The general construction of realizability toposes was first described in [3]. The data from which such a topos is constructed (called a partial applicative structure in [3]) consists of a set A equipped with a partial binary operation (called application and denoted by juxtaposition, with the convention that unbracketed expressions are evaluated from left to right) and two constants K, S satisfying

for all x, y, z ∈ A (where, as usual, an equality between possibly undefined terms means ‘one side is defined iff the other is, and then they are equal’). Following Lambek [6], we now call such a structure a partial Schönfinkel algebra, or a global Schönfinkel algebra if application is defined on the whole of A × A. (The name ‘combinatory algebra’ is also in common use.) We write for the realizability topos constructed from A.

Let E/F be a finite normal extension of number fields with Galois group G. For each virtual character χ of G, denote by L(s, χ) = L(s, χ, F) the Artin L-series attached to χ. It is defined for Re (s) > 1 by an Euler product which is absolutely convergent, making it holomorphic in this half plane. Artin's holomorphy conjecture asserts that, if χ is a character, L(s, χ) has a continuation to the entire s-plane, analytic except possibly for-a pole at s = 1 of multiplicity equal to 〈χ, 1〉, where 1 denotes the trivial character. A well-known group-theoretic result of Brauer implies that L(s, χ) has a meromorphic continuation for all s.

Let ψ be the Grossencharacter attached to an elliptic curve E defined over an imaginary quadratic field K ⊂ of discriminant −dK, and having complex multiplication by the maximal order of K. We denote the conductor of ψ by and fix a Weierstrass model for E with coefficients in ,

whose discriminant is divisible only by primes dividing 6. Let Kab be the abelian closure of K in and choose a fundamental period Ω ∈ for the above model of the curve.

In [8] and [9] S. J. Pride has initiated a study of group presentations in which each defining relator involves exactly two members of the generating set. The methods there involve the use of graphs and so-called edge groups – the building blocks of such presentations. In this paper we replace ‘graph’ by ‘set of finite subsets of a given set’, and ‘edge group’ by ‘face group’ in order to study a larger class of presentations. This way we are able to extend to this larger class a Freiheitssatz and a result on diagrammatic asphericity which appear in the references cited above.

A recurrent problem over many years in the study of linear groups has been the determination of the central height of a unipotent normal subgroup of some matrix group of specified type. In the theory of matrix groups over division rings, unipotent elements frequently present special difficulties and these have usually been by-passed by the addition of some suitable hypothesis. In this paper we make a start on the removal of these extraneous hypotheses. Our motivation for doing this now conies from [9], where by 3·7 of that paper the additional assumptions have finally reduced us to degree one, a situation where unipotent elements present few problems!

Hamiltonian circuits with associated word an n-cycle, in Schreier right coset graphs for symmetric groups Sn mod cyclic groups Zn, correspond to change ringing principles on n bells for which the plain course is the extent; that is, neither bobs nor singles are required. This connection is made explicit for the general case, and then specialized to the cases n = 4 (minimus) and n= 5 (doubles). In particular, all 102 no-call doubles principles on three generators are found and catalogued.

It can be very difficult to determine whether or not certain rings are really full matrix rings. For example, let p be an odd prime, let H be the ring of quaternions over the integers localized at p, and set

Then T is not presented as a full matrix ring, but there is a subring W of H such that T ≅ M2(W). On the other hand, if we take H to be the ring of quaternions over the integers and form T as above, then it is not known whether T ≅ M2(W) for some ring W. The significance of p being an odd prime is that H/pH is a full 2 x 2 matrix ring, whereas H/2H is commutative. Whether or not a tiled matrix ring such as T above can be re-written as a full matrix ring depends on the sizes of the matrices involved in T and H/pH. To be precise, let H be a local integral domain with unique maximal ideal M and suppose that every one-sided ideal of H is principal. Then H/M ≅ Mk(D) for some positive integer k and division ring D. Given a positive integer n. let T be the tiled matrix ring consisting of all n x n matrices with elements of H on and below the diagonal and elements of M above the diagonal. We shall show in Theorem 2.5 that there is a ring W such that T ≅ Mn(W) if and only if n divides k. An important step in the proof is to show that certain idempotents in T/J(T) can be lifted to idempotents in T, where J(T) is the Jacobson radical of T. This technique for lifting idempotents also makes it possible to show that there are (k + n − 1)!/ k!(n−1)! isomorphism types of finitely generated indecomposable projective right T-modules (Theorem 2·10).

