Problèmes linéaires dans le Compendy de la praticque des nombres de Barthélemy de Romans et Mathieu Préhoude (1471): Une approche nouvelle basée sur des sources proches du Liber abbaci de Léonard de Pise

The Compendy de la praticque des nombres (1471) is one of a number of commercial arithmetics produced in southern France during the medieval period. Its interest and originality rest in its treatment of problem-solving. The author of the text limited his treatment to an in-depth analysis of only a few types of problems, not treating particular cases but rather emphasizing general methods. The sources from which he drew were very close to the Liber abbaci of Leonardo Fibonacci and many of them were new to the southern French arithmetic tradition. Thus, the Compendy sheds new light on the transmission of arithmetic thought into Europe. Copyright 2000 Academic Press.Le Compendy de la praticque des nombres (1471) est un traité qui appartient au groupe des arithmétiques commerciales du Sud de la France. L'intérêt et l'originalité du texte résident dans la partie consacrée à la résolution de problèmes. L'auteur sélectionne quelques types de problèmes seulement, auxquels il consacre une longue étude, délaissant les cas particuliers pour privilégier les méthodes. Ce faisant, il utilise de nouvelles sources, proches du “Liber abbaci” de Léonard de Pise, étrangères aux autres arithmétiques françaises de la même famille. Le Compendy nous apporte ainsi un éclairage nouveau sur la transmission de l'algorisme. Copyright 2000 Academic Press.MSC subject classifications: 01A35; 01A40.     

Les débuts de l''histoire des mathématiques sur les scènes internationales et le cas de l''entreprise encyclopédique de Felix Klein et Jules Molk

At the beginning of the 20th century, the history of mathematics first appeared on different international scenes as a result of different initiatives of mathematicians, historians, and philosophers. By studying the way in which historians of mathematics took advantage of these opportunities, we have sought the contours, the lines of force, and the equilibria of this process of the internationalization of the history of mathematics on an institutional as well as intellectual plane.This search was focused through the study of the German- and French-language editions of the Encyklopädie der mathematischen Wissenschaften, which is one of the most remarkable examples dating from this period of an international collaboration for mathematics and its history. Copyright 1999 Academic Press.Au début du vingtième siècle l'histoire des mathématiques fait ses débuts sur différentes scènes internationales à l'occasion de différentes initiatives de mathématiciens, d'historiens, de philosophes. En étudiant la façon dont les historiens des mathématiques se saisirent de ces opportunités, nous avons cherché à comprendre les contours, les lignes de force, les équilibres de ce processus d'internationalisation de l'histoire des mathématiques, tant sur le plan institutionnel qu'intellectuel.Cette enquête a été précisée par l'étude des éditions en langues allemande et française de l'Encyklopä-die der mathematischen Wissenschaften qui est un des exemples les plus marquant de cette période d'une collaboration internationale, tant pour les mathématiques que leur histoire. Copyright 1999 Academic Press.MSC subject classifications: 01A60, 01A74.     

Ahlfors-régularité des quasi-minima de Mumford–Shah

Let be an open set. We consider on Ω the competitors (U,K) for the reduced Mumford–Shah functional, that is to say the Mumford–Shah functional in which the -norm of U term is removed, where K is a closed subset of Ω and U is a function on ΩK with gradient in  . The main result of this paper is the following: there exists a constant c for which, whenever (U,K) is a quasi-minimizer for the reduced Mumford–Shah functional and B(x,r) is a ball centered on K and contained in Ω with bounded radius, the -measure of is bounded above by cr^N−1 and bounded below by c⁻¹r^N−1.     

Asymptotic distributions of maxima of complete and incomplete samples from multivariate stationary Gaussian sequences

Let be a sequence of d-dimensional stationary Gaussian vectors, and let denote the partial maxima of . Suppose that there are missing data in each component of and let denote the partial maxima of the observed variables. In this note, we study two kinds of asymptotic distributions of the random vector where the correlation and cross-correlation satisfy some dependence conditions.     

