668 research outputs found
Symmetric Differentiation on Time Scales
We define a symmetric derivative on an arbitrary nonempty closed subset of
the real numbers and derive some of its properties. It is shown that
real-valued functions defined on time scales that are neither delta nor nabla
differentiable can be symmetric differentiable.Comment: This is a preprint of a paper whose final and definite form will be
published in Applied Mathematics Letters. Submitted 30-Jul-2012; revised
07-Sept-2012; accepted 10-Sept-201
On Diamond-Alpha Dynamic Equations and Inequalities
In view of the recently developed theory of calculus for dynamic equations on time scales (which unifies discrete and continuous systems), in this project we give some of the basics of the extension of the theory to the combined delta (forward) and nabla (backward) derivatives. In this set up the newly developed theory of diamond-alpha derivatives are analyzed through some equation and inequality properties. In particular Opial type Diamond-alpha dynamic Inequalities are discussed in this context and recently developed results and their improved versions are given in this work
Probability theory on time scales and applications to finance and inequalities
In this dissertation, the recently discovered concept of time scales is applied to probability theory, thus unifying discrete, continuous and many other cases. A short introduction to the theory of time scales is provided. Following this preliminary overview, the moment generating function is derived using a Laplace transformation on time scales. Various unifications of statements and new theorems in statistics are shown. Next, distributions on time scales are defined and their properties are studied. Most of the derived formulas and statements correspond exactly to those from discrete and continuous calculus and extend the applicability to many other cases. Some theorems differ from the ones found in the literature, but improve and simplify their handling. Finally, applications to finance, economics and inequalities of Ostrowski and Grüss type are presented. Throughout this paper, our results are compared to their well known counterparts in discrete and continuous analysis and many examples are given --Abstract, page iii
Algorithmes incrémentaux pour la théorie de la fonctionnelle de la densité
The ability to model molecular systems on a computer has become a crucial tool for chemists. Indeed molecular simulations have helped to understand and predict properties of nanoscopic world, and during the last decades have had large impact on domains like biology, electronic or materials development. Particle simulation is a classical method of molecular dynamic. In particle simulation, molecules are split into atoms, their inter-atomic interactions are computed, and their time trajectories are derived step by step. Unfortunately, inter-atomic interactions computation costs prevent large systems to be modeled in a reasonable time. In this context, our research team looks for new accurate and efficient molecular simulation models. One of our team's focus is the search and elimination of useless calculus in dynamical simulations. Hence has been proposed a new adaptively restrained dynamical model in which the slowest particles movement is frozen, computational time is saved if the interaction calculus method do not compute again interactions between static atoms. The team also developed several interaction models that benefit from a restrained dynamical model, they often updates interactions incrementally using the previous time step results and the knowledge of which particle have moved.In the wake of our team's work, we propose in this thesis an incremental First-principles interaction models. Precisely, we have developed an incremental Orbital-Free Density Functional Theory method that benefits from an adaptively restrained dynamical model. The new OF-DFT model keeps computation in Real-Space, so can adaptively focus computations where they are necessary. The method is first proof-tested, then we show its ability to speed up computations when a majority of particle are static and with a restrained particle dynamic model. This work is a first step toward a combination of incremental First-principle interaction models and adaptively restrained particle dynamic models.In the wake of our team's work, we propose in this thesis an incremental First-principles interaction models. Precisely, we have developed an incremental Orbital-Free Density Functional Theory method that benefits from an adaptively restrained dynamical model. The new OF-DFT model keeps computation in Real-Space, so can adaptively focus computations where they are necessary. The method is first proof-tested, then we show its ability to speed up computations when a majority of particle are static and with a restrained particle dynamic model. This work is a first step toward a combination of incremental First-principle interaction models and adaptively restrained particle dynamic models.L'informatique est devenue un outil incontournable de la chimie. En effet la capacité de simuler des molécules sur ordinateur a aidé à la compréhension du monde nanoscopic et à la prédiction de ses propriétés. La simulation moléculaire a eu ces dernières décennies un impact scientifique énorme