The notion of \emph{hyperdecidability} has been introduced by the first author as a tool to prove decidability of semidirect products of pseudovarieties of semigroups. In this paper we consider some stronger notions which lead to improved decidability results allowing us in turn to establish the decidability of some iterated semidirect products. Roughly speaking, the decidability of a semidirect product follows from a mild, commonly verified property of the first factor plus the stronger property for all the other factors. A key role in this study is played by intermediate free semigroups (relatively free objects of expanded type lying between relatively free and relatively free profinite objects). As an application of the main results, the decidability of the Krohn--Rhodes (group) complexity is shown to follow from non-algorithmic abstract properties likely to be satisfied by the pseudovariety of all finite aperiodic semigroups, thereby suggesting a new approach to answer (affirmatively) the question as to whether complexity is decidable.

1991 Mathematics Subject Classification: primary 20M05, 20M07, 20M35; secondary 08B20.

The energy minimization problem associated to uniform, isotropic, linearly elastic rods leads to a geometric variational problem for the rod centreline, whose solutions include closed, knotted curves. We give a complete description of the space of closed and quasiperiodic solutions.

The quasiperiodic curves are parametrized by a two-dimensional disc. The closed curves arise as a countable collection of one-parameter families, connecting the $m$-fold covered circle to the $n$-fold covered circle for any relatively prime $m$ and $n$. Each family contains exactly one self-intersecting curve, one elastic curve, and one closed curve of constant torsion. Two torus knot types are represented in each family, and all torus knots are represented by elastic rod centrelines.

1991 Mathematics Subject Classification: primary 53A04, 73C02; secondary 57M25.

We consider the stationary non-linear Schrödinger equation\begin{equation*}\Delta u + \{1 + \lambda g(x)\} u = f(u)\mbox{with}u \in H^{1} (\mathbb{R}^{N}), u \not\equiv 0,\end{equation*} where $\lambda >0$ and the functions $f$ and $g$ are such that\begin{equation*} \lim_{s \rightarrow 0}\frac{f(s)}{s} = 0 \mbox{and} 1 < \alpha + 1 = \lim _{|s| \rightarrow \infty}\frac{f(s)}{s} < \infty\end{equation*} and \begin{equation*} g(x)\equiv 0 \mbox{on} \bar{\Omega}, g(x)\in (0, 1] \mbox{on} {\mathbb{R}^{N}} \setminus {\overline{\Omega}} \mbox{and} \lim_{|x| \rightarrow + \infty} g(x) = 1 \end{equation*} for some bounded open set $\Omega \in \mathbb{R}^{N}$. We use topological methods to establish the existence of two connected sets $\mathcal{D}^{\pm}$ of positive/negative solutions in $\mathbb{R} \times W^{2, p} (\mathbb{R}^{N})$ where $p \in [2, \infty) \cap (\frac{N}{2},\infty)$ that cover the interval $(\alpha,\Lambda(\alpha))$ in the sense that \begin{align*} P \mathcal{D}^{\pm} & = (\alpha, \Lambda(\alpha)) \text{where}P(\lambda, u) = \lambda \text{and furthermore,} \\ \lim_{\lambda \rightarrow \Lambda(\alpha)-}\left\Vert u_{\lambda} \right\Vert _{L^{\infty} (\mathbb{R}^{N})} & = \lim_{\lambda \rightarrow \Lambda (\alpha )-} \left\Vert u_{\lambda} \right\Vert _{W^{2, p}(\mathbb{R}^{N})} = \infty \text{ for }(\lambda, u_{\lambda}) \in \mathcal{D}^{\pm}. \end{align*} The number $\Lambda(\alpha)$ is characterized as the unique value of $\lambda$ in the interval $(\alpha, \infty)$ for which the asymptotic linearization has a positive eigenfunction. Our work uses a degree for Fredholm maps of index zero.

For quantized irreducible flag manifolds the locally finite part of the dual coalgebra is shown to coincide with a natural quotient coalgebra $\overline{U}$ of $U_q ( \mathfrak{g} )$. On the way the coradical filtration of $\overline{U}$ is determined. A graded version of the duality between $\overline{U}$ and the quantized coordinate ring is established. This leads to a natural construction of several examples of quantized vector spaces.

As an application, covariant first order differential calculi on quantized irreducible flag manifolds are classified.

We develop a notion of Morita equivalence for general C$^{\ast}$-correspondences over C$^{\ast}$-algebras. We show that if two correspondences are Morita equivalent, then the tensor algebras built from them are strongly Morita equivalent in the sense developed by Blecher, Muhly and Paulsen. Also, the Toeplitz algebras are strongly Morita equivalent in the sense of Rieffel, as are the Cuntz--Pimsner algebras. Conversely, if the tensor algebras are strongly Morita equivalent, and if the correspondences are aperiodic in a fashion that generalizes the notion of aperiodicity for automorphisms of C$^{\ast}$-algebras, then the correspondences are Morita equivalent. This generalizes a venerated theorem of Arveson on algebraic conjugacy invariants for ergodic, measure-preserving transformations. The notion of aperiodicity, which also generalizes the concept of full Connes spectrum for automorphisms, is explored; its role in the ideal theory of tensor algebras and in the theory of their automorphisms is investigated. 1991 Mathematics Subject Classification: 46H10, 46H20, 46H99, 46M99, 47D15, 47D25.

