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"

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

The purpose of this paper is to prove irrationality results from Jacobi's triple product identity, which can be written, for x∈[Copf]*, y∈[Copf], [mid ]y[mid ]<1:

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There are various proofs of this identity; the classical one rests on the theory of theta functions ([3], theorem 6, p. 69). An alternative proof uses Heine's summation formula ([10], p. 12). An elementary, self-contained proof, can be found in [9], p. 227.

In this paper, we will use the same elementary methods as in [5] and [6], and prove the following theorems.

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.

1. It is known that the distribution of the vanishing coefficients in the power series of an integral function is closely connected with the analytic properties of the function. But the following theorem has apparently not yet been noticed, although simple and elegant both in statement and in proof.

We consider IA-endomorphisms (i.e. Identical in Abelianization) of a free metabelian group of finite rank, and give a matrix characterization of their fixed points which is similar to (yet different from) the well-known characterization of eigenvectors of a linear operator in a vector space. We then use our matrix characterization to elaborate several properties of the fixed point groups of metabelian endomorphisms. In particular, we show that the rank of the fixed point group of an IA-endomorphism of the free metabelian group of rank n[ges ]2 can be either equal to 0, 1, or greater than (n−1) (in particular, it can be infinite). We also point out a connection between these properties of metabelian IA-endomorphisms and some properties of the Gassner representation of pure braid groups.

Introduction. Dans cet article, K désigne un corps commutatif quelconque. Toutes les algèbres de Lie considérées sont des algèbres de Lie de dimension finie sur K. Si g est une algèbre de Lie, on désigne par (D0g, D1g, D2g, …) la suite des algèbres dérivées de g (D0g = g; Dig = [Di−1g, Di−1g]). Si h est un idéal de g, et si x ↦ g, on désigne par adh, x l'application y → [x, y] de h dans h.

1. Introduction. A number of authors (2,3,7) have recently quantized the motion of an inviscid fluid. The starting point has been the variational principle of Bateman which uses the Clebsch variables. The density turns out to be the canonical conjugate to the velocity potential, and the transition to the quantum theory is then made in the usual way. If the fluid is making small vibrations, ‘phonons’, which are scalar, appear as a result of the quantization.

It was shown by Sheffer that all functions of the two-valued propositional calculus can be denned in terms of a single primitive. This result has been extended by Post (3) and Webb (5) to the functionally complete m-valued propositional calculus. We shall consider here similar extensions for the extended propositional calculus and for the (functionally incomplete) systems of Łukasiewicz (1).

As a partial generalization of a theorem of Hurewicz (3), Whitehead gave an important theorem on deformation retracts ((9), p. 273) which was proved only for polytopes. A demonstration of a part of Whitehead's theorem for absolute neighbourhood retracts was published by Fox ((2), p. 42) with a remark given by Samelson(8).

1. Let ξ, η denote the rectangular Cartesian coordinates of a point in a plane. Let J (ξ, η) denote a harmonic function which is positive in the half-plane η > 0. In this paper, we first show (Theorem I) that every such function J determines a non-negative number d, and a bounded non-diminishing function G(x), such that

In this paper we give a new generating function for the Jacobi polynomials. The result obtained is of general character and includes as particular cases some of the results given earlier by Carlitz(2), Salam(1), Manocha and Sharma (3,4).

1. Let be a Banach algebra. We say that homomorphisms from are continuous if every homomorphism from into a Banach algebra is automatically continuous, and that derivations from are continuous if every derivation from into a Banach -bimodule is automatically continuous.

In [10] Loeb and Osswald develop a Daniell–Stone integration theory in topological vector lattices. As an application they obtain an extension of the Bochner integral for functions with values in a topological vector lattice. The Daniell–Stone approach requires an order structure for definition of a null function. Here we build on the ideas in [10], but abandon the lattice structure. The internal representatives of the extended integrable functions in [10] turn out to be S-integrable liftings. This is our starting point: we define the class of extended integrable functions as the class of functions that have S-integrable liftings; a definition that is independent of any order structure. As a consequence, several of the proofs in the construction of the Banach space of extended integrable functions are almost identical to the analogous proofs in [10] and we omit them. After defining the integral we show that each extended integrable function induces a countably additive vector measure of bounded variation. This permits us to use the tools of the theory of vector measures. If the nonstandard hull of the Banach space has the Radon–Nikodym property, then the vector measure induced by an extended integrable function has a Bochner integrable Radon–Nikodym derivative; this yields a characterization of the Banach space of extended integrable functions. On the other hand for some spaces we can construct internal functions and standardize them to get very ‘spiky’ extended integrable functions whose induced measure is not differentiable on any set of positive measure. With this method we show that if a Banach space is not superreflexive, then it does not have the super-Radon–Nikodym property.

We prove that for an action of a finite group G on a systolic complex X there exists a G–invariant subcomplex of X of diameter ≤5. For 7–systolic locally finite complexes we prove there is a fixed point for the action of any finite G. This implies that free products with amalgamation (and HNN extensions) of 7–systolic groups over finite subgroups are also 7–systolic.

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).

1. In this paper we shall use the methods of (6) to give a short proof of Wunderlich's result (7, 8) on the distribution of twin ‘primes’ for the Hawkins random sieve. We shall also obtain a sharp estimate on the gaps between successive ‘primes’.

A period of a function f(z) is defined to be a number ω (≠ 0) such that

is identically zero; and it can be shown that an integral function may either have no periods or else a single sequence kλ (k = ± 1, ± 2, …).

Given an embedding of an oriented surface F in an oriented three-manifold M and a homeomorphism h of F, one can construct another three-manifold Mn by cutting M along F and re-glueing the two boundary components using h. Let F be a genus two surface and r the involution which exhibits F as the 2-fold branched cover of S2 branched over six points. In this special case we call Mr the mutation of M along F.

Let be a finite group scheme over an algebraically closed field k of characteristic char(k) = p ≥ 3. In generalisation of the familiar notion from the modular representation theory of finite groups, we define the p-rank rkp() of and determine the structure of those group schemes of p-rank 1, whose linearly reductive radical is trivial. The most difficult case concerns infinitesimal groups of height 1, which correspond to restricted Lie algebras. Our results show that group schemes of p-rank ≤ 1 are closely related to those being of finite or domestic representation type.

In the second half of the last century the French mathematician Emil Mathieu discovered two quintuply transitive permutation groups, now labelled M12 and M24, acting on twelve and twenty-four letters respectively. With the classification of finite simple groups complete we now know that any other quintuply transitive permutation group, on any number of letters, must contain the corresponding alternating group. Indeed, the only quadruply transitive groups, other than the alternating and symmetric groups, are the point stabilizers in M12 and M24, which are denoted by M11 and M23 respectively. To put it another way, the study of multiply (≥ 4-fold) transitive groups now means the study of the symmetric groups and the Mathieu groups. Apart from their beauty and interest in their own right the Mathieu groups are involved in many of the other sporadic simple groups: see ([2], p. 238). Thus a detailed understanding of the other exceptional groups necessitates an intimate knowledge of M12 and M24.