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The proximal gradient method is a splitting algorithm for the minimization of the sum of two convex functions, one of which is smooth. It has applications in areas such as mechanics, inverse problems, machine learning, image reconstruction, variational inequalities, statistics, operations research, and optimal transportation. Its formalism encompasses a wide variety of numerical methods in optimization such as gradient descent, projected gradient, iterative thresholding, alternating projections, the constrained Landweber method, as well as various algorithms in statistics and sparse data analysis. This paper aims at providing an account of the main properties of the proximal gradient method and to discuss some of its applications.
Functions with bounded variations are particular integrable functions whose total variations are finite. They play an important role in modern mathematical analysis. Functions with bounded variations of a single variable and with several variables are presented, with examples and counterexamples. Functions with bounded local variations are introduced succinctly. Another part is devoted to finite distributional perimeter sets (i.e. Caccioppoli sets), as well as the presentation of generalizations, extensions and restrictions. Serveral concrete examples of practical applications in functional analysis, geometry, probability and statistics, physics and mathematical imaging are detailed.
The general problem of deterministic chaos is to predict the long-term behaviour of a physical system, knowing the deterministic laws that govern it. The difficulty lies in the fact that the system is ‘sensitive to initial conditions’, making precise prediction impossible but suggesting probabilistic predictions. The general scientific approach is presented here with Lorenz’s historical model, which initially describes the convective motion of a fluid. A study of the trajectories shows us the property of ‘sensitivity to initial conditions’ and we deduce some statistical properties consistent with observations, such as the attraction of the trajectories towards a fractal set called a ‘strange attractor’ and their statistical fluctuations.
The universal real Clifford algebra associated to a real linear space of dimension n contains this linear space and also R: It has the dimension 2n as a linear space and is currently a subject of interest of a fairly large scientific community, thanks to the fact that it offers opportunities of applications. In this article, starting from a concrete problem, it is showed how such algebra can be helpful for overcoming the insufficiency of computations when the latter are restricted only to linear spaces. In the fact, the multiplicative law allows doing products of the linear space’s vectors.
This article, the third in a series of three, deals with nonclassical logics that are extensions, restrictions, or significant variations of classical logics. They have made it possible to formalize many non-deductive reasoning. A broad panorama (although not exhaustive) of more than two hundred and eighty logics is drawn up, each being presented in a succinct manner. They have largely developed in the second half of the 20th century, mainly in computer science and especially in artificial intelligence.
This article presents the classical theory of asset pricing for financial derivatives. Moreover, the Black and Scholes model and, more generally local volatility models, are defined from Brownian motions that we introduce. Numerical procedures to compute prices are provided with Python scripts. At last, a new approach in discrete time is presented that avoids the risk neutral probability measures.
This article, the second in a series of three, deals with the classical logics which will give rise to mathematical logic at the end of the 19th century. The logic of propositions is first presented, which is the one introduced by Aristotle and which reigned for two thousand years. Next, the logic of predicates is exposed, which imposed itself at the turn of the 19th and 20th centuries, because it admitted greater expressive power. Many didactic and application examples illustrate many points. In the appendix are listed the properties of logical connectives and the logical forms used as axioms or rules of inference, as well as a list of notations.
The term "logic" is derived from ancient Greek meaning both "speech" and "reasoning". As an interdisciplinary field of philosophy, linguistics, mathematics, and more recently computer science and especially artificial intelligence, logic deals with inference, which is defined as a "cognitive operation", elementary form of reasoning from premises to a conclusion. This article, the first in a series of three, presents elements on languages and on reasoning, before approaching logical systems, then metalogic. A glossary in the appendix precisely summarizes the definitions of many concepts.
The finite element method is used to solve the Navier-Stokes equations. This method is very well adapted to the approximation of the equations governing the behavior of fluids and can also provide an approximation of the domains of definition of the equations to be solved, in particular for the consideration of curved boundaries. The solution of these equations with various boundary conditions and with various admissible finite elements is presented. Results for error maximization are given. Several different methods for solving discretized systems are presented. Finally, some applications as well as finite element codes compared to finite volume codes are given and criteria for the choice of industrial codes are identified.
The solution of the Navier-Stokes equations by finite differences method is presented. Various concepts are presented as well as the mathematical model governing the behavior of a fluid; particular cases of formulation of the Navier-Stokes equations are indicated. Two distinct formulations are considered to solve the target problem; on the one hand the current- vorticity formulation to compute a 2D flow where one has to solve simply a Poisson equation coupled to a convection-diffusion equation. Another method also allows to solve the target equations formulated in velocity-pressure. In both cases the numerical analysis of the algorithms is presented. The last part presents the solution of the Navier-Stokes equations in turbulent regime.
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