Pseudo[n,m]rotaxanes of Cucurbit[7/8]uril and Viologen‐Naphthalene Derivative: A Precise Definition of Rotaxane

Rotaxane is a kind of classic supramolecule, which is usually constructed from a number of macrocycles and one axis molecule. Herein, we have expanded the supramolecular structure of [n]rotaxane to offer a precise definition of (pseudo)[n,m]rotaxane for accurately describing the two kinds of (pseudo)rotaxanes structures, which are self‐assembled from cucurbit[7/8]uril (CB[7/8]) and viologen‐naphthalene derivative, respectively. Furthermore, these CB‐based pseudorotaxanes exhibit varied photophysical properties, stimuli‐responsive behavior triggered by competitive guest, and self‐sorting behavior.     

Novel silver-containing supramolecular frameworks constructed by combination of coordination bonds and supramolecular interactions

The hydrothermal reactions of AgNO(3), 4,4'-bipy, and carboxylate ligands gave rise to three supramolecular architectures, namely [Ag(bipy)].H(2)SIPA.1/2bipy.H(2)O (1), [Ag(bipy)].1/2H(2)btec.H(2)O (2), and [Ag(bipy)](2).H(2)dpstc.2H(2)O (3) (H(3)SIPA = 5-sulfoisophthalic acid, bipy = 4,4'-bipyridine, H(4)btec = 1,2,4,5-benzenetetracarboxylic acid, H(4)dpstc = 3,3',4,4'-tetracarboxydiphenyl sulfone). All complexes are extended from Ag-bipy linear chains by the combination of coordination bonds and supramolecular interactions in two different approaches. Complexes 1 and 3 comprise two-dimensional frameworks. In the two complexes, a one-dimensional ladderlike structure is first formed by the connection of a Ag-bipy chain through hydrogen bonding between a free carboxylate/bipy ligand and weak coordinative interactions between a free carboxylate ligand and silver ion. The ladderlike structure is then extended to a two-dimensional layer architecture by pi...pi interactions between bipy ligands of the Ag-bipy chains. Complex 2 possesses a three-dimensional framework. The free H(2)btec(2)(-) ligands form a two-dimensional layer network by hydrogen-bonding interactions between protonated and deprotonated carboxylate groups; meanwhile, pi.pi interactions between bipy ligands of Ag-bipy chains also result in a two-dimensional layer. The two layers are further connected by weak Ag-O interactions to generate a three-dimensional supramolecular structure.     

The effects of polymer chain rigidification, zeolite pore size and pore blockage on polyethersulfone (PES)-zeolite A mixed matrix membranes

The polyethersulfone (PES)-zeolite 3A, 4A and 5A mixed matrix membranes (MMMs) were fabricated with a modified solution-casting procedure at high temperatures close to the glass transition temperatures (T_g) of polymer materials. The effects of membrane preparation methodology, zeolite loading and pore size of zeolite on the gas separation performance of these mixed matrix membranes were studied. SEM results show the interface between polymer and zeolite in MMMs experiencing natural cooling is better (i.e., less defective) than that in MMMs experiencing immediate quenching. The increment of glass transition temperature (T_g) of MMMs with zeolite loading confirms the polymer chain rigidification induced by zeolite. The experimental results indicate that a higher zeolite loading results in a decrease in gas permeability and an increase in gas pair selectivity. The unmodified Maxwell model fails to correctly predict the permeability decrease induced by polymer chain rigidification near the zeolite surface and the partial pore blockage of zeolites by the polymer chains. A new modified Maxwell model is therefore proposed. It takes the combined effects of chain rigidification and partial pore blockage of zeolites into calculation. The new model shows much consistent permeability and selectivity predication with experimental data. Surprisingly, an increase in zeolite pore size from 3 to 5 Å generally not only increase gas permeability, but also gas pair selectivity. The O₂/N₂ selectivity of PES-zeolite 3A and PES-zeolite 4A membranes is very similar, while the O₂/N₂ selectivity of PES-zeolite 5A membranes is much higher. This implies the blockage may narrow a part of zeolite 5A pores to approximately 4 Å, which can discriminate the gas pair of O₂ and N₂, and narrow a part of zeolites 3A and 4A pores to smaller sizes. It is concluded that the partial pore blockage of zeolites by the polymer chains has equivalent or more influence on the separation properties of mixed matrix membranes compared with that of the polymer chain rigidification.     

Cartan’s Second Main Theorem and Mason’s Theorem for Jackson Difference Operator

Let f : C → Pⁿ be a holomorphic curve of order zero. The authors establish a Jackson difference analogue of Cartan’s second main theorem for the Jackson q-Casorati determinant and introduce a truncated second main theorem of Jackson difference operator for holomorphic curves. In addition, a Jackson difference Mason’s theorem is proved by using a Jackson difference radical of a polynomial. Furthermore, they extend the Mason’s theorem for m + 1 polynomials. Some examples are constructed to show that their results are accurate.     

A Space-Time Polynomial Collocation Method for Solving 3D Burgers Equations北大核心CSCD

Burgers方程是一类应用广泛的非线性偏微分方程,方程中的非线性项难以处理。该文提出一种新的时空多项式配点法——多项式特解法求解三维Burgers方程。求解过程分为两步:第一步,对三维Burgers方程中的线性导数项(包括时间导数项),求出相应的多项式特解。第二步,将求出的多项式特解作为基函数,对三维Burgers方程中剩余的非线性项进行迭代求解。与时空多项式函数作为基函数对三维Burgers方程进行直接求解相比,该算法简单易行,得到的近似解精度非常高,算法极其稳定,对于教学过程中提高学生的编程能力,加深对高维Burgers方程的理解能力以及Burgers方程的实际应用具有重要意义。     

The distribution of square-full integers

Let , wheref(n) is the characteristic function of square-full integers. A formula is proved that can be used to obtain better upper bound estimate for (x). We get (x)=O(x ^5/33+), which improves the exponent 11/72 obtained by R. Blasubramania and K. Ramachandra.