STABILITY OF EQUILIBRIA OF NEMATIC LIQUID CRYSTALLINE POLYMERS

We provide an analytical study on the stability of equilibria of rigid rodlike nematic liquid crystalline polymers (LCPs) governed by the Smoluchowski equation with the Maier-Saupe intermolecular potential. We simplify the expression of the free energy of an orientational distribution function of rodlike LCP molecules by properly selecting a coordinate system and then investigate its stability with respect to perturbations of orientational probability density. By computing the Hessian matrix explicitly, we are able to prove the hysteresis phenomenon of nematic LCPs: when the normalized polymer concentration b is below a critical value b*(6.7314863965), the only equilibrium state is isotropic and it is stable; when b*< b < 15/2, two anisotropic (prolate) equilibrium states occur together with a stable isotropic equilibrium state. Here the more aligned prolate state is stable whereas the less aligned prolate state is unstable. When b > 15/2, there are three equilibrium states: a stable prolate state, an unstable isotropic state and an unstable oblate state.     

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?     

一种场站设置问题中的几个结论

本文研究如下一种场站设置问题:设S是欧空间E~m中由有限个点A_1,A_2,…,A_n组成的集合.d(A_i,A_j)表示点A_i和A_j之间的距离.令σ(S)=Σ_(1≤i相似文献   

解决正态分布问题的两个着眼点

正态分布是加盟高中数学的“新成员”,给命题提供了新颖的背景,注入了新的活力,许多现实生活中的分布问题都可以用它来描述,自然受命题者的青睐．在高考中常以选择题,填空题的形式出现,虽然难度不大,但也不可忽视．对于正态分布问题的求解,主要从“数”与“形”两个基本方向去思考,可以比较轻松地完成求解．     

生产系统的检修时间间隔及关键备件采购的联合优化研究

研究了检修间隔时间与零备件库存采购的联合优化问题.利用带有吸收状态的连续时间马尔科夫链表示有m个生产状态以及一个损坏状态的生产系统,这个系统的再生点是生产系统的维修或检修完成的时刻点.利用更新过程报酬定理给出单位时间维修成本,利用概率分析技术给出备件库存的单位时间成本,进而得到整个生产系统的单位时间的总成本.最后,通过...     

一类离散的时滞脉冲切换系统的H_∞二次稳定性

研究了一类离散的时滞脉冲切换系统的H_∞控制问题,利用Lyapunov稳定性理论和线性矩阵不等式方法,给出了离散脉冲切换系统具有性能指标γ的充分条件.     

关于Erlang分布的一个注记

利用递推的方法得到了尺度参数不同的两个相互独立的Erlang分布的卷积的表达形式.作为应用得到了Beta分布的Laplace变换的表达式.     

多元Beta分布特性分析

利用小样本截尾序贯检验理论,在武器系统对空中目标的命中精度检验问题中,遇到了一类多元Beta概率分布函数,讨论分析了多维Beta概率分布函数的特性并给出了概率计算表.结果对武器精度检验具有重要意义和实用价值.     

具有染病者输入的离散SIR传染病模型分析

引入相应的概率建立了具有染病者输入的离散SIR传染病模型,确定了决定其动力学性态的阂值.在阈值之下模型仅存在无病平衡点,且无病平衡点是全局渐近稳定的;在阈值之上模型是一致持续的,有唯一的地方病平衡点存在,且地方病平衡点是局部渐近稳定的.