Showing papers on "Disjoint sets published in 1975"

Efficiency of a Good But Not Linear Set Union Algorithm

Abstract: TWO types of instructmns for mampulating a family of disjoint sets which partitmn a umverse of n elements are considered FIND(x) computes the name of the (unique) set containing element x UNION(A, B, C) combines sets A and B into a new set named C A known algorithm for implementing sequences of these mstructmns is examined It is shown that, if t(m, n) as the maximum time reqmred by a sequence of m > n FINDs and n -- 1 intermixed UNIONs, then kima(m, n) _~ t(m, n) < k:ma(m, n) for some positive constants ki and k2, where a(m, n) is related to a functional inverse of Ackermann's functmn and as very slow-growing

Social Choice Scoring Functions

Abstract: Let a committee of voters be considering a finite set $A = \{ {a_1 ,a_2 , \cdots ,a_m } \}$ of alternatives for election. Each voter is assumed to rank the alternatives according to his preferences in a strict linear order. A social choice function is a rule which, to every finite committee of voters with specified preference orders, assigns a nonempty subset of A, interpreted as the set of “winners”. A social choice function is consistent if, whenever two disjoint committees meeting separately choose the same winner(s), then the committees meeting jointly choose precisely these winner(s). The function is symmetric if it does not depend on the names of the various voters and the various alternatives. It is shown that every symmetric, consistent social choice function is obtained (except for ties) in the following way: there is a sequence $s_1 ,s_2 , \cdots $, $s_m $ of m real numbers such that if every voter gives score $s_i $ to his ith most preferred alternative, then the alternative with highest score ...

On edge‐disjoint branchings

Abstract: : Edmonds has given a complicated algorithmic proof of a theorem characterizing directed graphs that contain k edge-disjoint branchings having specified root sets. Tarjan has described a conceptually simple and good algorithm for finding such branchings when they exist. Tarjan's algorithm is based on a lemma implicit in Edmonds' results. A simple direct proof of this lemma is given, thereby providing a simpler proof of Edmonds' theorem and a simpler proof that Tarjan's algorithm works.

Isolated invariant sets for flows on vector bundles

Abstract: This paper studies isolated invariant sets for linear flows on the projective bundle associated to a vector bundle, e. g., the projective tangent flow to a smooth flow on a manifold. It is shown that such invariant sets meet each fiber, roughly in a disjoint union of linear subspaces. Isolated invariant sets which are intersections of attractors and repellers (Morse sets) are discussed. We show that, over a connected chain recurrent set in the base space, a Morse filtration gives a splitting of the projective bundle into a direct sum of invariant subbundles. To each factor in this splitting there corresponds an inferval of real numbers (disjoint from those for other factors) which measures the exponential rate of growth of the orbits in that factor. We use these results to see that, over a connected chain recurrent set, the zero section of the vector bundle is isolated if and only if the flow is hyperbolic. From this, it follows that if no equation in the hull of a linear, almost periodic differential equation has a nontrivial bounded solution then the solution space of each equation has a hyperbolic splitting.

A Fast Algorithm for Reliability Evaluation

Abstract: An algorithm is developed to obtain a simplified reliability expression for a general network. All the success paths of the network are determined; then they are modified to be mutually disjoint. The reliability expression follows directly from the disjoint paths. The algorithm is easy and computationally economical. The reliability expression involves fewer terms and arithmetic operations than any of the existing methods. This is an advantage, considering the size of the system a reliability engineer normally handles and the frequency with which the expression is used for reliability studies.

Packing and covering of the complete graph with $4$-cycles

Abstract: The maximal number of pairwise edge disjoint 4-cycles in the complete graph Kn and the minimal number of 4-cycles whose union is Kn are determined.

On the maximum number of disjoint triple systems

Abstract: Let D(λ;v) denote the maximum number of mutually disjoint S(λ;2,3,v). We prove that D(2;v)=v−2/2 for all v ≡ 0 or 4 (mod 6), v ≡ 0 and that D(6;v)=v−2/6 for all v ≡ 2 (mod 6).

