Elementary and Φ-Free Lie Triple Systems

We introduce elementary and Φ-free Lie triple systems and study the properties of these systems. In particular, structures of subsystems of an elementary Lie triple system and a class of Φ-free Lie triple systems are investigated.     

Some new perfect Steiner triple systems

In a Steiner triple system STS(v) = (V, B), for each pair {a, b} ⊂ V, the cycle graph G_a,b can be defined as follows. The vertices of G_a,b are V \ {a, b, c} where {a, b, c} ∈ B. {x, y} is an edge if either {a, x, y} or {b, x, y} ∈ B. The Steiner triple system is said to be perfect if the cycle graph of every pair is a single (v − 3)-cycle. Perfect STS(v) are known only for v = 7, 9, 25, and 33. We construct perfect STS (v) for v = 79, 139, 367, 811, 1531, 25771, 50923, 61339, and 69991. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 327–330, 1999     

三类有向三元系之间关系的一种猜想

一个Mendelsohn (Directed, 或Hybrid)三元系 MTS$(v, \lambda)$~(DTS$(v, \lambda)$,或HTS$(v,\lambda))$, 是由$v$元集$X$ 上的一些循环（可迁，或循环和可迁）三元组(简称区组)构成的集合${\cal B}$, 使得$X$上每个由不同元素组成的有序对都恰在 ${\cal B}$的$\lambda$个区组中出现.本文主要讨论了这三类有向三元系之间的一种关联关系，给出猜想：任意MTS$(v,\lambda)$的区组关联图$G(\     

自反DTS(υ,λ)的存在谱

一个Directed三元系DTS(υ,λ)=(X,B)是自反的,如果它与它的逆(X,B-1)同构,其中B-1=｛(z,y,x);(x,y,z)∈B｝.继[2]给出自反DTS(υ,1)的存在谱之后,本文将给出自反DTS(υ,λ)的存在谱为     

A proof of Lindner's conjecture on embeddings of partial Steiner triple systems

Lindner's conjecture that any partial Steiner triple system of order u can be embedded in a Steiner triple system of order v if and is proved. © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 63–89, 2009     

The inside outside intersection problem for hexagon triple systems

We give a complete solution of the following two problems:

1. (1)
   For which (n, x) does there exist a pair of hexagon triple systems of order n having x inside triples in common?
    
2. (2)
   For which (n, x) does there exist a pair of hexagon triple systems having x outside triples in common?
    

     

李三系的Frattini子系

李三系是从黎曼对称空间产生的三元运算的代数系统,近年来备受数学家们的重视.针对李三系的Frattini子系和基本李三系的问题进行了研究,给出了Frattini子系和基本李三系的一些性质,并证明了李三系的非嵌入定理,同时得到了幂零李三系是基本李三系的一个充要条件.     

A decomposition of (<Emphasis Type="Italic">λK</Emphasis><Subscript><Emphasis Type="Italic">υ</Emphasis></Subscript>)<Superscript>+</Superscript> with extended triangles

By an extended triangle, we mean a loop, a loop with an edge attached (known as a lollipop), or a copy of K ₃ (known as a triangle). In this paper, we completely solve the problem of decomposing the graph (λK _υ )⁺ into extended triangles for all possible number of loops. Dedicated to Professor Alex Rosa on the occasion of his 70th birthday     

不相交斯泰纳和柯克曼三元系大集两定理(一)

给定一个斯泰纳或柯克曼三元系,介绍其大集的生成方法;得到其大集存在条件的判据是一个位差各异的循环数;柯克曼三元系其大集判据是,不能分解的柯克曼三元系有大集,能分解的无大集.     

Steiner triple systems with two disjoint subsystems

It is shown that there exists a triangle decomposition of the graph obtained from the complete graph of order v by removing the edges of two vertex disjoint complete subgraphs of orders u and w if and only if u,w, and v are odd, (mod 3), and . Such decompositions are equivalent to group divisible designs with block size 3, one group of size u, one group of size w, and v – u – w groups of size 1. This result settles the existence problem for Steiner triple systems having two disjoint specified subsystems, thereby generalizing the well‐known theorem of Doyen and Wilson on the existence of Steiner triple systems with a single specified subsystem. © 2005 Wiley Periodicals, Inc. J Combin Designs     

Large sets of extended directed triple systems with odd orders

For three types of triples, unordered, cyclic and transitive, the corresponding extended triple, extended triple system and their large set are introduced. The spectrum of LEDTS(v) for even v has been given in our paper (Liu and Kang (2009) [9]). In this paper, we shall discuss the existence problem of LEDTS(v) for odd v and give the almost complete conclusion: there exists an LEDTS(v) for any positive integer v≠4 except possible v=95,143,167,203,215.     

