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.

We prove the existence of a family of Travelling Wave (TW) solutions for a large class of scalar reaction-diffusion equations with degenerate, nonlinear diffusion coefficients and monostable nonlinear reaction terms. We also investigate stability. Specifically, we show that, as in the linear diffusion case [6], the slowest TW in the family yields the asymptotic rate of the propagation of disturbances from the unstable rest state in these systems. In addition, we give conditions on the reaction term and diffusion coefficient ensuring the existence of interfaces.

We show that a twisted Hilbert space with an unconditional basis is isomorphic to a Hilbert space.

AMS 2000 Mathematics subject classification: Primary 46B10. Secondary 46C15

We exhibit a smooth complex affine 5-fold whose Witt group of quadratic forms does not inject into the Witt group of its function field. The dimension 5 is the smallest possible. The example depends on relating Witt groups, mod 2 Chow groups, and Steenrod operations.

AMS 2000 Mathematics subject classification: Primary 19G12. Secondary 19E15

An analytically irreducible hypersurface germ $(S,0)\subset(\bm{C}^{d+1},0)$ is quasi-ordinary if it can be defined by the vanishing of the minimal polynomial $f\in\bm{C}\{X\}[Y]$ of a fractional power series in the variables $X=(X_1,\dots,X_d)$ which has characteristic monomials, generalizing the classical Newton–Puiseux characteristic exponents of the plane-branch case ($d=1$). We prove that the set of vertices of Newton polyhedra of resultants of $f$ and $h$ with respect to the indeterminate $Y$, for those polynomials $h$ which are not divisible by $f$, is a semigroup of rank $d$, generalizing the classical semigroup appearing in the plane-branch case. We show that some of the approximate roots of the polynomial $f$ are irreducible quasi-ordinary polynomials and that, together with the coordinates $X_1,\dots,X_d$, provide a set of generators of the semigroup from which we can recover the characteristic monomials and vice versa. Finally, we prove that the semigroups corresponding to any two parametrizations of $(S,0)$ are isomorphic and that this semigroup is a complete invariant of the embedded topological type of the germ $(S,0)$ as characterized by the work of Gau and Lipman.

AMS 2000 Mathematics subject classification: Primary 14M25; 32S25

We prove the existence of a solution to a free boundary value problem arising in elastohydrodynamic lubrication. Previously, the existence was proved only under severe restrictions on the range of the physical parameters. We remove those restrictions here.