完全分配格与点格

本文第一部分利用完备格上的上拓扑子基,给出完全分配格与点格的若干新刻划,并讨论其上的 Scott 拓扑与 Lawson 拓扑的基与子基的构造.第二部分讨论点格与代数格的关系,证明了 L 是点格当且仅当 L 为代数格且 L~(op)为完全 Heyting代数,并证明了代数偏序集范畴与点格范畴是等价的.     

一阶线性和拟线性双曲型方程的格点模型

一阶线性和拟线性双曲型方程的格点模型李元香（武汉大学软件工程国家重点实验室）黄樟灿（武汉工学院）ＬＡＴＴＩＣＥＭＯＤＥＬＳＦＯＲＦＩＲＳＴＯＲＤＥＲＬＩＮＥＡＲＡＮＤＱＵＡＳＩ－ＬＩＮＥＡＲＨＹＰＥＢＯＬＩＣＥＱＵＡＴＩＯＮＳ￥ＬｉＹｕａｎ－ｘｉａｎ...     

一类完全正则半群簇子簇格的性质

本文研究了完全正则半群簇的子簇格［Ｖ＋∩ＰＶ，Ｖ＋∩ＰＶ］的某些格运算性质，我们证明了簇Ｖ＋∩ＰＶ可分解为Ｖ与Ｖ＋∩ＰＶ的并；对任意完全正则半群簇Ｗ，有Ｗ∩（Ｖ∨Ｖ＋∩ＰＶ）＝（Ｗ∩Ｖ）∨（Ｗ∩Ｖ＋∩ＰＶ）．特别地，我们得到了等式Ｖ＋∩ＰＶ＝Ｖ成立的若干条件．     

快件揽收实时车辆路径问题的一种贪婪算法

快递运营中，调派车辆前往随机发生的快件发件人处上门揽收快件，是一个实时编排行车路径的动态决策过程．本文针对该问题，采用了揽收所有快件的最后时刻最早和行车路径最短的目标，结合车辆揽收快件数平衡的要求，给出一种贪婪算法；然后，对Solomon设计的100个点规模的VRPTW算例做计算试验，分析了车辆数对目标的影响．     

格点动力系统中的异宿环

讨论格点动力系统中空间混沌出现的一个判据——异宿环.证明了若格点动力系统有渐近稳定的异宿环,则系统有渐近稳定的同宿点,从而系统有空间混沌.     

关于格点覆盖的一个注记

对于一个给定椭球,本文给出了它的任一全等椭球都包含一个整点的一个充分必要条件.     

格点形心问题的若干结果

设n(k)为满足如下条件的最小整数，给定平面上任意n个格点，其中必存在k个点的形心也是格点，文献[4]提出关于确定n(4)的未解问题，本文给出解答n(4)=13，并进一步给出相关的一些问题的结果。     

分子格的极大点的相通支及其应用

在一般的分子格中建立类似于Ｆ集中的“点的支撑”的概念－极大点的相通支，得到分子格的最大元的一种分解，并定义了ＴＭＬ中元的良聚点、良导元，讨论了它们的性质。结果表明它们具有Ｆ拓扑学中的导集的许多好的性质。     

与格点有关的问题的求解策略

所谓格点（又称整点），是指平面直角坐标系中横、纵坐标都是整数的点，这类问题因其与整数有关，因而在数学竞赛中多有出现，本文探讨这类问题的解法。     

The research and progress of the enumeration of lattice paths

The enumeration of lattice paths is an important counting model in enumerative combinatorics. Because it can provide powerful methods and technical support in the study of discrete structural objects in different disciplines, it has attracted much attention and is a hot research field. In this paper, we summarize two kinds of the lattice path counting models that are single lattice paths and family of nonintersecting lattice paths and their applications in terms of the change of dimensions, steps, constrained conditions, the positions of starting and end points, and so on. (1) The progress of classical lattice path such as Dyck lattice is introduced. (2) A method to study the enumeration of lattice paths problem by generating function is introduced. (3) Some methods of studying the enumeration of lattice paths problem by matrix are introduced. (4) The family of lattice paths problem and some counting methods are introduced. (5) Some applications of family of lattice paths in symmetric function theory are introduced, and a related open problem is proposed.     

