The construction of numerical integration rules of degree three for product regions

Numerical integration formulas in n-dimensional Euclidean space of degree three are discussed. In this paper, for the product regions a method is presented to construct numerical integration formulas of degree three with 2n real points and positive weights. The presented problem is a little different from those dealt with by other authors. All the corresponding one-dimensional integrals can be different from each other and they are also nonsymmetrical. In this paper an n-dimensional numerical integration problem is turned into n one-dimensional moment problems, which simplifies the construction process. Some explicit numerical formulas are given. Furthermore, a more generalized numerical integration problem is considered, which will shed light on the final solution to the third degree numerical integration problem.     

The maximum and minimum degrees of random bipartite multigraphs

In this paper the authors generalize the classic random bipartite graph model, and define a model of the random bipartite multigraphs as follows:let m = m（n） be a positive integer-valued function on n and ζ（n,m;{pk}） the probability space consisting of all the labeled bipartite multigraphs with two vertex sets A ={a₁,a₂,...,a_n} and B = {b₁,b₂,...,b_m}, in which the numbers t_ai,b_j of the edges between any two vertices a_i∈A and b_j∈ B are identically distributed independent random variables with distribution P{t_ai,b_j=k}=pk,k=0,1,2,...,where pk ≥0 and ∞Σk=0 pk=1. They obtain that X_c,d,A, the number of vertices in A with degree between c and d of G_n,m∈ζ（n, m;{pk}） has asymptotically Poisson distribution, and answer the following two questions about the space ζ（n,m;{pk}） with {pk} having geometric distribution, binomial distribution and Poisson distribution, respectively. Under which condition for {pk} can there be a function D（n） such that almost every random multigraph G_n,m∈ζ（n,m;{pk}） has maximum degree D（n）in A? under which condition for {pk} has almost every multigraph G（n,m）∈ζ（n,m;{pk}） a unique vertex of maximum degree in A?     

Existence of two periodic solutions for a non-autonomous SIR epidemic model

A non-autonomous SIR model with periodic transmission rate and a constant removal rate is formulated. By using the continuation theorem of coincidence degree theory, sufficient conditions for the existence of at least two positive periodic solutions are obtained. The stability of the periodic solution for small seasonality is investigated. Numerical simulations are done to show our theoretical results.     

共振条件下p-Laplace型方程多点边值问题的可解性

本文研究了共振条件下一类p-Laplace 型方程边值问题的可解性. 利用度理论, 得到至少存在一个解的充分条件, 并举例说明所得的结果是有效的.     

The existence of a 2‐factor in K1, n‐free graphs with large connectivity and large edge‐connectivity

In this article, we study the existence of a 2‐factor in a K_{1, n}‐free graph. Sumner [J London Math Soc 13 (1976), 351–359] proved that for n?4, an (n?1)‐connected K_{1, n}‐free graph of even order has a 1‐factor. On the other hand, for every pair of integers m and n with m?n?4, there exist infinitely many (n?2)‐connected K_{1, n}‐free graphs of even order and minimum degree at least m which have no 1‐factor. This implies that the connectivity condition of Sumner's result is sharp, and we cannot guarantee the existence of a 1‐factor by imposing a large minimum degree. On the other hand, Ota and Tokuda [J Graph Theory 22 (1996), 59–64] proved that for n?3, every K_{1, n}‐free graph of minimum degree at least 2n?2 has a 2‐factor, regardless of its connectivity. They also gave examples showing that their minimum degree condition is sharp. But all of them have bridges. These suggest that the effects of connectivity, edge‐connectivity and minimum degree to the existence of a 2‐factor in a K_{1, n}‐free graph are more complicated than those to the existence of a 1‐factor. In this article, we clarify these effects by giving sharp minimum degree conditions for a K_{1, n}‐free graph with a given connectivity or edge‐connectivity to have a 2‐factor. Copyright © 2010 Wiley Periodicals, Inc. J Graph Theory 68:77‐89, 2011     

平面图线性色数的一个上界

如果图G的一个正常染色满足染任意两种颜色的顶点集合导出的子图是一些点不交的路的并,则称这个正常染色为图G的线性染色．图G的线性色数用lc（G）表示,是指G的所有线性染色中所用的最少颜色的个数．证明了：若G是一个最大度△（G）≠5,6的平面图,则lc（G）≤2△（G）．     

一种新的自动聚焦算法的研究

聚焦评价函数和极点搜索策略是自动聚焦的重要模块,直接影响聚焦的效果.为同时满足自动聚焦速度和精度上的要求,提出了一种新的自动聚焦算法.该算法综合了快速改进的灰度差分法和高精度小波变换法的优点,根据各阶段聚焦评价函数的梯度选择合适的评价函数和镜头移动步长,并且实现了从大步长快速粗扫到小步长精确细调的过渡.实验表明,与传统...     

高斯涡旋光束的傍轴度

按单色光束傍轴度的定义,对高斯涡旋光束的傍轴度进行了研究.使用角谱表示法推导出高斯涡旋光束傍轴度的解析公式,用此研究了傍轴度和高斯涡旋光束参数之间的关系.结果表明,随背景高斯光束束腰宽度和光涡旋离轴参数的增加及随光涡旋拓扑电荷的减少,高斯涡旋光束的傍轴度增加.对所得结果用傍轴度与远场发散角间的关系做了物理解释.     

不同聚合机理的“平均聚合度”公式的分析讨论

平均聚合度和聚合度分布是决定聚合物产品质量的一个重要指标。平均聚合度直接表征聚合产品的分子量的大小,它的大小又决定高分子材料的性能,因此,研究平均聚合度具有重要的理论意义和实际意义。聚合反应按机理分类,可分为链型聚合和逐步聚合,链型聚合又分为自由基聚合、离子聚合、配位聚合,逐步聚合又分为缩聚和加聚反应。本文以自由基聚合、离子聚合、配位聚合以及缩聚的聚合机理为主线,对不同聚合机理对应的平均聚合度的公式进行推导和讨论,并加深理解,掌握其内涵。旨在提高学生对平均聚合度的认识和把握。     

Determination of Required HLB of Capryol 90

The required hydrophilic-lipophilic balance (HLB) of an oily substance indicates the HLB of the surfactants required to prepare a stable emulsion using the oil. This study utilizes empirical method to calculate required HLB (RHLB) of Capryol 90 using surfactant blends (Span 20, Tween 20, vitamin E TPGS, Pluronic F68, Span 60, and Tween 80). The methods used were assessment of degree of creaming after centrifugation and after shelf storage for 28 days at room temperature, turbidimetric method, accelerated stability study, and droplet size analysis. The droplet size was found to be in the range of 15 to 2 µm and v/v% separation after centrifugation was found to be 0–82%. The RHLB for Capryol 90 was found to be 15. The effect of various surfactant blends was investigated. Results revealed that a blend of tocopherol polyethylene glycol succinate (TPGS) and Pluronic F68 gave the most stable emulsion for Capryol 90.