Showing papers on "Disjoint sets published in 1970"

Some Applications of Almost Disjoint Sets

Abstract: Publisher Summary This chapter discusses some applications of disjoint sets. It is shown assuming the consistency of Zermelo-Fraenkel (ZFC), that there are models of ZFC in which there is a non-constructible set of integers, a , which is a Π 1 2 singleton, and in which every constructible set of integers is recursive. The first consistency result is due to Kripke and Martin. Using the minimal degree construction of Sacks, they constructed a model of ZF, in which there was a non-constructible Δ 1 4 real; the axiom of choice did not hold in their model. The chapter develops a method, due to Solovay, of producing subsets of ω in Cohen extensions. It is indicated that the modifications, due to Jensen, are needed to prove the full result.

Disjointness and weak mixing of minimal sets

Abstract: In [3, Problem G, p. 34] the following problem is proposed: characterize the class o' of all flows which are disjoint from every distal flow. We show here that ai consists precisely of the weakly mixing minimal flows. In arriving at this conclusion we make use of a result (Corollary to Theorem 1) which has been arrived at independently by Keynes and Robertson [4, Theorem 3.4, p. 366]. In the following all transformation groups will be assumed to have compact Hausdorff phase spaces. For any unexplained notation or terminology the reader is referred to [1 ] and [3 ]. This research is part of the author's doctoral dissertation prepared at Yale University under the guidance of Professor S. Kakutani, whose assistance the author gratefully acknowledges. We say that a transformation group (X, T) is weakly mixing if given nonempty open subsets A, B, C, D of X there is tET such that AtnCC0 and BtnDS 0.

Connectivity considerations in the design of survivable networks

Abstract: The problem of constructing networks that are "survivable" with respect to branch damage is considered. The networks are modeled by linear graphs and a square symmetric "redundancy" matrix R'=[r'_ij] is specified. Algorithms are given to construct an undirected graph G with a minimum number of branches such that 1) G contains no parallel branches, and 2) for all i, j there are at least r'_ij branch disjoint paths between the ith and jth vertices. These algorithms are complicated but may easily be applied to construct graphs with several hundred vertices.

Connectivity Considerations in Design Survivable Networks

Abstract: The problem of constructing networks that are "survivable" with respect to branch damage is considered. The networks are modeled by linear graphs and a square symmetric "redundancy" matrix R' = (rii) is specified. AIgorithms are given to construct au undirected graph G with a minimum number of branches such that 1) G contains no parallel branches, and 2) for all i, i there are at least r$i branch disjoint paths between the ith and $h vertices. These algorithms are complicated but may easily be applied to construct graphs with several hundred yertices.

Survivable communication networks and the terminal capacity matrix

Abstract: The problem of constructing networks that are "survivable" with respect to branch damage is considered. A square redundancy matrix R = [r_{i,j}] is specified. Algorithms are given to construct a graph with minimum number of branches so that for all i, j there are 1) at least r_{i,j} undirected branch disjoint paths between the ith and the j th vertices, or 2) there are exactly r_{i,j} undirected branch disjoint paths between the ith and the jth vertices. These algorithms are closely related to the optimal realization of terminal capacity matrices. Two of the algorithms are extended to the optimal realization of terminal capacity matrices for symmetric or pseudosymmetric graphs.

Exponential representations of the canonical commutation relations

Abstract: A class of representations of the canonical commutation relations is investigated. These representations, which are called exponential representations, are given by explicit formulas. Exponential representations are thus comparable to tensor product representations in that one may compute useful criteria concerning various properties. In particular, they are all locally Fock, and non-trivial exponential representations are globally disjoint from the Fock representation. Also, a sufficient condition is obtained for two exponential representations not to be disjoint. An example is furnished by Glimm's model for the :Φ4: interaction for boson fields in three space-time dimensions.

A direct proof that a linearly ordered space is hereditarily collectionwise normal

Abstract: Although it appears well known that a linearly ordered space is completely normal (=hereditarily normal), most available proofs (in, for instance, [1] and [2]) are very indirect. In this paper we present a direct proof of a stronger theorem, namely that the interval topology is hereditarily collectionwise normal.2 If X is linearly ordered, we will call a set SCX convex if a, b E S and a

A note on quantum probability spaces

Abstract: In general the collection cf sets closed with respect to countable disjoint unions and with respect to the complementa- tion, generated by a given collectien A does not coincide with the -field generated by A. In the present paper two necessary and sufficient conditions for the equality of the last mentioned systems are given. The coincidence of the above systems in cases when A is the collection of all open sets in a topological space is obtained as a corellary.

