Showing papers on "Disjoint sets published in 1969"

The Design of Minimum-Cost Survivable Networks

Abstract: We consider the problem of designing a network which satisfies a prespecified survivability criterion with minimum cost. The survivability criterion demands that there be at least r_{ij} node disjoint paths between nodes i and j , where (r_{ij}) is a given redundancy requirement matrix. This design problem appears to be at least as difficult as the traveling salesman problem, and present techniques cannot provide a computationally feasible exact solution. A heuristic approach is described, based on recent work on the traveling salesman problem, which leads to a practical design method. Algorithms are described for generating starting networks, for producing local improvements in given networks, and for testing the redundancy of networks at each stage. This leads to networks which are locally optimum with respect to the given transformation. Randomizing the starting solution ensures that the solution space is widely sampled. Two theorems are given which allow great reduction in the amount of computation required to test the redundancy of a network. Finally, some design examples are given.

A Dynamic Programming Algorithm for Cluster Analysis

Abstract: This paper considers the problem of partitioning N entities into M disjoint and nonempty subsets clusters. Except when both N and N-M are very small, a search for the optimal solution by total enumeration of all clustering alternatives is quite impractical. The paper presents a dynamic programming approach that reduces the amount of redundant transitional calculations implicit in a total enumeration approach. A comparison of the number of calculations required under each approach is presented in Appendix A. Unlike most clustering approaches used in practice, the dynamic programming algorithm will always converge on the best clustering solution. The efficiency of the dynamic programming approach depends upon the rapid-access computer memory available. A numerical example is given in Appendix B.

Weakly mixing transformations which are not strongly mixing

Abstract: 2. Preliminaries. The examples will be of an invertible measure preserving transformation of the unit interval (equipped with the usual Lebesgue sets and Lebesgue measure). The transformation will be constructed in a sequence of steps, each step enlarging the domain of definition of the transformation. The construction utilizes a geometric approach described in [1] which consists of mapping subintervals of the same length linearly onto each other, and of representing this geometrically by a figure in which each interval is placed below its image. Thus, if II, . . . , In are pairwise disjoint subintervals of the unit interval, if they have the same length, and if we define r on Un, Ik by mapping Ik linearly onto Ik+', k =1, * *, n -1 (and leave r undefined elsewhere), the geometric figure that corresponds to this map consists of the intervals I', . . . ,In arranged in a stack with I' on the bottom and In on the top, and with the remaining intervals arranged in order between them, so that each point is located below its image. The action of the transformation can thus be regarded as an upward flow through the stack which ends when the top layer is reached, since the transformation is undefined there. Note that we have not assumed that the union of I', * . *, In is the unit interval. The main advantage of the geometric approach is that many properties of transformations become much clearer when viewed in this way. Of course, the geometric figure associated with each stage of the definition is not necessary for the definition of the transformation nor for the proof of its properties. It serves simply as an aid in understanding the construction.

On Partitioning a Set of Normal Populations by Their Locations with Respect to a Control

Abstract: 0. The problem and the approaches. This paper is concerned with a problem of partitioning a set of normal populations into two subsets according to their locations with respect to a control population, based on indifference zone formulation. Let no, II, ... , Ilk be (k + 1) normal populations with means ,o , * * *, k and a common variance o-2; and let no denote the standard or control population. For arbitrary but fixed constants 61 * and 62* such that 61* < 82*, we define three disjoint and exhaustive subsets QB, Or and QG of the set

Intersection theorems for systems of sets (ii)

Abstract: In this paper we present the complete solution of the problem which was considered in [1], with the exception of the case in which both the given cardinal numbers are finite. The results of [1] will not be assumed. We begin by introducing some definitions, f A system Li = (Bv: veiV) of sets Bv, where v ranges over the index set N, is said to contain the system Eo = (A^ : n e M) if, for (j.o e M, the set A^o occurs in Sx at least as often as in I o , i.e. if |{v : v e JV; Bv = A,o}\\ > |{/<: /< e M; A, = AM}| (n0 e M). If Sj contains Zo and, at the same time, So contains Zx then we do not distinguish between the systems So and 5^. The system Sx is called a (a, < b)-system if |N| = a and |5V| < & for veN. The system So is called a A(c)-system if |M| = c and i4Mi4^, = Atl2 Afl3 whenever ô» J\"i. f*2> ^ 3 e M J Mo ̂ A*i; M2 ̂ M3The relation a A(b, c) (1)

