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.     

Perfect dexagon triple systems with given subsystems

The graph consisting of the six triples (or triangles) {a,b,c}, {c,d,e}, {e,f,a}, {x,a,y}, {x,c,z}, {x,e,w}, where a,b,c,d,e,f,x,y,z and w are distinct, is called a dexagon triple. In this case the six edges {a,c}, {c,e}, {e,a}, {x,a}, {x,c}, and {x,e} form a copy of K₄ and are called the inside edges of the dexagon triple. A dexagon triple system of order v is a pair (X,D), where D is a collection of edge disjoint dexagon triples which partitions the edge set of 3K_v. A dexagon triple system is said to be perfect if the inside copies of K₄ form a block design. In this note, we investigate the existence of a dexagon triple system with a subsystem. We show that the necessary conditions for the existence of a dexagon triple system of order v with a sub-dexagon triple system of order u are also sufficient.     

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     

The existence spectrum for overlarge sets of pure directed triple systems

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.     

Distortion Properties of Holomorphic Functions of Several Complex Variables

Let Q(D) be a class of functions q, q(0) = 0, |q(z)| < 1 holomorphic in the Reinhardt domain D ? C ⁿ, a and b — arbitrary fixed numbers satisfying the condition — 1 ≤ b < a ≤ 1. ??(a, b; D) — the class of functions p such that p ? ??(a, b; D) iff for some q ? Q(D) and every z ? D. S*(a, b; D) — the class of functions f such that f ? S*(a, g; D) iff S^c(a, b; D) — the class of functions q such that q ? S^c(a, b; D) iff , where p ε ??(a, b; D) and K is an operator of the form for z=z₁,z₂,…z_n. The author obtains sharp bounds on |p(z)|, f(z)| g(z)| as well as sharp coefficient inequalities for functions in ??(a, b; D), S*(a, b; D) and S^c(a, b; D).     

Large sets of pure directed triple systems with index λ

A directed triple system of order v with index λ, briefly by DTS（v,λ）, is a pair （X, B） where X is a v-set and B is a collection of transitive triples （blocks） on X such that every ordered pair of X belongs to λ blocks of B. A simple DTS（v, λ） is a DTS（v, λ） without repeated blocks. A simple DTS（v, ）,） is called pure and denoted by PDTS（v, λ） if （x, y, z） ∈ B implies （z, y, x）, （z, x, y）, （y, x, z）, （y, z, x）, （x, z, y） B. A large set of disjoint PDTS（v, λ）, denoted by LPDTS（v, λ）, is a collection of 3（v - 2）/λ disjoint pure directed triple systems on X. In this paper, some results about the existence for LPDTS（v, λ） are presented. Especially, we determine the spectrum of LPDTS（v, 2）.     

Rectangles in triangles

We establish necessary and sufficient conditions on the five positive real numbers a, b, c, u, v for a rectangle with sides u, v to fit in a triangle with sides a, b, c, and we note a few curious consequences.     

3-(v, 4, 1) covering designs with chromatic numbers 2 and 3

A t-(v, k, λ) covering design is a pair (X, B) where X is a v-set and B is a collection of k-sets in X, called blocks, such that every t element subset of X is contained in at least λ blocks of B. The covering number, C_λ(t, k, v), is the minimum number of blocks a t-(v, k, λ) covering design may have. The chromatic number of (X, B) is the smallest m for which there exists a map φ: X → Z_m such that ∣φ((β)∣ ≥2 for all β ∈ B, where φ(β) = {φ(x): x ∈ β}. The system (X, B) is equitably m-chromatic if there is a proper coloring φ with minimal m for which the numbers ∣φ^?1(c)∣ c ∈ Z_m differ from each other by at most 1. In this article we show that minimum, (i.e., ∣B∣ = C λ (t, k, v)) equitably 3-chromatic 3-(v, 4, 1) covering designs exist for v ≡ 0 (mod 6), v ≥ 18 for v ≥ 1, 13 (mod 36), v ≡ 13 and for all numbers v = n, n + 1, where n ≡ 4, 8, 10 (mod 12), n ≥ 16; and n = 6.5^a 13^b 17^c ?4, a + b + c > 0, and n = 14, 62. We also show that minimum, equitably 2-chromatic 3-(v, 4, 1) covering designs exist for v ≡ 0, 5, 9 (mod 12), v ≥ 0, v = 2.5^a 13^b 17^c + 1, a + b + c > 0, and v = 23. © 1993 John Wiley & Sons, Inc.     

