The authors of this paper consider the anti-ferromagnetic Potts model on the the integer lattice Z2. The model has two parameters: q, the number of spins, and λ = exp(−β), where β 4 is ‘inverse temperature’. It is known that the model has strong spatial mixing if q > 7, or if q = 7 and λ = 0 or λ > 1/8, or if q = 6 and λ = 0 or λ > 1/4. The λ = 0 case corresponds to the model in which configurations are proper q-colourings of Z2. It is shown that the system has strong spatial mixing for q ≥ 6 and any λ. This implies that Glauber dynamics is rapidly mixing (so there is a fully-polynomial randomised approximation scheme for the partition function), and also that there is a unique infinite-volume Gibbs state. We also show that strong spatial mixing occurs for a larger range of λ than was previously known for q = 3, q = 4 and q = 5.

We formulate a variational problem on a Riemannian manifold M whose solutions are piecewise smooth geodesies that best fit a given data set of time labelled points in M. By a limiting process, these solutions converge to a single point in M. which we prove to be the Riemannian mean of the given points for some particular Riemannian manifolds such as Euclidean spaces, connected and compact Lie groups, and spheres.

Let $G$ be a simple simply connected exceptional algebraic group of type $G_{2}$, $F_{4}$, $E_{6}$ or $E_{7}$ over an algebraically closed field $k$ of characteristic $p>0$ with $\mathfrak{g}=\text{Lie}(G)$. For each nilpotent orbit $G\cdot e$ of $\mathfrak{g}$, we list the Jordan blocks of the action of $e$ on the minimal induced module $V_{\text{min}}$ of $\mathfrak{g}$. We also establish when the centralizers $G_{v}$ of vectors $v\in V_{\text{min}}$ and stabilizers $\text{Stab}_{G}\langle v\rangle$ of $1$-spaces $\langle v\rangle \subset V_{\text{min}}$ are smooth; that is, when $\dim G_{v}=\dim \mathfrak{g}_{v}$ or $\dim \text{Stab}_{G}\langle v\rangle =\dim \text{Stab}_{\mathfrak{g}}\langle v\rangle$.

This paper illuminates the derivation, applicability, and efficiency of the multiplicative Runge–Kutta method, derived in the framework of geometric multiplicative calculus. The removal of the restrictions of geometric multiplicative calculus on positive-valued functions of real variables and the fact that the multiplicative derivative does not exist at the roots of the function are presented explicitly to ensure that the proposed method is universally applicable. The error and stability analyses are also carried out explicitly in the framework of geometric multiplicative calculus. The method presented is applied to various problems and the results are compared to those obtained from the ordinary Runge–Kutta method. Moreover, for one example, a comparison of the computation time against relative error is worked out to illustrate the general advantage of the proposed method.

We consider the 33 conjugacy classes of genus zero, torsion-free modular subgroups, computing ramification data and Grothendieck’s dessins d’enfants. In the particular case of the index 36 subgroups, the corresponding Calabi–Yau threefolds are identified, in analogy with the index 24 cases being associated to K3 surfaces. In a parallel vein, we study the 112 semi-stable elliptic fibrations over ${ \mathbb{P} }^{1} $ as extremal K3 surfaces with six singular fibres. In each case, a representative of the corresponding class of subgroups is identified by specifying a generating set for that representative.

We find all quadratic post-critically finite (PCF) rational functions defined over $\mathbb{Q}$, up to conjugation by elements of $\mathop{\rm PGL}_2(\overline{\mathbb{Q}})$. We describe an algorithm to search for possibly PCF functions. Using the algorithm, we eliminate all but 12 rational functions, all of which are verified to be PCF. We also give a complete description of all possible rational preperiodic structures for quadratic PCF functions defined over $\mathbb{Q}$.

We test R. van Luijk’s method for computing the Picard group of a K3 surface. The examples considered are the resolutions of Kummer quartics in ℙ3. Using the theory of abelian varieties, the Picard group may be computed directly in this case. Our experiments show that the upper bounds provided by van Luijk’s method are sharp when sufficiently many primes are used. In fact, there are a lot of primes that yield a value close to the exact one. However, for many but not all Kummer surfaces V of Picard rank 18, we have for a set of primes of density at least 1/2.

We introduce the computer algebra package PyCox, written entirely in the Python language. It implements a set of algorithms, in a spirit similar to the older CHEVIE system, for working with Coxeter groups and Hecke algebras. This includes a new variation of the traditional algorithm for computing Kazhdan–Lusztig cells and W-graphs, which works efficiently for all finite groups of rank ≤8 (except E8). We also discuss the computation of Lusztig’s leading coefficients of character values and distinguished involutions (which works for E8 as well). Our experiments suggest a re-definition of Lusztig’s ‘special’ representations which, conjecturally, should also apply to the unequal parameter case. Supplementary materials are available with this article.

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.

