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Topic: Pairwise disjoint

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disjoint union

Disjoint sets

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	Disjoint sets - Wikipedia, the free encyclopedia
		In mathematics, two sets are said to be disjoint if they have no element in common.
		Formally, two sets A and B are disjoint if their intersection is the empty set, i.e.
		A collection of sets is pairwise disjoint or mutually disjoint if any two distinct sets in the collection are disjoint.
	en.wikipedia.org /wiki/Disjoint (220 words)

	Disjoint union - Wikipedia, the free encyclopedia
		In set theory, a disjoint union (or discriminated union) is a union of a collection of sets whose members are pairwise disjoint.
		This notation is meant to be suggestive of the fact that the cardinality of the disjoint union is the sum of the cardinalities of the terms in the family.
		In the language of category theory, the disjoint union is the coproduct in the category of sets.
	en.wikipedia.org /wiki/Discriminated_union (344 words)

	PlanetMath: disjoint union
		This is also often called being pairwise disjoint and is a much stronger condition than that the intersection of all the images is empty.
		Of course, there are many categories where this usage is unnatural: In the category of pointed sets, the coproduct will be the disjoint union with the distinguished points identified.
		This is version 4 of disjoint union, born on 2004-03-01, modified 2004-03-23.
	planetmath.org /encyclopedia/DisjointUnion.html (218 words)

	PlanetMath: pairwise disjoint
		Then these sets are pairwise disjoint if for distinct
		This is version 4 of pairwise disjoint, born on 2004-02-29, modified 2004-05-01.
		Such collections are often referred to as "disjoint" (without qualification) or "pairwise disjoint".
	planetmath.org /encyclopedia/MutuallyDisjoint2.html (67 words)

	2.15. Overlap Inclusion and Disjointness of Types
		The Common Lisp data type hierarchy is tangled and purposely left somewhat open-ended so that implementors may experiment with new data types as extensions to the language.
		If types x and y are disjoint then no object (in any implementation) may be both of type x and of type y.
		pairwise disjoint from a number of other types as well; see note above.
	www.oopweb.com /LISP/Documents/cltl/Volume/node42.html (723 words)

	Constructing Pairwise Disjoint Paths with Few Links - Gupta, Wenger (ResearchIndex) (Site not responding. Last check: 2007-10-25)
		Abstract: Let P be a simple polygon and let f(u i ; u 0 i)g be m pairs of distinct vertices of P where for every distinct i; j m, there exist pairwise disjoint paths connecting u i to u 0 i and u j to u 0 j.
		We wish to construct m pairwise disjoint paths in the interior of P connecting u i to u 0 i for i = 1; : : : ; m, with minimal total number of line segments.
		We define the pairwise disjoint link paths problem as: given an untangled set, f(u i ; u 0 i)g, of m pairs of distinct vertices of P, find the...
	citeseer.ist.psu.edu /340852.html (411 words)

	Disjoint - The Maximum Edge-Disjoint Paths Problem in Bidirected Trees (Site not responding. Last check: 2007-10-25)
		Disjoint Routing in Telecommunication Networks In this project, we have developed an efficient algorithm for the Disjoint Routing Problem.
		Disjoint sets are also said to be mutually exclusive or independent.
		Given a graph GVE and two vertices stVst, the Menger problem is to find a maximum number of disjoint paths connecting s and t.
	chucang.com /chcn/disjoint.html (260 words)

	Abstract (Site not responding. Last check: 2007-10-25)
		Pairwise Disjoint Shortest Paths in the n-Cube and Related Problems
		An efficient algorithm for the p-pairwise node disjoint shortest paths problem when every source point is at a distance at most two from its target is presented and it is shown that for distance three pairs the problem is NP-complete.
		The p-pairwise edge disjoint shortest paths problem is shown to be solvable in polynomial time when every source point is at a distance at most two from its target and it is shown to be NP-complete for distance three pairs.
	www.cs.ucsb.edu /research/tech_reports/abstract.php?id=935 (173 words)

	[Alice-users] Re: expression for pairwise disjoint (Site not responding. Last check: 2007-10-25)
		Previous message: [Alice-users] Re: expression for pairwise disjoint
		Next message: [Alice-users] Re: expression for pairwise disjoint
		disjoint (t::tr) = > let > fun disjp ((dur1,start1),(_,start2))= > post (sp, start1 `+ dur1 `= start2, FD.BND) > in > foldl (fn (t,t') => (disjp(t,t'); t')) t tr; > () > end Maybe I'm just tired, but shouldn't this rather be foldl (fn (t,t') => (disjp(t',t); t)) t tr because the accumulator comes second?
	www.ps.uni-sb.de /pipermail/alice-users/2005/000394.html (132 words)

