We describe a numerical algorithm to compute surfaces that are approximately self-similar under mean curvature flow. The method restricts computation to a two-dimensional subspace of the space of embedded manifolds that is likely to contain a self-similar solution.

Using the algorithm, we recover the self-similar torus of Angenent and find several surfaces that appear to approximate previously unknown self-similar surfaces. Two of them may prove to be counterexamples to the conjecture of uniqueness of the weak solution for mean curvature flow for surfaces.

Euler expressed certain sums of the form

\sum_{k=1}^\infty \Bigl(1 + {1 \over 2^m} + \cdots + {1 \over k^m}\Bigr) (k + 1)^{-n}\hbox{,}

where m and n are positive integers, in terms of the Riemann zeta function. In [Borwein et al.\ 1993], Euler's results were extended to a significantly larger class of sums of this type, including sums with alternating signs.

This research was facilitated by numerical computations using an algorithm that can determine, with high confidence, whether or not a particular numerical value can be expressed as a rational linear combination of several given constants. The present paper presents the numerical techniques used in these computations and lists many of the experimental results that have been obtained.

Consider the collection of all integer partitions whose part sizes lie in a given set. Such a set is called monotone if the generating function has weakly increasing coefficients. The monotone subsets are classified, assuming an open conjecture.

Let $\L$ be a lattice in a Euclidean space of dimension n, containing n independant minimal vectors (this condition is verified in particular by perfect lattices in the sense of Voronoi). We study in this article the property of perfection for relative lattices spanned by minimal vectors of $\L$ for some "classical'' lattices $\L$, for the most part contained in the Leech lattice.

Soit $\L$ un réseau d'un espace euclidien de dimension n, contenant n vecteurs minimaux indépendants (une condition satisfaite en particulier par les réseaux parfaits au sens de Voronoï). Nous étudions dans cet article la perfection des réseaux relatifs engendrés par des vecteurs minimaux de $\L$ dans le cas de certains réseaux "classiques", le plus souvent contenus dans le réseau de Leech.

We report on numerical results for certain families of S-unimodal maps with flat critical point. For four one-parameter families, differing in their amount of flatness, we study the Feigenbaum limits $\alpha$ and $\delta$. There seems to be a finite $\delta$ and a finite $\alpha$ associated with each period doubling cascade in each family. Some rough numerical estimates are obtained, and our upper bound on $\delta$ is smaller than the corresponding supremum for families with nonflat critical point. One would expect that these numbers should only depend on the nature (flatness) of the maximum, and thus be constant in each family. Our data support this hypothesis for $\alpha$, but are inconclusive when it comes to $\delta$.

We consider pattern formation during the supercooling solidification of a pure material, using a phase field model. The model gives rise to a rich variety of \threed/ patterns, including very realistic dendritic crystal forms. We show how the strength of anisotropy has a crucial influence on the shape of crystals.