Capillary waves on fluid sheets are computed in the presence of a uniform electric field acting horizontally with respect to the undisturbed configuration. The fluid is taken to be inviscid, incompressible and nonconducting. In previous work (Papageorgiou & Vanden-Broeck [14]) symmetric travelling waves were investigated. In this paper we show that there are in addition antisymmetric waves. These waves are calculated numerically for arbitrary amplitudes and wavelengths and the effect of the electric field is studied. The numerical procedure is based on a reformulation of the problem as a system of nonlinear integro–differential equations.

The effects of the thin air layer entering play when a water droplet impacts on otherwise still water or on a fixed solid are studied theoretically with special attention on surface tension and on post-impact behaviour. The investigation is based on the small density and viscosity ratios of the two fluids. In certain circumstances, and in particular for droplet Reynolds numbers below a critical value which is about ten million, the air-water interaction depends to leading order on lubricating forces in the air coupled with potential flow dynamics in the water. The nonlinear integro-differential system for the evolution of the interface and induced pressure is studied for pre-impact surface tension effects, which significantly delay impact, and for post-impact interaction phenomena which include significant decrease of the droplet spread rate. Above-critical Reynolds numbers are also considered.

A class of exact mathematical solutions describing distributed regions of uniform vorticity attached to two solid walls meeting at an angle $2\alpha$ is derived. Exterior to the uniform vorticity region the flow is quiescent and irrotational. The mathematical method used is a generalization of ideas original presented in Crowdy [4] combined with elements of conformal mapping theory associated with a differential equation method due to Polubarinova-Kochina traditionally applied in finding the solution to various free boundary problems.

The modeling of the motion of a contact line, the triple point at which solid, liquid and air meet, is a major outstanding problem in the fluid mechanics of thin films [2, 9]. In this paper, we compare two well-known models in the specific context of Marangoni driven films. The precursor model replaces the contact line by a sharp transition between the bulk fluid and a thin layer of fluid, effectively pre-wetting the solid; the Navier slip model replaces the usual no-slip boundary condition by a singular slip condition that is effective only very near the contact line. We restrict attention to traveling wave solutions of the thin film PDE for a film driven up an inclined planar solid surface by a thermally induced surface tension gradient. This involves analyzing third order ODE that depend on several parameters. The two models considered here have subtle differences in their description, requiring a careful treatment when comparing traveling waves and effective contact angles. Numerical results exhibit broad agreement between the two models, but the closest comparison can be done only for a rather restricted range of parameters. The driven film context gives contact angle results quite different from the case of a film moving under the action of gravity alone. The numerical technique for exploring phase portraits for the third order ODE is also used to tabulate the kinetic relation and nucleation condition, information that can be used with the underlying hyperbolic conservation law to explain the rich combination of wave structures observed in simulations of the PDE and in experiments [3, 15].

We consider the dynamics of multi-component heat-conducting viscous incompressible flow in a plane domain when the viscosity and thermal conductivity of the medium depend on temperature. The dynamics of the flow is governed by an initial-boundary value problem for the Navier–Stokes system with heat conduction and heat transfer taken into account. The existence of a generalized global solution with velocity field and temperature of Hopf's class has been established in conjunction with the estimate of fractional smoothness of the order 1/2 in the time variable.

For a large class of vorticities we prove that a steady periodic deep-water wave must be symmetric if its profile is monotone between crests and troughs.

We construct algebras of pseudodifferential operators on a continuous family groupoid $\mathcal{G}$ that are closed under holomorphic functional calculus, contain the algebra of all pseudodifferential operators of order 0 on $\mathcal{G}$ as a dense subalgebra and reflect the smooth structure of the groupoid $\mathcal{G}$, when $\mathcal{G}$ is smooth. As an application, we get a better understanding on the structure of inverses of elliptic pseudodifferential operators on classes of non-compact manifolds. For the construction of these algebras closed under holomorphic functional calculus, we develop three methods: one using semi-ideals, one using commutators and one based on Schwartz spaces on the groupoid.

One of our main results is to reduce the construction of spectrally invariant algebras of order 0 pseudodifferential operators to the analogous problem for regularizing operators. We then show that, in the case of the generalized ‘cusp’-calculi$c_n$, $n\ge2$, it is possible to construct algebras of regularizing operators that are closed under holomorphic functional calculus and consist of smooth kernels. For $n=1$, this was shown not to be possible by the first author in an earlier paper.

