We show that there is initial order rather than chaos for the solution to a Diophantine difference equation when, and only when, the associated parameter takes the smaller values of prime twins.

In this paper we discuss several heuristic strategies which allow one to solve the Whitehead minimization problem much faster (on most inputs) than the classical Whitehead algorithm. The mere fact that these strategies work in practice leads to several interesting mathematical conjectures. In particular, we conjecture that the length of most nonminimal elements in a free group can be reduced by a Nielsen automorphism which can be identified by inspecting the structure of the corresponding Whitehead graph.

In this paper we describe a simple method for obtaining a classification of small-dimensional solvable Lie algebras. Using this method, we obtain the classification of three- and four-dimensional solvable Lie algebras (over fields of any characteristic). Precise conditions for isomorphism are given.

We give an algorithm for computing the special values of twisted standard zeta functions of elliptic modular forms by using the pullback formula for the Siegel-Eisenstein series of degree 2.

We study lower bounds of the packing density of a system of nonoverlapping equal spheres in $\rb^{n}, n \geq 2,$ as a function of the maximal circumradius of its Voronoi cells. Our viewpoint, using Delone sets, allows us to investigate the gap between the upper bounds of Rogers or Kabatjanskii-Levenstein and the Minkowski-Hlawka type lower bounds for the density of lattice-packings, without entering the fundamental problem of constructing Delone sets with Delone constants between $2^{-0.401}$ and $1$. As a consequence we provide explicit asymptotic lower bounds of the covering radii (holes) of the Barnes-Wall, Craig, and Mordell-Weil lattices, respectively $BW_{n},$ $\ab_{n}^{(r)},$ and $MW_{n}$, and of the Delone constants of the BCH packings, when $n$ goes to infinity.

In a recent article the first three authors proved that in dimension $4m+1$ all homotopy spheres that bound parallelizable manifolds admit Einstein metrics of positive scalar curvature which, in fact, are Sasakian-Einstein. They also conjectured that all such homotopy spheres in dimension $4m-1, m\geq2$ admit Sasakian-Einstein metrics, and proved this for the simplest case, namely dimension 7. In this paper we describe computer programs that show that this conjecture is also true for 11-spheres and 15-spheres. Moreover, a program is given that determines the partition of the 8,610 deformation classes of Sasakian-Einstein metrics into the 28 distinct oriented diffomorphism types in dimension 7.

We present an algorithm for the computation of logarithmic $\l$-class groups of number fields. Our principal motivation is the effective determination of the $\l$-rank of the wild kernel in the {\ASI K}-theory of number fields.

An important problem in the dynamics of surface homeomorphisms is determining the forcing relation between orbits. The forcing relation between periodic orbits can be computed using existing algorithms. Here we consider forcing relations between homoclinic orbits. We outline a general procedure for computing the forcing relation and apply this to compute the equivalence and forcing relations for homoclinic orbits of the Smale horseshoe map.

Let $f(x)$ be a polynomial with integral coefficients and let, for $c>0$, $S(f(x),c)=\sum_{j \!\! \pmod c} \exp(2\pi\imath\frac{f(j)}c)$. If $f$ is a cubic polynomial then it is expected that $\sum_{c\le X} S(f(x),c) \sim k(f)X^{4/3}$. In this paper, we consider the special case $f(x)=Ax^3+Bx$ and propose a precise formula for $k(f)$. This conjecture represents a refined version of the classical Kummer conjecture.

This paper revisits the Taylor method for the numerical integration of initial value problems of Ordinary Differential Equations (ODEs). The main goal is to present a computer program that outputs a specific numerical integrator for a given set of ODEs. The generated code includes a function to compute the jet of derivatives of the solution up to a given order plus adaptive selection of order and step size at run time. The package provides support for several extended precision arithmetics, including user-defined types. The paper discusses the performance of the resulting integrator in some examples, showing that it is very competitive in many situations. This is especially true for integrations that require extended precision arithmetic. The main drawback is that the Taylor method is an explicit method, so it has all the limitations of these kind of schemes. For instance, it is not suitable for stiff systems.

Serre's conjecture relates two-dimensional odd irreducible characteristic $p$ representations to modular forms. We discuss a generalization of this conjecture (due to Ash and Sinnott) to higher-dimensional Galois representations. In particular, we give a refinement of the conjecture in the case of wildly ramified Galois representations and we provide computational evidence for this refinement.