the unknown coefficients in this expansion by evaluating the functions and integral arising in the equation at chosen points xi. A similar method (known as collocation) is used here for cases where the kernel is not separable. From the properties of expansion of functions in Chebyshev series (see, for example, [2]), one expects greater accuracy in this case when compared with other polynomial expansions of the same order. This is well borne out in comparison with one of Crout's examples.

Two sets of r + 2 points, Pi, P'i, each spanning a projective space of r + 1 dimensions, [r + 1], which has no solid ([3]) common with that spanned by the other, are said to be projective from an [r — 1], if here is an [r — 1] which meets the r + 2 joins Pi ′i. It is to be proved that the two sets are projective, if and only if the r + 2 intersections Ai of their corresponding [r]s lie in a line a. Ai are said to be the arguesian points and a the arguesian line of the sets. When r= 1, the proposition becomes the well- known Desargues' two-triangle theorem (3) in a plane. Therefore in analogy with the same we name it as the Desargues' theorem in [2r]. Following Baker (1, pp. 8—39), we may prove this theorem in the same synthetic style by making use of the axioms and the corresponding proposition of incidence in [2r + 1] or with the aid of the Desargues' theorem in a plane and the axioms of [2r] only. But the use of symbols makes its proof more concise; the algebraic approach adopted here is due to the referee (Arts. 2, 3, 5, 6, 7). Pairs of sets of r + p points each projective from an [r— 1] are also introduced to serve as a basis for a much more thorough investigation.

A Room square is an arrangement of the k(2k−1) unordered pairs (ar, as), with r≠s, formed from 2k symbols a0, a1 …, a2k−1 in a square of 2k−1 rows and columns such that in each row and column there appear k pairs (and k−1 blanks) which among them contain all 2k symbols.

The incidence matrices for the finite projective planes, for the so-called λ-planes (where two lines meet in λ points instead of the usual 1) and for some other configurations, including those of Pappus and Desargues, are all cases of what are defined below as (v, k, t, λ)-matrices. These are shown here to possess arithmetical properties which reduce, in the case of cyclic projective planes, to properties remarked on by Marshall Hall [3]. And if a certain group hypothesis (which suggests itself in a natural way) is satisfied by a matrix of this type, the matrix is shown to be equivalent (by rearranging the rows and columns) to a direct sum of incidence matrices for λ-planes, each of which satisfies the same group hypothesis.

The main object of this paper is to show that an indefinite nonsingular quadratic form which is incommensurable (that is, is not a constant multiple of a form with integral coefficients) takes infinitely many distinct small values, for a suitable interpretation of the word small. This proves a conjecture made by Dr. Chalk in conversation with the writer. I believe that the theorems proved are new and of interest, though they are easy deductions from known results.

In [1] and [2] the flow of a conducting fluid past a magnetized sphere was considered. The magnetic distribution was an arbitrary axially symmetric one. The magnetic field, velocity, vorticity and drag were evaluated for the particular case of a dipole field when the dipole was in the free stream flow direction. Astronomically the more interesting case is that in which the dipole is perpendicular to the free stream flow. The axially symmetric nature of the problem is thereby lost.

In previous papers [1, 2], the authors have solved the Stokes flow problem for certain axially symmetric bodies, with the velocity at infinity uniform and parallel to the axis of symmetry. Each of the bodies considered possessed the property that the meridional section intercepted a segment of the axis of symmetry. In the present paper this assumption is removed; in addition, we shall consider the particular case of the Stokes flow about a torus.

Let Κ be a finite number field and let ο be the ring of algebraic integers in Κ. The algebraic integers in a finite extension field Λ of Κ form a ring . We shall be concerned here with the structure of such rings , viewed as modules over ο. It will be useful to begin with a brief discussion of a new concept of the discriminant of Λ/Κ, introduced in a preceding paper [1], which will be our principal tool (see also [2]).