λ-Terms as total or partial functions on normal forms

Abstract: In this paper the set of λ -terras is split into 2ω+1 disjoint classes Ph (−ω≤h≤ω). This classification takes into account the meaning of a λ-term F as function on normal forms, and more precisely: 1 iff when auccessively applied to any number of normal forms it gives a λ-term without normal form 2 (0ω) iff when successively applied to h arbitrary normal forms it gives a λ-term which possesses normal form, but there exist h+1 normal forms X1,...,Xh+1 such that FX1...Xh+1 possesses no normal form 4 iff when successively applied to any number of normal forms it gives a λ-term which possesses normal form.

Construction of large sets of almost disjoint Steiner triple systems

Abstract: A Steiner triple system (briefly STS) is a pair (S, t) where S is a set and t is a collection of 3-subsets of S (called triples) such that every 2-subset of S is contained in exactly one triple of t. The number |S| is called the order of the STS (S, t). It is well-known that there is an STS of order v if and only if v = 1 or 3 (mod 6). Therefore in saying that a certain property concerning STS is true for all v it is understood that v = 1 or 3 (mod 6).

The Monodromy of a Degenerating Family of Curves

Abstract: In this paper we analyze the homological monodromy of a degenerating family g: V-*D of complex curves over the disk, in particular developing a necessary and sufficient criterion for it to be of finite order. This criterion is in terms of what we call the number of cycles in the special fiber g-l(0). A one-dimensional analytic space contains cycles when its components intersect to form circular chains. We look at this in two cases, the special fiber of the degenerating family, and the exceptional locus in the resolution of a surface singularity. If the degenerating family is normal, that is, the special fiber is a reduced curve with at most ordinary double points, we show by a topological argument that the monodromy is the identity or of infinite order, the latter occuring precisely when g-l(0) contains cycles (Theorem 1). Mumford's semistable reduction theorem asserts that any degenerating family is dominated by a normal degenerating family (Proposition 2). The main result of this paper (Theorem 2) asserts that if the special fiber of an arbitrary degenerating family contains cycles, then its monodromy is of infinite order, and that its monodromy is of finite order precisely when the normal degenerating family dominating it contains no cycles. To prove this, we completely analyze dominating families (Proposition 1). Finally, we specialize to the case where the degenerating family consists of local curves, that is, the local monodromy of an analytic function f (x , y) of two variables. We show that it is of infinite order if the singularity f (x , y ) z" = 0 contains cycles for some n > 0, and that it is of finite order precisely when f ( x , y ) z N = 0 contains no cycles, where N is computed from the multiplicity sequence of f. I_~ [6] proved in the local case that the monodromy is of finite order when f is irreducible; A'Campo [I] provided another proof, and the first example of infinite order with f reducible. We present another example (Example 3). The methods of this paper allow one to easily decide whether the monodromy of any degenerating family of curves (local or not) is of finite order: One only need blow up at points of curves on the non-singular surface V, keeping track of multiplicities. Another criterion for infinite order by way of knot theory is presented in [3], and generalized in [13]. Throughout this paper we will use the term curve (surface) for a pure one (two) dimensional complex analytic space. We include the possibility that a curve or surface Z have boundary, namely that Z is a topological space with two subsets S z and B z such that (i) S z and B z are disjoint, (ii) Z S z is a smooth manifold with boundary B z, (iii) Z B z is a complex analytic space with singular

A Local Time for a Storage Process.

Abstract: : A storage system subject to a general release rule and an additive input process is considered. If (X sub t) is the content at time t, then the set X = X sub t; t> or = 0) is a standard Markov process, and the concern is the local time at x = 0 of this process X. Depending on the parameters of the system, namely the release rule and the Levy measure of the input process, there are four cases possible. In terms of the set E = (t : X sub t = 0), these are as follows: E is the union of countably many isolated points; E is the union of countably many disjoint intervals; E is a Cantor set (a perfect set with an empty interior) with positive Lebesgue measure; E is a Cantor set with Lebesgue measure zero. The last is the most interesting case, and the construction of the local time then is the main result. Local times in other cases are also considered along with time inverses and hitting times. (Author)

Integrals for one-sided confidence bounds: A general result

Abstract: SUMMARY Certain multivariate normal and t integrals have been calculated by Bohrer and Francis and others to find confidence bounds in regression analysis. Their technique of partitioning Rn into disjoint regions is generalized to bounds based on arbitrary polyhedral cones. A geometrical solution for cones in R3 results. The probabilities are also those arising from tests for a multivariate mean against alternatives lying in such cones, developed by Bartholomew and others.

