It is proved that the variety of all 4-Engel groups of exponent 4 is a maximal proper subvariety of the Burnside variety [Bfr ]4, and the consequences of this are discussed for the finite basis problem for varieties of groups of exponent 4. It is proved that, for r [ges ] 2, the 4-Engel verbal subgroup of the r-generator Burnside group B(r, 4) is irreducible as an [ ]2GL(r, 2)-module. It is observed that the variety of all 4-Engel groups of exponent 4 is insoluble, but not minimal insoluble.

The Schur algebra S(n, r) has a basis (described in [6, §2.3]) consisting of certain elements ζi,j, where i, j∈I(n, r), the set of all ordered r-tuples of elements from the set n={1, 2, …, n}. The multiplication of two such basis elements is given by a formula known as Schur's product rule. In recent years, a q-analogue Sq(n, r) of the Schur algebra has been investigated by a number of authors, particularly Dipper and James [3, 4]. The main result of the present paper, Theorem 3.6, shows how to embed the q-Schur algebra in the rth tensor power Tr(Mn) of the n×n matrix ring. This embedding allows products in the q-Schur algebra to be computed in a straightforward manner, and gives a method for generalising results on S(n, r) to Sq(n, r). In particular we shall make use of this embedding in subsequent work to prove a straightening formula in Sq(n, r) which generalises the straightening formula for codeterminants due to Woodcock [12].

We shall be working mainly with three types of algebra: the quantized enveloping algebra U(gln) corresponding to the Lie algebra gln, the q-Schur algebra Sq(n, r), and the Hecke algebra, [Hscr ](Ar−1). It is often convenient, in the case of the q-Schur algebra and the Hecke algebra, to introduce a square root of the usual parameter q which will be denoted by v, as in [5]. This corresponds to the parameter v in U(gln). We shall denote this ‘extended’ version of the q-Schur algebra by Sv(n, r), and we shall usually refer to it as the v-Schur algebra. All three algebras are associative and have multiplicative identities, and the base field will be the field of rational functions, ℚ(v), unless otherwise stated. The symbols n and r shall be reserved for the integers given in the definitions of these three algebras.

It is shown that a $\scriptstyle \overline {\mathbb Q}$-curve of genus $g$ and with stable reduction (in some generalized sense) at every finite place outside a finite set $S$ can be defined over a finite extension $L$ of its field of moduli $K$ depending only on $g$, $S$ and $K$. Furthermore, there exist $L$-models that inherit all places of good and stable reduction of the original curve (except possibly for finitely many exceptional places depending on $g$, $K$ and $S$). This descent result yields this moduli form of the Shafarevich conjecture: given $g$, $K$ and $S$ as above, only finitely many $K$-points on the moduli space ${\cal M}_g$ correspond to $\scriptstyle \overline {\mathbb Q}$-curves of genus $g$ and with good reduction outside $S$. Other applications to arithmetic geometry, like a modular generalization of the Mordell conjecture, are given.

Let [ ] denote the integer part. Among other results in [3] we gave a complete solution to the following problem.

PROBLEM. Given an increasing sequence an ∈ ℝ+, n = 1, 2, …, where an → ∞ as n → ∞, are there infinitely many primes in the sequence [αan] for almost all α?

Those $(2m-1)$-edge-colourings of a spanning subgraph of $K_{2m}$, consisting of $K_{r}$ and independent edges, that can be embedded in a $(2m-1)$-edge-colouring of $K_{2m}$ are characterised.

It is well known that the multiplicity of a complex zero ρ=β+iγ of the zeta-function is O(log[mid ]γ[mid ]). This may be proved by means of Jensen's formula, as in Titchmarsh [7, Chapter 9]. It may also be seen from the formula for the number N(T) of zeros such that 0<γ<T,

formula here

due to Backlund [1], in which E(T) is a continuous function satisfying E(T)=O(1/T) and

formula here

We assume here that T is not the ordinate of a zero; with appropriate definitions of N(T) and S(T) the formula is valid for all T. We have S(T)=O(logT). On the Lindelöf Hypothesis S(T)=o(logT), (Cramér [2]), and on the Riemann Hypothesis

formula here

(Littlewood [5]). These results are over 70 years old.

formula here

Because the multiplicity problem is hard, it seems worthwhile to see what can be said about the number of distinct zeros in a short T-interval. We obtain the following result, which is independent of any unproved hypothesis.

