Symmetric Graph Designs

We introduce the notion of a symmetric graph design which is a common generalization of symmetric BIBDs and of orthogonal double covers. Revised: January 21, 1999     

Jets in inclusive e+e− and leptoproduction

The assumption of jets with bounded p_T leads to the following consequences: (a) In e⁺e^? annihilation the coefficient of the cos²θ term (a) of single-particle inclusive distributions] exhibits a unique dependence on the observed particle momentum. This dependence provides a measure of 〈p_T²〉 in a jet. (b) In two-particle inclusive annihilation the average of the squared relative p_T is determined by the same parameter and by the momenta of the two particles. (c) The observation that in the current fragmentation region of leptoproduction the p_T distribution of fast hadrons is broader than that of show ones is explained.     

More on the Erdös-Ko-Rado theorem for integer sequences

For positive integers n, k₁, k₂,…, k_n, t the probem: how many integer sequences (x₁, x₂,…, x_n) does it take such that 0 ? x_i ? k_i for 1 ? i ? n and any two sequences agree in at least t positions is investigated. Moreover all maximal systems of sequences are described. Results of Livingston and Frankl and Füredi are generalized also.     

On orthogonal double covers of kn and a conjecture of chung and west

A collection ?? = {G₁, G₂,…,G_n} of spanning subgraphs of K_n is called an orthogonal double cover if (i) every edge of K_n belongs to exactly two of the G_i's and (ii) any two distinct G_i's intersect in exactly one edge. Chung and West conjectured that there exists an orthogonal double cover of K_n, for all n, in which each G_i has maximum degree 2, and proved this result for n in six of the residue classes modulo 12. In another context, Ganter and Gronau showed that for n ≡ 1 mod 3, n ≠ 10, there exists an orthogonal double cover of K_n in which each G_i consists of an isolated vertex and the vertex disjoint union of K₃'s (actually these orthogonal double covers result from the solution of the directed version of the problem, which reduces to the undirected case when the orientation of the arcs is ignored). Clearly the covers of Ganter and Gronau satisfy the Chung-West requirement. In this article, the configurations of Ganter and Gronau are generalized to include the cases n ≡ 0,2 mod 3, and the results are used to provide a unified solution of the Chung-West problem. For n ≠ 5 mod 6, all the spanning subgraphs in the collection ?? are isomorphic to each other; however, this is not the case when n ≡ 5 mod 6. In addition to solving the Chung-West problem, we also go on to show that for n ≡ 2 mod 3 and n > 287, there exists an orthogonal double cover of K_n in which each spanning subgraph G_i consists of the vertex-disjoint union of an isolated vertex, and quadrilateral, and (n?5)/3 triangles. Of the 96 cases with 2 ? n ? 287, 65 cases are resolved and 31 remain open. © 1995 John Wiley & Sons, Inc.     

B0-->D(*-)a(+)(1): chirality tests and resolving an ambiguity in the CP violation parameter 2beta+gamma

We point out that the decays of B mesons into a vector meson and an axial-vector meson can distinguish between left and right-handed polarized mesons, in contrast to decays into two vector mesons. Measurements in B0-->D(*-)a(+)(1) are proposed for testing factorization and the V-A structure of the b-->c current, and for resolving a discrete ambiguity in 2beta+gamma.     

On sperner families in which nok sets have an empty intersection III

LetR be anr-element set and ℱ be a Sperner family of its subsets, that is,X ⊈Y for all differentX, Y ∈ ℱ. The maximum cardinality of ℱ is determined under the conditions 1)c≦|X|≦d for allX ∈ ℱ, (c andd are fixed integers) and 2) nok sets (k≧4, fixed integer) in ℱ have an empty intersection. The result is mainly based on a theorem which is proved by induction, simultaneously with a theorem of Frankl.