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.     

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.     

Everything you wanted to know about phase and reference frequency in one- and two-dimensional NMR spectroscopy

The fundamental concept of phase discussed in this tutorial aimed at providing students with an explanation of the delays and processing parameters they may find in nuclear magnetic resonance (NMR) pulse programs. We consider the phase of radio-frequency pulses, receiver, and magnetization and how all these parameters are related to phases and offsets of signals in spectra. The impact of the off-resonance effect on the phase of the magnetization is discussed before presenting an overview of how adjustment of the time reference of the free induction decay avoids first-order correction of the phase of spectra. The main objective of this tutorial is to show how the relative phase of a pulse and the receiver can be used to change the reference frequency along direct and indirect dimensions of NMR experiments. Unusual of phase incrementation with non-90° angles will be illustrated on one- and two-dimensional NMR spectra.