Intervalle (mathématiques)

Résumé

En mathématiques, un intervalle (du latin intervallum) est étymologiquement un ensemble ordonné de points compris entre deux bornes. Cette notion première s'est ensuite développée jusqu'à aboutir à la notion topologique de boule d'un espace métrique. Intervalles de ℝ Inventaire Initialement, on appelle intervalle réel un ensemble de nombres délimité par deux nombres réels constituant une borne inférieure et une borne supérieure. Un intervalle contient tous les nombres réels compris entre ces deux bornes. Cette définition regroupe les intervalles des types suivants (avec a et b réels et a < b) :

{x\in\R \mid a < x < b } = ;]a;b[ (ouvert et non fermé)
{x\in\R \mid a \leq x \leq b } = [a;b] (fermé et non ouvert)
{x\in\R \mid a < x \leq b } = ;]a;b] (semi-ouvert à gauche, semi-fermé à droite)
{x\in\R \mid a \leq x < b } = [a;b[ (semi-fermé à gauche, semi-ouvert à droite) Les intervalles du premi

Source officielle

https://fr.wikipedia.org/wiki/Intervalle_(math%C3%A9matiques)

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Typicality results for weak solutions of the incompressible Navier-Stokes equations

Maria Colombo, Luigi De Rosa, Massimo Sorella

In this work we show that, in the class of L-infinity((0,T); L-2(T-3)) distributional solutions of the incompressible Navier-Stokes system, the ones which are smooth in some open interval of times are meagre in the sense of Baire category, and the Leray ones are a nowhere dense set.

EDP SCIENCES S A2022

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The subject of the present thesis is an optimal prediction problem concerning the ultimate maximum of a stable Lévy process over a finite interval of time. Such "optimal prediction" problems are of both theoretical and practical interest, in particular they have applications in finance. For instance, suppose that an investor has a long position in one financial asset, whose price is modelled by some stochastic process. The investor's objective is to determine a "best moment" at which to close out the position and to sell the asset at the highest possible price. This optimal decision must be based on continuous observations of the asset price performance and only on the information accumulated to date. Hence, the investor should use a prediction (forecasting) of the future evolution of the price of the financial security. We examine this problem in the case where the asset price is modelled by a Lévy process. Indeed, during the last several years, the application of Lévy processes in the modelling financial asset returns has become one of the active research directions in quantitative finance. Thus, this thesis contains suitable new results concerning Lévy processes. We derive the law of the supremum process associated with a strictly stable Lévy process with no negative jumps which is not a subordinator. We note that the latter problem dates back to 1973. In particular, we show that the probability density function of the supremum process can be expressed using an explicit power series representation or via an integral representation. We also derive the infinitesimal generator of the reflected process associated with a general strictly stable Lévy process. Throughout this thesis, we apply the theory of optimal stopping, the methods of fractional differential calculus, and some results from fluctuation theory. Implementing these theories in the context of Lévy processes requires the development of specific analytical results. In the case where the asset price is modelled by a spectrally positive stable Lévy process, we describe the optimal strategy under certain conditions on the model parameters. The optimal strategy is of the following form: the investor must stop the observation of the price process and sell the asset as soon as the associated reflected process crosses for the first time a particular stopping boundary. We also provide numerical estimates and simulation examples of the results obtained by using this strategy.

EPFL2007

Chargement

We consider a two-type contact process on Z in which both types have equal finite range and supercritical infection rate. We show that a given type becomes extinct with probability 1 if and only if, in the initial configuration, it is confined to a finite interval [-L, L] and the other type occupies infinitely many sites both in (-infinity, L) and (L, infinity). Additionally, we show that if both types are present in finite number in the initial configuration, then there is a positive probability that they are both present for all times. Finally, it is shown that, starting from the configuration in which all sites in (-infinity,0] are occupied by type 1 particles and all sites in (0, infinity) are occupied by type 2 particles, the process rho(t) defined by the size of the interface area between the two types at time t is tight.

2010

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EPFL2007