An algorithm for computing the elements of a given weight of the canonical basis of a quantized enveloping algebra is described. Subsequently, a similar algorithm is presented for computing the canonical basis of a finite-dimensional module.

Let {\ASIE K}\,/{\small $\Q$}({\ASIE t \!}) be a finite extension. We describe algorithms for computing subfields and automorphisms of {\ASIE K}\,/{\small $\Q$}({\ASIE t }\!). As an application we give an algorithm for finding decompositions of rational functions in {\small $\Q(\alpha)$}. We also present an algorithm which decides if an extension {\ASIE L}\,/{\small $\Q$}({\ASIE t \!}) is a subfield of {\ASIE K}. In case [{\ASIE K : \;}{\small$\Q$}({\ASIE t \!})] = [{\ASIE L : \;}{\small $\Q$}({\ASIE t \!})] we obtain a {\small $\Q$}({\ASIE t \!})-isomorphism test. Furthermore, we describe an algorithm which computes subfields of the normal closure of {\ASIE K}\,/{\small $\Q$}({\ASIE t \!}).

We describe algorithms to determine extensions of infinite polycyclic groups having certain properties. In particular, we are interested in torsion-free extensions and extensions whose Fitting subgroup has a minimal centre. Then we apply these methods in topological applications. We discuss the calculation of Betti numbers for compact manifolds, and we describe algorithmic approaches in classifying almost Bieberbach groups.

In this paper we present a new computational record: the aliquot sequence starting at 3630 converges to 1 after reaching a hundred decimal digits. Also, we show the current status of all the aliquot sequences starting with a number smaller than 10,000; we have reached at least 95 digits for all of them. in particular, we have reached at least 112 digits for the so-called "Lehmer five sequences," and 101 digits for the "Godwin twelve sequences." Finally, we give a summar showing the number of aliquot sequences of unknown end starting with a number less than or equal 10^6.

The Weierstrass data are derived--from geometric assumptions--for a family of screw-motion-invariant minimal surfaces asymptotic to the helicoid. The period problem for these data is solved numerically and the the surfaces are approximated using adaptive mesh methods. These simulations give strong evidence that the family exists, is continuous, consists of embedded surfaces, and limits to the genus-one helicoid.

We develop numerical implementations of several criteria to assess the regularity of functions. The criteria are based on the finite difference method and harmonic analysis: Littlewood-Paley theory, and wavelet analysis.

As a first application of the methods, we study the regularity of conjugacies between critical circle maps (i.e., differentiable homeomorphisms with a critical point) with a golden mean rotation number. These maps have a very well-developed mathematical theory as well as a wealth of numerical studies.

We compare the results produced by our methods among themselves and with theorems in the mathematical literature. We confirm that several of the features that are predicted by the mathematical results are observable by numerical computation. Some universal numbers predicted can be computed reliably. Our calculations suggest that several simple upper bounds are sharp in some cases, but not in others. This indicates that there may be conceptually different mechanisms at play.

We give a geometric interpretation for the Euler-Lagrange equation for the Möbius cross energy of (nontrivially linked) 2-component links in the euclidean 3-space. The minimizer of this energy is conjectured to be a Hopf link of 2 round circles. We prove some elementary properties of the minimizers using the Euler-Lagrange equations. In particular, we give a rigorous proof of the fact that the minimizer is topologically a Hopf link.

We determine the systoles for a family of closed hyperbolic triangle surfaces which admit a particularly simple combinatorial description. We show that, in this family, there are exactly four surfaces which are maximal, i.e., for which the length of the systole is a local maximum in Teichmüller space. One of these surfaces gives a new example of a maximal surface.

This paper presents evidence for a conjecture concerning the structure of the set of braid types of periodic orbits of Smale's horseshoe map, partially ordered by Boyland's forcing order. The braid types are partitioned into totally ordered subsets, which are defined by parsing the symbolic code of a periodic orbit into two segments, the prefix and the decoration: The set of braid types of orbits with each given decoration is totally ordered, the order being given by the unimodal order on symbol sequences. The conjecture is supported by computer experiment, by proofs of special cases, and by intuitive argument in terms of pruning theory.

The values that the first two Vassiliev invariants take on prime knots with up to fourteen crossings are considered. This leads to interesting fish-like graphs. Several results about the values taken on torus knots are proved.

We investigate the inequalities {\small $\pi(M+N)\leq a\pi(M/a)+\pi(N)$} and {\small $\pi(M+N)\leq a\left(\pi(M/a)+\pi(N/a)\right)$} with {\small $a\ge1$}.

In the proof by Wiles, completed by Taylor-Wiles, of the fact that all semistable elliptic curves over {\small $\QQ$} are modular, certain deformation rings play an important role. In this note, we explicitly compute these rings for the elliptic curve {\small $Y^2+XY=X^3-X^2-X-3$} of conductor 142.

We compute certain polynomial invariants for the finite reflection groups of the types {\small $H_3$, $H_4$ and $F_4$}. Using this result, we explicitly determine the solution space of functions satisfying a mean value property related to the exceptional regular polytopes, namely, the icosahedron and dodecahedron in three dimensions and the 24-cell, 600-cell, and 120-cell in four dimensions.