We describe an algorithm to compute subfields of an algebraic number field as block systems of its Galois group. It relies only on symbolic computations and avoids numerical approximations.

We present a set of generators for the sporadic group $J_3$ designed to allow easily reproducible computations in the group. We also discuss the relationship between $J_3$ and $J_3{:}2$, and the maximal subgroups of these two groups.

Chen and Gackstatter constructed two complete minimal surfaces of finite total curvature, each having one Enneper-type end and all the symmetries of Enneper's surface. Karcher generalized the genus-one surface by increasing the winding order of the end. We prove that a similar generalization of the Chen-Gackstatter genus-two surface also exists. We describe a collection of immersed minimal surfaces that generalize both Chen-Gackstatter's and Karcher's surfaces by increasing the genus and the winding order of the end. The period problem associated with each of these surfaces is explained geometrically, and we present numerical evidence of its solvability for surfaces of genus as high as 35. We also make conjectures concerning these surfaces, and explain their motivation. Our numerical results led us to the Weierstrass data for several infinite-genus, one-ended, periodic minimal surfaces.

We confirm a conjecture, due to Grötschel, regarding the intersection vertices of two longest cycles in a graph. In particular, we show that if G is a graph of circumference at least $k+1$, where $k\in\{6,7\}$, and G has two longest cycles meeting in a set W of k vertices, then W is an articulation set. Grötschel had previously proved this result for $k\in\{3,4,5\}$ and shown that it fails for $k > 7$. As corollaries, we obtain results regarding the minimum lengths of longest cycles in certain vertex-transitive graphs. Our proofs are novel in that they make extensive use of a computer, although the programs themselves are straightforward

For odd $v \geq 5$, Schmerl and Spiegel have proved that the $1$-additive sequence $(2,v)$ has precisely two even terms and, consequently, is regular. For $5 \leq v \equiv 1 \bmod{4}$, we prove, using a different approach, that the $1$-additive sequence $(4,v)$ has precisely three even terms. The proof draws upon the periodicity properties of a certain ternary quadratic recurrence.

Unlike the case of $(2,v)$, the regularity of $(4,v)$ can be captured by expressions in closed form. For example, its period can be written as an exponential sum of binary digit sums. Therefore the asymptotic density $\Delta (v)$ of $(4,v)$ tends to $0$ as $v \to \infty$, but is misbehaved in the sense that

$$ \eqalign{ \liminf_{\textstyle{\vrule height 5pt width 0pt v \to \infty\atop v \equiv 1 \bmod{4}}} \left(\frac{v}{2}\right)^{2-\log_23}\Delta (v)=\quarter,\cr \limsup_{\textstyle{v \to \infty\atop v \equiv 1 \bmod{4}}} \left(\frac{v}{2}\right)^{2-\log_23}\Delta (v)> 0.27164.} $$

This is proved using techniques adapted from Harborth and Stolarsky.

We construct by computer all W-graphs corresponding to irreducible representations of Hecke algebras $H(q,n)$ for $n$ up to 15, using a modification of a method proposed by Lascoux and Schützenberger (which fails for $n>13$).

Let $\alpha$ be a unit of degree d in an algebraic number field, and assume that $\alpha$ is not a root of unity. We conduct a numerical investigation that suggests that if $\alpha$ has small Mahler measure, there are many values of n for which $1-\alpha^n$ is a unit and also many values of m for which $\Phi_m(\alpha)$ is a unit, where $\Phi_m$ is the m-th cyclotomic polynomial. We prove that the number of such values of n and m is bounded above by $O(d^{\;1+0.7/\log\log d})$, and we describe a construction of Boyd that gives a lower bound of $\Omega(d^{\;0.6/\log\log d})$.