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

A module of a ring of differential operators $\mathcal{D}$ over a smooth surface has order $1$ if it is isomorphic to a factor module of $\mathcal{D}$ by a cyclic ideal generated by an operator of order $1$. Let $k$ be a positive integer. We give conditions under which an indecomposable $\mathcal{D}$-module of order $1$ is GK-critical of length $k$. We also give examples of indecomposable, non-critical, $\mathcal{D}$-modules whose subfactors have order $1$.

AMS 2000 Mathematics subject classification: Primary 16S32. Secondary 37F75

Let $G$ be a cyclic group of order $n$ and let $\mu = \{x_1,x_2, \dots, x_m\}$ be a sequence of elements of $G$. Let $k$ be the number of distinct values taken by the sequence $\mu$. Let $n\wedge \mu$ be the set of the $n$-subsequence sums.

We show that one of the following conditions holds:

1. $\mu$ has a value repeated $n-k+3$ times

2. $n\wedge \mu$ contains a non-null subgroup

3. $|n\wedge \mu|\geq m-n+k-2.$

We conjecture that the last condition could be improved to $|n\wedge \mu|\geq m-n+k-1$. This conjecture generalizes several known results. We also obtain a generalization of a recent result due to Bollobás and Leader.

A definition of a shock layer of thickness $\delta$ is proposed when a parabolic perturbation is applied to a scalar conservation law. The asymptotic equality $\delta\asymp\sqrt{\varepsilon}$ is established, where $\varepsilon$ denotes viscosity. This equality is proved to be optimal. Nevertheless, the equality $\delta\asymp\varepsilon$ is also proved to be valid for a class of shocks in accordance with the Mises conjecture.

AMS 2000 Mathematics subject classification: Primary 35L67. Secondary 35K65

Let G be a simple 3-connected graph with at least five vertices. Tutte [13] showed that G has at least one contractible edge. Thomassen [11] gave a simple proof of this fact and showed that contractible edges have many applications. In this paper, we show that there are at most $\frac{|V(G)|}{5}$ vertices that are not incident to contractible edges in a 3-connected graph G. This bound is best-possible. We also show that if a vertex v is not incident to any contractible edge in G, then v has at least four neighbours having degree three, and each such neighbour is incident to exactly two contractible edges. We give short proofs of several results on contractible edges in 3-connected graphs as well. We also study the contractible elements for k-connected matroids. We partially solve an open problem for regular matroids.

Let $\varOmega$ be a convex, open subset of $\mathbb{R}^n$ and let $\mathcal{D}'(\varOmega)$ be the space of distributions on $\varOmega$. It is shown that there exist linear embeddings of $\mathcal{D}'(\varOmega)$ into a differential algebra that commute with partial derivatives and that embed $\mathcal{C}^{\infty}(\varOmega)$ as a subalgebra. This embedding appears to be the first one after Colombeau’s to possess these properties. We show that many nonlinear operations on distributions can be defined that are not definable in the Colombeau setting.

AMS 2000 Mathematics subject classification: Primary 46F30. Secondary 13C11

We introduce an action of a discrete subgroup $\varGamma$ of $SL(2,\mathbb{R})^n$ on the space of pseudodifferential operators of $n$ variables, and construct a map from the space of Hilbert modular forms for $\varGamma$ to the space of pseudodifferential operators invariant under such a $\varGamma$-action, which is a lifting of the symbol map of pseudodifferential operators. We also obtain a necessary and sufficient condition for a certain type of pseudodifferential operator to be $\varGamma$-invariant.

AMS 2000 Mathematics subject classification: Primary 11F41; 35S05

In this paper we consider the existence of positive solutions to the boundary-value problems

\begin{align*} (p(t)u')'-q(t)u+\lambda f(t,u)\amp=0,\quad r\ltt\ltR, \\[2pt] au(r)-bp(r)u'(r)\amp=\sum^{m-2}_{i=1}\alpha_iu(\xi_i), \\ cu(R)+dp(R)u'(R)\amp=\sum^{m-2}_{i=1}\beta_iu(\xi_i), \end{align*}

where $\lambda$ is a positive parameter, $a,b,c,d\in[0,\infty)$, $\xi_i\in(r,R)$, $\alpha_i,\beta_i\in[0,\infty)$ (for $i\in\{1,\dots m-2\}$) are given constants satisfying some suitable conditions. Our results extend some of the existing literature on superlinear semipositone problems. The proofs are based on the fixed-point theorem in cones.

