A generalization of norm type cone-compression and -expansion results due to Krasnosel'skiì (see Dokl. Akad. Nauk SSSR NS 135 (1960) 527–530) is presented here for single-valued completely continuous maps defined on a Fréchet space. Applications to second-order differential equations on the half line are presented, and the existence of nontrivial solutions is established.

The paper presents an improvement of the author's earlier estimate for the average number of limit cycles of a planar polynomial vector field situated in a neighbourhood of the origin, provided that the field is close enough to a linear centre in a larger neighbourhood. The result follows from a new distributional inequality for the number of zeros of a family of univariate holomorphic functions depending holomorphically on a parameter.

This paper considers the cone multiplier operator which is defined by $\[\widehat{S^\mu f}(\xi,\tau)=m_\mu(\xi,\tau)\widehat f(\xi,\tau)$, $\qquad (\xi,\tau)\in \mathbb R^2\times \mathbb R\]$ where $m_\mu(\xi,\tau)=\phi(\tau)(1-|\xi|^2/\tau^2)_+^\mu/\Gamma(\mu+1)$ and $\phi\in C_0^\infty(1,2)$. For $-3/2<\mu<-3/14$, sharp $L^p-L^q$ estimates and endpoint estimates for $S^{\mu}$ are obtained.

The behaviour of the Ricci curvature along rays in a complete open manifold is examined.

Les études récentes sur les idéaux á droite de $A_{1}\left(k\right),$ la premiére algébre de Weyl sur un corps algébriquement clos et de caractéristique nulle $k$, nous montrent que : pour tout idéal $I\neq 0$ á droite de $A_{1}(k)$, il existe $x\in Q\,{=}\,\ffrac( A_{1}( k)),$ et $V\in \mathcal{V}$ tels que : $I\,{=}\,x\mathcal{D}(R,V)$ o\`{u} $\mathcal{V}$ est l'ensemble des sous-espaces primairement décomposables de $k[t]\,{=}\,R$, et $\mathcal{D}( R,V)$ l'idéal á droite $\{ d\in A_{1}( k) /d( R) \subset V\}$. Dans cet article nous montrerons principalement que: pour tout $0\neq I$ idéal á droite de $A_{1} (k), \exists !n\in N,\exists ( x,\sigma ) \in Q^{\ast }\,{\times}\,\Aut_{k} ( A_{1} (k)) :I\,{=}\,x\sigma (\mathcal{D} (R,O (X_{n})))$, oú $X_{n}$ est la courbe d'algébre des fonctions réguliéres : $O (X_{n})\,{=}\,k+t^{n+1}k [t]$. La forme des idéaux décrite ci-dessus permet de voir dans une hypothése de Letzter et Makar-Limanov, pour deux courbes algébriques affines $X$ et $X^{\prime}$ on a : $\mathcal{D} (X) \simeq \mathcal{D} ( X^{\prime}) \Leftrightarrow \co \dim \mathcal{D} (X)\,{=}\,\co \dim \mathcal{D}( X^{\prime})$.

Let $E$ be an elliptic curve over a finitely generated infinite field $K$. Let $K^{\rm s}$ denote a separable closure of $K$, $\sigma$ an element of the Galois group $G_{K}\,{=}\,\hbox{Gal}(K^{\rm s}/K)$, and $K^{\rm s}(\sigma)$ the invariant subfield of $K^{\rm s}$. If the characteristic of $K$ is not 2 and $\sigma$ belongs to a suitable open subgroup of $G_K$, then $E(K^{\rm s}(\sigma))$ has infinite rank.Partially supported by the Sloan Foundation and by NSF Grant DMS 97-27553.

In this paper, the behaviour of the positive eigenfunction $\phi$ of $\Lp u=\la |u|^{p-2}u$ in $\Om$, $u_{|\p \Om} =0$, $p>1$, is studied near its critical points. Under some convexity and symmetry assumptions on $\Om$, $\phi$ is seen to have a unique critical point at $x=0$; also, the behaviour of both $\phi$ and $\nabla\phi$ is determined nearby. Positive solutions $u$ to some general problems $\Lp u=f(u)$ in $\Om$, $u_{|\p \Om} =0$, are also considered, with some convexity restrictions on $u$.

This paper provides a method of generating infinitely many integer solutions of the simultaneous equations $x_1^r+x_2^r+x_3 ^r=y_1^r+y_2^r+y_3^r$ where $r$ is equal to $2$, $3$ or $4$. Three numerical solutions of this system of equations have also been given in this paper.

Global convergence is established in this paper for monotone and subhomogeneous discrete dynamical systems on product Banach spaces. This result is then used to obtain the asymptotic periodicity of solutions to a class of periodic and cooperative reaction-diffusion systems.

Let $X$ and $Y$ be affine nonsingular real algebraic varieties. A general problem in real algebraic geometry is to try to decide when a continuous map $f : X \rightarrow Y$ can be approximated by regular maps in the space of ${\cal C}_0$ mappings from $X$ to $Y$ , equipped with the ${\cal C}_0$ topology. This paper solves this problem when $X$ is the connected component containing the origin of the real part of a complex Abelian variety and $Y$ is the standard 2-dimensional sphere.

