In (2), Hall considered the question: for what varieties of soluble groups do all finitely generated groups satisfy max-n (the maximal condition for normal subgroups)? He has shown that the variety M of metabelian groups and more generally the variety of Abelian-by-nilpotent-of-class-c (c ≥ 1) groups has this property; whereas on the contrary, there are finitely generated groups in the variety V of centre-by-metabelian groups (i.e. defined by the law [x, y; u, v; z]) which do not satisfy max-n. One naturally raises the question: for what subvarieties of V do all finitely generated groups satisfy max-n?

Throughout this paper we deal only with complex and semi-simple algebras. Let B be such an algebra. We denote the socle of B as SB. B is a modular annihilator algebra if B/SB is a radical algebra, i.e. if every element of B is quasi-regular modulo the socle of B; see (1) or (12). Now assume that B is a modular annihilator algebra and a Banach algebra. Then any semi-simple closed subalgebra of B is a modular annihilator algebra by ((4), Cor. to Theorem 4·2,). It is not true, however, that any semi-simple subalgebra A of B is a modular annihilator algebra, even when A is a Banach algebra in some norm. We give a simple example to illustrate this. Let A be the algebra of all complex functions f, continuous on the closed unit disk D in the complex plane, analytic in the interior of D, and such that f(0) = 0. A is a Banach algebra in the usual sup norm over D. Now consider the norm on A defined by

Let B be the completion of A in this norm. A has an involution * defined by and also ‖ff*‖ = ‖f‖2 for all f ∈ A. Therefore B is a B*-algebra. It is not difficult to verify that the only non-zero multiplicative linear functionals on A which are continuous with respect to the norm ‖·‖, are the point evaluations at 1/n, n = 1, 2 …. It follows that every non-zero multiplicative linear functional on B is an extension of one of these point evaluations to B. Thus B can be identified with the algebra of all complex sequences which converge to zero.

We work throughout in the p.l. category (see (8)) consisting of polyhedra and p.l. (piecewise linear) maps. We are concerned with the following problem which is of importance in the theory of p.l. transversality.

The simplest definition of transversality in the PL or Top categories is the purely local one: Manifolds M and N are transverse in Q if, for each point x of their intersection, there is a (closed) neighbourhood U of x in Q and a (PL) homeomorphism

for suitable p, q, and r

Haefliger's theorem (3) that the group of isotopy classes of embeddings of Sx in Sn when n − x > 2 is isomorphic to the triad group πx+1(G; G(n − x),SO), where G(n − x) is the H-space of homotopy equivalences of Sn−x−1 of degree 1, , and SO is the stable special orthogonal group, is generalized in this paper by replacing Sx and Sn by arbitrary compact connected smooth manifolds Xx, Mn without boundary. The embedding and knot problems are reduced to homotopy theory. The question of P.L. manifolds is discussed in section 4. The case Mn = Sn will be considered first; the generalization is stated in section 3.

The existence and uniqueness of a positive self-adjoint nth. root of a positive, self-adjoint, not necessarily bounded operator on a Hilbert Space H can be readily demonstrated using the spectral representation of the transformation.

In this paper G will denote a compact metric Abelian group having finite topological dimension, dim G. The discrete dual group of G will be denoted by Γ. The metric on G will be denoted by d(x, y), or |x-y| since translation-invariance is assumed.

In a recent paper (1), I considered the summability method (D, α) defined, for α > 0, by the sequence-to-sequence transformation

We note that, as is easily verified (and as was pointed out in (1)) a necessary and sufficient condition for the convergence of (1), and thus for the applicability of (D, α), is that

should converge. It was proved in (1) that, provided that (2) converges, a sequence summable (C, r) for any r > − 1 is necessarily summable (D, α). We now show that we can strengthen this result by replacing Cesàro by Abel summability. Moreover, we can omit the hypothesis that (2) converges provided that we interpret (1) as an Abel sum.