Problems Concerning Intertheory Relations

Abstract: As with other metascientific problems, scientists as well as philosophers have contributed to the literature on the relations among theories. And, as usual, the two groups have done their best to ignore each other. In this case they have also managed to ignore a third group, which happens to be the most articulate of all: namely the logicians and mathematicians who have created the calculus of theories, model theory and categories, and have studied the formal relations among hypothetico-deductive systems. The unfortunate result of this lack of communication among the three groups is that we have three disjoint sets of studies. It is an urgent task of metascientists to intertwine these three separate threads with a view to producing a unified picture of intertheory relations.

An elementary proof of the triod theorem

Abstract: PROOF. Let cR = { Ri }I be a countable collection of open discs in the plane which forms a base for the usual topology. Let T be a simple triod in the plane and let R(T, 1) be an element of 6I which contains 0 such that R(T, 1) n {X, Y, Z} e0. Let XT, YT, and ZT be the first points of the sets (OX)Bd (R(T, 1)), (OY)flBd (R(T, 1)) and (OZ)fBd(R(T, 1)), respectively. The continuum (OXT)U(O YT) Ud(OZT) is a simple triod which is a subset of T. It follows from the Jordan Curve Theorem that R(T, 1)-[(OXT)U(OyT)U(OZT)] =I(T, 2) UI(T, 3) UI(T, 4), where {I(T, n) }4=2 are pairwise disjoint open sets. Let { R(T, n) } 4n2be elements of (R contained in { I(T, n) } 42, respectively. The triod T is thus associated with a quadruple of elements of (R. Let T' be a simple triod such that T'C T= 0 and let {R(T', n) }4nbe the quadruple of elements of (R associated with T'. Since T'(-) T = 0, it follows that if R(T, 1) =R(T', 1) then two of the sets {I(T', n) 2 must be a subset of one of the sets {I(T, n)}42. Because of the method used to select the quadruple associated with a given triod, it follows that the two quadruples {R(T, n) }4_ are not identical. It follows that any collection of mutually exclusive simple triods can be placed in 1-1 correspondence with a subset of the collection of all quadruples of elements of (R and hence is a countable collection.

Lower Bounds for the Disk Packing Constant

Abstract: An osculatory packing of a disk, U, is an infinite sequence of disjoint disks, f Un }, contained in U, chosen so that, for n _ 2, U,, has the largest possible radius, r,,, of all disks fitting in U\(U1 U... U U.-1). The exponent of the packing, S, is the least upper bound of numbers, t, such that E r = 0. Here, we present a number of methods for obtaining lower bounds for S, based on obtaining weighted averages of the curvatures of the U,,. We are able to prove that S > 1.28467. We use a number of well-known results from the analytic theory of matrices.

Two-dimensional polynomial splines

Abstract: {~i} and their derivatives of order k or less converge uniformly to u and its corresponding derivatives. We show that these splines can be used to obtain a numerical method for solving elliptic boundary value problems. The method of construction can be easily extended to higher dimensional Euclidean spaces. For one dimensional splines, see, for example, references [2] and [3JWe denote by G a bounded open domain of class C o (see reference It] for definition) in the Euclidean plane with boundary ~G and closure G. For a function uEC/(G) denote by Illuilli, the maximum norm max max l D~u(~)] and denote by the norm (f ~, I D~u(~)]ZdY)t, Here ~ = (~1, ~)is the multiple G I~l 1 and an integer p >--_ 0. Subdivide the domain ~ by a grid into a finite number of subdomains such that each subdomain is a polygon with infinitely differentiable curvilinear sides. We call any such subdomain a cell. Let {Si} be the set of grid lines forming the sides of cells. We require that the interior of any grid line Si must not contain a vertex. Let sup (Length of S¢) -- h. We denote by {Gt} the disjoint collection of interiors of the cells and let G 0---U Gi. Assume that each grid line, S t, is a non-singular curve, defined by an equation of the form x = fi (~3

