Z-cyclic triplewhist tournaments when the number of players involves primes of the form 8u + 5

When the number of players, v, in a whist tournament, Wh(v), is ≡ 1 (mod 4) the only instances of a Z-cyclic triplewhist tournament, TWh(v), that appear in the literature are for v = 21,29,37. In this study we present Z-cyclic TWh(v) for all v ∈ T = {v = 8u + 5: v is prime, 3 ≤ u ≤ 249}. Additionally, we establish (1) for all v ∈ T there exists a Z-cyclic TWh(vⁿ) for all n ≥ 1, and (2) if v_i ∈ T, i = 1,…,n, there exists a Z-cyclic TWh(v… v) for all ?_i ≥ 1. It is believed that these are the first instances of infinite classes of Z-cyclic TWh(v), v ≡ 1 (mod 4). © 1994 John Wiley & Sons, Inc.     

On the 3BD‐closed set B3({4,5,6})

Let B₃(K) = {v:? an S(3,K,v)}. For K = {4} or {4,6}, B₃(K) has been determined by Hanani, and for K = {4, 5} by a previous paper of the author. In this paper, we investigate the case of K = {4,5,6}. It is easy to see that if v ∈ B₃ ({4, 5, 6}), then v ≡ 0, 1, 2 (mod 4). It is known that B₃{4, 6}) = {v > 0: v ≡ 0 (mod 2)} ? B₃({4,5,6}) by Hanani and that B₃({4, 5}) = {v > 0: v ≡ 1, 2, 4, 5, 8, 10 (mod 12) and v ≠ 13} ? B₃({4, 5, 6}). We shall focus on the case of v ≡ 9 (mod 12). It is proved that B₃({4,5,6}) = {v > 0: v ≡ 0, 1, 2 (mod 4) and v ≠ 9, 13}. © 2003 Wiley Periodicals, Inc.     

Long cycles passing through a specified edge in a 3-connected graph

We prove the following theorem: For a connected noncomplete graph G, let τ(G): = min{d_G(u) + d_G(v)|d_G(u, v) = 2}. Suppose G is a 3-connected noncomplete graph. Then through each edge of G there passes a cycle of length ≥ min{|V(G)|, τ (G) − 1}. © 1997 John Wiley & Sons, Inc.     

Quintessential PBDs and PBDs with prime power block sizes ≥ 8

In this paper, we look at the existence of (v K) pairwise balanced designs (PBDs) for a few sets K of prime powers ≥ 8 and also for a number of subsets K of {5, 6, 7, 8, 9}, which contain {5}. For K = {5, 7}, {5, 8}, {5, 7, 9}, we reduce the largest v for which a (v, K)‐PBD is unknown to 639, 812, and 179, respectively. When K is Q_≥8, the set of all prime powers ≥ 8, we find several new designs for 1,180 ≤ v ≤ 1,270, and reduce the largest unsolved case to 1,802. For K =Q_0,1,5(8), the set of prime powers ≥ 8 and ≡ 0, 1, or 5 (mod 8) we reduce the largest unknown case from 8,108 to 2,612. We also obtain slight improvements when K is one of {8, 9} or Q_0,1(8), the set of prime powers ≡ 0 or 1 (mod 8). © 2004 Wiley Periodicals, Inc.     

Failure of cancellation conditions for additive linear orders

Coordinate independence assumptions, also known as cancellation conditions, play a central role in the representational theory of measurement for an ordering relation ≻ on a finite Cartesian product set A₁ × A₂ × ··· × A_m. A sequence of increasingly complex cancellation conditions is known to be sufficient for additive representability in the form (a₁, a₂, ⋖ a_m) ≻ (b₁, b₂, ⋖ b_m) ⇔ ∑_i v(a_i) > &sum_i v(b_i). A longstanding open problem is to determine the simplest subset of cancellation conditions as a function of the size of A₁ × ··· × A_m that is violated by every order ≻ that is not additively representable. This article proves a lower bound on minimum subset sufficiency when all A_i are binary. We conjecture that this lower bound, which is very near to a known upper bound, is the exact minimum. The binary-factors version of the problem is reformulated under a first-order independence assumption by a map from ≻ on {0,1}^m into a subset L of {1,0,−1}^m that is referred to as an additive linear order. The lower bound is then established by examples of additive linear orders on {1,0,−1}^m that exhibit worst-case failures of cancellation. © 1997 John Wiley & Sons, Inc. J Combin Designs 5:353–365, 1997     

