Questions concerning extensions of polynomials or analytic functions from a Banach space E to its bidual E", and others about reflexivity and related properties on spaces of homogeneous polynomials Pk(E), have recently spurred interest regarding the structure of the bidual of a space of polynomials [4, 14, 22].

In this paper we study the relationship between the bidual of Pk(E) and the space of polynomials over E". We define a map through which elements of the bidual of Pk(E) may be viewed as polynomials over E"

formula here

and study this map to obtain information about Pk(E)". We have tried as far as possible to avoid imposing restricting conditions on E.

After the characterization of 1-point topological compactifications by Alexandroff in 1924, n-point topological compactifications by Magill [4] in 1965, and 1-point order compactifications by McCallion [5] in 1971, spaces that admit an n -point order compactification are characterized in Section 2. If X* and X** are finite-point order compactifications of X, sup{X*, X**} is given explicitly in terms of X* and X** in § 3. In § 4 it is shown that if an ordered topological space X has an m-point and an n-point order compactification, then X has a k-point order compactification for each integer k between m and n. The author is indebted to Professor Darrell C. Kent, who provided assistance and encouragement during the preparation of this paper.

In this paper we develop a descent technique for generalized (co)-homology theories defined in the category of algebraic varieties. By such a theory we mean a functor Sch→C, where C is a closed model category in the sense of Quillen satisfying certain axioms (cf. §4). We have chosen to work in such a general context so as to include two situations for which the results of SGA 4 of Deligne and Saint-Donat are not applicable: descent of multiplicative structures (i.e. of differential graded algebras) and descent for generalized sheaf cohomology (such as the algebraic K-theory of coherent sheaves over a noetherian scheme).

In the first part of this paper we discuss the radiation from a single charged particle moving in an arbitrary central field of force and obeying Dirac's equation. We consider the electric quadripole and magnetic dipole radiation as well as the electric dipole. We derive the selection rules for the magnetic dipole radiation and collect together for reference the corresponding selection rules for the electric dipole and quadripole radiations. In the second part we discuss the relative intensities of the various types of radiation, treating in detail the cases where the selection rules for magnetic dipole and electric quadripole are simultaneously satisfied. Finally we show that these results have an important bearing on the theory of internal conversion of γ-rays. The internal conversion of soft γ-rays occurs with such high probability that the theory is unable to account for the experimental results unless it is assumed that the radiation is largely magnetic dipole in character. On the other hand, Fisk and Taylor (loc. cit.) were unable to account for the presence of magnetic dipole radiation in appreciable amounts. We show that this is due to the fact that, of the two possible transitions (a and e of § 2) in which both magnetic dipole and electric quadripole radiation can be emitted, Fisk and Taylor considered only the second. In the case of the second, corresponding to a transition between two distinct terms, we show that Fisk and Taylor were correct in predicting a negligible amount of magnetic dipole radiation, but in the case of the first, corresponding to a transition between two levels of one multiplet term, we find that there is indeed a high percentage of magnetic dipole radiation.

A subset, X, of R belongs to the family T(a) if |X| ≥ 2 and, for all x and y belonging to X, ax + (1 − a) y ∈ X, where a ∈ R.

I consider the problem of determining for which values of a > 1 all elements of T(a) are dense in R.

This paper studies the spectral properties of a class of probability measures on the circle. The key aim is to describe the local structure of the maximal ideal space on the L-subspaces generated by the measures and hence the spectral properties of these measures. In particular we give a necessary and sufficient condition for such measures to belong to M0 (T).

A continuous arc joining two points (x1, y1), (x2, y2) is the aggregate of points given by

where f and φ are continuous functions, and where

The generalized function (x + iO)λ is defined by

for λ ╪ -1, -2, … and (x + iO)λ is denned by

for n = 1, 2,…, see (l).

The object of this paper is to obtain some new generating functions for Jacobi polynomials. The results obtained, are of general character.

A microcalorimeter is described accurate to 0·0005 cal. The elevation of temperature was measured by a series of iron-constantan thermo-couples, with one set of junctions making good thermal contact with the tube-where the heat was liberated and the others in a brass ring outside, kept in a thermostat. They were connected in series with a very sensitive moving coil galvanometer. The Tian multiple walled thermostat was used—three concentric copper cylinders insulated with kapok, the inner filled with water and the outer controlled by a mercury regulator. The constancy of temperature in the inner vessel was rather better than 1/500,000° C.

Continuous heat evolutions could also be measured by passing a current through another set of thermo-couples, when the Peltier cooling compensated for the heating and the temperature was kept down to its original value, thus avoiding cooling corrections.

In this paper, a generalized predator-prey system with delays is considered. By using coincidence degree theory, we establish the existence result of multiple positive periodic solutions for the system.

We develop the theory of contractive Markov systems initiated in [8]. We construct a coding map for such systems and investigate some of its properties.

We discuss a confluence from Siegel–Whittaker functions to Whittaker functions on $Sp(2,{\mathbb{R})$ by using their explicit formulae. In our proof, we use expansion theorems of the good Whittaker functions by the secondary Whittaker functions.

The interaction between an external electromagnetic field and a nuclear system can be expressed in terms of the multipole moments. The electric quadripole and the magnetic dipole moments of the deuteron have been calculated, taking into account the exchange forces as given by the meson theory. The cross-section of the photomagnetic effect of the deuteron has been calculated.

This work was carried out under the guidance of Dr Heitler and Dr Fröhlich. The writer wishes to express his sincerest thanks to them for suggesting the problem and many valuable comments. The writer is also indebted to Dr Kahn for discussions during the early stages of this work.

We study a construction of an HNN extension for inverse semigroups with zero. We prove a normal form for the elements of the universal group of an inverse semigroup that is categorical at zero, and use it to establish structural results for the universal group of an HNN extension. Our main application of the HNN construction is to show that graph inverse semigroups –including the polycyclic monoids –admit HNN decompositions in a natural way, and that this leads to concise presentations for them.

Let Kn, p be a random subgraph of a complete graph Kn obtained by removing edges, each with the same probability q = 1 – p, independently of all other edges (i.e. each edge remains in Kn, p with probability p). Very detailed results devoted to probability distributions of the number of vertices of a given degree, as well as of the extreme degrees of Kn, p, have already been obtained by many authors (see e.g. [l]–[5], [7]–[9]). A similar subject for other models of random graphs has been investigated in [10]–[13], The aim of this note is to give some supplementary information about the distribution of the ith smallest (i ≥ 1 is fixed) and the ith largest degree in a sparse random graph Kn, p, i.e. when p = p(n) = o(1).

Using coupling techniques, we prove uniqueness in $G$-measures under a weak regularity condition and give estimates of the associated rates of convergence. We also show how to generate a random variable distributed according to the unique $G$-measure on cylinder sets for any fixed level of precision.

We prove monadic second order limit laws for classes of logarithmic adequate relational structures by using probabilistic methods.

1. Introduction. In a paper on the many nucleon problem (l), which we shall henceforth call I, the determination of the irreducible representations of the four-dimensional unitary group were found from a decomposition of its infinitesimal ring U04 The method of decomposition made use of the four primitive four-component idempotents (projection operators) of the Dirac ring each of which, as was recognized long ago by Eddington (2), can be identified with a possible charge-spin state of a Dirac particle. Some experimental justification for the representations was also provided, and it is the purpose of this paper to apply the same tools to the many electron problem. In particular, matrices will be derived for the spin multiplets of a system of r electrons, and it will be shown how the model can account for the atomic shell structure and orbital angular momentum.