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.     

On the existence of triplewhist tournaments TWh(v)

It is well known that a triplewhist tournament TWh(v) exists only if v ≡ 0 or 1 (mod 4) and v ¬ 5, 9. In this article, we introduce a new concept TWh-frame and use it to show that the necessary condition for the existence of a TWh(v) is also sufficient with a handful possible exceptions of v ∞ {12, 56} ∪ {13, 17, 45, 57, 65, 69, 77, 85, 93, 117, 129, 133, 153}. © 1997 John Wiley & Sons, Inc. J Combin Designs 5: 249–256, 1997     

Existence of directed triplewhist tournaments with the three person property

A directed triplewhist tournament on v players, briefly DTWh(v), is said to have the three person property if no two games in the tournament have three common players. We briefly denote such a design as a 3PDTWh(v). In this paper, we show that a 3PDTWh(v) exists whenever v>17 and with few possible exceptions.     

Authentication perpendicular arrays APA1(2, 5, v)

Authentication perpendicular arrays APA_λ (t, k, v), as a special kind of perpendicular arrays, are introduced by D. R. Stinson in constructing authentication and secrecy codes. In this article, we improve the existence results for APA₁(2, 5, v) and show that such a design exists if and only if v ≥ 5 is odd, except v = 7 and possibly excepting v = 9, 13, 15, 17, 33, 39, 49, 53, 57, 63, 69, 73, 87, 89, 97, 113, 137, and 213. © 1996 John Wiley & Sons, Inc.     

Reduction for primitive flag-transitive (<Emphasis Type="Italic">v</Emphasis>, <Emphasis Type="Italic">k</Emphasis>, 4)-symmetric designs

It has been shown that if a (v, k, λ)-symmetric design with λ ≤ 3 admits a flag-transitive automorphism group G which acts primitively on points, then G must be of affine or almost simple type. Here we extend the result to λ = 4.     

Some rigid Steiner 5‐designs

Hitherto, all known non‐trivial Steiner systems S(5, k, v) have, as a group of automorphisms, either PSL(2, v−1) or PGL(2, (v−2)/2) × C₂. In this article, systems S(5, 6, 72), S(5, 6, 84) and S(5, 6, 108) are constructed that have only the trivial automorphism group. © 2010 Wiley Periodicals, Inc. J Combin Designs 18:392–400, 2010     

Automorphisms of higher‐dimensional Hadamard matrices

This article derives from first principles a definition of equivalence for higher‐dimensional Hadamard matrices and thereby a definition of the automorphism group for higher‐dimensional Hadamard matrices. Our procedure is quite general and could be applied to other kinds of designs for which there are no established definitions for equivalence or automorphism. Given a two‐dimensional Hadamard matrix H of order ν, there is a Product Construction which gives an order ν proper n‐dimensional Hadamard matrix P⁽ⁿ⁾(H). We apply our ideas to the matrices P⁽ⁿ⁾(H). We prove that there is a constant c > 1 such that any Hadamard matrix H of order ν > 2 gives rise via the Product Construction to cν inequivalent proper three‐dimensional Hadamard matrices of order ν. This corrects an erroneous assertion made in the literature that ”P⁽ⁿ⁾(H) is equivalent to “P⁽ⁿ⁾(H′) whenever H is equivalent to H′.” We also show how the automorphism group of P⁽ⁿ⁾(H) depends on the structure of the automorphism group of H. As an application of the above ideas, we determine the automorphism group of P⁽ⁿ⁾(Hk) when Hk is a Sylvester Hadamard matrix of order 2^k. For ν = 4, we exhibit three distinct families of inequivalent Product Construction matrices P⁽ⁿ⁾(H) where H is equivalent to H₂. These matrices each have large but non‐isomorphic automorphism groups. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 507–544, 2008     

Some progress on the existence of 1-rotational Steiner triple systems

A Steiner triple system of order v (briefly STS(v)) is 1-rotational under G if it admits G as an automorphism group acting sharply transitively on all but one point. The spectrum of values of v for which there exists a 1-rotational STS(v) under a cyclic, an abelian, or a dicyclic group, has been established in Phelps and Rosa (Discrete Math 33:57–66, 1981), Buratti (J Combin Des 9:215–226, 2001) and Mishima (Discrete Math 308:2617–2619, 2008), respectively. Nevertheless, the spectrum of values of v for which there exists a 1-rotational STS(v) under an arbitrary group has not been completely determined yet. This paper is a considerable step forward to the solution of this problem. In fact, we leave as uncertain cases only those for which we have v =  (p ³−p)n +  1 ≡ 1 (mod 96) with p a prime, n \not o 0{n \not\equiv 0} (mod 4), and the odd part of (p ³ − p)n that is square-free and without prime factors congruent to 1 (mod 6).     

