Showing papers on "Disjoint sets published in 1978"

Vertex-critical subgraphs of Kneser-graphs

Abstract: We show that if the stable (independent) n-subsets of a circuit wit 2n+k vertices are split into k+1 classes, one of the classes contains two disjoint n-subsets; this yields a (k+2)-vertex-critical subgraph of Lovasz 's Kneser-graph KG n,k.

An Efficient Method for Reliability Evaluation of a General Network

Abstract: This paper presents a method for finding the terminal-pair reliability expression of a general network. First, the system success function S is found, beginning from the connection matrix for the logic diagram of the network. Second, using the concept of Exclusive operator, S is changed to its equivalent S (disjoint) form, and the reliability expression has been derived. The method has the advantage of not requiring step by step testing for disjointness. Examples illustrate the method.

On the representation and identification of structure systems

Abstract: In a number of recent papers a conceptual framework for structure modelling has been developed by Klir (1975, 1976). It is based upon a hierarchy of epistemological levels of systems. Also an operational procedure for solving the problem of structure identification has been proposed (see Klir, 1976, Klir and Uyttenhove 1976 a, b). It is the purpose of this paper to examine the applicability of information theory, or, rather, constraint analysis to the problem of structure identification. It will be shown that so-called structure candidates may conveniently be represented by means of the concept of total constraint, which is partitioned in terms of constraints existing within elements constituting the structure candidate. Since the decomposition of a structure system occurs in terms of subsystems (elements) which contain sets of variables that are not necessarily disjoint, a description of the decomposition is derived which may be considered as a generalized version of the way of partitioning a sy...

Limit theorems for weakly exchangeable arrays

Abstract: An array of random variables, indexed by a multidimensional parameter set, is said to be dissociated if the random variables are independent whenever their indexing sets are disjoint. The idea of dissociated random variables, which arises rather naturally in data analysis, was first studied by McGinley and Sibson(7). They proved a Strong Law of Large Numbers for dissociated random variables when their fourth moments are uniformly bounded. Silver man (8) extended the analysis of dissociated random variables by proving a Central Limit Theorem when the variables also satisfy certain symmetry relations. It is the aim of this paper to show that a Strong Law of Large Numbers (under more natural moment conditions), a Central Limit Theorem and in variance principle are consequences of the symmetry relations imposed by Silverman rather than the independence structure. To prove these results, reversed martingale techniques are employed and thus it is shown, in passing, how the well known Central Limit Theorem for U-statistics can be derived from the corresponding theorem for reversed martingales (as was conjectured by Loynes(6)).

The subgraph homeomorphism problem

Abstract: We investigate the problem of finding a homeomorphic image of a “pattern” graph H in a larger input graph G. We view this problem as finding specified sets of edge disjoint or node disjoint paths in G. Our main result is a linear time algorithm to determine if there exists a simple cycle containing three given nodes in G; here H is a triangle. No polynomial time algorithm for this problem was previously known. We also discuss a variety of reductions between related versions of this problem and a number of open problems.

Combinatorial geometry and convexity classes

Abstract: Contents Introduction § 1. Definition of d-convex sets and their simplest properties § 2. Properties of the unit ball § 3. Separability of d-convex sets § 4. Definition of H-convex sets and their simplest properties § 5. Support properties of H-convex sets § 6. Helly's theorem for H-convex sets § 7. A theorem of Szokefalvi-Nagy and its generalization § 8. Applications of Helly's theorem § 9. The Helly dimension of a normed space References

The two paths problem is polynomial

Abstract: Given an undirected graph G = (V,E) and vertices $s_1$,$t_1$;$s_2$,$t_2$, the problem is to determine whether or not G admits two vertex disjoint paths $P_1$ and $P_2$, connecting $s_1$ with $t_1$ and $s_2$ with $t_2$ respectively. This problem is solved by an O($n\cdot m$) algorithm (n = |V|, m = |E|). An important by-product of the paper is a theorem that states that if G is 4-connected and non-planar, then such paths $P_1$ and $P_2$ exist for any choice of $s_1$, $s_2$, $t_1$, and $t_2$, (as was conjectured by Watkins [1968]).

Introducing disjoint spectral translation in spectral multiple-valued logic design

Abstract: A new form of expansion of multiple-valued logical functions in generalised Fourier series in terms of the Chrestenson functions is presented. It is shown that this expansion exhibits the property of `disjoint spectral translation´ known in binary spectral logic design. This allows extending the possibility of low complexity realisation to a large class of multiple-valued logical functions.

A Method of Global Blockdiagonalization for Matrix-Valued Functions

Abstract: Let $A(x)$ be an $n \times n$ analytic matrix function of the vector variable x. Let the eigenvalues of $A(x)$ belong to two disjoint sets for every fixed x. Then there exists an invertible analytic matrix function $M(x)$ which takes $A(x)$ by a similarity transformation into a blockdiagonal form. Similar theorems for $A(x)$ being smooth are also proved.

