The study of k- sets is a very relevant topic in the research area of computational geometry. The study of the maximum and minimum number of k-sets in sets of points of the plane in general position, specifically, has...The study of k- sets is a very relevant topic in the research area of computational geometry. The study of the maximum and minimum number of k-sets in sets of points of the plane in general position, specifically, has been developed at great length in the literature. With respect to the maximum number of k-sets, lower bounds for this maximum have been provided by Erdaos et al., Edelsbrunner and Welzl, and later by Toth. Dey also stated an upper bound for this maximum number of k-sets. With respect to the minimum number of k-set, this has been stated by Erdos el al. and, independently, by Lovasz et al. In this paper the authors give an example of a set ofn points in the plane in general position （no three collinear）, in which the minimum number of points that can take part in, at least, a k-set is attained for every k with 1 ≤ k 〈 n/2. The authors also extend Erdos＇s result about the minimum number of points in general position which can take part in a k-set to a set ofn points not necessarily in general position. That is why this work complements the classic works we have mentioned before.展开更多
We introduce a new method to study the asymptotic behavior of solutions on the basis of the continuation theory for k-set contractions.We apply this technique to show the existence of nontrivial decaying solutions to ...We introduce a new method to study the asymptotic behavior of solutions on the basis of the continuation theory for k-set contractions.We apply this technique to show the existence of nontrivial decaying solutions to the sup-linear generalized Emden-Fowler equation and the existence of asymptotically linear solutions to the sub-linear one.展开更多
文摘The study of k- sets is a very relevant topic in the research area of computational geometry. The study of the maximum and minimum number of k-sets in sets of points of the plane in general position, specifically, has been developed at great length in the literature. With respect to the maximum number of k-sets, lower bounds for this maximum have been provided by Erdaos et al., Edelsbrunner and Welzl, and later by Toth. Dey also stated an upper bound for this maximum number of k-sets. With respect to the minimum number of k-set, this has been stated by Erdos el al. and, independently, by Lovasz et al. In this paper the authors give an example of a set ofn points in the plane in general position （no three collinear）, in which the minimum number of points that can take part in, at least, a k-set is attained for every k with 1 ≤ k 〈 n/2. The authors also extend Erdos＇s result about the minimum number of points in general position which can take part in a k-set to a set ofn points not necessarily in general position. That is why this work complements the classic works we have mentioned before.
基金Supported by the National Natural Science Foundation of China(Grant Nos.12001397,12171039)the Science&Technology Development Fund of Tianjin Education Commission for Higher Education(Grant No.2022KJ204).
文摘We introduce a new method to study the asymptotic behavior of solutions on the basis of the continuation theory for k-set contractions.We apply this technique to show the existence of nontrivial decaying solutions to the sup-linear generalized Emden-Fowler equation and the existence of asymptotically linear solutions to the sub-linear one.
