We define an unbased H-space to be a pair (A, m) where A is a space and

is a map such that the maps La: x → m(a, x), Ra: x → m(x, a) are homotopy equivalences for all a ∈ A. This is the same as James's definition of an H'-space in (3); we follow his notation as far as possible. (A, m) is homotopy-associative if the maps m(m × 1) and m(l × m) are homotopic. A left (right) a-inverse (a ∈ A) is a map w: A → A such that the composition m(w × 1) d (w(1 × w) d) is homotopic to ka, the constant map to a. d denotes the diagonal map a → (a, a).

Elastic work functions may be restricted by material symmetry requirements and/ or by the Cauchy symmetry of the first order moduli with respect to the Green strain. This investigation enables the construction of any work function so constrained. The crystallographic point groups are considered in detail, and examples of such work functions are given.

We study the action of the elements of the mapping class group of a surface of finite type on the Teichmüller space of that surface equipped with Thurston's asymmetric metric. We classify such actions as elliptic, parabolic, hyperbolic and pseudo-hyperbolic, depending on whether the translation distance of such an element is zero or positive and whether the value of this translation distance is attained or not, and we relate these four types to Thurston's classification of mapping class elements. The study is parallel to the one made by Bers in the setting of Teichmüller space equipped with Teichmüller's metric, and to the one made by Daskalopoulos and Wentworth in the setting of Teichmüller space equipped with the Weil–Petersson metric.

In Part I of this series it was shown that for CCl4 the approximation of smoothing out the Cl nuclei over the surface of a sphere did not lead to reasonable results for the Thomas–Fermi distribution of electrons in the molecule. An alternative method in which the inner-shell Cl electrons were first compressed into the nuclei was not theoretically satisfactory because of the neglect of the exclusion principle. In this paper therefore a general scheme is presented in which we describe the inner-shell electrons of the outer atoms wave mechanically, assuming them to be undisturbed by molecular binding, but still use a Thomas–Fermi description for the valence electrons. A method by means of which the exclusion principle can be imposed is formulated. The following paper of the series deals with the application of the method described here to obtain an improved valence electron distribution in CCl4.

Let Zn denote the integer lattice in Rn, let A be a non-singular n × n matrix and ʗ ∈ Rn. Then G = AZn + ʗ is called a grid (non-homogeneous lattice) and its determinant det G is defined to be |det A|.

Second order focusing can be obtained with an apparatus, which contains a circular magnetic field and differs little in its dimensions from the apparatus at present in use. The lay-out is shown to scale in Fig. 4: the essential data are collected at the end of Section V, and indications of the accuracy to be expected are given at the end of Section III.

The aim of this paper is to prove some results involving Weyl groups which are useful in an application to give a combinatorial construction of representations of Weyl groups which generalizes the well-known Young tableau method for symmetric groups. A preliminary version of these applications has been given in [12]. As these results seem to be of independent interest they are gathered together in the present paper.

We illustrate a minor error in the biadjointness result for 2-categories of traced monoidal categories and tortile monoidal categories stated by Joyal, Street and Verity. We also show that the biadjointness holds after suitably changing the definition of 2-cells.

This paper deals with the existence of a quadratic form as a Liapunov function for a linear homogeneous vector differential equation with constant coefficient matrix such that the total derivative of the Liapunov function is strictly negative semi-definite and not identically equal to 0 for every non-trivial solution of the given equation. A theorem is proved which guarantees the existence of a strictly semi-definite quadratic form which is not identically equal to 0 for every non-trivial solution if and only if the coefficient matrix of the equation is not proportional to the unit matrix. Furthermore, it is proved that if the equilibrium of the given linear equation is asymptotically stable and if the coefficient matrix is not proportional to the unit matrix, then there exists a positive definite quadratic form as a Liapunov function whose total derivative is strictly negative semi-definite and not identically equal to 0 for every non-trivial solution of the given equation. It is also shown how this theorem fits into the system of already known theorems concerned with quadratic forms as Liapunov functions for a linear equation with constant coefficient matrix.

This paper is a continuation of (F3). In its first part we shall expand and extend the general theory of the earlier paper, while in the second part we specialize to number fields. The theory of resolvents and of the trace form, presented here, complements the more arithmetic theory of module conductors and module resolvents as described elsewhere (cf. (F4)). Both these papers will be applied in work on the connexion, for tame extensions, between Galois module structure of algebraic integers on the one hand, and Artin conductors and root numbers on the other hand (cf. (F5)). The results of the present paper are however not restricted to the tame case and, it is hoped, will subsequently be applied in a more general context.

