In this paper we study some aspects of the integrability problem for polynomial vector fields $\dot{x}=P(x,y)$, $\skew1\dot{y}=Q(x,y)$. We analyze the possible existence of first integrals of the form $I(x,y)=e^{ h_1(x) \prod_{k=1}^r (y-a_k(x))/ \prod_{j=1}^s(y-f_j(x))} h_2(x)$$\prod_{i=1}^{\ell} (y-g_i(x))^{\alpha_i}$, where $g_i(x)$ and $f_j(x)$ are unknown particular solutions of $dy/dx=Q(x,y)/P(x,y)$, $\alpha_i \in \mathbb{C}$ are unknown constants, and $a_k(x)$, $h_1(x)$ and $h_2(x)$ are unknown functions. We give an algorithmic method to determine if the polynomial vector field has a first integral of the form above described. In the case when some of the particular solutions remain arbitrary and the other ones are explicitly determined or are functionally related to the arbitrary particular solutions, we will obtain a generalized nonlinear superposition principle, see [6]. In the case when all the particular solutions $g_i(x)$ and $f_j(x)$ are determined, they are algebraic functions and our algorithm gives an alternative method for determining such type of solutions.

The theory of generalized elliptic curves gives a moduli-theoretic compactification for modular curves when the level is a unit on the base, and the theory of Drinfeld structures on elliptic curves provides moduli schemes over the integers without a modular interpretation of the cusps. To unify these viewpoints it is natural to consider Drinfeld structures on generalized elliptic curves, but some of these resulting moduli problems have non-étale automorphism groups and so cannot be Deligne–Mumford stacks. Artin’s method as used in the work of Deligne and Rapoport rests on a technique of passage to irreducible fibers (where the geometry determines the group theory), and this does not work in the presence of non-étale level structures and non-étale automorphism groups. By making more efficient use of the group theory to bypass these difficulties, we prove that the standard moduli problems for Drinfeld structures on generalized elliptic curves are proper Artin stacks. We also analyze the local structure on these stacks and give some applications to Hecke correspondences.

A topological group $G$ is called extremely amenable if every continuous action of $G$ on a compact space has a fixed point. This concept is linked with geometry of high dimensions (concentration of measure). We show that a von Neumann algebra is approximately finite dimensional if and only if its unitary group with the strong topology is the product of an extremely amenable group with a compact group, which strengthens a result by de la Harpe. As a consequence, a $C^\ast$-algebra $A$ is nuclear if and only if the unitary group $U(A)$ with the relative weak topology is strongly amenable in the sense of Glasner. We prove that the group of automorphisms of a Lebesgue space with a non-atomic measure is extremely amenable with the weak topology and establish a similar result for groups of non-singular transformations. As a consequence, we prove extreme amenability of the groups of isometries of $L^p(0,1)$, $1\leq p<\infty$, extending a classical result of Gromov and Milman ($p=2$). We show that a measure class preserving equivalence relation $\mathcal{R}$ on a standard Borel space is amenable if and only if the full group $[\mathcal{R}]$, equipped with the uniform topology, is extremely amenable. Finally, we give natural examples of concentration to a non-trivial space in the sense of Gromov occurring in the automorphism groups of injective factors of type III.

Soit $G$ un groupe algébrique réductif sur la clôture algébrique d’un corps fini $\mathbb{F}_q$ et défini sur ce dernier. L’existence du support unipotent d’un caractère irréductible du groupe fini $G(\mathbb{F}_q)$, ou d’un faisceau caractère de $G$, a été établie dans différents cas par Lusztig, Geck et Malle, et le second auteur. Dans le présent article, nous démontrons que toute classe unipotente sur laquelle la restriction du caractère ou du faisceau caractère donné est non nulle est contenue l’adhérence de Zariski du support unipotent de ce dernier. Pour établir ce résultat, nous étudions certaines représentations des groupes de Weyl, dites “bien supportées”.

Let $G$ be a reductive algebraic group over the algebraic closure of a finite field $\mathbb{F}_q$, and defined over the latter. The existence of a unipotent support for an irreducible character of the finite group $G(\mathbb{F}_q)$, or for a character sheaf on $G$, has been established in several cases by Lusztig, Geck and Malle, and the second author. In this paper, we prove that any unipotent class on which the restriction of the given character or character sheaf does not vanish is contained in the Zariski closure of its unipotent support. In order to establish this result, we study some representations of Weyl groups, which are called ‘well supported’.