Let A be a regular local ring with quotient field K. Assume that 2 is invertible in A. Let W(A)→W(K) be the homomorphism induced by the inclusion A[rarrhk ]K, where W( ) denotes the Witt group of quadratic forms. If dim A[les ]4, it is known that this map is injective [6, 7]. A natural question is to characterize the image of W(A) in W(K). Let Spec1(A) be the set of prime ideals of A of height 1. For P∈Spec1(A), let πP be a parameter of the discrete valuation ring AP and k(P) = AP/PAP. For this choice of a parameter πP, one has the second residue homomorphism ∂P:W(K)→W(k(P)) [9, p. 209]. Though the homomorphism ∂P depends on the choice of the parameter πP, its kernel and cokernel do not. We have a homomorphism

formula here

A part of the so-called Gersten conjecture is the following question on ‘purity’. Is the sequence

formula here

exact? This question has an affirmative answer for dim(A)[les ]2 [1; 3, p. 277]. There have been speculations by Pardon and Barge-Sansuc-Vogel on the question of purity. However, in the literature, there is no proof for purity even for dim(A) = 3. One of the consequences of the main result of this paper is an affirmative answer to the purity question for dim(A) = 3.

We briefly outline our main result.

Le but de cet article est de démontrer une version locale de certains résultats classiques concernant l'espace de Hardy H1 et l'action sur cet espace des opérateurs de Calderón–Zygmund. Plus précisément, voici deux exemples de questions étudiées.

La première concerne l'invariance de l'espace H1 relativement aux fonctions maximales utilisées pour le définir. On sait que, si une fonction f est telle qu'une certaine fonction maximale raisonnable M1f soit intégrable dans ℝn, alors toute autre fonction maximale raisonnable M2f est également intégrable dans ℝn. Si on suppose seulement que M1f est intégrable sur une boule de ℝn, que peut-on dire de l'intégrabilitéde M2f?

La seconde concerne l'action sur l'espace H1 des opérateurs de Calderón–Zygmund. Un résultat classique de cette théorie affirme que, si T est un opérateur de Calderón–Zygmund (respectivement un opérateur de Calderón–Zygmund vérifiant la condition T*() = 0) et si f ∈ H1, alors T(f) ∈ L1 (respectivement T(f) ∈ H1). Que peut-on dire de T(f), si on suppose seulement qu'une certaine fonction maximale associée à f est intégrable sur une boule de ℝn?

La deuxième question s'est naturellement posée au cours de notre travail [3], dans lequel nous avons eu besoin de certains résultats de la Section 4 du présent article. A notre connaissance, ces résultats ne se trouvent pas dans la littérature et pour les obtenir, on doit “localiser” convenablement les méthodes utilisées classiquement pour établir les résultats globaux correspondants; nous en rappelons les grandes lignes, qui fixent le plan de notre article. La description de l'action des opérateurs de Calderón–Zygmund sur H1 est basée sur la décomposition atomique de cet espace; de façon un peu similaire, notre réponse à la deuxième question passe par le résultat de localisation de la Section 3 qui est, sans être aussi précis, dans l'esprit des décompositions atomiques. A son tour, la décomposition atomique de H1 est une conséquence du fait que cet espace puisse être défini au moyen de “grandes” fonctions maximales; parallèlement, notre résultat de localisation de la Section 3 utilise une version locale de ce résultat, qui fait l'objet de la Section 2.

This paper deals with the lack of compactness in the nonlinear elliptic problem $-\Delta u+u=|u|^{p-2}u$ in $\Omega,\ u>0$ in $\Omega,\ u=0$ on $\partial \Omega$, when $\Omega$ is un unbounded domain in $\mathbb{R}^n$ and $2<p<2n/(n-2)$.

In particular, a domain $\widetilde\Omega$ is provided where non-converging Palais–Smale sequences exist at every energy level. Nevertheless, it is proved that the problem has infinitely many solutions on $\widetilde\Omega$.

It is well known that the product of normal locally nilpotent (locally polycyclic) groups is again locally nilpotent (locally polycyclic). This is not true for locally solvable groups. We look at a class between locally polycyclic and locally solvable groups that is closed under normal products.

Every subalgebra of [Lscr ]([Hscr ]) which is isometrically weak*-homeomorphic with H∞(Ω) is reflexive, when Ω is a finitely connected region in the complex plane.

The structure of a collineation group G preserving a unital in a projective plane of order m2 with m ≡ 1(4) is investigated and several results are obtained about G, when G is a 2-group. These results are then used to investigate the structure of G/O(G) in the general case and, in particular, that of F*(G/O(G)).

The first person to consider the discrepancy of sequences of the type (αnσ)n[ges ]1 (where 0<σ<1) was H. Behnke. The subject was taken up again by one of the authors of this paper, who gave a detailed description of the discrepancy's behaviour if either 0<σ<½ or σ=½ and α2∉ℚ or σ=½ and α−2∈ℕ. In this paper, we study the case of sequences (α√n)n[ges ]1 where α>0 and α2∈ℚ. Both

formula here

are expressed as maxima of finitely many numbers which involve class numbers of imaginary quadratic fields in many cases.

