Showing papers on "Disjoint sets published in 1977"

Intersection theorems for systems of sets

Abstract: Let n and k be positive integers, k≥3. Denote by ϕ(n, k) the least positive integer such that if F is any family of more than ϕ(n, k) sets, each set with n elements, then some k members of F have pairwise the same intersection. In this paper we obtain a new asymptotic upper bound for ϕ(n, k), k fixed, n approaching infinity.

An Algorithm for Euclidean Sum of Squares Classification

Abstract: A ni analysis of surface pollen samples to discover iJ'they fall naturally in1to distinct groups oJ' simiiilar samiples is an example of a classification problem. In Euclidean classification, a set of n objects can be represented as n points in Euclidean space of p dimensions. The sum oJ'squares criterion defines the optimal partition of the points in1to g disjoint groups to be the partition which mninimzizes the total within-group sumn of squared distances about the g centroids. It is not usually Jeasible to examiine all possible partitions oJ'the objects into g groups. A critical review is mnade of algorithmiis which have been proposedfor seeking optimal partitions. The problem is reformulated itn non-linear programming terms, and a new algorithm for seeking the minimum sumi1 oJ'squares is described. The performance of this algorithm in analyzing the pollen data is Jound to compare vell vith the perJormance of three oJ the existing algorithms. An efficient hybrid algorithmi is introduced.

The eighty-one types of isohedral tilings in the plane

Abstract: 1. A tiling is a collection = {Ti|i = 1, 2, …} of closed topological discs which covers the Euclidean plane E2, and of which the individual tiles Ti have disjoint interiors. We shall assume throughout that the intersection of any two tiles is a connected set. If each tile is congruent (directly or reflectively isometric) to a given set T, then the tiling is called monohedral and T is called the prototile of . Clearly every monohedral tiling is locally finite.

Reference machines require non-linear time to maintain disjoint sets

Abstract: This paper describes a machine model intended to be useful in deriving realistic complexity bounds for tasks requiring list processing. As an example of the use of the model, the paper shows that any such machine requires non-linear time in the worst case to compute unions of disjoint sets on-line. All set union algorithms known to the author are instances of the model and are thus subject to the derived bound. One of the known algorithms achieves the bound to within a constant factor.

Enclaveless Sets and MK-Systems.

Abstract: A hypergraph H=(X,E) is called a Menger System if the maximum cardinality of a family of pairwise disjoint edges (v1(H)) is equal to the minimum cardinality of a subset of vertices which meets every edge (τ0(H)). A set S ⊆ X is defined to be enclaveless if each vertex in S is adjacent to at least one vertex in X - S. A parameter π0 related to the formation of maximal enclaveless sets is defined, and it is shown that if H has no singleton edges then v1(H) ≤ π0(H). MK-Systems are defined to be those hypergraphs H without singleton edges for which v1(H) = π0(H); simple graphs which are Menger Systems are shown also to be MK-Systems.

Definability of measures and ultrafilters

Abstract: Nonprincipal ultrafilters are harder to define in ZFC, and harder to obtain in ZF + DC, than nonprincipal measures.The function μ from P(X) to the closed interval [0, 1] is a measure on X if μ. is finitely additive on disjoint sets and μ(X) = 1. (P is the power set.) μ is nonprincipal if μ ({x}) = 0 for each x Є X. μ is an ultrafilter if Range μ= {0, 1}. The existence of nonprincipal measures and ultrafilters on any infinite X follows from the axiom of choice.Nonprincipal measures cannot necessarily be defined in ZFC. (ZF is Zermelo–Fraenkel set theory. ZFC is ZF with choice.) In ZF alone they cannot even be proved to exist. This was first established by Solovay [14] using an inaccessible cardinal. In the model of [14] no nonprincipal measure on ω is even ODR (definable from ordinal and real parameters). The HODR (hereditarily ODR) sets of this model form a model of ZF + DC (dependent choice) in which no nonprincipal measure on ω exists. Pincus [8] gave a model with the same properties making no use of an inaccessible. (This model was also known to Solovay.) The second model can be combined with ideas of A. Blass [1] to give a model of ZF + DC in which no nonprincipal measures exist on any set. Using this model one obtains a model of ZFC in which no nonprincipal measure on the set of real numbers is ODR. H. Friedman, in private communication, previously obtained such a model of ZFC by a different method. Our construction will be sketched in 4.1.

