We prove that there are only $O(H^{3+\epsilon})$ quartic integer polynomials with height at most $H$ and a Galois group which is a proper subgroup of $S_4$. This improves in the special case of degree four a bound by Gallagher that yielded $O(H^{7/2} \log H)$.

We study a partial differential equation on a bounded domain $\Omega\subset\mathbb{R}^N$ with a $p(x)$-growth condition in the divergence operator and we establish the existence of at least two nontrivial weak solutions in the generalized Sobolev space $W_0^{1,p(x)}(\Omega)$. Such equations have been derived as models of several physical phenomena. Our proofs rely essentially on critical point theory combined with corresponding variational techniques.

Let $R$ be a commutative Noetherian ring and $M$ a finite $R$-module. In this paper, we consider Zariski-openness of the FID-locus of $M$, namely, the subset of $\mathrm{spec}\,R$ consisting of all prime ideals ${\mathfrak p}$ such that $M_{\mathfrak p}$ has finite injective dimension as an $R_{\mathfrak p}$-module. We prove that the FID-locus of $M$ is an open subset of $\mathrm{spec}\,R$ whenever $R$ is excellent.

It is easy to imagine that a subvariety of a vector bundle, whose intersection with every fibre is a vector subspace of constant dimension, must necessarily be a sub-bundle. We give two examples to show that this is not true, and several situations in which the implication does hold. For example it is true if the base is normal and the field has characteristic zero. A convenient test is whether or not the intersections with the fibres are reduced as schemes.

It has been proved by D. E. Cohen [1] that the lattice of all varieties of metabelian groups is countable. In this paper, we show that the lattice of all varieties of completely simple semigroups with metabelian subgroups has the cardinality of the continuum. M. Petrich and N. R. Reilly have introduced in [6] the notion of near varieties of idempotent generated completely simple semigroups. The mapping assigning to every variety $\mathcal{V}$ of completely simple semigroups the class of all idempotent generated members of $\mathcal{V}$ is a complete lattice homomorphism of the lattice of all varieties of completely simple semigroups onto the lattice of all near varieties of idempotent generated completely simple semigroups. In this paper we show that, in fact, the lattice of all near varieties of idempotent generated completely simple semigroups with metabelian subgroups has itself the cardinality of the continuum.

We study the boundary value problem $\begin{array}{rcl} {\rm div}(|\n u|^{m-2}\n u) + u^av^b &=& 0\quad \mbox{ in } {\Omega}, \vspace{\jot}\\ {\rm div}(|\n v|^{m-2}\n v) + u^cv^d &=& 0 \quad \mbox{ in } {\Omega}, \vspace{\jot}\\ u =v &= & 0 \quad \mbox{ on } {\partial}{\Omega},\vspace{\jot}\\ \end{array}$ where ${\Omega}\subset\mathbb{R}^n$ ($n\ge2$) is a bounded connected smooth domain, and the exponents $m>1$ and $a,b,c,d\ge0$ are non-negative numbers. Under appropriate conditions on the exponents $m$, $a$, $b$, $c$ and $d$, a variety of results on a priori estimates and existence of positive solutions has been established.

We introduce and study a metric notion for trees and develop a fine structure theory for the corresponding class of Lipschitz trees. We also relate this structure theory to a conjecture of Shelah about the existence of a finite basis for a class of linear orderings and solve an old problem of Laver about well-quasi-ordering a certain class of trees.

Nous considérons l’intersection d’une sous-variété $X$ d’une variété abélienne $A$ avec l’union de tous les sous-groupes de $A$ obtenus comme somme d’un sous-groupe donné $\varGamma$ de rang fini et d’un sous-groupe algébrique de dimension donnée. Nous montrons que, quitte à retirer à $X$ un certain ensemble exceptionnel, l’intersection est de hauteur bornée. L’énoncé est optimal pour une courbe $X$. Nous étudions également les épaississements $\varGamma_{\varepsilon}$ introduits par Poonen. La démonstration repose sur une généralisation uniforme de la méthode de Vojta et sur des calculs de nombres d’intersection de cycles réels sur $A$.

We study the intersection of a subvariety $X$ of an abelian variety $A$ over $\bar{\mathbb{Q}}$ with the union of all the subgroups obtained as a sum of a given finite rank subgroup $\varGamma$ and an algebraic subgroup of $A$ of given dimension $d$. Our main result asserts that if we remove a suitable exceptional subset from $X$ then the intersection is a set of bounded height. In some cases, this combines with the output of Part I to yield finiteness. In terms of boundedness of the height, we get an optimal statement for a curve $X$ with $d=2$. We also deal with the fattenings $\varGamma_\varepsilon$ introduced by Poonen. The proof rests on a suitably uniform generalization of the method of Vojta and on computations of intersection numbers of real cycles on $A$.

