Steiner选择和算子的集值扩张(Ⅱ)——对逼近理论的应用

在(Ⅰ)的基础上,得出对集值函数逼近理论的某些应用:Korovkin型定理,一种将经典逼近算子扩张到集值族的方法,以及Jackson估算.     

Trifold arrangements and cevian dissections

We determine precisely how many triple points can be formed in the plane by an arrangement of n lines lying in three parallel families of p, q, r lines, respectively. Using this result we solve the Euclidean realization problem for such arrangements. We apply our results to solve an analogous problem in which a triangle is dissected by three families of cevians. We conclude by mentioning some related unsolved problems. Received 19 August 1999.     

λ_5-geometry中的Steiner树问题(Ⅰ)

本文首先提出了 λ5-geometry中的 Steiner最小树问题 .讨论了 λ5-ge-ometry中的 Steiner最小树的若干性质 ,并给出了给定点数为 3或 4时 Steiner最小树的基本结构 .     

Sequencing partial Steiner triple systems

A partial Steiner triple system of order $n$ is sequenceable if there is a sequence of length $n$ of its distinct points such that no proper segment of the sequence is a union of point‐disjoint blocks. We prove that if a partial Steiner triple system has at most three point‐disjoint blocks, then it is sequenceable.     

A complete solution to existence of H designs

An H design is a triple , where is a set of points, a partition of into disjoint sets of size , and a set of ‐element transverses of , such that each ‐element transverse of is contained in exactly one of them. In 1990, Mills determined the existence of an H design with . In this paper, an efficient construction shows that an H exists for any integer with . Consequently, the necessary and sufficient conditions for the existence of an H design are , , and , with a definite exception .     

Codes of Small Defect

The parameters of a linear code C over GF(q) are given by [n,k,d], where n denotes the length, k the dimension and d the minimum distance of C. The code C is called MDS, or maximum distance separable, if the minimum distance d meets the Singleton bound, i.e. d = n-k+1 Unfortunately, the parameters of an MDS code are severely limited by the size of the field. Thus we look for codes which have minimum distance close to the Singleton bound. Of particular interest is the class of almost MDS codes, i.e. codes for which d=n-k. We will present a condition on the minimum distance of a code to guarantee that the orthogonal code is an almost MDS code. This extends a result of Dodunekov and Landgev Dodunekov. Evaluation of the MacWilliams identities leads to a closed formula for the weight distribution which turns out to be completely determined for almost MDS codes up to one parameter. As a consequence we obtain surprising combinatorial relations in such codes. This leads, among other things, to an answer to a question of Assmus and Mattson 5 on the existence of self-dual [2d,d,d]-codes which have no code words of weight d+1. Actually there are more codes than Assmus and Mattson expected, but the examples which we know are related to the expected ones.     

A construction for large sets of disjoint Kirkman triple systems

A Steiner system S(t, k, v) is called i-resolvable, 0 < i < t, if its block set can be partitioned into S(i, k, v). In this paper, a 2-resolvable S(3, 4, v) is used to construct a large set of disjoint Kirkman triple systems of order 3v − 3 (briefly LKTS) and some new orders for LKTS are then obtained. Research supported by Tianyuan Mathematics Foundation of NSFC Grant 10526032 and Natural Science Foundation of Universities of Jiangsu Province Grant 05KJB110111.     

The design of the century

We construct a 2-chromatic Steiner system S(2, 4, 100) in which every block contains three points of one colour and one point of the other colour. The existence of such a design has been open for over 25 years.      

A dual ascent approach for steiner tree problems on a directed graph

The Steiner tree problem on a directed graph (STDG) is to find a directed subtree that connects a root node to every node in a designated node setV. We give a dual ascent procedure for obtaining lower bounds to the optimal solution value. The ascent information is also used in a heuristic procedure for obtaining feasible solutions to the STDG. Computational results indicate that the two procedures are very effective in solving a class of STDG's containing up to 60 nodes and 240 directed/120 undirected arcs. The directed spanning tree and uncapacitated plant location problems are special cases of the STDG. Using these relationships, we show that our ascent procedure can be viewed as a generalization ofboth the Chu-Liu-Edmonds directed spanning tree algorithm and the Bilde-Krarup-Erlenkotter ascent algorithm for the plant location problem. The former comparison yields a dual ascent interpretation of the steps of the directed spanning tree algorithm.