We present a new algorithm for constructing a Chevalley basis for any Chevalley Lie algebra over a finite field. This is a necessary component for some constructive recognition algorithms of exceptional quasisimple groups of Lie type. When applied to a simple Chevalley Lie algebra in characteristic p⩾5, our algorithm has complexity involving the seventh power of the Lie rank, which is likely to be close to best possible.

Modular symbols of weight 2 for a congruence subgroup Γ satisfy the identity {α,γ,(α)}={β,γ(β)} for all α,β in the extended upper half plane and γ ∊ Γ. The analogue of this identity is false for modular symbols of weight greater than 2. This paper provides a definition of transportable modular symbols, which are symbols for which an analogue of the above identity holds, and proves that every cuspidal symbol can be written as a transportable symbol. As a corollary, an algorithm is obtained for computing periods of cuspforms.

Many discretization methods for differential equations that evolve in Lie groups and homogeneous spaces advance the solution in the underlying Lie algebra. The main expense of computation is the calculation of commutators, a task that can be made significantly cheaper by the introduction of appropriate bases of function values and by the exploitation of redundancies inherent in a Lie-algebraic structure by means of graded spaces. In many Lie groups of practical interest a convenient alternative to the exponential map is a Cayley transformation, and the subject of this paper is the investigation of graded algebras that occur in this context. To this end we introduce a new concept, a hierarchical algebra, a Lie algebra equipped with a countable number of m-nary multilinear operations which display alternating symmetry and a ‘hierarchy condition’. We present explicit formulae for the dimension of graded subspaces of free hierarchical algebras and an algorithm for the construction of their basis. The paper is concluded by reviewing a number of applications of our results to numerical methods in a Lie-algebraic setting.

If $S$ is a quintic surface in $\mathbb{P}^{3}$ with singular set 15 3-divisible ordinary cusps, then there is a Galois triple cover ${\it\phi}:X\rightarrow S$ branched only at the cusps such that $p_{g}(X)=4$, $q(X)=0$, $K_{X}^{2}=15$ and ${\it\phi}$ is the canonical map of $X$. We use computer algebra to search for such quintics having a free action of $\mathbb{Z}_{5}$, so that $X/\mathbb{Z}_{5}$ is a smooth minimal surface of general type with $p_{g}=0$ and $K^{2}=3$. We find two different quintics, one of which is the van der Geer–Zagier quintic; the other is new.

We also construct a quintic threefold passing through the 15 singular lines of the Igusa quartic, with 15 cuspidal lines there. By taking tangent hyperplane sections, we compute quintic surfaces with singular sets $17\mathsf{A}_{2}$, $16\mathsf{A}_{2}$, $15\mathsf{A}_{2}+\mathsf{A}_{3}$ and $15\mathsf{A}_{2}+\mathsf{D}_{4}$.

In this paper we refine and extend the applicability of the algorithms in Bates and Rowley (Arch. Math. 92 (2009) 7–13) for computing part of the normalizer of a 2-subgroup in a black-box group.

Supplementary materials are available with this article.

In recent years, Com–Poisson has emerged as one of the most popular discrete models in the analysis of count data owing to its flexibility in handling different types of dispersion. However, in a stationary longitudinal Com–Poisson count data set-up where the covariates are time independent, estimation of regression and dispersion parameters based on a generalized quasi-likelihood (GQL) approach involves some major computational difficulties particularly in the inversion of the joint covariance matrix. On the other hand, in practical real-life longitudinal studies, time-dependent covariates leading to non-stationary responses are more frequently encountered. This implies that further computational problems will now arise when estimating parameters under non-stationary set-ups. This paper overcomes this problem by approximating the inverse of the ill-conditioned covariance matrix in the GQL approach through a multidimensional conjugate gradient method. The performance of this novel version of the GQL approach is then assessed on simulations of AR(1) stationary and AR(1) non-stationary longitudinal Com–Poisson counts and on real-life epileptic seizure counts. However, there is not yet an algorithm to generate non-stationary longitudinal Com–Poisson counts nor a GQL algorithm to estimate the parameters under non-stationary set-ups. Thus, the paper also provides a framework to generate non-stationary AR(1) Com–Poisson counts along with the construction of a GQL equation under non-stationary set-ups.

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?

Complete interpolating sequences for L2(−π,π) are considered under the condition that the real parts of the sequence are a subsequence of the scaled integers xℤ, x>0. It is found that this condition leads to very specific and restrictive conditions on the existence and structure of complete interpolating sequences for L2(−π,π). Further general results in the case of bunched sampling of Bernstein functions are also given.

