Extensive numerical experiments have been conducted by the authors, aimed at finding the admissible range of the ratios of the first three eigenvalues of a planar Dirichlet Laplacian. The results improve the previously known theoretical estimates of M. Ashbaugh and R. Benguria. Some properties of a maximizer of the ratio λ3/λ1 are also proved in the paper.

The Alekseev–Gröbner lemma is combined with the theory of modified equations to obtain an a priori estimate for the global error of numerical integrators. This estimate is correct up to a remainder term of order h2p, where h denotes the step size and p the order of the method. It is applied to nonlinear oscillators whose behaviour is described by the Emden–Fowler equation y″+tνyn=0. The result shows explicitly that later terms sometimes blow up faster than the leading term of order hp, necessitating the whole computation. This is supported by numerical experiments.

The authors present an algorithm to construct conjugacy class representatives of the solvable primitive subgroups of Sd for a given degree d. Using this method, they determine the solvable primitive permutation groups of degree at most 6560 (that is, 38 – 1), up to conjugacy.

In this paper, the study of the expressive power of certain classes of program schemes on finite structures is continued, in relation to more mainstream logics studied in finite model theory and to computational complexity. The author shows that there exists a program scheme – whose constructs are assignments and while-loops with quantifier-free tests and which has access to a stack – that can accept a P-complete problem, the deterministic path system problem, even in the absence of non-determinism, so long as problem instances are presented in a functional style. (The proof given here leans heavily on Cook's proof that the classes of formal languages accepted by deterministic and non-deterministic logspace auxiliary pushdown machines coincide.) However, whilst this result is of independent interest, in that it leads to a deterministic model of computation capturing P, whose non-deterministic variant also captures P, the program scheme can also be used to build a successor relation in certain classes of structures (namely: the class of strongly connected locally ordered digraphs, the class of connected planar embeddings, and the class of triangulations), with the consequence that on these classes of graphs, (a fragment of) path system logic (with no built-in relations) captures exactly the polynomial-time solvable problems.

This paper presents an algorithm for computing numerical evidence for a conjecture whose validity is predicted by the requirement that the equivariant Tamagawa number conjectures for Tate motives as formulated by Burns and Flach are compatible with the functional equation of the Artin L-series. The algorithm includes methods for the computation of Fitting ideals and projective lattices over the integral group ring.

The number of non-isomorphic n-fold branched coverings of a given closed surface can be determined by the number of nonisomorphic n-fold unbranched coverings of the surface and the number of nonisomorphic connected n-fold graph coverings of a suitable bouquet of circles. A similar enumeration can also be done for regular branched coverings. Some explicit enumerations are also possible.

The paper describes an algorithm to compute the canonical basis of an irreducible module over a quantized enveloping algebra of a finite-dimensional semisimple Lie algebra. The algorithm works for any module that is constructed as a submodule of a tensor product of modules with known canonical bases.

The authors construct faithful permutation representations of maximal 2-local subgroups and classify the radical chains of the Janko simple group J4; hence the Alperin weight conjecture and the Dade reductive conjecture for J4 are verified.

In this paper, the strong mean square convergence theory is established for the numerical solutions of stochastic functional differential equations (SFDEs) under the local Lipschitz condition and the linear growth condition. These two conditions are generally imposed to guarantee the existence and uniqueness of the true solution, so the numerical results given here were obtained under quite general conditions.

In this paper, the author presents a new algorithm to recognise, constructively, when a given black-box group is a homomorphic image of the unitary group SU(d, q) for known d and q. The algorithm runs in polynomial time, assuming the existence of oracles for handling SL(2, q) subgroups, and for computing discrete logarithms in cyclic groups of order q ± 1.

The proof of the relative consistency of the axiom of choice has been mechanized using Isabelle⁄ZF, building on a previous mechanization of the reflection theorem. The heavy reliance on metatheory in the original proof makes the formalization unusually long, and not entirely satisfactory: two parts of the proof do not fit together. It seems impossible to solve these problems without formalizing the metatheory. However, the present development follows a standard textbook, Kenneth Kunen's Set theory: an introduction to independence proofs, and could support the formalization of further material from that book. It also serves as an example of what to expect when deep mathematics is formalized.

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.

Positive results are proved here about the ability of numerical simulations to reproduce the exponential mean-square stability of stochastic differential equations (SDEs). The first set of results applies under finite-time convergence conditions on the numerical method. Under these conditions, the exponential mean-square stability of the SDE and that of the method (for sufficiently small step sizes) are shown to be equivalent, and the corresponding second-moment Lyapunov exponent bounds can be taken to be arbitrarily close. The required finite-time convergence conditions hold for the class of stochastic theta methods on globally Lipschitz problems. It is then shown that exponential mean-square stability for non-globally Lipschitz SDEs is not inherited, in general, by numerical methods. However, for a class of SDEs that satisfy a one-sided Lipschitz condition, positive results are obtained for two implicit methods. These results highlight the fact that for long-time simulation on nonlinear SDEs, the choice of numerical method can be crucial.

The algorithms described in this paper were developed to investigate three problems regarding polynomials with restricted coefficients: (i) determining whether there exist polynomials with {0, 1} coefficients and repeated noncyclotomic factors, (ii) searching for polynomials with {−1, 1} coefficients and small Mahler measure, and (iii) finding polynomials with {−1, 0, 1} coefficients with a root of high multiplicity off the unit circle. The results in the first problem presented here answer a question of Odlyzko and Poonen.

It is proved, within the constructive theory of apartness spaces, that a strongly continuous mapping from a totally bounded uniform space with a countable base of entourages to a uniform space is uniformly continuous. This lifts a result of Ishihara and Schuster from metric to uniform apartness spaces. The paper is part of a systematic development of computable topology using apartness as the fundamental notion.