Use of a new technique in homogeneous continua

Abstract: Recently Ungar has employed a theorem due to Effros [3] in the study of homogeneous continua. It is the purpose of this paper to use the same theorem to clarify and simplify Bing's proof that the only plane homogeneous continuum containing an arc is a simple closed curve [ 1 ]. Preliminary definitions and lemmas. Let M denote a continuum (= compact, connected, metric space). Then M is said to be homogeneous if any point x of M can be moved to any other point y of M by a homeomorphism of M onto M. The simplest examples are the circle, the torus and the Hilbert cube. Effros has recently proved a theorem which yields the following useful corollary: If M is a homogeneous continuum and e is a positive number, there exists a positive number b such that if x, y G M and d(x,y) < b there exists a homeomorphism h of M onto M such that h takes x to y and moves no point more than e[i.e., d(h(p),p) •< • for p G M]. This gives us a precision that Bing lacked and almost makes the proof of his theorem trivial. However, some care is still required and the following well known (or easily established) properties of plane homogeneous indecomposable continua are helpful. Let M denote a homogeneous indecomposable continuum lying in the plane E. (a) M is hereditarily unicoherent (otherwise M would contain a proper subcontinuum separating the plane and hence each of the uncountably many disjoint composants would contain a homeomorphic copy). (b) M is atriodic (for much the same reason as in (a) since the plane cannot contain uncountably many disjoint triods). (c) If two arcs A and B of M intersect then A U B is an arc, and in particular where they have a common end point and intersect somewhere else, then one of them is a subset of the other.

On Haar Measure in Locally Compact T 2 Spaces

Abstract: Let G be a group of homeomorphisms of a locally compact Hausdorff space X onto itself. If for each disjoint pair of compact sets there is a non-empty open set, no image of which meets both of the compact sets simultaneously, then there is a non-trivial regular Borel measure on X which is invariant under the action of G (a Haar measure). In particular if the group is equicontinuous, there exists such a (Haar) measure; in this case, Haar measure is unique if and only if the group is weakly transitive.

Free i-groups and vector lattices

Abstract: The purpose of this paper is to present three somewhat disparate results on free objects in three different classes of λ-groups. The first is that no proper ideal of a finitely generated free vector lattice can itself be a free vector lattice. Second, each free abelian l group is characteristically simple. The third result is that each disjoint subset of a free (non-abelian) l group is countable.

Rearrangements of conditionally convergent real series with preassigned cycle type

Abstract: For any conditionally convergent real series, any real number r, and any infinite cycle type, there is a permutation of the indices, of the given cycle type, which makes the series converge to r. One result usually mentioned and sometimes proved when conditionally convergent series are first discussed is Riemann's theorem that for any real number r there is a permutation of the indices so that the renumbered series converges to r. The usual method of renumbering gives no control over the type of the permutation (in the sense of disjoint cycle decomposition). John Kelly has asked whether we may, for example, take the permutations to be of order 2. In this note we investigate whether the job can be done with a permutation with prescribed cycle type, and show that, subject to the obvious condition that the cycle type allow for moving infinitely many indi-

Imbedding compact 3-manifolds in ³

Abstract: We show that in a large finite disjoint collection of compacta in a closed orientable 3-manifold there is a compactum that imbeds in E3. However, given a closed 3-manifold M3, there is a pair of compact 3-manifolds (L, N) such that L contains infinitely many disjoint copies of N but N does not imbed in M3. It is sometimes useful to know that a subset of a 3-manifold can be imbedded in E3. For example, it is sometimes desirable to use a linking argument in a neighborhood of an element of a decomposition. We will prove that in a large enough disjoint collection of compacta in a closed orientable 3-manifold there is a compactum with a neighborhood that imbeds in E3. We will then show that no closed 3-manifold is an imbedding manifold for nonorientable 3-manifolds in the sense that E3 is for orientable 3-manifolds. Definitions. A surface is a closed connected 2-manifold. A surface S is incompressible in the 3-manifold M3 means (1) S is not a 2-sphere and if D C M3 is a disk with D n S = BdD, then BdD bounds a disk in S; or (2) S is a 2-sphere that bounds no 3-cell in M3. The surfaces Si and 52 are said to bound a parallelity component U of M3 and the disjoint collection of surfaces 2J Sn} if U is a component of M3 n nS U is homeomorphic to S x I, and BdU = S1U 2 We assign some constants to a given compact 3-manifold M3. Let B31(H1(M3, G)) be the rank of H1(M3, G). When G = Z we sometimes write just /B1. Also let a be the maximum number of disjoint nonparallel incompressible 2-sided surfaces in M3. We know a exists by [1]. Theorem 1. Suppose M3 is a closed orientable 3-manifold and ICJ, ' Ca+,81+2} is a disjoint collection of compacta in M3. Then there is a C. with a neighborhood imbeddable in E3. Proof. Let U1,..., U be a disjoint collection of compact polyReceived by the editors November 26, 1973 and, in revised form, January 10, 1974. AMS (MOS) subject classifications (1970). Primary 57A10.

