Let $\Delta$ denote the (2,3,7)-group. We establish an upper bound for the number of congruence subgroups of index $n$ and a lower bound for the total number of subgroups of index $n$. Since the latter grows more quickly, there exist non-congruence subgroups of index $n$ for all $n$ greater than some $n_0$.

We consider a one-dimensional weakly damped wave equation, with a damping coefficient depending on the displacement. We prove the existence of a regular connected global attractor of finite fractal dimension for the associated dynamical system, as well as the existence of an exponential attractor.

Let $R$ be a ring. An $R$-module $M$ is called a (weak) duo module provided every (direct summand) submodule of $M$ is fully invariant. It is proved that if $R$ is a commutative domain with field of fractions $K$ then a torsion-free uniform $R$-module is a duo module if and only if every element $k$ in $K$ such that $kM$ is contained in $M$ belongs to $R$. Moreover every non-zero finitely generated torsion-free duo $R$-module is uniform. In addition, if $R$ is a Dedekind domain then a torsion $R$-module is a duo module if and only if it is a weak duo module and this occurs precisely when the $P$-primary component of $M$ is uniform for every maximal ideal $P$ of $R$.

A number of integral Hopf algebras have been studied that have, as their underlying modules, the free ${\bf Z}$-module generated by finite words in a certain alphabet. For example, the tensor algebra, the rings of quasisymmetric functions and of noncommutative symmetric functions, the Solomon descent algebra, the Malvenuto-Reutenauer algebra and the homology and cohomology of $\Omega\Sigma{\bf C} P^{\infty}$ are all of this type. Some of these are known to be isomorphic or dual to each other, some are known only to be rationally isomorphic, some have been stated in the literature to be isomorphic when they are only rationally isomorphic.

This paper is, in part, an attempt to find order in this chaos of word Hopf algebras. We consider three multiplications on such modules, and their dual comultiplications, and clarify which of these operations can be combined to obtain Hopf structures. We discuss when the results are isomorphic, integrally or rationally, and study the resulting structures. We are not attempting a classification of Hopf algebras of words, merely an organization of some of the Hopf algebras of this type that have been studied in the literature.

Pointing out the difference between the Discrete Nonlinear Schrödinger equation with the classical power law nonlinearity – for which solutions exist globally, independently of the sign and the degree of the nonlinearity, the size of the initial data and the dimension of the lattice – we prove either global existence or nonexistence, in time, for the Discrete Klein-Gordon equation with the same type of nonlinearity (but of “blow-up” sign), under suitable conditions on the initial data, and sometimes on the dimension of the lattice. The results consider both the conservative and the linearly damped lattice. Similarities and differences with the continuous counterparts are remarked. We also make a short comment on the existence of excitation thresholds, for forced solutions of damped and parametrically driven Klein-Gordon lattices.

Let $G$ be a finite $p$-group, where $p$ is an odd prime number, $H$ a subgroup of $G$ and s$\theta\in \hbox{\rm Irr}(H)$ an irreducible character of $H$. Assume also that $|G:H|=p^2$. Then the character $\theta^G$ of $G$ induced by $\theta$ is either a multiple of an irreducible character of $G$, or has at least $\frac{p\,{+}\,1}{2}$ distinct irreducible constituents.

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.