Showing papers on "Disjoint sets published in 1981"

Consistency of Single Linkage for High-Density Clusters

Abstract: High-density clusters are defined on a population with density f in r dimensions to be the maximal connected sets of form {x | f(x) ≥ c}. Single-linkage clustering is evaluated for consistency in detecting such high-density clusters—other standard hierarchical techniques, such as average and complete linkage, are hopelessly inconsistent for these clusters. The asymptotic consistency of single linkage closely depends on the percolation problem of Broadbent and Hammersley—if small spheres are removed at random from a solid, at which density of spheres will water begin to flow through the solid? If there is a single critical density such that no flow takes place below a certain density, and flow occurs through a single connected set above that density, then single linkage is consistent in separating high-density clusters (by disjoint single-linkage clusters that include a positive fraction of sample points in the respective clusters and pass arbitrarily close to all points in the respective clusters...

Decentralized control with overlapping information sets

Abstract: A decentralized control scheme is proposed for linear systems composed of overlapping subsystems. By expanding the state space of the system, a higher-dimensional space is formed where the subsystems appear as disjoint. In the expanded space, standard optimization techniques can be used to formulate a suboptimal decentralized control law for the overall system. A suitable contraction of the obtained control law, which is compatible with the information constraints imposed by the overlapping subsystems, can be implemented in the original system. The application of the proposed decentralized control scheme is illustrated using a 19th order model for load-frequency control of a two-area interconnected power system.

On the use of the geodesic metric in image analysis

Abstract: SUMMARY Let X be a phase in a specimen. Given two arbitrary points x and y of X, let us define the number dx(x, y) as follows: dx(x, y) is the greatest lower bound of the lengths of the arcs in X ending at points x and y, if such arcs exist, and + ∞ if not. The function dX is a distance function, called ‘geodesic distance’. Note that if x and y belong to two disjoint connected components of X, dx(x, y) = + ∞. In other words, dx seems to be an appropriate distance function to deal with connectivity problems. In the metric space (X, dx), all the classical morphological transformations (dilation, erosion, skeletonizations, etc.) can be defined. The geodesic distance dx also provides rigourous definitions of topological transformations, which can be performed by automatic image analysers with the help of iterative algorithms. All these notions are illustrated with several examples (definition of the length of a fibre; automatic detection of cells having at least one nucleus, or having exactly a single nucleus; definitions of the geodesic centre and of the ends of a particle without holes, etc.). As an application, a general problem of segmentation is treated (automatic separation of balls in a polished section).

A weakly infinite-dimensional compactum which is not countable-dimensional

Abstract: A compact metric space is constructed which is neither a countable union of zero-dimensional sets nor has an essential map onto the Hilbert cube. We consider only separable metrizable spaces and a compactum means a compact space. A space is countable-dimensional if X = U7 1 Xi with Xi zero-dimensional; a space X is weakly infinite-dimensional if for each countable family {(Ai, B,): i = 0, 1, . .} of pairs of disjoint closed sets in X there are partitions Si between A, and Bi (i.e., closed sets separating Ai and Bi in X) with f70 Si = 0 [A-P, Chapter 10, ?47], [N]. Countable-dimensional spaces are weakly infinite-dimensional2 and an old open question of P. S. Aleksandrov [Al, ?4, Hypothesis] (cf. also [A2], [A-P, Chapter 10], [S], [N, Problem 13-7]) asked whether the converse is true for compacta.3 In this note we present a counterexample, i.e., we describe a compactum X with the properties indicated in the title. The existence of such an X is an easy consequence of the following lemma. LEMMA. There exists a topologically complete space Y which is totally disconnected but not countable-dimensional (not even weakly infinite-dimensional). The existence of such a space Y follows immediately from a construction in [R-S-W] (see also Comment A). More specifically, if one performs the construction in Example 4.5 of [R-S-W] using, as indicated in Remark 4.4, the Hilbert cube instead of the n + I-dimensional cube, then one obtains a compactum M and a continuous map p: M -> A onto the Cantor set A such that each subset of M which maps onto A is not weakly infinite-dimensional (see Proposition 3.4 and Remark 4.1 of [R-S-W]). It is, however, well known that in this situation there exists a G6-set Y c M which intersects each fiber p l(t) in exactly one point [B, p. 144, Exercise 9a], [Ku2, Chapters IV, IX], and this is the space Y we need. Received by the editors April 21, 1980. 1980 Mathematics Subject Classification Primary 54F45. IThis paper was written while the author was visiting the University of Washington. 2This follows from the fact [H-W, Chapter II, ?2, F] that, given two closed disjoint sets A, B c X and a zero-dimensional set E c X, there is a partition in X between A and B disjoint from E; cf. also [H-W, Chapter IV, ?6, Al. 3For nonmetrizable compact spaces, a counterexample was recently constructed by Fedorcuk [F]. i 1981 American Mathematical Society 0002-9939/81 /0000-0374/$01.75

