The Hopf＇s maximum principles are utilized to obtain maximum principles for functions defined on solutions of nonlinear elliptic equations in divergence form （g（u）u,i）,i ＋f（x,u,q）=0（q=｜△↓u｜^2）, subject T...The Hopf＇s maximum principles are utilized to obtain maximum principles for functions defined on solutions of nonlinear elliptic equations in divergence form （g（u）u,i）,i ＋f（x,u,q）=0（q=｜△↓u｜^2）, subject The principles derived may be used to deduce bounds on the gradient q.展开更多
Gutin and Rafiey（Australas J. Combin. 34（2006）, 17-21） provided an example of an n-partite tournament with exactly n-m ＋ 1 cycles of length of m for any given m with 4 ≤ m ≤ n, and posed the following question....Gutin and Rafiey（Australas J. Combin. 34（2006）, 17-21） provided an example of an n-partite tournament with exactly n-m ＋ 1 cycles of length of m for any given m with 4 ≤ m ≤ n, and posed the following question. Let 3 ≤ m ≤n and n ≥ 4. Are there strong n-partite tournaments, which are not themselves tournaments, with exactly n-m ＋ 1 cycles of length m for two values of m？ In the same paper,they showed that this question has a negative answer for two values n-1 and n. In this paper, we prove that a strong n-partite tournament with exactly two cycles of length n-1 must contain some given multipartite tournament as subdigraph. As a corollary, we also show that the above question has a negative answer for two values n-1 and any l with 3 ≤ l ≤ n and l ≠n-1.展开更多
基金Supported by the National Natural Science Foundation of China （No.60174007） and PNSFS.
文摘The Hopf＇s maximum principles are utilized to obtain maximum principles for functions defined on solutions of nonlinear elliptic equations in divergence form （g（u）u,i）,i ＋f（x,u,q）=0（q=｜△↓u｜^2）, subject The principles derived may be used to deduce bounds on the gradient q.
基金supported by the Natural Science Young Foundation of China(No.11701349)by the Natural Science Foundation of Shanxi Province,China(No.201601D011005)by Shanxi Scholarship Council of China(2017-018)
文摘Gutin and Rafiey（Australas J. Combin. 34（2006）, 17-21） provided an example of an n-partite tournament with exactly n-m ＋ 1 cycles of length of m for any given m with 4 ≤ m ≤ n, and posed the following question. Let 3 ≤ m ≤n and n ≥ 4. Are there strong n-partite tournaments, which are not themselves tournaments, with exactly n-m ＋ 1 cycles of length m for two values of m？ In the same paper,they showed that this question has a negative answer for two values n-1 and n. In this paper, we prove that a strong n-partite tournament with exactly two cycles of length n-1 must contain some given multipartite tournament as subdigraph. As a corollary, we also show that the above question has a negative answer for two values n-1 and any l with 3 ≤ l ≤ n and l ≠n-1.