分裂的$\delta$-Jordan 李三系

本文研究了任意分裂的$\delta$-Jordan李三系的结构,其为分裂的李三系的结构的推广. 利用这种三系的根连通, 得到了带有对称根系的分裂的 $\delta$-Jordan 李三系可以表示成 $T=U+\sum_{[\alpha]\in \Lambda^{1}/\sim} I_{[\alpha]}$,其中$U$是0根空间$T_{0}$的子空间,任意$I_{[\alpha]}$为$T$的理想, 并且满足 当$[\alpha]\neq [\beta]$时, $\{I_{[\alpha]},T,I_{[\beta]}\}=\{I_{[\alpha]},I_{[\beta]},T\}=\{T,I_{[\alpha]},I_{[\beta]}\}=0$.     

可度量化李三系

研究域K上可定义非退化不变对称双线性形的李三系■(称这样的李三系■为可度量化的).对一个李三系■,我们定义了该李三系的T_ω~*-扩张,给出了一个度量化李三系(■,φ)同构于某李三系的T~*-扩张的充分必要条件,并且讨论了什么时候两个T~*-扩张是等价或度量等价的.最后证明当基域的特征为零时,偶数维幂零的度量化李三系与某李三系的T~*-扩张同构,而奇数维幂零的度量化李三系则同构于某李三系的T~*-扩张的余维数为1的非退化理想.     

The Existence of LHMTS(<Emphasis Type="Italic">m</Emphasis><Superscript><Emphasis Type="Italic">v</Emphasis></Superscript>) and LHDTS(<Emphasis Type="Italic">m</Emphasis><Superscript><Emphasis Type="Italic">v</Emphasis></Superscript>)

In this article, we establish the existence of an LHMTS(m^v) for v ≡ 2 (mod 6) and m≡ 3 (mod 6). Thus there exists an LHMTS(m^v) if and only if v(v-1)m² ≡ 0 (mod 3) except possibly for v=6, m≡ 1, 5 (mod 6) and m≠1. In the similar way, the existence of LHDTS(m^v) is completely determined, i.e., there exists an LHDTS(m^v) if and only if v(v-1)m² ≡ 0 (mod 3).     

Maximum Kirkman Signal Sets for Synchronous Uni-Polar Multi-UserCommunication Systems

A unipolar signalingsystem transmits using intensity or amplitude in multiple dimensions.Typical examples arise in optical transmission or radio communicationusing MT-MFSK as both the signaling and the modulation technique.There are dimensions which represent pulses ortones. Each codeword consists of a selection of kof these tones with unit intensity. Each user is assigned mof these binary codewords. In a synchronous multi-user environment,two codewords assigned to a single user have distance 2k,while two codewords assigned to different users have distanceat least 2k-2. Such an assignment of codewords tousers is called a Kirkman signal set when the number of usersaccommodated is the maximum. In this paper, the existence ofKirkman signal sets with k=3 and mas large as possible is settled for all values of .     

Doyen–Wilson theorem for perfect hexagon triple systems

The graph consisting of the three 3-cycles (or triples) (a,b,c), (c,d,e), and (e,f,a), where a,b,c,d,e and f are distinct is called a hexagon triple. The 3-cycle (a,c,e) is called an inside 3-cycle; and the 3-cycles (a,b,c), (c,d,e), and (e,f,a) are called outside 3-cycles. A hexagon triple system of order v is a pair (X,C), where C is a collection of edge disjoint hexagon triples which partitions the edge set of 3K_v. Note that the outside 3-cycles form a 3-fold triple system. If the hexagon triple system has the additional property that the collection of inside 3-cycles (a,c,e) is a Steiner triple system it is said to be perfect. In 2004, Küçükçifçi and Lindner had shown that there is a perfect hexagon triple system of order v if and only if and v≥7. In this paper, we investigate the existence of a perfect hexagon triple system with a given subsystem. We show that there exists a perfect hexagon triple system of order v with a perfect sub-hexagon triple system of order u if and only if v≥2u+1, and u≥7, which is a perfect hexagon triple system analogue of the Doyen–Wilson theorem.     

Some infinite families of large sets of Kirkman triple systems

A large set of Kirkman triple systems of order v, denoted by LKTS(v), is a collection {(X, B_i) : 1 ≤ i ≤ v ? 2}, where every (X,B_i) is a KTS(v) and all B_i form a partition of all triples on X. Many researchers have studied the existence of LKTS(v) for a long time. In [13], the author introduced a concept—large set of generalized Kirkman systems (LGKS), which plays an important role in the discussion of LKTS. In this article, we give a new construction for LGKS and obtain some new results of LKTS, that is, there exists an LKTS(6u + 3) for u = qⁿ, where n ≥ 1, q ≡ 7 (mod 12) and q is a prime power. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 202–212, 2008