In this paper we describe a way of representing varieties of algebras by algebras. That is, to each variety of algebras we assign an algebra of a certain type, such that two varieties are rationallv equivalent if and only if the assigned algebras are isomorphic.

The ellipsoid is characterised among all convex bodies in n-dimensional Euclidean space, Rn, by many different properties. In this paper we give a characterisation which generalizes a number of previous results mentioned in [2], p. 142. The major result will be used, in a paper yet to be published, to prove some results concerning generalizations of the Minkowski theory of reduction of positive definite quadratic forms.

Solutions of the iterated equation of generalized axially symmetric potential theory [1]

where the operator Lk is defined by

will be denoted by except that when n = 1, fk will be written instead of . It is easily shown [2, 3] that

by which is meant that any function is a solution of (1).

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.

Here we introduce a new integral transform whose kernel is the H-function. Since most of the important functions occurring in Applied Mathematics and Physics are special cases of the H-function, various integral transforms involving these functions as kernels follow as special cases of our transform. We mention some of them here and observe that a study of this transform gives general and useful results which serve as key formulae for several important integral transforms viz. Laplace transform, Hankel transform. Stieltjes transform and the various generalizations of these transforms. In the end we establish an inversion formula for the new transform and point out its special cases which are generalizations of results found recently.

Let name be a class of finite soluble groups with the properties: (1) is a Fitting class (i.e. normal subgroup closed and normal product closed) and (2) if N ≦ H ≦ G ∈, N ⊲ G and H/N is a p-group for some prime p, then H ∈. Then is called a Fischer class. In any finite soluble group G, there exists a unique conjugacy class of maximal -subgroups V called the -injectors which have the property that for every N◃◃G, N ∩ V is a maximal -subgroup of N [3]. 3. By Lemma 1 (4) [7] an -injector V of G covers or avoids a chief factor of G. As in [7] we will call a chief factor -covered or -avoided according as V covers or avoids it and -complemented if it is complemented and each of its complements contains some -injector. Furthermore we will call a chief factor partially-complemented if it is complemented and at least one of its complements contains some -injector of G.

The BFC number of a group G is defined to be the least upper bound n of the cardinals of the conjugacy classes of G, provided this is finite, and we then say that G is n-BFC. It was shown by B. H. Neumann [2] that the derived group G′ of such a group is finite, and J. Wiegold [5] proved that.This bound was sharpened by I. D. Macdonald [1] to, and P. M. Neumann has recently communicated the (unpublished) result that G′ ≦ nq(n) with q(n) a quadratic in log2w, an immense improvement on the above. J. A. H. Shepperd and J. Wiegold [4] improved the bound in two special cases, showing that if G is soluble, G′ ≦ np(n) with p(n) a quintic in Iog2n, and that if G is nilpotent of class 2, , It is conjectured that for any n-BFC group G, , Wiegold [5] having shown that this bound is attained by certain nilpotent groups of class 2.

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.

The theme of this paper is a conjecture whose origins go back to work of B. H. Neumann ([5], Theorem 3·1) and his former students I. Wiegold and I. D. Macdonald ([7], [4]). B. H. Neumann proved that if there is a bound to the sizes of conjugacy classes in the group G. that is. if G is a BFC group. then the derived group G′ is finite; Wiegold. and later Macdonald. produced explicit upper bounds for G′ in terms of the maximum n of the sizes of the conjugacy classes in G. The coniecture is that where l(n) should be the arithmetic function1 λ(n) for a best possible bound, or l(n) may be interpreted as log2n for a smooth, monotonicorderof-magnitude estimate. However, the bounds produced in [7] and [4], and even the vastly better upper bounds proved bv Sheooerd and Wieeold [6] for soluble groups, are very much bigger than , and it is mv aim in this paper to take a first step towards closing the gap. This first, step is study of BFC p-groups: in a sequel I hope to show how the results proved here can be used to obtain improved bounds for arbitrary BFC groups.

This note is concerned with the following question: What is the structure of those groups which possess two antiautomorphisms different from identity such that every element of the group is fixed by (at least) one of them?

