Duskin [1, 2] showed that gerbes over a Grothendieck topos E can be interpreted by certain internal groupoids of E which he called bouquets. He proved that for any bouquet B of E the fibred category F ors(B) of B-torsors is a gerbe over E, and conversely that any gerbe G gives rise to a bouquet B satisfying G ≅ F ors(B). We want to give here a somewhat different approach to these results, based on a construction of gerbes given in [3].

As in [2], we define a geometry Γ = (B2, …, Br; *) to be an ordered sequence of r pairwise disjoint non-empty sets Bi together with a symmetric incidence relation * on their union B = B1 ∪ … ∪ Br such that if F is any maximal set of pairwise incident elements (i.e. a maximal flag), then |F ∩ Bi| = 1 for i = 1,…, r. The number r is called the rank of Γ. The geometry Γ is called connected if the r-partite graph (Γ, *) is connected.

Let U = [{Mi: i ∈ I}; U; {øi: i ∈ I}] be a monoid amalgam of inclusions, i.e. U and each Mi is a monoid, and øi: U → Mi are inclusions. Let be the monoid free product of the amalgam U (see for example [10] for these concepts), and let β: B → H be a homomorphism of monoids. The type of question we seek to answer in this paper is under what conditions (on β, B and H) can we deduce that B is isomorphic to the free product of the amalgam

Boardman's stable category (see [5]) is a closed category ([4], VII·7), and in the best of all possible worlds, the category of spectra underlying the stable category would be closed as well; this would make life considerably easier for those doing calculations in stable homotopy theory. Unfortunately none of the categories of spectra introduced to date are closed; only S, the category introduced in [2], is even symmetric monoidal. The problem with making S closed is that it comes equipped with an augmentation to I, the category of universes and linear isometries (called Un in [2]), which preserves the symmetric monoidal structure. Since I is not closed, this makes it difficult to see how S might be closed.

We show that determining the Jones polynomial of an alternating link is #P-hard. This is a special case of a wide range of results on the general intractability of the evaluation of the Tutte polynomial T(M; x, y) of a matroid M except for a few listed special points and curves of the (x, y)-plane. In particular the problem of evaluating the Tutte polynomial of a graph at a point in the (x, y)-plane is #P-hard except when (x − 1)(y − 1) = 1 or when (x, y) equals (1, 1), (−1, −1), (0, −1), (−1, 0), (i, −i), (−i, i), (j, j2), (j2, j) where j = e2πi/3

The local change in an oriented link diagram which replaces by k positive half-twists is called a tk move. For k even, the local change replacing by is called a tk move. For an unoriented diagram define a k-move, replacing by for any k. The following conjecture was stated in [14] and [10].

The aim of this paper is twofold: first, to construct the compact regular coreflection of uniform frames, that is, the frame counterpart of the Samuel compactification of uniform spaces (Samuel [10]), and then to use this for a new description of the completion of a uniform frame, as an alternative to those previously given by Isbell [6] on the one hand and Kříž [8] on the other. In addition, we present a few further results, as well as new proofs of known ones, that are naturally connected with completions and arise particularly easily from our approach to them. Most prominently among these, we identify the uniform space of minimal Cauchy filters of a uniform frame as the spectrum of its completion.

The main aim of this paper is to study the role of the assumptions in the following

Theorem 0. Every ideal I of codimension 1 in an F-lattice X is closed†.

For X a Banach lattice this result appears in [11], ii.5·3, corollary 3. The proof carries over to the case where X is an. F-lattice, i.e. a topological linear lattice the topology of which is metrizable and complete, with the help of a result due to Klee ([10], theorem v.5·5; see also [3], theorem 16·6). We first note that the assumption of metric completeness is essential (Example 2). We then extend Theorem 0 to the case of ideals of finite codimension (Theorem 1), and show that non-closed ideals exist whenever X is infinite-dimensional (Corollary 2). Our main result commenting on Theorem 0 is, however, the existence of dense sublattices of arbitrary codimension between 1 and in every infinite-dimensional F-lattice (Theorem 6).

This article deals with several related questions in harmonic analysis which are well understood for non-degenerate curves in ℝn, but poorly understood in the degenerate case. These questions invariably involve a positive ‘reference’ measure on the curve. In the non-degenerate case the choice of measure is not particularly critical and it is usually taken to be the Euclidean arclength measure. Since the questions considered here are invariant under the group of affine motions (of determinant 1), the correct choice of reference measure is the affine arclength measure. We refer the reader to Guggenheimer [8] for information on affine geometry. When the curve has degeneracies, the choice of measure becomes critical and it is the affine arclength measure which yields the most powerful results. From the Euclidean point of view the affine arclength measure has correspondingly little mass near the degeneracies and thus compensates automatically for the poor behaviour there. This principle should also be valid for general submanifolds of ℝn but alas the affine geometry of submanifolds is itself not well understood in general.

For 0 < α < 1, let for 0 ≤ x < 1, where is the sequence of Rademacher functions. We give a class of fα so that their graphs have Hausdorff dimension 2 − α. The result is closely related to the corresponding unsolved question for the Weierstrass functions.

In the paper [3] we obtained a lower bound for the Hausdorff dimension of certain sets of singular n-tuples in ℝn. In this paper we will obtain an upper bound for the dimension of these sets. These two bounds are not sharp enough to yield the exact dimension of all the sets considered but they do yield the dimension of the set of ‘highly singular’ (in a sense to be made precise below) n-tuples.

We consider a 2-dimensional square lattice which is partitioned into a periodic array of rectangular cells, on which a nearest neighbour random walk with symmetric increments is defined whose transition probabilities only depend on the relative position within a cell. On the basis of a determinantal identity proved in this paper, we obtain a result for finite Markov chains which shows that the diffusion constants for the random walk are monotonic functions of the individual transition probabilities. We point out the similarity of this monotonicity property to Rayleigh's Monotonicity Law for electric networks or, equivalently, reversible random walks.

In this paper, we develop an abstract formulation of a problem which arises in the investigation of the number of limit cycles of systems of the form

where p and q are homogeneous polynomials. This is part of the much wider study of Hilbert's sixteenth problem, in which information is sought about the number of limit cycles of systems of the form

where P and Q are polynomials, and their possible configurations.

A three-dimensional constitutive equation governing the flow of an isotropic rigid/perfectly plastic granular material is presented. The equation relates the strain-rate tensor to the Cauchy stress tensor and to the co-rotational rate of the Cauchy stress. It contains scalar functions of the scalar invariants involving the stress, stress-rate and strain-rate tensors together with parameters which characterize the material. The model generalizes the double-shearing model and its relationship to existing theories is demonstrated.

The Rikitake two-disc system [10] has been proposed as a model of the Earth's magnetic dipole. Numerical integration of the system by Rikitake[10] and Allan [1] showed the phenomenon of polarity reversal which is seen in terrestrial paleomagnetic data [6]. Reversals in the simulations occurred at irregular and apparently unpredictable time intervals, in accord with the capricious character of the real data, though the agreement was qualitative rather than quantitative.