For a stationary random closed set Ξ in ℝd it is well known that the first-order characteristics volume fraction VV, surface intensity SV and spherical contact distribution function Hs(t) are related by

(1 – VV) Hs′(0) = SV.

The aim of this article is to present a general “large deviations approach” to the geometry of polytopes spanned by random points with independent coordinates. The origin of our work is in the study of the structure of ±1-polytopes, the convex hulls of subsets of the combinatorial cube . Understanding the complexity of this class of polytopes is important for the “polyhedral combinatorics” approach to combinatorial optimization, and was put forward by Ziegler in [20]. Many natural questions regarding the behaviour of ±1-polytopes in high dimensions are open, since, for many important geometric parameters, low-dimensional intuition does not help to identify the extremal ±1-polytopes. The study of random ±1-polytopes sheds light to some of these questions, the main reason being that random behaviour is often the extremal one.

Assume that n points P1,…,Pn are distributed independently and uniformly in the triangle with vertices (0, 1), (0, 0), and (1, 0). Consider the convex hull of (0, 1), P1,…,Pn, and (1, 0). The vertices of the convex hull form a convex chain. Let be the probability that the convex chain consists – apart from the points (0, 1) and (1, 0) – of exactly k of the points P1,…,Pn. Bárány, Rote, Steiger, and Zhang [3] proved that . The values of are determined for k = 1,…,n − 1, and thus the distribution of the number of vertices of a random convex chain is obtained. Knowing this distribution provides the key to the answer of some long-standing questions in geometrical probability.

Let Ω ⊂ Rn be open. Given a homeomorphism of finite distortion with |Df| in the Lorentz space Ln−1, 1 (Ω), we show that and f−1 has finite distortion. A class of counterexamples demonstrating sharpness of the results is constructed.

It is shown that every Banach space with a norming Markushevich basis has a strong Markushevich basis. It is also shown that the converse is false.

The lower dimensional Busemann-Petty problem asks whether origin-symmetric convex bodies in ℝ n with smaller i-dimensional central sections necessarily have smaller volume. A generalization of this problem is studied, when the volumes are measured with weights satisfying certain conditions. The case of hyperplane sections (i = n − 1) has been studied by A. Zvavitch.

Towards the end of the nineteenth century, Halphen studied a remarkable sequence of higher-order linear equations with doubly periodic coefficients, generalizations of a certain Lamé equation, having the property that quotients of solutions are single valued. Here we consider further generalizations where, instead of the Weierstrass ℘-function, the coefficients depend on the first Painlevé transcendent. Using these equations, we obtain new higher-order systems of nonlinear equations having the Painlevé property. We also give new results on the interpretation of the Painlevé tests with regard to the representations of solutions, general and particular, afforded by various branches, and to understanding the corresponding pattern of compatibility conditions.

Analogues of the classical inequalities from the Brunn-Minkowski theory for rotation intertwining additive maps of convex bodies are developed. Analogues are also proved of inequalities from the dual Brunn-Minkowski theory for intertwining additive maps of star bodies. These inequalities provide generalizations of results for projection and intersection bodies. As a corollary, a new Brunn-Minkowski inequality is obtained for the volume of polar projection bodies.

The aim of the paper is to study the asymptotic behaviour of the solution of a quasilinear elliptic equation of the formwith a high-contrast discontinuous coefficient aε(x), where ε is the parameter characterizing the scale of the microstucture. The coefficient aε(x) is assumed to degenerate everywhere in the domain Ω except in a thin connected microstructure of asymptotically small measure. It is shown that the asymptotical behaviour of the solution uε as ε → 0 is described by a homogenized quasilinear equation with the coefficients calculated by local energetic characteristics of the domain Ω.

We study the contact between nonlinearly elastic bodies by variational methods. After the formulation of the mechanical problem, we provide existence results based on polyconvexity and on quasiconvexity. We then derive the Euler—Lagrange equation as a necessary condition for minimizers. Here Clarke's generalized gradients are an essential tool for treating the nonsmooth obstacle condi

This paper deals with entire solutions of a bistable reaction—diffusion equation for which the speed of the travelling wave connecting two constant stable equilibria is zero. Entire solutions which behave as two travelling fronts approaching, with super-slow speeds, from opposite directions and annihilating in a finite time are constructed by using a quasi-invariant manifold approach. Such solutions are shown to be unique up to space and time translations.

A general theorem about the existence of periodic solutions for equations with distributed delays is obtained by using the linear chain trick and geometric singular perturbation theory. Two examples are given to illustrate the application of the general the general therom.