The problem which we discuss in this paper can be easily settled for a closed plane set; after a brief introduction giving the definitions and theorems used later we indicate how this may be done; we then proceed to the main problem.

Let A be a finite dimensional algebra with identity element over a field. A is generalised uniserial if every primitive left ideal and every primitive right ideal of A has only one compositions series. In the previous papers in this series (6, 7) generalised uniserial algebras have been characterised as algebras all of whose residue class algebras are of certain types. The purpose of this paper is to extend the earlier results by showing that in order that A be generalised uniserial it is sufficient to require weaker conditions on merely a finite sequence of residue class algebras of A.

The notion of a closed vector measure m, due to I. Kluv´;nek, is by now well established. Its importance stems from the fact that if the locally convex space X in which m assumes its values is sequentially complete, then m is closed if and only if its L1-space is complete for the topology of uniform convergence of indefinite integrals. However, there are important examples of X-valued measures where X is not sequentially complete. Sufficient conditions guaranteeing the completeness of L1(m) for closed X-valued measures m are presented without the requirement that X be sequentially complete.

In this note the volumes of certain regions in the n-sphere will be found in two ways: (a) by using a symmetry argument, (b) by expressing the volumes as repeated integrals over the (n-l)-cube. By considering the 4 and 5 spheres and equating the integrals obtained by method (b) to the solution obtained by method (a) we evaluate integrals of the form

for certain values of a, b and c; it does not appear easy (if indeed it is possible) to evaluate these integrals by direct methods.

Barrelled and quasibarrelled spaces form important classes of locally convex spaces. In (2), Husain considered a number of less restrictive notions, including infinitely barrelled spaces (these are the same as barrelled spaces), countably barrelled spaces and countably quasibarrelled spaces. A separated locally convex space E with dual E' is called countably barrelled (countably quasibarrelled) if every weakly bounded (strongly bounded) subset of E' which is the countable union of equicontinuous subsets of E' is itself equicontinuous. It is trivially true that every barrelled (quasibarrelled) space is countably barrelled (countably quasibarrelled) and a countably barrelled space is countably quasibarrelled. In this note we give examples which show that (i) a countably barrelled space need not be barrelled (or even quasibarrelled) and (ii) a countably quasibarrelled space need not be countably barrelled. A third example (iii)shows that the property of being countably barrelled (countably quasibarrelled) does not pass to closed linear subspaces.

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