Any integer is expressible as a sum of two squares and a cube, mixed signs being allowed. We study the analogous question for a square and two cubes, and obtain an affirmative answer in the range from $-$4,000,000 to 2,000,000. For two squares and a cube with everything positive, computations support the possibility that there are only finitely many exceptions. However, $x^2 + y^2 + z^9$ admits infinitely many positive exceptions.

We study even modular lattices having level $\ell$ and dimension $2(p-\nobreak 1)$, for p prime, and arising from the ideal class group of the p-th cyclotomic extension of $\Q(\sqrt{-\ell})$. After giving the basic theory we concentrate on Galois-invariant ideals, obtain computational results on minimal vectors and isometries, and identify several old or new extremal lattices.

The Surface Evolver has been used to minimise the surface area of various ordered structures for monodisperse foam. Additional features have enabled its application to foams of arbitrary liquid fraction. Early results for the case of dry foam (negligible liquid fraction) produced a structure having lower surface area, or energy, than Kelvin's 1887 minimal tetrakaidecahedron. The calculations reported here show that this remains the case when the liquid fraction is finite, up to about 11\%, at which point an f.c.c. arrangement of cells becomes preferable.

Backwards iteration of the $3x+1$ function starting from a fixed integer a produces a tree of preimages of a. Let $\T_{k} (a)$ denote this tree grown to depth k, and let $\T^*_{k}(a)$ denote the pruned tree resulting from the removal of all nodes $n \equiv 0 \pmod{3}$. We previously computed the maximal and minimal number of leaves in $\T^*_{k} (a)$ for all $a \not\equiv 0 \pmod{3}$ and all $k\le30$. Here we compare these data with predictions made using branching process models designed to imitate the growth of $3x+1$ trees, developed in [Lagarias and Weiss 1992]. We derive rigorous results for the branching process models. The range of variation exhibited by the $3x+1$ trees appears significantly narrower than that of the branching process models. We also study the variation in expected leaf-counts associated to the congruence class of $a \pmod{3^j}$. This variation, when properly normalized, converges almost everywhere as $j \to \infty$ to a limit function on the invertible 3-adic integers.

It is well known that the nontorsion part of the unit group of a real quadratic field $\K$ is cyclic. With no loss of generality we may assume that it has a generator $\eps_{0} > 1$, called the fundamental unit of $\K$. The natural logarithm of $\eps_{0}$ is called the regulator R of $\K$. This paper considers the following problems: How large, and how small, can R get? And how often?

The answer is simple for the problem of how small R can be, but seems to be extremely difficult for the question of how large R can get. In order to investigate this, we conducted several large-scale numerical experiments, involving the Extended Riemann Hypothesis and the Cohen--Lenstra class number heuristics. These experiments provide numerical confirmation for what is currently believed about the magnitude of R.

We propose an improvement upon the standard algorithm for computing the kernel of a polynomial map, assuming that the map sends monomials into monomials. Rather than computing a Gröbner basis in the joint polynomial ring, and then selecting only the elements of interest, we show that a moderate number of iterations of the Buchberger algorithm in the variables of the domain suffices.

We classify the primitive distance-transitive representations of the Fischer sporadic simple groups and their automorphism groups. It turns out that the only primitive distance-transitive representations of these groups are their rank 3 representations. In the process of our work, we also classify and study the primitive multiplicity-free permutation representations of these Fischer groups. Our methods, which we describe in some detail, demonstrate the use of computational and randomized techniques in the classification of distance-transitive graphs and the study of very large permutation representations.