Comparaison des deux composantes d''un subordinateur bivarié, puis étude de l''enveloppe supérieure d''un processus de Lévy

Soit (Y,Z) un subordinateur bivarié. Nous donnons une condition suffisante pour que Y_t/Z_t converge vers zéro quand t tend vers 0 ou +∞. Ceci généralise partiellement des résultats de Bertoin et de Kesten–Erickson.Soit X un processus de Lévy et S_t=sup{X_s: st}. Soit f une fonction sous-additive. En appliquant le résultat précédent au subordinateur bivarié d'échelle, nous donnons des conditions nécéssaires et suffisantes pour que et égalent 0 ou +∞.Let (Y,Z) be a bivariate subordinator. Generalizing theorems of Bertoin and Kesten–Erickson, we give a sufficient condition for Y_t/Z_t to converge to 0 when t tends either to 0 or +∞.Let X be a Lévy process. Denote by S_t=sup{X_s: st} and let f be any sub-additive function. Applying our first result to the bivariate ladder process, we give necessary and sufficient conditions for and to be either 0 or +∞.     

Principe de Hartogs dans les variétés CR

Let M be a CR manifold. The main results of this paper are the following:

When M is real analytic, a semi-global Hartogs extension phenomenon occurs for real analytic CR functions if and only if M is nowhere strictly pseudoconvex and .

When M is a standard manifold, the Hartogs–Bochner extension phenomenon occurs for non-CR-confined domains if and only if M is nowhere strictly pseudoconvex and dim_CRM2.

If M is a smooth submanifold of foliated by complex curves, a semi-global Hartogs–Bochner extension phenomenon occurs for smooth non-CR-confined domains if and only if dim_CRM2.

If M is a real analytic nowhere strictly pseudoconvex manifold and if Ω is a sufficiently small domain in M, a hyperfunction which is real analytic in a neighborhood of bΩ and CR in a neighborhood of is in fact real analytic on Ω.

Une étude mathématique d'un modèle de Rotenberg

In this paper, a cell proliferation model of Rotenberg is mathematically studied. Each cell of this population is distinguished by its degree of maturity μ and its maturation velocity v. We equipped this mathematical model by boundary conditions wich generalize all biological rules imposed by Rotenberg. We show that this model is governed by a strongly continuous semigroup. We also develop some properties of this semigroup and we study its asymptotic behaviour in the uniform topology.     

Deviations of discrete distributions and a question of Móri

In this note, we consider a question of Móri regarding estimating the deviation of the kth terms of two discrete probability distributions in terms of the supremum distance between their generating functions over the interval [0,1]. An optimal bound for distributions on finite support is obtained. Properties of Chebyshev polynomials are employed.     

Lower classes of the Riemann–Liouville process

Let be a Riemann–Liouville process with index H>0. We characterize the lower classes of its sup-norm statistic by a unique integral test and thus measure the influence of the non-stationarity of increments.     

The supremum of Brownian local times on Hölder curves

For , we consider L^f_t, the local time of space-time Brownian motion on the curve f. Let be the class of all functions whose Hölder norm of order α is less than or equal to 1. We show that the supremum of L^f₁ over f in is finite if α>1/2 and infinite if α<1/2.     

Forward–backward SDEs with random terminal time and applications to pricing special European-type options for a large investor

In this paper, we first discuss the solvability of coupled forward–backward stochastic differential equations (FBSDEs, for short) with random terminal time. We prove the existence and uniqueness of adapted solution to such FBSDEs under some natural assumptions. The method of proof adopted is to construct a contraction mapping related to the solutions of a sequence of backward SDEs. Our monotonicity-type assumptions are different from those in Hu and Peng (1995) [4], Peng and Shi (2000) [11], and so on. As a corollary of our main result, the solvability of FBSDEs with a finite time horizon is discussed. Finally, the existence and uniqueness theorem of the solution to FBSDEs with a finite time horizon is applied to price special European-type options for a large investor.     

