Extremal functions for singular Trudinger–Moser inequalities in the entire Euclidean space

In a previous work (Adimurthi and Yang, 2010 [2]), Adimurthi–Yang proved a singular Trudinger–Moser inequality in the entire Euclidean space ${\mathbb{R}}^N$$(N\ge 2)$. Precisely, if $0\le \beta <1$ and $0<\gamma \le 1?\beta$, then there holds for any $\tau >0$,

$\underset{u\in W^{1,N}({\mathbb{R}}^N),{\int }_{{\mathbb{R}}^N}(|\mathrm{?}u{|}^N+\tau |u{|}^N)dx\le 1}{\sup }?\underset{{\mathbb{R}}^N}{\int }\frac{1}{|x{|}^{N\beta }}(e^{{\alpha }_N\gamma |u{|}^{\frac{N}{N?1}}}?\sum _{k=0}^{N?2}\frac{{{\alpha }_N}^k{\gamma }^k|u{|}^{\frac{kN}{N?1}}}{k!})dx<\infty ,$

where ${\alpha }_N=N{\omega }_{N?1}^{1/(N?1)}$ and ${\omega }_{N?1}$ is the area of the unit sphere in ${\mathbb{R}}^N$. The above inequality is sharp in the sense that if $\gamma >1?\beta$, all integrals are still finite but the supremum is infinity. In this paper, we concern extremal functions for these singular inequalities. The regular case $\beta =0$ has been considered by Li and Ruf (2008) [12] and Ishiwata (2011) [11]. We shall investigate the singular case $0<\beta <1$ and prove that for all $\tau >0$, $0<\beta <1$ and $0<\gamma \le 1?\beta$, extremal functions for the above inequalities exist. The proof is based on blow-up analysis.     

含羟基基团取代的氮氧自由基构筑的三个双核稀土化合物的结构及磁性表征

以三氟乙酰丙酮(tfac)为共配体的稀土配合物分别与5-溴-2-羟基苯取代的自由基配体和5-硝基-2-羟基苯取代的自由基配体进行反应,合成3个稀土-自由基配合物[Ln2(tfac)4(NIT-5Br-2PhO)2](Ln=Gd (1),Dy (2))和[Dy2(tfac)4(NIT-5NO2-2PhO)2](3)(NIT-5Br-2PhOH=2-(2′-hydroxy-5′-bromophenyl)-4,4,5,5-tetramethylimidazoline-1-oxyl-3-oxide,NIT-5NO2-2PhOH=2-(2′-hydroxy-5′-nitrophenyl)-4,4,5,5-tetra-methylimidazoline-1-oxyl-3-oxide)。单晶结构分析表明这3个化合物中的稀土离子均通过自由基配体上的羟基氧基团连接为双核的结构。配合物1的直流磁化率表征揭示了Gd(Ⅲ)离子间的反铁磁耦合对其磁行为起主要作用。     

Morphological evolution and visible light-induced degradation of Rhodamine 6G by nanocrystalline bismuth tungstate prepared using a template-based approach

The cleaning of water contaminated with organic dyes is a crucial problem nowadays. The search for good catalysts is intense, and bismuth tungstates have attracted a lot of attention because of their catalytic properties which are related to their crystal structure and morphology. In this study, we show that Bi₂WO₆ (BWO) crystals synthesized by the surfactant-assisted hydrothermal method create a different morphology than non-assisted crystals. With the assistance of the PVP surfactant, even the BWO crystalline structure could change, crystallizing into a high-symmetry metastable phase. These changes in morphology imply a decrease in BWO catalytic activity, which shows that insightful control of BWO synthesis is necessary to improve the BWO properties.     

Decentralized coordination of autonomous swarms inspired by chaotic behavior of ants

In this paper, we propose a decentralized coordination algorithm for a group of mobile nodes, called an autonomous swarm, on a finite two-dimensional space, which can efficiently coordinate cooperatively the autonomous swarm to the optimal solution. Our algorithm is inspired by chaotic behavior of a single ant and self-organization behavior of the whole ant colony. To construct this algorithm, we firstly assume that each agent is a nonlinear oscillator presenting the chaotic behavior of a single ant. Then we establish a self-organization mechanism according to the self-organization behavior of the whole ant colony. Moreover, we analyze the convergence of the proposed algorithm. Finally, we experimentally evaluate the performance of our algorithm with the clustering and dispersion operations of a swarm. Comparison results of the proposed algorithm and the gradient-type one are also presented to illustrate the effectiveness of the proposed scheme in approximately global optimization for swarms.     

一道行列式证明题的多种证明方法

针对线性代数教材中一道行列式证明题，利用行列式的性质，给出多种证明方法，旨在启发学生对相关行列式计算或证明题的解题方法进行探索．     

Facile Synthesis of 2-Aryl-3H-indol-3-ones via Singlet Oxygenation of 2-Arylindoles

2-Aryl-3H-indol-3-ones have been synthesized via singlet oxygenation of 2-arylindoles in methanol followed by thermodemethoxylation of the resulting 2-aryl-2-methoxy-1, 2-dihydro-3H-indol-3-ones in one pot in good yields.     

Morphology and interpenetrated growth of spherulites in miscible poly(ethylene succinate)/poly(ethylene oxide) blends

Morphology development and growth process of spherulites in miscible poly(ethylene succinate)/poly(ethylene oxide) blends are studied by means of polarizing optical microscopy and atom force microscopy in this paper. Thin films with different film thicknesses were used to follow the growth processes of spherulites and dendrites. It is shown that, when one component spherulite grows, the other component in the melt is always excluded from the spherulite. The excluded component may reenter into the spherulite through diffusion depending on amorphous volume fraction of spherulite and segmental mobility of molecules, which leads to the occurrence of interpenetrated growth. This mechanism was analyzed in detail in this paper.     

CuI-catalyzed Suzuki coupling reaction of organoboronic acids with alkynyl bromides

A CuI-catalyzed Suzuki cross-coupling reaction of organoboron derivatives with alkynyl bromides has been developed. In the presence of CuI (10 mol %) and 8-hydroxyquinoline (20 mol %), organoboron derivatives including aromatic and alkenyl boronic acids, potassium aryltrifluoroborates, and sodium tetraphenylborate reacted smoothly with 1-bromo-2-substituted acetylene to generate the corresponding cross-coupling products in good to excellent yields in C₂H₅OH. It is important to note that aromatic N,O-ligand 8-hydroxyquinoline is the most effective ligand for the reaction.     

NiMo 催化剂上MoS2 形貌与其DBT 加氢脱硫选择性的关系

制备了两个系列不同镍钼负载量的NiMo 催化剂,并用X 衍射、N2 物理吸附和透射电镜进行了表征。以二苯并噻吩
为模型硫化物,在高压固定床微型反应器上对该NiMo 催化剂的加氢脱硫性能进行了评价,研究了MoS2 形貌与催化剂加氢脱
硫选择性之间的关系。结果表明,镍钼负载量对MoS2 形貌有明显的影响。Mo18Ni4 催化剂(含18% MoO3 和4% NiO)上
MoS2 呈多级层状结构,具有较高的加氢脱硫活性和优异的加氢脱硫选择性。加氢选择性与催化剂上活性组分MoS2 的堆积层
数相关联呈很好的线性关系;堆积层数越多,其加氢选择性也越高。