All elements mentioned herein are in a group G. A well-known definition states that x and y are conjugate if there exists an element a such that y = α–1xα. Conjugacy is an equivalence relation in G. In the present paper this will be called 1-conjugacy.

The equations considered are Fredholm integral equations of the second kind with regular kernels, whose argument depends only on the difference of the variables. Approximate solutions are sought for a given finite range of the eigenvalues, and for large values of the range of integration. Certain special conditions are imposed on the general form of the Fourier transforms of the kernel. Then it is shown that approximate solutions may be obtained in terms of the solutions of the corresponding (singular) Wiener-Hopf equations. Approximations to the eigenvalues are also found. It is shown that the eigenfunctions are unique, and that except possibly near the end points of the range, the solutions are of trigonometric type with the zeros of successive solutions interlacing.

M. L. Cartwright has given ((2), 180–181) the following theorem, together with a neat proof of it.

Theorem C.Suppose that f(z) is regular and

in the half-strip S

of the complex plane.

Suppose also that for some constant α in α<a<β

as y→∞. Then for every δ>0.

uniformly asy→∞ for α+δ≦x≦β–δ.

The problem considered is that of long gravity waves approaching, from an arbitrary direction, a semi-infinite barrier, the whole system being in rotation. It is shown that the rotation gives rise to a wave in the shadow region whose amplitude depends upon the angle of incidence, but whose form is independent of it and which travels along the barrier without attenuation in that direction. The work is an extension and simplification of previous work by Crease, involving the use of methods previously developed by the author.

In this paper we discuss a new class of integral transforms and their inversion formula. The kernel in the transform is a G-function (for a treatment of this function, see ((1), 5.3) and integration is performed with respect to the argument of that function. In the inversion formula, the kernel is likewise a G-function, but there integration is performed with respect to a parameter. Known special cases of our results are the Kontorovitch-Lebedev transform pair ((2), v. 2; (3))

and the generalised Mehler transform pair (7)

These transforms are used in solving certain boundary value problems of the wave or heat conduction equation involving wedge or conically-shaped boundaries, and are extensively tabulated in (6).

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.

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.

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.

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.

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.