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）.     

The fine triangle intersections for maximum kite packings

In this paper, the fine triangle intersection problem for a pair of maximum kite packings is investigated. Let Fin(v) = {(s,t) : a pair of maximum kite packings of order v intersecting in s blocks and s+t triangles}. Let Adm(v) = {(s, t) : s + t ≤ bv , s,t are non-negative integers}, where b v = v(v 1)/8 . It is established that Fin(v) = Adm(v)\{(bv-1, 0), (bv-1,1)} for any integer v ≡ 0, 1 (mod 8) and v ≥ 8; Fin(v) = Adm(v) for any integer v ≡ 2, 3, 4, 5, 6, 7 (mod 8) and v ≥ 4.     

Concerning multiplier automorphisms of cyclic Steiner triple systems

A cyclic Steiner triple system, presented additively over Z _v as a set B of starter blocks, has a non-trivial multiplier automorphism λ ≠ 1 when λB is a set of starter blocks for the same Steiner triple system. When does a cyclic Steiner triple system of order v having a nontrivial multiplier automorphism exist? Constructions are developed for such systems; of most interest, a novel extension of Netto's classical construction for prime orders congruent to 1 (mod 6) to prime powers is proved. Nonexistence results are then established, particularly in the cases when v = (2^β + 1)^α, when v = 9p with p ≡ 5 (mod 6), and in certain cases when all prime divisors are congruent to 5 (mod 6). Finally, a complete solution is given for all v < 1000, in which the remaining cases are produced by simple computations.     

On the construction of handcuffed designs

A handcuffed design with parameters ν, k, λ consists of a set of ordered k-subsets of a v-set, called handcuffed blocks; in a block (a₁, a₂, a_k) each element is assumed to be “handcuffed” to its neighbors. A block, therefore, contains k ? 1 handcuffed pairs, the pairs being considered unordered. Each element of the v-set appears in exactly r blocks, and each pair of distinct elements of the v-set is handcuffed in exactly A blocks of the design.These designs have been studied recently by Hung and Mendelsohn [1], who construct a number of families of such designs by recursive methods. In this paper we show how difference methods can be applied to the construction of handcuffed designs. The methods are powerful, and a number of families of designs are constructed. A main new result is the determination of necessary and sufficient conditions for the existence of handcuffed designs for all parameter sets in which v is an odd prime power.     

Berezin kernels and analysis on Makarevich spaces

Following ideas of van Dijk and Hille we study the link which exists between maximal degenerate representations and Berezin kernels.We consider the conformal group Conf(V) of a simple real Jordan algebra V. The maximal degenerate representations π_s (s ε ℂ) we shall study are induced by a character of a maximal parabolic subgroup P¯ of Conf(V). These representations π_s can be realized on a space I_s of smooth functions on V. There is an invariant bilinear form ℬ_s on the space Is. The problem we consider is to diagonalize this bilinear form ℬ_s, with respect to the action of a symmetric subgroup G of the conformal group Conf(V). This bilinear form can be written as an integral involving the Berezin kernel B_v an invariant kernel on the Riemannian symmetric space G/K, which is a Makarevich symmetric space in the sense of Bertram. Then we can use results by van Dijk and Pevzner who computed the spherical Fourier transform of B_v. From these, one deduces that the Berezin kernel satisfies a remarkable Bernstein identity: D(_ν)B_ν=b(ν)B_ν+1, where D(_ν) is an invariant differential operator on G/K and b(ν) is a polynomial. By using this identity we compute a Hua type integral which gives the normalizing factor for an intertwining operator from I_−s to I_s. Furthermore, we obtain the diagonalization of the invariant bilinear form with respect to the action of the maximal compact group U of the conformal group Conf(V).     

Hexavalent (G,s)-transitive graphs

Let X be a finite simple undirected graph with a subgroup G of the full automorphism group Aut(X). Then X is said to be (G, s)-transitive for a positive integer s, if G is transitive on s-arcs but not on (s + 1)-arcs, and s-transitive if it is (Aut(X), s)-transitive. Let G _v be a stabilizer of a vertex v ∈ V (X) in G. Up to now, the structures of vertex stabilizers G _v of cubic, tetravalent or pentavalent (G, s)-transitive graphs are known. Thus, in this paper, we give the structure of the vertex stabilizers G _v of connected hexavalent (G, s)-transitive graphs.     