The aim of this paper is to show the existence of solutions with an arbitrarily large number of bubbles for the slightly super-critical elliptic problem $-\Delta u=u^{{(N+2)/(N-2)} +\ve }$ in $\Omega$, subject to the conditions that $u>0$ in $\Omega$, and $u=0$ on $\partial \Omega$, where $\ve >0$ is a small parameter and $\Omega \subset \RR^N$ is a bounded domain with certain symmetries, for instance an annulus or a torus in $\RR^3$.

In this note a proof is given that in the case of finitely generated amenable groups, the classical zero divisor conjecture implies the analytic zero divisor conjecture of Linnell.

Star complements and associated quadratic functions are used to obtain a sharp upper bound for the order of a graph with an eigenspace of prescribed codimension. It is shown that for regular graphs the bound can be reduced by 1, and that this reduced bound is attained by a regular graph $G$ if and only if $G$ is an extremal strongly regular graph.

A homoclinic class of a vector field is the closure of the transverse homoclinic orbits associated to a hyperbolic periodic orbit. An attractor is a transitive set to which every positive nearby orbit converges; likewise, every negative nearby orbit converges to a repeller. It is shown in this paper that a generic $C^1$ vector field on a closed $n$ -manifold has either infinitely many homoclinic classes, or a finite collection of attractors (or, respectively, repellers) with basins that form an open-dense set. This result gives an approach to use in proving a conjecture by Palis. A proof is also given of the existence of a locally residual subset of $C^1$ vector fields on a 5-manifold having finitely many attractors and repellers, but infinitely many homoclinic classes.

It is shown that the method of estimation of exponential sums with nonlinear recurring sequences, invented by the authors in a recent series of works, can be applied to estimating sums of multiplicative characters as well. As a consequence, results are obtained about the distribution of power residues and primitive elements in such sequences.

This paper is concerned with the oscillatory behaviour of first-order delay differential equations of the form \begin{equation} x^{\prime}(t) + p(t)x(\tau(t)) = 0, \quad t \geqslant t_0, \end{equation} where $p, \tau \in C([t_0,\infty),{\rm I\!R}^+),{\rm I\!R}^+ = [0,\infty),\tau(t)$ is non-decreasing, $\tau(t) < t$ for $t \geqslant t_0$ and $\lim_{t\rightarrow\infty}\tau(t) = \infty$. Let the numbers $k$ and $L$ be defined by \[ k = \mathop{{\lim\inf}}_{{t\rightarrow\infty}} \int\nolimits^t_{\tau(t)} p(s)ds \quad\hbox{and}\quad L = \mathop{\lim\sup}_{{t\rightarrow\infty}} \int\nolimits^t_{\tau(t)} p(s)ds. \]

It is proved here that when $L < 1$ and $0 < k \leqslant 1/e$ all solutions of equation (1) oscillate in several cases in which the condition \[ L > \frac{\ln\lambda_1 - 1 + \sqrt{5 - 2\lambda_1 + 2k\lambda_1}}{\lambda_1} \] holds, where $\lambda_1$ is the smaller root of the equation $\lambda = e^{k\lambda}$.

It is proved here that a minimal isometric immersion of a Kähler-Einstein or homogeneous Kähler-manifold into an Euclidean space must be totally geodesic. As an application, it is shown that an open subset of the real hyperbolic plane ${\mathbb R}H^2$ cannot be minimally immersed into the Euclidean space. As another application, a proof is given that if an irreducible Kähler manifold is minimally immersed in a Euclidean space, then its restricted holonomy group must be $U(n)$, where $n = \dim_{\mathbb C}M$.Supported by an EPSRC Grant GR/R69174.

For arbitrary infinite directed graphs $E$, the characterisation of the (not necessarily simple) Cuntz–Krieger algebras $C^*(E)$ which are purely infinite in the sense of Kirchberg–Rørdam is given. It is also shown that $C^*(E)$ has real rank zero if and only if the graph $E$ satisfies Condition (K).

The purpose of this work is to show that the fractional maximal operator has somewhat unexpected regularity properties. The main result shows that the fractional maximal operator maps $L^p$-spaces boundedly into certain first-order Sobolev spaces. It is also proved that the fractional maximal operator preserves first-order Sobolev spaces. This extends known results for the Hardy–Littlewood maximal operator.

Let $X$ be the Shimura curve corresponding to the quaternion algebra over $\Q$ ramified only at 3 and 13. B. Jordan showed that $X_{\Q(\sqrt{-13})}$ is a counterexample to the Hasse principle. Using an equation of $X$ found by A. Kurihara, it is shown here, by elementary means, that $X$ has no $\Q(\sqrt{-13})$-rational divisor classes of odd degree. A corollary of this is the fact that this counterexample is explained by the Manin obstruction.