Weak convergence of conditioned sums of independent random vectors

Abstract: Conditions are given for the weak convergence of processes of the form (X"(i) | X"(l) 6 £■") to tied-down stable processes, where X"(i) is constructed from normalized partial sums of independent and identically distributed random vectors which are in the domain of attraction of a multidimensional stable law. The con- ditioning events are defined m terms of subsets E" of Rd which converge in an appro- priate sense to a set of measure zero. Assumptions which the sets E" must satisfy include that they can be expressed as disjoint unions of "asymptotically convex" sets. The assumptions are seen to hold automatically in the special case in which E" is taken to be a "natural" neighborhood of a smooth compact hypersurface in R". 1. Introduction and notation. Empirical distribution functions have been widely studied in probability theory and statistics. Often, the fact that they may be represented as conditioned sums of independent random variables has played an important role in these investigations. This has led to an interest in the behavior of these sums under various forms of conditioning. It is the purpose of this paper to present conditions under which certain stochastic processes, obtained from the partial sums {Sk, k^n} of independent identically distributed random vectors in the domain of attraction of a stable law by conditioning on information concerning Sn, converge to a limiting process.

Spaces of functions of one variable, analytic in open sets and on compacta

Abstract: is the space of functions analytic on the compactum of the extended complex plane with the usual locally convex topology; ; .The following assertions are proved: 1.For the spaces and to be isomorphic, it is necessary and sufficient that the set have no more than a finite number of connected components and that the compactum be regular (i.e. the Dirichlet problem is solvable in for any continuous function on ). 2.For and to be isomorphic, it is necessary and sufficient that the logarithmic capacity of the compactum be equal to zero. 3.For and to be isomorphic, it is necessary and sufficient that the compactum be represented in the form of the sum of two disjoint nonempty compacta, one of which has zero capacity and the other of which is regular and has a complement consisting of no more than a finite number of connected components. Dual results are obtained for the space , where is an open set.Bibliography: 20 entires.

Communication by smooth high order composites of trigonometric product functions

Abstract: Member functions of certain disjoint sets of harmonically related trigonometric product functions (the term ''''disjoint'''' is used herein to describe sets which have no common member functions and relatively distinct class properties K) are combined for transmission by simultaneously selecting plural subsets of a first one of the sets, in fundamental half-periods, and superposing the members of each subset by linear addition to form subset composites. These are individually multiplied (''''upconverted'''') by members of other sets and superposed in groups. Such cascaded multiplications and superpositions are continued convergently to provide at one central terminal a comprehensive high order composite transmission waveform which has smooth outline and contains, in a highly distinguishable form, all of the binary intelligence utilized in the initial selections of subsets of the first set. At receiving apparatus the composite transmission waveform is decomposed (down-converted) in divergently cascaded stages of multiplications by locally synthesized functions. Plural sets of higher order product waveforms, issuing from the last stages of such multiplication in parallel, are separately integrated over fundamental half-period intervals. The integrand functions correspond to distinct sums of products of pairs of high order trigonometric product functions having identical class and order. The terms of any sum all have distinct binary coefficients. The product functions form an orthogonal set with associated order and class properties respectively relating to sums and maxima of respective order and class properties of the disjoint sets containing the transmission components. Each integrand sum representation contains a unique term in which the paired product functions are identical and all other terms have unmatched functions. The function in the matching term is different for each integrand. Hence with appropriate timing of integration sampling and resetting functions a unique set of binary state pulse functions, which correspond to the binary coefficients of the matching terms of respective integrands, is sampled at outputs of respective integration stages. Normally these pulse functions correspond identically to the binary selection pulses utilized in the pretransmission subset selections.

Letter to the Editor-A Note on Pattern Separation

Abstract: This note describes a linear programming method of separating two, not necessarily disjoint convex polyhedral sets in En. If the sets are disjoint, a strictly separating hyperplane is generated that maximizes and equalizes the distance between the sets and the hyperplane. If the sets intersect a hyperplane is generated that minimizes the maximum error.

Compactness in l., spaces

Abstract: Conditions for compactness in LOO(S, 2, A) are known when A(S) is finite [1, p. 297]. The purpose of this note is to state a compactness criterion which does not depend on A(S). The proof uses the standard diagonal procedure. It should be noted that the criterion is not a necessary one unless the underlying space S is a-finite (see remark at the end). PRELIMINARIES. We assume that (S, 2, ,u) is a positive measure space [1, p. 126]. The triple (S, 2, ,u) is said to be a-finite whenever there is a sequence of sets in z of finite ,u-measure whose union is S. A finite decomposition (of S) is a finite collection of pairwise disjoint sets in 2 whose union is S. A set NC2 is called locally ,u-null if ,u(EO N) = 0 for all E 2 such that itu(E) a} C D. Let

