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.

We discuss a novel strategy for computing the eigenvalues and eigenfunctions of the relativistic Dirac operator with a radially symmetric potential. The virtues of this strategy lie in the fact that it avoids completely the phenomenon of spectral pollution and it always provides two-sided estimates for the eigenvalues with explicit error bounds on both eigenvalues and eigenfunctions. We also discuss convergence rates of the method and illustrate our results with various numerical experiments.

Toric log del Pezzo surfaces correspond to convex lattice polygons containing the origin in their interior and having only primitive vertices. Upper bounds on the volume and on the number of boundary lattice points of these polygons are derived in terms of the index ℓ. Techniques for classifying these polygons are also described: a direct classification for index two is given, and a classification for all ℓ≤16 is obtained.

If C is a curve of genus 2 defined over a field k and J is its Jacobian, then we can associate a hypersurface K in ℙ3 to J, called the Kummer surface of J. Flynn has made this construction explicit in the case when the characteristic of k is not 2 and C is given by a simplified equation. He has also given explicit versions of several maps defined on the Kummer surface and shown how to perform arithmetic on J using these maps. In this paper we generalize these results to the case of arbitrary characteristic.

In this paper we present computer-assisted proofs of a number of results in theoretical fluid dynamics and in quantum mechanics. An algorithm based on interval arithmetic yields provably correct eigenvalue enclosures and exclosures for non-self-adjoint boundary eigenvalue problems, the eigenvalues of which are highly sensitive to perturbations. We apply the algorithm to: the Orr–Sommerfeld equation with Poiseuille profile to prove the existence of an eigenvalue in the classically unstable region for Reynolds number R=5772.221818; the Orr–Sommerfeld equation with Couette profile to prove upper bounds for the imaginary parts of all eigenvalues for fixed R and wave number α; the problem of natural oscillations of an incompressible inviscid fluid in the neighbourhood of an elliptical flow to obtain information about the unstable part of the spectrum off the imaginary axis; Squire’s problem from hydrodynamics; and resonances of one-dimensional Schrödinger operators.

We use the technique of Fischer matrices to write a program to produce the character table of a group of shape (2×2.G):2 from the character tables of G, G:2, 2.G and 2.G:2.

We compute the conjugacy classes of elements and the character tables of the maximal parabolic subgroups of the simple Ree groups 2F4(q2). For one of the maximal parabolic subgroups, we find an irreducible character of the unipotent radical that does not extend to its inertia subgroup.

Two fundamental questions in the theory of Gröbner bases are decision (‘Is a basis G of a polynomial ideal a Gröbner basis?’) and transformation (‘If it is not, how do we transform it into a Gröbner basis?’) This paper considers the first question. It is well known that G is a Gröbner basis if and only if a certain set of polynomials (the S-polynomials) satisfy a certain property. In general there are m(m−1)/2 of these, where m is the number of polynomials in G, but criteria due to Buchberger and others often allow one to consider a smaller number. This paper presents two original results. The first is a new characterization theorem for Gröbner bases that makes use of a new criterion that extends Buchberger’s criteria. The second is the identification of a class of polynomial systems G for which the new criterion has dramatic impact, reducing the worst-case scenario from m(m−1)/2 S-polynomials to m−1.

Let β∈(1,2) be a Pisot number and let Hβ denote Garsia’s entropy for the Bernoulli convolution associated with β. Garsia, in 1963, showed that Hβ<1 for any Pisot β. For the Pisot numbers which satisfy xm=xm−1+xm−2+⋯+x+1 (with m≥2), Garsia’s entropy has been evaluated with high precision by Alexander and Zagier for m=2 and later by Grabner, Kirschenhofer and Tichy for m≥3, and it proves to be close to 1. No other numerical values for Hβ are known. In the present paper we show that Hβ>0.81 for all Pisot β, and improve this lower bound for certain ranges of β. Our method is computational in nature.

We study the automorphism groups of the reduction of a modular curve X0(N) over primes p∤N.

In this paper we construct the maximal subgroups of geometric type of the orthogonal groups in dimension d over GF(q) in O(d3+d2log q+log qlog log q) finite field operations.

Let k be a field of characteristic other than 2. There can be an obstruction to a principally polarized abelian threefold (A,a) over k, which is a Jacobian over , being a Jacobian over k; this can be computed in terms of the rationality of the square root of the value of a certain Siegel modular form. We show how to do this explicitly for principally polarized abelian threefolds which are the third power of an elliptic curve with complex multiplication. We use our numerical results to prove or refute the existence of some optimal curves of genus 3.

This paper demonstrates the application of a new higher-order weak approximation, called the Kusuoka approximation, with discrete random variables to non-commutative multi-factor models. Our experiments show that using the Heath–Jarrow–Morton model to price interest-rate derivatives can be practically feasible if the Kusuoka approximation is used along with the tree-based branching algorithm.

In this paper, we compute Ford fundamental domains for all genus-zero and genus-one congruence subgroups. This is a continuation of previous work, which found all such groups, including ones that are not subgroups of PSL(2,ℤ). To compute these fundamental domains, an algorithm is given that takes the following as its input: a positive square-free integer f, which determines a maximal discrete subgroup Γ0(f)+ of SL(2,ℝ); a decision procedure to determine whether a given element of Γ0(f)+ is in a subgroup G; and the index of G in Γ0(f)+. The output consists of: a fundamental domain for G, a finite set of bounding isometric circles; the cycles of the vertices of this fundamental domain; and a set of generators of G. The algorithm avoids the use of floating-point approximations. It applies, in principle, to any group commensurable with the modular group. Included as appendices are: MAGMA source code implementing the algorithm; data files, computed in a previous paper, which are used as input to compute the fundamental domains; the data computed by the algorithm for each of the congruence subgroups of genus zero and genus one; and an example, which computes the fundamental domain of a non-congruence subgroup.

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 describe an algorithm for computing successive quotients of the Schur multiplier M(G) of a group G given by an invariant finite L-presentation. As applications, we investigate the Schur multipliers of various self-similar groups, including the Grigorchuk super-group, the generalized Fabrykowski–Gupta groups, the Basilica group and the Brunner–Sidki–Vieira group.

We discuss the Mordell–Weil sieve as a general technique for proving results concerning rational points on a given curve. In the special case of curves of genus 2, we describe quite explicitly how the relevant local information can be obtained if one does not want to restrict to mod p information at primes of good reduction. We describe our implementation of the Mordell–Weil sieve algorithm and discuss its efficiency.

Supplementary materials are available with this article.

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.

Suppose that p is 3,5,7,11 or 13. We classify the radical p-chains of the Monster 𝕄 and verify the Alperin weight conjecture and Uno’s reductive conjecture for 𝕄, the latter being a refinement of Dade’s reductive conjecture and the Isaacs–Navarro conjecture.

We consider the finite W-algebra U(𝔤,e) associated to a nilpotent element e∈𝔤 in a simple complex Lie algebra 𝔤 of exceptional type. Using presentations obtained through an algorithm based on the PBW-theorem for U(𝔤,e), we verify a conjecture of Premet, that U(𝔤,e) always has a 1-dimensional representation when 𝔤 is of type G2, F4, E6 or E7. Thanks to a theorem of Premet, this allows one to deduce the existence of minimal dimension representations of reduced enveloping algebras of modular Lie algebras of the above types. In addition, a theorem of Losev allows us to deduce that there exists a completely prime primitive ideal in U(𝔤) whose associated variety is the coadjoint orbit corresponding to e.