An ideal I in a C*-algebra A is called primal if whenever n ≥ 2 and J1,…, Jn are ideals in A with zero product then Jk ⊆ I for at least one k. The topologized space of minimal primal ideals of A, Min-Primal (A), has been extensively studied by Archbold[3]. Very much in the spirit of Fell's work [14] it was shown in [3, theorem 5·3] (see also [5, theorem 3·4]) that if A is quasi-standard, then A is *-isomorphic to a maximal full algebra of cross-sections of Min-Primal (A). Moreover, if A is separable the fibre algebras are primitive throughout a dense subset. On the other hand, the complete regularization of the primitive ideal space of A gives rise to the space of so-called Glimm ideals of A, Glimm (A). It turned out that A is quasi-standard exactly when Min-Primal (A) and Glimm (A) coincide as sets and topologically [5, theorem 3·3].

In Parts I and II of this series (March(6); Ballinger and March(1)), we considered in detail the application of the Thomas-Fermi (T.F.) approximation to molecules with tetrahedral and octahedral symmetry. In these papers, following the work of Buckingham, Massey and Tibbs(2), who obtained results for CH4, we averaged the nuclear field over angles (the ‘smoothing approximation’) and considered the electrons as though they moved in the resulting central field. In this way, it was possible in (6) to carry through self-consistent (T.F.) calculations giving the electron distributions and potential fields in tetrahedral and octahedral molecules.

Malecot(4) under certain conditions derived the formula for the covariance of the genotypic values of a quantitative character in two individuals AI and AII, which were related but not by direct descent. This generalized some results of Fisher (l). Kempthorne (2) extended the theory to multiple allelic systems with any degree of epistacy (i.e. interlocular genie interaction) but without linkage. He gave the formula Here is the item in the population variance which can be attributed to the interaction of additive gene effects at r loci and dominance gene effects at s loci. φ and φ′ are the coefficients of relation between the two individuals. The various assumptions normally included under random mating equilibrium were made, i.e. no selection, mutations, maternal effects or differential viability. Kempthorne (2), (3) gave two rather different proofs of this important result. His second proof was the more straightforward, but it was somewhat condensed.

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 residual closure of a subgroup H of a group G is the intersection of all virtually normal subgroups of G containing H. We show that if G is generated by finitely many cosets of H and if H is commensurated, then the residual closure of H in G is virtually normal. This implies that separable commensurated subgroups of finitely generated groups are virtually normal. A stream of applications to separable subgroups, polycyclic groups, residually finite groups, groups acting on trees, lattices in products of trees and just-infinite groups then flows from this main result.

The purpose of this paper is to study the structure of locally free modules over the ring of differential operators on projective space. Let be a non-singular, complex, algebraic variety. Denote by the sheaf of rings of differential operators over and by its ring of global sections. A -module M is called locally free if the associated sheaf ⊗ M is locally free as a sheaf of -modules. Locally free modules arise naturally in -module theory as inverse images of determined modules; see [1] for definitions and examples.

In this paper we generalize a result of Hayman 4, lemma 4 on asymptotic values of meromorphic functions, which can be stated as follows.

According to the Frenkel theory of adsorption, the first molecules striking a surface move about on it like a two-dimensional gas before finally being adsorbed. This surface motion has been demonstrated by Estermann and by Cockcroft in the case of vaporised cadmium.

Nicod, using Sheffer's operation “│” as a primitive (undefined) idea, obtained a drastic reduction in the number of primitive propositions (postulates) used by Whitehead and Russell in their theory of deduction for “elementary” propositions. In this reduction Nicod accepts Sheffer's replacement of the primitive ideas “∼” and “v” and the primitive propositions *1·7 and *1·71 of the Principia by a single primitive idea and a single primitive proposition, and, by adding two ingeniously devised postulates of his own, derives from the three postulates the Principia's six primitive propositions *1·1, *1·2, *1·3, *1·4, *1·5, *1·6. But the author does not consider the consistency or the independence of his three postulates; does not discuss the Principia's remaining primitives—the ideas “φx” and “⊢. φx” and the propositions *1·11 and *1·72; and fails to prove that his postulates are derivable from those of the Principia.

