A 4n-dimensional Riemannian manifold (M, g) is hyperkähler if it possesses three anti-commuting complex structures I, J, K such that the metric g is Kähler with respect to each of them. The reduced holonomy group of such a manifold is necessarily a subgroup of Sp(n) so the Ricci tensor of g vanishes and (M, g) can be regarded as a positive definite solution to Einstein's equations in vacuum.

A Coxeter system is a pair (W, S) where W is a group and where S is a set of involutions in W such that W has a presentation of the form

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Here m(s, t) denotes the order of st in W and in the presentation for W, (s, t) ranges over all pairs in S × S such that m(s, t) ≠ ∞. We further require the set S to be finite. W is a Coxeter group and S is a fundamental set of generators for W.

Obviously, if S is a fundamental set of generators, then so is wSw−1, for any w∈W. Our main result is that, under certain circumstances, this is the only way in which two fundamental sets of generators can differ. In Section 3, we will prove the following result as Theorem 3.1.

On a large class of post-critically finite (finitely ramified) self-similar fractals with possibly little symmetry, we consider the question of existence and uniqueness of a Laplace operator. By considering positive refinement weights (local scaling factors) which are not necessarily equal, we show that for each such fractal, under a certain condition, there are corresponding refinement weights which support a unique self-similar Dirichlet form. As compared with previous results, our technique allows us to replace symmetry by connectivity arguments.

Let G be a group and P be a property of groups. If every proper subgroup of G satisfies P but G itself does not satisfy it, then G is called a minimal non-P group. In this work we study locally nilpotent minimal non-P groups, where P stands for ‘hypercentral’ or ‘nilpotent-by-Chernikov’. In the first case we show that if G is a minimal non-hypercentral Fitting group in which every proper subgroup is solvable, then G is solvable (see Theorem 1.1 below). This result generalizes [3, Theorem 1]. In the second case we show that if every proper subgroup of G is nilpotent-by-Chernikov, then G is nilpotent-by-Chernikov (see Theorem 1.3 below). This settles a question which was considered in [1–3, 10]. Recently in [9], the non-periodic case of the above question has been settled but the same work contains an assertion without proof about the periodic case.

The main results of this paper are given below (see also [13]).

In this paper, we prove that positive conditional entropy implies the existence of asymptotic pairs and scrambled sets on fibers. Moreover, we introduce the notion of conditional topological entropy for a subset using Bowen's definition of separated and spanning sets, and prove that the conditional topological entropy of a system relative to a factor is the supremum of conditional topological entropy of its scrambled sets on fibers.

Let α > 0. The operator of the form

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is considered, where the real weight function v(x) is locally integrable on R+ := (0, ∞). In case v(x) = 1 the operator coincides with the Riemann–Liouville fractional integral, Lp → Lq estimates of which with power weights are well known. This work gives Lp → Lq boundedness and compactness criteria for the operator Tα in the case 0 < p, q < ∞, p > max(1/α, 1).

The following autoduality theorem is proved for an integral projective curve $C$ in any characteristic. Given an invertible sheaf ${\cal L}$ of degree 1, form the corresponding Abel map $A_{\cal L}:C\longrightarrow \bar{J}$ , which maps $C$ into its compactified Jacobian, and form its pullback map $A^{\ast}_{\cal L}:{\rm Pic}^0_{\bar{J}}\longrightarrow J$ , which carries the connected component of $0$ in the Picard scheme back to the Jacobian. If $C$ has, at worst, points of multiplicity $2$ , then $A^{\ast}_{\cal L}$ is an isomorphism, and forming it commutes with specializing $C$ .

Much of the work in the paper is valid, more generally, for a family of curves with, at worst, points of embedding dimension $2$ . In this case, the determinant of cohomology is used to construct a right inverse to $A^{\ast}_{\cal L}$ . Then a scheme-theoretic version of the theorem of the cube is proved, generalizing Mumford's, and it is used to prove that $A^{\ast}_{\cal L}$ is independent of the choice of ${\cal L}$ . Finally, the autoduality theorem is proved. The presentation scheme is used to achieve an induction on the difference between the arithmetic and geometric genera; here, special properties of points of multiplicity $2$ are used.