AMS 2000 Mathematics subject classification: Primary 35S05. Secondary 35J15; 47G30; 58J40; 46L87

Nous étudions la nature arithmétique de $q$-analogues des valeurs $\zeta(s)$ de la fonction zêta de Riemann, notamment des valeurs des fonctions $\zeta_q(s)=\sum_{k=1}^{\infty}q^k\sum_{d\mid k}d^{s-1}$, $s=1,2,\dots$, où $q$ est un nombre complexe, $|q|<1$ (ces fonctions sont intimement liées au monde automorphe). Le théorème principal de cet article montre que, si $1/q$ est un nombre entier différent de $\pm1$ et si $M$ est un nombre impair suffisamment grand, alors la dimension de l’espace vectoriel engendré sur $\mathbb{Q}$ par $1,\zeta_q(3),\zeta_q(5),\dots,\zeta_q(M)$ est au moins $c_1\sqrt{M}$, avec $c_1=0,3358$. Ce résultat peut être considéré comme un $q$-analogue du résultat de Rivoal et de Ball et Rivoal, qui affirme que la dimension de l’espace vectoriel engendré sur $\mathbb{Q}$ par $1,\zeta(3),\zeta(5),\dots,\zeta(M)$ est au moins $c_2\log M$, avec $c_2=0,5906$. Pour les mêmes valeurs de $q$, une minoration similaire pour les valeurs $\zeta_q(s)$ aux entiers $s$ pairs nous permet de redémontrer un cas particulier d’un résultat de Bertrand qui affirme la transcendance sur $\mathbb{Q}$ de l’une des deux séries d’Eisenstein $E_4(q)$ et $E_6(q)$ pour tout nombre complexe $q$ tel que $0<|q|<1$.

We formulate a number of open problems for time-harmonic inverse electromagnetic scattering theory focusing on uniqueness theorems, the determination of the support of a scattering object and the determination of material parameters

Suppose that $K$ is a subfield of $\mathbb{C}$ for which the $\ell$-adic cyclotomic character has infinite image. Suppose that $C$ is a curve of genus $g\geq3$ defined over $K$, and that $\xi$ is a $K$-rational point of $C$. This paper considers the relation between the actions of the mapping class group of the pointed topological curve $(C^{\mathrm{an}},\xi)$ and the absolute Galois group $G_K$ of $K$ on the $\ell$-adic prounipotent fundamental group of $(C^{\mathrm{an}},\xi)$. A close relationship is established between

1. the image of the absolute Galois group of $K$ in the automorphism group of the $\ell$-adic unipotent fundamental group of $C$; and

2. the $\ell$-adic Galois cohomology classes associated to the algebraic $1$-cycle $C- C^{-}$ in the Jacobian of $C$, and to the algebraic $0$-cycle $(2g-2)\xi-K_C$ in $C$.

AMS 2000 Mathematics subject classification: Primary 11G30. Secondary 14H30; 12G05; 14C25; 14G32

We study several types of curves and higher-dimensional objects inside the moduli spaces of curves, insisting on their arithmetic properties in the perspective of Grothendieck–Teichmüller theory. On the way we explicitly identify those curves which were originally associated by W. Veech with certain rational polygonal billiards.

AMS 2000 Mathematics subject classification: Primary 14D22; 32G15; 14H30

Two integral structures on the $\mathbb{Q}$-vector space of modular forms of weight two on $X_0(N)$ are compared at primes $p$ dividing $N$ at most once. When $p=2$ and $N$ is divisible by a prime that is $3$ mod $4$, this comparison leads to an algorithm for computing the space of weight one forms mod $2$ on $X_0(N/2)$. For $p$ arbitrary and $N>4$ prime to $p$, a way to compute the Hecke algebra of mod $p$ modular forms of weight one on $\varGamma_1(N)$ is presented, using forms of weight $p$, and, for $p=2$, parabolic group cohomology with mod $2$ coefficients. Appendix A is a letter of October 1987 from Mestre to Serre in which he reports on computations of weight one forms mod $2$ of prime level. Appendix B, by Wiese, reports on an implementation for $p=2$ in Magma, using Stein’s modular symbols package, with which Mestre’s computations are redone and slightly extended.

We classify the five-dimensional $C^\infty$ Anosov flows which have $C^\infty$-Anosov splitting and preserve a smooth pseudo-Riemannian metric. Up to a special time change and finite covers, such a flow is $C^\infty$ flow equivalent either to the suspension of a symplectic hyperbolic automorphism of $\mathbb{T}^{4}$, or to the geodesic flow on a three-dimensional hyperbolic manifold.

