We prove that up to isomorphism $\langle a,b\,\vert\, ab=0\rangle$ is the unique principal Rees quotient of a free inverse semigroup that is not trivial or monogenic with zero, satisfying a nontrivial identity in signature with involution.

We continue the investigation of tabular algebras with trace (a certain class of associative $\mathbb{z}[v, v^{-1}]$-algebras equipped with distinguished bases) by determining the extent to which the tabular structure may be recovered from a knowledge of the structure constants. This problem is equivalent to understanding a certain category (the category of table data associated to a tabular algebra) which we introduce. The main result is that this category is equivalent to another category (the category of based posets associated to a tabular algebra) whose structure we describe explicitly.

We prove that for a negatively pinched ($-b^2\le\cK\le -1$) topologically tame 3-manifold $\skew5\tilde{M}/\Gamma$, all geometrically infinite ends are simply degenerate. And if the limit set of $\Gamma$ is the entire boundary sphere at infinity, then the action of $\Gamma$ on the boundary sphere is ergodic with respect to harmonic measure, and the Poincaré series diverges when the critical exponent is 2.

In this note, we give a new proof of the fact that an affine semiprime algebra $R$ of Gelfand-Kirillov dimension 1 satisfies a polynomial identity. Our proof uses only the growth properties of the algebra and yields an explicit upper bound for the pi degree of $R$.

In this paper we introduce simple multipliers, a special subclass of multipliers on a Banach module. We show that, from a local spectral point of view, these multipliers behave like multipliers on a commutative Banach algebra. Our definition of simple multipliers relies on the notion of point multipliers. These multipliers were studied earlier. However our approach gives new insight into this topic and therefore could be of some interest by itself.

We study Gevrey classes of holomorphic functions of several variables on a polysector, and their relation to classes of Gevrey strongly asymptotically developable functions. A new Borel-Ritt-Gevrey interpolation problem is formulated, and its solution is obtained by the construction of adequate linear continuous extension operators. Our results improve those given by Haraoka in this context, and extend to several variables the one-dimensional versions of the Borel-Ritt-Gevrey theorem given by Ramis and Thilliez, respectively. Some rigidity properties for the constructed operators are stated.

We show that locally supersoluble finitary groups over certain division rings (e.g. fields) have local systems of hypercyclic normal subgroups.

We compare the probability of generating with a given number of random elements two almost simple groups with the same socle $S$. In particular we analyse the case $S\,{=}\,{\psl}(2,p)$.

Tools from equivariant bifurcation theory are applied to the problem of the optimisation of horizontal planar arrays to minimise array noise gain in a 2D isotropic noise field. We specifically use the performance measure derived in the paper by Hayward [3], which is invariant under suitable actions of the symmetry group ${\bf S}_n\times{\bf O}(2)$, although we would expect most suitable measures to inherit such a symmetry due to the physical properties of the array. An analysis of bifurcations in the presence of this symmetry provides a list of array configurations which one would expect to (locally) optimise performance. This analysis provides a systematic way to search through $2n$-dimensional phase-space, via low dimensional searches, for solutions. We provide an example of the technique for an array with seven hydrophones and a maximum aperture size of $3m$.

The equations describing the pulsating output of a laser containing a saturable absorber are investigated numerically and analytically. The laser admits a singular Hopf bifurcation to a nearly vertical branch of periodic solutions. Using asymptotic methods, we determine a simplified problem that describes the transition from harmonic to pulsating oscillations as the bifurcation parameter is changed. This transition occurs in a layer bounded by the Hopf bifurcation point, and by a critical point near which the branch of solutions becomes vertical.

Calculations, based on exact solutions, of the viscous sintering of simple packings with unimodal and bimodal distributions of particle sizes with shrinking pores are performed. The case of square unit cells is considered in detail. It is found that, for the most part, pore shrinkage times have very weak dependence on the precise details of the pore shape and that accurate estimates of total pore shrinkage times can be obtained based only on knowledge of the initial relative density (green density) and unit cell size. An exception is found when the particle packing is loose, the enclosed pore having a large perimeter-to-area ratio. By studying two different square packings of equal-sized particles it is shown that the looser packing can exhibit dilation, rather than densification, in the early stage of sintering.

We introduce a differential model to study damage accumulation processes in the presence of chemical reactions. The influence of micro-structure leads to a nonlinear parabolic system characterized by the presence of a characteristic length. Here, we first present an analytical description of the qualitative behaviour of solutions which blow-up in finite time. Numerical simulations are given to describe the shape of solutions near the rupture time and the influence of the chemical reagents. As in the non-reactive model, the failure of the material occurs in a region of positive measure, due to the diffusive effects of the micro-structure, although some localization phenomena are observed. Moreover, if we increase the chemical concentration beyond a given threshold, which depends upon the specific conditions of the material, we observe a strong acceleration in the damage process.