Ordered tournaments and ordered triplewhist tournaments with the three person property

It is well known that an ordered tournament OWh(v) exists if and only if v ≡ 1 (mod 4), v ≥ 5. An ordered triplewhist tournament on v players is said to have the three person property if no two games in the tournament have three common players. We briefly denote such a design as a 3POTWh(v). In this article, we show that a 3POTWh(v) exists whenever v>17 and v ≡ 1 (mod 4) with few possible exceptions. We also show that an ordered whist tournament on v players with the three person property, denoted 3POWh(v), exists if and only if v ≡ 1 (mod 4), v ≥ 9. © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 39–52, 2009     

Automorphism group of exceptional symmetric domainsRV

The exceptional symmetric Siegel domainR _v(16) in ℂ¹⁶ is defined. The exceptional classical domain ℛ_v(16) = t(R_v(16)) is computed, where t is the Bergman mapping of the Siegel domain R_v(16). And holomorphical automorphism group Aut (R_v(16)) of the exceptional symmetric Siegel domainR _v(16) is presented     

New results on large sets of Kirkman triple systems

The existence problem on the large sets of Kirkman triple systems (LKTS) was posed by Sylvester in 1850’s as an extension of Kirkman’s 15 schoolgirls problem. An LKTS(15) was constructed by Denniston in 1974. However, up to now the smallest unknown order for the existence of LKTS is still 21. In this paper we construct the two smallest unknown LKTS(v)s with v = 21 and v = 39 by using multiplier automorphism groups. Applying known recursive constructions, we show the existence of more infinite classes of large sets of Kirkman triple systems.     

Twisted Conjugacy Classes for Polyfree Groups

Let G be a finitely generated polyfree group. If G has nonzero Euler characteristic then we show that Aut(G) has a finite index subgroup in which every automorphism has infinite Reidemeister number. For certain G of length 2, we show that the number of Reidemeister classes of every automorphism is infinite.     

Generalized Steiner systems GS4 (2, 4, v,g) for g = 2, 3, 6

Generalized Steiner systems GS_d(t, k, v, g) were first introduced by Etzion and used to construct optimal constant‐weight codes over an alphabet of size g + 1 with minimum Hamming distance d, in which each codeword has length v and weight k. Much work has been done for the existence of generalized Steiner triple systems GS(2, 3, v, g). However, for block size four there is not much known on GS_d(2, 4, v, g). In this paper, the necessary conditions for the existence of a GS_d(t, k, v, g) are given, which answers an open problem of Etzion. Some singular indirect product constructions for GS_d(2, k, v, g) are also presented. By using both recursive and direct constructions, it is proved that the necessary conditions for the existence of a GS₄(2, 4, v, g) are also sufficient for g = 2, 3, 6. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 401–423, 2001     

Large Sets and Overlarge Sets of Triangle-Decomposition

There are six types of triangles:undirected triangle,cyclic triangle,transitive triangle,mixed-1triangle,mixed-2 triangle and mixed-3 triangle.The triangle-decompositions for the six types of triangles havealready been solved.For the first three types of triangles,their large sets have already been solved,and theiroverlarge sets have been investigated.In this paper,we establish the spectrum of LT_i(v,λ),OLT_i(v)(i=1,2),and give the existence of LT_3(v,λ)and OLT_3(v,λ)with λ even.     

