Gilbert Baums lag has given sufficient conditions for a group G to be nilpotent in terms of properties of a normal subgroup H and the quotient G/H, and has shown these conditions to be necessary when G is a wreath product and H its base group. His results are extended here to give corresponding conditions for weaker properties ranging from “bounded local nilpotency” to “being an Engel group”. Again we ask whether the sufficient conditions found are necessary when G is non-trivially a wreath product, or is of the form the answer is “yes” for the stronger properties considered, but for the weaker ones the question remains open.

We prove that a non-degenerate homomorphic image of a projective plane is determined to within isomorphism by the inverse image of any one point. An application gives conditions for the preservation of central collineations by a homomorphism.

J.H. Michael recently proved that there exists a metric semigroup U such that every compact metric semigroup with property P is topologically isomorphic to a subsemigroup of U; where a semigroup S has property P if and only if for each x, y in S, x ≠ y, there is a z in S such that xs ≠ yz or zx ≠ zy

A stronger result is proved here more simply. It is shown that for any set A of metric semigroups there exists a metric semigroup U such that each S in A is topologically isomorphic to a subsemigroup of U. In particular this is the case when A is the class of all separable metric semigroups, or more particularly the class of all compact metric semigroups.

The rings of this paper are assumed to have relations of orthogonality defined on them. Such relations are uniquely determined by complete boolean algebras of ideals. Using the Stone space of these boolean algebras, and following J. Dauns and K.H. Hofmann, a sheaf-theoretic representation is obtained for rings with orthogonality relations, and the rings of global sections of these sheaves are characterized. Baer rings, f–rings and commutative semi-prime rings have natural orthogonality relations and among these the Baer rings are isomorphic to their associated rings of global sections. A special type of ideal is singled out in commutative semi-prime rings and following G. Spirason and E. Strzelecki, in an unpublished note, a characterization of a class of such rings is obtained.

This is proved: if B is a commutative Banach algebra with identity, then the non-invertible elements of B are weakly dense in B if and only if the maximal ideal space of B is infinite.

It is proved that if a class X of algebras of countable similarity type is closed under isomorphism and ultrapower, then the class of subalgebras of direct products of elements of X is of countable character.

It is known that in a B*-algebra every self-adjoint element is hermitian. We give an elementary proof that this condition characterizes B*-algetras among Banach*-algebras.

It is veil-known that every ring automorphism of the ring of all linear transformations of a real vector space into itself is inner. We shall show that, if this ring is regarded as a semigroup with respect to composition and the dimension of the vector space is not less than 2, every semigroup automorphism is inner.

Some properties of projective planes having a certain type of collineation group are proved, and a class of these planes which properly contains the class of all Hall planes of odd order is explicitly constructed.

Let be a Cross variety and let n be the least integer such that (n) is locally finite; then n ≤ 2d + 3 where d is an upper bound for the number of generators of certain critical groups in .

Maximal sum-free sets in elementary abelian 3-groups and groups G = Z3 ⊕ Z3 ⊕ Zp where p is a prime congruent to 1 modulo 3 are completely characterized.

This paper deals with the motion of a point test charge in an external electromagnetic field with the effect of electromagnetic radiation reaction included. The equation of motion applicable in a general Riemannian space-time is written as the geodesic equation of an affine connection. The connection is the sum of the Christoffel connection and a tensor which depends on, among other things, the external electromagnetic field, the charge and mass of the particle and the Ricci tensor. The affinity is not unique; a choice is made so that the covariant derivative of the metric tensor with respect to the connection vanishes. The special cases of conformally flat spaces and the space of general relativity are discussed.

An upper bound is given of the dimension of certain spaces of cusp harmonic forms of arithmetic subgroups Γ of semisimple algebraic groups G in terms of the multiplicities of corresponding irreducible unitary representations of the group GR of real rational points of G in the space 0L2(GR/Γ) of cusp forms.

Given qo ∈ [1, 2), functions and are constructed such that fgqAq for every q ∈ [1, qo]. In particular, if p ∈ (1, 2), AP is not an algebra.

The paper extends a theorem given by Entringer on the asymptotic expansion of a composite function.

The following convergence criterion of Fourier series is due to M. Izumi, S. Izumi and the author:

THEOREM. Let Δ ≥ 1. If

(i) , and

(ii) as t → 0

for an a, 0 < a < 1 and for a δ, 0 < δ < π, then the Fourier series of φ(t) is convergent at the origin.

The object of this paper is to generalize the above theorem in the Hardy-Littlewood direction.

In this paper we discuss the interrelations among various spaces of distributions and show that none of them can be linearly and differentiably homeomorphic to the space of Mikusiński operators. It is also shown that the distributions of Mikusiński-Sikorski can also be defined by the method described by Temple as the completion of the space of continuous functions after introducing a weaker notion of convergence in this space.

This paper contains some results on order paracompact spaces. Some of the results offer improvements of some of the results of McCandless in Canad. J. Math. 21 (1969). Some other results of McCandless are deduced as corollaries from our results. The concepts of order closure preserving and order cushioned collections are introduced and using these, characterizations of paracompactness in regular spaces are obtained.