A partition theorem for perfect sets

Abstract: Let P be a perfect subset of the real line, and let the «-element subsets of P be partitioned into finitely many classes, each open (or just Borel) in the natural topology on the collection of such subsets. Then P has a perfect subset whose «-element subsets he in at most (n — 1)! of the classes. Let C be the set of infinite sequences of zeros and ones, topologized as the product of countably many discrete two-point spaces, and ordered lexicographically. For X C C, let [X]n be the set of «-element subsets of X. When we describe a finite subset of C by listing its elements, we always assume that they are listed in increasing order. Thus, [C]" is identified with a subset of the product space C, from which it inherits a topology. A subset of C is perfect if it is nonempty and closed and has no isolated points. The purpose of this paper is to prove the following partition theorem, which was conjectured by F. Galvin who proved it [3] for n < 3. Theorem. Let P be a perfect subset of C and let [P]n be partitioned into a finite number of open (in [P]") pieces. Then there is a perfect set Q E P such that [Q]n intersects at most (n — 1)! of the pieces. Before setting up the machinery for the proof of this theorem, we point out some of its consequences. First, the therem remains true if C is replaced by the real line R with its usual topology and order. To see this, it suffices to observe that every perfect subset of R has a subset (a generalized Cantor set) homeomorphic to C via an order-preserving map and that any one-to-one continuous image in R of a perfect subset of C is perfect in R. Second, the hypothesis that the pieces of the partition are open can be greatly relaxed. Mycielski [6], [7] has shown that any meager set or any set of measure zero in [R]" is disjoint from [P]" for some perfect PçR. For the meager case, he obtains the same result with R replaced by any complete metric space X without isolated points. It follows that, if [R]" (or [X]") is partitioned into finitely many pieces that have the Baire property, then their intersections with [P]" are open in [P]" for some perfect P. Similarly, if the pieces are Lebesgue measurable, they become Gs sets when restricted to [P']n for suitable perfect 7"; since Gs sets have the Baire property, we can apply the preceding sentence, with 7" as X, to get a perfect PCP' such that the pieces intersected with [P]n are open in [P]". Thus, our theorem, as extended by the first remark above, implies the following partition theorem. Received by the editors April 27, 1979 and, in revised form, May 6, 1980; presented at a meeting on combinatorial set theory, Aachen, West Germany, June 1976. 1980 Mathematics Subject Classification. Primary 03E15, 03E05, 54H05. © 1981 American Mathematical Society 0002-9939/81/0000-0272/$02.75 271 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

On the Algorithmic Properties of Concurrent Programs

Abstract: A simple model of concurrent computations is presented in which disjoint instructions /processes/ of program are executed concurrently by processors /in a sufficiently large number/ under a shared memory environment. The semantics of such a program specifies the tree of configuration sequences which are acceptable as possible computations of the program.

Some $E$-Optimal Block Designs

Abstract: When a BIBD or a Group Divisible design with $\lambda_2 = \lambda_1 + 1$ is extended by certain disjoint and binary blocks the resulting structure is proved $E$-optimal. A BIBD abridged by a certain number of such blocks is also shown $E$-optimal. These optimality results hold over the class of all block designs (with the respective sets of parameters). Proofs rely mainly on averaging information matrices, which proves useful in many settings related to design optimality.

Generalized decompositions of dynamic systems and vector Lyapunov functions

Abstract: The notion of decomposition is generalized to provide more freedom in constructing vector Lyapunov functions for stability analysis of nonlinear dynamic systems. A generalized decomposition is defined as a disjoint decomposition of a system which is obtained by expanding the state-space of a given system. An inclusion principle is formulated for the solutions of the expansion to include the solutions of the original system, so that stabliity of the expansion implies stability of the original system. Stability of the expansion can then be established by standard disjoint decompositions and vector Lyapunov functions. The applicability, of the new approach is demonstrated using the Lotka-Voiterra equations.

Graphs as models of communication network vulnerability: Connectivity and persistence

Abstract: It is well known that the maximum connectivity k of a graph G with p points and q lines is given by [2 q/p]. This is restated in two useful alternative forms which minimize q given p and k, and which maximize p in terms of q and k. We define the persistence of a graph as the smallest number of points whose removal increases the diameter. It is shown that the persistence of a graph of diameter d is the minimum over all pairs of nonadjacent points of the maximum number of disjoint paths of length at most d joining them. A similar result is obtained for line-persistence and it is shown that these invariants are independent of each other.

