Logarithmic signatures are a special type of group factorizations, introduced as basic components of certain cryptographic keys. Thus, short logarithmic signatures are of special interest. We deal with the question of finding logarithmic signatures of minimal length in finite groups. In particular, such factorizations exist for solvable, symmetric, and alternating groups.

We show how to use the known examples to derive minimal length logarithmic signatures for other groups. Namely, we prove the existence of such factorizations for several classical groups and---in parts by direct computation---for all groups of order <175,560 ($=\ord(J_1)$, where $J_1$ is Janko's first sporadic simple group). Whether there exists a minimal length logarithmic signature for each finite group still remains an open question.

We introduce a finite boundary type condition on iterated function systems of contractive similitudes on $\R^d$ Under this condition, we compute the Hausdorff dimension of the boundary of the attractor in terms of the spectral radius of some finite offspring matrix. We describe how to construct such a matrix. We also show that, in this case, the box dimension equals the Hausdorff dimension. In particular, this allows us to compute the Hausdorff dimension of the boundary of a class of self-similar sets defined by expansion matrices with noninteger entries.

Soient k un corps de caractéristique 0 et E une courbe elliptique définie sur k. Il existe une courbe modulaire définie sur k, unique à k-isomorphisme près, qui classifie les courbes elliptiques $E'$ telles que les modules galoisiens des points de 7-torsion de E et $E'$ soient symplectiquement isomorphes. Dans cet article, nous explicitons une équation de cette courbe, et nous en précisons l'interprétation modulaire.

Let k be a field of characteristic O, and E be an elliptic curve defined over k. There exists a modular curve defined over k, unique up to k-isomorphism, which classifies the elliptic curves $E'$ such that the modules of the 7-torsion points of E and $E'$ are galois symplectically isomorphic. In this paper, we explicit an equation of this curve and precise its modular interpretation.

We study the solutions in s of a "Dirichlet polynomial equation'' $m_1r_1^s+\dots+m_Mr_M^s=1$. We distinguish two cases. In the lattice case, when $r_j=r^{k_j}$ are powers of a common base r, the equation corresponds to a polynomial equation, which is readily solved numerically by using a computer. In the nonlattice case, when some ratio $\log r_j/\log r_1$, $j\geq 2$, is irrational, we obtain information by approximating the equation by lattice equations of higher and higher degree. We show that the set of lattice equations is dense in the set of all equations, and deduce that the roots of a nonlattice Dirichlet polynomial equation have a quasiperiodic structure, which we study in detail both theoretically and numerically.

This question is connected with the study of the complex dimensions of self-similar strings. Our results suggest, in particular, that a nonlattice string possesses a set of complex dimensions with countably many real parts (fractal dimensions) which are dense in a connected interval. Moreover, we find dimension-free regions of nonlattice self-similar strings. We illustrate our theory with several examples.

In the long term, this work is aimed in part at developing a Diophantine approximation theory of (higher-dimensional) self-similar fractals, both qualitatively and quantitatively.

In this note, we give an example of a flat manifold having a trivial group of affinities by constructing a Bieberbach group with a trivial center and trivial outer automorphism group.

We present a conjecture, based on computational results, on the area-minimizing method of enclosing and separating two arbitrary volumes in the flat cubic three-torus, $T^3$. For comparable small volumes, we prove that the standard double bubble from $R^3$ is area-minimizing.

We describe an algorithm to compute homomorphism spaces between modules of finite dimensional algebras over finite fields. The algorithm is implemented in the C-Meat-Axe

Let h(p) denote the class number of the real quadratic field formed by adjoining $\sqp$} where p is a prime, to the rationals. The Cohen-Lenstra heuristics suggest that the probability that $h(p)=k$ (a given odd positive integer) is given by $Cw(k)/k$, where C is an explicit constant and w(k) is an explicit arithmetic function. For example, we expect that about 75.45 % of the values of h(p) are 1, 12.57% are 3, and 3.77% are 5. Furthermore, a conjecture of Hooley states that

$H(x):=\sum_{p\le x}h(p)\sim x/8,$

where the sum is taken over all primes congruent to 1 modulo 4. In this paper, we develop some fast techniques for evaluating h(p) where p is not very large and provide some computational results in support of the Cohen-Lenstra heuristics. We do this by computing h(p) for all p ($\equiv1\bmod{4}$) and $p<2\cdot10^{11}$. We also tabulate H(x) up to $2\cdot10^{11}$.

We discuss various methods to compute the Fourier coefficients of the elliptic modular function j(z). We present run times to compute the coefficients in practice. If possible, we discuss the theoretical complexity of the corresponding method, too. We conclude that, in practice, an approach due to Kaneko and Zagier turns out to be most efficient.

We show that the product replacement algorithm can be used to produce random elements of the Monster group. These random elements are shown to have the same distribution of element orders as uniformly distributed random elements after a small number of steps.