Topology and geometry of nematic braids

Topological analysis of disclinations in nematic liquid crystals is an interesting and diverse topic that goes from strict mathematical theorems to applications in elaborate systems found in experiments and numerical simulations. The theory of nematic disclinations is shown from both the geometric and topological perspectives. Entangled disclination line networks are analyzed based on their shape and the behavior of their cross section. Methods of differential geometry are applied to derive topological results from reduced geometric information. For nematic braids, systems of −1/2 disclination loops, created by inclusion of homeotropic colloidal particles, a formalism of rewiring is constructed, allowing comparison and construction of an entire set of different conformations. The disclination lines are described as ribbons and a new topological invariant, the self-linking number, is introduced. The analysis is generalized from a constant −1/2 profile to general profile variations, while retaining the geometric treatment. The workings of presented topological statements are demonstrated on simple models of entangled nematic colloids, estimating the margins of theoretical assumptions made in the formal derivations, and reviewing the behavior of the disclinations not only under topological, but also under free-energy driven constraints.     

公交车调度问题的数学模型

对 2 0 0 1年全国大学生数学建模竞赛的 B题——公交车调度问题进行了分析 ,建立了调度的目标规划模型及 0— 1规划模型 .在假设各站上、下车人数服从均匀分布的条件下 ,通过对模型的求解 ,求出了公交公司的最小运行车辆数 5 2辆 ,并给出了发车时刻表 ,其中上行方向运行 2 2 5班次 ,下行方向运行 2 2 0班次 .该模型简单 ,求解容易 ,能较好地考虑各方利益     

A PRECONDITIONER FOR COUPLING SYSTEM OF NATURAL BOUNDARY ELEMENT AND COMPOSITE GRID FINITE ELEMENT

1. IntroductionThe coupling of boundary elemellts and finite elements is of great imPortance for the nu-mercal treatment of boundary value problems posed on unbounded domains. It permits us tocombine the advanages of boundary elements for treating domains extended to infinity withthose of finite elemenis in treating the comP1icated bounded domains.The standard procedure of coupling the boundary elemeni and finite elemeni methods isdescribed as follows. First, the (unbounded) domain is divided…     

Hermite and Smith normal form algorithms over Dedekind domains

We show how the usual algorithms valid over Euclidean domains, such as the Hermite Normal Form, the modular Hermite Normal Form and the Smith Normal Form can be extended to Dedekind rings. In a sequel to this paper, we will explain the use of these algorithms for computing in relative extensions of number fields.

     

一维云粒子激光探测仪研制及探测数据分析

阐述基于Mie散射理论和激光技术而研制的云粒子探测仪的相关问题。利用m量级的小孔光阑模拟感应区域的散射光,并对系统的探测敏感区域面积进行测定;通过使用不同直径的标准粒子对系统进行标定,得到可靠的响应曲线,用于定量测量云粒子尺度谱及粒子数密度。在进行了一系列实验室内的实验之后,将仪器装载在飞机上进行穿云飞行测量实验,表明了该仪器在飞行过程中工作正常、稳定,并且能够即时地显示采样区内云粒子尺度谱分布和数浓度;通过分析探测得到的数据,并与云粒子谱分布进行比较,确认了探测数据有效可靠,反映了该仪器具有良好的测云能力。     

Domination number in graphs with minimum degree two

A set D of vertices of a graph G = （V, E） is called a dominating set if every vertex of V not in D is adjacent to a vertex of D. In 1996, Reed proved that every graph of order n with minimum degree at least 3 has a dominating set of cardinality at most 3n/8. In this paper we generalize Reed＇s result. We show that every graph G of order n with minimum degree at least 2 has a dominating set of cardinality at most （3n ＋IV21）/8, where V2 denotes the set of vertices of degree 2 in G. As an application of the above result, we show that for k ≥ 1, the k-restricted domination number rk （G, γ） ≤ （3n＋5k）/8 for all graphs of order n with minimum degree at least 3.     

The <Emphasis Type="Italic">k</Emphasis>-domatic number of a graph

Let k be a positive integer, and let G be a simple graph with vertex set V (G). A k-dominating set of the graph G is a subset D of V (G) such that every vertex of V (G)-D is adjacent to at least k vertices in D. A k-domatic partition of G is a partition of V (G) into k-dominating sets. The maximum number of dominating sets in a k-domatic partition of G is called the k-domatic number d _k (G). In this paper, we present upper and lower bounds for the k-domatic number, and we establish Nordhaus-Gaddum-type results. Some of our results extend those for the classical domatic number d(G) = d ₁(G).      

Bounds relating the weakly connected domination number to the total domination number and the matching number

Let G=(V,E) be a connected graph. A dominating set S of G is a weakly connected dominating set of G if the subgraph (V,E∩(S×V)) of G with vertex set V that consists of all edges of G incident with at least one vertex of S is connected. The minimum cardinality of a weakly connected dominating set of G is the weakly connected domination number, denoted . A set S of vertices in G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number γ_t(G) of G. In this paper, we show that . Properties of connected graphs that achieve equality in these bounds are presented. We characterize bipartite graphs as well as the family of graphs of large girth that achieve equality in the lower bound, and we characterize the trees achieving equality in the upper bound. The number of edges in a maximum matching of G is called the matching number of G, denoted α^′(G). We also establish that , and show that for every tree T.     

-Difference sequence spaces of fuzzy numbers

In this paper, using the difference operator of order m and an Orlicz function, we introduce and examine some classes of sequences of fuzzy numbers. We give the relations between the strongly Cesàro type convergence and statistical convergence in these spaces. Furthermore, we study some of their properties like completeness, solidity, symmetricity, etc. We also give some inclusion relations related to these classes.