Disjointness of Measure-Preserving Transformations, Minimal Self-Joinings and Category

Abstract: In the coarse topology on the group of measure-preserving transformations of a Lebesgue probability space, the class of transformations disjoint from a given ergodic transformation is a dense Gδ. The class of transformations T such that the family {Ti: i ϵ ℤ} is disjoint is also a dense Gδ. As a corollary there exists an uncountable family {Tα: α ϵ A} of weakly-mixing transformations such that the family \( \{ {\text{T}}_{\text{a}}^{\text{i}}:\alpha \in {\text{A,i}} \in {\Bbb Z} - \{ 0\} \} \) is disjoint.

A probability measure for character compatibility

Abstract: It is proved that two undirected binary cladistic characters are compatible iff their smaller states are disjoint or one is a subset of the other. The concept of a cladistic character as an ordered tree of subsets is defined. Cladistic characters that have the same number of elements in their corresponding states are defined to be “nesting equivalent.” The equivalence classes of this relation are called “nestings.” A certain class of n-tuples is shown to have a biunique correspondence with the n!-membered set of all nestings of n binary characters. The model of randomness proposed is that all characters that are nesting equivalent are equally likely. The probability that a pair of undirected binary characters is compatible is derived under this model. This result is extended to collections of undirected binary characters, to collections of directed binary characters, and finally to collections that may include multistate characters. Some proofs are presented which allow a more efficient use of the n-tuple representation of ordered trees of subsets.

Transitions and distribution functions for chaotic systems

Abstract: We study chaotic systems generated by deterministic or probablistic mappings. We introduce the density function which is an eigenfunction of a probability-preserving kernel $K$. We are able to show that all eigenvalues of $K$ have magnitude less than or equal to 1 and that the only magnitude-one eigenvalues are the $N\mathrm{th}$ roots of unity. We have also calculated the corresponding eigenfunctions associated with these magnitude-one eigenvalues: These eigenfunctions can be expanded in terms of $N$ positive functions having disjoint support. We then concentrate on a one-dimensional system, and study the behavior and mechanism for various chaotic transitions. We find that the mechanism associated with the 2 to 1 (or more generally, $2N$ to $N$) transition is different from those associated with other chaotic transitions. We then determine the conditions for these transitions, and express them in a universal form. We confirm the Huberman-Rudnick scaling in the large ${2}^{n}$ to ${2}^{n\ensuremath{-}1}$ chaotic-transition region, and determine the prefactor at these transitions. In addition, we establish a simple relation between the Lyapunov exponent and the folding of the distribution functions. We have also studied the chaotic regions of this system numerically.

On a generalisation of the root iterations for polynomial complex zeros in circular interval arithmetic

Abstract: Consider a polynomialP (z) of degreen whose zeros are known to lie inn closed disjoint discs, each disc containing one and only one zero Starting from the known simultaneous interval processes of the third and fourth order, based on Laguerre iterations, two generalised iterative methods in terms of circular regions are derived in this paper These interval methods make use of the definition of thek-th root of a disc The order of convergence of the proposed interval methods isk+2 (k≧1) Both procedures are suitable for simultaneous determination of interval approximations containing real or complex zeros of the considered polynomialP A criterion for the choice of the appropriatek-th root set is also given For one of the suggested methods a procedure for accelerating the convergence is proposed Starting from the expression for interval center, the generalised iterative method of the (k+2)-th order in standard arithmetic is derived

Refinement properties and extensions of filters in Boolean algebras

Abstract: We consider the question, under what conditions a given family A in a Boolean algebra I13 has a disjoint refinement. Of course, A cannot have a disjoint refinement if A is a dense subset of an atomless 03, or if 03 is complete and A generates an ultrafilter on S. We show in the first two sections that these two counterexamples can be the only possible ones. The third section is concerned with the question, how many sets must necessarily be added to a given filter in order to obtain an ultrafilter base. 0. Introduction. Let us recall the famous Disjoint Refinement Lemma due to Bernstein, Kuratowski, Sierpifnski and others: "Assume K to be an infinite cardinal and let A = {aa: a < K) be a family of sets, each of power K. Then there is a family D = {d a < K) such that for every a <,8 < K we have Idal = K, da C a,a, da nd, = 0". The family D is called a disjoint refinement of the family A. This lemma, first conceived as a mere technical tool, has turned out to be the birth cry of the following general disjoint refinement problem: "What are the conditions under which a given family has a disjoint refinement?" The power set of a given set, a factor algebra of a set modulo some ideal, a partially ordered set and most generally, the Boolean algebra, for all these structures the question is meaningful. The problem has an extensive literature (e.g. [B], [BF], [BHM], [BV], [C], [CH], [vD21, [H], [Hi], [KI, [KU2], [Kr], [P], [Si], [TI); the results concerning the problem have plenty of applications-in Boolean algebras, in the theory of filters and ultrafilters, in the theory of ultrapowers, in the descriptive set-theory and topology. The aim of the present paper is to study the refinement problem in Boolean algebras. The notion of the disjoint refinement is the natural one. 0. 1. DEFINITION. Let Si be a Boolean algebra, K an infinite cardinal and A = {aa: a < K) a family of nonzero elements of I-3. The family A has a disjoint refinement in 513 if there is a family D = {da a

