On divisors of pseudovarieties generated by some classes of full transformation semigroups

Algebra Colloquium, 15 (2008), p. 581–588

Generic Nekhoroshev theory without small divisors

In this article, we present a new approach of Nekhoroshev theory for a generic unperturbed Hamiltonian which completely avoids small divisors problems. The proof is an extension of a method introduced by P. Lochak which combines averaging along periodic orbits with simultaneous Diophantine approximation and uses geometric arguments designed by the second author to handle generic integrable Hamiltonians. This method allows to deal with generic non-analytic Hamiltonians and to obtain new results of generic stability around linearly stable tori.

Trisecció de divisors sobre corbes hiperel·líptiques de gènere 2

La criptografia amb corbes el·liptiques i hiperel·líptiques ha adquirit un gran interès en els últims anys. Per a determinar si una corba és bona criptogràficament és necessari, en el cas d'una corba el·liptica trobar el cardinal del grup de punts i, en el cas d'una corba hiperel·líptica trobar el cardinal de la varietat jacobiana. En aquest treball donem mètodes per trobar els trisecats d'un divisor d'una corba de gènere 2. Això ens permetrà determinar el subgrup de 3-Sylow i trobar la major potencia de 3 que divideix al cardinal de la seva Jacobiana, obtenin així informació sobre aquest cardinal.

Fraccions, múltiples i divisors. 'Fracciones, múltiplos y divisores'.

Paquetes de actividades bajo Windows sobre números múltiples y divisores, descomposición en números primos y operaciones con fracciones para educación primaria y secundaria obligatoria. Permite instalar el programa o hacer una copia en disquete.

Rational Hausdorff Divisors: A New approach to the Approximate Parametrization of Curves.

When applying computational mathematics in practical applications, even though one may be dealing with a problem that can be solved algorithmically, and even though one has good algorithms to approach the solution, it can happen, and often it is the case, that the problem has to be reformulated and analyzed from a different computational point of view. This is the case of the development of approximate algorithms. This paper frames in the research area of approximate algebraic geometry and commutative algebra and, more precisely, on the problem of the approximate parametrization.

Very special divisors on 4-gonal real algebraic curves

International audience

Mètodes de multiplicació escalar en corbes de gènere 2

En els últims anys la criptografia basada en el problema del logaritme discret ha despertat un gran interes. Un dels grups que s'utilitza per plantejar aquest problema és el conjunt de divisors de la varietat jacobiana d'una corba de gènere 2, i, l'operació bàsica és el producte escalar de divisions. En aquest treball estudiem mètodes per calcular aquest producte de forma eficient emprant tècniques de harving de divisors. Comparem també aquest mètode amb el de doblat de divisors, mostrant una comparativa entre ells.

Borcherds products and arithmetic intersection theory on Hilbert modular surfaces

We prove an arithmetic version of a theorem of Hirzebruch and Zagier saying that Hirzebruch-Zagier divisors on a Hilbert modular surface are the coefficients of an elliptic modular form of weight 2. Moreover, we determine the arithmetic selfintersection number of the line bundle of modular forms equipped with its Petersson metric on a regular model of a Hilbert modular surface, and we study Faltings heights of arithmetic Hirzebruch-Zagier divisors.

Rings, Group Rings, and Their Graphs

We associate some graphs to a ring R and we investigate the interplay between the ring-theoretic properties of R and the graph-theoretic properties of the graphs associated to R. Let Z(R) be the set of zero-divisors of R. We define an undirected graph ᴦ(R) with nonzero zero-divisors as vertices and distinct vertices x and y are adjacent if xy=0 or yx=0. We investigate the Isomorphism Problem for zero-divisor graphs of group rings RG. Let Sk denote the sphere with k handles, where k is a non-negative integer, that is, Sk is an oriented surface of genus k. The genus of a graph is the minimal integer n such that the graph can be embedded in Sn. The annihilating-ideal graph of R is defined as the graph AG(R) with the set of ideals with nonzero annihilators as vertex such that two distinct vertices I and J are adjacent if IJ=(0). We characterize Artinian rings whose annihilating-ideal graphs have finite genus. Finally, we extend the definition of the annihilating-ideal graph to non-commutative rings.

Prime Rational Functions and Integral Polynomials

Let f(x) be a complex rational function. In this work, we study conditions under which f(x) cannot be written as the composition of two rational functions which are not units under the operation of function composition. In this case, we say that f(x) is prime. We give sufficient conditions for complex rational functions to be prime in terms of their degrees and their critical values, and we derive some conditions for the case of complex polynomials. We consider also the divisibility of integral polynomials, and we present a generalization of a theorem of Nieto. We show that if f(x) and g(x) are integral polynomials such that the content of g divides the content of f and g(n) divides f(n) for an integer n whose absolute value is larger than a certain bound, then g(x) divides f(x) in Z[x]. In addition, given an integral polynomial f(x), we provide a method to determine if f is irreducible over Z, and if not, find one of its divisors in Z[x].

Distribution asymptotique du nombre de diviseurs premiers distincts inférieurs ou égaux à m

Le sujet principal de ce mémoire est l'étude de la distribution asymptotique de la fonction f_m qui compte le nombre de diviseurs premiers distincts parmi les nombres premiers $p_1,...,p_m$. Au premier chapitre, nous présentons les sept résultats qui seront démontrés au chapitre 4. Parmi ceux-ci figurent l'analogue du théorème d'Erdos-Kac et un résultat sur les grandes déviations. Au second chapitre, nous définissons les espaces de probabilités qui serviront à calculer les probabilités asymptotiques des événements considérés, et éventuellement à calculer les densités qui leur correspondent. Le troisième chapitre est la partie centrale du mémoire. On y définit la promenade aléatoire qui, une fois normalisée, convergera vers le mouvement brownien. De là, découleront les résultats qui formeront la base des démonstrations de ceux chapitre 1.

Exercise sheet 8

Exercises and solutions in LaTex

Exercise sheet 2

Exercises and solutions in LaTex

Exercise sheet 1

Exercises and solutions in LaTex

Exercise sheet 8 pdf

Exercises and solutions in PDF