Numerical Classification Method for deriving Natural Classes

Abstract: THE current approach in numerical taxonomy is directed towards the so-called “minimum-variance” solution, for which it is argued that a population should be partitioned into cluster subsets by minimizing the total within group variation. Several classification methods have been compared1 and shown to possess related variance constraints, and a case has been made1–3 for suggesting that such methods are not ideally suited to the taxonomic problem of resolving “natural” classes. Implicit in the minimum variance approach is the concept that cluster should have no significant overall variance or spread, and this implies that in the case of a unimodal swarm the distribution should be split into an arbitrary number of compact sections. By contrast, Forgey has argued2,3 that for a “natural” classification, clusters should correspond to data modes, and there can only be as many classes as there are distinct modes. No variance constraint is implied, or should be induced, for when a mode is elongated rather than spherical the distribution merely reflects some internal factor of variation for the corresponding class. Such factors will be present to some extent, depending on data transformations and the quality of the selected character set, and therefore a subsequent variable search is necessary to discover the hidden constant characteristics of the class. Furthermore, those characters which are non-constant for a cluster mode may be inter-correlated, suggesting that the original character choice was poor, and in such cases the consideration of correlations, ratio variables and regression coefficients is indicated. Forgey interprets2,3 a data mode as a continuous dense swarm of points, separated from other such modes by either empty space or a scattering of “noise” data. It has been suggested that “noise” data usually result from sampling errors, and while this is true, they can also be interpreted as those natural phenomena associated with the intersecting tails of disjoint continuous distributions. We can therefore expect a “natural” cluster to exhibit a dense centre (of any shape) which is surrounded by a haze or cloud of points, and the problem is to isolate the dense centres irrespective of this interference.

Some Results on Matching in Bipartite Graphs

Abstract: Let A, , Am and B1, ... , B, be partitions of A and B respectively into -nonempty disjoint subsets. Let G' denote the quotient bipartite graph with sets of vertices A' = {Al,... , Am} and B' = {B1, * * , B4} and a set of edges E' defined such that {Ai,Bj} eE' if and only if {a,b} eE for some aAi, bBj. Of course, VA, and VB' denote the natural induced measures on A' and B', and D' is defined in the obvious way. It is important to note that if all vertices have weight 1 and with a e A we associate the subset D({a}) c B, then by the marriage theorem of P. Hall (cf. [3]) VB dominates VA if and only if there exists a system of distinct representatives (cf. [3]) for these subsets.2 If b(a) is the representative chosen from D({a}), then the mapping a -b(a) is a matching of A into B, i.e., a 1 1 mapping of A into B such that {a, b(a)} is an edge of G. It is this application of our results which motivated the present study (cf. Examples 1 and 2). Our main object in this note is to develop several theorems which will enable one to show that VB dominates VA by examining the structure of the (hopefully much simpler) quotient graph G'.

On the Solid-Packing Constant for Circles

Abstract: A solid packing of a circular disk U is a sequence of disjoint open circular subdisks Ul, U2, . whose total area equals that of U. The Mergelyan- Wesler theorem asserts that the sum of radii diverges; here numerical evidence is presented that the sum of ath powers of the radii diverges for every a < 1.306951. This is based on inscribing a particular sequence of 19660 disks, fitting a power law for the radii, and relating the exponent of the power law to the above constant. U 1. We shall be concerned here with solid packings of a closed circular disk U. Such a packing P consists of a sequence of open pairwise disjoint circular disks U1, U2, which are subsets of U; P is called solid if the areas of U and U n= Un are the same. Let r be the radius of U and rn that of Un so that the condition for a

Realcompactifications and Normal Bases

Abstract: In a recent paper (see [2]), Orrin Frink introduced a method to provide Hausdorff compactifications for Tychonoff or completely regular T1 spaces X. His method utilized the notion of a normal base. A normal base ℒ for the closed sets of a space X is a base which is a disjunctive ring of sets, disjoint members of which may be separated by disjoint complements of members of ℒ.

