A. J. Weir [1] has found the maximal normal abelian subgroups of the Sylow p-subgroups of the general linear group over a finite field of characteristic p, and a theorem of J. L. Alperin [2] shows that the Sylow p-subgroups of the general linear group over finite fields of characteristic different from p have a unique largest normal abelian subgroup and that no other abelian subgroup has order as great.

In place of a real valued differentiable (C2) function on a closed n-dimensional differentiable manifold M, we may more generally consider a differentiable section s in any line bundle L on M, assumed to have structural group Z2, the group of integers modulo two. Since the usual definitions of a critical point and of a non-degenerate critical point are local in nature, and since composing a differentiable real valued function with the function t →—t does not change its set of critical points or its set of non-degenerate critical point, it is clear that we may speak of critical points and nondegenerate critical points of the section s. Unless the bundle has a fixed trivialization however, the index of a non-degenerate critical point must be thought of as a set of two numbers {k, n—k), corresponding to the two indices arising from the two trivializations possible for L restricted to a small enough neighborhood of the point, i.e. corresponding to the two possible ways of reading the index. With this understanding we extend the usual definitions, and call a differentiable (C2) section s of L a Morse section if each of its critical points is non-degenerate.

It is assumed throughout this paper that all topological spaces under consideration are Hausdorff. Since the notion of topological property is fundamental in this paper, we begin by making it precise. For our purposes here, it is sufficient to think of a topological property Q as being a class of space such that if X ∈ Q and Y is homeomorphic to X, then Y∈Q. To say that a space X has property Q would then be equivalent to saying that X ∈ Q.

Let ϕ be a continuous, decreasing, real-valued funtion on 0 ≦ r ≦ 1 with ϕ(1) = 0 and ϕ(r) > 0 for r < 1. Let E0 be the Banach space of analytic function f on the open unit disc D, such that f(z)φ(|z|) → 0 as |z| → 1, with norm , where we write ϕ(z) = ϕ(z) for z ∈ D. Let E be the Banach space of analytic functions f on D for which fφ is bounded in D, with the same norm as E0. It is easy to see that E is complete in this norm, and that E0 is a closed subspace of E.

Let {Ln} be a sequence of positive linear operators defined on C[a, b] of the form where xnk ∈ [a, b] for each k = 0, 1,…, n = 1, 2,…. The convergence properties of the sequences {Ln(f)} to for each f ∈ C[a, b] have been the object of much recent research (see e.g. [4], [8], [11], [13]). In many cases positive linear operators of the form (1) give rise to interesting summability matrices A = (ank(x)) and vice- versa.

R denotes always a radical algebra over a field φ. A left ring ideal of R which is also a subvector space over φ is called a left algebra ideal of R. R is said to be left algebra noetherian if it satisfies the ascending chain condition for left algebra ideals. If dim R < ∞, then(i) R is finitely generated(ii) R is left alehra noetherian(iii) R is algebraic.Since the radical of an algebraic algebra is nil ([4] P. 19), conditions (i), (ii), (iii) are also sufficient for R to be finite-dimensional.

Using the terminology in 2 (where the expression m-type is also explained) we will prove the following theorems:Theorem 1. If there exist(i) a skew-Hadamard matrix H = U+I of order h,(ii)m-type matrices M = W+I and N = NT of order m,(iii) three matrices X, Y, Z of order x = 3 (mod 4) satisfying(a) XYT, YZT and ZXT all symmetric, and(b) XXT = aIx+bJxthen is an Hadamard matrix of order mxh.

Let A be a commutative Banach algebra, D a closed derivation defined on a subalgebra Δ of A, and with range in A. The elements of Δ may be called derivable in the obvious sense. For each integer k ≦.l, denote by Δk the domain of Dk (so that Dgr;1 = Δ); it is a simple consequence of Leibniz's formula that each Δk is an algebra. The classical example of this situation is A = C(O, 1) under the supremum norm with D ordinary differentiation, and here Δk = Ck(0, 1) is a Banach algebra under the norm ∥.∥k: Furthermore, the maximal ideals of Ak are precisely those subsets of Δk of the form M ∩ Δk where M is a maximal ideal of A, and = M, the bar denoting closure in A. In the present note we show how this extends to the general case.

Let Tn denote a tree with n(≧ 2) labelled points: we assume Tn is rooted at a given point x, say the point labelled 1 (see [3] for definitions not given here). If we remove some edge e of Tn, then Tn falls into two subtrees one of which, say Tk, contains the root x. If k ≧ 2 we can remove some edge of Tk and obtain an even smaller subtree of Tn that contains x. If we repeat this process we will eventually obtain the subtree consisting of x itself. Let λ = λ(Tn) denote the number of edges removed from Tn before the root x is isolated. Our main object here is to determine the expected value μ(n) and variance σ2(n) of λ(Tn) under the assumptions (1) Tn is chosen at random from the set of nn−2 trees with n labelled points that are rooted at point x, and (2) at each stage the edge removed is chosen at random from the edges of the remaining subree containing x. It follows from our results that μ(n) ~ (½πn)½ and (2−½π)n ~ (2−½π)n as n tends to infinity. We also consider the corresponding problem for forests of rooted trees and for trees in which the degree of the root is specified. We are indebted to Professor Alistair Lachlan for suggesting the original problem to us.

