Showing papers on "Disjoint sets published in 1980"

Distributed Termination

Abstract: Discussed is a distributed system based on communication among disjoint processes, where each process is capable of achieving a post-condition of its local space in such a way that the conjunction of local post-conditions implies a global post-condition of the whole system. The system is then augmented with extra control communication in order to achieve distributed termination, without adding new channels of communication. The algorithm is applied to a problem of constructing a sorted partition.

On the adiabatic theorem of quantum mechanics

Abstract: Let H(t) be a Hamiltonian whose spectrum has for all t a finite number of disjoint components sigma j(t). It is proved that when the change of H(t) is made infinitely slow the system, when started from a state corresponding to sigma k(0), passes through states corresponding to sigma k(t), for all t.

On translating a set of rectangles

Abstract: Given a collection of disjoint objects in the plane, we are interested in translating them by a common vector. If we have a primitive for translating one object at a time, then the order in which the objects can individually be translated is often geometrically constrained. In this paper we study the nature of these constraints and exhibit optimal algorithms for finding valid motion ordering for several different classes of objects. These algorithms find use in computer display applications.

Detection is easier than computation (Extended Abstract)

Abstract: Perhaps the most important application of computer geometry involves determining whether a pair of convex objects intersect. This problem is well understood in a model of computation where the objects are given as input and their intersection is returned as output. However, for many applications, we may assume that the objects already exist within the computer and that the only output desired is a single piece of data giving a common point if the objects intersect or reporting no intersection if they are disjoint. For this problem, none of the previous lower bounds are valid and we propose algorithms requiring sublinear time for their solution in 2 and 3 dimensions.

Starlike domains of convergence for Newton's method at singularities

Abstract: Given a solutionx * of a system of nonlinear equationsf with singular Jacobian ?f(x *) we construct an open starlike domainR of initial points, from which Newton's method converges linearly tox *. Under certain conditions the union of those straight lines throughx *, that do not intersect withR is shown to form a closed set of measure zero, which is necessarily disjoint from any starlike domain of convergence. The results apply to first and higher order singularities.

Construction and properties of a useful two-dimensional random field

Abstract: A two-dimensional homogeneous and isotropic random field is described which is generated by a polygonal partition process on the plane. In particular, the plane is divided into elementary disjoint polygonal regions by a Poisson line process. Gray levels within these elementary polygonal regions are assigned to have specified correlation wilh gray levels in contiguous regions. Explicit evaluation of the resulting autocorrelation function and power spectral density is provided. Several applications are described.

Recursive Disjoint Products, Inclusion-Exclusion, and Min-Cut Approximations

Abstract: Three different ways to estimate the reliability of an s-coherent system are compared: 1) recursive disjoint products (DP), 2) recursive inclusion-exclusion (IE), and 3) minimal-cut approximations based on partial information. The following three points are made. 1. Recursive DP and recursive IE are mathematically identical and obtain the same numerical values at each step of the recursion, although the recursive system-reliability functions are different. 2. Recursive DP seems to result in fewer comparisons and a shorter polynomial than recursive IE, and therefore also less work, for small scale systems, such as the 2-out-of-4:G example discussed herein. I do not yet know if this comparative advantage persists for larger systems. 3. For complex highly reliable systems, min-cut approximations based on partial information, that sacrifice some accuracy for convenience and ease of computation are preferable to min-path exact methods since the approximations come very close to the true value of the reliability with comparably little effort, in some cases requiring only hand calculations.

Intersection Properties of Steiner Systems

Abstract: Publisher Summary Cayley has established in 1850 that there exist two but no more disjoint Steiner triple systems on a given 7-element set and Kirkman found in the same year that the largest number of disjoint Steiner triple systems of order 9 is seven. But virtually all results of substance in this area that was label “intersection properties of Steiner systems” have been obtained in the last decade. This chapter surveys the present state of affairs in this area. The chapter also discusses some related questions, such as disjoint triple systems with λ > 1, perpendicular Steiner systems and applications to the existence of designs with larger λ.

Covering 2ω with ω1 Disjoint Closed Sets

Abstract: It is shown that 2 ω is the ω 1 union of meager sets does not imply 2 ω is the ω 1 , union of disjoint non-empty closed sets and the latter does not imply CH.

An Extension of the Change-in-Ratio Method

Abstract: The traditional change-in-ratio (CIR) method for estimating the size of animal populations assumes that the population is split into two disjoint and exhaustive components. A method for obtaining maximum likelihood estimates of population component sizes is presented for the more general situation in which the CIR method is utilized by splitting the population into three disjoint components. The large sample variance-covariance matrix of these estimators is derived.

