Let $G$ be a finite group. The strong symmetric genus$\sigma ^{0}(G)$ is the minimum genus of any Riemann surface on which $G$ acts, preserving orientation. For any non-negative integer $g$, there is at least one group of strong symmetric genus $g$. For $g \neq 2$, one such group has the form $Z_{k} \times D_{n}$ for some $k$ and $n$.

Holomorphic principal bundles over a compact Riemann surface X that admits a flat connection are considered. A holomorphic G-bundle over X, where G is a connected semisimple linear algebraic group over ${\Bbb C}$, admits a flat connection if and only if the adjoint vector bundle admits one. More generally, for a complex reductive group G, the necessary and sufficient condition on a G-bundle to admit a flat connection is described. This simplifies the criterion obtained by the authors and given in Math. Ann. 322 (2002) 333–346.

Let $C$ be a germ at $O \in {\mathbb R}^2$ of a real analytic plane curve, and $C^{\mathbb C}$ its complexification; let $C_t \subset B_{\varepsilon}$ be a fiber of a real smooth deformation of $C$ in the ball $B_{\varepsilon} = B(O, \varepsilon)$. The following inequality is proved between the integrals of real curvature $k$ of $C_t$ and those of Gaussian curvature $K$ of $C_{t}^{\mathbb C}$: $$ 2 \lim_{\varepsilon, t \rightarrow 0} \int_{C_t^\varepsilon} \vert k \vert \leq \lim_{\varepsilon, t \rightarrow 0} \int_{C_t^{\varepsilon {\mathbb C}}} \vert K \vert.$$ The sharpness of this inequality is proved in the case where $C$ is a real irreducible germ. Similar results are proved for an affine algebraic curve $C \subset {\mathbb R}^2$ of degree $d$.

Wirtinger and Northcott introduced an inequality bounding the norm of a 2$\Omega$ periodic function by the norm of its derivatives. It is shown here that this type of result occurs for other operators, defined on various domains and spaces.

Let $G_0$ and $G$ be the heat kernels of $\Delta$ and $\Delta - V$ respectively. Under some sharp conditions on $V$, a proof is given that $G(x, t; y, 0)/G_0(x, t; y, 0)$ can be bounded from above and below by two positive constants. This largely improves the well-known bounds $C_1 e^{-c_1 |x-y|^2/t} \le G(x, t; y, 0)/G_0(x, t; y, 0) \le C_2 e^{c_2 |x-y|^2/t}$, $t \in (0, T]$. The result answers an open question of V. Liskevich and Yu. Semenov, in the case where $V \ge 0$. A sharp global bound in time is also obtained when $V \ge 0$.

A theorem of Beurling and Nevanlinna gives a lower bound for the harmonic measure of a compact set in the unit disk in terms of the harmonic measure of a projection of the set. In this paper, a generalization of that result is proved, involving the sum of harmonic measures evaluated at two points.

For any holomorphic map in the unit disk, the set of radial limits at a Borel set on the unit circle is a Suslin-analytic set. Here it is proved that, for a conformal map, this set is, in fact, Borel. As a consequence, the sets of accessible boundary points, of cut points and of transition points are Borel. In addition, it is shown that the set of end points is a $G_{\delta}$-set.

In this paper, some new results concerning the existence and uniqueness of nontrivial solutions to nonlinear Volterra integral equations with convolution kernels are presented.

Higher algebraic K-theoretic analogues of Hasse's norm theorem in class field theory are proved. The result gives a partial affirmative answer to a question posed by Bak and Rehmann.

If there are infinitely many p-Mersenne prime numbers, there is no bad field of positive characteristic p.

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 : $\mathcal{D} (X) \simeq \mathcal{D} ( X^{\prime}) \Leftrightarrow \co \dim \mathcal{D} (X)\,{=}\,\co \dim \mathcal{D}( X^{\prime})$.

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 $\RR^3$.

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.

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.

As well as the arc-sine laws for the time spent positive before time 1 and the time at which the maximum over [0, 1] occurs, Paul Lévy also established a third arc-sine law for Brownian motion; this relates to the time of the last visit to zero before time 1. For random walks, analogues of the first two of the these results are well known. Here it is shown that for any random walk satisfying Spitzer's condition there is an analogue of the third, but care has to be taken in its formulation. An associated conditional uniform law and some applications to stable processes are also given.

Let $V$ be a commutative valuation domain of arbitrary Krull-dimension, with quotient field $F$, let $K$ be a finite Galois extension of $F$ with group $G$, and let $S$ be the integral closure of $V$ in $K$. Suppose that one has a 2-cocycle on $G$ that takes values in the group of units of $S$. Then one can form the crossed product of $G$ over $S$, $S\ast G$, which is a $V$-order in the central simple $F$-algebra $K\ast G$. If $S\ast G$ is assumed to be a Dubrovin valuation ring of $K\ast G$, then the main result of this paper is that, given a suitable definition of tameness for central simple algebras, $K\ast G$ is tamely ramified and defectless over $F$ if and only if $K$ is tamely ramified and defectless over $F$. The residue structure of $S\ast G$ is also considered in the paper, as well as its behaviour upon passage to Henselization.

Jensen's operator inequality and Jensen's trace inequality for real functions defined on an interval are established in what might be called their definitive versions. This is accomplished by the introduction of genuine non-commutative convex combinations of operators, as opposed to the contractions considered in earlier versions of the theory by the authors, and by Brown and Kosaki. As a consequence, one no longer needs to impose conditions on the interval of definition. It is shown how this relates to the pinching inequality of Davis, and how Jensen's trace inequality generalizes to C*-algebras.

John Fauvel, who died on 12 May 2001 at the age of 53, was one of Britain's most distinguished historians of mathematics, and was emerging as a true and productive scholar in the subject when a long-running liver condition suddenly worsened and claimed his life.