M(22) is the sporadic simple group discovered by Fischer in 1969. The character tables of M(22) and the simple groups PSU(5, 2), PSU(6, 2), PSΩ+(6, 3) and PSΩ(7, 3) are presented. Only a brief description of the methods used to determine the tables is given.

It has recently emerged that the central limit theorem and iterated logarithm law for random walk processes have natural counterparts for Galton-Watson processes with or without immigration. Much of the work on these counterparts has previously involved the imposition of supplementary moment conditions. In this paper we show how to dispense with these supplementary conditions and in so doing make the analogy with the random walk results complete.

A space is said to be subparacompact if every open covering of it has a σ-discrete closed refinement. Subparacompactness is equivalent to Fσ -screenability of McAuley and also to σ-paracompactness of Arhangel'skiĭ. Some properties of these spaces have been obtained in this note. Countably subparacompact spaces, which can be defined in an analogous manner, have also been studied.

By constructing a special set of A-orthonormal functions, it is shown that, under certain smoothness assumptions, the variational and Fourier series representations for the solution of first initial boundary value problems for the simple parabolic differential equation coincide. This result is then extended in order to construct a variational representation for the solution of a very general first initial boundary value problem for this equation.

Let H be a complex Hilbert space and let {A1, A2, …} be a uniformly bounded sequence of invertible operators on H. The operator S on l2(H) = H ⊕ H ⊕ … given by S〈x0, x1, …〉 = 〈0, A1x0, A2x1, …〉 is called the invertibly veighted shift on l2(H) with weight sequence {An }. A matricial description of the commutant of S is established and it is shown that S is unitarily equivalent to an invertibly weighted shift with positive weights. After establishing criteria for the reducibility of S the following result is proved: Let {B1, B2, …} be any sequence of operators on an infinite dimensional Hilbert space K. Then there is an operator T on K such that the lattice of reducing subspaces of T is isomorphic to the corresponding lattice of the W* algebra generated by {B1, B2, …}. Necessary and sufficient conditions are given for S to be completely reducible to scalar weighted shifts.

Maximal sum-free sets in groups Zn, where n is any positive integer such, that every prime divisor of n is congruent to 1 modulo 3, are completely characterized.

Strongly differentiable solutions of the minimal surface equation are shown to be classical solutions and consequently locally analytic. A global regularity result is also proved.

Torsion topological groups with the additional property that the intersection of open sets is open are considered and their topological structure is determined:

1) the topology is uniquely determined by a normal subgroup;

2) each group is uniquely an extension of an indiscrete group by a discrete group;

3) the topology may be changed within limits without changing the dual group.

The paper discusses the differential equation

from a fresh point of view, to supplement an earlier discussion by Noye. In particular, for n = 1 the equation can be transformed to the equation for a pendulum with viscous damping, with corresponding to critical damping. At the end of the paper, some related equations are considered.

Since Paul J. Cohen's 1963 result, it has been possible to investigate the consequences of the axioms of Zermelo-Fraenkel set theory without the Axiom of Choice. In this paper we examine the relative strengths of a number of finiteness criteria against the background of ZF set theory without AC.

We consider the equation

εy″ + p(x)y′ + q(x)y = 0, where ε is a small positive parameter and p vanishes in the interval. Two asymptotic forms of solution are obtained and a rigorous estimate is made of the difference between the exact solutions and the asymptotic forms.

For every Banach space X there is a natural non-linear mapping from X into its dual X*. It is shown that if this mapping is a homeomorphism then it is onto X* and X is reflexive.

We show that each archimedean lattice-ordered group is contained in a unique (up to isomorphism) minimal archimedean vector lattice. This improves a result of Paul F. Conrad appearing previously in this Bulletin. Moreover, we show that this relationship between archimedean lattice-ordered groups and archimedean vector lattices is functorial.

For every positive number w, there exists an odd entire transcendental function

.

with rational integral coefficients ah such that f〈z+w〉 = f〈z〉.

In his paper in Mat. Sb. (N.S.) 61 (103) (1963), Zareckiĭ associated with any binary relation α an ordered pair, (Lα Mα), say, of lattices and showed that α is a left [right] divisor of β if and only if We provide an alternative proof of this result by embedding the category of relations in the category of sets. Our approach provides a unified treatment of several hitherto independent results, and gives new results for the category of partial transformations.

We establish the result that a finitely generated cyclic extension of a free group is residually finite. This is done, in part, by making use of the fact that a finitely generated module over a principal ideal domain is a direct sum of cyclic modules.

Let H be a subgroup of a finite group G. In this paper the Brauer correspondence between blocks of H and blocks of G is characterized in terms of a relationship among the block idempotents.

Let n, s, t be integers with s > t > 1 and n > (t+2)2S−t−l We prove that if n subsets of a set S with s elements have union S then some t of them have union S. The result is best possible.

We show that, under certain conditions, a soluble variety of groups which does not contain the variety of all metabelian groups is a finite exponent by nilpotent by finite exponent variety.

In Problem 1 of her book Varieties of groups, Hanna Neumann asked whether a fully invariant subsemigroup of a free group of infinite rank is necessarily a subgroup. This note presents an example which shows that the answer is negative.