Let G be a finite group. By Riemann's Existence Theorem, braid orbits of generating systems of G with product 1 correspond to irreducible families of covers of the Riemann sphere with monodromy group G. Thus, many problems on algebraic curves require the computation of braid orbits. In this paper, we describe an implementation of this computation. We discuss several applications, including the classification of irreducible families of indecomposable rational functions with exceptional monodromy group.

Recently, many new results have been found concerning algorithms for counting points on curves over finite fields of characteristic p, mostly due to the use of p-adic liftings. The complexity of these new methods is exponential in {\small $\log p$}, therefore they behave well when p is small, the ideal case being p=2. When applicable, these new methods are usually faster than those based on SEA algorithms, and are more easily extended to nonelliptic curves. We investigate more precisely this dependence on the characteristic, and in particular, we show that after a few modifications using fast algorithms for radix-conversion, Kedlaya's algorithm works in time almost linear in p. As a consequence, this algorithm can also be applied to medium values of p. We give an example of a cryptographic size genus 3 hyperelliptic curve over a finite field of characteristic 251.

We study regular and irregular sampling for functions defined on the Sierpinski Gasket (SG), where we interpret "bandlimited'' to mean the function has a finite expansion in the first {\small $d_m$} Dirichlet eigenfunctions of the Laplacian as defined by Kigami, and {\small $d_m$} is the cardinality of the sampling set. In the regular case, we take the sampling set to be the nonboundary vertices of the level m graph approximating SG. We prove that regular sampling is always possible, and we give an algorithm to compute the sampling functions, based on an extension of the spectral decimation method of Fukushima and Shima to include inner products. We give experimental evidence that the sampling functions decay rapidly away from the sampling point, in striking contrast to the classical theory on the line where the sinc function exhibits excruciatingly slow decay. Similar behavior appears to hold for certain Dirichlet kernels. We show by example that the sampling formula provides an appealing method of approximating functions that are not necessarily bandlimited, and so might be useful for numerical analysis. We give experimental evidence that reasonable perturbations of one of the regular sampling sets remains a sampling set. In contrast to what happens on the unit interval, it is not true that all sets of the correct cardinality are sampling sets.

In this paper, we give 14 examples of icosahedral representations for which we have numerically verified the Stark-Chinburg conjecture.

The hypermetric cone {\small $HYP_n$} is the set of vectors {\small $(d_{ij})_{1\leq i< j\leq n}$} satisfying the inequalities

\sum_{1\leq i<j\leq n} b_ib_jd_{ij}\leq 0\mbox{with}b_i\in\Z\mbox{and}\sum_{i=1}^{n}b_i=1\,.

A Delaunay polytope of a lattice is called extreme if the only affine bijective transformations of it into a Delaunay polytope, are the homotheties; there is correspondence between such Delaunay polytopes and extreme rays of {\small $HYP_n$}. We show that unique Delaunay polytopes of root lattices {\small $A_1$} and {\small $E_6$} are the only extreme Delaunay polytopes of dimension at most 6. We describe also the skeletons and adjacency properties of {\small $HYP_7$} and of its dual.

The computational technique used is polyhedral, i.e., enumeration of extreme rays, using the program cdd [Fukuda 03], and groups to reduce the size of the computations.

We show how to find series expansions for {\small $\pi$} of the form {\small $\pi=\sum_{n=0}^\infty {S(n)}\big/{\binom{mn}{pn}a^n}$}, where {\small $S(n)$} is some polynomial in n (depending on m, p, a). We prove that there exist such expansions for {\small $m=8k$, $p=4k$, $a=(-4)^k$}, for any k, and give explicit examples for such expansions for small values of m, p, a and a.

In this work, we propose two new methods devoted to provide a large list of new polynomials with high degree and small Mahler measure. First, by statistical considerations, we augment Mossinghoff's list of polynomials with degree at most 180, and then we give a new list of such polynomials of degree up to 300. The second idea is to perturb polynomials of Mossinghoff's list, and for higher degrees, of this new list, and to use them as initial polynomials for a minimization method, which converges to new polynomials with lower Mahler measure.

The aim of this article is the study of the circumstances under which a compact Riemann surface may contain two regular dessin d'enfants of different types. In terms of Fuchsian groups, an equivalent condition is the uniformizing group being normally contained in several different triangle groups.

The question is answered in a graph-theoretical way, providing algorithms that decide if a surface that carries a regular dessin (a quasiplatonic surface) can also carry other regular dessins.

The multiply quasiplatonic surfaces are then studied depending on their arithmetic character. Finally, the surfaces of lowest genus carrying a large number of nonarithmetic regular dessins are computed.

Based upon numerical experimentation, we claim that all caustics from any nonumbilical (nonpolar for ellipsoids of revolution) point p on any ellipsoid embedded in {\small $\mathbb{R}^3$} (except the 2-sphere) have exactly four cusps, all of which are on lines of curvature (meridians and parallels for ellipsoids of revolution) intersecting either p (even caustics) or {\small $-p$} (odd caustics). This is an extension of a statement usually attributed to Jacobi.

We produce families of irreducible cyclic presentations of the trivial group. These families comprehensively answer questions about such presentations asked by Dunwoody and by Edjvet, Hammond, and Thomas. Our theorems are purely theoretical, but their derivation is based on practical computations. We explain how we chose the computations and how we deduced the theorems.

We study the 1/2-Complex Bruno function and we produce an algorithm to evaluate it numerically, giving a characterization of the monoid {\small $\hat{\mathcal{M}}=\mathcal{M}_T\cup \mathcal{M}_S$}. We use this algorithm to test the Marmi-Moussa-Yoccoz Conjecture about the Holder continuity of the function {\small $z\mapsto -i\B(z)+ \log U\!\left(e^{2\pi i z}\right)$} on {\small $\{ z\in \C: \Im z \geq 0 \}$}, where {\small $\B$} is the 1/2-complex Bruno function and U is the Yoccoz function. We give a positive answer to an explicit question of S. Marmi et al [Marmi et al. 01].

We propose a new kind of Ramanujan-type formula for {\small $1/\pi^2$} and conjecture that it is related to the theory of modular functions.