An LRHTS（v） （or LARHTS（v）） is a collection of {（X, Bi） ： 1≤i ≤ 4（v - 2）}, where X is a v-set, each （X, Bi） is a resolvable （or almost resolvable） HTS（v）, and all Bis form a partition of all cycle tr...An LRHTS（v） （or LARHTS（v）） is a collection of {（X, Bi） ： 1≤i ≤ 4（v - 2）}, where X is a v-set, each （X, Bi） is a resolvable （or almost resolvable） HTS（v）, and all Bis form a partition of all cycle triples and transitive triples on X. An OLRHTS（v） （or OLARHTS（v）） is a collection {（Y／{y}, Ay^j） ： y ∈ Y,j = 0, 1, 2, 3}, where Y is a （v ＋ 1）-set, each （Y／{y}, Ay^j） is a resolvable （or almost resolvable） HTS（v）, and all Ay^js form a partition of all cycle and transitive triples on Y. In this paper, we establish some directed and recursive constructions for LRHTS（v）, LARHTS（v）, OLRHTS（v）, OLARHTS（v） and give some new results.展开更多
A idempotent quasigroup (Q, o) of order n is equivalent to an n(n-1)×3 partial orthogonal array in which all of rows consist of 3 distinct elements. Let X be a (n+1)-set. Denote by T(n+1) the set of (n+1)n(n-1) o...A idempotent quasigroup (Q, o) of order n is equivalent to an n(n-1)×3 partial orthogonal array in which all of rows consist of 3 distinct elements. Let X be a (n+1)-set. Denote by T(n+1) the set of (n+1)n(n-1) ordered triples of X with the property that the 3 coordinates of each ordered triple are distinct. An overlarge set of idempotent quasigroups of order n is a partition of T(n+1) into n+1 n(n-1)×3 partial orthogonal arrays A_x, x∈X based on X\{x}. This article gives an almost complete solution of overlarge sets of idempotent quasigroups.展开更多
A directed triple system of order v, denoted by DTS(v, λ), is a pair (X,B) where X is a v-set and B is a collection of transitive triples on X such that every ordered pair of X belongs to λ triples of B. An overlarg...A directed triple system of order v, denoted by DTS(v, λ), is a pair (X,B) where X is a v-set and B is a collection of transitive triples on X such that every ordered pair of X belongs to λ triples of B. An overlarge set of disjoint DTS(v, λ), denoted by OLDTS(v, λ), is a collection {(Y\{y}, Ai)}i,such that Y is a (v + 1)-set, each (Y\{y}, Ai) is a DTS(v, λ) and all Ai's form a partition of all transitive triples of Y. In this paper, we shall discuss the existence problem of OLDTS(v, λ) and give the following conclusion: there exists an OLDTS(v, λ) if and only if either λ = 1 and v = 0, 1 (mod 3), or λ = 3 and v≠2.展开更多
A hybrid triple system of order v and index A, denoted by HTS（v, λ）, is a pair （X,B） where X is a v-set and B is a collection of cyclic triples and transitive triples on X, such that every ordered pair of X belon...A hybrid triple system of order v and index A, denoted by HTS（v, λ）, is a pair （X,B） where X is a v-set and B is a collection of cyclic triples and transitive triples on X, such that every ordered pair of X belongs to A triples of B. An overlarge set of disjoint HTS（v, λ）, denoted by OLHTS（v, λ）, is a collection {（Y／{y}, .Ai）}i, such that Y is a （v ＋ 1）-set, each （Y／{y}, Ai） is an HTS（v, λ,） and all Ais form a partition of all cyclic triples and transitive triples on Y. In this paper, we shall discuss the existence problem of OLHTS（v, A） and give the following conclusion： there exists an OLHTS（v, λ） if and only if λ= 1, 2, 4, v = 0, 1 （rood 3） and v ≥ 4.展开更多