An imbedding problem

Abstract: If H is an uncountable collection of pairwise disjoint continua in En, each homeomorphic to M, then there exists a sequence from Hconverging homeomorphically to an element of H. In the present paper the auithors show that if { M, } is a sequence of continua in En which converges horneomorphically to Mo and such that for each i, Mi and Mo are disjoint and equivalently imbedded, then there exists an uncountable collection H of pairwise disjoint continua in En, each homeomorphic to M. For n=2, 3, and n25 it is shown that one cannot guarantee that the elements of H have the same imbedding as M1o. Introduction. Let Ml be a continuum in En. It is well known that if H is an uncountable collection of pairwise disjoint continua in En, each homeomorphic to M, then there exists MOCH and a sequence M1i } from H such that the sequence AM, } converges homeomorphically to Mo, that is, for E> 0 there exists N such that i ? N implies the existence of a homeomorphism hi of Mo onto Mi which moves no point more than E. The following question is immediately raised: suppose Mo is a continuum in En and { Mi } is a sequence of pairwise disjoint continua in En such that { Mi } converges homeomorphically to Mo and for each i, MinM0 = ,0. Does there necessarily exist an uncountable collection H of disjoint continua in En such that each M' in H is homeomorphic to Mlo? We remark that similar questions are discussed in [41]. The purpose of this note is to answer the above question in the affirmative under the additional condition that the Mi, i=O, 1, 2, , are equivalently imbedded in En and to note that the answer to the question is negative for n = 2, 3, and n > 5 under the condition that the elements of H have the same imbedding as Mo. Two continua M, and M2 are said to be equivalently imbedded in En provided there exists a homeomorphism of En onto En which carries M, onto 312. Finally, we note that if a nontopological imbedding property is imposed on the collection H in E2, then the answer is again negative. Received by the editors November 14, 1969. AMS Subject Classifications. Primary 5422, 5425, 5478.

Recognizing certain factors of

Abstract: It has been proved that for certain peculiar decomposition spaces Y of euclidean 3-space E3, YX El is homeomorphic to euclidean 4-space, E4. In this paper we prove that if a decomposition space Y of E3 is generated by a trivial defining sequence whose elements are cubes with handles, and this sequence can be replaced by a toroidal defining sequence, then YXE1 is homeomorphic to E4. For each natural number i, let Ai be a disjoint, locally finite set of cubes with handles imbedded in EK; let A>=U{aIa(Ai}. The components of X = fA7 are the nondegenerate elements of an upper semicontinuous decomposition G = G( { A }) of El and { A } will be called a defining sequence for G. If G({Bi})=G({Ai}) we shall say {A } can be replaced by {Bi}. In [1] the authors conjectured that if the defining sequence {A } is trivial, then E3/G is a factor of E. Theorem 2 below is a partial solution to that conjecture. If each Ai is a set of solid tori, then we say {As} is toroidal. It is our contention that if the defining sequence {A i} is trivial and can be replaced by a toroidal defining sequence {Bi}, then E3/G is a factor of E4. The main distinction to be made here is that { B i} need not be trivial. For related results see [2], [3], and [4]. A close examination of the proof of Theorem 2 of [I] will show that the requirement that { A } be trivial, i.e., that each inclusion j: A1iCA* be null homotopic could be replaced by the requirement that for each i, j:XCA* be null homotopic. This is stated in the following theorem. THEOREM 1. If {AI} is a toroidal defining sequence for G, and for each i, the inclusion j: XCA* is null homotopic, then E3/G is a factor of E4. It is easy to show that if a trivial defining sequence {Ai} can be replaced by {Bi}, then for each i, the inclusion j:XCB* is null homotopic. This can be seen by observing that if T(E& then TC'XC Int (T), TCAX is compact, and thus for some k there exists a finite set {S1, . . ., Sm} CAk such that TnXCUSiCT. Then since X is null homotopic in A* and the Si are components of A*, Received by the editors November 20, 1969. AMS 1969 subject classifications. Primary 5478, 5701, 5705.

