In (4) we showed how to stably split certain spaces CX built up from a ‘coefficient system’ and a ‘Π-space’ X. Via an approximation theorem relating particular examples to loop spaces, there resulted stable splittings of Ωn σnX for all n ≥ 1 and all (path) connected based spaces X.

We study the global behaviour of Gaussian curvature K and normal curvature K⊥ of zero mean curvature spacelike surfaces (stationary surfaces) in a four-dimensional Lorentzian space form L4(c). In particular, we show that the only complete stationary surfaces in Minkowski space E41 with K[ges ]0 are those with K≡0≡K⊥ and we give an explicit description of them. More general results are obtained for stationary surfaces in L4(c). We also discuss applications to Willmore surfaces in both Lorentzian and Riemannian three spaces. We give new examples of complete stationary surfaces in E41 with finite total curvature.

Under natural conditions it is shown that the rearrangement invariant hull of a weakly compact subset of a properly symmetric Banach space of measurable operators affiliated with a semi-finite von Neumann algebra is again relatively weakly compact.

The method of antithetic variates introduced by Hammersley and Morton (2) is one of the most widely used Monte Carlo techniques for estimating an unknown parameter θ. The basis for this method was established by Hammersley and Mauldon(l).in the case of unbiased estimators with the form

where each of the variates ξj is required to have a uniform marginal distribution over the unit interval [0,1]. By assuming that n = 2 and that the gj are bounded Borel functions, Hammersley and Mauldon showed that the greatest lower bound of var (t) over all admissible joint distributions for the variates ξj can be approached simply by arranging an appropriate strict functional dependence between the ξj. Handscomb(3) extended this result to the case of n > 2 bounded antithetic variates gj(ξj). In many experiments involving distribution sampling or the simulation of some stochastic process over time, the response functions gry are unbounded. This paper further extends the antithetic-variates theorem to include the case of n ≥ 2 unbounded antithetic variates gj(ξj) each with finite variance.

In this paper, we give an extension (Theorem 1) of the following well-known result of Banach ((l)): If X, Y are sets and Θ: X → Y, ψ: Y → X are injective mappings, then there exist partitions X = X1 ∪ X2, Y = Y1 ∪ Y2 such that Θ(X1) = Y1 and ψ(Y2) = X2. Here, as is usual, we say that X = X1 ∪ X2 is a partition of the set X = X1 ∩ X2 = π. Our theorem is applied in sections 2 and 3 to problems concerned with the existence of common representatives for two families of sets.

In 5, I provided a method whereby the study of an Artinian module A over a commutative ring R (throughout the paper, R will denote a commutative ring with identity) can, for some purposes at least, be reduced to the study of an Artinian module A' over a complete (Noetherian) local ring; in the latter situation, Matlis' duality 1 (alternatively, see 6, ch. 5) is available, and this means that the investigation can often be converted into a dual one about a finitely generated module over a complete (Noetherian) local ring.

We study compact Riemann surfaces of genus $g\geq2$ having a dihedral group of automorphisms. We find necessary and sufficient conditions on the signature of a Fuchsian group for it to admit a surface kernel epimorphism onto the dihedral group $D_N$. The question of extendability of the action of $D_N$ is considered. We also give an explicit parametrization of the moduli space of Riemann surfaces with maximal dihedral symmetry, showing that it is a one-dimensional complex manifold. Defining equations of all such surfaces and the formulae of their automorphisms are calculated. The locus of this moduli space consisting of those surfaces admitting some real structure is determined.

In the investigation of preference orders which are explanations of expenditure data which associates a quantity vector xr with a price vector pr (r = 1, …, k), in respect to some n goods, there is considered the class of functions φ, with gradient g, which are increasing and convex in some convex region containing the points xr, such that gr = g(xr) has the direction of pr. † Let ur = pr/er, where so that

The purpose of these remarks is to use elementary mathematics to describe some simple connexions between multi-valued Legendre transformations and certain elementary catastrophes. Multi-valued Legendre transforms appear in subjects such as non-linear elasticity and non-convex optimization.

This paper gives experimental results for the scattering of 4 and 2 volt electrons in nitrogen, which are compared with Stier's theoretical curves. An approximate theory of the scattering of electrons in nitrogen is given which is applicable when the wave length of the electron is small compared to the internuclear distance. The results are compared with experiment.