Constructions of optimal packing designs

Let v and k be positive integers. A (v, k, 1)-packing design is an ordered pair (V, B) where V is a v-set and B is a collection of k-subsets of V (called blocks) such that every 2-subset of V occurs in at most one block of B. The packing problem is mainly to determine the packing number P(k, v), that is, the maximum number of blocks in such a packing design. It is well known that P(k, v) ≤ ⌊v⌊(v − 1)/(k − 1)⌋/k⌋ = J(k, v) where ⌊×⌋ denotes the greatest integer y such that y ≤ x. A (v, k, 1)-packing design having J(k, v) blocks is said to be optimal. In this article, we develop some general constructions to obtain optimal packing designs. As an application, we show that P(5, v) = J(5, v) if v ≡ 7, 11 or 15 (mod 20), with the exception of v ∈ {11, 15} and the possible exception of v ∈ {27, 47, 51, 67, 87, 135, 187, 231, 251, 291}. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 245–260, 1998     

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     

Some results on (v,{5,w*})-pbds

In this article, we construct pairwise balanced designs (PBDs) on v points having blocks of size five, except for one block of size w ? {17,21,25,29,33}. A necessary condition for the existence of such a PBD is v ? 4w + 1 and (1) v ≡ 1 or 5 (mod 20) for w = 21, 25; (2) v ≡ 9 or 17 (mod 20) for w = 17,29; (3) v ≡ 13 (mod 20) for w = 33. We show that these necessary conditions are sufficient with at most 25 possible exceptions of (v,w). We also show that a BIBD B(5, 1; w) can be embedded in some B(5, 1; v) whenever v ≡ w ≡ 1 or 5 (mod 20) and v ? 5w ? 4, except possibly for (v, w) = (425, 65). © 1995 John Wiley & Sons, Inc.     

Acyclic 5‐choosability of planar graphs without small cycles

A proper vertex coloring of a graph G = (V,E) is acyclic if G contains no bicolored cycle. A graph G is acyclically L‐list colorable if for a given list assignment L = {L(v): v: ∈ V}, there exists a proper acyclic coloring ? of G such that ?(v) ∈ L(v) for all v ∈ V. If G is acyclically L‐list colorable for any list assignment with |L (v)|≥ k for all v ∈ V, then G is acyclically k‐choosable. In this article, we prove that every planar graph G without 4‐ and 5‐cycles, or without 4‐ and 6‐cycles is acyclically 5‐choosable. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 245–260, 2007     

Maximal resolvable packings and minimal resolvable coverings of triples by quadruples

Determination of maximal resolvable packing number and minimal resolvable covering number is a fundamental problem in designs theory. In this article, we investigate the existence of maximal resolvable packings of triples by quadruples of order v (MRPQS(v)) and minimal resolvable coverings of triples by quadruples of order v (MRCQS(v)). We show that an MRPQS(v) (MRCQS(v)) with the number of blocks meeting the upper (lower) bound exists if and only if v≡0 (mod 4). As a byproduct, we also show that a uniformly resolvable Steiner system URS(3, {4, 6}, {r₄, r₆}, v) with r₆≤1 exists if and only if v≡0 (mod 4). All of these results are obtained by the approach of establishing a new existence result on RH(6²ⁿ) for all n≥2. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 209–223, 2010     

Mean-Field Conditions for Percolation on Finite Graphs

Let {G _n } be a sequence of finite transitive graphs with vertex degree d = d(n) and |G _n | = n. Denote by p ^t (v, v) the return probability after t steps of the non-backtracking random walk on G _n . We show that if p ^t (v, v) has quasi-random properties, then critical bond-percolation on G _n behaves as it would on a random graph. More precisely, if $\mathop {\rm {lim\, sup\,}} \limits_{n} n^{1/3} \sum\limits_{t = 1}^{n^{1/3}} {t{\bf p}^t(v,v) < \infty ,}$ then the size of the largest component in p-bond-percolation with ${p =\frac{1+O(n^{-1/3})}{d-1}}Let {G _n } be a sequence of finite transitive graphs with vertex degree d = d(n) and |G _n | = n. Denote by p ^t (v, v) the return probability after t steps of the non-backtracking random walk on G _n . We show that if p ^t (v, v) has quasi-random properties, then critical bond-percolation on G _n behaves as it would on a random graph. More precisely, if

