Riemannian Langevin Monte Carlo schemes for sampling PSD matrices with fixed rank

This paper introduces two explicit schemes to sample matrices from Gibbs distributions on $\mathcal S^{n,p}_+$, the manifold of real positive semi-definite (PSD) matrices of size $n\times n$ and rank $p$. Given an energy function $\mathcal E:\mathcal S^{n,p}_+\to \mathbb{R}$ and certain Riemannian metrics $g$ on $\mathcal S^{n,p}_+$, these schemes rely on an Euler-Maruyama discretization of the Riemannian Langevin equation (RLE) with Brownian motion on the manifold. We present numerical schemes for RLE under two fundamental metrics on $\mathcal S^{n,p}_+$: (a) the metric obtained from the embedding of $\mathcal S^{n,p}_+ \subset \mathbb{R}^{n\times n}$; and (b) the Bures-Wasserstein metric corresponding to quotient geometry. We also provide examples of energy functions with explicit Gibbs distributions that allow numerical validation of these schemes

Manipulating coupling state and magnetism of Mn-doped ZnO nanocrystals by changing the coordination environment of Mn via hydrogen annealing

Mn-doped ZnO nanocrystals are synthesized by a wet chemical route and treated in H2/Ar atmosphere with different H2/Ar ratios. It is found that hydrogen annealing could change the coordination environment of Mn in ZnO lattice and manipulate the magnetic properties of Mn-doped ZnO. Mn ions initially enter into interstitial sites and a Mn3+O6 octahedral coordination is produced in the prepared Mn-doped ZnO sample, in which the nearest neighbor Mn3+ and O2 ions could form a Mn3+-O2--Mn3+ complex. After H2 annealing, interstitial Mn ions can substitute for Zn to generate the Mn2+O4 tetrahedral coordination in the nanocrystals, in which neighboring Mn2+ ions and H atoms could form a Mn2+-O2--Mn2+ complex and Mn-H-Mn bridge structure. The magnetic measurement of the as-prepared sample shows room temperature paramagnetic behavior due to the Mn3+-O2--Mn3+ complex, while the annealed samples exhibit their ferromagnetism, which originates from the Mn-H-Mn bridge structure and the Mn-Mn exchange interaction in the Mn2+-O2--Mn2+ complex

Effects of packing rates of cubic-shaped polyurethane foam carriers on the microbial community and the removal of organics and nitrogen in moving bed biofilm reactors

The effects of packing rates (20%, 30%, and 40%) of polyurethane foam (PUF) to the removal of organics and nitrogen were investigated by continuously feeding artificial sewage in three aerobic moving bed biofilm reactors. The results indicated that the packing rate of the PUF carriers had little influence on the COD removal efficiency (81% on average). However, ammonium removal was affected by the packing rates, which was presumably due to the different relative abundances of nitrifying bacteria. A high ammonium removal efficiency of 96.3% at a hydraulic retention time of 5 h was achieved in 40% packing rate reactor, compared with 37.4% in 20% packing rate. Microprofiles of dissolved oxygen and nitrate revealed that dense biofilm limits the DO transfer distance and nitrate diffusion. Pyrosequencing analysis of the biofilm showed that Proteobacteria, Bacteroidetes and Verrucomicrobia were the three most abundant phyla, but the proportions of the microbial community varied with the packing rate of the PUF carriers. (C) 2012 Elsevier Ltd. All rights reserved

Comparison Studies on Sub-Nanometer-Sized Ion Clusters in Aqueous Solutions: Vibrational Energy Transfers, MD Simulations, and Neutron Scattering

In this work, MD simulations with two different force fields, vibrational energy relaxation and resonant energy transfer experiments, and neutron scattering data are used to investigate ion pairing and clustering in a series of GdmSCN aqueous solutions. The MD simulations reproduce the major features of neutron scattering experimental data very well. Although no information about ion pairing or clustering can be obtained from the neutron scattering data, MD calculations clearly demonstrate that substantial amounts of ion pairs and small ion clusters (subnanometers to a few nanometers) do exist in the solutions of concentrations 0.5 M*, 3 M*, and 5 M* (M* denotes mole of GdmSCN per. 55.55 mole of water). Vibrational relaxation experiments suggest that significant amounts of ion pairs form in the solutions. Experiments measuring the resonant energy transfers among the thiocyanate anions in the solutions suggest that the ions form clusters and in the clusters the average anion distance is 5.6 angstrom (5.4 angstrom) in the 3 M* (5 M*) Gdm(-D)SCN/D2O solution

Ultrafast probes of electron–hole transitions between two atomic layers

The investigation into the dynamical transitions of charged quasiparticles on interfaces remains technically challenging. Here, the authors use ultrafast, mid-infrared micro-spectroscopy to unveil the formation of tightly bound interlayer excitons between conducting graphene and semiconducting MoSe2