AMS 2000 Mathematics subject classification: Primary 34Cxx; 34Dxx; 37-XX

Hadwiger's well known conjecture (see the survey of Toft [9]) states that any graph $G$ has a $K_{\chi(G)}$ minor, where $\chi(G)$ is the chromatic number of $G$. Let $\alpha(G)$ denote the independence (or stability) number of $G$, namely the maximum number of pairwise nonadjacent vertices in $G$. It was observed in [1], [4], [10] that via the inequality $\chi(G)\ge {|V(G)|\over \alpha(G)}$, Hadwiger's conjecture implies

Conjecture 1.1. Any graph G on n vertices contains a $K_{\lceil {n\over \alpha(G)}\rceil}$ as a minor.

Many applications of Szemerédi's Regularity Lemma for graphs are based on the following counting result. If ${\mathcal G}$ is an $s$-partite graph with partition $V({\mathcal G}) =\bigcup_{i=1}^{s} V_i$, $\vert V_i\vert =m$ for all $i\in [s]$, and all pairs $(V_i, V_j)$, $1\leq i < j\leq s$, are $\epsilon$-regular of density $d$, then $\mathcal{G}$ contains $(1\pm f(\epsilon))d^{({s\atop 2})}m^s$ cliques $K_{s}$, provided $\epsilon<\epsilon(d)$, where $f(\epsilon)$ tends to 0 as $\epsilon$ tends to 0.

Guided by the regularity lemma for 3-uniform hypergraphs established earlier by Frankl and Rödl, Nagle and Rödl proved a corresponding counting lemma. Their proof is rather technical, mostly due to the fact that the ‘quasi-random’ hypergraph arising after application of Frankl and Rödl's regularity lemma is ‘sparse’, and consequently difficult to handle.

When the ‘quasi-random’ hypergraph is ‘dense’ Kohayakawa, Rödl and Skokan (J. Combin. Theory Ser. A 97 307–352) found a simpler proof of the counting lemma. Their result applies even to $k$-uniform hypergraphs for arbitrary $k$. While the Frankl–Rödl regularity lemma will not render the dense case, in this paper, for $k=3$, we are nevertheless able to reduce the harder, sparse case to the dense case.

Namely, we prove that a ‘dense substructure’ randomly chosen from the ‘sparse $\delta$-regular structure’ is $\delta$-regular as well. This allows us to count the number of cliques (and other subhypergraphs) using the Kohayakawa–Rödl–Skokan result, and provides an alternative proof of the counting lemma in the sparse case. Since the counting lemma in the dense case applies to $k$-uniform hypergraphs for arbitrary $k$, there is a possibility that the approach of this paper can be adopted to the general case as well.

In a one-parameter model for evolution of random trees, which also includes the Barabási–Albert random graph [1], the law of large numbers and the central limit theorem are proved for the maximal degree. In the proofs martingale methods are applied.

We derive a generalization of a theorem of Raimi proving there is a partition of natural numbers with given densities of classes which meet structured translates of any other class of a partition of natural numbers.

The lamplighter group over $\Z$ is the wreath product $\Z_q \wr \Z$. With respect to a natural generating set, its Cayley graph is the Diestel–Leader graph $\mbox{\sl DL}(q,q)$. We study harmonic functions for the ‘simple’ Laplacian on this graph and, more generally, for a class of random walks on $\mbox{\sl DL}(q,r)$, where $q,r \geq 2$. The $\mbox{\sl DL}$-graphs are horocyclic products of two trees, and we give a full description of all positive harmonic functions in terms of the boundaries of these two trees. In particular, we determine the minimal Martin boundary, that is, the set of minimal positive harmonic functions.

A detachment of a graph $G$ is formed by splitting each vertex into one or more subvertices, and sharing the incident edges arbitrarily among the subvertices. In this paper we consider the question of whether a graph $H$ is a detachment of some complete graph $K_n$. When $H$ is large and restricted to belong to certain classes of graphs, for example bounded degree planar triangle-free graphs, we obtain necessary and sufficient conditions which give a complete characterization.

A harmonious colouring of a simple graph $G$ is a proper vertex colouring such that each pair of colours appears together on at most one edge. The harmonious chromatic number $h(G)$ is the least number of colours in such a colouring. The results on detachments of complete graphs give exact results on harmonious chromatic number for many classes of graphs, as well as algorithmic results.

Let $T$ denote a real function defined on random subsets of a given family of finite sets. The random variable $T$ is decomposed into the sum of the linear, the quadratic, the cubic etc. parts which are mutually uncorrelated. Applications of this decomposition to the asymptotics of the probability distribution of $T$ (as the sizes of random subsets and of finite sets increase) are discussed.