Throughout this note, A will denote a (commutative, Noetherian) local ring (with identity) having maximal ideal m and dimension d. Let x1, …, xd be a system of parameters (s.o.p.) for A. A (not necessarily finitely generated) A-module M is said to be a big Cohen–Macaulay A-module with respect to x1, …, xd, if x1, …, xd is an M-sequence. In the last ten or fifteen years there has been substantial interest in such modules, initially stemming from M. Hochster's discoveries that, if A contains a field as a subring, and x1, …,xd is any s.o.p. for A, then there exists a big Cohen-Macaulay A-module with respect to x1, …,xd, and that the existence of such modules has important consequences for the local homological conjectures in commutative algebra: see [6].

Let Σg be a closed orientable surface of genus g, which will be assumed to be greater than one throughout this paper. In our previous paper [11], we have associated to any oriented ∑g-bundle π: E → X with a cross-section s: X → E a flat T2g-bundle π: J → X and a fibre-preserving embedding j: E → J such that the restriction of j to any fibre Ep = π−1(p)(p ∈ X) is equivalent to the Jacobi mapping of Ep with respect to some conformal structure on it and relative to the base-point s(p) ∈ Ep. There is a canonical oriented S1-bundle over J and the main result of [11] is the identification of the Euler class of the pullback of this S1-bundle by the map j as an element of H2(E, ). In this paper we deal with the case of Σg-bundles without cross-sections. First of all we associate a flat T2g-bundle π′: J′ → X to any oriented Σg-bundle π: E → X and construct a fibre-preserving embedding j′: E → J′ such that the restriction of j′ to any fibre Ep is equivalent to some translation of the Jacobi mapping of it. Although our original motivation for the present work came from a different source, this should be considered as a topological version of Earle's embedding theorem [5] which states that any holomorphic family of compact Riemann surfaces over a complex manifold can be embedded in a certain associated family of the corresponding Jacobian varieties in an essentially unique way.

The polynomial invariants of classical links that relate to the Jones polynomial have various inter-connections. Here a new one is given: the square of the Jones polynomial is obtained from the F-polynomial by means of a certain substitution of variables. This explains and corrects some similarities between known evaluations of these polynomials. The proof uses the ‘Dubrovnik’ version of the F-polynomial; this is first proved to be equivalent to the F-polynomial. Finally a correlation between the F and P polynomials is included as a provocative curiosity.

The 2-jet of a Σ3 map-germ f:(3, 0) → (3, 0) determines a net of quadratic maps from 3 to 3; for nets of general type this jet is sufficient for equivalence. The classification of such nets involves a single parameter c. It is shown in [7], also in [3], that the versai unfolding of f is topologically trivial over the parameter space. However, there are 4 connected components of this space of nets. The main object of this paper is to show that the corresponding unfolded maps are of different topological types.

Several important classes of Banach spaces are characterized by means of convergence properties of sequences. For example, if X is a Banach space, then X belongs to the class Nl1 of spaces without copies of l1, the class R of reflexive spaces or the class F of finite-dimensional spaces if and only if each bounded sequence has respectively a weakly Cauchy (w-Cauchy), weakly convergent (w-convergent) or convergent subsequence. Similarly X is in the class WSC of weakly sequentially complete spaces, or the class SCH of spaces with the Schur property if and only if each w-Cauchy sequence is w-convergent, or convergent, respectively; note that X ∈ SCH if and only if each w-convergent sequence of X is convergent (see [12], p. 47).