Minimum embedding of Steiner triple systems into -designs I

A (K₄-e)-design on v+w points embeds a Steiner triple system (STS) if there is a subset of v points on which the graphs of the design induce the blocks of a STS. It is established that wv/3, and that when equality is met that such a minimum embedding of an STS(v) exists, except when v=15.     

On three-designs of small order

For positive integers t?k?v and λ we define a t-design, denoted B_i[k,λ;v], to be a pair (X,B) where X is a set of points and B is a family, (B_i:i?I), of subsets of X, called blocks, which satisfy the following conditions: (i) |X|=v, the order of the design, (ii) |B_i|=k for each i?I, and (iii) every t-subset of X is contained in precisely λ blocks. The purpose of this paper is to investigate the existence of 3-designs with 3?k?v?32 and λ>0.Wilson has shown that there exists a constant N(t, k, v) such that designs B_t[k,λ;v] exist provided λ>N(t,k,v) and λ satisfies the trivial necessary conditions. We show that N(3,k,v)=0 for most of the cases under consideration and we give a numerical upper bound on N(3, k, v) for all 3?k?v?32. We give explicit constructions for all the designs needed.     

Degree functions and projectively full ideals in 2-dimensional rational singularities that can be desingularized by blowing up the unique maximal ideal

Let (R,m) be a 2-dimensional rational singularity with algebraically closed residue field and for which the associated graded ring is an integrally closed domain. According to Göhner, (R,m) satisfies condition (N): given a prime divisor v, there exists a unique complete m-primary ideal A_v in R with T(A_v)={v} and such that any complete m-primary ideal with unique Rees valuation v, is a power of A_v. We use the theory of degree functions developed by Rees and Sharp as well as some results about regular local rings, to investigate the degree coefficients d(A_v,v). As an immediate corollary, we find that for a simple complete m₁-primary ideal I₁ in an immediate quadratic transform (R₁,m₁) of (R,m); the inverse transform of I₁ in R is projectively full.     

Resolvable balanced bipartite designs

A block B denotes a set of k = k₁ + k₂ elements which are divided into two subsets, B¹ and B², where ∣Bⁱ∣ = k_i, i = 1 or 2. Two elements are said to be linked in B if and only if they belong to different subsets of B. A balanced bipartite design, BBD(v, k₁, k₂, λ), is an arrangement of v elements into b blocks, each containing k elements such that each element occurs in exactly r blocks and any two distinct elements are linked in exactly λ blocks. A resolvable balanced bipartite design, RBBD(v, k₁, k₂, λ), is a BBD(v, k₁, k₂, λ), the b blocks of which can be divided into r sets which are called complete replications, such that each complete replication contains all the v elements of the design.Necessary conditions for the existence of RBBD(v, 1, k₂, λ) and RBBD(v, n, n, λ) are obtained and it is shown that some of the conditions are also sufficient. In particular, necessary and sufficient conditions for the existence of RBBD(v, 1, k₂, λ), where k₂ is odd or equal to two, and of RBBD(v, n, n, λ), where n is even and 2n ? 1 is a prime power, are given.     

Ternary codes of steiner triple systems

The code over a finite field F^q of a design ?? is the space spanned by the incidence vectors of the blocks. It is shown here that if ?? is a Steiner triple system on v points, and if the integer d is such that 3^d ≤ v < 3^d+1, then the ternary code C of ?? contains a subcode that can be shortened to the ternary generalized Reed-Muller code ?_F3(2(d ? 1),d) of length 3^d. If v = 3^d and d ≥ 2, then C_? ? ?_F3(1,d)? ? _F3(2(d ? 1),d) ? C. © 1994 John Wiley & Sons, Inc.     