Cutting families of convex sets

Abstract: A family of convex sets in the plane admits a common transversal if there is a straight line which intersects (cuts) each member of the family. It is shown that there is a positive integer k such that for any compact convex set C in the plane and for any finite family & of pairwise disjoint translates of C: If each 3-membered subfamily of & admits a common transversal then there is a subfamily $ of & such that i& admits a common transversal and \â \ 9> \ <,k.

A decidable fragment of the elementary theory of the lattice of recursively enumerable sets

Abstract: A natural class of sentences about the lattice of recursively enumerable sets modulo finite sets is shown to be decidable. This class properly contains the class of sentences previously shown to be decidable by Lachlan. New structure results about the lattice of recursively enumerable sets are proved which play an important role in the decision procedure. 0. Introduction. Much of the recent work dealing with S, the lattice of recursively enumerable sets, has dealt with global properties of S such as automorphisms and decidability, rather than local properties of S, i.e., properties of definable classes of recursively enumerable sets. Two of the major results are Lachlan's [3] decision procedure for a natural fragment of the elementary theory of S *, the quotient lattice of S by the ideal of finite sets, Soare's result [15] on the existence of automorphisms carrying any maximal set into any other maximal set. More recently, Shore [13] has determined the definable automorphism bases for £. These global results have inspired new local results, in that they have naturally led to the discovery of important new S-definable classes of recursively enumerable sets whose properties have been investigated. Lachlan's result led to the discovery of small sets, Soare's result led to the discovery of ¿/-simple sets, and Shore's result led to the discovery of nowhere simple sets; the first two of these classes have been studied by Lerman and Soare [8], and the third class by Shore [14]. The class of ¿-simple sets proved to be of particular importance, in that it led to the refutation of conjectures of Martin and Shoenfield which imply that the degrees of elements of any S-definable class can be characterized by a finite set of equalities and inequalities involving the jumps of those degrees. Evidence for these conjectures included Martin's result [9] that a is the degree of a maximal set if and only if a' = 0\", and results of Lachlan [2] and Shoenfield [12] that a is the degree of an atomless set if and only if a\" > 0\". In this paper, we give a decision procedure for a larger fragment of the Received by the editors April 14, 1978 and, in revised form, November 3, 1978. AMS (MOS) subject classifications (1970). Primary 02F25; Secondary 02G05.

Recognizing Intersection Patterns

Abstract: Publisher Summary Many combinatorial problems have the following form: given an n × n matrix A = ( a ij ) decide whether there are sets S 1 , S 2 , . . . , S n such that | S i S j |= a ij for all choices of i and j . If the answer is affirmative, then A is called an “intersection pattern.” Recognizing intersection patterns does not seem easy—for example, deciding whether there is a projective plane of order ten amounts to deciding whether a certain matrix of size 112 × 112 is an intersection pattern. This chapter discusses the recognizing intersection patterns with a ij = 3 for all i is an NP-complete problem. In a sense, the bound a ij ≤ 3 is as severe as one can impose and still expect NP-completeness: recognizing intersection patterns with a ij = 2 for all i amounts to recognizing line-graphs, which is known to be easy. It is convenient to represent each would-be intersection pattern A = (a ij ) with a ii = 3 for all i by a multigraph H in which every two distinct vertices w i , w j are joined by precisely a ij edges. By an admissible partition of H , it is shown in the chapter that partition of its edge-set into disjoint cliques such that every vertex belongs to at most three of these cliques.

Arithmetic of complex sets

Abstract: Let\(\mathbb{I}\)(ℝ) be the set of all real closed intervals and letΩ1:= {+, −, ×, /} be the set of arithmetic operators of ℝ. By extendingΩ1 from ℝ to\(\mathbb{I}\)(ℝ) as usual one finds that\(\mathbb{I}\)(ℝ) is closed with respect to the operations fromΩ1 (R. E. Moore [9]). In the literature several possibilities are discussed to go over from complex numbers to “complex intervals”: rectangles (Alefeld [1] et al.), discs (Henrici [4] et al.) or ellipses (Kahan [5] et al.). In all three cases the resulting sets are not closed with respect toΩ1, since the multiplication and division of such “intervals” does not lead to sets of the same kind. In what follows the question is treated whether there are classes of complex sets (“generalized intervals”) which are closed with respect toΩ1 or to subsets ofΩ1. One such class is easy to find. Also the shape of the sets involved is discussed. If it is assumed however that the sets under consideration are described by a finite number of parameters then there isno such class closed underΩ1.

