1. A few months ago, in the course of teaching an elementary class. I had occasion to discuss problems of the familiar type, ‘if α, β, γ are the roots of the equation x3 + qx + r = 0, form the equation whose roots are α2 + βγ, β2 + γα, γ2 + αβ'. After I had explained the device of replacing βγ by αβγ/α = −r/α, so that the roots of the new equation are the same rational function of the separate roots of the original equation, I was asked ‘whether such a transformation was always possible’. On investigation the answer to the question proved to be a surprising one.

1. In the paper [Kai] it was observed that if A is a Banach algebra (over R or C) then the dual space is not only an A-A-bimodule, but is also injective as a left (or right) A-module. Furthermore, if M is a left (or right) Banach module over the unital Banach algebra A, then there is a natural bilinear map, there denoted TrA, from M × M′ to A′, defined by

formula here

(or 〈TrA(m, m′), a〉 = 〈m′, ma〉). The map TrA can be extended to the projective tensor product M[otimesas]M′, which is also an A–A–bimodule.It is easy to see that the map TrA is a bimodule homomorphism, so that the image is an A–A–submodule of A′. This module was denoted EA (M) in [Kai] and is in general not closed as a subspace of M′. It does, however, have a natural norm (as a quotient space of M[otimesas]M′) and the unit ball can be used to define a new norm ∥a∥new = sup {|〈e, a〉 | e∈ the unit ball of EA(M)} on A, and it is easy to see that this new norm is simply the operator norm of a as an operator on M. The conclusion is that if a′∈A′ is not only continuous with respect to the norm ∥a∥L(M) (which is of course in general smaller thatn the given norm on A) but also with respect to the weak topology on A given by the set of all functionals of the form (0·1), then a′ has a representation of the form

formula here.

Let (E, C) be a partially ordered vector space with positive cone C. The order-bound topology Pb(6) (order topology in the terminology of Schaefer(9)) on E is the finest locally convex topology for which every order-bounded subset of E is topologically bounded.

§ 1. The general quaternary cubic

can be expressed in various interesting canonical forms involving suitably chosen linear forms Xi, Yi, Ai. Thus, referred to the pentahedron. X1X2X3X4X5 of Sylvester the cubic becomes the sum of five cubes

with its Hessian in the form

where the coefficients ai may if necessary be taken as equal to unity and the five linear forms Xi each contain four independent parameters, making a total of twenty parameters which is the number of coefficients aijk in the given cubic C3

It is shown that the well-known connection between the second order linear differential equation h″ + B(z)h = 0, with a solution base {h1, h2}, and the Schwarzian derivativeof f = h1/h2, can be extended to the equation h(k) + B(z) h = 0 where k ≥ 2. This generalization depends upon an appropriate definition of the generalized Schwarzian derivative Sk(f) of a function f which is induced by k−1 ratios of linearly independent solutions of h(k) + B(z) h = 0. The class k(Ω) of meromorphic functions f such that Sk(f) is analytic in a given domain Ω is also completely described. It is shown that if Ω is the unit disc or the complex plane , then the order of growth of f ∈ k(Ω) is precisely determined by the growth of Sk(f), and vice versa. Also the oscillation of solutions of h(k) + B(z) h = 0, with the analytic coefficient B in or , in terms of the exponent of convergence of solutions is briefly discussed.

The theory of surfaces and of manifolds of higher dimension is greatly handicapped by the lack of exact information concerning the extension of the theorem of Riemann-Roch for curves. Allied to this is the difficulty of finding an expression for the arithmetic genus Pa of primals in terms of their defining elements. This paper seeks to examine certain particular applications of the theorem.

We shall understand the norm, h(A) of a regular matrix A = (amn) to be

The method has a norm, ‖ ‖, ‖ ‖ ≥ 1, given by ‖ ‖ = inf h(A) where the inf is taken over all the matrix methods equivalent to for bounded sequences (b-equivalent). These definitions are due essentially to Brudno(1) though his definition of

1. The relation

in which we suppose each of a, b, c, a2 − c2 to be different from zero, leads, from a value θ, to two values of φ, which we may denote by θ1 and θ−1. Each of these, put in place of θ, leads, beside the value θ of φ, to another value of φ, say, respectively, θ2 and θ−2. If θ2 or θ−2 be put for θ, the same relation leads to two values of φ, say, θ1 and θ3 or θ−1 and θ−3, respectively. And so on. It may happen that θn is the same as θ, in which case also θ−n is the same as θ; this we may express by saying that the relation is closed, or that there is closure, after n links. It is the object of the present note to express in reduced form, in terms of a, b, c, the condition that this may be so. Evidently, if n = pq, the condition of closure after n links is satisfied when the condition for closure after p links (or also after q links) is satisfied. But there is a condition for closure after n links which is not satisfied for any less number of links; this is the condition which we call the reduced or proper condition for closure after n links, and it is this which we seek to express.