Let R be a commutative ring, Γ a finite group acting on R, and let k∈ℕ be invertible in R. Generalizing a definition of Kervaire, we construct an Adams operation ψk on the Grothendieck group and on the higher K-theory of projective modules over the twisted group ring R#Γ. For this, we generalize Atiyah's cyclic power operations and use shuffle products in higher K-theory. For the Grothendieck group, we show that ψk is multiplicative and that it commutes with base change, with the Cartan homomorphism, and with ψl for any other l which is invertible in R.

1. The series

is the simplest and most familiar power series whose radius of convergence is zero. It is natural to regard it as a development of the function G(z) defined, when

by

For

say;and

or

according as x is positive or negative. Thus the series (1·1) is an asymptotic series for G(z) in the sense of Poincaré.

The W.K.B. solution for radio waves in a plane stratified ionosphere was obtained by Budden ((4), §§ 18.10 and 18.11) by a process using approximate diagonalization of the governing equations. Alternatively the solution may be obtained by specializing to the one-dimensional case the result of Bennett (1) which employs frequency scaling. Formally similar results were obtained by Ludwig (8) without explicitly introducing frequency scaling.

By making use of a convergence-factor theorem of Bosanquet(3), Cooke((4), Theorem I) gave conditions for a regular sequence-to-sequence summability matrix B to be at least as strong as Cesàro summability (C, κ) (κ > 0), namely:

Theorem C. Let κ > 0. In order that the T-matrix B = (bρμ) shall satisfy B ⊇ (C, κ) it is necessary and sufficient that

If 0 < κ ≤ 1 then (2) alone is necessary and sufficient.

Suppose [Cscr ] and [Cscr ]′ are two sets of simple closed curves on a hyperbolic surface F. We give necessary and sufficient conditions for the existence of a pseudo-Anosov map g such that g([Cscr ]) ≅ [Cscr ]′.

For many questions, both in Mathematics and in Physics, the most important representations of a Lie algebra a are those which are unitarizable and highest weight (such representations are automatically irreducible). The classification of such representations when a is a finite-dimensional complex simple Lie algebra was completed only recently (see [3] for details and further references) and the corresponding question when a is an affine algebra was investigated by Jakobsen and Kac [5]. Theorem 3·1 of that paper contains a list of unitarizable highest weight representations which is claimed to be exhaustive. However, we shall show that this list is incomplete by constructing a further family of such representations. In fact, the classification problem in the affine case must be considered to be still open.

A two-generator group 〈f, g〉 of Möbius maps is determined, up to conjugacy, by the numbers β = tr2(f) − 4, β′ = tr2(g) − 4 and γ = tr (fgf−1g−1) − 2, provided that γ ≠ 0. We study the subset D of C3 of those (β, β′, γ) which arise from discrete groups. In particular, we identify precisely D[setmn ]D.

It is well known that a characteristically-simple finite group, that is, a group having no characteristic subgroup other than itself and the identity subgroup, must be either simple or the direct product of a number of isomorphic simple groups. It was suggested to the author by Prof. Hall that finite groups possessing exactly one proper characteristic subgroup would repay attention. We shall call a finite group having a unique proper characteristic subgroup a ‘UCS group’. In the present paper we first give some results on direct products of isomorphic UCS groups, and then we consider in more detail one of the types of UCS groups which can exist, that consisting of groups whose orders are divisible by exactly two distinct primes.

The object of this paper is to evaluate some integrals involving the product of Gauss's hypergeometric function and Meijer's G-function by expressing the G-function as a Mellin–Barnes type integral and interchanging the order of integrations. The integrals are important because on specializing the parameters they lead to many results for MacRobert's E-function, Bessel, Legendre, Whittaker functions and other related functions. In what follows δ is a positive integer and Δ(δ, α) represents the set of parameters

Approximate expressions are obtained, when the distance R between the nuclei is very large, for that portion of the wave function in the two-centre problem which depends on the hyperbolic coordinate μ. From these expressions the number and the approximate position of the nodes in μ can be deduced and hence the rules can be found by which that state of the combined atom at R = 0 can be determined which corresponds to a given state of the atom when the nuclei are completely separated. These rules are also applicable to the cases where the two atoms which can be formed when the nuclei are completely separated have the same energy. The converse problem of finding what state of the completely separated atom corresponds to a given state of the combined atom at R = 0 can also be solved by the use of the rules.

We explain and generalise row and column removal phenomena for Schur algebras via integral isomorphisms between subquotients of these algebras. In particular, we prove new reduction formulae for p-Kostka numbers.