Handcuffed designs

Handcuffed designs are a particular case of block designs on graphs. A handcuffed design with parameters v, k, λ consists of a system of ordered k-subsets of a v-set, called handcuffed blocks. In a block {A₁, A₂,…, A_k} each element is assumed to be handcuffed to its neighbors and the block contains k ? 1 handcuffed pairs (A₁, A₂), (A₂, A₁), …, (A_k?1, A_k). These pairs are considered unordered. The collection of handcuffed blocks constitutes a handcuffed design if the following are satisfied: (1) each element of the v-set appears amongst the blocks the same number of times (and at most once in a block) and (2) each pair of distinct elements of the v-set are handcuffed in exactly λ of the blocks. If the total number of blocks is b and each element appears in r blocks the following conditions are necessary for the handcuffed design to exist. (1) λv (v ? 1) = (k ? 1)b. (2) rv = kb. In this paper it is shown that the necessary conditions are also sufficient.     

Cyclic block designs with block size 3 from Skolem-type sequences

A Skolem-type sequence is a sequence (s ₁, . . . , s _t ) of positive integers \({i\in D}\) such that for each \({i\in D}\) there is exactly one \({j\in \{1, \ldots , t - i\}}\) such that s _j  = s _j+i  = i. Positions in the sequence not occupied by integers \({i\in D}\) contain null elements. In 1939, Peltesohn solved the existence problem for cyclic Steiner triple systems for v ≡ 1, 3(mod 6), v ≠ 9. Using the same technique in 1981, Colbourn and Colbourn extended the solution to all admissible λ > 1. It is known that Skolem-type sequences may be used to construct cyclic Steiner triple systems as well as cyclic triple systems with λ = 2. The main result of this paper is an extension of former results to cyclic triple systems with λ > 2. In addition we introduce a new kind of Skolem-type sequence.     

Exklusive Graphen und Hamiltonche Graphenn-ten Grades II

The graphs considered are finite and undirected, loops do not occur. An induced subgraphI of a graphX is called animitation ofX, if

1. the degreesd _I(v)≡d _X(v) (mod 2) for allv∈V(I)
2. eachu∈V(X)?V(I) is connected with the setv(I) by an even number of edges. If the set of imitations ofX consists only ofX itself, thenX is anexclusive graph. AHamiltonian graph of degree n (abbr.:HG ⁿ) in the sense ofA. Kotzig is ann-regular graph (n>1) with a linear decomposition and with the property, that any two of the linear components together form a Hamiltonian circuit of the graph.

In the first chapter some theorems concerning exclusive graphs and Euler graphs are stated. Chapters 2 deals withHG ^n′ s and bipartite graphs. In chapters 3 a useful concept—theX-graph of anHG ⁿ—is defined; in this paper it is the conceptual connection between exclusive graphs andHG ^n′ s, since a graphG is anHG ⁿ, if all itsX-graphs are exlusive. Furthermore, some theorems onX-graphs are proved. Chapter 4 contains the quintessence of the paper: If we want to construct a newHG ⁿ F from anotherHG ⁿ G, we can consider certain properties of theX-graphs ofG to decide whetherF is also anHG ⁿ.     

A note on the conditional moments of a multivariate normal distribution confined to a convex set

Let Y be an N(μ, Σ) random variable on R^m, 1 ≤ m ≤ ∞, where Σ is positive definite. Let C be a nonempty convex set in R^m with closure $\text{C}$. Let (·,-·) be the Eculidean inner product on R^m, and let μ_c be the conditional expected value of Y given Y ∈ C. For v ∈ R^m and s ≥ 0, let β_s(v) be the expected value of |(v, Y) ? (v, μ)|^s and let γ_s(v) be the conditional expected value of |(v, Y) ? (v, μ_c)|^s given Y ∈ C. For s ≥ 1, γ_s(v) < β_s(v) if and only if $\text{C}\text{+ Σ v ≠}\text{C}$, and γ_s(v) < β_s(v) for all v ≠ 0 if and only if $\text{C}\text{+ v ≠}\text{C}$ for any v ∈ R^m such that v ≠ 0.     