A rough fundamental domain for Teichmüller spaces

Abstract: Let T(S) be the Teichmuller space of Riemann surfaces of finite type and let M(S) be the corresponding modular group. In [11] we described T(S) in terms of real analytic parameters. In this paper we determine a subspace R(S) of T(S) which is a "rough fundamental domain" for M (S) acting on T(S). The construction of R(S) is a generalization of the constructions in [14] and [15]. The previous constructions depended heavily upon an analysis of the action of the elements of M (S) on parameters of T(S) corresponding to disjoint closed geodesies on S, and on a theorem of Bers [2] which gives bounds for the lengths of these curves. In the general case, the disjoint closed geodesies of Bers' theorem no longer always correspond directly to the parameters. Hence we must carefully study how their lengths are related to the parameters. In §1 we outline the basic preliminary notions relating hyperbolic geometry, Fuchsian groups and Teichmuller spaces. In §§2 and 3 we give the constructions of Teichmuller space and of a fundamental domain for the action of the modular group in the simplest cases; that is, where S has type (0; 3) and (1; 1). In §4 we give a detailed discussion of the Teichmuller space and of the fundamental domain in the case (0; 4). These constructions are the heart of the general constructions which follow. In §5 we discuss the special case of surfaces of genus 2 and state Bers* theorem. In §6 we give the construction of Teichmuller space in general. In §7 we analyze the topologically distinct sets of mutually disjoint geodesies which occur in Bers' theorem and determine their relationship to the moduli curves. Finally, in §8 we put all the pieces of the construction together and determine the rough fundamental domain. In §9 we use this construction to affirmatively settle a conjecture of Bers [3].

An algebraic classification of some links of codimension two

Abstract: For q > 2, J. Levine proved that two simple (2q 1)-knots are isotopic if and only if their Seifert matrices are equivalent. In this paper, we will prove the analogue of Levine's result for simple boundary (2q 1)links; we will show that: "For q > 3, two simple boundary (2q 1)-links are isotopic if and only if their Seifert matrices are I-equivalent (defined by some algebraic moves)." An n-link of multiplicity m, denoted by L = K1 U ... U Km is an embedding of m disjoint copies of the n-sphere (or homotopy spheres) Ki into the (n + 2)-sphere Sn+2. L is called boundary if it extends to an embedding of m disjoint orientable compact (n + 1)-manifolds Mi, called the Seifert manifolds, with aMi = Ki. Let X denote the link complement. Gutierrez [1] showed that an n-link of multiplicity m is boundary if and only if there is an epimorphism from 71(X) onto Fm, the free group in m generators, sending meridians to generators. An (2q 1)-link L is called simple if 7Ti(X) = 7Ti(VmS1) for i 2, Levine [5] proved that two simple (2q 1)-knots are isotopic if and only if their Seifert matrices are "equivalent" (defined by certain algebraic "moves" in [5], also called S-equivalent in [7]). In this paper, we will prove the analogue of Levine's Theorems 1-3 for simple boundary (2q 1)links, q > 3: two simple boundary (2q 1)-links are isotopic if and only if their "Seifert matrices" are related by certain algebraic "moves". Since our proofs are almost the same as those of [4] and [5], we will only give the outlines here. 1. For simplicity, we will consider only the (2q 1)-link of multiplicity 2. Everything considered here is in the smooth category. Let L = K1 u K2 be a boundary (2q 1)-link. According to [1], there exist two disjoint 2q-dimensional Seifert manifolds Ml and M2 for L, that is, WI= K1 and 3M2 = K2. Let A 1 be the corresponding Seifert matrix for the Received by the editors January 13, 1977 and, in revised form, February 28, 1977. AMS (MOS) subject classifications (1970). Primary 57C45, 57D40, 57D65.

Certain Duality Principles in Integer Programming

Abstract: This paper surveys some results of the following type: “If a linear program and some derived programs have integral solutions, so does its dual.” Several well-known minimax theorems in combinatorics can be derived from such general principles. Similar principles can be proved if integrality is replaced by a condition of the least common denominator of the entries of a solution. An analogy between Tutte's 1-factor-theorem and the Lucchesi-Younger Theorem on disjoint directed cuts is pointed out.

Time-optimal feedback control

Abstract: For the time-optimal problem of reaching the origin within linear, autonomous, finite-dimensional, bounded-control systems inn-space, it is shown that, in a generic case and for small time, each reachable set consists of a finite number of disjoint analytick-manifolds (at least two for eachk≥1, and precisely 2m n−1 fork=n, wherem is the control dimension). Parametric descriptions of these make it possible to construct an optimal feedback control.

Cautionary remark on applying symmetry techniques to the Coulomb problem

Abstract: We discuss the existence of a general Wigner-Eckart theorem for the physical hydrogen-atom problem. The group generators are not a derivation with respect to the physical variables (r,theta,phi), and thus the eigenfunctions do not qualify as tensor operators. The lack of the derivation property is reflected in disjoint subspaces which for the Coulomb problem are threefold. (AIP)

Compact LP bases for a class of IP problems

Abstract: A variety of IP problems in location, logical design, and project selection contain collections of constraints of the form $$x_{ij} \leqslant y_i ,j \in J_i ,i \in I$$ or more generally $$x_{ij} \leqslant y_i ,j \in J_i ,i \in I$$ where all coefficients are nonnegative, and the setsS j ,j∈J i , are pairwise disjoint. We show how to solve the associated LP problem for these and other related structures (simultaneously including upper bound restrictions) while keeping the tableau the same size as if such constraints were absent. Our procedure not only reduces the effective size of such problems, but bypasses many of the calculations ordinarily required by the simplex method.