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.

Assume $P$ is a family of primes, and let $()_P$ represent the $P$-localization functor. If $1\,{\to}\,N\stackrel{\iota}{\to} G\stackrel{\epsilon}{\to} Q\,{\to}\,1$ is an exact sequence of groups with $N$ finite, we prove that the sequence $N_P\stackrel{\iota_P}{\to} G_P\stackrel{\epsilon_P}{\to} Q_P\,{\to}\,1$ is exact. Moreover, we provide an explicit description of $\mbox{Ker}\ \iota_P$ when $Q$ belongs to a specific class of groups defined by a cohomological property. This class contains all nilpotent groups, all free groups and all $P$-local groups, as well as certain extensions formed from these three types of groups. In conclusion, we discuss the implications of our results for the study of finite-by-nilpotent groups.

Knopfmacher [13] introduced the idea of an additive arithmetic semigroup as a general setting for an algebraic analogue of number theory. Within his framework, Zhang [19] showed that the asymptotic distribution of the values taken by additive functions closely resembles that found in classical number theory, in as much as there are direct analogues of the Erdős–Wintner and Kubilius Main Theorems. In this paper, we use probabilistic arguments to show that similar theorems, and their functional counterparts, can be proved in a much wider class of decomposable combinatorial structures.

We study some fine arithmetic properties of the components of solutions of a decomposable form equation. Lower growth rates for the greatest prime factor of a component are obtained for density 1 of the solutions. Also, high pure powers are shown to occur rarely. Computations illustrate the applicability of our results; for example, to the study of units in abelian group rings.

It is proved that the exchange property, the Bass stable rank and the quasi-Bass property are all preserved under surjective inverse limits. This is then applied to multiplier rings by showing that these in many cases can be obtained as inverse limits.

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 $y^2=x^{5}+\cdots$.

We prove that the order of any subloop of a finite Moufang loop is a factor of the order of the loop, thus obtaining an analog of Lagrange's theorem for finite Moufang loops.

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 construct stably nontrivial vector bundles of dimension 5 and also stably nontrivial vector bundles of dimension 6 on real projective spaces of arbitrarily high dimensions. Consequently, we answer a problem posed by J. F. Adams.

Let $G_{\bb C}$ be the algebraic group over ${\bb C}$ corresponding a simply connected complex Lie group $G$. The algebraic cobordism $\Omega^*(G_{\bb C)$ defined by Levine and Morel is computed explicitly.

We study two systems of tangle equations that arise when modeling the action of the Integrase family of proteins on DNA. These two systems—direct and inverted repeats—correspond to two different possibilities for the initial DNA sequence. We present one new class of solutions to the tangle equations. In the case of inverted repeats we prove that any solution not in these, or 2 previously known classes, would have to belong to one specific class. In the case of direct repeats we prove that the three classes are the only solutions possible.

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.

In this paper we report on a surprising relation between the transfer operators for the congruence subgroups $\Gamma_{0}(nm),n,m\in \mathbb{N}$, and some kind of Hecke operators on the space of vector valued period functions for the groups $\Gamma_{0}(n)$. We study special eigenfunctions of the transfer operators for the groups $\Gamma_{0}(nm)$ with eigenvalues $\mp$1 which are also solutions of the Lewis equations for these groups and which are determined by eigenfunctions of the transfer operator for the congruence subgroup $\Gamma_{0}(n)$. In the language of the Atkin–Lehner theory of old and new forms one should hence call them old eigenfunctions or old solutions of the Lewis equation for $\Gamma_{0}(n)$. It turns out that certain linear combinations of the components of these old solutions for the group $\Gamma_{0}(nm)$ determine for any $m$ a solution of the Lewis equation for the group $\Gamma_{0}(n)$ and hence also an eigenfunction of the transfer operator for this group.

Our construction gives linear operators $\tilde{T}_{n}$ in the space of vector valued period functions for the group $\Gamma_{0}(n)$ which are rather similar to the Hecke operators. Indeed, in the case of the group $\Gamma_{0}(1)\,{=}\,\SL(2,\mathbb{Z})$ these operators are just the well-known Hecke operators on the space of period functions for the modular group, derived previously using the Eichler–Manin–Shimura correspondence between period polynomials and modular forms for this group, and its extension to Maass wave forms by Lewis and Zagier.

This paper presents some new results on algebraic K-theory with finite coefficients. The argument is based on a topological construction of a space $F_mK(R)$, for any ring $R$ and any integer $m\geq 2$, having the property that the ordinary homotopy theory of $F_mK(R)$ is isomorphic to the algebraic K-theory of $R$ with coefficients in $\bb {Z}/m$: $\pi_n(F_mK(R))\cong K_n(R;\bb {Z}/m)$ for $n\geq 1$. This space $F_mK(R)$ is called the $\mod m$ K-theory space of $R$. The paper is devoted to the investigation of several properties of the groups $K_n(R;\bb {Z}/m)$, for $n\in\bb {Z}$, and to some calculations of the integral homology of finite K-theory spaces.

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.

We study the dynamics of non-entire transcendental meromorphic functions with a finite asymptotic value mapped after some iterations onto a pole. This situation does not appear in the case of rational or entire functions. We consider the family of non-entire functions \[ f(z)=\frac{a\exp(z^p)+b\exp(-z^p)}{c\exp(z^p)+d\exp(-z^p)} \] with this property, i.e. there exists a finite asymptotic value $\xi$ and a positive natural number $q$ such that $f^q(\xi)\,{=}\,\infty$, where $p\,{\geq}\,1$, $a,b,c,d\,{\in}\,\mathbb{C}$. We show that the Hausdorff dimension of the set of points in the Julia set with bounded trajectory is strictly greater than one. We then impose two conditions on functions from this family. If the first condition holds, then the Lebesgue measure of the Julia set is zero, but the Hausdorff dimension is equal to two. On the other hand, if a function $f$ satisfies the second condition, then the Lebesgue measure of the Julia set is positive. Finally, we describe the measurable dynamics of these functions and prove that there does not exist an invariant measure on the Julia set absolutely continuous with respect to the Lebesgue measure and finite on compact subsets of $\mathbb{C}$.

We prove the following characterization of positive knots: a knot is positive if and only if it is homogeneous and strongly quasipositive.

Meromorphic solutions of the Schröder equation $f(sz)\,{=}\,R(f(z)),$ where $|s|\,{>}\,1$ and $R(w)$ is a rational function with $\deg[R]\,{\geq}\,2$, are studied. We will show that, if $\arg[s]\notin 2\pi {\mathbb Q}$, then $f(z)$ has any Borel direction, without exceptional values other than Picard values, which depend on $R(w)$. Further the case $\arg[s]\,{\in}\,2 \pi {\mathbb Q}$ is also considered. We investigate the relation between Julia directions of $f(z)$ and the Julia set of $R(w)$.

Given a compact, connected, oriented 3-manifold $M$ with boundary, and epimorphism $\chi$ from $H_1M$ to a free abelian group $\Pi$, two invariants $\beta$, $\tau \in \bb {Z}\Pi$ are defined. If $M$ embeds in another such 3-manifold $N$ such that $\chi_N$ factors through $\chi$, then the product $\beta\tau$ divides $\Delta_0(H_1\tilde {N})$.

A theorem of D. Krebes concerning 4-tangles embedded in links arises as a special case. Algebraic and skein theoretic generalizations for $2n$-tangles provide invariants that persist in the corresponding invariants of links in which they embed. An example is given of a virtual 4-tangle for which Krebes's theorem does not hold.