The purpose of this note is to present two characterizations of conjugate Banach spaces. More precisely, we present two conditions, each necessary and sufficient for a (real or complex) Banach space to be isomorphic to the conjugate space of a Banach space, and two corresponding conditions for to be equivalent to the conjugate space of a Banach space. Other characterizations, in terms of weak topologies, have been given by Alaoglu ((1), Theorem 2:1, p. 256, and Corollary 2:1, p. 257) and Bourbaki ((4), Chap, IV, §5, exerc. 15c, p. 122). Here, by the conjugate space* of a Banach space we mean ((2), p. 188) the space of continuous linear functionals over . Two Banach spaces and are said to be isomorphic if there is a one-one continuous linear mapping of onto (its inverse is necessarily continuous by the inversion theorem ((2), Théorème 5, p. 41; (6), Theorem 2·13·7, Corollary, p. 29)); they are said to be equivalent if there is a norm-preserving linear mapping of onto .((2), p. 180).

Tensor operators under any group are defined and the theory is developed for semi-simple continuous groups. Coupled tensor operators are introduced and the matrix elements of tensor operators are expressed in terms of the coupling coefficients. The structure of generalized Casimir operators is investigated.

The compact–open topology for function spaces is usually attributed to R. H. Fox in 1945 [16]; and indeed, there is no earlier publication to attribute it to. But it is clear from Fox's paper that the idea of the compact–open topology, and its notable success in locally compact spaces, were already familiar. The topology of course goes back to Riemann; and to generalize to locally compact spaces needs only a definition or two. The actual contributions of Fox were (1) to formulate the partial result, and the problem of extending it, clearly and categorically; (2) to show that in separable metric spaces there is no extension beyond locally compact spaces; (3) to anticipate, partially and somewhat awkwardly, the idea of changing the category so as to save the functorial equation. (Scholarly reservations: Fox attributes the question to Hurewicz, and doubtless Hurewicz had a share in (1). As for (2), when Fox's paper was published R. Arens was completing a dissertation which gave a more general result [1] – though worse formulated.)

Absorption experiments have been carried out on the β- and γ-rays of RaC”. The range of the β-rays in aluminium corresponds to an energy limit of (1·95 ± 0·15) × 106 e.v.; the absorption coefficient of the γ-rays in copper corresponds to a fairly homogeneous radiation of quantum energy equal to (5 ± 1) × 106 e.v. The energy of disintegration is, within the limits of error, in agreement with the value 6·1 × 106 e.v. to be expected from the known energies of the other disintegrations in the RaC branches.

The fact that the rate of production of atomic hydrogen at a tungsten surface at a given temperature is proportional to the square root of the hydrogen pressure means either that the important process is the evaporation of atoms from an adsorbed film which over the whole range of experimental conditions is sparsely occupied, or that the production of atoms is in the main due to a process in which a hydrogen molecule strikes a bare tungsten atom in the surface, one atom being adsorbed and the other evaporating and the surface being almost completely covered over the whole range of experimental conditions. Either process leads to a temperature variation in the rate of atom production in agreement with experiment. A definite decision between the two processes cannot yet be made.

The ‘static-geometric analogy’ in thin shell structures is a formal correspondence between equilibrium equations on the one hand and geometric compatibility equations on the other. It is well known as a fact, but no satisfactory explanation of its basis has been given. The paper gives an explanation for the analogy, within the framework of shallow-shell theory. The explanation is facilitated by two innovations: (i) separation of the shell surface conceptually into separate stretching (S) and bending (B) surfaces; (ii) use of change of Gaussian curvature as a prime variable. Various limitations of the analogy are pointed out, and a scheme for numerical calculation which embodies the most useful features of the analogy is outlined.

The transmission of a quantal particle through a barrier obtained by inverting a Morse potential is studied by means of the ‘symmetric’ phase-integral approximations introduced by N. Fröman and further developed by N. Fröman and P. O. Fröman. The calculations are performed up to the fifth-order approximation. Approximate expressions for important quantities such as transmission and reflexion coefficients, phases of the wave function and also for quantities related to the phenomenon of resonances, which may occur when the wave function is given as a standing wave, are compared to the corresponding exact quantities, and this comparison shows the great accuracy of these phase-integral approximations.

The thermodynamic behaviour of a body in a static electromagnetic field depends only on the field over the region of the body itself, whence its own thermodynamic behaviour is determined even in the presence of other polarizable media. Its behaviour depends only on the entropy density, for in fact no real energy density need exist. Also new formulae are developed for the energy and for the entropy of a body in electric and magnetic fields, including those due to permanent magnets.

Following LeVan (6), we were led in (3) to investigate a purely categorical formulation of shape theory, which had hitherto been confined to certain rather circumscribed areas of topology. However, several authors (6, 7, 8, 9, 10, 11) had noted that, within this topological context, shape theory could be related to the process of expressing objects of the larger category T as filtered limits of objects of the given full subcategory P.

The first result of this paper was proved in January 1975 in order to engage the interest of Professor J. E. Littlewood, who was in hospital at the time. This theorem is of some independent interest and we present it here with some related results.