A family of planar discrete electrostatic systems on the unit circle with finitely atomic external fields is considered. The geometry of particles in the external field yielding a given minimum energy configuration is studied. As an application, the wandering vectors of the shift operator in the Dirichlet spaces associated with finitely atomic measures are also studied. In particular, the zero locus of a wandering vector is discussed.

Let G be a locally finite group of cardinality ℵn where n is a natural number. Let π(G) be the set of primes p for which G has an element of order p. In [5], Holt conjectures that if k is a finite field with char k ∉ π(G) then

(1) G has cohomological dimension n+1 over k;

(2) Hn+1(G, kG) has cardinality 2ℵn;

(3) Hi(G, kG) = 0 for 0 [les ] i [les ] n.

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?

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.

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.

Part of the Choquet theory of integral representations on a universal cap is presented here in a slightly improved form. This makes possible a rapid proof of a generalised Bochner theorem for non-separable locally compact groups and of two extensions to such groups of the classical Lévy–Cramér continuity theorem concerning the convergence of sequences of probabilities.

The definiteness of the Peano kernel is proved for a functional associated with the mean-value property of Picone and Bramble and Payne for polyharmonic functions in the ball. An important corollary of this is that if a function f satisfying (−1)pΔpf>0 vanishes on p concentric spheres centered at 0, then f(0)>0. This generalizes a well-known property of subharmonic functions (which arise in the special case p = 1).

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.

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.

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.

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 $t\to \infty$.

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.

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 $\widetilde\Omega$.

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.

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.

In order to obtain the linear relations among the 24-solutions for the p-differential equation of Riemann, Barnes [12] discovered an elegant method which uses the following identity for the classical Γ function.

For complex numbers α, β, γ, δ the equality

formula here

holds and the path of integration from −i∞ to +i∞ is curved so that the poles of Γ(γ−s) Γ(δ−s) lie on the right and the poles of Γ(α+s) Γ(β+s) lie on the left.

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.

We consider the iterates of the heat operator

formula here

on Rn+1={(X, t); X =(x1, x2, …, xn) ∈Rn, t∈R}. Let Ω⊂Rn+1 be a domain, and let m[ges ]1 be an integer. A lower semi-continuous and locally integrable function u on Ω is called a poly-supertemperature of degree m if

formula here

If u and −u are both poly-supertemperatures of degree m, then u is called a poly-temperature of degree m. Since H is hypoelliptic, every poly-temperature belongs to C∞(Ω), and hence

formula here

For the case m=1, we simply call the functions the supertemperature and the temperature.

In this paper, we characterise a poly-temperature and a poly-supertemperature on a strip

formula here

by an integral mean on a hyperplane. To state our result precisely, we define a mean A[·, ·]. This plays an essential role in our argument.

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)).