A directed triple system of order v,denoted by DTS(v),is a pair (X,B) where X is a v-set and B is a collection of transitive triples on X such that every ordered pair of X belongs to exactly one triple of B.A DTS(v) (...A directed triple system of order v,denoted by DTS(v),is a pair (X,B) where X is a v-set and B is a collection of transitive triples on X such that every ordered pair of X belongs to exactly one triple of B.A DTS(v) (X,A) is called pure and denoted by PDTS(v) if (a,b,c) ∈ A implies (c,b,a) ∈/ A.An overlarge set of PDTS(v),denoted by OLPDTS(v),is a collection {(Y \{yi},Aij) : yi ∈ Y,j ∈ Z3},where Y is a (v+1)-set,each (Y \{yi},Aij) is a PDTS(v) and these Ais form a partition of all transitive triples on Y .In this paper,we shall discuss the existence problem of OLPDTS(v) and give the following conclusion: there exists an OLPDTS(v) if and only if v ≡ 0,1 (mod 3) and v 】 3.展开更多
In this paper, we introduce a new concept -- overlarge sets of generalized Kirkman systems （OLGKS）, research the relation between it and OLKTS, and obtain some new results for OLKTS. The main conclusion is： If ther...In this paper, we introduce a new concept -- overlarge sets of generalized Kirkman systems （OLGKS）, research the relation between it and OLKTS, and obtain some new results for OLKTS. The main conclusion is： If there exist both an OLKF（6^k） and a 3-OLGKS（6^k-1,4） for all k ∈{6,7,...,40}／{8,17,21,22,25,26}, then there exists an OLKTS（v） for any v ≡ 3 （mod 6）, v ≠ 21. As well, we obtain the following result： There exists an OLKTS（6u ＋ 3） for u = 2^2n-1 - 1, 7^n, 31^n, 127^n, 4^r25^s, where n ≥ 1,r＋s≥ 1.展开更多
There are six types of triangles： undirected triangle, cyclic triangle, transitive triangle, mixed-1 triangle, mixed-2 triangle and mixed-3 triangle. The triangle-decompositions for the six types of triangles have al...There are six types of triangles： undirected triangle, cyclic triangle, transitive triangle, mixed-1 triangle, mixed-2 triangle and mixed-3 triangle. The triangle-decompositions for the six types of triangles have already been solved. For the first three types of triangles, their large sets have already been solved, and their overlarge sets have been investigated. In this paper, we establish the spectrum of LTi（v,λ）, OLTi（v）（i = 1, 2）, and give the existence of LT3（v, λ） and OLT3（v, λ） with λ even.展开更多
基金Supported by the National Natural Science Foundation of China(Grant No.11471096)
文摘An LRHTS（v） （or LARHTS（v）） is a collection of {（X, Bi） ： 1≤i ≤ 4（v - 2）}, where X is a v-set, each （X, Bi） is a resolvable （or almost resolvable） HTS（v）, and all Bis form a partition of all cycle triples and transitive triples on X. An OLRHTS（v） （or OLARHTS（v）） is a collection {（Y／{y}, Ay^j） ： y ∈ Y,j = 0, 1, 2, 3}, where Y is a （v ＋ 1）-set, each （Y／{y}, Ay^j） is a resolvable （or almost resolvable） HTS（v）, and all Ay^js form a partition of all cycle and transitive triples on Y. In this paper, we establish some directed and recursive constructions for LRHTS（v）, LARHTS（v）, OLRHTS（v）, OLARHTS（v） and give some new results.
基金Supported by NSFC grant No. 10371002 (Y. Chang) and No.19901008 (J. Lei)
文摘A idempotent quasigroup (Q, o) of order n is equivalent to an n(n-1)×3 partial orthogonal array in which all of rows consist of 3 distinct elements. Let X be a (n+1)-set. Denote by T(n+1) the set of (n+1)n(n-1) ordered triples of X with the property that the 3 coordinates of each ordered triple are distinct. An overlarge set of idempotent quasigroups of order n is a partition of T(n+1) into n+1 n(n-1)×3 partial orthogonal arrays A_x, x∈X based on X\{x}. This article gives an almost complete solution of overlarge sets of idempotent quasigroups.
