In this paper, the authors re-examine the reduction of Maurer and Wolf of the discrete logarithm problem to the Diffie-Hellman problem. They give a precise estimate for the number of operations required in the reduction, and then use this to estimate the exact security of the elliptic curve variant of the Diffie-Hellman protocol for various elliptic curves defined in standards.

The object of this article is to examine a conjecture of Zassenhaus and certain variations of it for integral group rings of sporadic groups. We prove the ℚ-variation and the Sylow variation for all sporadic groups and their automorphism groups. The Zassenhaus conjecture is established for eighteen of the sporadic simple groups, and for all automorphism groups of sporadic simple groups G which are different from G. The proofs are given with the aid of the GAP computer algebra program by applying a computational procedure to the ordinary and modular character tables of the groups. It is also shown that the isomorphism problem of integral group rings has a positive answer for certain almost simple groups, in particular for the double covers of the symmetric groups.

We present experimental evidence to support the widely held belief that one half of all elliptic curves have infinitely many rational points. The method used to gather this evidence is a refinement of an algorithm due to the author which is based upon rigid and crystalline cohomology.

This paper announces the creation of a database for number fields. It describes the contents and the methods of access, indicates the origin of the polynomials, and formulates the aims of this collection of fields.

We consider the problem of strong approximations of the solution of stochastic differential equations of Itô form with a constant lag in the argument. We indicate the nature of the equations of interest, and give a convergence proof in full detail for explicit one-step methods. We provide some illustrative numerical examples, using the Euler–Maruyama scheme.

Constructing numerical models of noisy partial differential equations is a very delicate task. Our long-term aim is to use modern dynamical systems theory to derive discretisations of dissipative stochastic partial differential equations. As a second step, we consider here a small domain, representing a finite element, and derive a one-degree-of-freedom model for the dynamics in the element; stochastic centre manifold theory supports the model. The approach automatically parametrises the microscale structures induced by spatially varying stochastic noise within the element. The crucial aspect of this work is that we explore how a multitude of microscale noise processes may interact in nonlinear dynamical systems. The analysis finds that noise processes with coarse structure across a finite element are the significant noises for the modelling. Further, the nonlinear dynamics abstracts effectively new noise sources over the macroscale time-scales resolved by the model.

In this paper we study number fields which are Euclidean with respect to functions that are different from the absolute value of the norm, namely weighted norms that depend on a real parameter c. We introduce the Euclidean minimum of weighted norms as the set of values of c for which the function is Euclidean, and we show that the Euclidean minimum may be irrational and not isolated. We also present computational results on Euclidean minima of cubic number fields, and present a list of norm-Euclidean complex cubic fields that we conjecture to be complete.

The authors of this paper consider the anti-ferromagnetic Potts model on the the integer lattice Z2. The model has two parameters: q, the number of spins, and λ = exp(−β), where β 4 is ‘inverse temperature’. It is known that the model has strong spatial mixing if q > 7, or if q = 7 and λ = 0 or λ > 1/8, or if q = 6 and λ = 0 or λ > 1/4. The λ = 0 case corresponds to the model in which configurations are proper q-colourings of Z2. It is shown that the system has strong spatial mixing for q ≥ 6 and any λ. This implies that Glauber dynamics is rapidly mixing (so there is a fully-polynomial randomised approximation scheme for the partition function), and also that there is a unique infinite-volume Gibbs state. We also show that strong spatial mixing occurs for a larger range of λ than was previously known for q = 3, q = 4 and q = 5.

We study the p-form spectrum of the Laplace-Beltrami operator acting on lens spaces as considered by Ikeda [Geometry of manifolds (Academic Press, Boston, MA, 1989) 383–417]. Ikeda gave examples of such spaces that are non-isometric but isospectral for all p ≤ p0. In this paper we exhibit examples of such spaces that are not isometric, and are isospectral for various, but not for all. values of p. In particular, examples are given of non-isometric lens spaces that are isospectral for some values of p but not for the case p = 0.

The Verheul homomorphism is a group homomorphism from a finite subgroup of the multiplicative group of a field to an elliptic curve. The hardness of computation of the Verheul homomorphism was shown by Verheul to be closely related to the hardness of the computational Diffie-Hellman problem. Let p ≥ 5 be a prime, and let N be a prime satisfying √(12p) < N < 2p / √3, where N ≠ p. Let E be an ordinary elliptic curve over Fp, and let C ⊂ E be a cyclic subgroup of order N. Let H be the group of all Nth roots of unity (contained in the algebraic closure of Fp), and let phi be the Verheul isomorphism from H to C.

We consider a polynomial P such that P(z) is the X-coordinate of phi(z) for all z ∈ H – {1}. We show that, for at least approximately 58% of pairs (E, C), none of the coefficients of the non-constant terms of P vanishes.