We consider the cosine functional equation (see [1, 2, 3]) , where f(z) is an entire function of a complex variable z and x, y are complex variables.

A normal matrix A = (aij) with complex elements is a matrix such that AACT = ACTA where ACT denotes the (complex) conjugate transpose of A. In an article by K. Morita [2] a quasi-normal matrix is defined to be a complex matrix A which is such that AACT = ATAC, where T denotes the transpose of A and AC the matrix in which each element is replaced by its conjugate, and certain basic properties of such a matrix are developed there. (Some doubt might exist concerning the use of ‘quasi’ since this class of matrices does not contain normal matrices as a sub-class; however, in deference to the original paper and the normal canonical form of Theorem 1 below, the terminology in [2] is used.)

We shall be concerned with problem 7 of Hanna Neumann [7], which asks: Prove or disprove that if U and B are varieties, and neither of U and B is contained in the other, then U ∪ B and [U, B] are decomposable if and only if U and B have a common non-trivial right hand factor.

Hadamard matrices exist for infinitely many orders 4m, m ≧ 1, m integer, including all 4m < 100, cf. [3], [2]. They are conjectured to exist for all such orders. Skew Hadamard matrices have been constructed for all orders 4m < 100 except for 36, 52, 76, 92, cf. the table in [4]. Recently Szekeres [6] found skew Hadamard matrices of the order 2(pt +1)≡ 12 (mod 16), p prime, thus covering the case 76. In addition, Blatt and Szekeres [1] constructed one of order 52. The present note contains a skew Hadamard matrix of order 36 (and one of order 52), thus leaving 92 as the smallest open case.

Semirings of endomorphisms of semigroups form an important class of semirings. All examples which can be found in the literature concern semigroups with a subcommutative operation. We show that there exists a non-subcommutative semigroup whose endomorphisms form a semiring (this answers a question raised by Professor A. H. Clifford). We also give an example of a semigroup whose set of endomorphisms is not embeddable into a semiring; however it is the disjoint union of two semirings, but one of these semirings is not embeddable into a semiring with identity.

For a finite semigroup S and a field Φ, denote by Φ [S] the semigroup algebra of S over Φ and when S has a zero element, denote by Φ0[S] the contracted semigroup algebra of S over Φ (see § 5.2 [1]). Then theorem 5.31 [1], due to E.Hewitt and H. S. Zuckerman, gives a determination of the radical of Φ0[S] when S is commutative and the characteristic of Φ does not divide the order of any subgroup of S; and on page 168 [1] some results concerning the radical of Φ0[S] when S is completely 0-simple are given, the determination of the radical in the general case remaining open.

D. E. Cohen has shown in [1] that a variety consisting of metabelian groups has a finite basis for its laws. In this paper I make use of Cohen's result to prove the following theorem.

In this paper we give a necessary and sufficient condition for decomposition (as a direct sum of fields) of a ring R in which for every x ∈ R there exists a (and hence the smallest) natural number n(x) > 1 such that . We would like to emphasize that in what follows R stands for a ring every element x of which satisfies (1).

In a previous paper [1] J. Aczél has shown the followingTheorem 1. If in the (closed, half-closed or open, finite or infinite) interval 〈A, B〉and there f, F are there, f F are continuous, F intern (the value F(x, y) lies strictly between x and y) and u → H(u, v, x, y) or v → H(u, v, x, y) are injective (i.e. or , then the functional equation (*) with the initial conditionshas at most one solution.

It is common practice for research workers in a great number of widely differing fields, to gather vast amounts of experimental data (see [1 – 3], for example). These data are then analysed, using various statistical or other techniques, in an attempt to obtain information concerning the phenomenon being studied. One important source of such information is evidence of oscillations in the data collected. Various techniques for revealing frequencies of oscillations are available (see [4], for example); they usually involve some form of data processing. In many cases the collected data are records of one or more realisations of a stationary stochastic process, and the frequencies of interest appear as local maxima or peaks in the spectrum of the process [5]. In this paper a method for determining the position of the maximum spectral value directly from the autocovariance function is presented and discussed.

A near-ring is a triple (R, +, · such that (R, +) is a group, (R, ·) is a semigroup, and is left distributive work on near-rings is [1]. A near-ring R is distributively generated if there exists S ⊂ R such that (S, ·) is sub-semigroup of (R, ·), each element of S is right distributive, and S is an additive generating set for (R, +). Distributively generated near-rings, first treated in [3], arise out of consideration of the system generated by the endomorphisms of an (not necessarily commutative) additive group. A near-field is a near-ring such that the nozero elements form a group under multiplication. Near fields are discussed in [9]. An element x ≠ 0 in R is a left (right) zero divisor if there is a ≠ 0 in R such that xa = 0 (ax = 0). A zero divisor is an element that is either a left or a right zero divisor. In a near-ring R it will be assumed that Ox = 0 for each x ∈ R.