The computation and bounding of rate-distortion functions

Abstract: Methods are given for the numerical computation of Shannon's rate-distortion function R(D) for certain memoryless message sources. It is first assumed that U , the set of possible message-source outputs, and V , the set of possible destination symbols, are countable. The computation of R(D) for this case is reduced to a minimization problem in which the variables are the destination-symbol probabilities. For arbitrary U and V , upper and lower bounds on R(D) are derived by partitioning U and V each into a countable collection of disjoint subsets and employing the results derived previously for the case of countable U and V . Conditions are then discussed under which these bounds can be made arbitrarily close to each other by choosing sufficiently fine partitions of U and V . Two examples are included to illustrate the results in detail.

ON THE EXISTENCE OF c-POINTS IN flN\N

Abstract: closures. (A cozero set in a space X is a preimage by a continuous real valued function of an open set.) It is well known that if ax is any cardinal greater than c then there do not exist a-points in 3N\N, since in fact, if 91 is a collection of pairwise disjoint open sets in /3N\N then CI LI c. 1. Results without the continuum hypothesis. Let AP be the ultrafilter on a discrete space D associated with the point p in OD. The ultrafilter AP on D is called uniform if each element of AP has cardi

Cardinal algebras and measures invariant under equivalence relations.

Abstract: Introduction. There have been discussions from time to time of "abstract measures" the values of which need not be numerical (e.g. [2], [3], [4], [6], [7]). One of the purposes of this paper is to present arguments in favor of the use of cardinal algebras as values for these measures. Cardinal algebras were introduced and developed by A. Tarski in [8]. They have many of the good properties of real numbers and arise naturally in situations like the following: A (pseudo) group G of one-one functions is given with domain and range in a u-ring of sets X2 An equivalence relation between members of 1 is defined as follows: A B iffthere are Ai, Bi E 1[,fi E G for i

A locally connected, complete moore space on which every real-valued continuous function is constant'

Abstract: Armentrout [l] has shown that there is a Moore space on which every real-valued continuous function is constant. While this space is connected, it is not known to be complete. In answer to questions raised by F. B. Jones [2] it is shown in this paper that there is a connected, locally-connected, complete, separable Moore space on which every continuous real-valued function is constant. This space is very similar to one whose existence was announced by P. Roy [5]. A topological space is called a M\oore space if it satisfies Axiom 0 and the first three parts of Axiom 1 of [4]. A Moore space is said to be complete if it satisfies all of Axiom 1 in [4]. In a Moore space, domains are open sets and the set of regions whose existence is assured by Axiom 1 is a base for the topology. The construction uses a space which is essentially Armentrout's modification of a Moore space constructed by F. B. Jones [3]. It is redescribed here to make the construction more amenable to geonmetric intuition. The author is indebted to H. Cook for suggesting this geometric realization of the space 2. Let C be a planar disc topologized as follows. If P is on the rim of C, regions containing P shall be interiors of circles lying in C and having only P in common with the rim of C together with P. If P is in C but not on the rim of C, regions containing P shall be interiors of circles lying in C having P in their interior and containing no point of the rim of C. With this topology, C becomes a separable, connected, locally connected, complete Moore space S. The rim of C is a discrete set in S and is the union of two disjoint uncountable sets A and B with the property that any domain in S which contains an uncountable subset of one of them has an uncountable subset of the other in its closure [3]. Let M be a countable dense subset of S containing no point of the rim of C. Let T be the point set in E3 obtained by rotating the graph of y=x2-1, (-I

Index Sets of Finite Classes of Recursively Enumerable Sets

Abstract: Let q0, q1,… be a standard enumeration of all partial recursive functions of one variable. For each i, let wi = range qi and for any recursively enumerable (r.e.) set α, let θα = {n | wn = α}. If A is a class of r.e. sets, let θA = the index set of A = {n | wn ∈ A}. It is the purpose of this paper to classify the possible recursive isomorphism types of index sets of finite classes of r.e. sets. The main theorem will also provide an answer to the question left open in [2] concerning the possible double isomorphism types of pairs (θα, θβ) where α ⊂ β.

