A problem at the interface of differential geometry and dynamical systems gives rise to the question of what control of solutions of the Riccati equation {$\dot x+x^2=k(t)$} with positive right-hand side can be obtained from control of the forcing term k. We show that a known result about "relative'' pinching is optimal and refine two known theorems. This gives improved regularity of horospheric foliations and may be of interest in control or filtering theory.

Let {$f(x)$} be a polynomial with integral coefficients and let, for {$c>0$, $S(f(x),c)=\sum_{j \pmod c} \exp(2\pi\imath\frac{f(j)}c)$}. It has been possible, for a long time, to estimate these sums efficiently. On the other hand, when the degree of {$f(x)$} is greater than 2 very little is known about their asymptotic distribution, even though their history goes back to C. F. Gauss and E. E. Kummer. The purpose of this paper is to present both experimental and theoretic evidence for a very regular asymptotic behaviour of {$S(f(x),c)$}.

We review the main conjecture for an elliptic curve on {$\Q$} having good supersingular reduction at p and give some consequences of it. Then we define notions of {$\lambda$}-invariant and {$\mu$}-invariant in this situation, generalizing a work of Kurihara and deduce the behaviour of the order of the Shafarevich-Tate group up the cyclotomic {$\Z_p$}-extension. On examples, we give some arguments which, by combining theorems and numeral calculations, allow to calculate the order of the p-primary part of the Shafarevich-Tate group in cases that are not yet known (nontrivial Shafarevich-Tate group, curves of rank greater than 1).

Nous faisons le point sur la conjecture principale pour une courbe elliptique sur {$\Q$} ayant bonne réduction supersingulière en p et en donnons quelques conséquences. Puis nous définissons la notion de {$\lambda$} invariant et de {$\mu$} invariant dans cette situation, généralisant un travail de Kurihara et en déduisons la forme de l'ordre du groupe de Shafarevich-Tate le long de la {$\Z_p$}-extension cyclotomique. Par des exemples, nous donnons quelques arguments qui, en alliant théorèmes et calculs numériques, permettent de calculer l'ordre de la composante p-primaire du groupe de Shafarevich-Tate dans des cas non connus jusqu'à présent (groupe de Shafarevich-Tate non trivial, courbes de rang {$\geq 1$}).

A heuristic model is presented for a solution of the planar Newtonian four-body problem which has a noncollision singularity.

We study four different methods for distributing points on the sphere and numerically analyze their relative merits with respect to certain metrics.

This paper presents algorithms for computing the two fundamental units and the regulator of a cyclic cubic extension of a rational function field over a field of order {$q \equiv 1 \pmod{3}$}. The procedure is based on a method originally due to Voronoi that was recently adapted to purely cubic function fields of unit rank one. Our numerical examples show that the two fundamental units tend to have large degree, and frequently, the extension has a very small ideal class number.

The spectrum of the damped wave operator for a bounded domain in {$R^2$} is shown to be related to the asymptotic average of the damping function by the geodesic flow. This allows the calculation of an asymptotic expression for the distribution of the imaginary parts of the eigenvalues for a radially symmetric geometry. Numerical simulations confirm the theoretical model. In addition, we are able to exhibit the beautiful structure of the spectrum and the close links between the eigenfunctions, the rays of geometrical optics, and the geometry of the damping region. The MATLAB code used in this paper is provided.

In this paper, we report the computation and tabulation, using MAGMA, of all congruence subgroups of {$PSL(2,\Z)$} of genus less than or equal to 24. We include full tables of the congruence groups of genus 0, 1, 2, and 3 and a summary of the remaining cases.