The representation theory of Clifford algebras has been used to obtain information on the possible orders of amicable pairs of orthogonal designs on given numbers of variables. If, however, the same approach is tried on more complex systems of orthogonal designs, such as product designs and amicable triples, algebras which properly generalize the Clifford algebras are encountered. In this paper a theory of such generalizations is developed and applied to the theory of systems of orthogonal designs, and in particular to the theory of product designs.

What we cail here the “construction principle” is a principle on the ground of which some functionals can be defined; the domain and the range of such a functional consist of some “computable” functionals of various finite types. The principle above is considered here as the basis of the functional interpretation of transfinite induction up to ε0. It is concretely repesented as the “term-forms”, where every term-form is shown to be “computable” in some sense.

It is shown that if R is a 2-torsion-free semi-prime ring such that [xy, [xy, yx]] = 0 for all x, y ∈ R, then R is commutative.

Two local nilpotent properties of an associative or alternative ring A containing an idempotent are shown. First, if A = A11 + A10 + A01 + A00 is the Peirce decomposition of A relative to e then if a is associative or semiprime alternative and 3-torsion free then any locally nilpotent ideal B of Aii generates a locally nilpotent ideal 〈B〉 of A. As a consequence L(Aii) = Aii ∩ L(A) for the Levitzki radical L. Also bounds are given for the index of nilpotency of any finitely generated subring of 〈B〉. Second, if A(x) denotes a homotope of A then L(A) ⊆ L(A(x)) and, in particular, if A(x) is an isotope of A then L(A) = L(A(x)).

A celebrated result, due to Jacobi, says that the number of representations of the positive integer n as a sum of four squares is equal to eight times the sum of the divisors of n which are not divisible by 4. We give a new and simple proof of this result which depends only on Jacobi's triple product identity.

Suppose that f(z) is a function of one complex variable satisfying

where ρ is an integer larger than 1 and a(z) and b(z) are rational functions. We consider f evaluated at the algebraic point a and develop a transcendence measure for f(α) under suitable conditions on f and α.

Schmidt has shown that if r and s are positive integers and there is no positive integer power of r which is also a positive integer power of s, then there exists an uncountable set of reals which are normal to base r but not even simply normal to base s. We give a structurally simple proof of this result

In an earlier paper the author investigated the properties of a class of multivanable polynomial vectors which generalise the multivariable Chebyshev polynomial vectors. In this paper the behaviour of these polynomials over rings of the typeZ/(m) is investigated, and conditions are determined for such an n-variable polynomial vector to induce a permutation of (Z/(m))n. More detailed results on the Chebyshev polynomial vectors follow. The composition properties of these vectors imply that the permutations induced by certain subsets of them form groups under composition of mappings, and the structure of these groups is investigated.

Let S be a regular semigroup and D(S) its associated category as defined in Loganathan (1981). We introduce in this paper the notion of an extension of a D(S)-module A by S and show that the set Ext(S, A) of equivalence classes of extensions of A by S forms an abelian group under a Baer sum. We also study the functorial properties of Ext(S, A).

In this paper we characterize the structure of finite inverse perfect semigroups and study congruences on those semigroups, in particular we study those semigroups that have modular lattice of congruences.

The object of the paper is to establish an instability theorem for sixth order differential equations of the form (1.1). The proof is based on the use of Krasovskii criteria.

A Banach space is an Asplund space if every continuous convex function on an open convex subset is Fréchet differentiable on a dense G8 subset of its domain. The recent research on the Radon-Nikodým property in Banach spaces has revealed that a Banach space is an Asplund space if and only if every separable subspace has separable dual. It would appear that there is a case for providing a more direct proof of this characterisation.