A gap in the proof of the main theorem of (1) has been pointed out by C. D. Papakyriakopoulos and R. H. Fox. That proof was incomplete in that the table of homeomorphisms on page 776 of (1) did not cover all the possible cases (of which there are an infinite number). The corrected proof, given below, is intended to be a re-written version of the first half of section 3 of (1), the notation and references being those of (1).

1. We work in the category of compact polyhedral spaces and polyhedral maps ((5)). Given spaces M and Q and a map f: M → Q, we define Ur(f) to be the set of points x ∈ M such that f−1f(x) contains at least r points. Sr(f) is the closure of Ur(f) in M. f is an embedding if it is a homeomorphism into, and an immersion if it is locally an embedding. We shall call f a simple immersion if S3(f) = ø and the connected components of S2(f) are individually embedded by f. Obviously a simple immersion is an immersion. If M and Q are manifolds (as they will be for the rest of the paper)f: M → Q is proper if it takes ∂M, the boundary of M, into ∂Q. In (2) the following result was proved:

Theorem 1. Let Mmbe(2m−q)-connected and Qq be(2m−q + 1)-connected, m ≤ q−3. Then any proper map f: M → Q which embeds ∂M is homotopic rel ∂M to an embedding.

A theory of quadratic forms is developed over R(t), the field of rational functions with real coefficients, instead of over an algebraic number field: some results are analogous to the classical ones, but many, in particular the non-finiteness of the class number, whose more detailed treatment will be the subject of a later paper, are not analogous.

We use the notations and definitions of the introduction to the previous paper ((1)), and say that 2 lattices K and L on a quadratic space over R(t) with its Dedekind set C are in the same genus if, for all , the lattices and are in the same class. We mean to count the number of classes of lattices in a given genus over a binary space: we shall find that it is either 1 or .

In a recent paper, Smith ((l)), considering a statement or conjecture of Isbell ((2)), proved that the Hausdorff uniformities arising from two different uniformities on the same set must induce different topologies on the ‘hyperspace’ of subsets, except possibly in one case which remained open, namely, when the two given uniformities induce the same proximity and neither is precompact. I now give an example to show that different uniformities can (in this case) induce the same topology on the hyperspace (in the example one of the uniformities is metric). This disproves Isbell's conjecture.

The purpose of this paper is to prove theorems for subharmonic functions analogous to those stated below for regular functions. We make the following definition: D(∈, R) is the domain in the Euclidean plane defined by

Introduction. In this paper, a study is made of the solutions of Lamé's differential equation as series of associated Legendre functions. The particular feature studied is the representation of the second solution corresponding to the case when the first solution is a Lamé polynomial.

We refer to a beautiful and important result of Tutte(1), in the theory of graphs; that a linear graph G is prime if and only if it contains a set ∑ of vertices, such that u(G∑) > n(∑); where n(∑) is the number of vertices in ∑, G∑ is the graph obtained from G by deleting the star of ∑ (all the vertices of G in ∑, together with all the edges of G meeting vertices of ∑), and u(G∑) is the number of connected components of G∑ having an odd number of vertices.

Spectral theorems and a functional calculus for families of permutable self-adjoint operators in a separable Hilbert space are developed in a form convenient for physical applications. A recent paper of Jauch and Misra is discussed.

1. The determination of potentials and wave functions defined in regions bounded by the natural coordinate surfaces of a cylindrical polar coordinate system leads to eigenvalue problems connected with Bessel's differential equation

We shall consider the annulus 0 < a ≤ r ≤ b and take first the boundary condition to be that of vanishing on both surfaces. It is then convenient to introduce the function

This function will automatically satisfy the condition of vanishing for r = a. It will also vanish on r = b if u and k are related by the condition that ψ(u, k, b) = 0. If we regard u as being fixed and real then there will be an infinite number of zeros kn(u)(n = 1,2,…).

The integral, involving the functions Jn+½ and J−n−½(x) where n is a positive integer, arises in a problem in electrical network theory and represents a measure of the overall error in the diagonal Padé approximant to the function e−s. A method of evaluation based on Euler's transformation of series is described, values of the integral are tabulated to five decimal places for n = 1(1)10(5)40, and the asymptotic behaviour for large n is determined.

I asserted in (1), p. 35, that the topology of the hyperspace H(μX)of a uniform space μX determines the uniformity μ, and sketched a proof. D. Hammond Smith has shown (2) that the proof indicated is wrong, even for metric spaces. However, Smith proved the assertion for metric spaces and some others, and left open the question whether it was true in general. This note gives a counter example. I should note that I began with a transfinite construction based on Smith's example; whether that would have worked or not, the following construction is simpler.

This is one of a number of papers in which the author studies the properties of the transform Hα(f) and allied operators. In this case, identities involving these operators and the Fourier transform are obtained, and these are applied in giving alternative proofs to certain inversion processes derived by the author elsewhere.

Rohlin (7) has shown that Kolmogorov automorphisms are mixings of all degrees. Later Sucheston(9) introduced regular automorphisms and after showing that they are mixings of all degrees asks if the converse of either of the above theorems holds. Since the equivalence of Kolmogorov and regular automorphisms follows from (1) and (8) we have only one problem. We show that the converse of either of the above theorems is false by constructing a stationary Gaussian process which is a mixing of all degrees but which is not a Kolmogorov automorphism.

Introduction. The object of this paper is to prove some formulae for Jacobi polynomials. The formulae (6), (7), (9), (10) and (13) are believed to be new.

Introduction. In this paper we consider the two cross-product combinations of Bessel functions

where δ = (k − 1) z and (') denotes differentiation with respect to the argument. Here Jν and Yν designate respectively the Bessel functions of the first and second kind of order ν.

The Feynman integral is evaluated for classical waves using a Hamiltonian approach to the summation over histories.

An important and much-investigated class of measures is the class of Hausdorff measures, first defined by Hausdorff (1). These measures form a subclass of the class of translation invariant measures, but just how wide a class they form is not known.

Certain physical theories are short-wave limits of more general theories. Thus ray optics is the short-wave limit of wave optics, and classical mechanics is the short-wave limit of wave mechanics. In principle it must be possible to deduce the former from the latter theories by a rigorous mathematical limiting process; in fact the arguments found in the literature are formal, plausible and non-rigorous. (We are here concerned with linear wave equations and time-periodic phenomena.) For some wave equations there are, however, a few explicit rigorous canonical solutions relating to simple geometrical configurations, e.g. to conics in two dimensions for the equations of acoustics, and for these the asymptotics can be found rigorously. For more general configurations the solution of a typical boundary-value problem can be reduced to the solution of a Fredholm integral equation of the second kind.

The evaluation of the eigenvalues and corresponding solutions of a Schrödinger equation is an ever-present problem in atomic and nuclear physics. For the numerical evaluation of the Schrödinger equation for a particle in a central potential, a useful corrector formula was derived by Douglas and Ridley ((5)) some years ago. Given a first estimate of the eigenvalue, this formula enables one to obtain a better estimate using any standard integration procedure for the differential equation. In a particular example, it was found ((5)) that it was necessary to start with an initial value which was accurate to about 10% in order that the procedure should converge to the desired eigenvalue, and that few iterations were required to obtain the eigenvalue with good precision.