Showing papers on "Disjoint sets published in 1973"
IBM1
TL;DR: In this paper, it was shown that the right-hand side is a concave function of the diagonal matrix U such that the sum of the adjacency matrix of the graph plus all the elements of the sum matrix is zero.
Abstract: Let a k-partition of a graph be a division of the vertices into k disjoint subsets containing m1 ≥ m2,..., ≥mk vertices. Let Ec be the number of edges whose two vertices belong to different subsets. Let λ1 ≥ λ2, ..., ≥ λk, be the k largest eigenvalues of a matrix, which is the sum of the adjacency matrix of the graph plus any diagonal matrix U such that the suomf all the elements of the sum matrix is zero. Then Ec ≥ 1/2Σr=1k-mrλr.
A theorem is given that shows the effect of the maximum degree of any node being limited, and it is also shown that the right-hand side is a concave function of U.C omputational studies are madoef the ratio of upper bound to lower bound for the two-partition of a number of random graphs having up to 100 nodes.
693 citations
TL;DR: The main result is that all Borei sets are Ramsey, which was discovered independently by Andrzej Ehrenfeucht, Paul Cohen, and probably many others, but no proof has been published.
Abstract: Definition 1. For a set S and a cardinal κ,In particular, 2ω denotes the power set of the natural numbers and not the cardinal 2ℵ0. We regard 2ω as a topological space with the usual product topology.Definition 2. A set S ⊆ 2ω is Ramsey if there is an M ∈ [ω]ω such that either [M]ω ⊆ S or else [M]ω ⊆ 2ω − S.Erdos and Rado [3, Example 1, p. 434] showed that not every S ⊆ 2ω is Ramsey. In view of the nonconstructive character of the counterexample, one might expect (as Dana Scott has suggested) that all sufficiently definable sets are Ramsey. In fact, our main result (Theorem 2) is that all Borei sets are Ramsey. Soare [10] has applied this result to some problems in recursion theory.The first positive result on Scott's problem was Ramsey's theorem [8, Theorem A]. The next advance was Nash-Williams' generalization of Ramsey's theorem (Corollary 2), which can be interpreted as saying: If S1 and S2 are disjoint open subsets of 2ω, there is an M ∈ [ω]ω such that either [M]ω ⋂ S1 = ∅ or [M]ω ∩ S2 = ⊆. (This is halfway between “clopen sets are Ramsey” and “open sets are Ramsey.”) Then Galvin [4] stated a generalization of Nash-Williams' theorem (Corollary 1) which says, in effect, that open sets are Ramsey; this was discovered independently by Andrzej Ehrenfeucht, Paul Cohen, and probably many others, but no proof has been published.
283 citations
TL;DR: This work has proposed a new approach to the computation of the terminal reliability of a network by transforming a Boolean sum of products into an equivalent form in which all terms are disjoint.
Abstract: Given the set of all simple paths between two nodes in a network, the terminal reliability can be symbolically computed by transforming a Boolean sum of products into an equivalent form in which all terms are disjoint. This new approach seems to be promising in respect to existing 4[ 6 methods both for the exact and for the approximate computation of the terminal reliability.
211 citations
TL;DR: The binding number of a graph G, bind(G), is defined in this article, and some examples of its calculation are given, and some upper bounds for it are proved.
208 citations
TL;DR: A proof is given of the result that, if real n-dimensional Euclidean space Rn is covered by any n + 1 sets, then at least one of these sets is such that each distance d(0) is integers.
78 citations
74 citations
01 Jan 1973
TL;DR: In this article, the authors present an approach that consists of the application of a complex version of the computations of ANOVA to the components of the discrete Fourier transforms of the observed stretches of series.
Abstract: Publisher Summary This chapter discusses the frequency analysis of stretches of time series resulting from runs made in accordance with an experimental design. The analyses proposed may be viewed as extensions of the familiar ANOVA procedures employed when each run of a design leads to the measurement of a real-valued variate. The chapter presents an approach that consists of the application of a complex version of the computations of ANOVA to the components of the discrete Fourier transforms of the observed stretches of series. It presents a case where the observed series form sections of stationary point processes. It also discusses two different procedures available for effectively increasing the number of observations; one can evaluate the finite Fourier transform at a number of frequencies in the neighborhood of λ, or one can evaluate the finite Fourier transform for a number of disjoint stretches of the series.
48 citations
TL;DR: It is proved that D(3v)>=2v+D(v) for every v = 1 or 3 (mod 6), v>=3.