We describe an algorithm to determine whether a matrix group over a finite field, generated by a given set of matrices, contains one of the classical groups or the special linear group. The algorithm is based on finding elements with certain properties by independent random selection from the group. Very few matrix groups other than the classical groups contain subsets of elements with the required properties, and a classification of such matrix groups is the principal theoretical basis of the algorithm. The classification, which is proved here, builds on work of Guralnick, Penttila, Saxl and the second author, and depends on the finite simple group classification. Moreover, probability estimates for finding such elements in classical groups are derived, and these are used to give a theoretical analysis of the performance of the algorithm. The algorithm has been implemented in the computer systems {\sf GAP} and{\scMagma}.

1991 Mathematics Subject Classification: primary 20G40, 11E57, 20-04, 60B99; secondary 15A30.

We consider the model theory of the real and complex fields with a multiplicative group having the Mann property. Among these groups are the finitely generated multiplicative groups in these fields. As a by-product we obtain some results on groups with the Mann property in rings of Witt vectors and in fields of positive characteristic.k

We introduce orbifold Euler numbers for normal surfaces with boundary $\mathbb{Q}$-divisors. These numbers behave multiplicatively under finite maps and in the log canonical case we prove that they satisfy the Bogomolov–Miyaoka–Yau type inequality. Existence of such a generalization was earlier conjectured by G. Megyesi [Proc. London Math. Soc. (3) 78 (1999) 241–282]. Most of the paper is devoted to properties of local orbifold Euler numbers and to their computation.

As a first application we show that our results imply a generalized version of R. Holzapfel's ‘proportionality theorem’ [Ball and surface arithmetics, Aspects of Mathematics E29 (Vieweg, Braunschweig, 1998)]. Then we show a simple proof of a necessary condition for the logarithmic comparison theorem which recovers an earlier result by F. Calderón-Moreno, F. Castro-Jiménez, D. Mond and L. Narváez-Macarro [Comment. Math. Helv. 77 (2002) 24–38].

Then we prove effective versions of Bogomolov's result on boundedness of rational curves in some surfaces of general type (conjectured by G. Tian [Springer Lecture Notes in Mathematics 1646 (1996) 143–185)]. Finally, we give some applications to singularities of plane curves; for example, we improve F. Hirzebruch's bound on the maximal number of cusps of a plane curve.

2000 Mathematical Subject Classification: 14J17, 14J29, 14C17.

The low-dimensional projective irreducible representations in cross characteristics of the projective special linear group $\mbox{PSL}_{n}(q)$ are investigated.

If $n \geq 3$ and $(n,q) \neq (3,2)$, $(3,4)$, $(4,2)$, $(4,3)$, all such representations of the first degree (which is $(q^{n}-q)/(q-1) - \kappa$ with $\kappa = 0$ or $1$) and the second degree (which is $(q^{n}-1)/(q-1)$) come from Weil representations. We show that the gap between the second and the third degree is roughly $q^{2n-4}$.

We provide an analogue of Gundy's decomposition for $L_1$-bounded non-commutative martingales. An important difference from the classical case is that for any $L_1$-bounded non-commutative martingale, the decomposition consists of four martingales. This is strongly related with the row/column nature of non-commutative Hardy spaces of martingales. As applications, we obtain simpler proofs of the weak type $(1,1)$ boundedness for non-commutative martingale transforms and the non-commutative analogue of Burkholder's weak type inequality for square functions. A sequence $(x_n)_{n \ge 1}$ in a normed space $\mathrm{X}$ is called 2-co-lacunary if there exists a bounded linear map from the closed linear span of $(x_n)_{n \ge 1}$ to $l_2$ taking each $x_n$ to the $n$th vector basis of $l_2$. We prove (using our decomposition) that any relatively weakly compact martingale difference sequence in $L_1 (\mathcal{M}, \tau)$ whose sequence of norms is bounded away from zero is 2-co-lacunary, generalizing a result of Aldous and Fremlin to non-commutative $L_1$-spaces.

We study modules for the general linear group (over an infinite field of arbitrary characteristic) which are direct summands of tensor products of exterior powers and symmetric powers of the natural module. These modules, which we call listing modules, include the tilting modules and the injective modules for Schur algebras. The modules are studied via their relationship to linear source modules for symmetric groups on the one hand, and simple modules for Schur superalgebras on the other. Listing modules are parametrized by certain pairs of partitions. They are used to describe, by generators and relations, the Grothendieck ring of polynomial functors generated by the symmetric and exterior powers. We also (continuing work of J. Grabmeier) describe the vertices and sources of linear source modules for symmetric groups. 2000 Mathematical Subject Classification: 20G05, 20C30.