In this paper we investigate the model for the motion of a contact line over a smooth solid surface developed by Shikhmurzaev, [24]. We show that the formulation is incomplete as it stands, since the mathematical structure of the model indicates that an additional condition is required at the contact line. Recent work by Bedeaux, [4], provides this missing condition, and we examine the consequences of this for the relationship between the contact angle and contact line speed for Stokes flow, using asymptotic methods to investigate the case of small capillary number, and a boundary integral method to find the solution for general capillary number, which allows us to include the effect of viscous bending. We compare the theory with experimental data from a plunging tape experiment with water/glycerol mixtures of varying viscosities [11]. We find that we are able to obtain a reasonable fit using Shikhmurzaev's model, but that it remains unclear whether the linearized surface thermodynamics that underlies the theory provide an adequate description for the motion of a contact line.

We study the propagation of a crack in critical equilibrium for a brittle material in a Mode III field. The energy variations for small virtual extensions of the crack are handled in a novel way: the amount of energy released is written as a functional over a family of univalent functions on the upper half plane. Classical techniques developed in connection to the Bieberbach Conjecture are used to quantify the energy-shape relationship. By means of a special family of trial paths generated by the so-called Löwner equation we impose a stability condition on the field which derives in a local crack propagation criterion. We called this the anti-symmetry principle, being closely related to the well known symmetry principle for the in-plane fields.

Rational solutions and rational-oscillatory solutions of the defocusing nonlinear Schrödinger equation are expressed in terms of special polynomials associated with rational solutions of the fourth Painlevé equation. The roots of these special polynomials have a regular, symmetric structure in the complex plane. The rational solutions verify results of Nakamura and Hirota [J. Phys. Soc. Japan, 54 (1985) 491–499] whilst the rational-oscillatory solutions appear to be new solutions of the defocusing nonlinear Schrödinger equation.

Soient $\bm{G}$ un groupe réductif connexe sur un corps local non archimédien $F$ et $\bm{H}$ un groupe endoscopique de $\bm{G}$. On suppose $\bm{G}$ et $\bm{H}$ non ramifiés. Le {\og}lemme fondamental{\fg} affirme l’égalité de certaines sommes pondérées d’intégrales orbitales sur $\bm{G}(F)$ et sur $\bm{H}(F)$. On peut descendre cette assertion en un {\og}lemme fondamental pour les algèbres de Lie{\fg} affirmant l’égalité de sommes pondérées analogues sur $\bm{\mathfrak{g}}(F)$ et sur $\bm{\mathfrak{h}}(F)$, où $\bm{\mathfrak{g}}$ et $\bm{\mathfrak{h}}$ sont les algèbres de Lie de $\bm{G}$ et $\bm{H}$. Cette assertion est cruciale pour la théorie de l’endoscopie de Langlands. D’importants cas particuliers ont été prouvés récemment, sous l’hypothèse que la caractéristique de $F$ est positive. Nous donnons un sens précis à l’assertion suivante, et nous la prouvons : soient $F$ et $F'$ deux corps locaux de même corps résiduel $\mathbb{F}_{q}$ et de caractéristique résiduelle $p$ assez grande ; supposons le lemme fondamental (pour les algèbres de Lie) vrai sur le corps de base $F$ ; alors ce lemme est vrai sur le corps de base $F'$. Cela permet de relever ce lemme de la caractéristique positive à la caractéristique nulle. Une grande partie de l’article est consacrée à reformuler des théories bien connues (endoscopie, immeubles, réseaux de Moy–Prasad) de sorte que $F$ n’y intervienne que via le corps résiduel $\mathbb{F}_{q}$.