43 citations
TL;DR: Let q be a number all whose prime factors divide integers of the form 2s − 1, s odd, if n = q + 2, the (3n) triples on n marks can be partitioned into q sets, each forming a Steiner triple system.
39 citations
TL;DR: A family T of k-subsets of an n-set such that no more than r have pairwise fewer than s elements in common is maximum (for sufficiently large n) only if T consists of all the k-sets containing at least one of r fixed disjoint s- Subsets.
34 citations
TL;DR: In this article, it was shown that O 5 is an edge disjoint union of two Hamiltonian circuits and a 1-factor and O 6 is a edge disjunctive union of three Hamiltonian ciruits.
TL;DR: In this paper, the Hartree-Fock electric polarizabilities have been calculated for H2, He, Li, Be, LiH, and N2, using a variety of basis sets for each.
Abstract: Hartree-Fock electric polarizabilities have been calculated for H2, He, Li, Be, LiH, and N2. Perturbation theory with all the coupling terms was employed variationally for the first five, using a variety of basis sets for each. Each basis for the perturbation calculations was composed of a zero-order set, plus a first-order set (appropriate to the direction of polarization, for the molecules). The two sets are disjoint to ensure identical zero-order functions for the two molecular polarizability components and, hence, reliable anisotropy values. Nonorthogonal theory as formulated by Das and Duff [Phys. Rev. 168, 43 (1968)], assuming exact zero-order orbitals, was used for LiH. For practical reasons, the nitrogen molecule was treated by the fully self-consistent approach which does not distinguish orders of perturbation. The results for all six species are in very good agreement with experiment, reflecting both a reliable choice of polarization functions and, more significantly, the basic accuracy of the Hartree-Fock method for the static charge distributions, both unperturbed and perturbed by an electric field.
TL;DR: It is shown that the class of Post algebras of finite order n is equationally definable where the only unary operators are the disoint operators Ci, i = 0, 1,..., n -1.
Abstract: It is shown that the class of Post algebras of finite order n is equationally definable where the only unary operators are the disoint operators C i , i = 0, 1,..., n -1.
01 Feb 1973
TL;DR: In this paper, an upper semicontinuous decomposition of En (n_3) into points and tame arcs such that the associated decomposition space is distinct from En was constructed.
Abstract: In this paper we construct an upper semicontinuous decomposition of En (n_3) into points and tame arcs such that the associated decomposition space is distinct from En. The purpose of this paper is to construct an upper semicontinuous decomposition G of En, n>3, into points and tame arcs such that the decomposition space EnIG is topologically distinct from En. For n=3, the example is a modification of R. H. Bing's dog bone space [2]. Unlike the dog bone space, it is not difficult to distinguish our decomposition spaces from Euclidean space. The main idea for the construction of these decomposition spaces was communicated to us by R. D. Anderson during his visit to the University of Texas in March, 1972. Anderson's idea was roughly the following: Take two disjoint wild Cantor sets in En. To each point in one Cantor set correspond a unique point in the other Cantor set and join the two points with an arc that is locally polygonal modulo its end points. The collection of arcs thus obtained can be constructed so that its union is homeomorphic to the product of a Cantor set and an arc. Such arcs for n>4 are tame and, hopefully, if the Cantor sets are wild enough and the pairings are chosen cleverly, the resulting upper semicontinuous decomposition of En will have a decomposition space distinct from En. The difficulty of proving that such decomposition spaces are not En lies in finding a topological property of En not shared by the decomposition spaces. We use the following elementary property of En. THEOREM 1. If C is a Cantor set in En (n _ 3), U is an open set containing C, andf and g are maps from a 2-cell D into En, then there exist maps f' and g' from D into En such that f'lf-1(EnU)=f If`1(EnU), g'lg-1(En U) =gjg`(EnU), fI(f(U)) c U, g'(g-(U)) c U and f'(D ng'(D) nx C_ = Received by the editors July 24, 1972. AMS (MOS) subject class/ifcations (1970). Primary 57A15, 57A10, 54A30; Secondary 57A35, 57A45.
TL;DR: Several theorems on families of sets with the same prescribed intersection numbers are proved, one of which is that the integer m = 2 m ′ + 1 is odd and apart from the labeling of sets and elements there exist exactly m ″ + 1 such families of set.
TL;DR: In this study the authors are only concerned with the minimax over a particular class of merging algorithms, which includes the tape merge algorithm, the simple binary algorithm, and the generalized binary algorithm.
