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.

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.

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.

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.

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

.

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

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.

There is a skew-Hadamard matrix of order 92.

Previously the smallest order for which a skew-Hadamard matrix was not known was 92. We construct such a matrix below. The orders < 200 which are now undecided are 100, 116, 148, 156, 172, 188, 196; see [2], [3]. The existence of any Hadamard matrix of order 92 was unknown until 1962 [1].

Ralph P. Boas, Jr, proved the following theorem: Let g be an odd function, integrable on (0, π) and periodic with period 2π, and its Fourier series be Σ bn sinnt. If 0 < r < 1 and bn ≥ 0 b > for all n, then t−rg(t) ∈ L(0, π) if and only if the series Σ bn/n1−r converges. Philip Heywood asked whether the conditions g(t) ∈ L(0, π) and bn ≥ 0 can be replaced by tg(t) ∈ L(0, π) and bn ≥ −A/nr or not. We prove this problem affirmatively.

Let A be a hermitian transformation on an n-dimensional unitary space En, with proper values a1 ≥ … ≥ an. Let M be a proper subspace of En. Suppose b1 ≥ … ≥ bh are the proper values of A/M and c1 ≥ … ≥ ck are the proper values of . Let i1. < … < ir and j1 < … < jr be sequences of positive integers, with ir ≤ k and jr ≤ h. then

.

This is a special case of one of the Thompson-Freede theorems which is proved by use of certain invariants.

A general mean value theorem, for real valued functions, is proved. This mean value theorem contains, as a special case, the result that for any, suitably restricted, function f defined on [a, b], there always exists a number c in (a, b) such that f(c) − f(a) = f′(c)(c−a). A partial converse of the general mean value theorem is given. A similar generalized mean value theorem, for vector valued functions, is also established.

An example is given of a finitely based variety of groups such that is not finitely based.

Let be the variety of groups determined by the laws (1) [[x1, x2], [x3, x4, [x5, x6]], (2) [[x1, x2, x3], [x4, x5]] [[x1x2], [x4, x5, x3]]−1, [[x1, x2, x3], [x1, x2]]. Then is not finitely based.

We study the notion (for categories of modules) dual to that of torsion radical and its connections with projective modules.

Torsion radicals in categories of modules have been studied extensively in connection with quotient categories and rings of quotients. (See [8], [12] and [13].) In this paper we consider the dual notion, which we have called a cotorsion radical. We show that the cotorsion radicals of the category RM correspond to the idempotent ideals of R. Thus they also correspond to TTF classes in the sense of Jans [9].

It is well-known that the trace ideal of a projective module is idempotent. We show that this is in fact a consequence of the natural way in which every projective module determines a cotorsion radical. As an application of these techniques we study a question raised by Endo [7], to characterize rings with the property that every finitely generated, projective and faithful left module is completely faithful. We prove that for a left perfect ring this is equivalent to being an S-ring in the sense of Kasch. This extends the similar result of Morita [15] for artinian rings.

The two-dimensional flow around an infinite cylinder at low Reynolds number has interested fluid dynamicists for many years. In this paper it is shown that an infinite cylinder can propel itself through a viscous fluid (for example micro-organisms) if it has certain undulations on its surface.

Let α be a generalized character of a (not necessarily finite) group G over an arbitrary field, and n a positive integer. It is shown that the function α(n) defined by α(n) (g) = α(gn) is also a generalized character of G. An application confirms a conjecture of Robert Higgins and David Ballew: if G is finite and k is also a positive integer, the k-th power of the number of n-th roots of g, summed over all g in G, is divisible by the order of G.

It is well-known that every finite ring with non-zero-divisors has order not exceeding the square of the order n of its left zero-divisor set. Unital rings whose order is precisely n2 have been described already. Here we discuss finite rings with relatively larger zero-divisor sets, namely those of order greater than n3/2. This is achieved by describing the class of all finite rings with left composition length two at most, and using a theorem relating the left composition length of a finite ring to the size of its left zero-divisor set.

The sub-near-field structure of finite near-fields is analogous to the sub-field structure of finite fields. A finite near-field of order pln contains a unique sub-near-field of order pλ if and only if λ divides ln.

Let p denote a prime, G a finite p′-group and A an elementary atelian group of operators on G. Suppose that A has order p3 and that if w ∈ A# then CG(w) is nilpotent. It is proved that G is nilpotent.