This paper contains results related to Titchmarsh's convolution theorem and valid for , the additive group of Rn with the discrete topology. The method of proof consists in transferring the problem to Rn with the usual topology by a procedure which has been used earlier, for instance in Helson [3].

In Section 1, the classical support theorems are generalized to . In [1], Titchmarsh's convolution theorem [6] on R was generalized to convolutions of functions belonging to certain weighted Lp-spaces on R. Section 2 contains a corresponding generalization to weighted l2(Rd).

It should be observed that convolutions of elements f and g in l1() can be interpreted as convolutions of bounded discrete measures on Rn. Hence, in that case the support theorem (Theorem 4.33 of Hörmander [5]) is directly applicable to give the results of our Theorems 1 and 3. So the novelty in our theorems lies in the fact that they apply for instance to the case when it is only assumed f, g ∈l2(), together with support conditions. It is not known whether it suffices to assume f∈l1(), g∈lp(), when p > 2.

The following paper is a continuation of one read before the Society some years ago, and published in the Proceedings, Vol. XXXIV. (Part 2), Session 1915–1916. The results of that paper, more especially those summarised in Art. 14, and those of Arts. 17, 18 will be assumed.

I consider here the second solutions corresponding to the solutions of the above equation (when n is an integer) in finite terms for special values of B. If Un be such a solution, and Fn the corresponding second solution, we know that

The generalised Whittaker vector is Λμ which is prevented from vanishing by rejection of the constancy of ω, previously assumed by all writers. It is shown that (1) the null divergence of Λμ is equivalent to Dirac's equation, (2) the length of Λμ measures the probability of occurrence of the electron (3) components of Λμ are connected with the Dirac wave functions and.possible transformations of x5 are probably related to the Uncertainty Principle.

This paper is a continuation of (4). The main aim of this paper is the introduction of the concept of sex-linked duplication. In addition, we shall give several equivalent definitions for the concept of a genetic algebra and make several remarks on overlapping of generations.

For any positive integer n and positive real variables x1, x2, …, xn write

where xr is defined for all integers r by the relations

If

and

it is known that (1, 3, 4)

and

The motion of a rigid sphere embedded in an adhering medium and subjected to an external force is analysed exactly in the context of classical elastodynamics. For the limiting case of an incompressible medium it is possible to write down a simple second order differential equation relating displacement of the ball to the external force.

The theory is generalised to the case of a viscoelastic solid and the results obtained are pertinent to recently developed experimental methods for testing the dynamical mechanical properties of very low modulus polymer gels.

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 bmn be a positive function of m and n which decreases steadily with n, so that bmn ≥ bm, n+1 for all values of m and n. Assume also that |am1 + am+2 + … + bmn| < K for all values of m and n, K being finite. Denote bySmn the sum of the first n terms in the first m rows of the double series