Return times in nearly-completely decomposable stochastic processes

Abstract: Consider a finite irreducible aperiodic Markov chain with nearly-completely decomposable stochastic matrix: i.e. a Markov chain for which the states can be grouped into disjoint aggregates, in such a way that the probabilities of transition between states of the same aggregate are large compared to the probabilities of transition between states belonging to different aggregates. Let Ω be a subset of one of the aggregates. Second-order approximations are determined for the first and second moments of the time to reach Ω and the return time to Ω.

Another Criterion for Marriage in Denumerable Societies

Abstract: A society is an ordered triple (M, W, K) of sets such that M, W are disjoint and K⊆M × W. . An espousal of (M, W, K) is a subset of K of the form {(a, E(a)): a ∈ e M } where E(a 1 ) ≠ E(a 2 ) whenever a 1 ≠ a 2 . For every transfinite sequence f of distinct elements of W , we define (in a somewhat complicated manner) a number q(f). We prove that a necessary and, if M is countable, sufficient condition for (M, W, K) to have an espousal is that q(f)⩾ )0 for every countable transfinite sequence f of distinct elements of W.

The Hirsch Conjecture Fails for Triangulated 27-Spheres

Abstract: A triangulation Z of a 27-sphere is generated in which every sequence of adjacent 27-simplices joining a specified pair of disjoint 27-simplices revisits at least one vertex previously left behind. This shows thai the Hirsch conjecture, which asserts the existence of a nonredundant feasible pivot sequence between any pair of bases of a linear program, or equivalently a nonrevisiting sequence of facets of a simplicial polytope, is false when generalized in the obvious way to triangulated spheres. The complex Z is constructed from a complex with 400 4-simplices which has been shown to be shellable, and hence a 4-ball, by a computer computation.

Set representation and set intersection

Abstract: : This work discusses the representation and manipulation of sets based on two different concepts: tries, and hashing functions. The sets considered here are assumed to be static: once created, there will be no further insertions or deletions. For both trie- and hash-based strategies, a series of representations is introduced which together with the availability of preprocessing reduces the average sizes of the sets to nearly optimal values, yet retains the inherently good retrieval characteristics. The intersection procedure for trie-based representations is based on the traversal in parallel of the tries representing the sets to be intersected, and it behaves like a series of binary searches when the sets to be intersected are of very different sizes. Hashed intersection runs very fast. The average time is proportional to the size of the smallest set to be intersected and is independent of the number of sets (except for the intersection set itself which has to be checked for every set). (Author)

A note on the combinatorial principles

Abstract: Shelah has proved that 0 does not imply that 0(E) holds for every stationary set E 5 wl. We prove that, in the other direction, whenever 0(E) holds there are disjoint stationary sets F, G C E such that both 0(F) and 0(G) hold.

The probability that two samples in the plane will have disjoint convex hulls

Abstract: Suppose given an absolutely continuous distribution on the plane, and points P,, " ", P!, I, , Hk chosen independently according to the given distribution. Denoting by G0 the convex hull of {P,, -, Pi}, and by F, the convex hull of {i,, -, lH}, and writing pjk for the probability that G, and rk are disjoint, certain properties of the array {pk;j, k = 1,2, } are established, including a recurrence generating the array in terms of {p,,; n = 1, 2, }, and asymptotic results for {p,,; n = 1,2, }. Some examples are considered. BIVARIATE NORMAL DISTRIBUTION; CLOCKWISE CRITICAL LINE; CONVEX HULL

An inequality for generalized quadrangles

Abstract: Let S be a generalized quadrangle of order (s, t). Let X and Y be disjoint sets of pairwise noncollinear points of S such that each point of X is coUinear with each point of Y. If m = \X\ and n = | Y\, then (m — l)(n - 1) 1, t > 1, is a point-line incidence geometry S = (??, £, 7) with point set 1, t > 1. Let X = (xx, . . . , xm) and Y = (y\> ■ • • >yn) De disjoint sets of pairwise noncollinear points of §, m > 2 and n > 2. Let ki be the number of x/s with which v, is collinear, 1 < i < n, 0 < k, < m. Our main results consist of the following two theorems.

Generalizations of almost disjointness, c-sets, and the baire number of βN-N

Abstract: We consider certain combinatorial problems and their consequences in βN-N. Let F be any family of infinite subsets of N. Then we are interested in conditions under which it is possible to find an almost disjoint family A = {AF:Fϵ F } such that Aϵ⊆F. If such a family A exists, we call F separable. We note that any family of cardinality less than c is separable, but that it is independent of the negation of the Continuum Hypothesis as to whether or not a union of ℵ1 separable families is separable. We consider the separability of unions of ultrafilters and use our results to show that it is consistent that if S is any nowhere dense subset of βN-N, then there exists a family of c pairwise disjoint open sets each of which is disjoint from S but contains S in its closure. We also consider rectangular arrays of subsets of N subject to the condition that each row and each column is an almost disjoint family, and we consider various maximality questions concerning these. We then note that if there exists such an array with K rows and K + columns and the columns are all maximal, then βN-N may be covered by K + nowhere dense sets.

