Logarithmic signatures (LS) are a kind of factorization of finite groups which are used as a main component of cryptographic keys for secret key cryptosystems such as PGM and public key cryptosystems like MST1. As such, logarithmic signatures of short length are of special interest. In the present paper we deal with the fundamental question of the existence of logarithmic signatures of shortest length, called minimal logarithmic signatures (MLS), for finite groups. Studies of the problem can be found in several articles, especially in “On Minimal Length Factorizations of Finite Groups,” where González Vasco, Rö}tteler, and Steinwandt show that minimal logarithmic signatures exist for all groups of order less than 175,560 by direct computation using the method of factorization of a group into "disjoint'' subgroups. We introduce new approaches to deal with the question. The first method uses the double coset decomposition to construct minimal logarithmic signatures. This method allows one to prove, for instance, that if {\small $\gcd(n,q-1)\in \{1,4, p \ | \ p \ \mbox{prime} \}$}, then the projective special linear groups {\small $L_n(q)$} have an MLS. Another main goal of this paper is to construct MLS for all finite groups of order {\small $\leq 10^{10}$}. Surprisingly, the method of double coset decomposition turns out to be very effective, as we can construct MLS for all groups in the range except eight groups. We are also able to prove that if an MLS for any of these eight groups exists, then it cannot be constructed by the method of double coset decomposition. We further discuss a method of construction of MLS for groups of the form {\small $G=A.B$} with subgroups {\small $A$}, {\small $B$} and {\small $A\cap B \not= 1$}, by building suitable MLS for {\small $A$} and {\small $B$} and "gluing'' them together.

We prove that in Conway's Game of Life cellular automaton with initial configuration irrat2, the linear growth rate of the number of live cells is irrational, as conjectured in 1991 by Dean Hickerson. Our proof uses intertwined substitution sequences.

We describe how certain permutation actions of large symmetric groups can be efficiently implemented on a computer. Using a specially tailored adaptation of a general technique to enumerate huge orbits and substantial distributed computation on a cluster of workstations, we collect further evidence related to the approach to Foulkes' conjecture suggested in “A Note on Plethysm,” by S. C. Black and R. J. List.

We present numerical visualizations of Ricci flow of surfaces and three-dimensional manifolds of revolution. {\tt Ricci\_rot} is an educational tool that visualizes surfaces of revolution moving under Ricci flow. That these surfaces tend to remain embedded in {\small $\mathbb{R}^3$} is what makes direct visualization possible. The numerical lessons gained in developing this tool may be applicable to numerical simulation of Ricci flow of other surfaces. Similarly for simple three-dimensional manifolds like the 3-sphere, with a metric that is invariant under the action of {\small $SO(3)$} with 2-sphere orbits, the metric can be represented by a 2-sphere of revolution, where the distance to the axis of revolution represents the radius of a 2-sphere orbit. Hence we can also visualize the behaviour of such a metric under Ricci flow. We discuss briefly why surfaces and 3-manifolds of revolution remain embedded in {\small $\mathbb{R}^3$} and {\small $\mathbb{R}^4$}, respectively, under Ricci flow and finally indulge in some speculation about the idea of Ricci flow in the larger space of positive definite and indefinite metrics.

In this work we address the question of proving the stability of elliptic 2-periodic orbits for strictly convex billiards. Even though it is part of a widely accepted belief that ellipticity implies stability, classical theorems show that the certainty of stability relies upon more precise conditions. We present a review of the main results and general theorems and describe the procedure to fulfill the supplementary conditions for strictly convex billiards.

Using fast algorithms, we compute the Iwasawa invariants of {\small $\Bbb{Q}(\sqrt{f},\zeta_p)$} in the range {\small $1 < f < 200$} and {\small $3 \le p < 100,000$}. From these computational results, we obtain concrete information on the higher {\small $K$}-groups of the ring of integers of {\small $\Bbb{Q}(\sqrt{f})$}.

The authors have implemented three numerical quadrature schemes, using the Arbitrary Precision (ARPREC) software package. The objective here is a quadrature facility that can efficiently evaluate to very high precision a large class of integrals typical of those encountered in experimental mathematics, relying on a minimum of a priori information regarding the function to be integrated. Such a facility is useful, for example, to permit the experimental identification of definite integrals based on their numerical values. The performance and accuracy of these three quadrature schemes are compared using a suite of 15 integrals, ranging from continuous, well-behaved functions on finite intervals to functions with infinite derivatives and blow-up singularities at endpoints, as well as several integrals on an infinite interval. In results using 412-digit arithmetic, we achieve at least 400-digit accuracy, using two of the programs, for all problems except one highly oscillatory function on an infinite interval. Similar results were obtained using 1,012-digit arithmetic.

We give two numerical methods for computing the first bifurcation point for Delaunay nodoids. With regard to methods for constructing constant mean curvature surfaces, we conclude that the bifurcation point in the analytic method of Mazzeo-Pacard is the same as a limiting point encountered in the integrable systems method of Dorfmeister-Pedit-Wu.

We give 20 new Ramanujan-type formulae and 20 quadratic approximations to {\small $\pi$}, parameterized by extremal modular lattices of minimal square norm 2 and 4, at the ten special levels corresponding to square-free orders of the Mathieu group {\small $M_{23}$}. An algorithm for uncovering rational relations between two given power series that we used to discover some of the explicit relations is also given. Explicit relations parameterized by modular lattices between McKay Thompson series for the group {\small $\Gamma_0(\ell)^+$} and {\small $\Gamma_0(2\ell)+p_j$}, where {\small $p_j$} range over all odd primes dividing the special level {\small $\ell$}, are uncovered.

A semi-presentation for a group $G$ is a set of relations which characterises a set of generators of $G$ up to automorphism. We discuss some techniques for finding semi-presentations and illustrate them by exhibiting semi-presentations on standard generators for the $26$ sporadic simple groups and their automorphism groups. We then show how these semi-presentations were used to check the data in the World Wide Web Atlas of Group Representations.

A (geometric) hyperplane of a geometry is a proper subspace meeting every line. We present a complete list of the hyperplane classes of the symplectic dual polar space {\small $DW(5,2)$}. Theoretical results from Shult, Pasini and Shpectorov, and the author guarantee the existence of certain hyperplanes. To complete the list, we use a backtrack algorithm implemented in the computer algebra system GAP. We finally investigate what hyperplane classes arise from which projective embeddings of {\small $DW(5,2)$}.