Let $\bm{G}$ be a connected reductive group over a non-archimedean local field $F$ and let $\bm{H}$ be an endoscopic group of $\bm{G}$. We suppose that $\bm{G}$ and $\bm{H}$ are unramified. The fundamental lemma asserts an equality between certain linear combinations of integral orbitals over $\bm{G}(F)$ and $\bm{H}(F)$. We can translate this assertion in a ‘fundamental lemma for Lie algebras’, that is, a conjectural equality between linear combinations of integral orbitals over $\bm{\mathfrak{g}}(F)$ and $\bm{\mathfrak{h}}(F)$, where $\bm{\mathfrak{g}}$ and $\bm{\mathfrak{h}}$ are the Lie algebras of $\bm{G}$ and $\bm{H}$. Important particular cases of this lemma were recently proved, assuming the characteristic of $F$ to be positive. We give a precise meaning to the following assertion and we prove it: let $F$ and $F'$ be two local fields with the same residue field $\mathbb{F}_{q}$ and with ‘big’ residual characteristic; suppose that the fundamental lemma (for Lie algebras) is true over $F$; then it is true over $F'$. In particular, we can lift the lemma to $0$-characteristic if it is known for positive characteristic. A large part of the article reformulates well-known constructions (endoscopy, buildings, Moy–Prasad filtrations) so that $F$ appears only via his residual field $\mathbb{F}_{q}$.

It is still an open question whether a compact embedded hypersurface in the Euclidean space $\mathbb{R}^{n+1}$ with constant mean curvature and spherical boundary is necessarily a hyperplanar ball or a spherical cap, even in the simplest case of surfaces in $\mathbb{R}^3$. In a recent paper, Alías and Malacarne (Rev. Mat. Iberoamericana18 (2002), 431–442) have shown that this is true for the case of hypersurfaces in $\mathbb{R}^{n+1}$ with constant scalar curvature, and more generally, hypersurfaces with constant higher-order $r$-mean curvature, when $r\geq2$. In this paper we deal with some aspects of the classical problem above, by considering it in a more general context. Specifically, our starting general ambient space is an orientable Riemannian manifold $\bar{M}$, where we will consider a general geometric configuration consisting of an immersed hypersurface into $\bar{M}$ with boundary on an oriented hypersurface $P$ of $\bar{M}$. For such a geometric configuration, we study the relationship between the geometry of the hypersurface along its boundary and the geometry of its boundary as a hypersurface of $P$, as well as the geometry of $P$ as a hypersurface of $\bar{M}$. Our approach allows us to derive, among others, interesting results for the case where the ambient space has constant curvature (the Euclidean space $\mathbb{R}^{n+1}$, the hyperbolic space $\mathbb{H}^{n+1}$, and the sphere $\mathbb{S}^{n+1}$). In particular, we are able to extend the symmetry results given in the recent paper mentioned above to the case of hypersurfaces with constant higher-order $r$-mean curvature in the hyperbolic space and in the sphere.

The usual definition of average degree for a non-regular lattice has the disadvantage that it takes the same value for many lattices with clearly different connectivity. We introduce an alternative definition of average degree, which better separates different lattices.

These measures are compared on a class of lattices and are analysed using a Markov chain describing a random walk on the lattice. Using the new measure, we conjecture the order of both the critical probabilities for bond percolation and the connective constants for self-avoiding walks on these lattices.

Let $s$ and $t$ be integers satisfying $s \geq 2$ and $t \geq 2$. Let $S$ be a tree of size $s$, and let $P_t$ be the path of length $t$. We show in this paper that, for every edge-colouring of the complete graph on $n$ vertices, where $n=224(s-1)^2t$, there is either a monochromatic copy of $S$ or a rainbow copy of $P_t$. So, in particular, the number of vertices needed grows only linearly in $t$.

For any partition of $\{1, 2, \ldots{,}\, n\}$ we define its increments$X_i, 1 \leq i \leq n$ by $X_i = 1$ if $i$ is the smallest element in the partition block that contains it, $X_i = 0$ otherwise. We prove that for partially exchangeable random partitions (where the probability of a partition depends only on its block sizes in order of appearance), the law of the increments uniquely determines the law of the partition. One consequence is that the Chinese Restaurant Process CRP($\theta$) (the partition with distribution given by the Ewens sampling formula with parameter $\theta$) is the only exchangeable random partition with independent increments.

We consider the problem of reorienting an oriented matroid so that all its cocircuits are ‘as balanced as possible in ratio’. It is well known that any oriented matroid having no coloops has a totally cyclic reorientation, a reorientation in which every signed cocircuit $B = \{B^+, B^-\}$ satisfies $B^+, B^- \neq \emptyset$. We show that, for some reorientation, every signed cocircuit satisfies \[1/f(r) \leq |B^+|/|B^-| \leq f(r)\], where $f(r) \leq 14\,r^2\ln(r)$, and $r$ is the rank of the oriented matroid.

