Abstract:
We consider a pair of probability measures $\mu,\nu$ on the unit circle such that $\Sigma_{\lambda}(\eta_{\nu}(z))=z/\eta_{\mu}(z)$. We prove that the same type of equation holds for any $t\geq 0$ when we replace $\nu$ by $\nu\boxtimes\lambda_t$ and $\mu$ by $\mathbb{M}_t(\mu)$, where $\lambda_t$ is the free multiplicative analogue of the normal distribution on the unit circle of $\mathbb{C}$ and $\mathbb{M}_t$ is the map defined by Arizmendi and Hasebe. These equations are a multiplicative analogue of equations studied by Belinschi and Nica. In order to achieve this result, we study infinite divisibility of the measures associated with subordination functions in multiplicative free Brownian motion and multiplicative free convolution semigroups. We use the modified $\mathcal{S}$-transform introduced by Raj Rao and Speicher to deal with the case that $\nu$ has mean zero. The same type of the result holds for convolutions on the positive real line. We also obtain some regularity properties for the free multiplicative analogue of the normal distributions.

Abstract:
We obtain a formula for the density of the free convolution of an arbitrary probability measure on the unit circle of $\mathbb{C}$ with the free multiplicative analogues of the normal distribution on the unit circle. This description relies on a characterization of the image of the unit disc under the subordination function, which also allows us to prove some regularity properties of the measures obtained in this way. As an application, we give a new proof for Biane's classic result on the densities of the free multiplicative analogue of the normal distributions. We obtain analogue results for probability measures on $\mathbb{R}^+$. Finally, we describe the density of the free multiplicative analogue of the normal distributions as an example and prove unimodality and some symmetry properties of these measures.

Abstract:
Given a probability measure $\mu$ on the real line, there exists a semigroup $\mu_t$ with real parameter $t>1$ which interpolates the discrete semigroup of measures $\mu_n$ obtained by iterating its free convolution. It was shown in \cite{[BB2004]} that it is impossible that $\mu_t$ has no mass in an interval whose endpoints are atoms. We extend this result to semigroups related to multiplicative free convolution. The proofs use subordination results.

Abstract:
Serotonin exerts a powerful influence on neuronal excitability. In this study, we investigated the effects of serotonin on different neuronal populations in prefrontal cortex (PFC), a major area controlling emotion and cognition. Using whole-cell recordings in PFC slices, we found that bath application of 5-HT dose-dependently increased the firing of FS (fast spiking) interneurons, and decreased the firing of pyramidal neurons. The enhancing effect of 5-HT in FS interneurons was mediated by 5-HT2 receptors, while the reducing effect of 5-HT in pyramidal neurons was mediated by 5-HT1 receptors. Fluoxetine, the selective serotonin reuptake inhibitor, also induced a concentration-dependent increase in the excitability of FS interneurons, but had little effect on pyramidal neurons. In rats with chronic fluoxetine treatment, the excitability of FS interneurons was significantly increased, while pyramidal neurons remained unchanged. Fluoxetine injection largely occluded the enhancing effect of 5-HT in FS interneurons, but did not alter the reducing effect of 5-HT in pyramidal neurons. These data suggest that the excitability of PFC interneurons and pyramidal neurons is regulated by exogenous 5-HT in an opposing manner, and FS interneurons are the major target of Fluoxetine. It provides a framework for understanding the action of 5-HT and antidepressants in altering PFC network activity.

Abstract:
Terminal alkynes react with Cp2Zr(H)Cl (Cp = h5-C5H5) to give organozirconium complexes, which are trapped with alkylsulfenyl chlorides to afford (E)-vinyl alkyl sulfides in good yield.

Abstract:
The effect of reactant/product concentrations, reaction temperature and contact time on CO oxidation was investigated, using the perovskite-like oxide LaSrNiO4 as the catalyst. It was found that the reaction order of CO (reactant), as well as that of CO2 (product), is negative, the reaction orders for CO and CO2 being –0.32 and –0.51, respectively. However, the reaction order for O2 is positive, having a value of 0.62. The negative reaction order of CO and CO2 might be due to their competitive adsorption with O2, preventing the proceeding of oxygen dissociation (the rate-determining step of the reaction). The activation energy (Ea) of the reaction was calculated to be 49.3 kJ mol-1; this small activation energy suggests that LaSrNiO4 is a potential candidate for the CO oxidation reaction. The optimum weight hourly space velocity (WHSV) of the reaction was found to be 0.6 g s cm-3. The reaction conditions in the present case were (0.5–1 % CO + 0.5–2 % O2 + 0–2 % CO2), with He as the balance gas.

Abstract:
For a series of free $R$-diagonal operators, we prove an analogue of the three series theorem. We show that a series of free $R$-diagonal operators converges almost uniformly if and if two numerical series converge.

Abstract:
Metrics based on percentile ranks (PRs) for measuring scholarly impact involves complex treatment because of various defects such as overvaluing or devaluing an object caused by percentile ranking schemes, ignoring precise citation variation among those ranked next to each other, and inconsistency caused by additional papers or citations. These defects are especially obvious in a small-sized dataset. To avoid the complicated treatment of PRs based metrics, we propose two new indicators - the citation-based indicator (CBI) and the combined impact indicator (CII). Document types of publications are taken into account. With the two indicators, one would no more be bothered by complex issues encountered by PRs based indicators. For a small-sized dataset with less than 100 papers, special calculation is no more needed. The CBI is based solely on citation counts and the CII measures the integrate contributions of publications and citations. Both virtual and empirical data are used so as to compare the effect of related indicators. The CII and the PRs based indicator I3 are highly correlated but the former reflects citation impact more and the latter relates more to publications.

Abstract:
In this article, we propose a penalized clustering method for large scale data with multiple covariates through a functional data approach. In the proposed method, responses and covariates are linked together through nonparametric multivariate functions (fixed effects), which have great flexibility in modeling a variety of function features, such as jump points, branching, and periodicity. Functional ANOVA is employed to further decompose multivariate functions in a reproducing kernel Hilbert space and provide associated notions of main effect and interaction. Parsimonious random effects are used to capture various correlation structures. The mixed-effect models are nested under a general mixture model, in which the heterogeneity of functional data is characterized. We propose a penalized Henderson's likelihood approach for model-fitting and design a rejection-controlled EM algorithm for the estimation. Our method selects smoothing parameters through generalized cross-validation. Furthermore, the Bayesian confidence intervals are used to measure the clustering uncertainty. Simulation studies and real-data examples are presented to investigate the empirical performance of the proposed method. Open-source code is available in the R package MFDA.

Abstract:
We demonstrate the existence of a broad class of higher-order Laguerre--Gaussian asymmetric spatial optical solitons in strongly nonlocal nonlinear media. Furthermore, we discuss specific values (q=0) of the modulation depth parameter for different rational values of the topological charge in detail. Ourresults show that higher-order asymmetry spatial soliton family can exist in various forms, such as two-dimensional defect half-solitons, asymmetric single-layer and multi-layer necklace solitons.