Point Code Minimum Steiner Triple Systems

The point code of a Steiner triple system uniquely determines the system when the number of vectors whose weight equals the replication number agrees with the number of points. The existence of a Steiner triple system with this minimum point code property is established for all v 1,3 (mod 6) with v 15.     

泛系局整关系分析与三元复盖系统

我们于1985年8月至1986年2月,在我校试验站进行了卷帘式、有窗式和无窗式(不装空调)三种猪舍的对比试验和观测,结果表明: 1.正北京地区,用上述三种猪舍饲养30～90kg的猪,各舍试猪日增重和饲料利用率,在夏秋冬三季均无显著差异; 2.卷帘舍夏季通风好,有利于防暑,但冬季保温较差;无窗舍夏季通风差,较闷热,冬季保温虽好,但湿度较大;有窗舍夏季防暑和冬季保温均较好; 3.无窗舍除大跨度空调猪舍外,小跨度自然通风光照猪舍不易采用;卷帘舍在北京及其以南地区可用于小跨度育肥及成年猪舍;有窗舍可用于饲养各种猪     

Solving large Steiner Triple Covering Problems

Computing the 1-width of the incidence matrix of a Steiner Triple System gives rise to highly symmetric and computationally challenging set covering problems. The largest instance solved so far corresponds to a Steiner Tripe System of order 81. We present optimal solutions for systems of orders 135 and 243. These are computed by a tailored implementation of constraint orbital branching, a method designed to exploit symmetry in integer programs.     

Rigid Steiner Triple Systems Obtained from Projective Triple Systems

It was shown by Babai in 1980 that almost all Steiner triple systems are rigid; that is, their only automorphism is the identity permutation. Those Steiner triple systems with the largest automorphism groups are the projective systems of orders . In this paper, we show that each such projective system may be transformed to a rigid Steiner triple system by at most n Pasch trades whenever .     

The Minimum Covering l pb -Hypersphere Problem

The minimu vering hypersphere problem is defined as to find a hypersphere of minimum radius enclosing a finite set of given points in ⁿ. A hypersphere is a set S(c,r)={x ⁿ : d(x,c) r}, where c is the center of S, r is the radius of S and d(x,c) is the Euclidean distance between x and c, i.e.,d(x,c)=l ₂ (x-c). We consider the extension of this problem when d(x, c) is given by any l _pb -norm, where 1<p and b=(b ₁,...,b _n ) with b _j >0, j=1,...,n, then S(c,r) is called an l _pb -hypersphere, in particular for p=2 and b _j =1, j=1,..., n, we obtain the l ₂-norm. We study some properties and propose some primal and dual algorithms for the extended problem , which are based on the feasible directions method and on the Wolfe duality theory. By computational experiments, we compare the proposed algorithms and show that they can be used to approximate the smallest l _pb -hypersphere enclosing a large set of points in ⁿ.     

A New Efficient Heuristic for the Minimum Set Covering Problem

We solve approximately the minimum set covering problem by developing a new heuristic, which is essentially based on the flow algorithm originally developed by Ford and Fulkerson. We perform a comparative study of the performances (concerning solution qualities and execution times) of the flow algorithm as well as of the natural greedy heuristic for set covering originally studied by Johnson and Lovász.     

Equitable Specialized Block-Colourings for Steiner Triple Systems

We continue the study of specialized block-colourings of Steiner triple systems initiated in [2] in which the triples through any element are coloured according to a given partition π of the replication number. Such colourings are equitable if π is an equitable partition (i.e., the difference between any two parts of π is at most one). Our main results deal with colourings according to equitable partitions into two, and three parts, respectively.     

Construction Techniques for Anti-Pasch Steiner Triple Systems

Four methods for constructing anti-Pasch Steiner triple systemsare developed. The first generalises a construction of Stinsonand Wei to obtain a general singular direct product construction.The second generalises the Bose construction. The third employsa construction due to Lu. The fourth employs Wilson-type inflationtechniques using Latin squares having no subsquares of order2. As a consequence of these constructions we are able to produceanti-Pasch systems of order v for v 31 or 7 (mod 18), for v 49 (mod 72), and for many other values of v.     

Tricyclic Steiner Triple Systems

A Steiner triple system of order ν, denoted STS(ν), is said to be tricyclic if it admits an automorphism whose disjoint cyclic decomposition consists of three cycles. In this paper we give necessary and sufficient conditions for the existence of a tricyclic STS(ν) for several cases. We also pose conjectures concerning their existence in two remaining cases.     

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.     

Triple Systems are Eulerian

An Euler tour of a hypergraph (also called a rank‐2 universal cycle or 1‐overlap cycle in the context of designs) is a closed walk that traverses every edge exactly once. In this paper, using a graph‐theoretic approach, we prove that every triple system with at least two triples is eulerian, that is, it admits an Euler tour. Horan and Hurlbert have previously shown that for every admissible order >3, there exists a Steiner triple system with an Euler tour, while Dewar and Stevens have proved that every cyclic Steiner triple system of order >3 and every cyclic twofold triple system admits an Euler tour.     

