Comparison of activities of homogeneous Ti-based catalysts in the presence of methylaluminoxane (MAO) for synthesis of syndiotactic polystyrene by UV/visible spectroscopic study

The catalytic activities in syndiospecific polymerization of styrene in hydrocarbon using homogeneous Ti-based catalysts in the presence of methylaluminoxane (MAO) were investigated through UV/visible spectroscopic analysis. A strong UV absorption band of CpTiCl₃, itself, incipiently appeared at λ_max = 400 nm in toluene, followed by a bathochromic shift with its remarkable decrease by the addition of MAO. The absorption band intensity at λ_max = 400 nm arising from delocalization of π-electrons on the cyclopentadienyl ring decreased by methylation in the presence of MAO with regard to the mechanism for production of an active center (“cation-like”), for example, the change of the ionic nature. The intensity decrease at λ_max = 400 nm was suppressed over 2000 of the [Al]/[Ti] ratio. In the case of Ti(OC₄H₉)₄ having a σ-ligand, new and broad UV absorption bands were developed at λ_max = 340 nm and 410 nm in the presence of MAO in contrast with the CpTiCl₃/MAO system. Comparison between the relative absorption intensities at λ_max = 340 nm and 410 nm led to the determination of a maximum catalytic activity of Ti(OC₄H₉)₄ in the presence of MAO related to the polymerization yield. The maximum polymerization yield was observed with regard to the relative maximum value of the absorption intensity at λ_max = 410 nm with the [Al]/[Ti] ratio (500). From observation for polymorphism of the final products via differential scanning calorimetric analysis (DSC), the thermally unstable β-form seemed to be produced by the CpTiCl₃/MAO system independent of the MAO concentration, the Ti(OC₄H₉)₄/MAO system produced a thermally stable α-form in the low MAO concentration (up to 100 of the [Al]/[Ti] ratio), and a mixture of α- and/or β-forms over 200 of the [Al]/[Ti] ratio under our experimental conditions. © 1998 John Wiley & Sons, Inc. J Polym Sci A: Polym Chem 36: 1733–1741, 1998     

Random minimal spanning tree and percolation on the N-cube

The N-cube is a graph with 2^N vertices and N2^N−1 edges. Suppose independent uniform random edge weights are assigned and let T be the spanning tree of minimal (total) weight. Then the weight of T is asymptotic to N⁻¹2^N ∑^∞_i=1 i⁻³ as N→∞. Asymptotics are also given for the local structure of T and for the distribution of its kth largest edge weight, k fixed. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 12, 63–82, 1998     

A study of random Weyl trees

We study binary search trees constructed from Weyl sequences {nθ}, n≥1, where θ is an irrational and {·} denotes “mod 1.” We explore various properties of the structure of these trees, and relate them to the continued fraction expansion of θ. If H_n is the height of the tree with n nodes when θ is chosen at random and uniformly on [0, 1], then we show that in probability, H_n∼(12/π²)log n log log n. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 12, 271–295, 1998     

Linear Elliptic Boundary Value Problems with Non‐Smooth Data: Campanato Spaces of Functionals

In this paper linear elliptic boundary value problems of second order with non‐smooth data L^∞‐coefficients, sets with Lipschitz boundary, regular sets, non‐homogeneous mixed boundary conditions) are considered. It will be shown that such boundary value problems generate isomorphisms between certain Sobolev‐Campanato spaces of functions and functionals, respectively.     

高阶Broer-Kaup(BK)方程组的新精确解

利用扩展齐次平衡法,我们得到了1+1维和2+1维高阶BK方程组新的孤子解和无穷多有理函数解,尤其是求出了初始值问题解的封闭形式.