In this article, we propose to use the character theory of compact Lie groups and their orthogonality relations for the study of Frobenius distribution and Sato–Tate groups. The results show the advantages of this new approach in several aspects. With samples of Frobenius ranging in size much smaller than the moment statistic approach, we obtain very good approximation to the expected values of these orthogonality relations, which give useful information about the underlying Sato–Tate groups and strong evidence of the correctness of the generalized Sato–Tate conjecture. In fact, $2^{10}$ to $2^{12}$ points provide satisfactory convergence. Even for $g=2$, the classical approach using moment statistics requires about $2^{30}$ sample points to obtain such information.

The point-line collinearity graph ${\mathcal{G}}$ of the maximal 2-local geometry for the largest simple Fischer group, $Fi_{24}^{\prime }$, is extensively analysed. For an arbitrary vertex $a$ of ${\mathcal{G}}$, the $i\text{th}$-disc of $a$ is described in detail. As a consequence, it follows that ${\mathcal{G}}$ has diameter $5$. The collapsed adjacency matrix of ${\mathcal{G}}$ is given as well as accompanying computer files which contain a wealth of data about ${\mathcal{G}}$.

Supplementary materials are available with this article.

We describe an algorithm for finding the coefficients of $F(X)$ modulo powers of $p$, where $p\neq 2$ is a prime number and $F(X)$ is the power series associated to the zeta function of Kubota and Leopoldt. We next calculate the 5-adic and 7-adic ${\it\lambda}$-invariants attached to those cubic extensions $K/\mathbb{Q}$ with cyclic Galois group ${\mathcal{A}}_{3}$ (up to field discriminant ${<}10^{7}$), and also tabulate the class number of $K(e^{2{\it\pi}i/p})$ for $p=5$ and $p=7$. If the ${\it\lambda}$-invariant is greater than zero, we then determine all the zeros for the corresponding branches of the $p$-adic $L$-function and deduce ${\rm\Lambda}$-monogeneity for the class group tower over the cyclotomic $\mathbb{Z}_{p}$-extension of $K$.

Supplementary materials are available with this article.

In this paper we present an application of the read-once coupling from the past algorithm to problems in Bayesian inference for latent statistical models. We describe a method for perfect simulation from the posterior distribution of the unknown mixture weights in a mixture model. Our method is extended to a more general mixture problem, where unknown parameters exist for the mixture components, and to a hidden Markov model.

We characterize nonempty open subsets of the complex plane where the sum $\zeta (s, \alpha )+ {e}^{\pm i\pi s} \hspace{0.167em} \zeta (s, 1- \alpha )$ of Hurwitz zeta functions has no zeros in $s$ for all $0\leq \alpha \leq 1$. This problem is motivated by the construction of fundamental cardinal splines of complex order $s$.

Efficient methods for computing with matrices over finite fields often involve randomised algorithms, where matrices with a certain property are sought via repeated random selection. Complexity analyses for such algorithms require knowledge of the proportion of relevant matrices in the ambient group or algebra. We introduce a method for estimating proportions of families $N$ of elements in the algebra of all $d\times d$ matrices over a field of order $q$, where membership of a matrix in $N$ depends only on its ‘invertible part’. The method is based on the availability of estimates for proportions of certain non-singular matrices depending on $N$, so that existing estimation techniques for non-singular matrices can be used to deal with families containing singular matrices. As an application, we investigate primary cyclic matrices, which are used in the Holt–Rees MEATAXE algorithm for testing irreducibility of matrix algebras.

The authors present a nearly linear-time Las Vegas algorithm that, given a large-base primitive permutation group, constructs its natural imprimitive representation. A large-base primitive permutation group is a subgroup of a wreath product of symmetric groups Sn and Sr in product action on r-tuples of k-element subsets of {1, …, n}, containing Anr. The algorithm is a randomised speed-up of a deterministic algorithm of Babai, Luks, and Seress.

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.

In this paper we give a new formula for adding $2$-coverings and $3$-coverings of elliptic curves that avoids the need for any field extensions. We show that the $6$-coverings obtained can be represented by pairs of cubic forms. We then prove a theorem on the existence of such models with integer coefficients and the same discriminant as a minimal model for the Jacobian elliptic curve. This work has applications to finding rational points of large height on elliptic curves.

We describe an algorithm for enumerating the set of level one systems of Hecke eigenvalues arising from modular forms (mod $p$).

Supplementary materials are available with this article.

For each solvable Galois group which appears in degree $9$ and each allowable signature, we find polynomials which define the fields of minimum absolute discriminant.

We study the elliptic curves in Cremona’s tables that are predicted by the Birch–Swinnerton-Dyer conjecture to have elements of order $7$ in their Tate–Shafarevich group. We show that in many cases these elements are visible in an abelian surface or abelian 3-fold.