Numerical integration of vector fields over curves with zero area

construction of vector fields with positive lyapunov exponents

in this thesis our aim is to construct vector field in r3 for which the corresponding one-dimensional maps have certain discontinuities. two kinds of vector fields are considered, the first the lorenz vector field, and the second originally introced here. the latter have chaotic behavior and motivate a class of one-parameter families of maps which have positive lyapunov exponents for an open in...

Integration of Complex Vector Fields

(1.3) Lku = 0, k = 1,. . . , m. Any function u satisfying all the above equations must also satisfy the equations (1.4) [Lk9 Lh]u = LkLhu LhLku = 0. Thus it is reasonable to assume that the space spanned by the vector fields Lu..., Lm is closed under the bracket operation. Condition A. This condition is satisfied if (1.5) [Lk,IJ = I 4 ^ j where the a^6C°°((7). From (1.2) we obtain [Lk,Lh]u = Lk...

Integration of Brownian vector fields

Using the Wiener chaos decomposition, we show that strong solutions of non Lipschitzian S.D.E.'s are given by random Markovian kernels. The example of Sobolev flows is studied in some detail, exhibiting interesting phase transitions. Résumé. Intégration de champs de vecteurs browniens. En utilisant la décomposition en chaos de Wiener, nous montrons que les solutions fortes d'E.D.S. non Lipschit...

Elliptic Curves over Finite Fields

In this chapter, we study elliptic curves defined over finite fields. Our discussion will include the Weil conjectures for elliptic curves, criteria for supersingularity and a description of the possible groups arising as E(Fq). We shall use basic algebraic geometry of elliptic curves. Specifically, we shall need the notion and properties of isogenies of elliptic curves and of the Weil pairing....

Elliptic Curves over Finite Fields