AMS 2000 Mathematics subject classification: Primary 34B10, 34B18, 34B15

The characterization theorem for the Banach-space-valued local Laplace transform established by Keyantuo, Müller and Vieten is used to obtain a real variable characterization of generators of local convoluted semigroups. The concept of local convoluted semigroups extends that of distribution as well as ultradistribution semigroups. Complete characterizations existed only for exponentially bounded semigroups integrated $\alpha$ times, whereas for the non-exponential case generation results had been obtained in terms of complex conditions only.

AMS 2000 Mathematics subject classification: Primary 47D03; 47D06; 44A10

In this paper we develop general Minkowski-type formulae for compact spacelike hypersurfaces immersed into conformally stationary spacetimes, that is, Lorentzian manifolds admitting a timelike conformal field. We apply them to the study of the umbilicity of compact spacelike hypersurfaces in terms of their $r$-mean curvatures. We derive several uniqueness results, for instance, compact spacelike hypersurfaces are umbilical if either some of their $r$-mean curvatures are linearly related or one of them is constant.

AMS 2000 Mathematics subject classification: Primary 53C42. Secondary 53B30; 53C50; 53Z05; 83C99

For a random graph on n vertices where the edges appear with individual rates, we give exact formulas for the expected time at which the number of components has gone down to k and the expected length of the corresponding minimal spanning forest.

For a random bipartite graph we give a formula for the expected time at which a k-assignment appears. This result has a bearing on the random assignment problem.

We compute the (generalized) Poincaré series of the multi-index filtration defined by a finite collection of divisorial valuations on the ring $\mathcal{O}_{\mathbb{C}^2,0}$ of germs of functions of two variables. We use the method initially elaborated by the authors and Campillo for computing the similar Poincaré series for the valuations defined by the irreducible components of a plane curve singularity. The method is essentially based on the notions of the so-called extended semigroup and of the integral with respect to the Euler characteristic over the projectivization of the space of germs of functions of two variables. The last notion is similar to (and inspired by) the notion of the motivic integration.

AMS 2000 Mathematics subject classification: Primary 14B05; 16W70

We consider the factorization properties of block monoids on $\mathbb{Z}_n$ determined by subsets of the form $S_a=\{\bar{1},\bar{a}\}$. We denote such a block monoid by $\mathcal{B}_a(n)$. In §2, we provide a method based on the division algorithm for determining the irreducible elements of $\mathcal{B}_a(n)$. Section 3 offers a method to determine the elasticity of $\mathcal{B}_a(n)$ based solely on the cross number. Section 4 applies the results of §3 to investigate the complete set of elasticities of Krull monoids with divisor class group $\mathbb{Z}_n$.

AMS 2000 Mathematics subject classification: Primary 20M14; 20D60; 13F05

We establish two characterizations of local Laplace transforms in Banach spaces. The first result follows the classic approach of Widder, while the second is in terms of vector-valued moment sequences. As a consequence, we derive characterizations of nilpotent semigroups.

AMS 2000 Mathematics subject classification: Primary 44A10; 47D03

Nous démontrons une nouvelle minoration de la hauteur normalisée d’une sous-variété algébrique d’un tore multiplicatif (et par conséquent des petits points d’une telle sous-variété). Si dans le cas torique, une preuve effective de la conjecture de Bogomolov généralisée était déjà connue ainsi que des estimations “pluri-exponentielles” en le degré de la variété (Schmidt et Bombieri–Zannier), puis monomiales inverses (par le deuxième auteur et Philippon), notre approche qui est entièrement nouvelle, permet de démontrer à un $\varepsilon$-près les conjectures les plus précises (en fonction du degré) que l’on peut formuler dans ce cadre. On obtient ainsi pour ce problème l’exact analogue de ce que l’on sait obtenir dans le cadre du problème de Lehmer. Enfin, nous démontrons pour les sous-variétés de codimension au moins 2 une conjecture du deuxième auteur et Philippon.

When individuals move together in large groups, as seen in schools of fish, they adapt their speed and direction to that of their neighbours. We present and analyse a model for the speed adaptation process in the case in which all individuals move in the same or in two opposite directions. The model consists of a hyperbolic conservation law for the density of individuals coupled to a parabolic or elliptic equation for speed. A detailed linear analysis reveals several mechanisms for the appearance of instabilities of the homogeneous steady state, which trigger the formation of schools, herds, flocks, etc. Long-term existence of weak solutions is shown using the vanishing viscosity approach.