Let X be a curve with an involution T which fixes r points. We show that the Weierstrass weight of a fixed point is at least (r−2)(r−4)/8. Our proof is independent of the recent result of Torres.

We consider the case where X=Fn, the nth Fermat curve, and T is any of the involutions of Fn. We find that our bound is equal to the actual weight in all known cases (n[les ]7) and compute then n=8 case to demonstrate that the equality continues to hold.

The purpose of this paper is to prove irrationality results from Jacobi's triple product identity, which can be written, for x∈[Copf]*, y∈[Copf], [mid ]y[mid ]<1:

formula here

There are various proofs of this identity; the classical one rests on the theory of theta functions ([3], theorem 6, p. 69). An alternative proof uses Heine's summation formula ([10], p. 12). An elementary, self-contained proof, can be found in [9], p. 227.

In this paper, we will use the same elementary methods as in [5] and [6], and prove the following theorems.

We prove Du's positivity conjecture for the canonical basis of the q-Schur algebra, using elementary arguments and the positivity result for Lusztig's canonical basis for U+q(sln). We also describe a family of subalgebras of the q-Schur algebra, each of which is spanned by the canonical basis elements it contains.

Let [Sfr ]n and [Afr ]n denote the symmetric and alternating groups of degree n∈ℕ respectively. Let p be a prime number and let F be an arbitrary field of characteristic p. We say that a partition of n is p-regular if no p (non-zero) parts of it are equal; otherwise we call it p-singular. Let SλF denote the Specht module corresponding to λ. For λ a p-regular partition of n let DλF denote the unique irreducible top factor of SλF. Denote by ΔλF=DλF↓[Afr ]n its restriction to [Afr ]n. Recall also that, over F, the ordinary quiver of the modular group algebra FG is a finite directed graph defined as follows: the vertices are labelled by the set of all simple FG-modules, L1, ..., Lr, and the number of arrows from Li to Lj equals dimFExtFG(Li, Lj). The quiver gives important information about the block structure of G.

The theory of reductive monoids has been well developed by Renner and the author, cf. [8, 13]. This paper concerns the finite reductive monoids MF where M is a reductive monoid and F[ratio ]M→M is a Frobenius map. We show that the Deligne–Lusztig characters RT, θ of finite reductive groups GF have natural extensions to MF. We accomplish this by first showing that the action of a finite monoid (by partial transformations) on an algebraic variety gives rise to a virtual character of the monoid via étale cohomology. We then find the correct analogue of the algebraic set L−1(U) (where L is the Lang map) for M. The action of MF on this algebraic variety gives rise to virtual characters RT¯, θ where T is an F-stable maximal torus of G, T¯ the Zariski closure of T in M and θ an irreducible character of the finite commutative monoid T¯F. We go on to show that any irreducible character of MF is a component of some RT¯, θ.

A classical fact is that Seifert manifolds with non-empty boundary are covered by surface bundles over the circle S1 and closed Seifert manifolds may or may not be covered by surface bundles over S1. Some closed graph manifolds are not covered by surface bundles over S1 ([LW] and [N]). Thurston asked if complete hyperbolic 3-manifolds of finite volume are covered by surface bundles [T]. J. Luecke and Y. Wu asked if graph manifolds with non-empty boundary are covered by surface bundles over S1 ([LW]). In this paper we prove:

THEOREM 0·1. Each graph manifold with non-empty boundary is finitely covered by a surface bundle over the circle S1.

Questions concerning extensions of polynomials or analytic functions from a Banach space E to its bidual E", and others about reflexivity and related properties on spaces of homogeneous polynomials Pk(E), have recently spurred interest regarding the structure of the bidual of a space of polynomials [4, 14, 22].

In this paper we study the relationship between the bidual of Pk(E) and the space of polynomials over E". We define a map through which elements of the bidual of Pk(E) may be viewed as polynomials over E"

formula here

and study this map to obtain information about Pk(E)". We have tried as far as possible to avoid imposing restricting conditions on E.

We give a hyperkähler analogue of the classification of compact homogeneous Kähler manifolds. Namely we show that, for a compact semisimple Lie group G, the complete G-invariant hyperkähler manifolds which are also locally Gc-homogeneous with respect to one of the complex structures are precisely the hyperkähler metrics of Kronheimer, Biquard and Kovalev on coadjoint semisimple orbits of Gc. As a consequence the only complete hyperkähler metrics which are of cohomogeneity one with respect to a triholomorphic action of compact semisimple Lie group are the flat metric on ℍn and the Calabi metric on T*[Copf]Pn.

Let μ and ν be Radon probability measures on ℝ with compact supports. We will consider natural intersections of μ and the image of ν under an isometry f[ratio ]ℝn→ℝn. Our main result gives a lower bound for the packing dimensions of these intersection measures, if μ and ν satisfy some regularity assumptions. Our method, which is originally from [1], is the same as used in [3] where similarities were considered instead of isometries.

