The minimal size of a completely separating system on n points is found.
The minimal size of a completely separating system on n points is found.
The convex polytope of all stochastic and symmetric matrices is considered and its extreme points are determined. A method is given for counting these extreme points.
Nous montrons que le nombre de systèmes triples de Steiner non isomorphes d'ordre n qui contiennent un sous-système d'ordre 7 tend vers l'infini avec n, ce qui généralise un résultat récent de Assmus et Mattson.
On donne l'énoncé d'un théorème sur les réarrangements d'applications d'un ensemble fini dans lui-même, qui généralise un résultat connu sur les permutations, et qui permet de retrouver le dénombrement des classes d'applications ultimement idempotentes et indécomposables, ainsi que celui des arbres de n sommets et des graphes connexes de n arêtes et n sommets.
We investigate properties of common transversals of two families of sets. Some structure theorems are proved, and we settle affirmatively a conjecture of L. Mirsky and H. Perfect concerning the existence of a common transversal for two families of sets.
Let E be a finite set. Call a family of mutually noncomparable subsets of E a clutter on E. It is shown that for any clutter on E, there exists a unique clutter on E such that, for any function f from E to real numbers,
Specifically, consists of the minimal subsets of E that have non-empty intersection with every member of . The pair is called a blocking system on E. An algorithm is described and several examples of blockings systems are discussed.
Patterned after theorems of Mal'cev and Jónsson, certain types of conditions for equational classes of algebras are named “Mal'cev type.” Regularity and weak regularity of equational classes of algebras are proved to be of “Mal'cev type.”
Let A be a set and let be a collection of subsets of A. Conditions are given that must hold if a partition of A is a subset of . The main idea presented is a generalization of several methods that have been used to prove certain packing theorems.
Let N(n) be the maximal number of mutually orthogonal Latin squares of order n and let nr be the smallest integer such that N(n)≥r for every n>nr. It is known that N(n)→∞ as n→∞ and that n2=6. A proof is given for n3≤51, n5≤62 and n29≤34, 115, 553.
An m×n (0, 1) matrix (aij) is said to be a * matrix iff aij=1 implies ai′j′=1 for all (i′, j′) satisfying 1≤i′<i, 1≤j′≤j. * matrices with certain additional restrictions are counted and enumerations of random walks and decision patterns are obtained.
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