Strongly divisible 1-designs

If a 1-design, D, admits a tactical decomposition such that the number of blocks through two distinct points depends only on their point classes and further that the number of blocks through any two distinct points of the same point class is a constant, then the decomposition is called a tactical division. In the case of D being a 2-design the terms tactical division and tactical decomposition are synonymous. If the division has c block classes and d point classes then b+dv+c where b is the number of blocks of D and v is the number of points of D. Tactical divisions for which b+d=v+c are of special interest and are called strong. A 1-design admitting a strong tactical division is called strongly divisible.All symmetric and affine 2-designs are strongly divisible and I shall indicate some of the properties of strongly divisible designs that are similar to those of symmetric and affine designs.     

Oriented list colorings of graphs

A 2‐assignment on a graph G = (V,E) is a collection of pairs L(v) of allowed colors specified for all vertices v ∈V. The graph G (with at least one edge) is said to have oriented choice number 2 if it admits an orientation which satisfies the following property: For every 2‐assignment there exists a choice c(v)∈L(v) for all v ∈V such that (i) if c(v) = c(w), then vw ∉ E, and (ii) for every ordered pair (a,b) of colors, if some edge oriented from color a to color b occurs, then no edge is oriented from color b to color a. In this paper we characterize the following subclasses of graphs of oriented choice number 2: matchings; connected graphs; graphs containing at least one cycle. In particular, the first result (which implies that the matching with 11 edges has oriented choice number 2) proves a conjecture of Sali and Simonyi. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 217–229, 2001     

On tortkara triple systems

The commutator [a,b] = ab?ba in a free Zinbiel algebra (dual Leibniz algebra) is an anticommutative operation which satisfies no new relations in arity 3. Dzhumadildaev discovered a relation T(a,b,c,d) which he called the tortkara identity and showed that it implies every relation satisfied by the Zinbiel commutator in arity 4. Kolesnikov constructed examples of anticommutative algebras satisfying T(a,b,c,d) which cannot be embedded into the commutator algebra of a Zinbiel algebra. We consider the tortkara triple product [a,b,c] = [[a,b],c] in a free Zinbiel algebra and use computer algebra to construct a relation TT(a,b,c,d,e) which implies every relation satisfied by [a,b,c] in arity 5. Thus, although tortkara algebras are defined by a cubic binary operad (with no Koszul dual), the corresponding triple systems are defined by a quadratic ternary operad (with a Koszul dual). We use computer algebra to construct a relation in arity 7 satisfied by [a,b,c] which does not follow from the relations of lower arity. It remains an open problem to determine whether there are further new identities in arity n≥9.     

Additive property of Drazin invertibility of elements in a ring

In this article, we investigate additive properties on the Drazin inverse of elements in rings. Under the commutative condition of ab?=?ba, we show that a?+?b is Drazin invertible if and only if 1?+?a ^D b is Drazin invertible. Not only the explicit representations of the Drazin inverse (a?+?b) ^D in terms of a, a ^D , b and b ^D , but also (1?+?a ^D b) ^D is given. Further, the same property is inherited by the generalized Drazin invertibility in a Banach algebra and is extended to bounded linear operators.     

Direct construction methods for incomplete perfect Mendelsohn designs with block size four

Let v, k, λ, and n be positive integers. An incomplete perfect Mendelsohn design, denoted by (v,n,k,λ)-IPMD, is a triple (X,Y,B) where X is a v-set (of points), Y is an n-subset of X, and B is a collection of cyclically ordered k-subsets of X (called blocks) such that every ordered pair (a,b) E (X × X)\(Y × Y) appears t-apart in exactly λ blocks of B and no ordered pair (a,b) E Y × Y appears in any block of B for any t, where 1 ≤ t ≤ k − 1. In this article, we introduce an effective and easy way to construct IPMDs for k = 4 and even v − n, and use it to construct some small examples for λ = 1 and 2. Obviously, these results will play an important role to completely solve the existence of (v,n,4,λ)-IPMDs. Furthermore, we also use this method to construct some small examples for HPMDs. © 1996 John Wiley & Sons, Inc.     

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     

Surface area inequalities for ellipsoids using Minkowski sums

Given the convex body E=E(a,b,c) bounded by the ellipsoid with principal axes of lengths 2a, 2b, and 2c, its surface area, S(a,b,c), is a non-elementary integral unless a=b=c, (E is a ball) or two values of a,b, and c are equal (E is a solid spheroid). This leads to upper and lower estimates for S(a,b,c) in terms of the surface areas of balls or spheroids. We derive many of the known inequalities and some new inequalities for the surface areas of ellipsoids using Minkowski sums of ellipsoids and Minkowski's mixed volumes.     

