Approximations by Generalized Discrete Singular Operators

Here, we give the approximation properties with rates of generalized discrete versions of Picard, Gauss-Weierstrass, and Poisson-Cauchy singular operators. We cover both the unitary and non-unitary cases of the operators above. We present quantitatively the point-wise and uniform convergences of these operators to the unit operator by involving the higher modulus of smoothness of a uniformly continuous function. We also establish our results with respect to L_p norm, 1≤p\u3c∞. Additionally, we state asymptotic Voronovskaya type expansions for these operators. Moreover, we study the fractional generalized smooth discrete singular operators on the real line regarding their convergence to the unit operator with fractional rates in the uniform norm. Then, we give our results for the operators mentioned above over the real line regarding their simultaneous global smoothness preservation property with respect to L_p norm for 1≤p≤∞, by involving higher order moduli of smoothness. Here we also obtain Jackson type inequalities of simultaneous approximation which are almost sharp, containing neat constants, and they reflect the high order of differentiability of involved function. Next, we cover the approximation properties of on the general complex-valued discrete singular operators over the real line regarding their convergence to the unit operator with rates in the L_p norm for 1≤p≤∞. Finally, we establish the approximation properties of multivariate generalized discrete versions of these operators over R^N,N≥1. We give pointwise, uniform, and L_p convergence of the operators to the unit operator by involving the multivariate higher order modulus of smoothness

Mean Field Limit for Coulomb-Type Flows

We establish the mean-field convergence for systems of points evolving along the gradient flow of their interaction energy when the interaction is the Coulomb potential or a super-coulombic Riesz potential, for the first time in arbitrary dimension. The proof is based on a modulated energy method using a Coulomb or Riesz distance, assumes that the solutions of the limiting equation are regular enough and exploits a weak-strong stability property for them. The method can handle the addition of a regular interaction kernel, and applies also to conservative and mixed flows. In the appendix, it is also adapted to prove the mean-field convergence of the solutions to Newton's law with Coulomb or Riesz interaction in the monokinetic case to solutions of an Euler-Poisson type system.Comment: Final version with expanded introduction, to appear in Duke Math Journal. 35 page

Classical and Quantum Mechanical Models of Many-Particle Systems

This workshop was dedicated to the presentation of recent results in the field of the mathematical study of kinetic theory and its naturalextensions (statistical physics and fluid mechanics). The main models are the Vlasov(-Poisson) equation and the Boltzmann equation, which are obtainedas limits of many-body equations (Newton’s equations in the classical case and Schrödinger’s equation in the quantum case) thanks to the mean-field and Boltzmann-Grad scalings. Numerical aspects and applications to mechanics, physics, engineering and biology were also discussed