Generalized P(N)-graded Lie superalgebras

We generalize the P(N)-graded Lie superalgebras of Martinez-Zelmanov. This generalization is not so restrictive but suffcient enough so that we are able to have a classification for this generalized P(N)-graded Lie superalgebras. Our result is that the generalized P(N)-graded Lie super-algebra L is centrally isogenous to a matrix Lie superalgebra coordinated by an associative superalgebra with a super-involution. Moreover, L is P(N)-graded if and only if the coordinate algebra R is commutative and the super-involution is trivial. This recovers Martinez-Zelmanov's theorem for type P(N). We also obtain a generalization of Kac's coordinatization via Tits-Kantor-Koecher construction. Actually, the motivation of this generalization comes from the Fermionic-Bosonic module construction.     

算子代数上2-局部Lie三重导子的结构

设X是维数大于2的Banach空间,映射δ:B(X)→B(X)是2-局部Lie三重导子,则对所有A∈B(X)有δ(A)=[A,T]+φ(A),这里T∈B(X),φ是从B(X)到FI的齐次映射且满足对所有A,B∈B(X)有φ(A+B)=φ(A),其中B是交换子的和.     

Epidemiological models with quadratic equation for endemic equilibria—A bifurcation atlas

The existence and occurrence, especially by a backward bifurcation, of endemic equilibria is of utmost importance in determining the spread and persistence of a disease. In many epidemiological models, the equation for the endemic equilibria is quadratic, with the coefficients determined by the parameters of the model. Despite its apparent simplicity, such an equation can describe an amazing number of dynamical behaviors. In this paper, we shall provide a comprehensive survey of possible bifurcation patterns, deriving explicit conditions on the equation's parameters for the occurrence of each of them, and discuss illustrative examples.     

Quantizer–dequantizer operators as a tool for formulating the quantization procedure

We consider the quantization procedure and investigate the application of the quantizer–dequantizer method and star-product technique to construct associative products and the associative algebras formed by the quantizer–dequantizer operators and their symbols. The corresponding Lie algebras are also constructed. We study the case where the quantizer–dequantizer operators form a self-dual system and show that the structure constants of the Lie algebras satisfy some identity, in addition to the Jacobi identity. Using tomographic quantizer–dequantizer operators and their symbols, we construct the continuous associative algebra and the corresponding Lie algebra.     

Co-hydrolysis and Seed-Induced Synthesis of Basic Mesoporous ZSM-5 Zeolites with Enhanced Catalytic Performance

For zeolite catalysts, the regulation of active site and pore structure plays an important role in the enhancement of their catalytic performance. In this work, a one-pot and organic template-free co-regulation route is proposed to straightforwardly synthesize basic mesoporous ZSM-5 zeolites with adjustable alkaline-earth metal species. The synthesis pathway combines two decisive strategies: 1) the seed-induced interface assembly growth method and 2) the acidic co-hydrolysis/condensation of aluminosilicate species and alkaline-earth metal (e.g., Mg, Ca, Sr, or Ba) sources. It is interesting that the mesoporous structure was self-evolved through particle-attached seed-interfacial crystallization without the assistance of any template. Meanwhile, the incorporation of alkaline-earth metals species is homogeneous and highly dispersed in the solid products during the whole crystallization process, and finally generate the superior basicity. Catalysis tests of the as-synthesized samples displayed their novel performance in the typical base reaction of Knoevenagel condensation, even for bulky substrates owing to the enhanced diffusion arising from the meso/microporous network. This finding opens new possibilities for facile, cost-effective, and environmentally friendly synthesis of mesoporous high-silica zeolites with tunable acid/base properties, and deepens our understanding of the particle-attached crystallization.