Systems of Quotients of Lie Triple Systems

In this article, we introduce the notion of system of quotients of Lie triple systems and investigate some properties which can be lifted from a Lie triple system to its systems of quotients. We relate the notion of Lie triple system of Martindale-like quotients with respect to a filter of ideals and the notion of system of quotients, and prove that the system of quotients of a Lie triple system is equivalent to the algebra of quotients of a Lie algebra in some sense, and these allow us to construct the maximal system of quotients for nondegenerate Lie triple systems.     

On Upper Bounds for Minimum Distance and Covering Radius of Non-binary Codes

We consider upper bounds on two fundamental parameters of a code; minimum distance and covering radius. New upper bounds on the covering radius of non-binary linear codes are derived by generalizing a method due to S. Litsyn and A. Tietäväinen lt:newu and combining it with a new upper bound on the asymptotic information rate of non-binary codes. The upper bound on the information rate is an application of a shortening method of a code and is an analogue of the Shannon-Gallager-Berlekamp straight line bound on error probability. These results improve on the best presently known asymptotic upper bounds on minimum distance and covering radius of non-binary codes in certain intervals.     

Covering Complete Hypergraphs with Cuts of Minimum Total Size

A cut [X, V − X] in a hypergraph with vertex-set V is the set of all edges that meet both X and V − X. Let s _r (n) denote the minimum total size of any cover of the edges of the complete r-uniform hypergraph on n vertices K_n^r{K_n^r} by cuts. We show that there is a number n _r such that for every n > n _r , s _r (n) is uniquely achieved by a cover with ?\fracn-1r-1?{\lfloor \frac{n-1}{r-1}\rfloor} cuts [X _i , V − X _i ] such that the X _i are pairwise disjoint sets of size at most r − 1. We show that c₁r2^r < n_r < c₂r⁵2^r{c_1r2^r < n_r < c_2r^52^r} for some positive absolute constants c ₁ and c ₂. Using known results for s ₂(n) we also determine s ₃(n) exactly for all n.     

The Fine Intersection Problem for Steiner Triple Systems

The intersection of two Steiner triple systems and is the set . The fine intersection problem for Steiner triple systems is to determine for each v, the set I(v), consisting of all possible pairs (m, n) such that there exist two Steiner triple systems of order v whose intersection satisfies and . We show that for v ≡ 1 or 3 (mod 6), |I(v)| = Θ(v ³), where previous results only imply that |I(v)| = Ω(v ²). Received: January 23, 2006. Final Version received: September 2, 2006     

Frame Self-orthogonal Mendelsohn Triple Systems

A Mendelsohn triple system of order ν, MTS(ν) for short, is a pair (X, B) where X is a ν-set (of points) and B is a collection of cyclic triples on X such that every ordered pair of distinct points from X appears in exactly one cyclic triple of B. The cyclic triple (a, b, c) contains the ordered pairs (a, b), (b, c) and (c, a). An MTS(ν) corresponds to an idempotent semisymmetric Latin square (quasigroup) of order ν. An MTS(ν) is called frame self-orthogonal, FSOMTS for short, if its associated semisymmetric Latin square is frame self-orthogonal. It is known that an FSOMTS(1 ⁿ ) exists for all n≡1 (mod 3) except n=10 and for all n≥15, n≡0 (mod 3) with possible exception that n=18. In this paper, it is shown that (i) an FSOMTS(2 ⁿ ) exists if and only if n≡0,1 (mod 3) and n>5 with possible exceptions n∈{9, 27, 33, 39}; (ii) an FSOMTS(3 ⁿ ) exists if and only if n≥4, with possible exceptions that n∈{6, 14, 18, 19}. *Research supported by NSFC 10371002 *Partially supported by National Science Foundation under Grant CCR-0098093     

On Infinite Disjoint Congruence Covering Systems

In this paper we show that for any α∈(0,l] there exists an infinite disjoint covering system {a1(modni)}i=1∞ such that     

Linearly Derived Steiner Triple Systems

We attach a graph to every Steiner triple system. The chromatic number of this graph is related to the possibility of extending the triple system to a quadruple system. For example, the triple systems with chromatic number one are precisely the classical systems of points and lines of a projective geometry over the two-element field, the Hall triple systems have chromatic number three (and, as is well-known, are extendable) and all Steiner triple systems whose graph has chromatic number two are extendable. We also give a configurational characterization of the Hall triple systems in terms of mitres.     

Simple Signed Steiner Triple Systems

Let X be a v‐set, be a set of 3‐subsets (triples) of X, and be a partition of with . The pair is called a simple signed Steiner triple system, denoted by ST, if the number of occurrences of every 2‐subset of X in triples is one more than the number of occurrences in triples . In this paper, we prove that exists if and only if , , and , where and for , . © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 332–343, 2012     

Covering Method for Trajectory Generation and Orbital Decomposition of Systems

Based on previously stated approaches, we propose a method for solving point-topoint steering problems in the case where a 2n-parametric family of solutions of a nonlinear system is known and for Liouville systems. Two examples of the helicopter motions in the vertical plane and the Kapitsa pendulum are considered to demonstrate the efficiency of the proposed method.