The present paper is a sequel to a previous paper (1) in which I proved a conjecture made by R. M. Robinson concerning sums of three roots of unity. For any algebraic integer α, we denote by the maximum absolute value of α and all its algebraic conjugates. We say that α and β are equivalent if α = εβ′, where ε is some root of unity and β′ is some conjugate of β; when this happens we obviously have

Let p ≡ 1 (mod 3) or ≡ 1 (mod 5) be a rational prime and g a primitive root mod p. The non-zero residues g, g2,…,gp−1 (mod p) can then be divided into 3 or 5 classes , , , or , , , , respectively, by letting gν ∈ , , (respectively , , , , ) according as ν ≡ 0, 1, 2, (mod 3) (respectively 0, 1, 2, 3, 4 (mod 5)). Problems regarding the distribution of 1, 2, …, p − 1 amongst the 3 (respectively 5) classes are many. In this paper we consider the following problem in some detail (we shall state it here for the case p ≡ 1 (mod 3)). Let α, β, γ be typical members of , , respectively. Let the number of solutions of the congruences

be

Widder(1) obtained an inversion of the convolution transform

by the method of the Laplace transform, Ln(x) being the Laguerre polynomial. Buschman (2) inverted a similar transform with a generalized Laguerre polynomial as kernel and also solved (3) the singular integral equation

using Mikusinski operators. Srivastava(4, 4a) solved singular integral equations with kernels involving and Whittaker functions Mk,μ(x).

In this note it is shown that the modified Bessel function of arbitrary order may be represented as the sum of two products where one product involves Airy's function and the other product involves the first-order derivative of Airy's function.

In this paper, using a generalized hypergeometric polynomial defined by

where Δ(m, − n) denotes the set of m-parameters:

and m, n are positive integers, we have established some infinite series, transformations, integrals and expansion formulae for generalized hypergeometric polynomials. The polynomial is in a generalized form which yields many known polynomials with proper choice of parameters. Special cases have also been given.

In paper(1) it has been proved that

where the Jacobi polynomial is denned as ((3), p. 255)

Many mathematical problems which do not yield a closed-form solution admit of a solution in the form of a power series; differential equations are an obvious example. The direct use of this power series is limited to the interior of its circle of convergence, and this places a restriction—often a severe restriction—on its usefulness. The method described in this paper enables this restriction to be alleviated in many cases; it also enables the convergence of a power series within its circle of convergence to be improved. The method is based on the Euler transformation.

The paper deals with the existence of a quadratic Lyapunov function V = x′P(t)x for an exponentially stable linear system with varying coefficients described by the vector differential equation The derivative dV/dt is allowed to be strictly semi-(F) and the locus dV/dt = 0 does not contain any arc of the system trajectory. It is then shown that the coefficient matrix A(t) of the exponentially stable system is not identically equal to a unit matrix multiplied by a scalar. The result subsumes that of Lehnigk(1).

A method is given for deriving refractive index profiles which yield wave functions expressed in terms of the standard transcendental functions for fields of electric type in spherically stratified isotropic media. Some new profiles, as well as all the previously known ones, are derived by applying the method.

The equations describing transverse propagation of electromagnetic waves in a cylindrically stratified plasma, in which a static magnetic field is imposed parallel to the axis of cylindrical symmetry, are examined. A systematic method developed previously for isotropic media, Heading (3), Westcott (6) is adapted to yield generalized proffles relating to the plasma properties. These proffles are capable of producing exact solutions for the field distribution in terms of standard transcendental functions. Previously known proffles appear as special cases of the analysis.

The NUT metric is interpreted as the field of (a) a mass around the origin of coordinates, and (b) a semi-infinite massless source of angular momentum.

In a recent paper (1) it has been established that a spherically symmetric space-time of class one cannot describe an electromagnetic distribution of energy. In this investigation the inconsistency of class one conditions with Rainich algebraic conditions was shown in case of spherical symmetry. In the present paper it is shown that the result is true for space-time of class one, in general.