	98-22: Geometric Graphs with Few Disjoint Edges (Site not responding. Last check: 2007-10-25)
		Improving a result of Pach and T\"or\H ocsik, we show that a geometric graph on $n$ vertices with no $k+1$ pairwise disjoint edges has at most $k^3(n+1)$ edges.
		On the other hand, we construct geometric graphs with $n$ vertices and approximately ${3\over 2}(k-1)n$ edges, containing no $k+1$ pairwise disjoint edges.
		We also improve both the lower and upper bounds of Goddard, Katchalski and Kleitman on the maximum number of edges in a geometric graph with no four pairwise disjoint edges.
	dimacs.rutgers.edu /TechnicalReports/abstracts/1998/98-22.html (113 words)

	Vitali's Theorem
		The recursion ends with a finite, pairwise disjoint sequence of intervals I
		Remark 5.4 It is straightforward to generalize the previous lemma to the case of a Vitali covering of the n-cube [0,1]
		By disjoint countable additivity (Corollary 2.5), we have
	www.math.psu.edu /simpson/papers/vitali-l2h/node5.html (504 words)

	Measure Theory in
		For instance, it is straightforward to show that the Lebesgue measure of the union of a finite set of pairwise disjoint open intervals is equal to the sum of the lengths of the intervals.
		An open set is said to be connected if it is not the union of two disjoint nonempty open sets.
		and this would be a decomposition of B into two disjoint nonempty open sets, a contradiction.
	www.math.psu.edu /simpson/papers/vitali-l2h/node2.html (686 words)

	NP-Completeness Publications
		Pairwise Disjoint Shortest Paths in the n-Cube and Related Problems, (with D. Serena), (submitted for journal publication).
		Complexity of the Pairwise Shortest Path Routing in the Grid, (with D. Serena), Theoretical Computer Science, Vol.
		Complexity of k-Pairwise Disjoint Shortest Paths in the Undirected Hypercubic Network and Related Problems, (with F. Serena) Proceedings of the 14th IASTED International Conference on Parallel and Distributed Computing and Systems PDCS '02, (2002), MIT Cambridge, MA, 61 -- 66.
	www.cs.ucsb.edu /~teo/publications/NPC.html (869 words)

	Steiner Triple Systems Intersecting in Pairwise Disjoint Blocks (ResearchIndex) (Site not responding. Last check: 2007-10-25)
		Abstract: Two Steiner triple systems (X,A)and(X,B) are said to intersect in m pairwise disjoint blocks if A # B = m and all blocks in A#B are pairwise disjoint.
		For each v, we completely determine the possible values of m such that there exist two Steiner triple systems of order v intersecting in m pairwise disjoint blocks.
		9 An existence theory for pairwise balanced designs I: Composi..
	citeseer.ist.psu.edu /673424.html (330 words)

	Sets
		Two sets are disjoint if and only if there is no object that is a member of both sets.
		Note that mutually-disjoint sets need not be pairwise disjoint; in fact, every pair of sets might be overlapping.
		The first is the axiom of regularity - every non-empty set has an element with which it has no members in common.
	logic.stanford.edu /kif/hypertext/node23.html (511 words)

	Polygonizing Sets of Line Segments in the Plane
		We are given a full-dimensional set S of pairwise disjoint line segments with the property that at least one endpoint of each segment is on the boundary of the convex hull of the union of all the segments in S. Full-dimensional means that not all line segments in S are collinear.
		Our task is to construct a simple polygon P where all the line segments in S are either edges or diagonals of P. Introduction
		Given a full-dimensional set S of n pairwise disjoint segments.
	www.cs.mcgill.ca /~cschmi/507 (1773 words)

	[Alice-users] expression for pairwise disjoint (Site not responding. Last check: 2007-10-25)
		I want to set the constraint such that: - tasks are in the order given by the list.
		- tasks don't overlap fun disjoint tasks = let fun disjp((dur1, start1), (_, start2)) = post (sp, start1 `+ dur1 `= start2, FD.BND) fun disj2 (task1, nil) = ()
		I also have this nagging feeling that this would be a very common problem in scheduling and that there is likely to be a much simpler approach.
	www.ps.uni-sb.de /pipermail/alice-users/2005/000392.html (209 words)

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