Two different numerical methods for solving a non-self-adjoint evolution equation are compared in this paper. If the intial function lies in the domain of the operator, a recently proposed method that combines pseudospectral ideas and semigroup theory is shown to be considerably more accurate than a standard discretization method. One example is worked out in detail, but the methodsa used are of much wider applicability.

In this paper, the components in the stable model of X0(125) over C5 are determined by constructing (in the language of R. Coleman's ‘Stable maps of curves’, to appear in the Kato Volume of Doc. Math.) an explicit semi-stable covering. Empirical data is then offered regarding the placement of certain CM j-invariants in the supersingular disk of X(1) over C5, which suggests a moduli-theoretic interpretation for the components of the stable model. The paper then concludes with a conjecture regarding the stable model of X0(p3) for p > 3, which is as yet unknown.

This paper deals with the problem of finding the least length of a product of n binomials. A theorem of R. Maltby has shown that the problem is algorithmically solvable for any fixed n. Here, a different proof is presented for this result, and yields improved complexity. The author reports the results of computations of the upper bounds on the least length or Euclidean norm of a product of binomials.

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.

A k-multiset is an unordered k-tuple, perhaps with repetitions. If x is an r-multiset {x1, …, xr} and y is an s-multiset {y1, …, ys} with elements from an abelian group A the tensor product x ⊗ y is defined as the rs-multiset {xi yj | 1 ≤ i ≤ r, 1 ≤ j ≤ s}. The main focus of this paper is a polynomial-time algorithm to discover whether a given rs-multiset from A can be factorised. The algorithm is not guaranteed to succeed, but there is an acceptably small upper bound for the probability of failure. The paper also contains a description of the context of this factorisation problem, and the beginnings of an attack on the following division-problem: is a given rs-multiset divisible by a given r-multiset, and if so, how can division be achieved in polynomially bounded time?

The process of classifying possible symmetry-breaking bifurcations requires a computation involving the subgroups and irreducible representations of the original symmetry group. It is shown how this calculation can be automated using a group theory package such as GAP. This enables a number of new results to be obtained for larger symmetry groups, where manual computation is impractical. Examples of symmetric and alternating groups are given, and the method is also applied to the spatial symmetry-breaking of periodic patterns observed in experiments.

Suppose that p is 3, 5 or 7. In this paper, faithful permutation representations of maximal p-local subgroups are constructed, and the radical p-chains of the Baby Monster B are classified. Hence, the Alperin weight conjecture and the Uno reductive conjecture can be verified for B, the latter being a refinement of Dade's reductive conjecture and the Isaacs-Navarro conjecture.

The Weil descent construction of the GHS attack on the elliptic curve discrete logarithm problem (ECDLP) is generalised in this paper, to arbitrary Artin-Schreier extensions. A formula is given for the characteristic polynomial of Frobenius for the curves thus obtained, as well as a proof that the large cyclic factor of the input elliptic curve is not contained in the kernel of the composition of the conorm and norm maps. As an application, the number of elliptic curves that succumb to the basic GHS attack is considerably increased, thereby further weakening curves over GF2155.

Other possible extensions or variations of the GHS attack are discussed, leading to the conclusion that they are unlikely to yield further improvements.

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).

The decision Diffie-Hellman problem (DDH) is a central computational problem in cryptography. It is known that the Weil and Tate pairings can be used to solve many DDH problems on elliptic curves. Distortion maps are an important tool for solving DDH problems using pairings, and it is known that distortion maps exist for all super-singular elliptic curves. An algorithm is presented here to construct suitable distortion maps. The algorithm is efficient on the curves that are usable in practice, and hence all DDH problems on these curves are easy. The issue of which DDH problems on ordinary curves are easy is also discussed.

This paper contains a variety of results about the action of Con way‘s largest simple group upon the crosses in the Leech lattice. These results are tailor-made for use in ‘A Monster Graph, I'(Proc. London Math. Soc. (3) 90 (2005) 42-60), where a graph related to the Monster simple group is studied.

Prior to this paper, all small simple groups were known to be efficient, but the status of four of their covering groups was unknown. Nice, efficient presentations are provided in this paper for all of these groups, resolving the previously unknown cases. The authors‘presentations are better than those that were previously available, in terms of both length and computational properties. In many cases, these presentations have minimal possible length. The results presented here are based on major amounts of computation. Substantial use is made of systems for computational group theory and, in partic-ular, of computer implementations of coset enumeration. To assist in reducing the number of relators, theorems are provided to enable the amalgamation of power relations in certain presentations. The paper concludes with a selection of unsolved problems about efficient presentations for simple groups and their covers.

In this paper, efficient algorithms are given to test the intersection property and some of its variations on flag-transitive coset geometries. These algorithms are then applied to geometries of some sporadic groups, namely the Mathieu groups M11, M12, M22 and M23, the Janko groups J1, J2 and J3 and the Higman-Sims group HS.

This paper is concerned with (2, 3, 7)-generated linear groups of ranks less than 287. In particular, sixty new values of n are found, such that the groups SLn (q) are Hurwitz for any prime power q. This result provides the next step in deciding which classical groups are Hurwitz.

In this paper, the authors determine the suborbits of Conways largest simple group in its conjugation action on each of its three conjugacy classes of involutions. Matrix representatives of these suborbits are also provided in an accompanying computer file.