Some rapidly convergent formulae for special values of the Riemann zeta function are given. We obtain a generating function formula for \zet$(4n+3)$ that generalizes Apéry's series for \zet$(3)$, and appears to give the best possible series relations of this type, at least for n{\mathversion{normal}$\,<\,$}12. The formula reduces to a finite but apparently nontrivial combinatorial identity. The identity is equivalent to an interesting new integral evaluation for the central binomial coefficient. We outline a new technique for transforming and summing certain infinite series. We also derive a formula that provides strange evaluations of a large new class of nonterminating hypergeometric series.

[Editor's Note: The beautiful formulas in this paper are no longer conjectural. See note on page 194.]

A normalized modular eigenform f is said to be ordinary at a prime p if p does not divide the p-th Fourier coefficient of f. We take f to be a modular form of level $1$ and weight $k\in\{12$,$\,16$,$\,18$,$\,20$,$\,22$,$\,26\}$ and search for primes where f is not ordinary. To do this, we need an efficient way to compute the reduction modulo p of the p-th Fourier coefficient. A convenient formula was known for $k=12$; trying to understand it leads to generalized Rankin-Cohen brackets and thence to formulas that we can use to look for non-ordinary primes. We do this for $p\leq 1\,000\,000$.

Threshold Growth is a cellular automaton on an integer lattice in which the occupied set grows according to a simple local rule: a site becomes occupied if and only if it sees at least a threshold number of previously occupied sites in its prescribed neighborhood. We study the minimal number of sites that these dynamics need for persistent growth in two dimensions.

We construct an algorithm for estimating the capacity of nice compact subsets of the plane. Our method is based on a relation between capacity and Green's function together with a uniform harmonic approximation theorem. We use the simple least-square technique and, what is most important, we can obtain a bound on the error made in the estimated capacity.

The concept of nilpotency for a topological space is a generalization of simple connectivity. That it is a fruitful generalization was shown by Dror, Kan, Bousfield, Hilton, and others. In 1977 Brown and Kahn proved that the dimension of a nilpotent complex can be read from the ordinary homology groups, just as in the case of a simply connected complex. They also showed that if a nilpotent complex has finite and nontrivial fundamental group, its dimension must be at least 3.

In 1985 Lewis showed that for any finite nilpotent group there is a (not necessarily finite) three-dimensional nilpotent complex with that fundamental group. The smallest finite nilpotent group for which it was unknown whether a finite three-dimensional nilpotent complex exists was $\bold Z_2 \oplus \bold Z_6$.

The authors, together with a team of undergraduate students at Fordham University, used computers to search for three-dimensional finite nilpotent complexes over groups of the form $\bold Z_n \oplus \bold Z_m$. Such complexes were eventually found for $\bold Z_2 \oplus \bold Z_6$, $\bold Z_2 \oplus \bold Z_{10}$, and $\bold Z_3 \oplus \bold Z_6$.

This article describes the strategy for constructing nilpotent complexes of dimension three, and some of the issues in implementing the computer search. The main computational issues are "normalizing'' matrices, especially to the Smith normal form, and mapping matrices over $\bold Z$ to matrices over $\bold Z_p$ for various primes p. We conclude with a summary of the complexes discovered and open questions.

Let G be a finite group. The table of marks of G arises from a characterization of the permutation representations of G by certain numbers of fixed points. It provides a compact description of the subgroup lattice of G and enables explicit calculations in the Burnside ring of G. In this article we introduce a method for constructing the table of marks of G from tables of marks of proper subgroups of G. An implementation of this method is available in the GAP language. These computer programs are used to construct the table of marks of the sporadic simple Mathieu group $M_{24}$. The final section describes how to derive information about the structure of G from its table of marks via the investigation of certain Möbius functions and the idempotents of the Burnside ring of G. Tables with detailed information about $M_{24}$ and other groups are included.