We classify maximal finite irreducible subgroups of GL$_{\sans{24}} (\Q)$, together with their natural lattices. There are 65 conjugacy classes of such groups, 41 of which consist of primitive groups. New methods for finding the maximal finite supergroups of irreducible cyclic groups are developed and applied.

The existence of an invariant surface in high-dimensional systems greatly influences the behavior in a neighborhood of the invariant surface. We prove theorems that predict the behavior of periodic orbits in the vicinity of an invariant surface on which the motion is conjugate to a Diophantine rotation for symplectic maps and quasiperiodic perturbations of symplectic maps. Our results allow for efficient numerical algorithms that can serve as an indication for the breakdown of invariant surfaces.

We study the behavior of invariant sets of a volume-preserving map that is a quasiperiodic perturbation of a symplectic map, using approximation by periodic orbits. We present numerical results for analyticity domains of invariant surfaces, behavior after breakdown, and a critical function describing breakdown of invariant surfaces as a function of their rotation vectors. We discuss implications of our results to the existence of a renormalization group operator describing breakdown of invariant surfaces.

The Number Field Sieve (NFS) is the asymptotically fastest known factoring algorithm for large integers. This article describes an implementation of the NFS, including the choice of two quadratic polynomials, both classical sieving and a special form of lattice sieving (line sieving), the block Lanczos method and a new square root algorithm. Finally some data on factorizations obtained with this implementation are listed, including the record factorization of $12^{151}-1$.