This paper is a continuation of [15]. In that paper, I introduced a general framework which allows one to produce ‘weak’ counterexamples to Torelli for Calabi–Yau threefolds: deformation families containing non-isomorphic varieties Yt, Y+t with isomorphic Hodge theory on the third cohomology. The varieties arise as deformations of threefolds Y that are resolutions of singular varieties X with rather special properties (cf. Section 1). In [15], I discussed two families containing suitable X that do provide a counterexample and a third family with remarkably similar properties where however the existence of a nontrivial generic automorphism destroys the counterexample. This shows that explicit examples are necessary; there is a precise set of conditions one needs to check carefully in order to obtain counterexamples to Torelli (Theorem 1·1). In fact, there exist several families with renitent automorphisms (Remark 4·5).

In this paper, we study involutions of Godeaux surfaces whose bicanonical system has no base components. We classify all possible fixed loci and find corresponding examples to the classification.

This paper continues our study started in [5] of the group G defined by the presentation

formula here

where m [ges ] 2, n [ges ] 2 and ∈i = ±1 for 1 [les ] i [les ] 2. The problem under consideration can be stated as: precisely when does both a have order m and b have order n in the group G? This question arose as a result of our investigation into equations over groups and the connection is explained in Section 2.

We consider the bond percolation model on the three-dimensional cubic lattice, in which individual edges are retained independently with probability p. We shall describe a subgraph of the lattice as ‘entangled’ if, roughly speaking, it cannot be ‘pulled apart’ in three dimensions. We shall discuss possible ways of turning this into a rigorous definition of entanglement. For a broad class of such definitions, we shall prove that for p sufficiently close to zero, the graph of retained edges has no infinite entangled subgraph almost surely, thereby establishing that there is a phase transition for entanglement at some value of p strictly between zero and unity.

Let p be a prime number. Given a p-group G, denote by H*(G) the mod-p cohomology algebra of G. For every subgroup K of G, let ResGK be the restriction map from H*(G) to H*(K). Consider the case where K = Ker u, with 0 ≠ u ∈ H1 (G) = Hom(G,[ ]p). It was known that, for p = 2, Ker ResGK is the principal ideal (u) generated by u. However, for p > 2, a corresponding statement does not hold: while an argument used by Quillen-Venkov in [14] shows that (Ker ResGK)2 ⊂ (βu) (with β the Bockstein homomorphism), it turns out that, in general, (u, βu) is strictly smaller that Ker ResGK; an example is the case where G is any extraspecial p-group of order p3 and K is maximal elementary abelian in G (see [5, 8]).

The Dickson algebra Wn+1 of invariants in a polynomial algebra over [ ]2 is an unstable algebra over the mod 2 Steenrod algebra [Ascr ], or equivalently, over the Kudo–Araki–May algebra [Kscr ] of ‘lower’ operations. We prove that Wn+1 is a free unstable algebra on a certain cyclic module, modulo just one additional relation. To achieve this, we analyse the interplay of actions over [Ascr ] and [Kscr ] to characterize unstable cyclic modules with trivial action by the subalgebra [Ascr ]n−2 on a fundamental class in degree 2n – a. This involves a new family of left ideals [Iscr ]a in [Kscr ], which play the role filled by the ideals [Ascr ][Ascr ]n−2 in the Steenrod algebra.

This note is intended to correct the proof of a lemma in the article ‘Hit polynomials and conjugation in the dual Steenrod algebra’, which appeared in 1998 in Volume 123 of these Proceedings.

This paper gives an algebraic conjecture which is shown to be equivalent to Thurston's Geometrization Conjecture for closed, orientable 3-manifolds. It generalizes the Stallings–Jaco theorem which established a similar result for the Poincaré Conjecture. The paper also gives two other algebraic conjectures; one is equivalent to the finite fundamental group case of the Geometrization Conjecture and the other is equivalent to the union of the Geometrization Conjecture and Thurston's Virtual Bundle Conjecture.

0·1. Conical domains

The aim of this paper is to improve the results of my previous work in the area [1]. Let us start by recalling some of the notations and definitions of [1].

Let Ω ⊂ ℝd be some convex open cone with vertex at 0 such that [emptyv ] &≠; Ω &≠; ℝd. Let u ∈ C(Ω) be the reduite, i.e.

formula here

and let α = degΩ[ges ] 1 be the homogeneity degree of u. Let us consider the random walk

formula here

controlled by some compactly supported measure

formula here

that is centred with covarience = I, i.e.

formula here

We then denote:

formula here

Let X be a Banach space. In this paper we show that, assuming that X does not contain a copy of [lscr ]∞, then ba (Σ,X) contains a complemented copy of c0 whenever X contains a copy of c0. Conversely, if ba (Σ,X) contains a complemented copy of c0, then X contains a copy of c0.

The purpose of this short note is to reformulate theorem 9·3 in [5] which is not correct as stated. We note that all other results in [5] are independent of that statement.

The notation is the same as [5] with the sole exception that C0(S) will always denote the space of all real-valued continuous functions on the locally compact space S vanishing at infinity. As usual, αS stands for the one-point compactification of S. Recall that a norm ‖ · ‖ on a Banach space X is said to be maximal if there is no equivalent norm on X whose isometry group contains properly that of ‖ · ‖.

A general theory of piecewise multiharmonic splines is constructed for a class of fractals (post-critically finite) that includes the familiar Sierpinski gasket, based on Kigami's theory of Laplacians on these fractals. The spline spaces are the analogues of the spaces of piecewise Cj polynomials of degree 2j + 1 on an interval, with nodes at dyadic rational points. We give explicit algorithms for effectively computing multiharmonic functions (solutions of Δj+1u = 0) and for constructing bases for the spline spaces (for general fractals we need to assume that j is odd), and also for computing inner products of these functions. This enables us to give a finite element method for the approximate solution of fractal differential equations. We give the analogue of Simpson's method for numerical integration on the Sierpinski gasket. We use splines to approximate functions vanishing on the boundary by functions vanishing in a neighbourhood of the boundary.

The purpose of this paper to characterize the existence of nontrivial solutions of nonlinear Volterra integral equations and to prove the uniqueness of this kind of solutions. To this end, different properties of the solutions are presented, in particular some referred to their attracting character.

A theorem is established for generating by differentiation and integration the Airy stress function due to the presence of any arbitrary two-dimensional singularity near a thick layer separating two other dissimilar semi-infinite solids directly from the corresponding Airy stress function for the unbounded homogeneous solid. A systematic application of the theorem yields the dominant distant effect associated with any arbitrary influencing line singularity in a well-structured physically interpretable form. As an illustration, the problem of a concentrated line force of arbitrary orientation near the interface layer is examined from this standpoint. It is found that a force aligned normal to the interface layer produces a far-field effect of an edge dislocation with Burgers vector parallel to the interface layer, together with a semi-infinite plane of centres of dilatation, whereas a force aligned parallel to the interface layer produces a far-field effect of an edge dislocation with Burgers vector perpendicular to the interface layer together with a semi-infinite plane of concentrated couples.