K.A. Bush has asked whether there is a symmetric Hadamard matrix of order m2, m even, which can be partitioned into an m × m array of m × m blocks, such that:

(i) each diagonal block has every entry 1;

(ii) each non-diagonal block has every row-sum zero?

We give two ways of constructing such matrices.

Let B be a topological algebra with Fréchet space topology, A an algebra with locally convex topology and an algebra of formal power series over A in n commuting indeterminates which carries a Fréchet space topology. In a previous paper the author showed, for the case n = 1, that a homomorphism of B into whose range contains polynomials is necessarily continuous provided the coordinate projections of into A satisfy a certain equicontinuity condition. This result is here extended to the case of general n, and also to weaker topological assumptions.

The paper is concerned with an extension of the inequality 1 - u(2nt) ≤ 4n[1-u(t)] for u(t) the real part of a characteristic function. The main result is that the inequality in fact holds for all positive integer n and not only powers of 2. Certain consequences are deduced and a brief discussion is given of the circumstances under which equality holds.

An Oseén type linearization of the Navier-Stokes equations is made with respect to a uniform shear flow at the trailing edge of a flat plate. Asymptotic expansions are obtained to describe a symmetrical merging flow for distances from the trailing edge that are, in a certain sense, large. Expansions for three regions are found:

(i) a wake region,

(ii) an inviscid region, and

(iii) an upstream lower order boundary layer.

The results are compared with those of Hakkinen and O'Neil (Douglas Aircraft Co. Report, 1967) and Stewartson (Proc. Roy. Soc. Ser. A 306 (1968)). They are further related to the results of Stewartson (Mathematika 16 (1969)) and Messiter (SIAM J. Appl. Math. 18 (1970)).

A number of questions of Philip Hall concerning complemented normal subgroups of finite relatively free groups are considered.

Let f(x) ε Lipα, 0 < α < 1, in the range (-π, π), and periodic with period 2π, outside this range. Also let

.

We define the norm as

and let the degree of approximation be given by

where Tn (x) is some n–th trigonometric polynomial.

The objectives of this paper are the following:

(1) to show that a theorem of Ficker is incorrect;

(2) to show that a stronger version of Ficker's Theorem is valid for a certain class of measures;

(3) characterize all σ-algebras on which every measure is a countable sum of finite measures.

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

The class of ordered topological spaces which are projective limits of finite partially ordered sets (equipped with the restriction of the product of the discrete topologies) is shown to coincide with the class of compact totally order-disconnected ordered topological spaces. Hence this is another category of spaces equivalent to the category of distributive lattices with zero and unit.

For a noncompact riemannian manifold R, let MP (R) be the P-algebra, and R*P the P-compactification, with the assumption that ∫RPdV = ∞. If s is the P-singular point of the P-harmonic boundary ΔP, and Δ is the harmonic boundary of Royden's compactification R*, we construct a continuous mapping π: R* → R*P such that π(Δ) = Δp or π(Δ) = ΔP – s.

It is shown how a result of S.R. Caradus on the approximation problem can be obtained from known theorems.

Terms used here are standard (see [1] or [3]).

Let X denote a Banach space, S its unit ball in the weak topology, and X* the dual of X. In [1], the following theorem is proved: (I) If X is reflexive and X* (considered as a subspaoe of the continuous scalar-valued functions C(S) in the canonical way) is complemented in C(S), then X has the approximation property.

It is our purpose to point out that (I) is a corollary to some known theorems that yield the stronger conclusion (II) below.

This note establishes the impossibility of certain inequalities of the form holding for all trigonometric polynomials f on an infinite compact abelian group G. From this is deduced the impossibility of corresponding inclusion relations of the type or where FS denotes the Fourier image of the set S of integrable functions on G.

The skeleton of a variety of groups is defined to be the intersection of the section closed classes of groups which generate . If m is an integer, m > 1, is the variety of all abelian groups of exponent dividing m, and , is any locally finite variety, it is shown that the skeleton of the product variety is the section closure of the class of finite monolithic groups in . In particular, S) generates . The elements of S are described more explicitly and as a consequence it is shown that S consists of all finite groups in if and only if m is a power of some prime p and the centre of the countably infinite relatively free group of , is a p–group.

The topology of a topological group in which the intersection of open sets is open is uniquely determined by a normal subgroup, and the group is uniquely an extension of an indiscrete group by a discrete group. This was proved by M.C. Thornton under the additional hypothesis that the group is a torsion group. The proofs here given make the more general facts almost trivial.

The least upper bound for the nilpotent lengths of soluble linear groups of degree n is calculated. For each n it is , where r(n) = [log3 (2n–1)/4] and [x] is the integral part of x.

A ring is a left Q-ring if all of its left ideals are quasi-injective. For an integer m ≤ 2, a sfield D, and a null D-algebra V whose left and right D-dimensions are both equal to one, let H(m, D, V) be the ring of all m x m matrices whose only non-zero entries are arbitrary elements of D along the diagonal and arbitrary elements of V at the places (2, 1), …, (m, m-l) and (l, m). We show that the only indecomposable non-local left Q-rings are the simple artinian rings and the rings H(m, D, V). An arbitrary left Q-ring is the direct sum of a finite number of indecomposable non-local left Q-rings and a Q-ring whose idempotents are all central.

We prove that the solution set of a generalized differential equation is connected and that points on the boundary of the solution funnel are peripherally attainable. This is done without the additional assumption of continuity in the state variable required in previous results. The result on upper semicontinuity of the solution set with respect to initial conditions is extended to include variations of initial time.

It is shown that there exists a logarithmic upper bound on the Fitting length h(G) of a finite soluble group G in terms of the number ν(G) of the conjugacy classes of its maximal nilpotent subgroups. For ν(G) = 3, the best possible bound on h(G) is shown to be 4.

P. Hall defined the k–index of a variety , of groups to be the least cardinal number r such that if a group G is generated by a set S and every subset of S of cardinality at most r generates a group in then G ∈ . We show that the only variety which has finite k–index and contains a product of two non-trivial varieties is the variety of all groups. As a consequence of this and P. Hall's result that nilpotent varieties have finite k–index we show that a soluble variety or a variety generated by a finite group has finite k–index if and only if it is nilpotent.

In J. Math. Mech. 15 (1966), 877–898, Bonic and Frampton have laid the foundation for a general theory of smoothness of Banach spaces. In this paper, we shall study one aspect of the smoothness of topological vector spaces, namely, the relationship between smoothness and inductive and protective limits of topological vector spaces. As a consequence, we obtain smoothness results for nuclear spaces and some Montei spaces.