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 $\OO(c^{-0.16} \sqrt[4]{n})$

The topology of certain weighted inductive limits of Fréchet spaces of holomorphic functions on the unit disc can be described by means of weighted sup-seminorms in case the weights are radial and satisfy certain natural assumptions due to Lusky; in the sense of Shields and Williams the weights have to be normal. It turns out that no assumption on the (double) sequence of normal weights is necessary for the topological projective description in the case of o-growth conditions. For O-growth conditions, we give a necessary and sufficient condition (in terms of associated weights) for projective description in the case of (LB)-spaces and normal weights. This last result is related to a theorem of Mattila, Saksman and Taskinen.

AMS 2000 Mathematics subject classification: Primary 46E10. Secondary 30H05; 46A13; 46M40

Let $S$ and $T$ be symmetric unbounded operators. Denote by $\overline{S+T}$ the closure of the symmetric operator $S+T$. In general, the deficiency indices of $\overline{S+T}$ are not determined by the deficiency indices of $S$ and $T$. The paper studies some sufficient conditions for the stability of the deficiency indices of a symmetric operator $S$ under self-adjoint perturbations $T$. One can associate with $S$ the largest closed $^*$-derivation $\delta_{S}$ implemented by $S$. We prove that if the unitary operators $\exp(\ri tT)$, for $t\in\mathbb{R}$, belong to the domain of $\delta_{S}$ and $\delta_{S}(\exp(\ri tT))\rightarrow0$ in the strong operator topology as $t\rightarrow0$, then the deficiency indices of $S$ and $\overline{S+T}$ coincide. In particular, this holds if $S$ and $\exp(\ri tT)$ commute or satisfy the infinitesimal Weyl relation.

We also study the case when $S$ and $T$ anticommute: $\exp(-\ri tT)S\subseteq S\exp(\ri tT)$, for $t\in\mathbb{R}$. We show that if the deficiency indices of $S$ are equal, or if the group $\{\exp(\ri tT):t\in\mathbb{R}\}$ of unitary operators has no stationary points in the deficiency space of $S$, then $S$ has a self-adjoint extension which anticommutes with $T$, the operator $S+T$ is closed and the deficiency indices of $S$ and $S+T$ coincide.

AMS 2000 Mathematics subject classification: Primary 47B25

A set $S\subset \R^d$ is $C$-Lipschitz in the$x_i$-coordinate, where $C>0$ is a real number, if, for every two points $a,b\in S$, we have $|a_i-b_i|\leq C \max\{|a_j-b_j|\sep j=1,2,\ldots,d,\,j\neq i\}$. Motivated by a problem of Laczkovich, the author asked whether every $n$-point set in $\Rd$ contains a subset of size at least $cn^{1-1/d}$ that is $C$-Lipschitz in one of the coordinates, for suitable constants $C$ and $c>0$ (depending on $d$). This was answered negatively by Alberti, Csörnyei and Preiss. Here it is observed that a combinatorial result of Ruzsa and Szemerédi implies the existence of a 2-Lipschitz subset of size $n^{1/2}\varphi(n)$ in every $n$-point set in $\R^3$, where $\varphi(n)\to\infty$ as $n\to\infty$.

In this paper we consider a bipartite version of Schütte's well-known tournament problem. A bipartite tournament $T=(A,B,E)$ with teams $A$ and $B$, and set of arcs $E$, has the property $S_{k,l}$ if for any subsets $K\subseteq A$ and $L\subseteq B$, with $|K| =k$ and $| L | =l$, there exist conquerors of $K$ and $L$ in opposite teams. The task is to estimate, for fixed $k$ and $l$, the minimum number $f(k,l)=| A | + | B | $ of players in a tournament satisfying property $S_{k,l}$. We achieve this goal by reformulating the problem in terms of intersecting set families and applying probabilistic as well as constructive methods. Intriguing connections with some famous problems of this area have emerged in this way, leading to new open questions.

In this paper we introduce two classes of operators on spaces of continuous functions with values in $F$-spaces under the action of which many functions behave chaotically near the boundary. Several examples—including onto linear operators, left and right composition operators, multiplication operators, and operators with pointwise dense range or with some stability property—are given. This new theory extends one recently developed on spaces of holomorphic functions.

AMS 2000 Mathematics subject classification: Primary 47B38. Secondary 30D40; 46E10; 54D45

An analogue of the Stein embedding theorem for $C^\infty$ manifolds endowed with two equidimensional supplementary foliations is proved.

AMS 2000 Mathematics subject classification: Primary 30G35. Secondary 16P10; 26E05; 32E10; 53C12; 57R30.

We show that the support of a (possibly) coated anisotropic medium is uniquely determined by the electric far-field patterns corresponding to incident time-harmonic electromagnetic plane waves with arbitrary polarization and direction. Our proof avoids the use of a fundamental solution to Maxwell’s equations in an anisotropic medium and instead relies on the well-posedness and regularity properties of solutions to an interior transmission problem for Maxwell’s equations.

AMS 2000 Mathematics subject classification: Primary 35R30; 35Q60. Secondary 35P25; 78A45

We discuss the problem of the uniqueness of the solution to the Cauchy problem for second-order, linear, uniformly parabolic differential equations. For most uniqueness theorems the solution must be uniformly bounded with respect to the time variable $t$, but some authors have shown an interest in relaxing the growth conditions in time.

In 1997, Chung proved that, in the case of the heat equation, uniqueness holds under the restriction: $|u(x,t)|\leq C\exp[(a/t)^{\alpha}+a|x|^2]$, for some constants $C,a>0$, $0\lt\alpha\lt1$. The proof of Chung’s theorem is based on ultradistribution theory, in particular it relies heavily on the fact that the coefficients are constants and that the solution is smooth. Therefore, his method does not work for parabolic operators with arbitrary coefficients. In this paper we prove a uniqueness theorem for uniformly parabolic equations imposing the same growth condition as Chung on the solution $u(x,t)$. At the centre of the proof are the maximum principle, Gaussian-type estimates for short cylinders and a boot-strapping argument.

AMS 2000 Mathematics subject classification: Primary 35K15