Doeblin's and Harris' theory of Markov processes

Abstract: Our notation and definitions are taken from (Chung, K. L.: The general theory of Markov processes according to Doeblin. Z. Wahrscheinlichkeitstheorie und verw. Gebiete 2, 230–254 (1964)). A closed set H is called recurrent in the sense of Harris if there exists a σ-finite measure ϕ such that for E=H, ϕ(E) >0 implies Q(x, E)=1 for all tx∃H. Theorem 1. Let X be absolutely essential and indecomposable. Then there exists a closed set B⫅X. such that B contains no acountable disjoint collection of perpetuable sets if and only if X=H+1 where H is recurrent in the sense of Harris and I is either inessential or improperly essential. Theorem 2. If there exists no uncountable disjoint collection of closed sets, then there exists a countable disjoint collection {Dn} n=1 ∞ of absolutely essential and indecomposable closed sets such that $$I = X - \sum olimits_{n = 1}^\infty {D_n } $$ . Under the additional assumption that Suslin's Conjecture holds, Theorem 2 was proved by Jamison (Jamison, B.: A Result in Doeblin's Theory of Markov Chains implied by Suslin's Conjecture. Z. Wahrscheinlichkeitstheorie verw. Gebiete 24, 287–293 (1972)).

On C- and C∗-embeddings

Abstract: Referring to the separation axioms of Van Est and Freudenthal [ W. T. van Est and H. Freudenthal , Trenning durch Stetigkeit Funktionen in topologischen Raumen, Indagationes Math. 15, 359–368 (1951)] the following are included in the results. A set is C-embedded in every space satisfying pτS q that it is embedded in iff it is closed in every space satisfying pτS q that it is embedded in. A pseudocompact set X is C-embedded in a pτS A space Y which it is embedded in iff X with its weak topology is C-embedded in Y with its weak topology. Consequently a space X is C-embedded in every pτS A space that it is embedded in iff given two disjoint zero sets in X at least one is closed in every pτS q space that it is embedded in and furthermore a closed pseudocompact space with pseudocompact zero sets is C-embedded in a pτS A space Y if Y with its weak topology satisfies AτS B. A C-embedded subset is closed in a space X if X satisfies pτS A and every point is a Gδ. A C∗-embedded subset is closed in a space X if X is sequential T1 and hereditarily weakly normal and completely regular.

Disjoint Representation of Three Tree Realizable Sequences. I

Abstract: In this paper we provide a simple sufficient condition for the existence of an edge-disjoint packing in a graph for three trees of specified degree sequence. The condition is similar to the one given earlier for packing of two trees of specified degree sequence. The proof of the main theorem is by giving a procedure for constructing the packing.

Uniform absolute continuity in spaces of set functions

Abstract: Let X be a regular topological space, K a collection of bounded regular measures defined on the Borel sets of X. The following conditions are equivalent. (1) Let M(X) denote the Borel measures, M(X) the nonnegative members of M(X). There is a A EM(X) + such that K is uniformly X-continuous. (2) If IUnIn = 1, 2,. .. I is a disjoint sequence of open sets, then limn-4Un) = 0 uniformly for ,u E K. (3) If E is a Borel subset of X and E > 0, there is a compact set F C E such that 1L|(E F) 0 we can find a compact set K C E such that Iy(E-K)I 0, there is a 8 > 0 such that if E is a Borel set with A(E) < 8, then tL(E)I < E for