1. A “Fourier kernel” means here a function K(x) which gives rise to a formula

of the Fourier type. Thus

are Fourier kernels. If K(x) is a Fourier kernel, λ is real, and a positive, then

are Fourier kernels.

1.1. This paper is a theoretical investigation in the real domain of the existence of subharmonic solutions of non-linear differential equations of the form

where F is analytic and of least period 2π/ω in t; ε = (ε1, …, εn) is small; and F(x, ẋ, 0, t) is not linear in x and ẋ.

The usual method of calculating the probability of a switch between stationary states by the Quantum Mechanics is really equivalent to finding the intensity of the dipole radiation. Other switches of the system not involved in this radiation may be possible but are not taken account of by the calculation. A method of calculating the probability of switches due to radiation by the quadripole moment seems to be supplied by Dirac's recent theory of the interaction between matter and radiation in his paper “The Quantum Theory of Dispersion.” Equations from this paper will be denoted by a D. The results are probably of a purely theoretical interest and the intensities found probably too faint to be observed; but the present considerations do appear to emphasise the need for including the radiation in the exact theoretical treatment of a Quantum Mechanical system.

The Langmuir-Frenkel theory of surface action has been shown to involve a simple relation between the partial pressure of the catalyst poison, and the reduction in reaction velocity which results. Very interesting observations have been made by Beebe showing that the heat of adsorption of carbon monoxide on a reduced copper catalyst falls considerably and reaches a limiting value as the surface approaches saturation. Pease and Stewart have shown that the presence of less than one percent, of the total carbon monoxide that a given copper catalyst can adsorb will reduce the reaction velocity of the combination of hydrogen and ethylene at the surface of the copper by 88 per cent.

An orthonormal basis of analytic functions in the Bergman space is a doubly orthogonal system if it simultaneously is an orthogonal system in a larger domain. The case when one of the basis elements is a monomial is characterized as belonging to a pair of domains of concentric discs.

Suppose a group H is given as a free product with amalgamation

determined by groups A0, A1, A2 and homomorphisms α1: A0 → A1, α2: A0 → A2. Thus H may be described as the quotient of the free product A * A2 by the relations i1 α1 (α0) = i2α2 (α0) for all α0 ∈ A0, where i1, i2 are the two injections of A1, A2 into A1 * A2. We do not assume that α1, α2 are injective, so the canonical homomorphisms α′i: Ai → H, i = 0,1,2, also need not be injective.

Let G and G′ be reductive algebraic groups defined over infinite fields k and k′ respectively. The purpose of this paper is to show that G and G′ have isomorphic root systems if their rational subgroups G(R) and G′(R′), where R and R′ are integral domains with R ⊇ k and R′ ⊇ k′, are isomorphic to each other, except in one particular case (see Theorem 3·4). This has been proved by R. Steinberg in [6, theorem 31] for simple Chevalley groups over perfect fields. In particular, when G and G′ are semisimple and adjoint, every isomorphism between G(R) and G′(R′) induces an isomorphism between their irreducible components (see Proposition 3·3). These results imply that, when G and G′ are semisimple k-groups and when both are either simply connected or adjoint, then they are isomorphic to each other as algebraic groups if and only if their rational subgroups over an integral domain that contains k are isomorphic to each other, except in one particular case (see Corollary 3·5).

Let Sn denote the n-dimensional unit sphere. Recently, Bolton, Pedit and Woodward [1, 3] have begun to study a class of minimal surfaces in Sn, which are called superconformal. In this paper we will discuss the rigidity of superconformal minimal surfaces lying fully in S2m−1 among all superconformal minimal surfaces lying fully in odd-dimensional unit spheres.

Using a suitable notion of principal G-bundle, defined relative to an arbitrary cartesian category, it is shown that principal bundles can be characterised as adjunctions that stably satisfy Frobenius reciprocity. The result extends from internal groups to internal groupoids. Since geometric morphisms can be described as certain adjunctions that are stably Frobenius, as an application it is proved that all geometric morphisms, from a localic topos to a bounded topos, can be characterised as principal bundles.

In 1933 Orlicz proved various results concerning unconditional convergence in Banach spaces (4), which were noted by Banach ((l), p. 240) who remarked that absolute and unconditional convergence are equivalent in finite dimensional Banach spaces, but that whether or not the two are non-equivalent in all infinite dimensional spaces was still an open question. MacPhail (3) gave a criterion for the equivalence of the two notions of convergence in a general Banach space and used it to prove non-equivalence in the spaces l1 and L1. In 1950 Dvoretzky and Rogers demonstrated the non-equivalence of the two types of convergence in any infinite dimensional normed linear space, using an elegant and instructive geometrical approach (2). The result has also been proved by a different method by Grothendieck (5).