lim sup  n n1/3 ?t = 1n1/3 tpt(v,v) < ￥,\mathop {\rm {lim\, sup\,}} \limits_{n} n^{1/3} \sum\limits_{t = 1}^{n^{1/3}} {t{\bf p}^t(v,v) < \infty ,}      12. <Emphasis Type="Italic">v</Emphasis>-Adic maximal extensions,spectral norms and absolute Galois groups      Victor Alexandru  Angel Popescu  Elena Liliana Popescu  Sobia Sultana 《Monatshefte für Mathematik》2009,38(3):223-233 Let (K, v) be a perfect rank one valued field and let ([`(Kv)],[`(v)]){(\overline{K_{v}},\overline{v})} be the canonical valued field obtained from (K, v) by the unique extension of the valuation [(v)\tilde]{\widetilde{v}} of K v , the completion of K relative to v, to a fixed algebraic closure [`(Kv)]{\overline{K_{v}}} of K v . Let [`(K)]{\overline{K}} be the algebraic closure of K in [`(Kv)]{\overline {K_{v}}}. An algebraic extension L of K, L ì [`(K)]{L\subset\overline{K}}, is said to be a v-adic maximal extension, if [`(v)] | L{\overline{v}\mid_{L}} is the only extension of v to L and L is maximal with this property. In this paper we describe some basic properties of such extensions and we consider them in connection with the v-adic spectral norm on [`(K)]{\overline{K}} and with the absolute Galois groups Gal([`(K)]/K){(\overline{K}/K)} and Gal([`(Kv)] /Kv){(\overline{K_{v}} /K_{v})}. Some other auxiliary results are given, which may be useful for other purposes.      13. The spectrum of equitable, 4‐chromatic Steiner quadruple systems      Vaclav Linek 《组合设计杂志》2007,15(5):369-392 A Steiner quadruple system of order v (briefly an SQS(v)) is a pair (X, ) with |X| = v and a set of quadruples taken from X such that every triple in X is in a unique quadruple in . Hanani [Canad J Math 12 (1960), 145–157] showed that an SQS(v) exists if and only if v is {admissible}, that is, v = 0,1 or v ≡ 2,4 (mod 6). Each SQS(v) has a chromatic number when considered as a 4‐uniform hypergraph. Here we show that a 4‐chromatic SQS(v) exists for all admissible v ≥ 20, and that no 4‐chromatic SQS(v) exists for v < 20. Each system we construct admits a proper 4‐coloring that is equitable, that is, any two color classes differ in size by at most one. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 369–392, 2007      14. A sufficient condition for planar graphs to be acyclically 5‐choosable      Min Chen  André Raspaud 《Journal of Graph Theory》2012,70(2):135-151 A proper vertex coloring of a graph G = (V, E) is acyclic if G contains no bicolored cycle. Given a list assignment L = {L(v)|v∈V} of G, we say G is acyclically L‐list colorable if there exists a proper acyclic coloring π of G such that π(v)∈L(v) for all v∈V. If G is acyclically L‐list colorable for any list assignment with |L(v)|≥k for all v∈V, then G is acyclically k‐choosable. In this article we prove that every planar graph without 4‐cycles and without intersecting triangles is acyclically 5‐choosable. This improves the result in [M. Chen and W. Wang, Discrete Math 308 (2008), 6216–6225], which says that every planar graph without 4‐cycles and without two triangles at distance less than 3 is acyclically 5‐choosable. © 2011 Wiley Periodicals, Inc. J Graph Theory      15. A general oscillation theorem for self-adjoint differential systems with applications to Sturm-Liouville eigenvalue problems and quadratic functionals      G. Baur  W. Kratz 《Rendiconti del Circolo Matematico di Palermo》1989,38(3):329-370 For given 2n×2n matricesS 13,S 24 with rank(S 13,S 24)=2n we consider the eigenvalue problem:u′=A(x)u+B(x)v,v′=C 1(x;λ)u-A T(x)v with       16. When the v-Operation Distributes over Intersections      D. D. Anderson  Sharon M. Clarke 《代数通讯》2013,41(12):4327-4337 Let D be an integral domain. We investigate when (∩ Aα) ?1 = ∑ Aα ?1 or (∩ Aα) ?1 =(∑ Aα ?1)v (equivalently, (∩ A α) v  = ∩(A α) v ) for certain families {A α} of nonzero fractional ideals of D.      17. Triplewhist tournaments with the three person property      Gennian Ge 《Journal of Combinatorial Theory, Series A》2007,114(8):1438-1455 The necessary conditions for existence of a triplewhist tournament TWh(v) are . By the efforts of many authors through a century, these conditions are shown to be sufficient except for v=5,9,12,13 and possibly for v=17. A triplewhist tournament Wh(v) is said to have the three person property if any two games in the tournament do not have three common players. We briefly denote such a design as a 3PTWh(v). In this paper, we extend the known existence result for TWh(v)s and show that the necessary conditions for existence of a 3PTWh(v), namely, v?8 and , are also sufficient except for v=9,12,13 and possibly for v=17.      18. Planar cyclic difference packings      Christopher N. Swanson 《组合设计杂志》2000,8(6):426-434 A planar cyclic difference packing modulo v is a collection D = {d1, d2,…,dk} of distinct residues modulo v such that any residue α ≢ 0 (mod v) has at most one representation as a difference di − dj (mod v). This paper develops various constructions of such designs and for a fixed positive integer v presents upper and lower bounds on Ψ (v), the maximal number of elements that a planar cyclic difference packing modulo v can contain. This paper also presents the results of calculating Ψ (v) for v ≤ 144, including the fact that 134 is the smallest value of v for which the elementary upper bound of exceeds Ψ (v) by two. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 426–434, 2000      19. On algorithms to compute some H V-groups      Joongsoo Park  Sang-Cho Chung 《Journal of Applied Mathematics and Computing》2000,7(2):433-453 In this paper, we consider hyperstructures (H, ·) whenH = {e, a, b}. We put a condition on (H, ·) wheree is a unit. We obtain minimal and maximalH v-groups, semigroups and quasigroups, using Mathematica 3.0 computer programs.      20. Existence results for near resolvable designs      Steven Furino 《组合设计杂志》1995,3(2):101-113 A near resolvable design, NRB(v, k), is a balanced incomplete block design whose block set can be partitioned into v classes such that each class contains every point of the design but one, and each point is missing from exactly one class. The necessary conditions for the existence of near resolvable designs are v ≡ 1 mod k and λ = k ? 1. These necessary conditions have been shown to be sufficient for k ? {2,3,4} and almost always sufficient for k ? {5,6}. We are able to show that there exists an integer n0(k) so that NRB(v,k) exist for all v > n0(k) and v ≡ 1 mod k. Using some new direct constructions we show that there are many k for which it is easy to compute an explicit bound on n0(k). These direct constructions also allow us to build previously unknown NRB(v,5) and NRB(v,6). © 1995 John Wiley & Sons, Inc.

