In this paper we point out how some recent developments in the theory of constant scalar curvature Kähler metrics can be used to clarify the existence issue for such metrics in the special case of (geometrically) ruled complex surfaces.

It is proved that the existence of zero modes of Weyl–Dirac operators is a rare phenomenon. An estimate of the multiplicity is given in term of the magnetic potential.

Let F be a field, and let Σn be the symmetric group on n letters. In this paper we address the following question: given two irreducible FΣn-modules D1 and D2 of dimensions greater than 1, can it happen that D1 [otimes ] D2 is irreducible? The answer is known to be ‘no’ if char F = 0 [12] (see also [2] for some generalizations). So we assume from now on that F has positive characteristic p. The following conjecture was made in [4].

CONJECTURE. Let D1and D2be two irreducible FΣn-modules of dimensions greater than 1. Then D1 [otimes ] D2 is irreducible if and only if p = 2, n = 2 + 4l for some positive integer l; one of the modules corresponds to the partition (2l + 2, 2l) and the other corresponds to a partition of the form (n − 2j − 1, 2j + 1), 0 [les ] j < l. Moreover, in the exceptional cases, one has

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The main result of this paper is the following theorem, which establishes a big part of the conjecture.

For any $n \geq 3$, let $F \in \mathbb{Z}[X_0,\ldots,X_n]$ be a form of degree $d\geq 5$ that defines a non-singular hypersurface $X \subset \mathbb{P}^{n}$. The main result in this paper is a proof of the fact that the number $N(F;B)$ of $\mathbb{Q}$-rational points on $X$ which have height at most $B$ satisfies $N(F;B)=O_{d,\varepsilon,n}(B^{n-1+\varepsilon}),$ for any $\varepsilon >0$. The implied constant in this estimate depends at most upon $d$, $\varepsilon$ and $n$. New estimates are also obtained for the number of representations of a positive integer as the sum of three $d$th powers, and for the paucity of integer solutions to equal sums of like polynomials.

This paper proves bounds for the commutator width of a wreath product of two groups. As a corollary, it is shown that the commutator width of finite perfect linear groups of dimension 15 is unbounded. It follows that the covering number of these groups is unbounded. On the other hand, the commutator width of iterated wreath products of nonabelian finite simple groups is bounded by an absolute constant.

The classical Mikhlin–Hörmander multiplier theorem is generalised to the context of discrete groups of polynomial volume growth.

For the classical groups, Kraft and Procesi have resolved the question of which nilpotent orbits have closures that are normal and which do not, with the exception of the very even orbits in $D_{2l}$ that have partitions of the form $(a^{2k}, b^2)$ for $a\,{>}\,b$ even natural numbers satisfying $a k\,{+}\,b\,{=}\,2 l$.

In this paper, these orbits are shown to have normal closure.

In this paper, we give some complete characterizations of the primitive of a Henstock integrable function in Euclidean space.

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.

Upper bounds are obtained for the heat content and the torsional rigidity of a complete Riemannian manifold $M$, assuming a Hardy inequality for the Dirichlet Laplacian on $M$.

Let $D$ be a bounded domain in ${\bb R}^n$ . For a function $f$ on the boundary $\partial D$ , the Dirichlet solution of $f$ over $D$ is denoted by $H_D f$ , provided that such a solution exists. Conditions on $D$ for $H_D$ to transform a Hölder continuous function on $\partial D$ to a Hölder continuous function on $D$ with the same Hölder exponent are studied. In particular, it is demonstrated here that there is no bounded domain that preserves the Hölder continuity with exponent 1. It is also also proved that a bounded regular domain $D$ preserves the Hölder continuity with some exponent $\alpha$ , $0 < \alpha < 1$ , if and only if $\partial D$ satisfies the capacity density condition, which is equivalent to the uniform perfectness of $\partial D$ if $n = 2$ .

