We study blocks of the Iwahori–Hecke algebra ${\cal H}_q (\frak{S}_n)$ of weight two over a field of characteristic two. Using techniques and notation developed by Scopes, Richards, Chuang and Tan for the case of odd characteristic, we find the decomposition numbers and classify extensions between simple modules for these blocks.

D. Grant gave a set of defining equations and the addition formulae on the Jacobian varieties of hyperelliptic curves of the form $y^2=x^{5}+\cdots$, using the $\sigma$-function and hyperelliptic functions. In this paper, using these functions, we give the n-multiplication formulae on the Jacobian varieties of hyperelliptic curves of the form $y^2=x^{5}+\cdots$.

Let d be a Jordan derivation from a ring $\cal{A}$ into an $\cal{A}$-bimodule $\cal{M}$. Our main result shows that the restriction of d to the ideal of $\cal{A}$ generated by certain higher commutators of $\cal{A}$ is a derivation. This general statement is used for proving that under various additional conditions d must be a derivation on $\cal{A}$. Furthermore, several examples of proper Jordan derivations are given, $C^{\ast}$-algebras admitting proper additive jordan derivations are characterized, and the connections with the related problems on jordan homomorphisms and jordan $\cal{A}$-module homomorphisms are discussed.

We characterise the topological spaces which arise as the primitive ideal spaces of the Cuntz–Krieger algebras of graphs satisfying condition (K): directed graphs in which every vertex lying on a loop lies on at least two loops. We deduce that the spaces which arise as ${\rm Prim}\;C^*(E)$ are precisely the spaces which arise as the primitive ideal spaces of AF-algebras. Finally, we construct a graph $\wt{E}$ from E such that $C^*(\wt{E})$ is an AF-algebra and ${\rm Prim}\;C^*(E)$ and ${\rm Prim}\;C^*(\wt{E})$ are homeomorphic.

A categorical account is given of the Hofmann–Mislove theorem, describing the Scott open filters on a frame. The account is stable under an order duality and so is shown to also cover Bunge and Funk's constructive description of the points of the lower power locale.

The categorical axioms offered are based on a representation theorem for dcpo homomorphism between frames in terms of certain natural transformations; this allows for a categorical account to be given of dcpo homomorphisms. This specializes to give a new categorical description of the upper and lower power locale constructions.

Following Quillen's programme, one can read off a lot of information on the cohomology of a finite group $G$ by studying the restriction homomorphism from this cohomology to the cohomology of all maximal elementary abelian subgroups of $G$. This leads to a natural question on how much information on $\cal A$-generators of $H^*(G)$ one can read off from using the restriction homomorphism, where $\cal A$ denotes the steenrod algebra. In this paper, we show that the restriction homomorphism gives, in some sense, very little information on $\cal A$-generators of $H^*(G)$ at least in the important three cases, where G is either the symmetric group, the alternating group, or a certain type of iterated wreath products.

Motivated by the work of Farber, Tabachnikov and Yuzvinsky on the motion planning problem for projective spaces, we give an estimate for the topological complexity (TC) of lens spaces in terms of certain generalized “skew” maps between spheres. This last concept turns out to be closely related to that for a generalized axial map developed by Astey, Davis and the author to characterize the smallest Euclidean dimension where (2-torsion) lens spaces can be immersed. As a result, this suggests an alternative simpler “TC-approach” to the classical immersion problem for real projective spaces, whose initial stages we settle by means of techniques in obstruction theory.

We determine the Thurston–Bennequin invariant of graph divide links, which include all closed positive braids, all divide links and certain negative twist knots. As a corollary of this and a result of P. Lisca and A. I. Stipsicz, we prove that the 3-manifold obtained from $S^3$ by Dehn surgery along a non-trivial graph divide knot K with coefficient r carries positive, tight contact structures for every r except the Thurston-Bennequin invariant of K.

The concept of generalised (in the sense of Colombeau) connection on a principal fibre bundle is introduced. This definition is then used to extend results concerning the geometry of principal fibre bundles to those that only have a generalised connection. Some applications to singular solutions of Yang–Mills theory are given.

Different types of loss of stability of elastic structures are usually illustrated by simple models. This paper presents a family of structures which can demonstrate four cases of the double cusp catastrophe by the variation of a parameter. The path defined by this parameter is calculated in the diagram of the classes of the double cusp catastrophe. In one of the four cases the primary equilibrium path of the perfect structure intersects a secondary equilibrium surface at the critical point. In the other cases there are two secondary equilibrium paths, one of which belongs to critical state in all points. In all four cases the equilibrium paths of the imperfect structure are determined as well as the critical load in terms of the value of the simplest imperfections.

Multitype Moran sets are introduced in this paper. They appear naturally in the study of the structure of the quasi-crystal spectrum, and they generalize some known fractal structures such as self-similar sets, graph-direct sets and Moran sets. It is known that for any Moran set E with a bounded condition on contracting ratios, one has \[\dim_H E=s_*\le s^*=\dim_P E=\dimB E,\] where $s_*$ and $s^*$ are the lower and upper pre-dimension according to the natural coverings. For any multitype Moran set E with a bounded condition on contracting ratios, we prove $\dimB E{=}s^*$. With an additional assumption of primitivity, we prove that for multitype Moran sets, the above formula still holds. We also give some examples to show that if the condition of primitivity is not fulfilled, then it may happen that $\dim_H E<s_*$ or $\dim_P E<s^*$.

A sufficient condition is derived for a finite-time $L_2 $ singularity of the 3d incompressible Euler equations, making appropriate assumptions on eigenvalues of the Hessian of pressure. Under this condition $ \ \lim_{ t \uparrow T_*} \sup \|\frac{ D \omega} { Dt}\|_{L_2(\Omega)} = \infty $, where $\Omega \subset \mathbb{R}$ moves with the fluid. In particular, $| \omega | $, $| \S_{ij} | $, and $| \P_{ij} | $ all become unbounded at one point $(x_1, T_1) $, $T_1 $ being the first blow-up time in $L_2 $.