We obtain a simple new proof of Bonsall's characterization theorem for type 1 semi-algebras of continuous functions. An extension of our methods produces several new results on stability properties of wedges.

1. Introduction: In what follows, X will be a real inner-product linear space (inner product (x, y), norm |x| = (x, x)½); En will be n-dimensional real Euclidean space. We shall be dealing with sets {as: 1 ≤ s ≤ rk; r ≥ 1, k ≥ 2} of elements of X; we write a for the vector (a1; …, ark), and put . We suppose that |as| ≤ ∈ (< 0) for every s.

In computational matrix algebra, Gershgorin's Theorem [8] and Ostrowski's Theorem[14] play key roles in establishing regularity conditions for complex n × n matrices. As a result of this importance as well as possible applications in other areas, several extensions of these results have been made (Kotelyanski[9], Ky Fan [5], Brauer[2], Nwokah[10]), for point set matrices. More recently, further extensions have been made to include regularity conditions for composite matrices (matrices partitioned into blocks) (Brenner [3], Van der Sluis[16], Feingold and Varga[6], Cook [4]). Cook's result was motivated by engineering applications but is rather restrictive since, like Feingold and Varga, it requires that the matrices under consideration be diagonally dominant. In this paper, the diagonal dominance condition is replaced by a more general condition (the H-matrix condition). Estimates for the diagonal blocks of block-partitioned matrix are then obtained. The results on inclusion regions for eigenvalues unifiex the previous results of Feingolg and Varga [6], kotelyanski [9], Brauer [2], and generalizes the earlier results of Nwokah [10], from point matrices to block-partitioned matrices. In a sense, these results complement those of brenner [3] Van der Sluis [16], but like those of Cook [4], have a more immediate direct application to control engineering in the area of computer-aided design of composite feedback control systems (Nwokah [12]), which will be briefly reviewed towards the end of the paper.

I have to correct the proof of the assertion on p. 471 of my paper that if f (t) is integrable over (0, a) and ν ≥ – ½, then

Let G be a locally compact topological group with left Haar measure, m; let M(G) denote the bounded regular Borel measures on G and let Lp(G) denote the equivalence classes of pth power integrable functions on G with respect to the left Haar measure.

Let a cyclic group of odd prime order p act on a ℤ(p)-Poincaré duality space X. We prove a relation between the Witt classes associated to the [ ]p-cohomology rings of the fixed point set of this action and of X. This is applied to show a similar result for actions of finite p-groups on ℤ(p)-homology manifolds.

A β-ring is supplied with operations βH where Hruns over the conjugacy classes of subgroups of the symmetric groups Sn. In an earlier paper we introduced a second set of operations λH and we show here that the two sets are related by the isomorphism

formula here

We then consider the operations βH and λH as combinatorial species, in the sense of Joyal, and express their molecular decomposition as a finite sum of products of the exponential species with molecular species of degree at most n. We give combinatorial interpretations for βSn-structures and λSn-structures and derive various species isomorphisms.

We study some fine arithmetic properties of the components of solutions of a decomposable form equation. Lower growth rates for the greatest prime factor of a component are obtained for density 1 of the solutions. Also, high pure powers are shown to occur rarely. Computations illustrate the applicability of our results; for example, to the study of units in abelian group rings.

Let A denote the mod 2 Steenrod algebra. Let ℤ2[x, x−l] be the (graded) ring of finite Laurent series over ℤ2 in the variable x with dim (x) = 1. ℤ2[x, x−1] is a module over the Steenrod algebra A by

where are binomial coefficients modulo 2 and m > 0 is large compared with |k| and i. Let M be the A-submodule of ℤ2[x, x−1 ] generated by all powers xi with i ≠ −1. It is easy to see that ℤ2 [,x, x−1]/M ≅ σ−1ℤ2 (means ℤ2 on dimension − 1). Let ρ: ℤ2[x, x−1] → σ−1 ℤ2 be the projection map.