Abstract: Suppose we are given two disjoint linearly ordered subsets A and B of a linearly ordered set C, say A = {a1
TL;DR: For every regular uncountable cardinal κ, it was shown in this article that there is a unique κ-superuniversal metric space of cardinality κ iff κ = \kappa.
Abstract: For every uncountable cardinal κ define a metric spaceS to be κ-superuniversal iff for every metric spaceU of cardinality κ, every partial isometry intoS from a subset ofU of cardinality less than κ can be extended to all ofU. We prove that any such space must have cardinality at least\(2^{\bar \kappa } = \sum _{\lambda< \kappa } 2^\lambda \), and for each regular uncountable cardinal κ, we construct a κ-superuniversal metric space of cardinality\(2^{\bar \kappa } \), It is proved that there is a unique κ-superuniversal metric space of cardinality κ iff\(2^{\bar \kappa } = \kappa \). Several decomposition theorems are also proved, e.g., every κ-superuniversal space contains a family of\(2^{\bar \kappa } \) disjoint κ-superuniversal subspaces. Finally, we consider some applications to more general topological spaces, to graph theory, and to category theory, and we conclude with a list of open problems.
01 Jan 1973
TL;DR: In this article, the elements of a set are assumed to be exhaustive and pairwise disjoint, and the formal expression of this assumption is easily obtained if the elements are interpreted as sentences.
Abstract: Let S be a set. The elements of S will be called ‘states (of the world)’. They are assumed to be exhaustive and pairwise disjoint (the formal expression of this assumption is easily obtained if the elements of S are interpreted as sentences).
01 Feb 1973
TL;DR: In this article, it was shown that every sequence of closed approximable measures defined on the Borelfield of a normal topological space with values in an abelian topological group is Cauchy convergent for all Borel sets.
Abstract: The purpose of this paper is to prove that every sequence of closed approximable measures defined on the Borelfield of a normal topological space with values in an abelian topological group is Cauchy convergent for all Borel sets if it is Cauchy convergent for all regular open sets. In particular every sequence of measures on the Borel-field of a perfectly normal topological space which is Cauchy convergent for all regular open sets is Cauchy convergent for all Borel sets, too. 1. Preliminaries. In this paper a topological group G is always assumed to be abelian. The system of neighborhoods of the zero element of G is denoted by 5i7(O). A sequence an cG, n c N, is Cauchy convergent iff it is Cauchy convergent with respect to the uniformity: {{(a, b) E G x G:a-b E F}:Fe Y (O)}. Let (X, Y7) be a topological space. The closure of a set A c X be denoted by AC, and its interior by int A. A set A c X is regular open iff A=int Ac. The system of regular open sets is denoted by ST. The function T E -T*: int Tc c ST has the following properties: (i) T* * = T*; (ii) Tc: U implies T* c U* (iii) Tr) U= 0 implies T* rn U* = 0; (iv) Tc rn Uc= 0 implies (Tu U)* = T* u U*. Let X be a a-field on X and G be a topological group. A function It: --G is a measure iff for all disjoint sets Ai E M, i E N, the sequence (n=1 [L(Ai))neN converges to [L(UieN Ai). Let Mc X; a measure It: 4--G is Sr-regular iff for each A E X and each F E Y(O) there exists K E , Kc A such that f( r, (A K)): = {(B n (A K)):B E c F. For the case of a Banach space G, regularity in the sense of this definition is the usual regularity, defined in terms of the semivariation. Received by the editors May 3, 1971 and, in revised form, May 23, 1972. AMS (MOS) subject classifications (1970). Primary 60B10; Secondary 28A45.
TL;DR: In this article, the properties of certain numerical characteristics in a normed lattice that characterize its conjugate space are studied. But the properties are not restricted to normals.
Abstract: We study the properties of certain numerical characteristics in a normed lattice that characterize its conjugate space. A typical result is as follows: let X be a KσN-space or a KB-lineal. If for every sequence {xn} ⊂ X of pairwise disjoint positive elements with norms not exceeding 1 we have
$$\mathop {\underline {\lim } }\limits_{n \to \infty } \frac{1}{n} x_1 \vee x_2 \vee \ldots \vee x_n \parallel = 0,$$
then all the odd conjugate spaces X*, X***, ... are KB-spaces.
01 Jan 1973
TL;DR: In this paper, the authors characterize the local behavior of arbitrary functions of two real variables in terms of their cluster sets along various approach curves and find a "small" family F 0 of approach curves such that the cluster set along PO determine total cluster sets.