In his book Abelian groups, L. Fuchs raised the question as to whether, in general, in the factorization of a finite (cyclic) abelian group one factor may always be replaced by some subgroup. The answer turned out to be negative in general, but positive in certain cases. In this paper the complete answer for cyclic groups is given. In all previously unsolved cases, the answer turns out to be positive. It is shown that a cyclic group has the property that in every factorization, one factor may be replaced by a subgroup if and only if the group has order equal to the product of a prime and a square-free integer. Certain results are also given in non-cyclic cases.

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.

Let $X$ be a non-singular algebraic curve of genus $g\ge 2,\;n\ge 2$ an integer, $\xi$ a line bundle over $X$ of degree $d>2n(g-1)$ with $(n,d)=1$ and ${\cal M}_\xi$ the moduli space of stable bundles of rank $n$ and determinant $\xi$ over $X$ . It is proved that the Picard bundle ${\cal W}_\xi$ is stable with respect to the unique polarisation of ${\cal M}_\xi$ .

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.

Let $2\le p <\infty$, and let $X$ be a complex Banach space. It is shown that $X$ is $p$-uniformly PL-convex if and only if there exists $\lambda >0$ such that $ \|f\|_{H^p(X)}\,{\ge}\, (\|f(0)\|^p+\lambda\int_{\mathbb D} (1-|z|^2)^{p-1}\|f^{\prime}(z)\|^p\,dA(z))^{1/p}$, for all $f\in H^p(X)$. Applications to embeddings between vector-valued BMOA spaces defined via Poisson integral or Carleson measures are provided.

Write $G^{\ast} = G^{{\cal L}{\cal U}{\cal C}}\setminus G$ where $G^{{\cal L}{\cal U}{\cal C}}$ is the largest semigroup compactification of the locally compact group $G$ . Then the set of points of $G^{\ast}$ which are right cancellable in $G^{{\cal L}{\cal U}{\cal C}}$ is large; in fact it has an interior in $G^{\ast}$ which is dense in $G^{\ast}$ . Corollaries are given about the number of left ideals in $G^{{\cal L}{\cal U}{\cal C}}$ and the size of right ideals in the algebra ${{\cal L}{\cal U}{\cal C}}(G)^{\ast}$ .

In [1], Hirano gives a method for constructing families of curves with a large number of singularities. The idea is to consider an abelian covering of ℙ2 ramified along three lines in general position, and to take the pull-back of a curve C intersecting the lines non-generically. Similar constructions are used by Shimada in [10] and Oka in [8]. We apply this method for the case where C is a conic, constructing a family of curves with the following asymptotic behaviour (see [9]):

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The goal of this paper is to calculate the fundamental group for the curves in this family as well as their Alexander polynomial.

A remarkable theorem of Birch [2] shows that a system of homogeneous polynomials with rational coefficients has a non-trivial zero, provided only that these polynomials are of odd degree, and the system has sufficiently many variables in terms of the number and degrees of these polynomials. Despite four decades of effort, the problem of obtaining a reasonable bound for the latter number of variables has proved to be one of great difficulty. When the system consists of a single cubic form, Davenport [4] has succeeded in showing that 16 variables suffice, and Schmidt [17, 18, 19, 20] has devoted a series of papers to systems of cubic forms, showing in particular that 5140 variables suffice for pairs of cubic forms, and that (10r)5 variables suffice for systems of r cubic forms. The current state of knowledge for forms of higher degree is, by comparison, extremely weak (but see [21, 22]), and so it seems worthwhile expending further effort on the case of systems of cubic forms. In this paper we improve on Schmidt's result for pairs of cubic forms. In contrast with the sophisticated versions of the Hardy–Littlewood method employed by Davenport and Schmidt, our approach is based on an elementary idea of Lewis [12], and is applicable in arbitrary number fields. This method also has consequences for the existence of linear spaces of rational solutions on cubic hypersurfaces, thereby improving on work of Lewis and Schulze-Pillot [14] on this topic.

Given a field k and a finite group G acting on the rational function field k(X1, . . ., Xn) as a group of k-automorphisms, an important Noether's problem asks whether the invariant subfieldformula here is purely transcendental over k.