Search

International audience; We revisit the seminal Brill-Noether algorithm in the rather generic situation of smooth divisors over a nodal plane projective curve. Our approach takes advantage of fast algorithms for polynomials and structured matrices. We reach sub-quadratic time for computing a basis of a Riemann-Roch space. This improves upon previously known complexity bounds.

In this paper we study the dynamical behavior of the Chebyshev–Halley methods on the family of degree n polynomials z n + c . We prove that, despite increasing the degree, it is still possible to draw the parameter space by using the orbit of a single critical point. For the methods having z = ∞ as an attracting fixed point, we show how the basins of attraction of the roots become smaller as the value of n grows. We also demonstrate that, although the convergence order of the Chebyshev–Halley family is 3, there is a member of order 4 for each value of n. In the case of quadratic polynomials, we bound the set of parameters which correspond to iterative methods with stable behaviour other than the basins of attraction of the roots.

International audience; We develop and generalize the theory of extreme value for non-stationary stochastic processes, mostly by weakening the uniform mixing condition that was previously used in this setting. We apply our results to non-autonomous dynamical systems, in particular to sequential dynamical systems, both given by uniformly expanding maps and by maps with a neutral fixed point, and to a few classes of random dynamical systems. Some examples are presented and worked out in detail.

Consider a system of polynomials in many variables over the ring of integers of a number field $K$. We prove an asymptotic formula for the number of integral zeros of this system in homogeneously expanding boxes. As a consequence, any smooth and geometrically integral variety $X\subseteq \mathbb{P}_K^m$ satisfies the Hasse principle, weak approximation and the Manin-Peyre conjecture, if only its dimension is large enough compared to its degree. This generalizes work of Skinner, who considered the case where all polynomials have the same degree, and recent work of Browning and Heath-Brown, who considered the case where $K=\mathbb{Q}$. Our main tool is Skinner's number field version of the Hardy-Littlewood circle method. As a by-product, we point out and correct an error in Skinner's treatment of the singular integral., Comment: 23 pages; minor revision; to appear in Mathematika

The purpose of this paper is two fold: we study the behaviour of the forgetful functor from S-modules to graded vector spaces in the context of algebras over an operad and derive from this theory the construction of combinatorial Hopf algebras. As a byproduct we obtain freeness and cofreeness results for these Hopf algebras. Let O denote the forgetful functor from S-modules to graded vector spaces. Left modules over an operad P are treated as P-algebras in the category of S-modules. We generalize the results obtained by Patras and Reutenauer in the associative case to any operad P: the functor O sends P-algebras to P-algebras. If P is a Hopf operad then O sends Hopf P-algebras to Hopf P-algebras. If the operad P is regular one gets two different structures of Hopf P-algebras in the category of graded vector spaces. We develop the notion of unital infinitesimal P-bialgebra and prove freeness and cofreeness results for Hopf algebras built from Hopf operads. Finally, we prove that many combinatorial Hopf algebras arise from our theory, as Hopf algebras on the faces of the permutohedra and associahedra., Section 4.3 removed. To appear in Annales Math\'ematiques Blaise Pascal

International audience; We propose a general framework for the study of L1 contractive semigroups of solutions to conservation laws with discontinuous flux: (CL) u_t + f(x; u)_x = 0; f(x; u) = f^l(u), x < 0 and f(x;u)= f^r(u); x > 0; where the fluxes f^l and f^r are mainly assumed to be continuous. Developing the ideas of a number of preceding works (Baiti and Jenssen [14], Audusse and Perthame [12], Garavello et al. [35], Adimurthi et al. [3], Buerger et al. [21]), we claim that the whole admissibility issue is reduced to the selection of a family of "elementary solutions", which are piecewise constant weak solutions of the form c(x) = c^l 1l_{x0}. We refer to such a family as a "germ". It is well known that (CL) admits many different L1 contractive semigroups, some of which reflects different physical applications. We revisit a number of the existing admissibility (or entropy) conditions and identify the germs that underly these conditions. We devote specic attention to the "vanishing viscosity" germ, which is a way to express the "Gamma-condition" of Diehl [32]. For any given germ, we formulate "germ-based" admissibility conditions in the form of a trace condition on the flux discontinuity line {x = 0} (in the spirit of Vol'pert [80]) and in the form of a family of global entropy inequalities (following Kruzhkov [50] and Carrillo [22]). We characterize those germs that lead to the L1-contraction property for the associated admissible solutions. Our approach offers a streamlined and unifying perspective on many of the known entropy conditions, making it possible to recover earlier uniqueness results under weaker conditions than before, and to provide new results for other less studied problems. Several strategies for proving the existence of admissible solutions are discussed, and existence results are given for fluxes satisfying some additional conditions. These are based on convergence results either for the vanishing viscosity method (with standard viscosity or with specic viscosities "adapted" to the choice of a germ), or for specific germ-adapted finite volume schemes.