Orthogonal polynomials and continued fractions arising from contiguous relations and generalizations of Kummer and q-Kummer identities

The Ramanujan Journal - In this paper, we show formally how can we generate continued fractions starting from three-term contiguous relations. Then as examples, starting from certain contiguous...     

DG polynomial algebras and their homological properties

In this paper, we introduce and study differential graded(DG for short) polynomial algebras. In brief, a DG polynomial algebra A is a connected cochain DG algebra such that its underlying graded algebra A~# is a polynomial algebra K[x_1, x_2,..., x_n] with |xi| = 1 for any i ∈ {1, 2,..., n}. We describe all possible differential structures on DG polynomial algebras, compute their DG automorphism groups, study their isomorphism problems, and show that they are all homologically smooth and Gorenstein DG algebras. Furthermore, it is proved that the DG polynomial algebra A is a Calabi-Yau DG algebra when its differential ?_A≠ 0 and the trivial DG polynomial algebra(A, 0) is Calabi-Yau if and only if n is an odd integer.     

Self-similar asymptotic behavior for the solutions of a linear coagulation equation

In this paper we consider the long-time asymptotics of a linear version of the Smoluchowski equation which describes the evolution of a tagged particle moving in a random distribution of fixed particles. The volumes v of these particles are independently distributed according to a probability distribution which decays asymptotically as a power law $v^{?\sigma }$. The validity of the equation has been rigorously proved in [22] taking as a starting point a particle model and for values of the exponent $\sigma >3$, but the model can be expected to be valid, on heuristic grounds, for $\sigma >\frac{5}{3}$. The resulting equation is a non-local linear degenerate parabolic equation. The solutions of this equation display a rich structure of different asymptotic behaviors according to the different values of the exponent σ. Here we show that for $\frac{5}{3}<\sigma <2$ the linear Smoluchowski equation is well-posed and that there exists a unique self-similar profile which is asymptotically stable.     

Asymmetric Reactions involving Lewis Base Catalyst Tethered Dearomatized Intermediates

Numerous protocols have been developed for the functionalization of aromatic substances. Among them, the strategy by which aromatic substrates are activated in situ to generate dearomatized intermediates is highly efficient but challenging, especially in the field of asymmetric catalysis. In this Concept article, the application of some well-established chiral Lewis base catalysis, including primary/secondary amines and N-heterocyclic carbenes, that can covalently form catalyst-tethered dearomatized ortho/para-quinodimethane species with diverse heteroaryl and aryl carbonyl substrates is summarized in a number of asymmetric cycloaddition and addition reactions with diverse reagents generally having electrophilic properties. As a result, a variety of enantioenriched aromatic products with higher molecular complexity are constructed effectively through a rearomatization process.     

Imide-Functionalized Polymer Semiconductors

Imide-functionalized π-conjugated polymer semiconductors have received a great deal of interest owing to their unique physicochemical properties and optoelectronic characteristics, including excellent solubility, highly planar backbones, widely tunable band gaps and energy levels of frontier molecular orbitals, and good film morphology. The organic electronics community has witnessed rapid expansion of the materials library and remarkable improvement in device performance recently. This review summarizes the development of imide-functionalized polymer semiconductors as well as their device performance in organic thin-film transistors and polymer solar cells, mainly achieved in the past three years. The materials mainly cover naphthalene diimide, perylene diimide, and bithiophene imide, and other imide-based polymer semiconductors are also discussed. The perspective offers our insights for developing new imide-functionalized building blocks and polymer semiconductors with optimized optoelectronic properties. We hope that this review will generate more research interest in the community to realize further improved device performance by developing new imide-functionalized polymer semiconductors.