Distribution dependent stochastic differential equations

Due to their intrinsic link with nonlinear Fokker-Planck equations and many other applications, distribution dependent stochastic differential equations (DDSDEs) have been intensively investigated. In this paper, we summarize some recent progresses in the study of DDSDEs, which include the correspondence of weak solutions and nonlinear Fokker-Planck equations, the well-posedness, regularity estimates, exponential ergodicity, long time large deviations, and comparison theorems.     

Encapsulated Neutral Ruthenium Catalyst for Substrate-Selective Oxidation of Alcohols

The neutral complex dichloro-{diethyl[(5-phenyl-1,3,4-oxadiazol-2-ylamino)-(4-trifluoro-methylphenyl)methyl]phosphonate} (p-cymene)-ruthenium(II) was encapsulated inside a self-assembled hexameric host obtained upon reaction of 2,8,14,20-tetra-undecyl-resorcin[4]arene and water. The formation of an inclusion complex was inferred from a combination of spectral measurements (MS, UV/Vis spectroscopy, ¹H and DOSY NMR). The ³¹P and ¹⁹F NMR spectra are consistent with motions of the ruthenium complex inside the self-assembled capsule. Molecular dynamics simulations carried out on the inclusion complex confirmed these intra-cavity movements and highlighted possible supramolecular interactions between the ruthenium first coordination sphere ligands and the inner part (aromatic rings) of the capsule. The embedded ruthenium complex was assessed in the catalytic oxidation (using NaIO₄ as oxidant) of mixtures of three arylmethyl alcohols into the corresponding aldehydes. The reaction kinetics were shown to vary as a function of the substrates’ size, with the oxidation rate varying in the order benzylalcohol >4-phenyl-benzylalcohol >9-anthracenemethanol. Control experiments realized in the absence of hexameric capsule did not allow any discrimination between the substrates.     

Differentiable spaces that are subcartesian

We show that the differential structure of the orbit space of a proper action of a Lie group on a smooth manifold is weakly reflexive. This implies that the orbit space is a differentiable space in the sense of Smith, which ensures that the orbit space has an exterior algebra of differential forms, that satisfies Smith’s version of de Rham’s theorem. Because the orbit space is a locally closed subcartesian space, it has vector fields and their flows.     

Linearizability of planar polynomial Hamiltonian systems

Isochronicity and linearizability of two-dimensional polynomial Hamiltonian systems are revisited and new results are presented. We give a new computational procedure to obtain the necessary and sufficient conditions for the linearization of a polynomial system. Using computer algebra systems we provide necessary and sufficient conditions for linearizability of Hamiltonian systems with homogeneous non-linearities of degrees 5, 6 and 7. We also present some sufficient conditions for systems with nonhomogeneous nonlinearities of degrees two, three and five.     

Analysis of a micro–macro acceleration method with minimum relative entropy moment matching

We analyse convergence of a micro–macro acceleration method for the simulation of stochastic differential equations with time-scale separation. The method alternates short bursts of path simulations with the extrapolation of macroscopic state variables forward in time. After extrapolation, a new microscopic state is constructed, consistent with the extrapolated macroscopic state, that minimises the perturbation caused by the extrapolation in a relative entropy sense. We study local errors and numerical stability of the method to prove its convergence to the full microscopic dynamics when the extrapolation time step tends to zero and the number of macroscopic state variables tends to infinity.