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On crossintersecting families of sets
 Graphs Combin
"... Abstract. A family A of ‘element subsets and a family B of kelement subsets of an nelement set are crossintersecting if every set from A has a nonempty intersection with every set from B. We compare two previously established inequalities each related to the maximization of the product jAjjBj, a ..."
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Abstract. A family A of ‘element subsets and a family B of kelement subsets of an nelement set are crossintersecting if every set from A has a nonempty intersection with every set from B. We compare two previously established inequalities each related to the maximization of the product j
CrossIntersecting Families of Vectors
"... Abstract. Given a sequence of positive integers p = (p1,..., pn), let Sp denote the family of all sequences of positive integers x = (x1,..., xn) such that xi ≤ pi for all i. Two families of sequences (or vectors), A, B ⊆ Sp, are said to be rcrossintersecting if no matter how we select x ∈ A and ..."
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and y ∈ B, there are at least r distinct indices i such that xi = yi. We determine the maximum value of A  · B  over all pairs of rcrossintersecting families and characterize the extremal pairs for r ≥ 1, provided that min pi> r + 1. The case min pi ≤ r + 1 is quite different. For this case
CrossIntersecting Sets of Vectors
"... Given a sequence of positive integers p = (p1,..., pn), let Sp denote the set of all sequences of positive integers x = (x1,..., xn) such that xi ≤ pi for all i. Two families of sequences (or vectors), A,B ⊆ Sp, are said to be rcrossintersecting if no matter how we select x ∈ A and y ∈ B, there ar ..."
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, there are at least r distinct indices i such that xi = yi. We show that for any pair of 1crossintersecting families, A,B ⊆ Sp, we have A·B  ≤ Sp2/k2, where k = mini pi. We also determine the minimal value of A  · B  for any pair of rcrossintersecting families and characterize the extremal pairs for r
On families of weakly crossintersecting setpairs
"... Let F be a family of pairs of sets. We call it an (a, b)set system if for every setpair (A,B) in F we have that A  = a, B  = b, A ∩ B = ∅. The following classical result on families of crossintersecting setpairs is due to Bollobás [6]. Let F be an (a, b)set system with the crossintersec ..."
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Let F be a family of pairs of sets. We call it an (a, b)set system if for every setpair (A,B) in F we have that A  = a, B  = b, A ∩ B = ∅. The following classical result on families of crossintersecting setpairs is due to Bollobás [6]. Let F be an (a, b)set system with the crossintersecting
On crossintersecting families of independent sets in graphs
"... Let A1,...,Ak be a collection of families of subsets of an nelement set. We say that this collection is crossintersecting if for any i, j ∈ [k] with i = j, A ∈Ai and B ∈Aj implies A ∩ B = ∅. We consider a theorem of Hilton which gives a best possible upper bound on the sum of the cardinalities o ..."
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Let A1,...,Ak be a collection of families of subsets of an nelement set. We say that this collection is crossintersecting if for any i, j ∈ [k] with i = j, A ∈Ai and B ∈Aj implies A ∩ B = ∅. We consider a theorem of Hilton which gives a best possible upper bound on the sum of the cardinalities
Graphical models, exponential families, and variational inference
, 2008
"... The formalism of probabilistic graphical models provides a unifying framework for capturing complex dependencies among random variables, and building largescale multivariate statistical models. Graphical models have become a focus of research in many statistical, computational and mathematical fiel ..."
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Cited by 800 (26 self)
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of probability distributions — are best studied in the general setting. Working with exponential family representations, and exploiting the conjugate duality between the cumulant function and the entropy for exponential families, we develop general variational representations of the problems of computing
FAST VOLUME RENDERING USING A SHEARWARP FACTORIZATION OF THE VIEWING TRANSFORMATION
, 1995
"... Volume rendering is a technique for visualizing 3D arrays of sampled data. It has applications in areas such as medical imaging and scientific visualization, but its use has been limited by its high computational expense. Early implementations of volume rendering used bruteforce techniques that req ..."
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Cited by 541 (2 self)
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family of volume rendering algorithms that reduces rendering times to one second. First we present a scanlineorder volume rendering algorithm that exploits coherence in both the volume data and the image. We show that scanlineorder algorithms are fundamentally more efficient than commonlyused ray
Semantic Similarity in a Taxonomy: An InformationBased Measure and its Application to Problems of Ambiguity in Natural Language
, 1999
"... This article presents a measure of semantic similarityinanisa taxonomy based on the notion of shared information content. Experimental evaluation against a benchmark set of human similarity judgments demonstrates that the measure performs better than the traditional edgecounting approach. The a ..."
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Cited by 601 (9 self)
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This article presents a measure of semantic similarityinanisa taxonomy based on the notion of shared information content. Experimental evaluation against a benchmark set of human similarity judgments demonstrates that the measure performs better than the traditional edgecounting approach
Results 1  10
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