A sharp eigenvalue theorem for fractional elliptic equations |
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Authors: | Giovanni Molica Bisci Vicen?iu D R?dulescu |
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Institution: | 1.Department of Applied Mathematics and Institute for Theoretical Computer Science,Charles University, Faculty of Mathematics and Physics,Prague,Czech Republic;2.Department of Mathematics and Statistics, Faculty of Science,Masaryk University,Brno,Czech Republic;3.University of Szeged,Bolyai Institute,Szeged,Hungary |
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Abstract: | The invisibility graph I(X) of a set X ? R d is a (possibly infinite) graph whose vertices are the points of X and two vertices are connected by an edge if and only if the straight-line segment connecting the two corresponding points is not fully contained in X. We consider the following three parameters of a set X: the clique number ω(I(X)), the chromatic number χ(I(X)) and the convexity number γ(X), which is the minimum number of convex subsets of X that cover X.We settle a conjecture of Matou?ek and Valtr claiming that for every planar set X, γ(X) can be bounded in terms of χ(I(X)). As a part of the proof we show that a disc with n one-point holes near its boundary has χ(I(X)) ≥ log log(n) but ω(I(X)) = 3.We also find sets X in R5 with χ(X) = 2, but γ(X) arbitrarily large. |
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