Let f[ratio ]X→[Copf]ℙp be a finite complex analytic map into complex projective space, with dimX<p. We obtain a result on the equivalence of low homotopy groups between the image of f and [Copf]ℙp, the level of comparison is a function of p, the maximal number of preimages of f and how bad the singularities of X are. This global result is deduced from a generalisation of a theorem of H. Hamm on the local structure of singularities, see [7].

Let X be a smooth projective manifold over the complex number field [Copf] and L a Cartier divisor on X. Then (X, L) is called a polarized (resp. quasi-polarized) manifold if L is ample (resp. nef and big). For a polarized manifold (X, L), Takao Fujita introduced the notion of the Kodaira energy κε(X, L). The Kodaira energy of (X, L) is thought to have some interesting phenomena if κ(X)=−∞ (for example, Spectrum Conjecture which was proposed by T. Fujita). In order to study the Kodaira energy of (X, L), we consider the case in which X has a fibre space, that is, there exist a smooth projective manifold Y with dimX>dimY[ges ]1 and a surjective morphism f[ratio ]X→Y with connected fibres. In this case we introduce the notion of relative Kodaira energy and study some property of it. By using some results of relative Kodaira energy, we study some properties of the Kodaira energy.

In a recent paper, K. B. Lee introduced the notion of an infra-solvmanifold of type (R). These manifolds are completely determined by their fundamental group Π. Such a Π is a finite extension of a lattice Γ of a solvable Lie group of type (R) and this lattice Γ is called the translational part of Π.

Having fixed an abstract group Π occurring as the fundamental group of an infra-solvmanifold of type (R), it seems to be hard to describe, in a formal algebraic language, which subgroup of Π is the translational part. In his paper Lee formulated a conjecture which would solve this problem, however, we show that this conjecture fails. Nevertheless, by defining a concept of eigenvalues for automorphisms of certain solvable groups (both Lie groups and discrete groups), we are able to prove a new theorem, characterizing completely the translational part of the fundamental group of an infra-solvmanifold of type (R).

The Fourier binest algebra is defined as the intersection of the Volterra nest algebra on L2(ℝ) with its conjugate by the Fourier transform. Despite the absence of nonzero finite rank operators this algebra is equal to the closure in the weak operator topology of the Hilbert–Schmidt bianalytic pseudo-differential operators. The (non-distributive) invariant subspace lattice is determined as an augmentation of the Volterra and analytic nests (the Fourier binest) by a continuum of nests associated with the unimodular functions exp(−isx2/2) for s>0. This multinest is the reflexive closure of the Fourier binest and, as a topological space with the weak operator topology, it is shown to be homeomorphic to the unit disc. Using this identification the unitary automorphism group of the algebra is determined as the semi-direct product ℝ2×κℝ for the action κt(λ, μ)=(etλ, e−t μ).

A locally non-trivial holomorphic family of Riemann surfaces over a hyperbolic Riemann surface B determines a homomorphism (monodromy) of the fundamental group of B to the modular group. We give a necessary condition for a homomorphism to be a mondromy of a holomorphic family of Riemann surfaces. And we estimate the number of holomorphic families in terms of topological information for holomorphic families over Riemann surfaces of genus 0.

We describe a method of constructing explicitly eigenfunctions of the Laplacian with a prescribed boundary behaviour on a class of manifolds of hyperbolic type. These are manifolds of the form M=X×R×+, where X is an n-dimensional Riemannian manifold and the metric of M is a perturbation of the hyperbolic one.

Let X(t) (t ∈ RN) be a fractional Brownian motion in Rd of index α. Let GrX([0, 1]N) be the graph of X and let

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and

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It is proved that if N<αd, then almost surely

formula here

and if N>αd, then almost surely

formula here

where ϕ-m is the ϕ-Hausdorff measure and K1, K2 are positive finite constants. The exact Hausdorff measure of the image and graph of certain Gaussian random fields with independent fractional Brownian motion components are also obtained.

14 October 1996 In the Cockcroft Lecture Theatre, New Museums Site.

28 October 1996 In the Cockcroft Lecture Theatre, New Museums Site.

ANNUAL GENERAL MEETING 28 October 1996 In the Arts School.

11 November 1996 In the Cockcroft Lecture Theatre, New Museums Site.

25 November 1996 In the Cockcroft Lecture Theatre, New Museums Site.

20 January 1997 In the Cockcroft Lecture Theatre, New Museums Site.

3 February 1997 In the Cockcroft Lecture Theatre, New Museums Site.

17 February 1997 In the Cockcroft Lecture Theatre, New Museums Site.

3 March 1997 In the Cockcroft Lecture Theatre, New Museums Site.

19 May 1997 In the Arts School.

21 July 1997 In the Arts School.