Full cross-diffusion limit in the stationary Shigesada-Kawasaki-Teramoto model

This paper studies the asymptotic behavior of coexistence steady-states of the Shigesada-Kawasaki-Teramoto model as both cross-diffusion coefficients tend to infinity at the same rate. In the case when either one of two cross-diffusion coefficients tends to infinity, Lou and Ni [18] derived a couple of limiting systems, which characterize the asymptotic behavior of coexistence steady-states. Recently, a formal observation by Kan-on [10] implied the existence of a limiting system including the nonstationary problem as both cross-diffusion coefficients tend to infinity at the same rate. This paper gives a rigorous proof of his observation as far as the stationary problem. As a key ingredient of the proof, we establish a uniform $L^{\infty }$ estimate for all steady-states. Thanks to this a priori estimate, we show that the asymptotic profile of coexistence steady-states can be characterized by a solution of the limiting system.     

Photocatalytic Cr(VI) sequestration and photo-Fenton bisphenol A decomposition over white light responsive PANI/MIL-88A(Fe)

Polyaniline (PANI)/MIL-88A(Fe) (Px@M88) composites were constructed through a simple one-pot hydrothermal method. The photocatalytic and photo-Fenton activities of Px@M88 composites toward reduction of Cr(VI) and degradation organic pollutants were explored by white light irradiation. PANI, as a conductive polymer, can improve MIL-88A(Fe)’s conductivity and the efficiency of photogenerated e⁻–h⁺ pair separation. In the presence of H₂O₂, a photo-Fenton reaction occured to boost the degradation efficiency of organic pollutants like bisphenol A. In addition, P9@M88 showed excellent recycling and stability in cycling experiments. Finally, a possible reaction mechanism for photocatalytic degradation was proposed and verified by X-ray photoelectron spectroscopy and electron spin resonance determination and electrochemical characterizations.     

Characterization of embeddings of Sobolev-type spaces into generalized Hölder spaces defined by Lp-modulus of smoothness

We prove a sharp estimate for the k-modulus of smoothness, modelled upon a $L^p$-Lebesgue space, of a function f in $W^k{L}^{\frac{pn}{n+kp},p}(\mathrm{\Omega })$, where Ω is a domain with minimally smooth boundary and finite Lebesgue measure, $k,n\in \mathbb{N}$, $k<n$ and $\frac{n}{n?k}<p<+\infty$. This sharp estimate is used to establish necessary and sufficient conditions for continuous embeddings of Sobolev-type spaces into generalized Hölder spaces defined by means of the k-modulus of smoothness. General results are illustrated with examples. In particular, we obtain a generalization of the classical Jawerth embeddings.     

Stevia rebaudiana targeting α-amylase: An in-vitro and in-silico mechanistic study

Diabetes mellitus (DM) is the fastest growing metabolic disorder in the world. Recently, more attention is paid to the study of natural products due to side effects of synthetic drugs. Stevia rebaudiana (Bertoni) is considered an encouraging starting point for the antidiabetic lead development. In the present study, the in vitro α-amylase inhibitory activity of the extracts of S. rebaudiana is investigated. In order to understand the molecular mechanism and future pharmacophore development, in silico study of secondary metabolites isolated from S. rebaudiana was carried out. Results indicated that water extract shows highest α-amylase inhibitory activity as compared to other extracts. Moreover, compound 20 (rebaudioside A) which has been previously reported and isolated from water extract showed the impressive binding profile with α-amylase. Therefore, our study suggests that S. rebaudiana could be used in the development of therapeutic drugs for the treatment of diabetes.     

Synthetic studies toward swinhoeisterol A,a novel steroid with an unusual carbon skeleton

Swinhoeisterol A is a novel steroid with unusual 6/6/5/7 tetracyclic skeleton. The model compound with BCD rings is constructed by Friedel–Crafts acylation and an oxidative dearomatization as key steps.     

A priori bounds and multiplicity for fully nonlinear equations with quadratic growth in the gradient

We consider fully nonlinear uniformly elliptic equations with quadratic growth in the gradient, such as

$?F(x,u,Du,D^{2}u)=\lambda c(x)u+〈M(x)Du,Du〉+h(x)$

in a bounded domain with a Dirichlet boundary condition; here $\lambda \in \mathbb{R}$, $c,h\in L^p(\mathrm{\Omega })$, $p>n\ge 1$, $c?0$ and the matrix M satisfies $0<{\mu }_{1}I\le M\le {\mu }_{2}I$. Recently this problem was studied in the “coercive” case $\lambda c\le 0$, where uniqueness of solutions can be expected; and it was conjectured that the solution set is more complex for noncoercive equations. This conjecture was verified in 2015 by Arcoya, de Coster, Jeanjean and Tanaka for equations in divergence form, by exploiting the integral formulation of the problem. Here we show that similar phenomena occur for general, even fully nonlinear, equations in nondivergence form. We use different techniques based on the maximum principle.We develop a new method to obtain the crucial uniform a priori bounds, which permit to us to use degree theory. This method is based on basic regularity estimates such as half-Harnack inequalities, and on a Vázquez type strong maximum principle for our kind of equations.