For any finite field {\small $k=\fq$}, we explicitly describe the k-isogeny classes of abelian surfaces defined over k and their behavior under finite field extension. In particular, we determine the absolutely simple abelian surfaces. Then, we analyze numerically what surfaces are k-isogenous to the Jacobian of a smooth projective curve of genus 2 defined over k. We prove some partial results suggested by these numerical data. For instance, we show that every absolutely simple abelian surface is k-isogenous to a Jacobian. Other facts suggested by these numerical computations are that the polynomials {\small $t^4+(1-2q)t^2+q^2$} (for all q) and {\small $t^4+(2-2q)t^2+q^2$} (for q odd) are never the characteristic polynomial of Frobenius of a Jacobian. These statements have been proved by E. Howe. The proof for the first polynomial is attached in an appendix.

We identify and investigate a class of complex Hénon maps {\small $H:\C^2\rightarrow\C^2$} that are reversible, that is, each H can be factorized as RU where {\small $R^2=U^2=\id_{\C^2}$}. Fixed points and periodic points of order two or three are classified in terms of symmetry, with respect to R or U, and as either elliptic or saddle points. We report on experimental investigation, using a Java applet, of the bounded orbits of H.

Soit {\small $\B$} le sous-espace de {\small $\Hi=L^2(0,+\infty)$} composé des fonctions {\small $f$} telles que {\small $f(t)=\sum_{k=1}^{n} c_{k} \rho \left( \frac{\theta_{k}}{t} \right),\; n\in\N,\; c_{k}\in\C,\; 0<\theta_{k}\le 1,\;\mbox{ pour } 1 \leq k \le n$}, où {\small $\rho(t)$} désigne la partie fractionnaire de t. Notons aussi {\small $\chi$} la fonction caractéristique de l'intervalle ]0,1]. Un résultat bien connu de Nyman et Beurling implique que l'hypothèse de Riemann est vraie si et seulement si {\small $d(\chi,\B)=0$}. Nous présentons ici divers résultats numériques concernant l'approximation de {\small $\chi$} par des éléments de {\small$ \B$}.

Let {\small $\B$} be the subspace of {\small $\Hi=L^2(0,+\infty)$} consisting of the functions {\small $f$} such that {\small $f(t)=\sum_{k=1}^{n} c_{k} \rho \left( \frac{\theta_{k}}{t} \right), \; n\in\N,\; c_{k}\in\C,\; 0<\theta_{k}\le 1,\;\mbox{ for } 1 \leq k \le n $}, where {\small $\rho(t)$} denotes the fractional part of {\small $t$}. We also denote by $\chi$ the characteristic function of $(0,1]$. A well known result of Nyman and Beurling implies that the Riemann hypothesis holds if and only if {\small $d(\chi,\B) = 0$}. We present several numerical results about the approximation of $\chi$ by elements of {\small $\B$}.

A Chaitin Omega number is the halting probability of a universal Chaitin (self-delimiting Turing) machine. Every Omega number is both computably enumerable (the limit of a computable, increasing,converging sequence of rationals) and random (its binary expansion is an algorithmic random sequence). In particular, every Omega number is strongly noncomputable. The aim of this paper is to describe a procedure, that combines Java programming and mathematical proofs, to compute the exact values of the first 64 bits of a Chaitin Omega:

0000001000000100000110001000011010001111110010111011101000010000.

Full description of programs and proofs will be given elsewhere.

Let (W,S) be an arbitrary Coxeter system, {\small $y\in S^*$}. We describe an algorithm which will compute, directly from {\small $y$} and the Coxeter matrix of W, the interval from the identity to {\small $y$} in the Bruhat ordering, together with the (partially defined) left and right actions of the generators. This provides us with exactly the data that are needed to compute the Kazhdan-Lusztig polynomials {\small $P_{x,z}$, $x\leq z\leq y$}. The correctness proof of the algorithm is based on a remarkable theorem due to Matthew Dyer.

Nous étudions le système dynamique défini par la transformation {\small $\Phi:]0,1]\fle ]0,1]$} où {\small $\Phi(x)=px-1$} si {\small $x\in ]1/p,1/q],$ $q$} et {\small $p$} étant deux nombres premiers consécutifs. La question de l'existence d'une mesure absolument continue invariante par {\small $\Phi$} est reliée par un argument de chaîne de Markov à une conjecture concernant un ensemble de suites de nombres premiers. Cette hypothèse est corroborée par des simulations de type Monte-Carlo. Nous montrons que cela entraîne la stabilité statistique de {\small $\Phi$} sur l'intervalle {\small $]0,2/3].$} En utilisant des arguments heuristiques nous définissons des versions simplifiées de l'opérateur de Perron-Frobenius associé à {\small $\Phi.$} Cela nous permet de construire à l'aide de Maple une densité de probabilité présentant une bonne adéquation expérimentale avec les histogrammes des orbites issues de constantes fondamentales.

We study the dynamical system defined by the transformation {\small $\Phi:]0,1]\fle ]0,1]$} where {\small $\Phi(x)=px-1$} if {\small $x\in ]1/p,1/q],$ $q$} and {\small $p$} being two consecutive prime numbers. The problem of the existence of an invariant absolutely continuous measure by {\small $\Phi$} is related via a Markov chain argument to a conjecture concerning a set of prime number sequences. This hypothesis is corroborated by Monte Carlo simulations. We prove that this implies the statistical stability of the transformation {\small $\Phi$} on the interval {\small $]0,2/3].$} By using heuristical arguments, we define simplified versions of the Perron-Frobenius operator associated to {\small $\Phi.$} Using Maple, we construct a probability density presenting a good experimental fit with the histograms of orbits stemming from fundamental constants.

We describe an algorithm to compute the trace of Hecke operators acting on the space of Hilbert cusp forms defined relative to a real quadratic field with class number greater than one. Using this algorithm, we obtain numerical data for eigenvalues and characteristic polynomials of the Hecke operators. Within the limit of our computation, the conductors of the orders spanned by the Hecke eigenvalue for any principal split prime ideal contain a nontrivial common factor, which is equal to a Hecke {\small$L$}-value.

R. M. Kashaev conjectured that the asymptotic behavior of the link invariant he introduced, which equals the colored Jones polynomial evaluated at a root of unity, determines the hyperbolic volume of any hyperbolic link complement. We observe numerically that for knots {\small $6_3$, $8_9$ and $8_{20}$} and for the Whitehead link, the colored Jones polynomials are related to the hyperbolic volumes and the Chern--Simons invariants and propose a complexification of Kashaev's conjecture.

The EKG or electrocardiogram sequence is defined by {\small $a(1) = 1$, $a(2) =2$} and, for {\small $n \ge 3$, $a(n)$} is the smallest natural number not already in the sequence with the property that {\small ${\rm gcd} \{a(n-1), a(n)\} > 1$}. In spite of its erratic local behavior, which when plotted resembles an electrocardiogram, its global behavior appears quite regular. We conjecture that almost all {\small $a(n)$} satisfy the asymptotic formula {\small $a(n) = n (1+ 1/(3 \log n)) + o(n/ \log n)$} as {\small $n \to \infty$}; and that the exceptional values {\small $a(n)=p$} and {\small $a(n)= 3p$}, for {\small $p$} a prime, produce the spikes in the EKG sequence. We prove that {\small $\{a(n): n \ge 1 \}$} is a permutation of the natural numbers and that {\small $c_1 n \le a (n) \le c_2 n$} for constants {\small $c_1, c_2$}. There remains a large gap between what is conjectured and what is proved.