Enumeration of structure-sensitive graphical subsets: Calculations

Abstract: Numerical calculations are presented, for all connected graphs on six and fewer vertices, of the numbers of independent sets, connected sets, point and line covers, externally stable sets, kernels, and irredundant sets. With the exception of the number of kernels, the numbers of such sets are all highly structure-sensitive quantities.

The uniqueness of the near hexagon on 759 points

Abstract: We show that the unique near hexagon with s = 2 and t = 14 and t 2 = 2 is the one with the blocks of the Steiner system S(5,8,24) as vertices and sets of three pairwise disjoint blocks as lines.

A Group Testing Problem on Two Disjoint Sets

Abstract: Recently the following group testing problem has been studied. We have two disjoint sets of items with cardinalities m and n respectively, where each set is known to contain exactly one defective item. The problem is to find the two defective items with a worst-case minimum number of group tests. It was conjectured that $\lceil \log _2 mn \rceil $ tests ($\lceil x \rceil $ denotes the smallest integer not less than x) always suffice. In this paper we prove that the conjecture is true.

Zur Einführung der Eulerschen Charakteristik

Abstract: For relatively open convex polyhedra (cells)Q ⊂ ℝd we put χ(Q):=(−1)dimQ. Any polyhedronQ ⊂ ℝd is the disjoint union of a finite number of cells:\(P = \bigcup\limits_i {Q_i } \). We show that\(\chi (P): = \sum\limits_i \chi (Q_i )\) is independent of the specific decomposition ofP into disjoint cells and therefore is uniquely determined byP. Since every closed convex polyhedron is the disjoint union of its relatively open faces of all dimensions, χ(P) is the Euler characteristic ofP. We finally present a new and elementary proof of the theorem of Euler-Schlafli.

Products of conjugate permutations.

Abstract: If s, pβSv, \s\ S \p\ and \p\ is infinite, then s is a product of 4 elements each conjugate to p. Furthermore, 4 is minimal with this property. The latter follows by examining s = (123) and any permutation p containing only transpositions (without fixed points) in its disjoint cycle decomposition, cf. G. Moran [6, p. 76] and [4, p. 288, 289]. If p is odd and s is even (with finite supports), then obviously s $ (pή\ and similar examples with finite | p | show

On the number of P-isomorphism classes of NP-complete sets

Abstract: All known NP-complete sets are P-isomorphic (i.e. there are polynomial time, one-to-one and onto, polynomial time invertible reductions between any two known NP-complete sets) [BH]. If all NP-complete sets are P-isomorphic, then. P ≠ NP. However it is not known if the existence of more than one P-isomorphism class of NP-complete sets has implications for the P = NP? problem. In the main result of this paper we prove: Theorem: If there is an NP-complete set that is not P-isomorphic to SAT, then there are infinitely many NP-complete sets that are mutually non-P-isomorphic. Thus, the number of P-isomorphism classes of NP-complete sets is either one or (countably) infinite. Two proof techniques are developed in this paper: we use delayed diagonalization [BCH, L] to construct new sets that are not P-isomorphic to existing sets; the diagonalization conditions are used to defeat P-isomorphisms. We also examine certain properties of 'generic' NP-complete sets and introduce techniques based on padding functions to assure that the sets constructed will be NP-complete. The results on P-isomorphisms and constructing non-P-isomorphic sets apply also to sets complete for PTAPE, EXPTIME, and EXPTAPE and other classes. We also examine the structure of NP-complete sets based on size increasing, invertible reductions, The degrees are P-isomorphism classes [BH]. We show that if there is more than one degree, then there is an ω chain of degrees with SAT representing a maximal element.

Random secants of a convex body generated by surface randomness

Abstract: Length distributions for random secants through a convex region K are derived for three types of randomness. The results are formulated in terms of geometric properties of K, e.g. the overlap surface content of K with its translated self. The distribution of distance between two random points in K, expressed in terms of the overlap volume, is shown to extend to non-convex (including disjoint) regions. GEOMETRICAL PROBABILITY; CONVEXITY; RANDOM SECANTS