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<Emphasis Type="Italic">v</Emphasis>-Adic maximal extensions,spectral norms and absolute Galois groups

Let (K, v) be a perfect rank one valued field and let ([`(K<sub>v</sub>)],[`(v)]){(\overline{K_{v}},\overline{v})} be the canonical valued field obtained from (K, v) by the unique extension of the valuation [(v)\tilde]{\widetilde{v}} of K _v , the completion of K relative to v, to a fixed algebraic closure [`(K<sub>v</sub>)]{\overline{K_{v}}} of K <sub>v</sub> . Let [`(K)]{\overline{K}} be the algebraic closure of K in [`(K<sub>v</sub>)]{\overline {K_{v}}}. An algebraic extension L of K, L ì [`(K)]{L\subset\overline{K}}, is said to be a v-adic maximal extension, if [`(v)] | <sub>L</sub>{\overline{v}\mid_{L}} is the only extension of v to L and L is maximal with this property. In this paper we describe some basic properties of such extensions and we consider them in connection with the v-adic spectral norm on [`(K)]{\overline{K}} and with the absolute Galois groups Gal([`(K)]/K){(\overline{K}/K)} and Gal([`(K_v)] /K_v){(\overline{K_{v}} /K_{v})}. Some other auxiliary results are given, which may be useful for other purposes.     