1. In a recent paper [6], the spectrum of the Cesàro operator C on c0 (the space of null sequences of complex numbers with the sup norm) was obtained by finding the eigenvalues of the adjoint operator on and showing that the operator (C–λI)−1 lies in B(c0) for all λ outside the closure of this set of eigenvalues. In this paper we apply a similar method to find the spectrum of the two-dimensional Cesàro operator on a space of double sequences c0(c0) (defined in §2). We shall introduce a simplification to the proof in [6] by observing that (C – λI)−1, when it exists, is a Hausdorff summability method (see page 288 of [11] for the single variable case on the space of convergent sequences c), and the crux of our proof is to show that the moment constant associated with the method (C – λI)−1 is regular for the space c0(c0) and the set of λ under consideration. It turns out that c0(c0) ≅ c0c0 (see page 237 of [7]) and that the two-dimensional Cesàro operator on c0(c0) is the tensor product C ⊗ C of the Cesàro operator C on c0. Thus our result gives a direct proof that the spectrum σ(C ⊗ C) equals σ(C)σ(C), which is a special case of the result of Schechter in [8].

Complex (associative) H*-algebras were introduced and studied in detail by Ambrose[1]; it was proved that every complex H*-algebra with zero annihilator is the l2-sum of a suitable family of topologically simple complex H*-algebras and that the H*-algebras (H) of all Hilbert-Schmidt operators on any complex Hilbert space H are the only topologically simple complex H*-algebras. In a recent paper [2] Balachandran and Swaminathan observe that the reduction of the theory of real H*-algebras to the topologically simple case follows easily with minor changes of the complex argument, and they prove a theorem describing topologically simple real H*-algebras. This theorem can be equivalently reformulated as follows.

By a complex function space A we will mean a uniformly closed linear space of continuous complex-valued functions on a compact Hausdorff space X, such that A contains constants and separates the points of X. We denote by S the state-space

endowed with the w*-topology. If A is self-adjoint then it is well known (cf. [1]) that A is naturally isometrically isomorphic to , and re A is naturally isometrically isomorphic to A(S), where (respectively A(S)) denotes the Banach space of all complex-valued (respectively real-valued) continuous affine functions on S with the supremum norm.

F. Riesz's ‘Representation Theorem’ has been proved by methods classical [11, 12], category-theoretic [7], and functional-analytic [2, 9]. (Garling's remarkable proofs [5, 6] owe their brevity to the combined strength of these and other methods.) These proofs often reveal a connection between the Riesz theorem and some unexpected area of mathematics.

An easy way to construct the representing vector measures of weakly compact operators on C(Ω) is given by using the Loeb measure technique. This construction is not based on the Riesz representation theorem for linear functionals; thus we have a uniform way to treat the scalar and vector cases. Also the star finite representations of regular vector measures follow from the proof.

The tauberian theorems concerning power-bounded elements of Banach algebras studied by Katznelson and Tzafriri, Allan, O'Farrell and Ransford and Allan are considered, and it is shown that (almost) exactly the same results are true for power-bounded elements in a very large class of locally convex topological algebras, the pseudo-complete algebras. The submultiplicativity of the Banach algebra norm is, for once, inessential to the proof of these theorems.

A classical theorem of Weyl [7] guarantees that the eigenvalues, ordered according to decreasing absolute values, of a symmetric kernel of class Cm (m ≥ 0) satisfy λn = o(n−m−½). Reade [5, 6] recently proved that if K is, in addition, positive definite, then λn = o(n−m−1;). He has also in [4] made similar improvements of classical spectral estimates for kernels of class Lip α. James Cochran pointed out to me that allied theorems for trigonometric Fourier coefficients seem to have been neglected in the literature. The trigonometric versions turn out to be elementary; nevertheless, in their conclusions concerning the decreasing rearrangement {f^*(n)} they generalize known results about the behaviour of monotone trigonometric transforms. Furthermore they suggest that the Cm hypothesis of Reade's theorem could be relaxed.

Reade[10] has recently improved Weyl's classical estimate λn = o(n−3/2) for the eigenvalues of a symmetric kernel K∈C1 by relaxing the Cl hypothesis to the assumptions that K∈L2[0, 2π]2, that K is absolutely continuous in each variable separately, and that both ∂K/∂s and ∂K/t belong to L2[0, 2π]2. The conclusion of his theorem, that is, of course, stronger than λn = o(n−3/2).