The singular homology groups of compact CW-complexes are finitely generated, but the groups of compact metric spaces in general are very easy to generate infinitely and our understanding of these groups is far from complete even for the following compact subset of the plane, called the Hawaiian earring:

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Griffiths [11] gave a presentation of the fundamental group of ℍ and the proof was completed by Morgan and Morrison [15]. The same group is presented as the free σ-product [smashp ]σℕℤ of integers ℤ in [4, Appendix]. Hence the first integral singular homology group H1(ℍ) is the abelianization of the group [smashp ]σℕℤ. These results have been generalized to non-metrizable counterparts ℍI of ℍ (see Section 3).

In Section 2 we prove that H1(X) is torsion-free and Hi(X) = 0 for each one-dimensional normal space X and for each i [ges ] 2. The result for i [ges ] 2 is a slight generalization of [2, Theorem 5]. In Section 3 we provide an explicit presentation of H1(ℍ) and also H1(ℍI) by using results of [4].

Throughout this paper, a continuum means a compact connected metric space and all maps are assumed to be continuous. All homology groups have the integers ℤ as the coefficients. The bouquet with n circles [xcup ]nj=1Cj is denoted by Bn. The base point (0, 0) of Bn is denoted by o for simplicity.

We study the boundedness of the Hardy–Littlewood maximal function in weighted Lorentz spaces, using some new techniques on distribution and rearrangement functions. Our goal is to give a unified version of some well-known weighted inequalities in ℝn and ℝ+, showing the relation between the theories.

The paper gives fundamental results on the universal abelian covers of rational surface singularities. Let $(X,o)$ be a normal complex surface singularity germ with a rational homology sphere link. Then $(X,o)$ has the universal abelian cover $(Y,o) \longrightarrow (X,o)$. It is shown that if $(X,o)$ is rational or minimally elliptic, and if it has a star-shaped resolution graph, then $(Y,o)$ is a complete intersection (a partial answer to the conjecture of Neumann and Wahl). A way is given to compute the multiplicity and the embedding dimension of $(Y,o)$ from the resolution graph of $(X,o)$ in the case when $(X,o)$ is rational.

Let [Gscr ] be a Burnside 3-group constructed by Gupta-Sidki. The group algebra ℤ3[[Gscr ]] has a proper ring-quotient which embeds [Gscr ] and is just-infinite and primitive.

Since the remarkable discovery of the relevance of derived equivalences in the theory of p-blocks of finite groups, where p is a prime, by J. Rickard in [15, 16], various attempts have been made to understand this phenomenon. In particular, J. Rickard defines in [18] a certain class of derived equivalences between the derived module categories of p-blocks of finite groups that he calls splendid equivalences (and that we are going to call splendid derived equivalences in this paper) which take into account the local structure, that is, which under suitable hypotheses induce a family of derived equivalences at all ‘local levels’ of the considered p-blocks (see [18] for a more detailed motivation). The main condition for a derived equivalence to be splendid is that it is given by a two-sided tilting complex consisting of p-permutation bimodules (see Definitions 1.3 and 1.4 below for the precise terminology).

The pressure function $p(t)$ of a non-recurrent map is real analytic on some interval ($0,t_\ast$) with $t_\ast$ strictly greater than the dimension of the Julia set. The proof is an adaptation of the well known tower techniques to the complex dynamics situation. In general, $p(t)$ need not be analytic on the whole positive axis.