基金This work was partially supported by the National Natural Science Foundation of China(Grant No.10671055)Tianyuan Mathematics Foundation of NSFC(Grant No.10526032)the Natural Science Foundation of Universities of Jiangsu Province(Grant No.05KJB110111)
文摘A directed triple system of order v, denoted by DTS(v, λ), is a pair (X,B) where X is a v-set and B is a collection of transitive triples on X such that every ordered pair of X belongs to λ triples of B. An overlarge set of disjoint DTS(v, λ), denoted by OLDTS(v, λ), is a collection {(Y\{y}, Ai)}i,such that Y is a (v + 1)-set, each (Y\{y}, Ai) is a DTS(v, λ) and all Ai's form a partition of all transitive triples of Y. In this paper, we shall discuss the existence problem of OLDTS(v, λ) and give the following conclusion: there exists an OLDTS(v, λ) if and only if either λ = 1 and v = 0, 1 (mod 3), or λ = 3 and v≠2.
基金Supported by the National Natural Science Foundation of China(No.10971051 and 11071056)
文摘A hybrid triple system of order v and index A, denoted by HTS（v, λ）, is a pair （X,B） where X is a v-set and B is a collection of cyclic triples and transitive triples on X, such that every ordered pair of X belongs to A triples of B. An overlarge set of disjoint HTS（v, λ）, denoted by OLHTS（v, λ）, is a collection {（Y／{y}, .Ai）}i, such that Y is a （v ＋ 1）-set, each （Y／{y}, Ai） is an HTS（v, λ,） and all Ais form a partition of all cyclic triples and transitive triples on Y. In this paper, we shall discuss the existence problem of OLHTS（v, A） and give the following conclusion： there exists an OLHTS（v, λ） if and only if λ= 1, 2, 4, v = 0, 1 （rood 3） and v ≥ 4.
基金supported by National Natural Science Foundation of China (Grant No.10971051)Natural Science Foundation of Hebei Province,China (Grant No.A2010000353)
文摘A directed triple system of order v,denoted by DTS(v),is a pair (X,B) where X is a v-set and B is a collection of transitive triples on X such that every ordered pair of X belongs to exactly one triple of B.A DTS(v) (X,A) is called pure and denoted by PDTS(v) if (a,b,c) ∈ A implies (c,b,a) ∈/ A.An overlarge set of PDTS(v),denoted by OLPDTS(v),is a collection {(Y \{yi},Aij) : yi ∈ Y,j ∈ Z3},where Y is a (v+1)-set,each (Y \{yi},Aij) is a PDTS(v) and these Ais form a partition of all transitive triples on Y .In this paper,we shall discuss the existence problem of OLPDTS(v) and give the following conclusion: there exists an OLPDTS(v) if and only if v ≡ 0,1 (mod 3) and v 】 3.
基金supported by NSFC Grant 10671055NSFHB A2007000230Foundation of Hebei Normal University L2004Y11, L2007B22
文摘In this paper, we introduce a new concept -- overlarge sets of generalized Kirkman systems （OLGKS）, research the relation between it and OLKTS, and obtain some new results for OLKTS. The main conclusion is： If there exist both an OLKF（6^k） and a 3-OLGKS（6^k-1,4） for all k ∈{6,7,...,40}／{8,17,21,22,25,26}, then there exists an OLKTS（v） for any v ≡ 3 （mod 6）, v ≠ 21. As well, we obtain the following result： There exists an OLKTS（6u ＋ 3） for u = 2^2n-1 - 1, 7^n, 31^n, 127^n, 4^r25^s, where n ≥ 1,r＋s≥ 1.
基金Supported by the National Natural Science Foundation of China(No.10371031) Doctor fund of Hebei Normal University
文摘There are six types of triangles： undirected triangle, cyclic triangle, transitive triangle, mixed-1 triangle, mixed-2 triangle and mixed-3 triangle. The triangle-decompositions for the six types of triangles have already been solved. For the first three types of triangles, their large sets have already been solved, and their overlarge sets have been investigated. In this paper, we establish the spectrum of LTi（v,λ）, OLTi（v）（i = 1, 2）, and give the existence of LT3（v, λ） and OLT3（v, λ） with λ even.