Disjoint common partial transversals of two families of sets

Abstract: : The theory of flows in networks is applied to obtain necessary and sufficient conditions on two finite families of subsets of a finite set in order that there exist k mutually disjoint common partial transversals, each of size p, of the two families.

Building cartesian products of surfaces with (0, 1)

Abstract: THEOREM. Suppose that M is a 3-manifold with boundary, S is a compact 2-manifold with boundary in M such that S f Bd (M) = Bd (S) n Bd (M) = R either a 1-manifold with boundary or the empty set, and e>0. There is a 8 > 0 such that iffo andf1 are homeomorphisms of S onto disjoint locally tame surfaces SO and S, in M where fe(S) n Bd (M) =fe(R) andfe moves no point

A lower bound for the dimension of certain _{} sets in completely normal spaces

Abstract: Nagami and Roberts have proved [3, Theorem l] that if X is a normal space of dimension at least n satisfying certain conditions, then dim (X — U?TM 1 Ai)^n — i if the Ai are disjoint closed subsets of X. In this paper we allow the Ai to intersect provided that the dimension of the pairwise intersections is known. (The dimension of the remainder is reduced accordingly.) By dimension (denoted dim) we mean the covering dimension. The symbols bdy and int are used to denote the topological boundary and interior. We will say that a Ti space X is w-solid (n a positive integer) if for every point x of X and every open neighborhood U of x there is a connected open neighborhood V of x such that VQU, F is compact, dim V^n, and no (n — 2)-dimensional subset of V separates it. Such a space is locally compact and locally connected. This property is hereditary on open subsets.

Potentials of Families of Unit Masses on Disjoint Jordan Curves

Abstract: Let D be a bounded open set in the complex plane with connected complement, ∂D = C its boundary. When we speak of limits, convergence or approximation in D, we always mean limits, convergence or approximation which are uniform on every compact subset of D.

On the equivalence of a countable disjoint class of sets of positive measure and a weaker condition than total σ–finiteness of measures

Abstract: Let ( X, S ) be a measurable space and S be a σ-algebra of subsets of X . A nonempty class M is said to be a class of null sets if M ⊂ S, M is closed under countable unions of sets and E ∩ F ∈ M whenever E ∈ M and F ∈ S . It is possible to show that such concepts as absolute continuity, singularity and independence of measures can be studied simply by classes of null sets and that similar results can be obtained under the condition that each disjoint subclass of S – M is countable, denoted ( S – M ) C . If ( X, S ,μ) is a measure space then M = { E ∈ S : μ( E ) = O} is a class of null sets of S and S – M the class of all sets of positive measure. We say that a measure μ has the property σ if there exists a sequence of totally finite measures on S such that σ( E ) = for all E ∈ S . This property of measures is weaker than total σ–finiteness of measures. The main result of the present paper is as follows: Let ( X, S ,μ) be a measure space and M = { E ∈ S : μ( E ) = O}. Then ( S – M ) C if and only if μ has the property σ.