The spectrum of equitable, 4‐chromatic Steiner quadruple systems

A Steiner quadruple system of order v (briefly an SQS(v)) is a pair (X, ) with |X| = v and a set of quadruples taken from X such that every triple in X is in a unique quadruple in . Hanani [Canad J Math 12 (1960), 145–157] showed that an SQS(v) exists if and only if v is {admissible}, that is, v = 0,1 or v ≡ 2,4 (mod 6). Each SQS(v) has a chromatic number when considered as a 4‐uniform hypergraph. Here we show that a 4‐chromatic SQS(v) exists for all admissible v ≥ 20, and that no 4‐chromatic SQS(v) exists for v < 20. Each system we construct admits a proper 4‐coloring that is equitable, that is, any two color classes differ in size by at most one. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 369–392, 2007     

A sufficient condition for planar graphs to be acyclically 5‐choosable

A proper vertex coloring of a graph G = (V, E) is acyclic if G contains no bicolored cycle. Given a list assignment L = {L(v)|v∈V} of G, we say G is acyclically L‐list colorable if there exists a proper acyclic coloring π of G such that π(v)∈L(v) for all v∈V. If G is acyclically L‐list colorable for any list assignment with |L(v)|≥k for all v∈V, then G is acyclically k‐choosable. In this article we prove that every planar graph without 4‐cycles and without intersecting triangles is acyclically 5‐choosable. This improves the result in [M. Chen and W. Wang, Discrete Math 308 (2008), 6216–6225], which says that every planar graph without 4‐cycles and without two triangles at distance less than 3 is acyclically 5‐choosable. © 2011 Wiley Periodicals, Inc. J Graph Theory     

A general oscillation theorem for self-adjoint differential systems with applications to Sturm-Liouville eigenvalue problems and quadratic functionals

For given 2n×2n matricesS ₁₃,S ₂₄ with rank(S ₁₃,S ₂₄)=2n we consider the eigenvalue problem:u′=A(x)u+B(x)v,v′=C ₁(x;λ)u-A ^T(x)v with

When the v-Operation Distributes over Intersections

Let D be an integral domain. We investigate when (∩ A_α) ?¹ = ∑ A_α ?¹ or (∩ A_α) ?¹ =(∑ A_α ?¹)_v (equivalently, (∩ A _α) _v  = ∩(A _α) _v ) for certain families {A _α} of nonzero fractional ideals of D.     

Triplewhist tournaments with the three person property

The necessary conditions for existence of a triplewhist tournament TWh(v) are . By the efforts of many authors through a century, these conditions are shown to be sufficient except for v=5,9,12,13 and possibly for v=17. A triplewhist tournament Wh(v) is said to have the three person property if any two games in the tournament do not have three common players. We briefly denote such a design as a 3PTWh(v). In this paper, we extend the known existence result for TWh(v)s and show that the necessary conditions for existence of a 3PTWh(v), namely, v?8 and , are also sufficient except for v=9,12,13 and possibly for v=17.     

Planar cyclic difference packings

A planar cyclic difference packing modulo v is a collection D = {d₁, d₂,…,d_k} of distinct residues modulo v such that any residue α ≢ 0 (mod v) has at most one representation as a difference d_i − d_j (mod v). This paper develops various constructions of such designs and for a fixed positive integer v presents upper and lower bounds on Ψ (v), the maximal number of elements that a planar cyclic difference packing modulo v can contain. This paper also presents the results of calculating Ψ (v) for v ≤ 144, including the fact that 134 is the smallest value of v for which the elementary upper bound of exceeds Ψ (v) by two. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 426–434, 2000     

On algorithms to compute some H V-groups

In this paper, we consider hyperstructures (H, ·) whenH = {e, a, b}. We put a condition on (H, ·) wheree is a unit. We obtain minimal and maximalH _v-groups, semigroups and quasigroups, using Mathematica 3.0 computer programs.     

Existence results for near resolvable designs

A near resolvable design, NRB(v, k), is a balanced incomplete block design whose block set can be partitioned into v classes such that each class contains every point of the design but one, and each point is missing from exactly one class. The necessary conditions for the existence of near resolvable designs are v ≡ 1 mod k and λ = k ? 1. These necessary conditions have been shown to be sufficient for k ? {2,3,4} and almost always sufficient for k ? {5,6}. We are able to show that there exists an integer n₀(k) so that NRB(v,k) exist for all v > n₀(k) and v ≡ 1 mod k. Using some new direct constructions we show that there are many k for which it is easy to compute an explicit bound on n₀(k). These direct constructions also allow us to build previously unknown NRB(v,5) and NRB(v,6). © 1995 John Wiley & Sons, Inc.