Mad families and ultrafilters

Abstract: For each almost disjoint family X let F(X) = (a C w: card {s E X: s\a is finite) = 2w), I(X) = (a C w: card {s E X: card (s n a) = w) = 21). Assuming P(2W) we show that for each nonprincipal ultrafilter p there exist a maximal almost disjoint family X and an almost disjoint family Y with F(X) = I(Y) =p.

A Group Testing Problem

Abstract: Suppose we have two disjoint sets of items of cardinalities m and n where each set contains exactly one defective item. A group test is a simultaneous test on an arbitrary group of items with two possible outcomes: The group is identified as good if it contains no defective items; otherwise it is identified as defective. The problem is to find the two defective items with a worst-case minimum number of (group) tests.The first question is whether there is anything to be gained by considering the two sets together. The answer is a somewhat surprising “yes.” The second question is this: Since there are $mn$ possible pairs of defective items,can we always solve the problem with $[ \log_2 mn ]$ tests—the information-theoretic lower bound where $[ x ]$ denotes the smallest integer not less than x. We conjecture that the answer is yes again and we provide partial evidence in favor of this conjecture. We also discuss several other related problems.

Two sets of graceful graphs

Abstract: We give graceful numberings to the following graphs: (a) the union of n K4 having one edge in common, in other words the join of K2 and the union of n disjoint K2 and (b) the union of n C4 having one edge in common, in other words the product of K2 and K1,n, with n + 1 not a multiple of 4.

Generating the k-Subsets of an n-Set

Abstract: While discussing with colleagues certain questions about generalized measures on finite measure spaces, the author came upon a combinatorial lemma-on generating all of the k-subsets of an n-set. In the context of these discussions, one is given an n-set, say X= {x,x2, ... .,xn,, together with a collection 9 of its subsets that is closed under disjoint union and under complementation. A generalized measure It is a nonnegative real-valued function on these "measurable" subsets, one however that is additive over disjoint unions. Of course, there are any number of such generalized measure spaces (g.m.s.), some interesting and some not, depending on one's point of view. Two useful but quite different introductions to the subject are those of Rado [1] and Gudder [2]. In the present context, we would like to be assured, first of all, that there are generalized measures that are not induced by an ordinary signed measure on the points of X. For this purpose, we let X = (1,2,3,4,5,6), and as the measurable sets we take

The generalized Očan topology on sets of subsets and topological Boolean rings

Abstract: In this article the study of OCAN spaces is continued. In a space (, ℬ) some topological properties are not disturbed if and ℬ are enlarged. The SORGENFREY plane can be identified with some OCAN space (Example 1). By use of systems of almost disjoint subsets some special topological rings on (X) can be constructed (Propositions 8 and 9). A metrisable or a locally compact OCAN ring has a simple structure (Propositions 10 and 11). If (, ℬ) neither discrete nor compact, then the closedness of all simple maps is a very strong condition (Theorem 1). The space of VIETORIS is in general not σ-extremally disconnected space (Theorem 2). At the end of the article some generalizations are made and some bibliographical references are given.

The Asymptotic Behavior of a Class of Counting Functions

Abstract: For a class C of finite relational structures (with prescribed finite signature) the profile ϕcof C is the function counting the number ϕc(n) of isomorphism types of relational structures from C on n-elements set domains. We recall our result: Theorem.If C is extensive (i.e. each element of C can be embedded in another one having a bigger domain) and closed under substructures, then: (1) ϕcis not decreasing, and (2) The asymptotic behavior of ϕCis polynomial or greater than every polynomial. Our first proof of (1) (in a similar form) occurs in [2] and, (in this form) in [4]; we give a more general result in [3]. The fact that unbounded profiles are at least linear occurs in [4] and the general result is in [5]. Here we precise a fragment of (2) containing the minimal frame needed for applications in permutation groups. Theorem.If C is closed under substructures and has the Disjoint-Embedding-Property (i.e. each pair of structures of C can be embedded in another one with disjoint images), then ϕc(n)/nk is bounded iff ϕc(n)≤(n+k/k. Because the bound is independent of the signature, an extension of the main theorem to certain relational structures with infinite signature can be considered. in particular the main theorem for relational structures associated to a permutation group G: the asymptotic behaviour of the orbital profile of G, (Function counting the number of n-element orbits of G), is polynomial or greater than every polynomial. Finally from a characterization of structures having automorphism groups with polynomial orbital profile we obtain decidable results for their complete theories (The simplest case is the nice theorem of [1]).