A set whose square can map onto a perfect set

Abstract: Assuming the continuum hypothesis, a set X of real numbers will be constructed such that every compact continuous image of X is countable, but XXX admits a uniformly continuous mapping onto the Cantor set. This completes (modulo the continuum hypothesis and the Ulam measure problem) the determination of whether a product or coproduct of m Boolean algebras or fields of sets can have an infinite free, or projective, or injective subor quotient object when the factors do not [1]. The concluding construction (modulo the hypothesis) is embarrassingly easy, but the result is unlike the others [1 ]; this is the only instance of a finite coproduct creating a remarkable subobject (viz. an infinite free subfield), and there is no instance among these of a finite product creating a remarkable quotient. The use made of the continuum hypothesis seems slight. However, Sierpinski has pointed out that the weaker result that there is a set of real numbers of the power of the continuum admitting no continuous mapping onto [0, 1] (C5 in [2]) has not been established without the hypothesis. On the continuum hypothesis, that is a by-product of an important construction of Lusin, the subject of Chapter II of [2]. We shall find X in the Cantor set C considered as an infinite power of the compact group Z2 (written additively). By a lemma of Lavrentiev (Theorem 99 of [3]), any continuous mapping of a subset of C onto [0, 1] can be extended over a Ga subset of C. There are only c (the power of the continuum) Ga sets, and each has only c continuous mappings to [0, 1]. Hence we can index all these mappingsfa by ordinals of smaller cardinal than c, and index similarly the points xa of C. For each fa, the inverse images of points are c disjoint subsets of C each closed in their union; so except for countably many, they are nowhere dense. By the continuum hypothesis, C is not a union of fewer than c such sets. There is no difficulty in building up sets S, T, no subset of either of which is mapped onto [0, 1] by any fa, but with every xa representable as sa+ta, Sa in S and ta in T. Carry along expanding sets Ha disjoint from S, Ka disjoint from T (Ho and Ko empty). Arrived

On the number of complete subgraphs and circuits in a graph

Abstract: Erdos (1964) conjectured that a graph with nk nodes each of valence at least ( n - 1) k contains k disjoint complete subgraphs each with n nodes. This and a related conjecture of Grunbaum are discussed and proved in some special cases. Some similar results are obtained showing the existence of disjoint circuits in graphs with sufficiently high valences.

Harmonic analysis in anSU1,1 basis

Abstract: The theory of harmonic analysis onSL2,σ is developed using anSU1,1 basis. It is found that regarded as an integral kernel, the Fourier transform (FT) of a function defined over the group acquires two additional labels τ representing the two disjoint subspaces needed in the analysis. The representation functions are used to discussSU1,1-bicovariant distributions: we find that the FT cannot be carried out using them alone, because of parametrization difficulties. The problem of equivalence is discussed; conditions upon the transforms obtained; and functions «of the second kind» shown to be just the representation functions themselves. The work is applied to analysis of the two-particle elastic-scattering amplitude at and away fromtμ=0, and nonunitary representations used to give a natural description of daughter trajectories and their properties.

A note on σ-finite invariant measures

Abstract: We consider a one-to-one, bi-measurable, non-singular transformation Φ of a finite measure space onto itself. We obtain two conditions which are equivalent to the existence of a σ-finite measure Μ which is invariant with respect to Φ and equivalent to the given measure m. The first is a generalization of a condition used by Ornstein in his construction of a transformation for which there does not exist any measure Μ as above. The second condition asserts that the entire space is the union of a countable collection {F¦ of subsets, each of which has the following property: if we countably decompose F in such a way that each set in the decomposition of F has an image (under some power of Φ) which is also a subset of F, then the sum of the m-measures of the images is finite (even though the images need not be disjoint).

On Existence of Reversible Adiabatic Surfaces

Abstract: From the notion of equivalence relation and classes induced by them, the sets of points related by reversible processes restricted suitably are shown to be disjoint. By definition, each of thr above sets is arcwise-connected and so connected by HAUSDORFF'S theorem. Adiabatic processes are defined to be those in which, changes of states of a system are entirely by changing the deformation coordinates. First and second laws of thermodynamics are introduced just after CARATHEODORY. Then by arguments of abstract mathematics every points in an r. a. set in an (n + 1)-dimensional space has been shown to gave a neighbourhood homeomorphics to a n-dimensional sphere, i. e., an r. a. set is an n-dimensional manifold having one-to-one correspondence with the deformation space. Thus, the existance of reversible adiabatic surfaces has been established.