Suppose that K is a closed, total cone in a real Banach space X, that A[ratio ]X→X is a bounded linear operator which maps K into itself, and that A′ denotes the Banach space adjoint of A. Assume that r, the spectral radius of A, is positive, and that there exist x0≠0 and m[ges ]1 with Am(x0) =rmx0 (or, more generally, that there exist x0∉(−K) and m[ges ]1 with Am(x0) [ges ]rmx0). If, in addition, A satisfies some hypotheses of a type used in mean ergodic theorems, it is proved that there exist u∈K−{0} and θ∈K′−{0} with A(u)=ru, A′(θ)=rθ and θ(u)>0. The support boundary of K is used to discuss the algebraic simplicity of the eigenvalue r. The relation of the support boundary to H. Schaefer's ideas of quasi-interior elements of K and irreducible operators A is treated, and it is noted that, if dim(X)>1, then there exists an x∈K−{0} which is not a quasi-interior point. The motivation for the results is recent work of Toland, who considered the case in which X is a Hilbert space and A is self-adjoint; the theorems in the paper generalize several of Toland's propositions.

Recently, systematic studies of mappings of finite distortion have emerged as a key area in geometric function theory. The connection with deformations of elastic bodies and regularity of energy minimizers in the theory of nonlinear elasticity is perhaps a primary motivation for such studies, but there are many other applications as well, particularly in holomorphic dynamics and also in the study of first order degenerate elliptic systems, for instance the Beltrami systems we consider here.

Recently Bost and Connes have studied an interesting C*-algebraic Hecke algebra arising in number theory. Here it is shown that this algebra can be realised as a semigroup crossed product, and be profitably studied using methods developed by the authors for analysing Toeplitz algebras. One main result is a characterisation of faithful representations of the Hecke algebra.

The vanishing ideal of an arrangement of linear subspaces in a vector space is considered, and the paper investigates when this ideal can be generated by products of linear forms. A combinatorial construction (blocker duality) is introduced which yields such generators in cases with a great deal of combinatorial structure, and examples are presented that inspired the work. A construction is given which produces all elements of this type in the vanishing ideal of the arrangement. This leads to an algorithm for deciding if the ideal is generated by products of linear forms. Generic arrangements of points in ${\bf P}^2$ and lines in ${\bf P}^3$ are also considered.

Let $F$ be an algebraically closed field of characteristic $0$ , and let $A$ be a $G$ -graded algebra over $F$ for some finite abelian group $G$ . Through $G$ being regarded as a group of automorphisms of $A$ , the duality between graded identities and $G$ -identities of $A$ is exploited. In this framework, the space of multilinear $G$ -polynomials is introduced, and the asymptotic behavior of the sequence of $G$ -codimensions of $A$ is studied.

Two characterizations are given of the ideal of $G$ -graded identities of such algebra in the case in which the sequence of $G$ -codimensions is polynomially bounded. While the first gives a list of $G$ -identities satisfied by $A$ , the second is expressed in the language of the representation theory of the wreath product $G \wr S_n$ , where $S_n$ is the symmetric group of degree $n$ .

As a consequence, it is proved that the sequence of $G$ -codimensions of an algebra satisfying a polynomial identity either is polynomially bounded or grows exponentially.

It is proved that the 3-part of the class number of a quadratic field $\mathbb{Q}(\sqrt{D})$ is $O(|D|^{55/112 + \ep})$ in general and $O(|D|^{5/12+\ep})$ if $|D|$ has a divisor of size $|D|^{5/6}$. These bounds follow as results of nontrivial estimates for the number of solutions to the congruence $x^a \,{\con}\, y^b$ modulo $q$ in the ranges $x \,{\leqslant}\,X$ and $y\,{\leqslant}\, Y$, where $a,b$ are nonzero integers and $q$ is a square-free positive integer. Furthermore, it is shown that the number of elliptic curves over $\Q$ with conductor $N$ is $O(N^{55/112 + \ep})$ in general and $O(N^{5/12 + \ep})$ if $N$ has a divisor of size $N^{5/6}$. These results are the first improvements to the trivial bound $O(|D|^{1/2 + \ep})$ and the resulting bound $O(N^{1/2 + \ep})$ for the 3-part and the number of elliptic curves, respectively.

It is shown that if a plane of PG(3,q), q even, meets an ovoid in a conic, then the ovoid must be an elliptic quadric. This is proved by using the generalized quadrangles T2([Cscr ]) ([Cscr ] a conic), W(q) and the isomorphism between them to show that every secant plane section of the ovoid must be a conic. The result then follows from a well-known theorem of Barlotti.