Limits of areas under lattice paths

We consider sequences of polynomials which count lattice paths by area. In some cases the reversed polynomials approach a formal power series as the length of the paths tend to infinity. We find the limiting series for generalized Schröder, Motzkin, and Catalan paths. The limiting series for Schröder paths and their generalizations are shown to count partitions with restrictions on the multiplicities of odd parts and no restrictions on even parts. The limiting series for generalized Motzkin and Catalan paths are shown to count generalized Frobenius partitions and some related arrays.     

Some properties of complete congruence lattices

For a complete lattice C, we consider the problem of establishing when the complete lattice of complete congruence relations on C is a complete sublattice of the complete lattices of join- or meet-complete congruence relations on C. We first argue that this problem is not trivial, and then we show that it admits an affirmative answer whenever C is continuous for the join case and, dually, co-continuous for the meet case. As a consequence, we prove that if C is continuous then each principal filter generated by a continuous complete congruence on C is pseudocomplemented. Received January 6, 1998; accepted in final form July 2, 1998.     

Multiple partitions, lattice paths and a Burge-Bressoud-type correspondence

A bijection is presented between (1): partitions with conditions f_j+f_j+1≤k−1 and f₁≤i−1, where f_j is the frequency of the part j in the partition, and (2): sets of k−1 ordered partitions (n⁽¹⁾,n⁽²⁾,…,n^(k−1)) such that and , where m_j is the number of parts in n^(j). This bijection entails an elementary and constructive proof of the Andrews multiple-sum enumerating partitions with frequency conditions. A very natural relation between the k−1 ordered partitions and restricted paths is also presented, which reveals our bijection to be a modification of Bressoud’s version of the Burge correspondence.     

Overpartitions, lattice paths, and Rogers-Ramanujan identities

We extend partition-theoretic work of Andrews, Bressoud, and Burge to overpartitions, defining the notions of successive ranks, generalized Durfee squares, and generalized lattice paths, and then relating these to overpartitions defined by multiplicity conditions on the parts. This leads to many new partition and overpartition identities, and provides a unification of a number of well-known identities of the Rogers-Ramanujan type. Among these are Gordon's generalization of the Rogers-Ramanujan identities, Andrews' generalization of the Göllnitz-Gordon identities, and Lovejoy's “Gordon's theorems for overpartitions.”     

Correction to "Some properties of complete congruence lattices"

Received April 26, 2000; accepted in final form July 19, 2001.     

Cumulants, lattice paths, and orthogonal polynomials

A formula expressing free cumulants in terms of Jacobi parameters of the corresponding orthogonal polynomials is derived. It combines Flajolet's theory of continued fractions and the Lagrange inversion formula. For the converse we discuss Gessel–Viennot theory to express Hankel determinants in terms of various cumulants.     

Rainbow and orthogonal paths in factorizations of Kn

For n even, a factorization of a complete graph K_n is a partition of the edges into n?1 perfect matchings, called the factors of the factorization. With respect to a factorization, a path is called rainbow if its edges are from distinct factors. A rainbow Hamiltonian path takes exactly one edge from each factor and is called orthogonal to the factorization. It is known that not all factorizations have orthogonal paths. Assisted by a simple edge‐switching algorithm, here we show that for n?8, the rotational factorization of K_n, GK_n has orthogonal paths. We prove that this algorithm finds a rainbow path with at least (2n+1)/3 vertices in any factorization of K_n (in fact, in any proper coloring of K_n). We also give some problems and conjectures about the properties of the algorithm. © 2010 Wiley Periodicals, Inc. J Combin Designs 18: 167–176, 2010     

The number of lattice paths below a cyclically shifting boundary

We count the number of lattice paths lying under a cyclically shifting piecewise linear boundary of varying slope. Our main result can be viewed as an extension of well-known enumerative formulae concerning lattice paths dominated by lines of integer slope (e.g. the generalized ballot theorem). Its proof is bijective, involving a classical “reflection” argument. Moreover, a straightforward refinement of our bijection allows for the counting of paths with a specified number of corners. We also show how the result can be applied to give elegant derivations for the number of lattice walks under certain periodic boundaries. In particular, we recover known expressions concerning paths dominated by a line of half-integer slope, and some new and old formulae for paths lying under special “staircases.”     

The genesis of involutions (Polarizations and lattice paths)

The number of Borel orbits in the symmetric space $S{L}_n∕S(G{L}_p\times G{L}_q)$ is analyzed, various (bivariate) generating functions are found. Relations to lattice path combinatorics are explored.