Paris Dauphine University

About: Paris Dauphine University is a education organization based out in Paris, France. It is known for research contribution in the topics: Context (language use) & Population. The organization has 1766 authors who have published 6909 publications receiving 162747 citations. The organization is also known as: Paris Dauphine & Dauphine.

Spectral analysis of semigroups and growth-fragmentation equations

Abstract: The aim of this paper is twofold: (1) On the one hand, the paper revisits the spectral analysis of semigroups in a general Banach space setting. It presents some new and more general versions, and provides comprehensible proofs, of classical results such as the spectral mapping theorem, some (quantified) Weyl's Theorems and the Krein-Rutman Theorem. Motivated by evolution PDE applications, the results apply to a wide and natural class of generators which split as a dissipative part plus a more regular part, without assuming any symmetric structure on the operators nor Hilbert structure on the space, and give some growth estimates and spectral gap estimates for the associated semigroup. The approach relies on some factorization and summation arguments reminiscent of the Dyson-Phillips series in the spirit of those used in [87,82,48,81]. (2) On the other hand, we present the semigroup spectral analysis for three important classes of ''growth-fragmentation" equations, namely the cell division equation, the self-similar fragmentation equation and the McKendrick-Von Foerster age structured population equation. By showing that these models lie in the class of equations for which our general semigroup analysis theory applies, we prove the exponential rate of convergence of the solutions to the associated remarkable profile for a very large and natural class of fragmentation rates. Our results generalize similar estimates obtained in \cite{MR2114128,MR2536450} for the cell division model with (almost) constant total fragmentation rate and in \cite{MR2832638,MR2821681} for the self-similar fragmentation equation and the cell division equation restricted to smooth and positive fragmentation rate and total fragmentation rate which does not increase more rapidly than quadratically. It also improves the convergence results without rate obtained in \cite{MR2162224,MR2114413} which have been established under similar assumptions to those made in the present work.

The State of the Art in Enhancing Trust in Machine Learning Models with the Use of Visualizations

Abstract: Machine learning (ML) models are nowadays used in complex applications in various domains such as medicine, bioinformatics, and other sciences. Due to their black box nature, however, it may someti ...

The Metropolis–Hastings Algorithm

Abstract: This chapter is the first of a series on simulation methods based on Markov chains. However, it is a somewhat strange introduction because it contains a description of the most general algorithm of all. The next chapter (Chapter 8) concentrates on the more specific slice sampler, which then introduces the Gibbs sampler (Chapters 9 and 10), which, in turn, is a special case of the Metropolis–Hastings algorithm. (However, the Gibbs sampler is different in both fundamental methodology and historical motivation.)

Renormalized solutions of some transport equations with partially W 1,1 velocities and applications

Abstract: We prove existence and uniqueness of renormalized solutions of some transport equations with a vector field that is not W1,1 with respect to all variables but is of a particular form. Two specific applications of this new result are then treated, based upon the equivalence between transport equations and ordinary differential equations. The first one consists of a result about the dependance upon initial conditions for solutions of ODEs. The second one is related to some stochastic differential equations arising in the modelling of polymeric fluid flows.