We prove short-time well-posedness of a Hele–Shaw system with two fluids and no surface tension (this is also known as the Muskat problem). We restrict our attention here to the stable case of the problem. That is, in order for the motion to be well-posed, the initial data must satisfy a sign condition which is a generalization of a condition of Saffman and Taylor. This sign condition essentially means that the more viscous fluid must displace the less viscous fluid. The proof uses the formulation introduced in the numerical work of Hou, Lowengrub, and Shelley, and relies on energy methods.

We prove the uniqueness for the inverse problem of determining a coefficient $q(x)$ in $\partial _t^2 u(x,t) = \uDelta u(x,t) - q(x)u(x,t)$ for $x \in R^n$ and $t > 0$, from observations of $u\vert_{\Gamma\times(0,T)}$ and the normal derivative $\frac{\partial u}{\partial \nu}\vert_{\Gamma\times(0,T)}$ where $\Gamma$ is an arbitrary $C^{\infty}$-hypersurface. Our main result asserts the uniqueness of $q$ over $R^n$ provided that $T > 0$ is sufficiently large and $q$ is analytic near $\Gamma$ and outside a ball. The proof depends on Fritz John's global Holmgren theorem and the uniqueness by a Carleman estimate.

We present a mathematical model describing the distribution of monomer and micellar surfactant in a steady straining flow beneath a fixed free surface. The model includes adsorption of monomer surfactant at the surface and a single-step reaction whereby $n$ monomer molecules combine to form each micelle. The equations are analysed asymptotically and numerically and the results are compared with experiments. Previous studies of such systems have often assumed equilibrium between the monomer and micellar phases, i.e. that the reaction rate is effectively infinite. Our analysis shows that such an approach inevitably fails under certain physical conditions and also cannot accurately match some experimental results. Our theory provides an improved fit with experiments and allows the reaction rates to be estimated.

The steady-state viscous quantum hydrodynamic model in one space dimension is studied. The model consists of the continuity equations for the particle and current densities, coupled to the Poisson equation for the electrostatic potential. The equations are derived from a Wigner–Fokker–Planck model and they contain a third-order quantum correction term and second-order viscous terms. The existence of classical solutions is proved for “weakly supersonic” quantum flows. This means that a smallness condition on the particle velocity is still needed but the bound is allowed to be larger than for classical subsonic flows. Furthermore, the uniqueness of solutions and various asymptotic limits (semiclassical and inviscid limits) are investigated. The proofs are based on a reformulation of the problem as a fourth-order elliptic equation by using an exponential variable transformation.

Movement of biological organisms is frequently initiated in response to a diffusible or otherwise transported signal, and in its simplest form this movement can be described by a diffusion equation with an advection term. In systems in which the signal is localized in space the question arises as to whether aggregation of a population of indirectly-interacting organisms, or localization of a single organism, is possible under suitable hypotheses on the transition rules and the production of a control species that modulates the transition rates. It has been shown [25] that continuum approximations of reinforced random walks show aggregation and even blowup, but the connections between solutions of the continuum equations and of the master equation for the corresponding lattice walk were not studied. Using variational techniques and the existence of a Lyapunov functional, we study these connections here for certain simplified versions of the model studied earlier. This is done by relating knowledge about the shape of the minimizers of a variational problem to the asymptotic spatial structure of the solution.

The imaginary powers of the Laplace operator over the circle give a $C_0$ group of bounded linear operators on $\mathsf{L}_{\theta}^p(0,2\pi)$ ($1\ltp\lt\infty$). Whereas the group is unbounded on $\mathsf{L}^4$, this paper shows that the $\mathsf{L}^4$ long-time averages of each $f$ in $\mathsf{L}^2$ are bounded. This is a Fourier restriction phenomenon.

AMS 2000 Mathematics subject classification: Primary 42A15; 42B15

This paper continue to study the interrelation and hierarchy of the spaces of operator-Lipschitz functions and the spaces of functions closed to them: commutator bounded and operator stable. It also examines various properties of symmetrically normed ideals, introduces new classes of ideals: regular and Fuglede, and investigates them.

AMS 2000 Mathematics subject classification: Primary 47A56; 47L20

For a large class of subsets $\varOmega\subset\mathbb{R}^{N}$ (including unbounded domains), we discuss the Fredholm and properness properties of second-order quasilinear elliptic operators viewed as mappings from $W^{2,p}(\varOmega;\mathbb{R}^{m})$ to $L^{p}(\varOmega;\mathbb{R}^{m})$ with $N\ltp\lt\infty$ and $m\geq1$. These operators arise in the study of elliptic systems of $m$ equations on $\varOmega$. A study in the case of a single equation ($m=1$) on $\mathbb{R}^{N}$ was carried out by Rabier and Stuart.

AMS 2000 Mathematics subject classification: Primary 35J45; 35J60. Secondary 47A53; 47F05

For a subset $E\subseteq\mathbb{R}^{d}$ and $x\in\mathbb{R}^{d}$, the local Hausdorff dimension function of $E$ at $x$ is defined by

$$ \mathrm{dim}_{\mathrm{loc}}(x,E)=\lim_{r\searrow0}\mathrm{dim}(E\cap B(x,r)), $$

where ‘dim’ denotes the Hausdorff dimension. Using some of our earlier results on so-called multifractal divergence points we give a short proof of the following result: any continuous function $f:\mathbb{R}^{d}\to[0,d]$ is the local dimension function of some set $E\subseteq\mathbb{R}^{d}$. In fact, our result also provides information about the rate at which the dimension $\mathrm{dim}(E\cap B(x,r))$ converges to $f(x)$ as $r\searrow0$.