On the decomposition of measures with applications to continuum physics

Abstract: In this note we show that some concepts of modern measure theory may be relevant to the formulation of theories in continuum physics. In particular we observe that a large class of measure spaces possesses a decomposition property which is useful in applications. This decomposition property is obtained as a corollary to a theorem of Kelley [10] concerning the "Lebesgue decomposition property," which result is based upon the investigations of Segal [3]. The mathematical notation and definitions (following Kelley [10]), are presented first; then a Lemma and the Decomposition are given. The relationship between these results and physical theories is presented in a series of remarks at the end . As a preliminary we establish some useful definitions and notation. I Let be a set and let sg/a be a ring of subsets of M whose union is ~ . We further assume that d / ~ has the property: if {d~ } is a sequence of sets in d¢ "~ such that if ~¢. _ d E d¢ '~, for all n and some d in d /~ , then U . d . ~ d¢ 'a~. By a measure v in we mean a non-negative, finite valued, countably additive set function defined on ~ ' a with the following additional property: if {~ ' . } is a disjoint sequence in ~.¢/a such that g . v ( d . ) converges, then U . d . a jfa~.2 A subfamily q / i n ~ ' a is called a a-ideal in d¢ 'a~ if and only if it is a ring of sets, the intersection of a member of q /wi th a member o f ~ / a is a member of q/, and any member of~¢¢ '~ which is a countable union of sets in ~ is itself a set in q/. A set ~ _ ~ is (locally) measurable if and only if ~¢ n ~ e J / fa for all ~¢ in J/¢'~. We denote the class of measurable sets in ~ by ~//¢ and observe that J g is a or-algebra of subsets of ~ containing ~ ,a .3 I f~¢ and cg are in J r , we write ~¢Ro~ if and only i f c ~ ~ ¢ contains no set

Measure-theoretic properties of arbitrary point sets.

Abstract: Consider an arbitary point set S on the real number line or in Euclidean n-space (the limitation to Euclidean space is unessential). The set S has an interior Lebesgue measure mi(S) and an exterior Lebesgue measure me(S). There are the following well known inequalities for two disjoint sets S1 and S2: mi(S1US2) [unk] mi(S1) + mi(S2), me(S1US2) ≤ me(S1) + me(S2). These state the superadditivity of interior measure for disjoint sets, and the subadditivity of exterior measure. The question is posed as to what conditions, besides these, are there on the six quantities mi() and me() for the disjoint sets S1,S2 and for their union S1US2. The present paper obtains a complete set of conditions on these six quantities. These are in the form of inequalities. The variability can be expressed in terms of six independent nonnegative quantities, and the six quantities mi() and me() for the disjoint sets S1, S2 and for S1US2 can be expressed as suitable sums of these. There appears the interesting property that the „average measure” ma(S), defined by ma(S) = (mi(S) + me(S))/2, is subadditive. Also, any linear combination αmi(S) + βme(S) is discussed.

The $n$-separated-arc property for homeomorphisms

Abstract: Let f be a function defined in the open unit disk D whose range is in the Riemann sphere W, and let C denote the unit circle. We show that if f is a homeomorphism of D onto a Jordan domain, then the set of points pE C at which f has the n-separatedarc property (n >2) is a subset of the set of ambiguous points of f and is thus countable. Let p be a point on C. If ois an arc in D for which oUJ{p} is a Jordan arc, then ois said to be an arc at p and the cluster set C(f, p, o-) of f at p along ais defined to be the set of all points wE W for which there exists a sequence { Zk } of points on awith Zk-*p and f(Zk)-*W. We say thatf possesses the n-separated-arc property at p, for some integer n (n > 2), if there exist n mutually disjoint arcs o-1, * * *, On at p for which the intersection of all n of the sets C(f, p, oj) (j= 1, * * *, n) is empty while the intersection of any n-1 of them is nonempty. A point p E C at which f has the 2-separated-arc property is called an ambiguous point of f. By a Jordan domain we mean an open connected subset of W whose boundary is a Jordan curve. Since each analytic homeomorphism of D onto a Jordan domain is necessarily continuous on DUJC, the set of points at which such a function has the n-separated-arc property (n > 2) is empty. The purpose of this note is to determine the nature of this set for nonanalytic homeomorphisms. THEOREM. Let f be a homeomorphism of the open unit disk D onto a Jordan domain U. If f has the n-separated-arc property (n >2) at the point pEC, then p is an ambiguous point of f. PROOF. Let o-1, * * *, a-, be n mutually disjoint arcs at p for which the intersection of all n of the sets C(f, p, aj) (j = 1, * * * , n) is empty while the intersection of any n -1 of them is nonempty. Then C(f, p,i 1) CC(f, p, 0n) = ai U a2 Received by the editors May 8, 1969. AMS Subject Classification. Primary 3062. Key Phrases. Homeomorphism of the disk, n-separated-arc property, ambiguous point. 1 This paper is part of the author's doctoral thesis written at Michigan State University under the direction of Professor Peter Lappan.