If L is a distributive lattice in which every element is the join of finitely many join-irreducible elements, and if the set of join-irreducible elements of L satisfies the descending chain condition, then L satisfies the descending chain condition: this follows easily from the results of Chapter VIII, Section 2, in the Third (New) Edition of Garrett Birkhoff's ‘Lattice Theory’ (Amer. Math. Soc., Providence, 1967). Certain investigations (M. S. Brooks, R. A. Bryce, unpublished) on the lattice of all subvarieties of some variety of algebraic systems require a similar result without the assumption of distributivity. Such a lattice is always join-continuous: that is, it is complete and (∧X) ∨ y = ∧ {x ∨ y: x ∈ X} whenever X is a chain in the lattice (for, the dual of such a lattice is complete and ‘algebraic’, in Birkhoff's terminology). The purpose of this note is to present the result:

If ε > 0, a subset M of a metric space is said to be ε-connected if for each pair p, q ∈ M there is a finite sequence a0, …, an such that each ai ∈ M, a0 = ρ an = q and the distance from ai−1 to ai is less than ε whenever 0 < i ≦n. It is known [1, p. 117, Satz 1] that a compact metric space is connected if and only if for each ε > 0 it is ε-connected. We present here a proof of an analogous characterization of locally connected unicoherent compacta.

Let L be a Lie algebra over a field κ of any characteristic, and consider the lattice ℒ(L) of all subalgebras of L. In this paper we prove that if L and M are lattice isomorphic Lie algebras, over a field of any characteristic, and L′ and M′ are nilpotent, then the difference between the orders of solvability of L and M differs by at most one.

The theory of discrete, abstract groups, as presented in current texts, consists of investigations of various special classes of groups: it has very few completely general results. For some classes (say, for finite groups) the investigations have been extensive and successful; in a few cases (say, for finitely generated abelian groups) they have even reached a sense of completeness. The choice of some of these classes was dictated by the needs of other branches of mathematics. Many more were introduced with the view of extending the scope of certain powerful but special results, and a large part of the literature is taken up by elaborate counterexamples which mark the limits of these generalizations. In so far as one is looking for some kind of classification theory, it is immediately evident that the classes investigated were chosen by historical accident rather than by any master plan, and so far do not appear to form the initial part of a pattern which could be enlarged and completed in the future.

Iseki and Kasahara (see [3]) have given a Hahn-Banach type theorem for semifield-valued linear functionals on real linear spaces. We shall generalize their result by considering linear spaces over semifields.

Exact tests, based on factorization of Wilks' Λ were derived by Williams (1952, 1955) and Bartlett (1951), for testing the goodness of fit of a single hypothetical discriminant function, in the case of several groups. An analytical derivation of the distributions associated with these tests was given by the author (1964 a), after expressing the test statistics in canonical forms. Wilks' Λ to cover the case of s (s > 1) hypothetical discriminant functions. Radcliffe, in his paper, states that an analytical derivation of the distributions, as well as of the independence of the factors of Λ is desirable. This is done in the present paper, by expressing the test criteria in simpler forms and using matrix transformations in the multivariate Beta distribution (Kshirsager 1961). This is a straightforward extension of the author's paper (1964 a), which dealt with only one hypothetical discriminant function.

Consider the classical Hardy spaces Hp(T) (1 ≦ p ≦ ∞) on the unit circle T. We shall ignore completely the fact that the elements of Hp(T) can be extended via the Poisson formula to certain types of functions analytic inside the unit disc. For our purposes, Hp(T) is the closed ideal in Lp(T) consisting of those functions f ∈ Lp(T) for which (n) = 0 (n= –1, –2,…).

The height (as in [3] and [4]) of a point in a rooted tree is the length of the path (that is, the number of lines in the path) from it to the root; the total height of a rooted tree is the sum of the heights of its points. The latter arises naturally in studies of random neural networks made by one of us (N.J.A.S.), where the enumeration of greatest interest is that of trees with all points distinctly labeled.

Let ε > 0 be a small real parameter, let y, z be real m-dimensional and n-dimensional vectors respectively and let f, g be respectively real m-dimensional and n-dimensional vector functions of their arguments. This paper aims to discuss the following two problems in singular perturbations.

Consider the simultaneous real differential equations , where the dot indicates differentiation with respect to time, and a, b, c, d are functions about which the only thing that we know is that they are uniformly bounded, .

In this paper we aim to give a characterization of the finite simple group L4(3) (i.e. PSL(4, 3)) by the structure of the centralizer of an involution contained in the centre of its Sylow 2-subgroup. More precisely, we shall prove the following result.

A topological semiring is a system (S, +, ⋅) where S is a Hausdorff space, (S, +) and (S, ⋅) are topological semigroups (i.e., + and ⋅ are continuous associative binary operations on S) and the distributive laws , hold for all x, y, z in S. The operations + and ⋅ are called addition and multiplication respectively.

The aim of this paper is to give a characterization of the finite simple group U4(3) i.e. the 4-dimensional projective special unitary group over the field of 9 elements. More precisely, we shall prove the following result.

A topological semiring is a system (S, +, ⋅) where (S, +) and (S, ⋅) are topological semigroups and the distributive laws , hold for all x, y, z in S; + and ⋅ are called addition and multiplication respectively.

Throughout this note Rn denotes the n-dimensional Euclidean space. Addition and multiplication in Rn are defined component-wise. If k ≦ n is a positive integer and x ∈ Rn, we write xk for the k-th component of x. The set {x ∈ Rn: xk ≠ 0 for each k ≦ n} is designated by R#.

In this paper we study open mappings of the sphere, Sn, onto itself. In particular, sufficient conditions are given that such a mapping be a homeomorphism. For the cases n≦ 2 many of the results could be obtained from the work of G. T. Whyburn [7], [8], and [10]. For the cases n ≦ 3 the useful results of A. V. Cernavskii, [1], [4], proved to be sufficient. An application is made concerning a finite to one open mapping of one n cell onto itself. It is interesting to note that for n ≦ 2 that we could use similar proofs to show that certain quasi-monotone mappings of Sn onto Sn are necessarily monotone mappings.

A set of polynomials p0(z), p1(z), … is said to form a basic set if every polynomial can be expressed in one and only one way as a finite linear combination of them.

A result contained in a previous paper [1] of the author is Theorem 1. If (G, K, H) is a T3-triple, G is finite, a is an involution contained in G–N(K) and H ∩ Ka=1, then the factor group of G over its centre is isomorphic to a group of similarities over a finite field.

It is well known that a real symmetric matrix can be diagonalised by an orthogonal transformation. This statement is not true, in general, for a symmetric matrix of complex elements. Such complex symmetric matrices arise naturally in the study of damped vibrations of linear systems. It is shown in this paper that a complex symmetric matrix can be diagonalised by a (complex) orthogonal transformation, when and only when each eigenspace of the matrix has an orthonormal basis; this implies that no eigenvectors of zero Euclidean length need be included in the basis. If the matrix cannot be diagonalised, then it has at least one invariant subspace which consists entirely of vectors of zero Euclidean length.

We here study permutational products of groups, a study which was initiated by B. H. Neumann [8], [9].