In this note we show that if a real square matrix P fulfils certain rather general conditions then there exists a real diagonal matrix D such that the characteristic equation of DP is stable and, furthermore, aperiodic. (A characteristic equation is called stable if the real parts of its roots are all negative. If the roots are all real and simple the equation is said to be aperiodic; see Fuller(3).)

We show that a locale possesses the localic analogue of the property of realcompactness if and only if it is regular Lindelöf. Thus, the localic version of the Hewitt real-compactification, originally defined by G.Reynolds using σ-frames, is the regular Lindelöf reflection. An immediate consequence is that a space is realcompact if and only if it is the point space of a regular Lindelöf local (3·2). We point out a nice analogy between a theorem of Reynolds and Stone's classical representation theorem for boolean algebras. Finally, we show that the quasi-F cover of a compact Hausdorff space is the Stone–čech compactifications of the smallest dense Lindelöf sublocale.

This paper defines the concept of sequential Monte Carlo and outlines the principal modes of approach which may be expected to yield useful sequential processes. Three workable sequential processes, derived from a non-sequential method of J. von Neumann and S. M. Ulam for solving systems of linear algebraic equations, are described and analysed in detail.

We introduce the notion of character amenable Banach algebras. We prove that character amenability for either of the group algebra L1(G) or the Fourier algebra A(G) is equivalent to the amenability of the underlying group G. Character amenability of the measure algebra M(G) is shown to be equivalent to G being a discrete amenable group. We also study functorial properties of character amenability. For a commutative character amenable Banach algebra A, we prove all cohomological groups with coefficients in finite-dimensional Banach A-bimodules, vanish. As a corollary we conclude that all finite-dimensional extensions of commutative character amenable Banach algebras split strongly.

When k is a 2-bridge knot with group πK, there are parabolic representations (p-reps) θ: πK → PSL(): = PSL(2, ). The most obvious problem that this suggests is the determination of a presentation for an image group πKθ. We shall settle the easiest outstanding case in section 2 below, viz. k the figure-eight knot 41, which has the 2-bridge normal form (5, 3). We shall prove that the (two equivalent) p-reps θ for this knot are isomorphisms of πK on πKθ. Furthermore, the universal covering space of S3\k can be realized as Poincaré's upper half space 3, and πKθ is a group of hyperbolic isometries of 3 which is also the deck transformation group of the covering 3 → S3\k. The group πKθ is a subgroup of two closely related groups that we study in section 3. We shall give fundamental domains, presentations, and other information for all these groups.

In (1) Atkinson and Reuter prove the existence of wave solutions with velocities strictly greater than some critical velocity ʗ0 of an equation associated with the deterministic model for the spread of an epidemic. The existence of wave solutions with velocity ʗ0 is left as an open question. In this note we show that Atkinson and Reuter's results and a simple limiting argument imply the existence of such wave solutions.

Braid groups on the plane were denned by Artin(1) in 1925. More recently Fox(8) defined braid groups on arbitrary topological spaces, the situation being particularly interesting if the space is a 2-manifold. Presentations of the braid groups on R2, S2 and P2 can be found in (2), (6) and (13) respectively, and some general results on braid groups of compact 2-manifolds can be found in (7). In section 1 of this paper, we give finite presentations for the braid groups on all closed 2-manifolds except S2 and P2.

In this paper, we define M24 from scratch as the subgroup of S24 preserving a Steiner system S(5, 8, 24). The Steiner system is produced and proved to be unique and the group emerges naturally with many of its properties apparent.

The main theme of this paper is that under wide conditions the product of a large number of square non-negative matrices is close to a positive matrix of rank 1. (A non-negative matrix is one whose elements are non-negative real numbers.)

It is usual in the theory of queues to assume that customers are served in the order of their arrival. In some applications (e.g. telephone engineering), however, other forms of queue discipline are more realistic. The precise effect of any such change on the waiting-time distribution of a customer will depend on the procedure envisaged (random service, “last come, first served”, etc.), but it is possible to make certain general statements. Thus it is well known that, under certain conditions, the mean is independent of the queue discipline. The purpose of the present note is to consider the variance of waiting time, and we shall prove that this is a minimum when the customers are served in order of arrival. Thus this is, in a sense, the “fairest” queue discipline. This does not, of course, mean that other procedures may not be justified when different criteria are taken into account.