We study the hole-filling problem for the porous medium equation $u_t= \frac{1}{m} \UDelta u^m$ with $m>1$ in two space dimensions. It is well known that it admits a radially symmetric self-similar focusing solution $u=t^{2\beta-1}F(|x|t^{-\beta})$, and we establish that the self-similarity exponent $\beta$ is a monotone function of the parameter $m$. We subsequently use this information to examine in detail the stability of the radial self-similar solution. We show that it is unstable for any $m>1$ against perturbations with 2-fold symmetry. In addition, we prove that as $m$ is varied there are bifurcations from the radial solution to self-similar solutions with $k$-fold symmetry for each $k=3,4,5,\dots.$ These bifurcations are simple and occur at values $m_3>m_4>m_5> \cdots\to1$.

The eddy-current problem for the time-harmonic Maxwell equations in domains and with conductors of general topology is considered. The existence of a unique magnetic field is proved for a suitable weak formulation. An equivalent strong formulation is then derived, where the conditions related to the specific geometry of the domain are made explicit. In particular, a new condition that must be satisfied by the magnetic field on the interface between a multiply-connected conductor and the non-conducting region is determined. Finally, the strong formulation of the problem for the electric field in the non-conducting region is derived, and the existence of a unique solution is proved. In conclusion, this leads to the determination of the complete set of equations describing the eddy-current problem in terms of the magnetic and the electric fields. Whether some commonly-used formulations satisfy the additional condition on the interface is also checked.

We consider the dynamic process of an elastic body in unilateral frictional contact with a rigid foundation. Friction is modelled with the Coulomb law with a coefficient that depends on the slip velocity. To allow for velocity discontinuities we use the elastodynamic (hyperbolic) framework. Nevertheless, this does not lead to a well-posed problem. To remedy this, we perturb the solution of the elastodynamic problem in a thin layer next to the contact boundary. This is a generalisation of an approach previously studied in a one-dimensional case. We establish existence and uniqueness results for the perturbed and regularised problem and provide an interpretation of this perturbation.

We prove that the separating space of an epimorphism from a Lie–Banach algebra onto the (continuous) derivation algebra $\mathfrak{Der}(A)$ of a Banach algebra $A$ consists of derivations which map into the radical of $A$.

AMS 2000 Mathematics subject classification: Primary 46H40; 46H70. Secondary 47B47; 47B48

This paper is concerned with path techniques for quantitative analysis of the logarithmic Sobolev constant on a countable set. We present new upper bounds on the logarithmic Sobolev constant, which generalize those given by Sinclair [20], in the case of the spectral gap constant involving path combinatorics. Some examples of applications are given. Then, we compare our bounds to the Hardy constant in the particular case of birth and death processes. Finally, following the approach of Rosenthal in [18], we generalize our bounds to continuous sets.

We prove that the space $L_\infty(\mu,X)$ has the same numerical index as the Banach space $X$ for every $\sigma$-finite measure $\mu$. We also show that $L_\infty(\mu,X)$ has the Daugavet property if and only if $X$ has or $\mu$ is atomless.

AMS 2000 Mathematics subject classification: Primary 46B20; 47A12

We show that the representations of the Cuntz $C^*$-algebras $\mathcal{O}_n$ which arise in wavelet analysis and dilation theory can be classified through a simple analysis of completely positive maps on finite-dimensional space. Based on this analysis, we find an application in quantum information theory; namely, a structure theorem for the fixed-point set of a unital quantum channel. We also include some open problems motivated by this work.

AMS 2000 Mathematics subject classification: Primary 46L45; 47A20; 46L60; 42C40; 81P15

In this article we introduce combinatorial multicolour discrepancies and generalize several classical results from $2$-colour discrepancy theory to $c$ colours ($c \geq 2$). We give a recursive method that constructs $c$-colourings from approximations of $2$-colour discrepancies. This method works for a large class of theorems, such as the ‘six standard deviations’ theorem of Spencer (1985), the Beck–Fiala (1981) theorem, the results of Matoušek, Wernisch and Welzl (1994) and Matoušek (1995) for bounded VC-dimension, and Matoušek and Spencer's (1996) upper bound for the arithmetic progressions. In particular, the $c$-colour discrepancy of an arbitrary hypergraph ($n$ vertices, $m$ hyperedges) is \[ \OO\Bigl(\sqrt{\tfrac n c\,\log m}\Bigr). \] If $m = \OO(n)$, then this bound improves to \[ \OO\Bigl(\sqrt{\tfrac n c\,\log c}\Bigr). \]

On the other hand there are examples showing that discrepancy in $c$ colours can not be bounded in terms of two-colour discrepancies in general, even if $c$ is a power of 2. For the linear discrepancy version of the Beck–Fiala theorem, the recursive approach also fails.

Here we extend the method of floating colours via tensor products of matrices to multicolourings, and prove multicolour versions of the Beck–Fiala theorem and the Bárány–Grinberg theorem. Using properties of the tensor product we derive a lower bound for the $c$-colour discrepancy of general hypergraphs. For the hypergraph of arithmetic progressions in $\{1, \ldots, n\}$ this yields a lower bound of $\frac{1}{25 \sqrt c} \sqrt[4]{n}$ for the discrepancy in $c$ colours. The recursive method shows an upper bound of $\OO(c^{-0.16} \sqrt[4]{n})$