Molecular QTAIM Topology Is Sensitive to Relativistic Corrections

The topology of the molecular electron density of benzene dithiol gold cluster complex Au₄−S−C₆H₄−S′−Au′₄ changed when relativistic corrections were made and the structure was close to a minimum of the Born–Oppenheimer energy surface. Specifically, new bond paths between hydrogen atoms on the benzene ring and gold atoms appeared, indicating that there is a favorable interaction between these atoms at the relativistic level. This is consistent with the observation that gold becomes a better electron acceptor when relativistic corrections are applied. In addition to relativistic effects, here, we establish the sensitivity of molecular topology to basis sets and convergence thresholds for geometry optimization.     

In vitro anticomplementary activity and quality evaluation of dried blossoms of Inula nervosa Wall. from different geographical origins

Blossoms of Inula nervosa Wall. (BINW) are traditionally used as an analgesic and antitussive in China. In this study, in vitro anticomplementary activities of crude extract from BINW in 21 batches and of extracts of four monomeric compounds were evaluated by the classical pathway. The effect of the region of origin on the quality of BINW was evaluated by fingerprint analysis for the first time. Furthermore, chemometric methods including similarity analysis and principal component analysis were employed to evaluate the quality of BINW. The nine major monomeric compounds were quantitated by ultra‐high‐performance liquid chromatography. All nine analytes demonstrated excellent linearity with recoveries ranging from 97.25% to 102.76%. The limits of detection and quantification were 0.07–12.20 μg/mL and 0.22–40.27 μg/mL, respectively. Results indicate that different regions of origin have a significant effect on the quality of BINW. Fingerprint analysis in combination with chemometrics and multi‐ingredient determination is an efficient and reliable approach for quality evaluation. The BINW samples from Yunnan had the highest ratio of 1,5‐dicaffeoylquinic acid and thymol; they also exhibited significantly higher anticomplementary activity than those from three other areas. This study successfully established a rapid and efficient method to evaluate the quality and biological activity of BINW.     

New lower bounds on the radius of spatial analyticity for the KdV equation

The radius of spatial analyticity for solutions of the KdV equation is studied. It is shown that the analyticity radius does not decay faster than $t^{?1/4}$ as time t goes to infinity. This improves the works of Selberg and da Silva (2017) [30] and Tesfahun (2017) [34]. Our strategy mainly relies on a higher order almost conservation law in Gevrey spaces, which is inspired by the I-method.     

Convergence of solutions for the fractional Cahn–Hilliard system

This paper deals with the Cauchy–Dirichlet problem for the fractional Cahn–Hilliard equation. The main results consist of global (in time) existence of weak solutions, characterization of parabolic smoothing effects (implying under proper condition eventual boundedness of trajectories), and convergence of each solution to a (single) equilibrium. In particular, to prove the convergence result, a variant of the so-called ?ojasiewicz–Simon inequality is provided for the fractional Dirichlet Laplacian and (possibly) non-analytic (but $C^1$) nonlinearities.     

Going Off the Grid: Iterative Model Selection for Biclustered Matrix Completion

We consider the problem of performing matrix completion with side information on row-by-row and column-by-column similarities. We build upon recent proposals for matrix estimation with smoothness constraints with respect to row and column graphs. We present a novel iterative procedure for directly minimizing an information criterion to select an appropriate amount of row and column smoothing, namely, to perform model selection. We also discuss how to exploit the special structure of the problem to scale up the estimation and model selection procedure via the Hutchinson estimator, combined with a stochastic Quasi-Newton approach. Supplementary material for this article is available online.     

Multifold factorizations of cyclic groups into subsets

Let $(G,+)$ be an abelian group. A finite multiset A over G is said to give a λ-fold factorization of G if there exists a multiset B over G such that each element of G occurs λ times in the multiset $A+B:=\{a+b:a\in A,b\in B\}$. In this article, restricting G to a cyclic group, we will provide sufficient conditions on a given multiset A under which the exact value or an upper bound of the minimum multiplicity λ of a factorization of G can be given by introducing a concept of ‘lcm-closure’. Furthermore, a couple of properties on a given factor A will be shown when A has a prime or prime power order (cardinality). A relation to multifold factorizations of the set of integers will be also glanced at a general perspective.