AMS 2000 Mathematics subject classification: Primary 28A80

Suppose we are given $n$ coloured balls and an integer $k$ between 2 and $n$. How many colour-comparisons $Q(n,k)$ are needed to decide whether $k$ balls have the same colour? The corresponding problem when there is an (unknown) linear order with repetitions on the balls was solved asymptotically by Björner, Lovász and Yao, the complexity being \smash{$\theta (n\log\frac{2n}{k})$}. Here we give the exact answer for \smash{$k>\frac{n}{2}: Q(n,k)=2n-k-1$}, and the order of magnitude for arbitrary \smash{$k:Q(n,k)=\theta(\frac{n^2}{k})$}.

We give a closed formula for the Conway function of a splice in terms of the Conway function of its splice components. As corollaries, we refine and generalize results of Seifert, Torres and Sumners-Woods.

AMS 2000 Mathematics subject classification: Primary 57M25

Randomized search heuristics like evolutionary algorithms and simulated annealing find many applications, especially in situations where no full information on the problem instance is available. In order to understand how these heuristics work, it is necessary to analyse their behaviour on classes of functions. Such an analysis is performed here for the class of monotone pseudo-Boolean polynomials. Results depending on the degree and the number of terms of the polynomial are obtained. The class of monotone polynomials is of special interest since simple functions of this kind can have an image set of exponential size, improvements can increase the Hamming distance to the optimum and, in order to find a better search point, it can be necessary to search within a large plateau of search points with the same fitness value.

Finite graph homology may seem trivial, but for infinite graphs things become interesting. We present a new ‘singular’ approach that builds the cycle space of a graph not on its finite cycles but on its topological circles, the homeomorphic images of $S^1$ in the space formed by the graph together with its ends.

Our approach permits the extension to infinite graphs of standard results about finite graph homology – such as cycle–cocycle duality and Whitney's theorem, Tutte's generating theorem, MacLane's planarity criterion, the Tutte/Nash-Williams tree packing theorem – whose infinite versions would otherwise fail. A notion of end degrees motivated by these results opens up new possibilities for an ‘extremal’ branch of infinite graph theory.

Numerous open problems are suggested.

Walter Deuber died on 16th July 1999 at the age of 56 after a one-and-a-half year struggle with cancer. On 6th October 2002 he would have celebrated his 60th birthday. In order to commemorate this date several of his friends, former students and colleagues came together in the evening of this day in Berlin and started a two-day conference on Combinatorics in honour of Walter Deuber.

This paper is an extended version of the lecture given by the second author delivered at the conference ‘Combinatorics: Walter Deuber Memorial Meeting’ held on 7–8 October 2002 at Humboldt-Universität zu Berlin. Regretfully, this topic was to be the last that fascinated Walter Deuber's mind. We wrote this article in remembrance of that.

Let ${\cal T}(n,m)$ denote the set of all labelled triangle-free graphs with $n$ vertices and exactly $m$ edges. In this paper we give a short self-contained proof of the fact that there exists a constant $C>0$ such that, for all $m\geq Cn^{3/2}\sqrt{\log n}$, a graph chosen uniformly at random from ${\cal T}(n,m)$ is with probability $1-o(1)$ bipartite.

We consider several extremal problems concerning representations of graphs as distance graphs on the integers. Given a graph $G=(V,E)$, we wish to find an injective function $\phi:V\to{\mathbb Z}^+=\{1,2,\dots\}$ and a set ${\mathcal D}\subset{\mathbb Z}^+$ such that $\{u,v\}\in E$ if and only if $|\phi(u)-\phi(v)|\in{\mathcal D}$.

Let $s(n)$ be the smallest $N$ such that any graph $G$ on $n$ vertices admits a representation $(\phi_G,{\mathcal D}_G)$ such that $\phi_G(v)\leq N$ for all $v\in V(G)$. We show that $s(n)=(1+o(1))n^2$ as $n\to\infty$. In fact, if we let $s_r(n)$ be the smallest $N$ such that any $r$-regular graph $G$ on $n$ vertices admits a representation $(\phi_G,{\mathcal D}_G)$ such that $\phi_G(v)\leq N$ for all $v\in V(G)$, then $s_r(n)=(1+o(1))n^2$ as $n\to\infty$ for any $r=r(n)\gg\log n$ with $rn$ even for all $n$.

Given a graph $G=(V,E)$, let $D_{\rm e}(G)$ be the smallest possible cardinality of a set ${\mathcal D}$ for which there is some $\phi\:V\to{\mathbb Z}^+$ so that $(\phi,{\mathcal D})$ represents $G$. We show that, for almost all $n$-vertex graphs $G$, we have \begin{equation*} D_{\rm e}(G)\geq\frac{1}{2}\binom{n}{2}-(1+o(1))n^{3/2}(\log n)^{1/2}, \end{equation*} whereas for some $n$-vertex graph $G$, we have \begin{equation*} D_{\rm e}(G)\geq\binom{n}{2}-n^{3/2}(\log n)^{1/2+o(1)}.\end{equation*} Further extremal problems of similar nature are considered.

We present a programme of characterizing Ramsey classes of structures by a combination of the model theory and combinatorics. In particular, we relate the classification programme of countable homogeneous structures (of Lachlan and Cherlin) to the classification of Ramsey classes. As particular instances of this approach we characterize all Ramsey classes of graphs, tournaments and partial ordered sets. We fully characterize all monotone Ramsey classes of relational systems (of any type). We also carefully discuss the role of (admissible) orderings which lead to a new classification of Ramsey properties by means of classes of order-invariant objects.

The aim of this paper is to point to a difference between binary and hyperary structures. The modular counting functions of a class of structures defined by a sentence of second-order monadic logic with equality, based on binary relations, are ultimately periodic. However, this is not the case for sentences based on quaternary relations.