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Our investigations in the 1980's of Thue's method yielded determinants that we were only able to analyse successfully in part. We explain the context of our work, recount our experiences, mention our conjectures, and allude to a number of open questions.
This note contains a report of a proof by computer that the Fibonacci group $F(2,9)$ is automatic. The automatic structure can be used to solve the word problem in the group. Furthermore, it can be seen directly from the word-acceptor that the group generators have infinite order, which of course implies that the group itself is infinite.
We describe algorithms to compute self-similar measures associated to iterated function systems (i.f.s.) on an interval, and more general self-replicating measures that include Hausdorff measure on the attractor of a nonlinear i.f.s. We discuss a variety of error measurements for these algorithms. We then use the algorithms to study density properties of these measures experimentally. By density we mean the behavior of the ratio $\mu(B_r(x))/(2r)^\alpha$ as $r \rightarrow 0$, were $\alpha$ is an appropriate dimension. It is well-known that a limit usually does not exist. We have found an intriguing structure associated to these ratios that we call density diagrams. We also use density computations to approximate the exact Hausdorff measure of the attractor of an i.f.s.
In recent years I have computed versal deformations of various singularities, partly by hand, but mostly with the program Macaulay. I explain here how to do these computations. As an application I discuss the smoothability of a certain curve singularity, a case I had not been to settle with general methods. As a result I find an example of a reduced curve singularity with several smoothing components.
We study to what extent a rational point of infinite order on an extension of an abelian variety by a linear group can lie in the extension's maximal compact subgroup (for the real topology). The theory of modular forms allows us to construct such points in the toric case, and this gives a counterexample to a density property recently introduced by Waldschmidt. In contrast, we show that there are no such points in the universal extension of an elliptic curve.
On étudie dans quelle mesure un point rationnel d'ordre infini d'une extension d'une variété abelienne par un groupe linéaire peut être situé sur son sous-groupe compact maximal (pour la topologie réelle). La théorie des formes modulaires permet de construire de tels points dans le cas torique, et cela fournit un contre-exemple a une propriété de densité récemment introduite par Waldschmidt. On démontre en revanche qu'il n'en existe pas sur l'extension vectorielle universelle d'une courbe elliptique.
We present an algorithm for computer verification of the global structure of structurally stable planar vector fields. Constructing analytical proofs for the qualitative properties of phase portraits has been difficult. We try to avoid this barrier by augmenting numerical computations of trajectories of dynamical systems with error estimates that yield rigorous proofs. Our approach lends itself to high-precision estimates, because the proofs are broken into independent calculations whose length in floating-point operations does not increase with increasing precision. The algorithm is tested on a system that arises in the study of Hopf bifurcation of periodic orbits with 1:4 resonance.