Eccentric sequences and eccentric sets in graphs

By a graph we mean a finite undirected connected graph of order p, p ? 2, with no loops or multiple edges. A finite non-decreasing sequence S: s₁, s₂, …, s_p, p ? 2, of positive integers is an eccentric sequence if there exists a graph G with vertex set V(G) = {v₁, v₂, …, v_p} such that for each i, 1 ? i ? p, s, is the eccentricity of v₁. A set S is an eccentric set if there exists a graph G such that the eccentricity ρ(v₁) is in S for every v₁ ? V(G), and every element of S is the eccentricity of some vertex in G. In this note we characterize eccentric sets, and we find the minimum order among all graphs whose eccentric set is a given set, to obtain a new necessary condition for a sequence to be eccentric. We also present some properties of graphs having preassigned eccentric sequences.     

Diagonal norm hermitian matrices

If v is a norm on $\text{C}$ⁿ, let H(v) denote the set of all norm-Hermitians in $\text{C}$ⁿⁿ. Let S be a subset of the set of real diagonal matrices D. Then there exists a norm v such that S=H(v) (or S = H(v)∩D) if and only if S contains the identity and S is a subspace of D with a basis consisting of rational vectors. As a corollary, it is shown that, for a diagonable matrix h with distinct eigenvalues λ₁,…, λ_r, r?n, there is a norm v such that h ∈ H(v), but h^s?H(v), for some integer s, if and only if λ₂–λ₁,…, λ_r–λ₁ are linearly dependent over the rationals. It is also shown that the set of all norms v, for which H(v) consists of all real multiples of the identity, is an open, dense subset, in a natural metric, of the set of all norms.     

Partitioning the planes of AG2m(2) into 2-designs

A t-design S_λ(t, k, v) is an arrangement of v elements in blocks of k elements each such that every t element subset is contained in exactly λ blocks. A t-design S_λ(t, k, v) is called t′-resolvable if the blocks can be partitioned into families such that each family is the block system of a S_λ(t′, k, v). It is shown that the S₁(3, 4, 2^2m) design of planes on an even dimensional affine space over the field of two elements is 2-resolvable. Each S₁(2, 4, 2^2m) given by the resolution is itself 1-resolvable. As a corollary it is shown that every odd dimensional projective space over the field of two elements admits a 1-packing of 1-spreads, i.e. a partition of its lines into families of mutually disjoint lines whose union covers the space. This 1-packing may be generated from any one of its spreads by repeated application of a fixed collineation.     

Cantor-Lebesgue theorem for double trignometric series

Let ∥·∥ be a norm in R² and let γ be the unit sphere induced by this norm. We call a segment joining points x,y ε R² rational if (x₁ ? y₁)/(x₂ ? y₂) or (x₂ ? y₂)/(x₁ ? y₁) is a rational number. Let γ be a convex curve containing no rational segments. Satisfaction of the condition $$T_\nu (x) = \sum\nolimits_{\parallel n\parallel = \nu } {c_n e^{2\pi i(n_1 x_1 + n_2 x_2 )} } \to 0(\nu \to \infty )$$ in measure on the set e? [- 1/2,1/2)×[- 1/2, 1/2) =T² of positive planar measure implies ∥T _v ∥L₄ (T²) → 0(v → ∞). if, however, γ contains a rational segment, then there exist a sequence of polynomials {T _v } and a set E ? T², ¦E¦ > 0, such that T _v (x) → 0(v → ∞) on E; however, ¦c_n¦ ? 0 for ∥n∥ → ∞.     

Decomposing triples into cyclic designs

Motivated by constructing cyclic simple designs, we consider how to decomposing all the triples of Z_v into cyclic triple systems. Furthermore, we define a large set of cyclic triple systems to be a decomposition of triples of Z_v into indecomposable cyclic designs. Constructions of decompositions and large sets are given. Some infinite classes of decompositions and large sets are obtained. Large sets of small v with odd v<97 are also given. As an application, the results are used to construct cyclic simple triple systems.