C. Ayoub [1] stated this problem after having proved a statement equivalent to the following: The group G is a non-abelian extension of an abelian group by a group of order two, if, and only if, there is an automorphism α ≠ 1 and an antiautomorphism β ≠ 1 such that every element of G is fixed by a α by β. The Theorem at the end of the paper will show that the class of groups considered by C. Ayoub coincides with the class considered here.

If velocity and temperature profiles are known at a particular distance along a vertical heated plate, the equations of motion determine the velocity and temperature at points downstream, for a given varition of plate temperature. The problem of continuing the boundary layer solution for given initial conditions was investigated by Goldstein [2], for the isothermal case of the laminar, incompressible flow past plate, with a given streamwise varitain of pressure gradient outside the boundary layer. He showed that the solution is not always free from singularities and developed an expansion procedure to calculate the flow downstream when these occurred. Typical singularities occur, for instance, near the leading edge of the plate where the no-slip condition is imposed on the plate surface and near the trailing edge, where this condition is relaxed to one of zero stress along the axis of symmetry of the wake.

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 g (u) be a regular Hausdorff weight function, and let hm (Ψ x m) denote the mthe corresponding Hausdorff transform, evaluated at x m, of the sequence of partial sums of the Fourier series of Ψ (x), where . In [3], Szász investigated the Gibbs phenomenon for Ψ(x) for these means. His main results are contained in the following two theorems: . (3) THEOREM 2. Taking the limit superior as. If this maximum is attained for τ = τ′ then.

Let ℜc denote the variety of all nilpotent groups of class ≦ c, that is, ℜc is the class of all groups satisfying the law, where we define, as usual, and, inductively, . Further, let Fk(ℜc) denote a free group of ℜe of rank k. In her book Hanna Neumann ([4], Problem 14) poses the following problem: Determine d(c), the least k such that Fk(ℜc) generates ℜc. Further, she suggests, incorrectly, that d(c) = [c/2] + l. However, as we shall prove here, the correct answer is d(c) = c—1, for c ≦ 3. 2 More generally, we shall prove the following result.

Let , be regular in the unit disc . G. Szegö [5] and Y. Miki [3] proved that if f(z), given by (1), is univalent (starlike with respect to the origin; convex; close-to-convex in E) then any one of the partial sums , is also univalent (starlike with respect to the origin; convex; close-to-convex) in |z| < ¼and that the constant ¼ cannot be replaced by a larger one.

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.

We refer the reader to Hanna Neumann [7] for notation and other undefined terms. Let and denote the varieties of groups defined by the laws (xy)n=xnyn, [x, y]n=1 and [x, yn]respectively, where n is an integer. (n)-groups were termed “n-abelian” by R. Baer [1] and have been a subject matter of investigation by various authors (see [3], [5], [6] and the references therein). Recently KaluŽnin [5] has shown that , thus clarifying the relationship between U(n) and the familiar varieties. From the elementary inequalities(n ≠ 0,1)it is easily deduced that(see for instance [5]). If G = Cm Wr C∞, then cleary but for any Thus and . It is also easy to see that in general (see for instance [6] § 5.1) and we are led to ask

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.

In this paper we are concerned with the uniqueness of solutions of functional equations of the form Some conditions for (1) or (2) to have at most one real continuous solution f which satisfies two given initial conditions are contained in [2], [3], [4] and [7]. Conditions sufficient for the equation to determine at most one continuous solution f with values in a Banach algebra are contained in [5]. It is well known (see [1] ch. 2) that one initial condition suffices for Cauchy's equation or two for Jensen's equation to uniquely determine a real solution f which is bounded on an interval or majorized on a set of positive measure by a measurable function. We place conditions on F and H so that similar statements can be made about solutions of (1) or (2). The corresponding results for solutions which are functions of many real variables follow as for Cauchy's and Jensen's equations (see [1] ch. 5).

The family (R) of all endomorphisms of a ring R is a semigroup under composition. It follows easily that if R and T are isomorphic rings, then (R) and (T) are isomorphic semigroups. We devote ourselves here to the converse question: ‘If (R) and (T) are isomorphic, must R and T be isomorphic?’ As one might expect, the answer is, in general, negative. For example, the ring of integers has precisely two endomorphisms – the zero endomorphism and the identity automorphism. Since the same is true of the ring of rational numbers, the two endomorphism semigroups are isomorphic while the rings themselves are certainly not.