In (10), (11) Wall determined the structure of the cobordism ring introduced by Thom in (9). Among Wall's results is a certain exact sequence relating the oriented and unoriented cobordism groups. There is also another exact sequence, due to Rohlin(5), (6) and Dold(3) which is closely connected with that of Wall. These exact sequences are established by ad hoc methods. The purpose of this paper is to show that both these sequences are ‘cohomology-type’ exact sequences arising in the well-known way from mappings into a universal space. The appropriate ‘cohomology’ theory is constructed by taking as universal space the Thom complex MSO(n), for n large. This gives rise to (oriented) cobordism groups MSO*(X) of a space X.

Error bounds are derived and examined for approximate solutions in terms of elementary functions of the differential equations

in which u is a positive parameter, the functions f and p are free from singularities and p does not vanish. Bounds are also obtained for the remainder terms in the asymptotic expansions of the solutions in descending powers of u. The variable x ranges over a real interval, finite or infinite or over a region of the complex plane, bounded or unbounded.

Applications are made to parabolic cylinder functions of large orders, and modified Bessel functions of large orders.

The problem of the enumeration of the different arrangements of n letters in an n × n Latin square, that is, in a square in which each letter appears once in every row and once in every column, was first discussed by Euler(1). A complete algebraic solution has been given by MacMahon(3) in two forms, both of which involve the action of differential operators on an expanded operand. If MacMahon's algebraic apparatus be actually put into operation, it will be found that different terms are written down, corresponding to all the different ways in which each row of the square could conceivably be filled up, that those arrangements which conflict with the conditions of the Latin square are ultimately obliterated, and those which conform to these conditions survive the final operation and each contribute unity to the result. The manipulation of the algebraic expressions, therefore, is considerably more laborious than the direct enumeration of the possible squares by a systematic and exhaustive series of trials. It is probably this circumstance which has introduced inaccuracies into the numbers of 5 × 5 and 6 × 6 Latin squares published in the literature.

A method of finding explicit expressions for the roots of a certain class of transcendental equations is discussed. In particular it is shown by determining a canonical solution of an associated Riemann boundary-value problem that expressions for the roots may be derived in closed form. The explicit solutions to two transcendental equations, tan β = ωβ and β tan β = ω, are discussed in detail, and additional specific results are given.

Computation of integrals of the type is of frequent occurrence in applied problems. Here λ is real and i is the imaginary unit. Tables of the integrals are often required for a wide range of λ values, some of which may be quite large. Tabulation of the integrand for each λ, followed by ordinary methods of numerical integration, is slow and tedious because the number of points required increases with increasing λ. In this paper formulas are given so that once f(y) is tabulated over the range of interest, the integral in question can be easily evaluated for any λ.

The question has arisen in the theory of queues whether the product of two independent exponentially distributed random variables is infinitely divisible. In this note it is proved that the product of any non-negative random variable with an independent exponentially distributed variable is infinitely divisible. Such random variables are members of a wider class of infinitely divisible random variables, which will be exhibited first.

In a recent communication to this Society Dr Hartree has put forward a method for calculating the field of anatom containing many electrons. Each orbit—to borrow a metaphor from the old quantum theory—is related to a wave-function Ψ which obeys Schrödinger's equation. The potential energy used in this equation is due partly to the field of the nucleus, and partly to the fields of the electrons in the other orbits. The latter are calculated upon Schrödinger's interpretation of the wave-function, that |Ψ|2 is the density of charge, measured in electronic charges per unit volume. It is not the purpose of this paper to discuss the practical methods of obtaining wave-functions which reproduce the fields from which they are derived; but to relate these wave-functions and their energy parameters to those of the accepted theory.

The exterior problems of acoustics can be reduced to the solution of Fredholm integral equations of the second kind in many different ways. Some of these equations break down at certain values of the wavenumber but this difficulty can be reduced or even removed altogether by choosing a different (usually more complicated) integral equation. The present paper is concerned with a class of equations proposed recently by D. S. Jones. It is shown that the proof of one of his principal results can be much shortened.

In a recent paper (1), Paria has discussed the propagation through a solid of a certain type of magneto-thermo-elastic plane wave. The analysis is essentially a reconciliation of the equations governing three fields: the electromagnetic field, the thermal field and the elastic field, which interact one with another. The principal result which was obtained was the dispersion equation connecting the frequency and the wavelength of waves of this type.

The rotation number ρ of an orientation preserving homeomorphism of a circle was originally defined by Poincaré It measures the asymptotic rotation of the orbit of each point under iterations of the homeomorphism (see, for example, (2)).