en biologie, en électronique, en science des matériaux ... La simulation de particules est une des méthodes classiques de dynamique moléculaire, les molécules y sont divisées en atomes, leurs interactions relatives calculées et leurs trajectoires déduites pas à pas. Malheureusement un calcul précis des interactions entre atomes demande énormément d'opérations et donc de temps, ce qui limite la portée de la simulation moléculaire à des systèmes de taille raisonnable. C'est dans ce contexte que notre équipe recherche de nouveaux modèles de simulation moléculaire rapide et précis. Un des angles de recherche est l'élimination des calculs inutiles des simulations. L'équipe a ainsi proposé un modèle de dynamique moléculaire dite restreinte de manière adaptative dans lequel le mouvement des particules les plus lentes est bloqué. Si la simulation ne recalcule pas les interactions inchangées entre atomes bloqués, le calcul des interactions est plus rapide. L'équipe a aussi développé plusieurs modèles d'interactions plus efficaces pour des modèles de dynamique restreinte de particules, ils mettent à jour les interactions de façon incrémentale en utilisant les résultats du pas de temps précédent et la liste des particules mobiles. Dans le sillage des travaux de notre équipe de recherche, nous proposons dans cette thèse une méthode incrémentale pour calculer des interactions interatomique basées sur les modèles de Théorie de la Fonctionnelle de la Densité Sans Orbitale. La nouvelle méthode garde les calculs dans l'espace réel et peut ainsi concentrer les calculs où cela est nécessaire. Dans ce manuscrit nous vérifions cette méthode, puis nous évaluons les gains de vitesse lorsqu'une majorité de particule est bloquée, avec un modèle de dynamique restreinte. Ces travaux sont un pas vers la l'intégration de modèles d'interactions Premier-principes pour des modèles dynamiques restreint de manière adaptative
Small divisor problem in the theory of three-dimensional water gravity waves
We consider doubly-periodic travelling waves at the surface of an infinitely
deep perfect fluid, only subjected to gravity and resulting from the
nonlinear interaction of two simply periodic travelling waves making an angle
between them. \newline Denoting by the dimensionless
bifurcation parameter ( is the wave length along the direction of the
travelling wave and is the velocity of the wave), bifurcation occurs for
. For non-resonant cases, we first give a large family of
formal three-dimensional gravity travelling waves, in the form of an expansion
in powers of the amplitudes of two basic travelling waves. "Diamond waves" are
a particular case of such waves, when they are symmetric with respect to the
direction of propagation.\newline \emph{The main object of the paper is the
proof of existence} of such symmetric waves having the above mentioned
asymptotic expansion. Due to the \emph{occurence of small divisors}, the main
difficulty is the inversion of the linearized operator at a non trivial point,
for applying the Nash Moser theorem. This operator is the sum of a second order
differentiation along a certain direction, and an integro-differential operator
of first order, both depending periodically of coordinates. It is shown that
for almost all angles , the 3-dimensional travelling waves bifurcate
for a set of "good" values of the bifurcation parameter having asymptotically a
full measure near the bifurcation curve in the parameter plane Comment: 119
Reinforcement learning for sequential decision-making: a data driven approach for finance
This work presents a variety of reinforcement learning applications to the
domain of nance. It composes of two-part. The rst one represents a technical
overview of the basic concepts in machine learning, which are required
to understand and work with the reinforcement learning paradigm and are
shared among the domains of applications. Chapter 1 outlines the fundamental
principle of machine learning reasoning before introducing the neural
network model as a central component of every algorithm presented in this
work. Chapter 2 introduces the idea of reinforcement learning from its roots,
focusing on the mathematical formalism generally employed in every application.
We focus on integrating the reinforcement learning framework with the
neural network, and we explain their critical role in the eld's development.
After the technical part, we present our original contribution, articulated
in three di erent essays. The narrative line follows the idea of introducing
the use of varying reinforcement learning algorithms through a trading application
(Brini and Tantari, 2021) in Chapter 3. Then in Chapter 4 we
focus on one of the presented reinforcement learning algorithms and aim at
improving its performance and scalability in solving the trading problem by
leveraging prior knowledge of the setting. In Chapter 5 of the second part,
we use the same reinforcement learning algorithm to solve the problem of
exchanging liquidity in a system of banks that can borrow and lend money,
highlighting the
exibility and the e ectiveness of the reinforcement learning
paradigm in the broad nancial domain. We conclude with some remarks
and ideas for further research in reinforcement learning applied to nance
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