In a recent investigation of a conjecture on an upper bound for permanents of (0, 1)-matrices (2) we obtained some inequalities involving the function (r!)1/r which are of interest in themselves. Probably the most interesting of them, and certainly the hardest to prove, is the inequality

where ø(r = (r!)1/r. In the present paper we prove (1) and other inequalities involving the function ø(r.

A previous note (2) showed how the integral of f(1x1+2x2++nxn) over the interior of a simplex could be reduced to a contour integral. The same idea is applied here in Theorems 1 and 2 to give a generalisation of Dirichlet's multiple integral ((1), pp. 169172). These results are then used in Theorem 3 to reduce an integral over all real n-dimensional space to a contour integral. In Theorem 4 an integral over the group of all 3 3 orthogonal matrices of determinant 1 is reduced to a contour integral. This result can be extended formally to the case of 4 4 matrices; beyond this it seems difficult to go.

1. The problem discussed is the temperature distribution in a semi-infinite solid, initially at zero, when the surface is suddenly raised to and maintained at a temperature which is an arbitrary function of the distance from some line which lies in that surface.

Let f(x) be integrable L(0, 2π) and periodic with period 2π, and let ψ(t) be the conjugate function of with respect to the variable t, where x is onsidered as an arbitrary constant. The following theorems are due to K. K. Chen (1), (2), pp. 111–124.

In this paper we give a method for the solution of the dual integral equations

where Jv and Yv are Bessel functions of the first and second kind, −½≦α≦½, f1(ρ) and f2(ρ) are known functions and ψ(ξ) is to be determined. Such equations arise in the discussion of boundary value problems for half-spaces containing a cylindrical cavity. For example, let us take the problem of finding a potential function φ(ρ, θ, z) which satisfies Laplace's equation for

subject to the usual regularity conditions and the following boundary conditions:

We prove the following theorem, which was established by Abel (1) for the case u = 1.

Theorem. If u, k are positive integers and x, 1, , u, are real numbers, then

where the sum is taken over all distinct ordered solutions (s1, , su) in non-negative integers of the equation .

Finding the distribution of stress in earth dams containing cracks is an outstanding problem of soil mechanics. Even the simplest mathematical model, viz., that of a wedge containing a plane crack which is symmetrically situated along the bisector plane of the angle of the wedge, with the plane strain assumption of the infinitesimal theory of elasticity, presents a difficult problem of solving the bi-harmonic equation subject to mixed boundary conditions. While elasticity problems related to wedge-shaped bodies have been investigated, it appears little attention has been paid to the mixed boundary-value problems.As a first step towards the solution of the mixed boundary value problem for the biharmonic equation, we discuss in this paper the solution of Laplace's equation

for wedge-shaped regions subject to mixed type of conditions on the boundary. If we assume that φ does not depend on z, the equation (1.1) is reduced to the equation

E. C. Posner (5) has shown that a ring R is primitive if and only if the corresponding matrix ring Mn(R) is primitive. From this result he is able to deduce that the primitive ideals in Mn(R) are precisely those ideals of the form Mn(P), where P is a primitive ideal in R. This affords an alternative proof that the Jacobson radical of Mn(R) is Mn(J), where J is the Jacobson radical of R. But Patterson (3, 4) has shown that this last result does not hold in general for rings of infinite matrices and thus that the above result concerning primitive ideals cannot be extended to the infinite case. Nevertheless in this paper we are able to show that Posner's result on primitive rings does extend to infinite matrix rings. Patterson's result depends on showing that if the Jacobson radical J of R is not right vanishing then a certain matrix with entries from J does not lie in the Jacobson radical of the infinite matrix ring. In the final section of this paper we consider a ring R with this property and exhibit a primitive ideal in the infinite matrix ring, which does not arise, as above, from a primitive ideal in R. Finally the Jacobson radical of this ring is determined.

Kilmister (1) has considered dynamical systems specified by coordinates q( = 1, 2, , n) and a Lagrangian

(with summation convention). He sought to determine generally covariant conditions for the existence of a first integral, , linear in the velocities. He showed that it is not, as is usually stated, necessary that there must exist an ignorable coordinate (equivalently, that b must be a Killing field:

where covariant derivation is with respect to a). On the contrary, a singular integral, in the sense that for all time if satisfied initially, need not be accompanied by an ignorable coordinate.

Let R be a ring and let J be the set of all integers. In the set M(R) of all mappings A: J×J→R, let addition and multiplication be defined by

Here aij denotes the image of (i, j) under A and bij, cij, dij are similarly defined for the mappings B, C, D. In (2) we require A, B to be such that the sum is defined and is in R. Thus, in general, M(R) is not closed with respect to multiplication.

A new finite integral transformation (an extension of those given by Sneddon (1)), whose kernel is given by cylindrical functions, is used to solve the problem of finding the temperature at any point of a hollow cylinder of any height, with boundary conditions of radiation type on the outside and inside surfaces, with independent radiation constants. It is to be noticed that all possible problems on boundary conditions in hollow cylinders can be solved by particularising the method described here.

We shall extend some of the results of (7) to the case of multiple alleles, our primary concern being that of polyploidy combined with multiple alleles. Generalisations often tend to make the computations more involved as is expected. Fortunately here, the attempt to generalise has led to a new method which not only handles the case of multiple alleles, but is an improvement over the method used in (7) for the special case of polyploidy with two alleles. This method which consists essentially of expressing certain elements of the algebra in a so-called “ factored ” form, gives greater insight into the structure of a polyploidy algebra, and avoids a great deal of the computation with binomial coefficients, e.g. see (7), p. 46.

An equation is derived for the strains of an arbitrary elastic field in an infinite matrix perturbed by several inclusions. The equation is solved exactly when the shear moduli of the inclusions and matrix are identical, and also when only a single ellipsoidal inclusion perturbs a field uniform at infinity. Mean-values of the strains are then calculated for non-uniform fields perturbed either by an ellipsoid or by a system of weakly-interacting spheres.

In my paper on the continuous (α + β) – theorem (these Proceedings 12 (1961), 209–211) the proof of Lemma 6 assumes that if A and B are closed sets in the relative topology as subsets of the space P of positive reals then C = A + B is closed.

A congruence ρ on a semigroup will be called idempotent-separating if each ρ-class contains at most one idempotent. It is shown below that there exists a maximum such congruence µ on every inverse semigroup S. Two characterisations of µ are found, and it is shown (a) that S/µ⋍E, the semilattice of idempotents of S, if and only if E is contained in the centre of S; (b) that µ is the identical congruence on S if and only if E is self-centralising, in a sense explained below.