Showing papers on "Monotone polygon published in 1969"

On Dedekind’s problem: The number of monotone Boolean functions

Abstract: with an = ce~nli, /3„ = e'(Iog w)/«1'2.The number \p(n) is equal to the number of ideals, or of antichains,or of monotone increasing functions into 0 and 1 definable on thelattice of subsets of an «-element set Sn. Here an ideal is a collectionI of subsets such that BEI, AEB implies ^4G£ an antichain is acollection of subsets no two of which are ordered by inclusion. An

Starshaped Transformations and the Power of Rank Tests

Abstract: It is known (Lehmann (1959), p. 187, and Bell, Moser and Thompson (1966), p. 134) that for the two-sample problem, the closer the two samples are together stochastically, the smaller is the power of monotone rank tests. Here it is shown that if one uses the ideas of van Zwet (1964) to define "skewness" and "heavy tails," then the more skew the distributions of the two samples are, the smaller is the power of monotone rank tests; and heavy tails similarly leads to smaller power of monotone rank tests. Skewness and heavy tails are defined using convex and star-shaped transformations of random variables. These are the same transformations used in reliability theory (Barlow and Proschan (1965), Birnbaum, Esary and Marshall (1966), and others) to describe the concept of "wear-out." Thus if $X$ is a random variable that represents "time to failure," and if failure is caused by wear-out or by the environment, then there exists a convex or starshaped function $g$ such that $Z = g(X)$ is an exponential $(1 - \exp \lbrack -\lambda z \rbrack)$ random variable. The distributions of these variables $X$ are called increasing failure rate (IFR) distributions when $g$ is convex and (IFRA) distributions when $g$ is starshaped. It turns out that if one restricts attention to such distributions, then the results of this paper can be used to construct a simple optimality theory for rank tests. This is done in a later paper [6]. The power inequalities related to skewness and heavy tails readily extends to sequential rank tests. It is shown (Example 5.1) that the sequential probability ratio test based on ranks for exponential scale alternatives (e.g. [11] and [12]) also is valid for the class of IFRA scale alternatives.

Upper and Lower Probability Inferences for Families of Hypotheses with Monotone Density Ratios

Abstract: Models of the second kind, described in Section 2, are applied to define inferences about single parameter families having monotone density ratio structure. A definition and some properties of this structure are given in Section 3 along with basic general formulas for inferences. Mixture families always have the monotone density ratio property, and some theory and illustrations are given in Section 4. Location and scale parameter families sometimes have the property, as shown and illustrated in Section 5 where the examples consist of uniform and normal location parameters and exponential and normal scale parameters.

Convexity properties of nonlinear maximal monotone operators

Abstract: Minty [9] has shown that, when X is finite-dimensional and T is a maximal monotone operator, the sets D(T) and R(T) are almost convex, in the sense that each contains the relative interior of its convex hull. The purpose of this note is to announce some generalizations of Minty's result to infinite-dimensional spaces. A subset C of X will be called virtually convex if, given any relatively (strongly) compact subset K of the convex hull of C and any €>0 , there exists a (strongly) continuous single-valued mapping from K into C such that ||#(x)— *|| =* f ° r every xÇ:K. I t can be shown that , in the finite-dimensional case, C is virtually convex if and only if C is almost convex, so that the following result contains Minty's result as a special case.

The use and utility of the monotone criterion in multidimensional scaling

Abstract: The monotone criterion as a multidimensional scaling technique is theoret- ically and empirically evaluated using the algorithms of Kruskal (MDSCAL) and Guttman-Lingoes (SSA-1). Geometric configurations are used to test the recovery capabaty and other aspects of MDSCAL and SSA-1. In addition to theoretical shortcomings, the monotone criterion permits results which do not correspond to the shape of the input data. Because of its deficiencies, alterna- tive goodness-of-fit criteria are suggested in preference to the monotone criterion.

On weak and monotone σ-closures ofC*-algebras

Abstract: It is proved that the monotone σ-closure of the self-adjoint part of anyC*-algebraA is the self-adjoint part of aC*-algebra ℬ IfA is of type I it is proved that ℬ is weakly σ-closed, ie ℬ is aΣ*-algebra The physical importance ofΣ*-algebras was explained in [1] and [7]

Computation of minimal length full Steiner trees on the vertices of a convex polygon

Abstract: A Steiner minimal tree is a tree of minimal length whose vertices are a given set of points ax, ■ •, an in E2 and any set of additional points Sx, • ■ -, Sk(k è 0). In general, the introduction of extra points makes possible shorter trees than the minimal length tree whose vertices are precisely ax, ■ • •, a„ and for which practical algorithms are known. A Steiner minimal tree is the union of special subtrees, known as full Steiner trees. This paper demonstrates the use of the computer in generating minimal length full Steiner trees on sets of points in E2 which are the vertices of convex polygons. The procedure given is a basis from which further research might proceed towards an ultimate practical algorithm for the construction of Steiner minimal trees.