Assuming the Riemann Hypothesis, Montgomery showed by means of his pair correlation method that at least two-thirds of the zeros of Riemann's zeta-function are simple. Later he and Taylor improved this to 67.25 percent and, more recently, Cheer and Goldston increased the percentage to 67.2753. Here we prove by a new method that if the Riemann and Generalized Lindel\"of Hypotheses hold, then at least 70.3704 percent of the zeros are simple and at least 84.5679 percent are distinct. Our method uses mean value estimates for various functions defined by Dirichlet series sampled at the zeros of the Riemann zeta-function.

1991 Mathematics Subject Classification: 11M26.

In this paper we study the eigenvalues and eigenfunctions of metric measure manifolds. We prove that any eigenfunction is $C^{1,\alpha}$ at its critical points and $C^{\infty}$ elsewhere. Moreover, the eigenfunction corresponding to the first eigenvalue in the Dirichlet problem does not change sign. We also discuss the first eigenvalue, the Sobolev constants and their relationship with the isoperimetric constants. 2000 Mathematics Subject Classification: 47J05, 47J10, 53C60, 58E05, 58C40.

The summatory function of the Möbius function is denoted $M(x)$. In this article we deduce conditional results concerning $M(x)$ assuming the Riemann hypothesis and a conjecture of Gonek and Hejhal on the negative moments of the Riemann zeta function. Assuming these conjectures, we show that $M(x)$, when appropriately normalized, possesses a limiting distribution, and also that a strong form of the weak Mertens conjecture is true. Finally, we speculate on the lower order of $M(x)$ by studying the constructed distribution function.

We give several sufficient conditions on a pair of Banach spaces X and Y under which each Lipschitz mapping from a domain in X to Y has, for every $\epsilon > 0$, a point of $\epsilon$-Fréchet differentiability. Most of these conditions are stated in terms of the moduli of asymptotic smoothness and convexity, notions which have appeared in the literature under a variety of names. We prove, for example, that for $\infty > r > p \ge 1$, every Lipschitz mapping from a domain in an $\ell_r$-sum of finite-dimensional spaces into an $\ell_p$-sum of finite-dimensional spaces has, for every $\epsilon > 0$, a point of $\epsilon$-Fréchet differentiability, and that every Lipschitz mapping from an asymptotically uniformly smooth space to a finite-dimensional space has such points. The latter result improves, with a simpler proof, an earlier result of the second and third authors. We also survey some of the known results on the notions of asymptotic smoothness and convexity, prove some new properties, and present some new proofs of existing results.

2000 Mathematical Subject Classification: 46G05, 46T20.

A vector $x$ in a Banach space $\cal B$ is called hypercyclic for a bounded linear operator $T: \cal B \rightarrow \cal B $ if the orbit $\{T^n x : n \geq 1 \}$ is dense in $\cal B$. We prove that if $T$ is hereditarily hypercyclic and its essential spectrum meets the closed unit disk, then there is an infinite-dimensional closed subspace consisting, except for zero, entirely of hypercyclic vectors for $T$. The converse is also true even if $T$ is not hereditarily hypercyclic. In this way, we improve and extend to Banach spaces a recent result for Hilbert spaces. We also investigate the corresponding problem for supercyclic operators. In this case we obtain a description of the norm closure of the class of all supercyclic operators that have an infinite-dimensional closed subspace of hypercyclic vectors. Moreover, for certain kinds of supercyclic operators we can characterize when they have an infinite-dimensional closed subspace of supercyclic vectors. 1991 Mathematics Subject Classification: 47B38, 47B99.

For $K$ a connected finite complex and $G$ a compact connected Lie group, a finiteness result is proved for gauge groups ${\mathcal G}(P)$ of principal $G$-bundles $P$ over $K$: as $P$ ranges over all principal $G$-bundles with base $K$, the number of homotopy types of ${\mathcal G}(P)$ is finite; indeed this remains true when these gauge groups are classified by $H$-equivalence, that is, homotopy equivalences which respect multiplication up to homotopy. A case study is given for $K = S^4$, $G = \text{SU}(2)$: there are eighteen $H$-equivalence classes of gauge group in this case. These questions are studied via fibre homotopy theory of bundles of groups; the calculations in the case study involve $K$-theories and $e$-invariants. 1991 Mathematics Subject Classification: 54C35, 55P15, 55R10.

Let ${\cal M} =\langle M,<,\ldots\rangle$ be alinearly ordered structure. We define ${\cal M}$ to be{\em o-minimal} if every definable subset of $M$ is a finite union of intervals. Classical examples are ordered divisible abelian groups and real closed fields. We prove a trichotomy theorem for the structure that an arbitraryo-minimal ${\cal M}$ can induce on a neighbourhood of any $a$in $M$. Roughly said, one of the following holds: \begin{enumerate}\item[(i)] $a$ is trivial (technical term), {\em or}\item[(ii)] $a$ has a convex neighbourhood on which ${\cal M}$ induces the structure of an ordered vector space, {\em or}\item[(iii)] $a$ is contained in an open interval on which ${\cal M}$ induces the structure of an expansion of a real closed field. \end{enumerate} The proof uses ‘geometric calculus’ which allows one to recover a differentiable structure by purely geometric methods.

1991 Mathematics Subject Classification: primary 03C45; secondary 03C52, 12J15, 14P10.