In this paper we describe how to compute smallest monic polynomials that define a given number field $\mathbb{K}$. We make use of the one-to-one correspondence between monic defining polynomials of $\mathbb{K}$ and algebraic integers that generate $\mathbb{K}$. Thus, a smallest polynomial corresponds to a vector in the lattice of integers of $\mathbb{K}$ and this vector is short in some sense. The main idea is to consider weighted coordinates for the vectors of the lattice of integers of $\mathbb{K}$. This allows us to find the desired polynomial by enumerating short vectors in these weighted lattices. In the context of the subexponential algorithm of Biasse and Fieker for computing class groups, this algorithm can be used as a precomputation step that speeds up the rest of the computation. It also widens the applicability of their faster conditional method, which requires a defining polynomial of small height, to a much larger set of number field descriptions.

The exterior square of a multiset is a natural combinatorial construction which is related to the exterior square of a vector space. We consider multisets of elements of an abelian group. Two properties are defined which a multiset may satisfy: recognisability and involution-recognisability. A polynomial-time algorithm is described which takes an input multiset and returns either (a) a multiset which is either recognisable or involution-recognisable and whose exterior square equals the input multiset, or (b) the message that no such multiset exists. The proportion of multisets which are neither recognisable nor involution-recognisable is shown to be small when the abelian group is finite but large. Some further comments are made about the motivating case of multisets of eigenvalues of matrices.

The spread of a group G is the greatest number r such that, for every set of non-trivial elements {x1,…,xr}, there exists an element y with the property that 〈xi, y〉 = G for 1 ≤ i ≤ r. In this paper we obtain good upper bounds for the spread of fourteen sporadic simple groups computationally, and we determine the value of the spread of M11 by hand.

For each group G of order up to 30 we compute a small 3-dimensional CW-space X with π1X≌ G and π2X = 0, and we quantify the ‘efficiency’ of X. Furthermore, we give a theoretical result for treating the case when G is a semi-direct product of two groups for which 3-presentations are known. We also describe the ZG-module structure on the second homotopy group π2X2 of the 2-skeleton of X. This module structure can in principle be used to determine the co-homology groups H2(G, A) and H3(G, A) with coefficients in a ZG-module A. Our computations, which involve the Todd–Coxeter procedure for coset enumeration and the LLL algorithm for finding bases of integer lattices, are rather naive in that the LLL algorithm is applied to matrices of dimension a multiple of |G|. Thus, in their present form, our techniques can be used only on small groups (say of order up to several hundred). They can in principle be used to construct (crossed) ZG-resolutions of Z, but again, only for small G. The paper is accompanied by two attachment files. The first of these is a summary of our computations in HTML format. The second contains various GAP programs used in the computations.

In the original article [LMS J. Comput. Math. 15 (2012) 71–83], the authors use a discrete form of the Itô formula, developed by Appleby, Berkolaiko and Rodkina [Stochastics 81 (2009) no. 2, 99–127], to show that the almost sure asymptotic stability of a particular two-dimensional test system is preserved when the discretisation step size is small. In this Corrigendum, we identify an implicit assumption in the original proof of the discrete Itô formula that, left unaddressed, would preclude its application to the test system of interest. We resolve this problem by reproving the relevant part of the discrete Itô formula in such a way that confirms its applicability to our test equation. Thus, we reaffirm the main results and conclusions of the original article.

We consider proper 5-colourings of the kagome lattice. Proper q-colourings correspond to configurations in the zero-temperature q-state anti-ferromagnetic Potts model. Salas and Sokal have given a computer assisted proof of strong spatial mixing on the kagome lattice for q ≥ 6 under any temperature, including zero temperature. It is believed that there is strong spatial mixing for q ≥ 4. Here we give a computer assisted proof of strong spatial mixing for q = 5 and zero temperature. It is commonly known that strong spatial mixing implies that there is a unique infinite-volume Gibbs measure and that the Glauber dynamics is rapidly mixing. We give a proof of rapid mixing of the Glauber dynamics on any finite subset of the vertices of the kagome lattice, provided that the boundary is free (not coloured). The Glauber dynamics is not necessarily irreducible if the boundary is chosen arbitrarily for q = 5 colours. The Glauber dynamics can be used to uniformly sample proper 5-colourings. Thus, a consequence of rapidly mixing Glauber dynamics is that there is fully polynomial randomised approximation scheme for counting the number of proper 5-colourings.

We show that for an elliptic divisibility sequence on a twist of the Fermat cubic, u3 + v3 = m, with m cube-free, all the terms beyond the first have a primitive divisor.

We determine the character table of the endomorphism ring of the permutation module associated with the multiplicity-free action of the sporadic simple Baby Monster group B on its conjugacy class 2B, where the centraliser of a 2B-element is a maximal subgroup of shape 21+22.Co2. This is one of the first applications of a new general computational technique to enumerate big orbits.

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.

We present an algorithm for computing Borcherds products, which has polynomial runtime. It deals efficiently with the bounds on Fourier expansion indices originating in Weyl chambers. Naive multiplication has exponential runtime due to inefficient handling of these bounds. An implementation of the new algorithm shows that it is also much faster in practice.

A comprehensive new perturbation theorem is posed and proven to estimate the magnitudes of roots of polynomials. The theorem successfully determines the magnitudes of roots for arbitrary degree of polynomial equations with no restrictions on the coefficients. In the previous papers ‘Pakdemirli and Elmas, Appl. Math. Comput. 216 (2010) 1645–1651’ and ‘Pakdemirli and Yurtsever, Appl. Math. Comput. 188 (2007) 2025–2028’, the given theorems were valid only for some restricted coefficients. The given theorem in this work is a generalization and unification of the past theorems and valid for arbitrary coefficients. Numerical applications of the theorem are presented as examples. It is shown that the theorem produces good estimates for the magnitudes of roots of polynomial equations of arbitrary order and unrestricted coefficients.

To support formal reasoning in mathematical and software engineering applications, it is desirable to have a generic prover that can be instantiated with a range of logics. This allows the prover to be applied to a wider variety of reasoning tasks than a fixed-logic prover. This paper describes the design principles and the architecture of the latest version of the Ergo proof engine, Ergo 6. Ergo 6 is a generic interactive theorem prover, similar to Isabelle, but with better support for proving schematic theorems with user-defined constraints, and with a different approach to handling variable scoping. A major theme of the paper is that Prolog implementation technology can be generalized to obtain efficient implementations of generic proof engines. This is demonstrated via a Qu-Prolog implementation of Ergo 6.

This article considers the positive integers $N$ for which ${\it\zeta}_{N}(s)=\sum _{n=1}^{N}n^{-s}$ has zeroes in the half-plane $\Re (s)>1$. Building on earlier results, we show that there are no zeroes for $1\leqslant N\leqslant 18$ and for $N=20,21,28$. For all other $N$ there are infinitely many such zeroes.

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

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.

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 consider a smooth system of two homogeneous quadratic equations over $\mathbb{Q}$ in $n\geqslant 13$ variables. In this case, the Hasse principle is known to hold, thanks to the work of Mordell in 1959. The only local obstruction is over $\mathbb{R}$. In this paper, we give an explicit algorithm to decide whether a nonzero rational solution exists and, if so, compute one.

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 chamber graph of the maximal 2-local geometry for M24, the Mathieu group of degree 24, is analysed extensively. In addition to determining the discs around a fixed chamber of the chamber graph, the geodesic closure of an opposite pair of chambers is investigated.

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.