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Ann N Trenk - One of the best experts on this subject based on the ideXlab platform.
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the total weak discrepancy of a Partially Ordered Set
Ars Mathematica Contemporanea, 2011Co-Authors: Alan Shuchat, Randy Shull, Ann N TrenkAbstract:We define the total weak discrepancy of a poSet P as the minimum nonnegative integer k for which there exists a function f : V ! Z satisfying (i) if a b then f(a) + 1 f(b) and (ii) P jf(a) f(b)j k, where the sum is taken over all unOrdered pairsfa;bg of incomparable elements. If we allow k and f to take real values, we call the minimum k the fractional total weak discrepancy of P . These concepts are related to the notions of weak and fractional weak discrepancy, where (ii) must hold not for the sum but for each individual pair of incomparable elements of P . We prove that, unlike the latter, the total weak and fractional total weak discrepancy of P are always the same, and we give a polynomial-time algorithm to find their common value. We use linear programming duality and complementary slackness to obtain this result.
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the fractional weak discrepancy of a Partially Ordered Set
Discrete Applied Mathematics, 2007Co-Authors: Alan Shuchat, Randy Shull, Ann N TrenkAbstract:In this paper we introduce the notion of the fractional weak discrepancy of a poSet, building on previous work on weak discrepancy in [J.G. Gimbel and A.N. Trenk, On the weakness of an Ordered Set, SIAM J. Discrete Math. 11 (1998) 655-663; P.J. Tanenbaum, A.N. Trenk, P.C. Fishburn, Linear discrepancy and weak discrepancy of Partially Ordered Sets, ORDER 18 (2001) 201-225; A.N. Trenk, On k-weak orders: recognition and a tolerance result, Discrete Math. 181 (1998) 223-237]. The fractional weak discrepancywd"F(P) of a poSet P=(V,@?) is the minimum nonnegative k for which there exists a function f:V->R satisfying (1) if a@?b then f(a)+1=
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Linear Discrepancy and Weak Discrepancy of Partially Ordered Sets
Order, 2001Co-Authors: Paul J. Tanenbaum, Ann N Trenk, Peter C. FishburnAbstract:The linear discrepancy of a Partially Ordered Set P=(X,≺) is the least integer k for which there exists an injection f: X→Z satisfying (i) if x≺y then f(x)
Asma Rashid Butt - One of the best experts on this subject based on the ideXlab platform.
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fixed point theorems for Set valued mappings in Partially Ordered metric spaces
World Academy of Science Engineering and Technology International Journal of Mathematical Computational Physical Electrical and Computer Engineering, 2013Co-Authors: Ismat Beg, Asma Rashid ButtAbstract:Let (X, ) be a Partially Ordered Set and d be a metric on X such that (X, d) is a complete metric space. Assume that X satisfies; if a non-decreasing sequence xn → x in X , then xn x, for all n. Let F be a Set valued mapping from X into X with nonempty closed bounded values satisfying; (i) there exists κ ∈ (0, 1) with D(F (x), F (y)) ≤ κd(x, y), for all x y, (ii) if d(x, y) < e < 1 for some y ∈ F (x) then x y, (iii) there exists x0 ∈ X, and some x1 ∈ F (x0) with x0 x1 such that d(x0, x1) < 1. It is shown that F has a fixed point. Several consequences are also obtained. Keywords—Fixed point, Partially Ordered Set, metric space, Set valued mapping.
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fixed point for Set valued mappings satisfying an implicit relation in Partially Ordered metric spaces
Nonlinear Analysis-theory Methods & Applications, 2009Co-Authors: Asma Rashid ButtAbstract:Abstract Let ( X , ⪯ ) be a Partially Ordered Set and d be a complete metric on X . Let F , G be two Set-valued mappings on X . We obtained sufficient conditions for the existence of common fixed point of F and G satisfying an implicit relation in Partially Ordered Set X .
Peter C. Fishburn - One of the best experts on this subject based on the ideXlab platform.
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Linear Discrepancy and Weak Discrepancy of Partially Ordered Sets
Order, 2001Co-Authors: Paul J. Tanenbaum, Ann N Trenk, Peter C. FishburnAbstract:The linear discrepancy of a Partially Ordered Set P=(X,≺) is the least integer k for which there exists an injection f: X→Z satisfying (i) if x≺y then f(x)
M R Koushesh - One of the best experts on this subject based on the ideXlab platform.