Abstract: An attempt to characterize the local behavior of arbitrary functions of two real variables in terms of their cluster sets along various approach curves leads to two main problems: (1) Finding a "small" family F0 of approach curves such that cluster sets along PO determine total cluster sets; (2) Finding a "large" family of approach curves along which cluster sets can be preassigned. A nice solution of (1) is found and (2) is partially solved. Conjectures are made concerning a link between (2) and sets which are always of the first category.
TL;DR: Given two continuous Wiener processes, the decomposition into weakly equivalent and mutually singular parts is given.
TL;DR: In this article, the main results of this paper can be formulated as follows: if there exists a countably-centered free ultrafilter, then there are dense in themselves T1-spaces whose product is irresolvable.
Abstract: A space dense in itself is said to be k-resolvable if there exists a system of cardinality k of disjoint dense subsets. The main results of the paper can be formulated as follows: If there exists a countably-centered free ultrafilter, then there are dense in themselves T1-spaces whose product is irresolvable. Any sets X and Y support irresolvable T1-topologies whose product is maximally resolvable. Assuming the continuum hypothesis, an ultrafilter whose cartesian square is dominated by only three ultrafilters is constructed on a countable set. If a set of uncountable cardinality supports an ultrafilter whose square is dominated by exactly three ultrafilters, then its cardinality is measurable. A number of problems are posed. Bibliography: 9 items.
TL;DR: The representation of the Weyl group in the renormalized space for the Yukawa model without cut-off is obtained in this article, and it is shown that this representation is globally disjoint from the Fock representation, but is locally Fock.
Abstract: The representation of the Weyl group in the renormalized space for the Yukawa model without cut-off is obtained. It is shown that this representation is globally disjoint from the Fock representation, but is locally Fock.
01 Feb 1973
TL;DR: Proizvolov as mentioned in this paper proved that the density of a peripherally compact tree-like space equals its weight and showed that any two disjoint closed subsets of such a space can be separated by a closed set.
Abstract: The theorem stated in the title is proved, and examples are given to show the need for both peripheral compactness and tree-likeness. Two recent papers by Proizvolov ([2], [3]) provide much interesting and useful information about the structure of tree-like spaces. Among other facts, Proizvolov proves [2] that the density of a peripherally compact tree-like space equals its weight; each tree-like peripherally compact space X has a unique tree-like compactification which must have the same weight as that of X and [3] any two disjoint closed subsets of a peripherally compact tree-like space can be separated by a closed set; tree-like peripherally compact spaces are hereditarily normal and dyadic tree-like compact spaces are metrizable. Basic to the proofs of all these theorems is the theorem of the title. A paper by Gurin [1] is the reference cited, but it is not easy to see how to derive this assertion from his results. This note is devoted to an elementary proof of the assertion and some relevant examples. A topological space is said to be tree-like if it is connected, Hausdorff and every two points are separated by a third point. A peripherally compact space is a space with an open basis whose elements have compact boundaries. It is easy to see that a peripherally compact tree-like space has an open basis consisting of sets with finite boundaries. Therefore, each peripherally compact tree-like space has an open base consisting of connected sets with finite boundaries (see [4, p. 19]). PROOF OF THE THEOREM. Let X be a peripherally compact tree-like space and let p and q be two points of X. Define M to be the subset of X consisting of p, q and the points which separate p from q. Arguments given by Whyburn [4, pp. 42, 43 and 51] show that M is compact and Received by the editors September 6, 1971. AMS (MOS) subject class.fications (1970). Primary 54D05, 54D30, 54F15.
TL;DR: The constructible orthomodular lattices defined by Janowitz in this article are embeddable into Boolean lattices, where the subdirect products are taken in the class of orthomodes.
Abstract: It is our purpose to show that the constructible orthomodular lattices defined by Janowitz in [2], are embeddable into Boolean lattices. In fact they are subdirect products of Boolean lattices, where the subdirect products are taken in the class of orthomodular posets. We shall make these notions precise. Other concepts, such as disjoint sum and constructible lattice, are defined in [1] and [2].
01 Jan 1973
TL;DR: The union of two disjoint sets of interpolation need not be of the same type, even if every finite subset of the set is of the type with bounded constant.
Abstract: The union of 2 disjoint sets of interpolation need not be of interpolation even if every finite subset is of interpolation with bounded constant.