In geometry, this problem corresponds to bounding the discrepancies (in ratio) that occur among the Radon partitions of a dependent set of vectors. For graphs, this result corresponds to bounding the chromatic number of a connected graph by a function of its Betti number (corank) $|E|-|V|+1$.

A dominating set $\cal D$ of a graph $G$ is a subset of $V(G)$ such that, for every vertex $v\in V(G)$, either in $v\in {\cal D}$ or there exists a vertex $u \in {\cal D}$ that is adjacent to $v$. We are interested in finding dominating sets of small cardinality. A dominating set $\cal I$ of a graph $G$ is said to be independent if no two vertices of ${\cal I}$ are connected by an edge of $G$. The size of a smallest independent dominating set of a graph $G$ is the independent domination number of $G$. In this paper we present upper bounds on the independent domination number of random regular graphs. This is achieved by analysing the performance of a randomized greedy algorithm on random regular graphs using differential equations.

In this paper, we study percolation on finite Cayley graphs. A conjecture of Benjamini says that the critical percolation $p_c$ of any vertex-transitive graph satisfying a certain diameter condition can be bounded away from one. We prove Benjamini's conjecture for some special classes of Cayley graphs. We also establish a reduction theorem, which allows us to build Cayley graphs for large groups without increasing $p_c$.

Let $c$ be a constant and $(e_1,f_1), (e_2,f_2), \dots, (e_{cn},f_{cn})$ be a sequence of ordered pairs of edges on vertex set $[n]$ chosen uniformly and independently at random. Let $A$ be an algorithm for the on-line choice of one edge from each presented pair, and for $i= 1,\hellip,cn$ let $G_A(i)$ be the graph on vertex set $[n]$ consisting of the first $i$ edges chosen by $A$. We prove that all algorithms in a certain class have a critical value $c_A$ for the emergence of a giant component in $G_A(cn) (ie$, if $c \gt c_A$, then with high probability the largest component in $G_A(cn)$ has $o(n)$ vertices, and if $c > c_A$ then with high probability there is a component of size $\Omega(n)$ in $G_A(cn))$. We show that a particular algorithm in this class with high probability produces a giant component before $0.385 n$ steps in the process ($ie$, we exhibit an algorithm that creates a giant component relatively quickly). The fact that another specific algorithm that is in this class has a critical value resolves a conjecture of Spencer.

In addition, we establish a lower bound on the time of emergence of a giant component in any process produced by an on-line algorithm and show that there is a phase transition for the off-line version of the problem of creating a giant component.

A 3-connected graph $G$ is weakly 3-connected if, for every edge $e$ of $G$, at most one of $G\backslash e$ and $G/e$ is 3-connected. The main result of this paper is that any weakly 3-connected graph can be reduced to $K_4$ by a sequence of simple operations. This extends a result of Dawes [5] on minimally 3-connected graphs.

The notion of conductance introduced by Jerrum and Sinclair [8] has been widely used to prove rapid mixing of Markov chains. Here we introduce a bound that extends this in two directions. First, instead of measuring the conductance of the worst subset of states, we bound the mixing time by a formula that can be thought of as a weighted average of the Jerrum–Sinclair bound (where the average is taken over subsets of states with different sizes). Furthermore, instead of just the conductance, which in graph theory terms measures edge expansion, we also take into account node expansion. Our bound is related to the logarithmic Sobolev inequalities, but it appears to be more flexible and easier to compute.

In the case of random walks in convex bodies, we show that this new bound is better than the known bounds for the worst case. This saves a factor of $O(n)$ in the mixing time bound, which is incurred in all proofs as a ‘penalty’ for a ‘bad start’. We show that in a convex body in $\mathbb{R}^n$, with diameter $D$, random walk with steps in a ball with radius $\delta$ mixes in $O^*(nD^2/\delta^2)$ time (if idle steps at the boundary are not counted). This gives an $O^*(n^3)$ sampling algorithm after appropriate preprocessing, improving the previous bound of $O^*(n^4)$.

The application of the general conductance bound in the geometric setting depends on an improved isoperimetric inequality for convex bodies.

The problem we consider here is based on the following question posed by François Jaeger in 1981 at a combinatorics meeting in Eger (Hungary):