If is a variety of groups which are nilpotent of class c then is generated by its free group of rank c. It is proved that under certain general conditions cannot be generated by its free group of rank c - 2, and that under certain other conditions is generated by its free group of rank c - 1. It follows from these results that if is the variety of all groups which are nilpotent of class c, then the least value of k such that the free group of of rank k generates is c - 1. This extends known results of L.G. Kovács, M.F. Newman, P.P. Pentony (1969) and F. Levin (1970).

Let G be a group and M a G-module; then d(G) denotes the minimal number of generators of G and dG(M) the minimal number of generators over ZG of M. For G a finite nilpotent group let G = F/R, F free, be a presentation for G; then it is shown that d(R/[F, R]) = dG(R/[R, R]), that is d(G) + d(M(G)) = dG(R/R′), where M(G) denotes the Schur multiplicator of G.

An elementary proof is given for the Lp conjecture, p > 2, which states that for a locally compact group G, Lp (G) (p > 2) is closed under convolution if and only if G is compact.

The aim of this paper is concerned with studying the stability properties of an integro-differential system by reducing it into a scalar integro-differential equation. A theorem is stated about the existence of a maximal solution of such systems and a basic result on integro-differential inequalities. Utilizing these results we obtain sufficient conditions for uniform asymptotic stability of the trivial solution of the integro-differential system of the form where , with , , C(J) denotes the space of continuous functions, A a continuous operator such that A maps C(J) into C(J). The fruitfulness of the results of the paper are illustrated with two applications.

Some of the results in the theory of Möbius functions of finite partially ordered sets are extended to arbitrary non-singular binary relations on finite sets.

Let C = (Aij)1≤i, j≤t be a hermitian matrix in partitioned form; here Aij, is an ni × nj. block. The purpose of this paper is to obtain inequalities linking the eigenvalues of C to those of the main diagonal blocks A11, …, Att of C.

These inequalities include, as special cases, inequalities due to N. Aronszajn and A. Hoffman.

It is of relevance to studies in the logic of quantum mechanics whether or not every separable completely orthomodular poset admits a normed σ-ortho-valuation. A finite orthomodular poset is constructed which is a counter-example to this proposition.

Apart from some (insoluble) subvarieties of , the jnC (just-non-Cross) varieties known so far comprise the following list: , , , , where p, q and r are any three distinct primes. In a recent paper I gave a partial confirmation of the conjecture that the soluble jnC varieties all appear in this list. Here I show that a jnC variety is reducible if and only if it is soluble of finite exponent; this reduces the problem of classifying jnC varieties to finding the irreducibles of finite exponent. I observe that these fall into three distinct classes, and show that the questions of whether or not two of these classes are empty have some bearing on some apparently difficult problems of group theory.

This note studies cardinal numbers κ which have a partition property which amounts to the following. Let ν be a cardinal, η an ordinal limit number and m a positive integer. Let the m-length sequences of finite subsets of κ be partitioned into ν parts. Then there is a sequence H1, … Hm of subsets of κ, each having order type η, such that for each choice of non-zero numbers n1, …, nm there is some class of the partition inside which fall all sequences having in their i-th place (for i = 1, …, m) a subset of Hi which contains exactly ni elements. The case when m = 1 is thus seen to be the well known property κ . The most interesting results obtained relate to the ordering of the least cardinals with the appropriate properties as m and η vary.

If the maximal ideal space of a commutative complex unitary Banach algebra, X, is equipped with a nonnegative, finite, regular Borel measure, m, then for each element, x, in X, a. complex regular Borel measure, mx, can be obtained by integrating the Gelfand transform of x with respect to m over the Borel sets. This paper considers the possibility of direct sum decompositions of the form X = Ax ⊕ Px where Ax = {z ε X: mz ≪ mx} and Px = {z ε X: mz ┴ mx}.

We prove a partial confirmation of Kovács and Newman's conjecture that a just-non-Cross variety is soluble if and only if it is in the following list: , , , , where p, q and r are any three distinct primes.

This paper deals with the motion of incoherent matter, and hence of test particles, in the presence of fields with an arbitrary energy-momentum tensor. The equations of motion are obtained from Einstein's field equations and are written in the form of geodesic equations of an affine connection. The special cases of the electromagnetic field, the Proca field and a scalar field are discussed.

In 1951 Mohanty established the following theorem.

If Φ(t)log is of bounded variation in (0, π), wherek ≥ πe2and, then is summable, for however large positive Δ. In this present note we have generalised the above theorem by taking a more general type of Riesz means and under the condition, is of bounded variation in (0, π), where c is finite, imposed upon the generating function of Fourier series.

Let H be a finite metabelian p-group which is nilpotent of class c. In this paper we will prove that for any prime p > 2 there exists a finite metacyclic p-group G which is nilpotent of class c such that H is isomorphic to a section of a finite direct product of G.

A topological space is E0 (resp. E1) provided every point is the countable intersection of neighborhoods (resp. closed neighborhoods). For i = 0 and i = 1, characterizations of minimal Ei. spaces (Ei. spaces with no strictly coarser Ei. topology) and Ei-closed spaces (Ei. spaces which are closed in every Ei. space containing them) are given; for example, the properties of minimal Ei. and minimal first countable Ti+1 are shown to be equivalent. Minimal E0 spaces are characterized as countable spaces with the cofinite topology, and minimal E1 spaces are characterized as E1-closed and semiregular spaces. E0-closed spaces are shown to be precisely the finite discrete spaces.

The method of interlacing of modules, like amalgamation of groups, is a way of getting new objects from old. Briefly, the interlacing module we consider is a certain factor module of a direct sum of copies (finite or infinite) of an original module M. The conditions given in a previous paper by the first author in order that the interlacing module (using finitely many copies) be indecomposable are here greatly weakened, and we further allow the number of copies of the original to be infinite. R. Colby has shown that if R is a left artinian ring, the existence of a bound on the number of generators required for any indecomposable finitely-generated left R-module implies that R has a distributive lattice of two-sided ideals. This result is extended to rings whose identity is a sum of orthogonal local idempotents.

For these rings the same distributivity is proved in case every indecomposable interlacing module of the above type which begins with an indecomposable projective M is finitely-generated. A consequence is that any finite-dimensional algebra over a field having infinitely many two-sided ideals has infinite-dimensional indecomposables.

By generalizing a technique of Landau, the authors prove that the excess of the number of primes of the form 10x ± 3 over the number of primes of the form 10x ± 1 is infinite.

It is remarked that, if A is a complete boolean algebra and δ is an A-valued equivalence relation on a non-empty set I, then the set of δ-extensional functions from I to A can be regarded as a complete boolean algebra extension of A and a characterization is given of the complete extensions which arise in this way.

The problem of unicity in the uniform approximation of vector-valued functions is considered. A recent result of Cheney and Wulbert concerning unicity in the uniform approximation of real-valued functions is extended and some “point-wise” criteria for unicity are given.