1D quintic nonlinear Schrödinger equation with white noise dispersion

In this article, we improve the Strichartz estimates obtained in A. de Bouard, A. Debussche (2010) [12] for the Schrödinger equation with white noise dispersion in one dimension. This allows us to prove global well posedness when a quintic critical nonlinearity is added to the equation. We finally show that the white noise dispersion is the limit of smooth random dispersion.     

Compactness of Schrödinger semigroups with unbounded below potentials

By using the super Poincaré inequality of a Markov generator L₀ on L²(μ) over a σ-finite measure space (E,F,μ), the Schrödinger semigroup generated by L₀−V for a class of (unbounded below) potentials V is proved to be L²(μ)-compact provided μ(V?N)<∞ for all N>0. This condition is sharp at least in the context of countable Markov chains, and considerably improves known ones on, e.g., Rd under the condition that V(x)→∞ as |x|→∞. Concrete examples are provided to illustrate the main result.     

Asymptotics of weighted empirical processes of linear fields with long-range dependence

We discuss the asymptotic behavior of weighted empirical processes of stationary linear random fields in with long-range dependence. It is shown that an appropriately standardized empirical process converges weakly in the uniform-topology to a degenerated process of the form fZ, where Z is a standard normal random variable and f is the marginal probability density of the underlying random field.     

General existence results for reflected BSDE and BSDE

In this paper, we are concerned with the problem of existence of solutions for generalized reflected backward stochastic differential equations (GRBSDEs for short) and generalized backward stochastic differential equations (GBSDEs for short) when the generator is continuous with general growth with respect to the variable y and stochastic quadratic growth with respect to the variable z. We deal with the case of a bounded terminal condition ξ and a bounded barrier L as well as the case of unbounded ones. This is done by using the notion of generalized BSDEs with two reflecting barriers studied in Essaky and Hassani (submitted for publication) [14]. The work is suggested by the interest the results might have in finance, control and game theory.     

Lower and upper bounds of large deviation for sums of subexponential claims in a multi-risk model

In view of the actual condition of the insurance company, a multi-risk model is proposed. The lower and upper bounds for the sums of subexponential claims in this model are given. The proof method is based on the results of the total claim amount under subexponential class.     

An excursion approach to maxima of the Brownian bridge

Distributions of functionals of Brownian bridge arise as limiting distributions in non-parametric statistics. In this paper we will give a derivation of distributions of extrema of the Brownian bridge based on excursion theory for Brownian motion. The idea of rescaling and conditioning on the local time has been used widely in the literature. In this paper it is used to give a unified derivation of a number of known distributions, and a few new ones. Particular cases of calculations include the distribution of the Kolmogorov–Smirnov statistic and the Kuiper statistic.     

Sur les bonnes valeurs initiales de la suite de Lucas–Lehmer

In this article we give an infinite number of good initial values for the Lucas–Lehmer sequence.     

Approximation of stationary solutions of Gaussian driven stochastic differential equations

We study sequences of empirical measures of Euler schemes associated to some non-Markovian SDEs: SDEs driven by Gaussian processes with stationary increments. We obtain the functional convergence of this sequence to a stationary solution to the SDE. Then, we end the paper by some specific properties of this stationary solution. We show that, in contrast to Markovian SDEs, its initial random value and the driving Gaussian process are always dependent. However, under an integral representation assumption, we also obtain that the past of the solution is independent of the future of the underlying innovation process of the Gaussian driving process.     

Reflected backward stochastic differential equation with jumps and RCLL obstacle

In this paper we study one-dimensional reflected backward stochastic differential equation when the noise is driven by a Brownian motion and an independent Poisson point process when the solution is forced to stay above a right continuous left limits obstacle. We prove existence and uniqueness of the solution by using a penalization method combined with a monotonic limit theorem.