Refinements in Boundary Complexes of Polytopes

Abstract: A complex °K is said to be a refinement of a complex L if there exists a homeomorphism Ψ Set K set L such that for each face L of L Ψ-1(L) is a union of faces of K. A face K of K is said to be principal if Ψ(K) is a face of L. Some results concerning 3-polytopes are shown not to extend to higher dimensions. For ≥ 4, there exist simple d-pclytopes with d+8 facets whose boundary complex cannot be expressed as a refinement of the boundary complex of v. d-polytope with d+7 facets. For ≥ 4, there exist simple d-polytopes whose graphs do not con¬tain refinements of the complete graph on d+1 vertices, if three particular vertices are preassigned as principal. A conjecture of Grunbaum is answered in the negative by constructing, for ≥ 4 simple d-polytopes P with d+4 facets in which two particular vertices may not be preassigned as principal if the boundary complex of P is expressed as a refinement of the boundary complex of a d-simplex; for d≥6, non-simple d-polytopes with d+3 facets having the same property are constructed. The main positive result is that the boundary complex of a d-polytope with d+2 facets, (d+3 facets if d = 4,5), may be ex¬pressed as a refinement of the boundary complex of the d-simplex with any two preassigned vertices principal. Several conjectures are made, among them the following genera¬lization of Balinsky's theorem on the d-connectedness of the graph of a d-polytope. If di+...+dk = ds di E e N, then between any two vertices of ad-polytepe exist strong chains of di-faces, I = 1,...,k, disjoint except for the chosen vertices.

Peak points for algebras on circled sets

Abstract: LetK be a compact, connected, circled subset of ℂ n . It is shown that each nonpeak point forH(K) lies on an analytic disc in the rational hull ofK. The peak points forH(K) that are disjoint from the coordinate axes are characterized in terms of the logarithmically extreme points of |K|.

The nonexistence of universal invariant measures

Abstract: It is shown that for arbitrary nontrivial o-finite measure defined on all subsets of a group there are at most countably many left translations of that measure that are mutually equivalent in the sense of the absolute continuity. This note was inspired by the theorem of Erdos and Mauldin [3] on the nonexistence of invariant universal measures on uncountable groups. Slightly modifying the reasonings of [3] we get a fairly general lemma on invariance properties of a-ideals of subsets of an arbitrary group. As a corollary we obtain a refinement of the theorem from [3], as follows: for arbitrary nontrivial a-finite measure defined on all subsets of a group there are at most countably many left translations of that measure that are mutually equivalent in the sense of the absolute continuity. Besides we mention results of a similar kind for semigroups and for finitely additive measures. One says that a a-ideal / of subsets of a set G satisfies the countable chain condition (CCC) if every uncountable family of pairwise disjoint subsets of G contains an element of /. The next assertion is essentially due to S. Ulam [5]: Proposition. If I is a proper o-ideal of subsets of a set G satisfying CCC, then G is not the union of N, sets from I. The proof is based on an application of the famous Ulam matrix of type K0 X K,. Now let us assume that the set G is a group. We shall say that an element g of G is left-side admissible with respect to / (in short: admissible) if for each E e G we have E E I iff gE E I. It is easy to see that the set A consisting of all admissible elements of G is a subgroup of G. Lemma. Under the above assumptions (Proposition) A is countable. Proof. Suppose, a contrario, that A is not countable. Then A contains a subgroup H of the cardinality N,. Let S be a selector from the family {Hg: g E G} of right-side cosets of H. We have hxS n h2S — 0 for A, h2, A,, A2 E H. Hence {hS: h E H) is an uncountable family of pairwise disjoint Received by the editors July 5, 1977. AMS (MOS) subject classifications (1970). Primary 28A70; Secondary 04A10.

On the Edge-Coloring Property for the Closure of the Complete Hypergraphs

Abstract: Given a set N of n elements, we address the question of when do there exist disjoint partitions of N, each partition containing only subsets of h or fewer elements, such that every subset of N having h or fewer elements is in exactly one partition. The question is answered, at least partly, in the negative for n = kh + l, l = 1, 2,…,h - 2, and in the affirmative for l = 0.

Some aspects of the Smoluchowski process

Abstract: This article deals with the generalized Smoluchowski process, { n (t), t ≧ 0}, defined by the temporal fluctuating of the numbers of randomly moving particles contained in m < ∞ disjoint regions of space. The relationship of the Smoluchowski process { n (t), t ≧ 0} to the emigration–immigration process is discussed and conditions for their equivalence are presented.