In order to assess the security of cryptosystems based on the discrete logarithm problem in non-prime finite fields, as are the torus-based or pairing-based ones, we investigate thoroughly the case in $\mathbb{F}_{p^{6}}$ with the number field sieve. We provide new insights, improvements, and comparisons between different methods to select polynomials intended for a sieve in dimension 3 using a special-$\mathfrak{q}$ strategy. We also take into account the Galois action to increase the relation productivity of the sieving phase. To validate our results, we ran several experiments and real computations for various polynomial selection methods and field sizes with our publicly available implementation of the sieve in dimension 3, with special-$\mathfrak{q}$ and various enumeration strategies.

We exhibit a numerical method to compute three-point branched covers of the complex projective line. We develop algorithms for working explicitly with Fuchsian triangle groups and their finite-index subgroups, and we use these algorithms to compute power series expansions of modular forms on these groups.

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.

This paper [1], which was published online on 1 June 2011, has been retracted by agreement between the authors, the journal’s Editor-in-Chief Derek Holt, the London Mathematical Society and Cambridge University Press. The retraction was agreed to prevent other authors from using incorrect mathematical results. (In this paper, we compute and verify the positivity of the Li coefficients for the Dirichlet $L$-functions using an arithmetic formula established in Omar and Mazhouda, J. Number Theory 125 (2007) no. 1, 50–58; J. Number Theory 130 (2010) no. 4, 1109–1114. Furthermore, we formulate a criterion for the partial Riemann hypothesis and we provide some numerical evidence for it using new formulas for the Li coefficients.)

The Verheul homomorphism is a group homomorphism from a finite subgroup of the multiplicative group of a field to an elliptic curve. The hardness of computation of the Verheul homomorphism was shown by Verheul to be closely related to the hardness of the computational Diffie-Hellman problem. Let p ≥ 5 be a prime, and let N be a prime satisfying √(12p) < N < 2p / √3, where N ≠ p. Let E be an ordinary elliptic curve over Fp, and let C ⊂ E be a cyclic subgroup of order N. Let H be the group of all Nth roots of unity (contained in the algebraic closure of Fp), and let phi be the Verheul isomorphism from H to C.

We consider a polynomial P such that P(z) is the X-coordinate of phi(z) for all z ∈ H – {1}. We show that, for at least approximately 58% of pairs (E, C), none of the coefficients of the non-constant terms of P vanishes.

We give a natural geometric condition that ensures that sequences of interpolation polynomials (of fixed degree) of sufficiently differentiable functions with respect to the natural lattices introduced by Chung and Yao converge to a Taylor polynomial.

We use an invariant-theoretic method to compute certain twists of the modular curves $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}X(n)$ for $n=7,11$. Searching for rational points on these twists enables us to find non-trivial pairs of $n$-congruent elliptic curves over ${\mathbb{Q}}$, that is, pairs of non-isogenous elliptic curves over ${\mathbb{Q}}$ whose $n$-torsion subgroups are isomorphic as Galois modules. We also find a non-trivial pair of 11-congruent elliptic curves over ${\mathbb{Q}}(T)$, and hence give an explicit infinite family of non-trivial pairs of 11-congruent elliptic curves  over ${\mathbb{Q}}$.

Supplementary materials are available with this article.

Elliptic boundary value problems are well posed in suitable Sobolev spaces, if the boundary conditions satisfy the Shapiro–Lopatinskij condition. We propose here a criterion (which also covers over-determined elliptic systems) for checking this condition. We present a constructive method for computing the compatibility operator for the given boundary value problem operator, which is also necessary when checking the criterion. In the case of two independent variables we give a formulation of the criterion for the Shapiro–Lopatinskij condition which can be checked in a finite number of steps. Our approach is based on formal theory of PDEs, and we use constructive module theory and polynomial factorisation in our test. Actual computations were carried out with computer algebra systems Singular and MuPad.

We construct explicit $K3$ surfaces over $\mathbb{Q}$ having real multiplication. Our examples are of geometric Picard rank 16. The standard method for the computation of the Picard rank provably fails for the surfaces constructed.

We propose to generalize the work of Régis Dupont for computing modular polynomials in dimension $2$ to new invariants. We describe an algorithm to compute modular polynomials for invariants derived from theta constants and prove heuristically that this algorithm is quasi-linear in its output size. Some properties of the modular polynomials defined from quotients of theta constants are analyzed. We report on experiments with our implementation.

For an elliptic curve $E/\mathbb{Q}$ without complex multiplication we study the distribution of Atkin and Elkies primes $\ell$, on average, over all good reductions of $E$ modulo primes $p$. We show that, under the generalized Riemann hypothesis, for almost all primes $p$ there are enough small Elkies primes $\ell$ to ensure that the Schoof–Elkies–Atkin point-counting algorithm runs in $(\log p)^{4+o(1)}$ expected time.