Monotone semigroups of operators on cones

Abstract: In this paper we consider a special class of linear operators defined on a cone K in a Banach space X. This class of operators is the natural generalization of a class of operators which has applications in the theory of interpolation spaces. In particular, using the criteria developed in Theorem 1, it is possible to characterize those sequence spaces X such that every linear operator A of weak types (p, p) and (q, q) is a continuous mapping of X into itself. For details of this we refer the reader to [3].

On Monotone Decomposable Operators

Abstract: This paper deals with the numerical s()lution of linear systems. Two monotonic sequences are generated which enclose the solution from above and below, respectively. Conditions for the existence of the sequences are established, and a numerical example is given which shows that the method in question is suitable for the solution of ill-conditioned systems.

Open mappings on spheres

Abstract: In this paper we study open mappings of the sphere, S n , onto itself. In particular, sufficient conditions are given that such a mapping be a homeomorphism. For the cases n ≦ 2 many of the results could be obtained from the work of G. T. Whyburn [7], [8], and [10]. For the cases n ≦ 3 the useful results of A. V. Cernavskii, [1], [4], proved to be sufficient. An application is made concerning a finite to one open mapping of one n cell onto itself. It is interesting to note that for n ≦ 2 that we could use similar proofs to show that certain quasi-monotone mappings of S n onto S n are necessarily monotone mappings.

Monotone mapping properties of hereditarily infinite dimensional spaces

Abstract: In a previous paper [7], we studied the structure of HID spaces. In this paper, we consider the behavior of HID spaces under monotone mappings. The principal result of this paper is that, given an arbitrary compact metric space Y, there is an HID space X and a monotone map f: X-) Y. We also show that an arbitrary HID space can be mapped monotonically onto a space of any preassigned dimension, and that, given an HID space X and a positive integer n, there is an n-dimensional space Y and a monotone map f: Y->X. R. H. Bing showed in [2 ] that each nondegenerate monotone image of a pseudo-arc is a pseudo-arc. The results of this paper show that no similar monotone invariance property holds for spaces of dimension greater than 1. In this paper, all spaces will be compact metric spaces (compacta). We will be dealing with the Hilbert cube, which we regard as being the product of a countably infinite collection of straight line intervals IX = 11 X 12 X 13 X ... , where Ij = [-1/2i, 1/2'].

Monotone Homomorphisms of Compact Semigroups

Abstract: The problem of determining the class of homomorphic images of a given class of topological semigroups seems to have received little attention in the literature. In [4] Cohen and Krule determined the homomorphic images of a semigroup with zero on an interval. Anderson and Hunter in [1] proved several theorems in this direction. In general, the problem seems to be rather difficult. However, the difficulty is lessened somewhat if all of the homomorphisms of the semigroups in question must be monotone. Phillips, [7], showed that every homomorphism of a standard thread is monotone and hence every homomorphic image of a standard thread is either a standard thread or a point. In this paper a larger class of topological semigroups which admit only monotone homomorphisms is given. These results are used to determine the topological nature of the homomorphic images of certain classes of topological semigroups. These include products of standard threads with min threads, certain semilattices on a two-cell, and compact connected lattices in the plane.

Slowly oscillating functions and a generalization of quasi-monotone coefficients

Abstract: We first state a conjecture due to Yong concerning equality of two classes of functions, and then give examples to disprove the conjecture. Later we extend some theorems concerning integrability of Fourier series.

Monotone $n$-frames are tame

Abstract: An n-frame Y,, is the union of n arcs which are disjoint except for a common end point called the branch point of 'in. An n-frame of E3 is tamely imbedded provided there is an autohomeomorphism of E3 which carries it onto a polygonal n-frame. We will say that an n-frame 9. in El is monotone provided each geometric 2-sphere centered at the branch point of 9Fn intersects each of the n defining arcs of 9n in at most one point. Because each monotone n-frame is of the same imbedding type as a monotone n-frame having its branch point at the origin and its end points on the unit sphere, we will assume that a monotone n-frame has these properties as well as a prescribed ordering, say a1, a2, * , a", of its defining arcs. Since for each s C (0, 1 ], the arc ai intersects the sphere of radius s centered at the origin in a single point pi(s), the equation +(s) = (p1(s), *. . , P,(s)), 0

On asymptotic behaviours of trigonometric series with δ-Quasi-monotone coefficients

Abstract: LetThe asymptotic behaviours of ƒ(x) and g(x), as x→+0, were first given by G. H. Hardy in (4), (5). In his papers {an}; is a monotone decreasing sequence. Further results on the asymptotic behaviours of ƒ(x) and g(x), as x→+0, for monotone coefficients have been given in (9) and (1). Recently, the results have been generalized to quasi-monotone coefficients.This paper is concerned with asymptotic behaviours of ƒ(x) and g(x) for δ-quasi-monotone coefficients.In what follows, we shall denote by L(x) a slowly varying function in the sense of Karamata (6), i.e.