In order to avoid the discretization in the classical rough set theory, a generlization rough set theory is proposed. At first, the degree of general importance of an attribute and attribute subsets are presented. The...In order to avoid the discretization in the classical rough set theory, a generlization rough set theory is proposed. At first, the degree of general importance of an attribute and attribute subsets are presented. Then, depending on the degree of general importance of attribute, the space distance can be measured with weighted method. At last, a generalization rough set theory based on the general near neighborhood relation is proposed. The proposed theory partitions the universe into the tolerant modules, and forms lower approximation and upper approximation of the set under general near neighborhood relationship, which avoids the discretization in Pawlak's rough set theory.展开更多
The bounds of the general M and J sets were analytically offered. Some of, the bounds were optimal in certain meaning. It not only solved the primary problem of the construction of fractal sets by escape time algorith...The bounds of the general M and J sets were analytically offered. Some of, the bounds were optimal in certain meaning. It not only solved the primary problem of the construction of fractal sets by escape time algorithm, and followed from the conclusion, but also offered two estimations of some special Julia set's Hausdorff's dimension by approximate linearization method.展开更多
Letα(F_(q)^(d),p)denote the maximum size of a general position set in a p-random subset of F_(q)^(d).We determine the order of magnitude ofα(F_(q)^(2),p)up to polylogarithmic factors for all possible values of p,imp...Letα(F_(q)^(d),p)denote the maximum size of a general position set in a p-random subset of F_(q)^(d).We determine the order of magnitude ofα(F_(q)^(2),p)up to polylogarithmic factors for all possible values of p,improving the previous results obtained by Roche-Newton and Warren(2022)and Bhowmick and Roche-Newton(2024).For d≥3,we prove upper bounds forα(F_(q)^(d),p)that are essentially tight within certain ranges for p.We establish the upper bound 2^((1+o(1))q) for the number of general position sets in F_(q)^(d),which matches the trivial lower bound 2q asymptotically in the exponent.We also refine this counting result by proving an asymptotically tight(in the exponent)upper bound for the number of general position sets with a fixed size.The latter result for d=2 improves the result of Roche-Newton and Warren(2022).Our proofs are grounded in the hypergraph container method.In addition,for d=2,we also leverage the pseudorandomness of the point-line incidence graph of F_(q)^(2).展开更多
The paper generalizes the classical Caristi＇s fixed point theorem. As an application. the classical Ekeland variational principle is generalized. In addition, it is proved that the generalized Caristi＇s fixed point ...The paper generalizes the classical Caristi＇s fixed point theorem. As an application. the classical Ekeland variational principle is generalized. In addition, it is proved that the generalized Caristi＇s fixed point theorem is equivalent to the generalized Ekeland variational principle.展开更多
基金Natural Science Foundation of Jiangsu Province of China ( No.BK2006176)High-Tech Key Laboratory of Jiangsu,China (No.BM2007201)
文摘In order to avoid the discretization in the classical rough set theory, a generlization rough set theory is proposed. At first, the degree of general importance of an attribute and attribute subsets are presented. Then, depending on the degree of general importance of attribute, the space distance can be measured with weighted method. At last, a generalization rough set theory based on the general near neighborhood relation is proposed. The proposed theory partitions the universe into the tolerant modules, and forms lower approximation and upper approximation of the set under general near neighborhood relationship, which avoids the discretization in Pawlak's rough set theory.
文摘The bounds of the general M and J sets were analytically offered. Some of, the bounds were optimal in certain meaning. It not only solved the primary problem of the construction of fractal sets by escape time algorithm, and followed from the conclusion, but also offered two estimations of some special Julia set's Hausdorff's dimension by approximate linearization method.
基金supported by European Research Council Advanced Grant(Grant No.101020255)Leverhulme Research Project Grant(Grant No.RPG-2018-424)+3 种基金supported by National Natural Science Foundation of China(Grant No.123B2012)supported by European Research Council Advanced Grants“GeoScape”(Grant No.882971)and“ERMiD”(Grant No.101054936)Jonathan Tidor for stimulating discussions at 2023 University of California San Diego Workshop on Ramsey Theoryinitiated while Ji Zeng was visiting Shanghai Center for Mathematical Sciences at the kind invitation of Hehui Wu.
文摘Letα(F_(q)^(d),p)denote the maximum size of a general position set in a p-random subset of F_(q)^(d).We determine the order of magnitude ofα(F_(q)^(2),p)up to polylogarithmic factors for all possible values of p,improving the previous results obtained by Roche-Newton and Warren(2022)and Bhowmick and Roche-Newton(2024).For d≥3,we prove upper bounds forα(F_(q)^(d),p)that are essentially tight within certain ranges for p.We establish the upper bound 2^((1+o(1))q) for the number of general position sets in F_(q)^(d),which matches the trivial lower bound 2q asymptotically in the exponent.We also refine this counting result by proving an asymptotically tight(in the exponent)upper bound for the number of general position sets with a fixed size.The latter result for d=2 improves the result of Roche-Newton and Warren(2022).Our proofs are grounded in the hypergraph container method.In addition,for d=2,we also leverage the pseudorandomness of the point-line incidence graph of F_(q)^(2).
基金Tianyuan Young Mathematics of China under Grant (10526025)Science Foundation of Nanjing Normal University under Grant(2002SXXXGQ2B20)
文摘The paper generalizes the classical Caristi＇s fixed point theorem. As an application. the classical Ekeland variational principle is generalized. In addition, it is proved that the generalized Caristi＇s fixed point theorem is equivalent to the generalized Ekeland variational principle.