Constructions for overlarge sets of disjoint pure directed triple systems

Let X be a v-set, v≥3. A transitive triple (x,y,z) on X is a set of three ordered pairs (x,y),(y,z) and (x,z) of X. A directed triple system of order v, denoted by DTS(v), is a pair (X,?), where X is a v-set and ? is a collection of transitive triples on X such that every ordered pair of X belongs to exactly one triple of ?. A DTS(v) is called pure and denoted by PDTS(v) if (x,y,z)∈? implies (z,y,x)??. An overlarge set of disjoint PDTS(v) is denoted by OLPDTS(v). In this paper, we establish some recursive constructions for OLPDTS(v), so we obtain some results.     

A New Approach to the Creation and Propagation of Exponential Moments in the Boltzmann Equation

We study the creation and propagation of exponential moments of solutions to the spatially homogeneous d-dimensional Boltzmann equation. In particular, when the collision kernel is of the form |v ? v _*|^β b(cos (θ)) for β ∈ (0, 2] with cos (θ) = |v ? v _*|^?1(v ? v _*)·σ and σ ∈ 𝕊 ^d?1, and assuming the classical cut-off condition b(cos (θ)) integrable in 𝕊 ^d?1, we prove that there exists a > 0 such that moments with weight exp (amin {t, 1}|v|^β) are finite for t > 0, where a only depends on the collision kernel and the initial mass and energy. We propose a novel method of proof based on a single differential inequality for the exponential moment with time-dependent coefficients.     

Travelling waves of auto-catalytic chemical reaction of general order—An elliptic approach

In this paper we study the existence and non-existence of travelling wave to parabolic system of the form at=axx−af(b), bt=Dbxx+af(b), with f a degenerate nonlinearity. In the context of an auto-catalytic chemical reaction, a is the density of a chemical species called reactant A, b that of another chemical species B called auto-catalyst, and D=DB/DA>0 is the ratio of diffusion coefficients, DB of B and DA of A, respectively. Such a system also arises from isothermal combustion. The nonlinearity is called degenerate, since f(0)=f^′(0)=0. One case of interest in this article is the propagating wave fronts in an isothermal auto-catalytic chemical reaction of order with 1<n<2, and D≠1 due to different molecular weights and/or sizes of A and B. The resulting nonlinearity is f(b)=bn. Explicit bounds v_∗ and v^∗ that depend on D are derived such that there is a unique travelling wave of every speed v?v^∗ and there does not exist any travelling wave of speed v<v_∗. New to the literature, it is shown that v_∗∝v^∗∝D when D<1. Furthermore, when D>1, it is shown rigorously that there exists a vmin such that there is a travelling wave of speed v if and only if v?vmin. Estimates on vmin improve significantly that of early works. Another case in which two different orders of isothermal auto-catalytic chemical reactions are involved is also studied with interesting new results proved.     

Strongly Singular Sturm – Liouville Problems

We consider a Sturm – Liouville operator Lu = —(r(t)u′)′ + p (t)u , where r is a (strictly) positive continuous function on ]a, b [ and p is locally integrable on ]a, b[. Let r₁(t) = (1/r) ds andchoose any c ∈]a, b[. We are interested in the eigenvalue problem Lu = λm(t)u, u (a) = u (b) = 0,and the corresponding maximal and anti .maximal principles, in the situation when 1/r ∈ L¹ (a, c),1 /r ∉ L¹ (c, b), pr₁ ∉ L¹ (a, c) and pr₁ ∉ L¹(c, b).     

Incomplete perfect mendelsohn designs with block size 4 and holes of size 2 and 3

Let v, k, and n be positive integers. An incomplete perfect Mendelsohn design, denoted by k-IPMD(v, n), is a triple (X, Y, ??) where X is a v-set (of points), Y is an n-subset of X, and ?? is a collection of cyclically ordered k-subsets of X (called blocks) such that every ordered pair (a, b) ∈ (X × X)\(Y × Y) appears t-apart in exactly one block of ?? and no ordered pair (a,b) ∈ Y × Y appears in any block of ?? for any t, where 1 ≤ t ≤ k ? 1. In this article, the necessary conditions for the existence of a 4-IPMD(v, n), namely (v ? n) (v ? 3n ? 1) ≡ 0 (mod 4) and v ≥ 3n + 1, are shown to be sufficient for the case n = 3. For the case n = 2, these conditions are sufficient except for v = 7 and with the possible exception of v = 14,15,18,19,22,23,26,27,30. The latter result provides a useful application to the problem relating to the packing of perfect Mendelsohn designs with block size 4. © 1994 John Wiley & Sons, Inc.