Using Jacobi field arguments, this paper describes an iterative procedure for finding the Riemannian barycentres of a class of probability measures on complete, simply connected Riemannian manifolds with a finite upper bound on their sectional curvatures. This, in particular, generalises an earlier result of the author's (‘Locating Fréchet means with application to shape spaces’, Adv. Appl. Probab. 33 (2001) 324-338).

A sequence of finite-dimensional quotients of affine Hecke algebras is studied. Each element of the sequence is constructed so as to have a weight space labelling scheme for Specht⁄standard modules. As in the weight space formalism of algebraic Lie theory, there is an action of an affine reflection group on this weight space that fixes the set of labelling weights. A linkage principle is proved in each case. Further, it is shown that the simplest non-trivial example may essentially be identified with the blob algebra (a physically motivated quasihereditary algebra whose representation theory is very well understood by Lie-theory-like methods). An extended role is hence proposed for Soergel's tilting algorithm, away from its algebraic Lie theory underpinning, in determining the simple content of standard modules for these algebras. This role is explicitly verified in the blob algebra case. A tensor space representation of the blob algebra is constructed, as a candidate for a full tilting module (subsequently proven to be so in a paper by Martin and Ryom-Hansen), further evidencing the extended utility of Lie-theoretic methods. Possible generalisations of this representation to other elements of the sequence are discussed.

An algorithm is given tjat recognises (in O(lN2 log N) time, where N is the size of the input and l the depth of a precalculated Schreier tree) when a transitive group, (G, Ω) is the action on one orbit of the action of G on the set Γ(2) of ordered pairs of distinct elements of some G-set Γ (that us, Ωis isomorphic to an orbital of (G,Γ)). This may be adapted to list all essentially different such actions in O(lN4log N)time, where N is the sum of sizes of the input and output. This will be a useful tool for reducing the degree of a permutation group as an aid to further study of the group.

This algorithm is then extended to provide an algorithm that will (in O(lN3 log N) time) recognise when a transiteve group is the action on one orbit of the action of G on the set Γ{2} ofunorderd pairs of distinct elements of some G-set Γ. An algorithm for finding all essentially different such actions is also provided, running in O(lN4logN) time. (again, N is the sum of the input and output sizes.) It is also indicated how these results may be applied to the more general problem of recognising when an intransitive group (G,Ω) is isomorphic to (G, Γ{2}) for some G-set Γ.

All the algorithms are practical; most have been implementd in GAP, and the code is made available with this paper. In some cases the algorithms are considerably more practical than their asymptotic analyses would suggest.

In the study of the rational cohomology of Hilbert schemes of points on a smooth surface, it is particularly interesting to understand the characteristic classes of the tautological bundles and the tangent bundle. In this note we pursue this study. We first collect all results appearing separately in the literature and prove some new formulas using Ohmoto's results on orbifold Chern classes on Hilbert schemes. We also explain the algorithmic counterpart of the topic: the cohomology space is governed by a vertex algebra that can be used to compute characteristic classes. We present an implementation of the vertex operators in the rewriting logic system MAUDE, and address observations and conjectures obtained after symbolic computations.

We give a mathematically rigorous analysis which confirms the surprising results in a recent paper of Benilov, O‘Brien and Sazonov [J. Fluid Mech. 497 (2003) 201-224] about the spectrum of a highly singular non-self-adjoint operator that arises in a problem in fluid mechanics. We also show that the set of eigenvectors does not form a basis for the operator.

The paper describes an approach to the computation of the zero energy thresholds for the appearance of negative energy eigenvalues of Schrödinger operators.

In a paper of 1933, D. H. Lehmer continued Pierce's study of integral sequences associated to polynomials generalizing the Mersenne sequence. He developed divisibility criteria, and suggested that prime apparition in these sequences — or in closely related sequences — would be denser if the polynomials were close to cyclotomic, using a natural measure of closeness.

We review briefly some of the main developments since Lehmer's paper, and report on further computational work on these sequences. In particular, we use Mossinghoff's collection of polynomials with smallest known measure to assemble evidence for the distribution of primes in these sequences predicted by standard heuristic arguments.

The calculations lend weight to standard conjectures about Mersenne primes, and the use of polynomials with small measure permits much larger numbers of primes to be generated than in the Mersenne case.

Sturm–Liouville potentials of the form xa ƒ(∈x) are considered, where a > 0, ƒ decays sufficiently rapidly at infinity, and ∈ is a small positive parameter. It is shown that there are a finite number N(∈) of spectral concentration points, and computational evidence is given to support the conjecture that N(∈) increases to infinity as ∈ decreases to zero.

We describe an algorithm to decompose rational functions from which we determine the poset of groups fixing these functions.

The question of what categorical structure is required to give semantics to O‘Hearn et al.'s type system Syntactic Control of Interference Revisited (SCIR) is considered. The previously proposed notion of bireflective model is rejected as being too restrictive to accommodate important concrete models based on game semantics and object spaces; furthermore it is argued that the existing proof-sketch of the important property of coherence for these models is incorrect. A new, more general notion of model is proposed and the coherence property proved.