Gelfand–Kirillov Dimension for Automorphism Groups of Relatively Free Algebras

Using ideas of our recent work on automorphisms of residually nilpotent relatively free groups, we introduce a new growth function for subgroups of the automorphism groups of relatively free algebras Fⁿ(V) over a field of characteristic zero and the related notion of Gelfand-Kirillov dimension, and study their behavior. We prove that, under some natural restrictions, the Gelfand-Kirillov dimension of the group of tame automorphisms of Fⁿ(V) is equal to the Gelfand-Kirillov dimension of the algebra Fⁿ(V). We show that, in some cases, the Gelfand-Kirillov dimension of the group of tame automorphisms of Fⁿ(V) is smaller than the Gelfand-Kirillov dimension of the whole automorphism group, and calculate the Gelfand-Kirillov dimension of the automorphism group of Fⁿ(V) for some important varieties V.Partially supported by Grant MM605/96 of the Bulgarian Foundation for Scientific Research.2000 Mathematics Subject Classification: primary 16R10, 16P90; secondary 16W20, 17B01, 17B30, 17B40     

Automorphisms of -subshifts of finite type

Let (Σ, σ) be a ^d-subshift of finite type. Under a strong irreducibility condition (strong specification), we show that Aut(Σ) contains any finite group. For ^d-subshift of finite type without strong specification, examples show that topological mixing is not sufficient to give any finite group in the automorphism group in general: in particular, End(Σ) may be an abelian semigroup. For an example of a topologically mixing ²-subshift of finite type, the endomorphism semigroup and automorphism group are computed explicitly. This subshift has periodic-point permutations that do not extend to automorphisms.     

Some Hadamard designs with parameters (71,35,17)

Up to isomorphisms there are precisely eight symmetric designs with parameters (71, 35, 17) admitting a faithful action of a Frobenius group of order 21 in such a way that an element of order 3 fixes precisely 11 points. Five of these designs have 84 and three have 420 as the order of the full automorphism group G. If |G| = 420, then the structure of G is unique and we have G = (Frob₂₁ × Z5):Z4. In this case Z(G) = 〈1〉, G′ has order 35, and G induces an automorphism group of order 6 of Z7. If |G| = 84, then Z(G) is of order 2, and in precisely one case a Sylow 2‐subgroup is elementary abelian. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 144–149, 2002; DOI 10.1002/jcd.996     

Degenerate triply nonlinear problems with nonhomogeneous boundary conditions

The paper addresses the existence and uniqueness of entropy solutions for the degenerate triply nonlinear problem: b(v) _t − div α(v, ▽g(v)) = f on Q:= (0, T) × Ω with the initial condition b(v(0, ·)) = b(v ₀) on Ω and the nonhomogeneous boundary condition “v = u” on some part of the boundary (0, T) × ∂Ω”. The function g is continuous locally Lipschitz continuous and has a flat region [A ₁, A ₂,] with A ₁ ≤ 0 ≤ A ₂ so that the problem is of parabolic-hyperbolic type.     

Brooks-type theorems for choosability with separation

We consider the following type of problems. Given a graph G = (V, E) and lists L(v) of allowed colors for its vertices v ∈ V such that |L(v)| = p for all v ∈ V and |L(u) ∩ L(v)| ≤ c for all uv ∈ E, is it possible to find a “list coloring,” i.e., a color f(v) ∈ L(v) for each v ∈ V, so that f(u) ≠ f(v) for all uv ∈ E? We prove that every of maximum degree Δ admits a list coloring for every such list assignment, provided p ≥ . Apart from a multiplicative constant, the result is tight, as lists of length may be necessary. Moreover, for G = K_n (the complete graph on n vertices) and c = 1 (i.e., almost disjoint lists), the smallest value of p is shown to have asymptotics (1 + o(1)) . For planar graphs and c = 1, lists of length 4 suffice. ˜© 1998 John Wiley & Sons, Inc. J Graph Theory 27: 43–49, 1998     

Chromatic neighborhood sets

For each vertex v in a graph G, we denote by χ_v the chromatic number of the subgraph induced by its neighborhood, and we set χ_N(G) = {χ_v: v ∈ V(G)}. We characterize those sets X for which there exists some G of prescribed size with X = χ_N(G), and prove a related conjecture of Fajtlowicz. We also discuss those graphs that are extremal with respect to χ_N(G). © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 303–311, 1999