The object of this paper is to extend a result of Agnew in [1] for single series to double series, and so enable Tauberian constants between matrix transformations of double sequences to be calculated using the technique that many authors have used previously for single sequences (see, for instance, [2, 3, 8]). This method (see below) gives the best possible Tauberian constants in the form of certain upper limits, and we attempt to calculate these for Cesàro summability with various known Tauberian conditions. In Section 2, we introduce our notation and review earlier work. In Section 3, we prove our main result, and we discuss applications to Cesàro summability in Section 4.

This paper is devoted to the long-time behavior of solutions to the Cauchy problem of the porous medium equation $u_t=\Delta(u^m)-u^p$ in $\mathbb{R}^n\times (0,\infty)$ with $(1-2/n)_{+}<m<1$ and the critical exponent $p=m+2/n$. For the strictly positive initial data $u(x,0)=O(1+|x|)^{-k}$ with $n+mn(2-n+nm)/(2[2-m+mn(1-m)])\leq k<2/(1-m)$, we prove that the solution of the above Cauchy problem converges to a fundamental solution of $u_t=\Delta(u^m)$ with an additional logarithmic anomalous decay exponent in time as $t\to \infty$.

It is proved that the number of countable models of a countable supersimple theory is either 1 or infinite. This result is an extension of Lachlan‘s theorem on a superstable theory.

Let K be the field of real or complex numbers. Let $(X \cong {\bf K}^{2n}, \omega)$ be a symplectic vector space and take $0 < k < n,N = ({2n \atop 2k})$. Let $L_1,\ldots,L_N \subset X$ be $2k$-dimensional linear subspaces which are in a sufficiently general position. It is shown that if $F : X \longrightarrow X$ is a linear automorphism which preserves the form $\omega^k$ on all subspaces $L_1,\ldots,L_N$, then $F$ is an $\epsilon_k$-symplectomorphism (that is, $F^*\omega = \epsilon_k\omega$, where $\epsilon_k^k = 1$). In particular, if ${\bf K} = \R$ and $k$ is odd then $F$ must be a symplectomorphism. The unitary version of this theorem is proved as well. It is also observed that the set ${\cal A}_{l,2r}$ of all $l$-dimensional linear subspaces on which the form $\omega$ has rank $\le 2r$ is linear in the Grassmannian $G(l,2n)$, that is, there is a linear subspace $L$ such that ${\cal A}_{l,2r} = L \cap G(l, 2n)$. In particular, the set ${\cal A}_{l,2r}$ can be computed effectively. Finally, the notion of symplectic volume is introduced and it is proved that it is another strong invariant.

Let $A$ be a Banach algebra and let $p$ and $q$ be two positive integers. We show that if

$A$ has a left bounded sequential approximate identity $(e_n)_{n\ge1}$ such that ${\rm lim}\,{\rm inf}_{n\to+\infty}\|e^p_n-e^{p+q}_n\| < ({p \over {p+q}})^{p\over q}{q\over{p+q}}$ then

$A$ has a left-bounded sequential identity $(f_n)_{n\ge1}$ such that $f^2_n = f_n$ for $n\ge1$. A simple example shows that the constant $({p\over {p+q}})^{p\over q}{q\over{p+q}}$ is best possible.

This result is based on some algebraic or integral formulae which associate an idempotent to elements of a Banach algebra satisfying some inequalities involving polynomials or entire functions.

This paper is devoted to the study of different types of twisting points of conformal maps. We define the sets of gyration, spiral and oscillation points and we prove, in the case that f is conformal almost nowhere, that the above sets have Hausdorff dimension one. Also we define points of bounded radial oscillation. It is proved that there are always points of π-bounded radial oscillation but there exists a conformal map without points of small bounded radial oscillation.

The paper investigates the vectorial Dirichlet problem defined by $$\begin{cases}\sigma_j(\nabla u(x))=1, \backslash, x\in\Omega \text{a.e.},\, j=1,\ldots, n u(x)=\varphi(x),\,x\in\partial\Omega. \end{cases}$$ Here $\Omega$ is an open bounded subset of $\mathbb{R}^n$ with boundary $\partial\Omega$, and $\sigma_j(A)$ ($j=1, \ldots , n$) denote the singular values of the gradient $\nabla u(x)$. The existence of solutions is established under one of the following assumptions: $\varphi: \overline{\Omega} \longrightarrow \mathbb{R}^n$ is continuous on $\overline{\Omega}$ and locally contractive on $\Omega$, or $\varphi: \partial\Omega \longrightarrow \mathbb{R}^n$ is contractive on $\partial\Omega$. This extends a result due to Dacorogna and Marcellini. The approach is based on the Baire category method developed earlier by the authors.