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the Partially Ordered Set of one point extensions
arXiv: General Topology, 2012Co-Authors: M R KousheshAbstract:A space $Y$ is called an {\em extension} of a space $X$ if $Y$ contains $X$ as a dense subspace. Two extensions of $X$ are said to be {\em equivalent} if there is a homeomorphism between them which fixes $X$ point-wise. For two (equivalence classes of) extensions $Y$ and $Y'$ of $X$ let $Y\leq Y'$ if there is a continuous function of $Y'$ into $Y$ which fixes $X$ point-wise. An extension $Y$ of $X$ is called a {\em one-point extension} of $X$ if $Y\backslash X$ is a singleton. Let ${\mathcal P}$ be a topological property. An extension $Y$ of $X$ is called a {\em ${\mathcal P}$-extension} of $X$ if it has ${\mathcal P}$. One-point ${\mathcal P}$-extensions comprise the subject matter of this article. Here ${\mathcal P}$ is subject to some mild requirements. We define an anti-order-isomorphism between the Set of one-point Tychonoff extensions of a (Tychonoff) space $X$ (Partially Ordered by $\leq$) and the Set of compact non-empty subSets of its outgrowth $\beta X\backslash X$ (Partially Ordered by $\subSeteq$). This enables us to study the order-structure of various Sets of one-point extensions of the space $X$ by relating them to the topologies of certain subspaces of its outgrowth. We conclude the article with the following conjecture. For a Tychonoff spaces $X$ denote by ${\mathscr U}(X)$ the Set of all zero-Sets of $\beta X$ which miss $X$. \noindent{\bf Conjecture.} {\em For locally compact spaces $X$ and $Y$ the Partially Ordered Sets $({\mathscr U}(X),\subSeteq)$ and $({\mathscr U}(Y),\subSeteq)$ are order-isomorphic if and only if the spaces ${\em cl}_{\beta X}(\beta X\backslash\upsilon X)$ and ${\em cl}_{\beta Y}(\beta Y\backslash\upsilon Y)$ are homeomorphic.}
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the Partially Ordered Set of one point extensions
Topology and its Applications, 2011Co-Authors: M R KousheshAbstract:Abstract A space Y is called an extension of a space X if Y contains X as a dense subspace. Two extensions of X are said to be equivalent if there is a homeomorphism between them which fixes X point-wise. For two (equivalence classes of) extensions Y and Y ′ of X let Y ⩽ Y ′ if there is a continuous function of Y ′ into Y which fixes X point-wise. An extension Y of X is called a one-point extension of X if Y ∖ X is a singleton. Let P be a topological property. An extension Y of X is called a P -extension of X if it has P . One-point P -extensions comprise the subject matter of this article. Here P is subject to some mild requirements. We define an anti-order-isomorphism between the Set of one-point Tychonoff extensions of a (Tychonoff) space X (Partially Ordered by ⩽) and the Set of compact non-empty subSets of its outgrowth β X ∖ X (Partially Ordered by ⊆). This enables us to study the order-structure of various Sets of one-point extensions of the space X by relating them to the topologies of certain subspaces of its outgrowth. We conclude the article with the following conjecture. For a Tychonoff spaces X denote by U ( X ) the Set of all zero-Sets of βX which miss X . Conjecture For locally compact spaces X and Y the Partially Ordered Sets ( U ( X ) , ⊆ ) and ( U ( Y ) , ⊆ ) are order-isomorphic if and only if the spaces cl β X ( β X ∖ υ X ) and cl β Y ( β Y ∖ υ Y ) are homeomorphic.
Jayme Luiz Szwarcfiter - One of the best experts on this subject based on the ideXlab platform.
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generating all forest extensions of a Partially Ordered Set
International Conference on Algorithms and Complexity, 2003Co-Authors: Jayme Luiz SzwarcfiterAbstract:Let P and P′ be Partially Ordered Sets, with ground Set E, |E| = n, and relation Sets R and R′, respectively. Say that P′ is an extension of P when R ⊆ R′. A Partially Ordered Set is a forest when the Set of ancestors of any given element forms a chain. We describe an algorithm for generating the complete Set of forest extensions of an order P. The algorithm requires O(n2) time between the generation of two consecutive forests. The initialization of the algorithm requires O(n|R|) time.
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on the generation of extensions of a Partially Ordered Set
International Conference on Algorithms and Complexity, 2003Co-Authors: Jayme Luiz SzwarcfiterAbstract:A Partially Ordered Set (or simply order) P is a Set of elements E, together with a Set R of relations of E, satisfying reflexivity, anti-symmetry and transitivity. The Set E is called the ground Set of P, while R is the relation Set of it. There are many special orders. For example, when any two elements of E are related, the order is a chain. Similarly, we can define tree orders, forest orders and many others. An extension P' of P is an order P' having the same ground Set as P, and such that its relation Set contains R. When P is a chain then P' is a linear extension of P. Similarly, when P is a forest then it is a forest extension of P. We consider the algorithmic problem of generating all extensions of a given order and also extensions of a special kind. The subject will be introduced by a general discussion on Partially Ordered Sets.
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CIAC - On the generation of extensions of a Partially Ordered Set
Lecture Notes in Computer Science, 2003Co-Authors: Jayme Luiz SzwarcfiterAbstract:A Partially Ordered Set (or simply order) P is a Set of elements E, together with a Set R of relations of E, satisfying reflexivity, anti-symmetry and transitivity. The Set E is called the ground Set of P, while R is the relation Set of it. There are many special orders. For example, when any two elements of E are related, the order is a chain. Similarly, we can define tree orders, forest orders and many others. An extension P' of P is an order P' having the same ground Set as P, and such that its relation Set contains R. When P is a chain then P' is a linear extension of P. Similarly, when P is a forest then it is a forest extension of P. We consider the algorithmic problem of generating all extensions of a given order and also extensions of a special kind. The subject will be introduced by a general discussion on Partially Ordered Sets.