In this paper, we establish an estimate for the solutions of small-divisor equation of higher order with large variable coefficient. Then by formulating an infinite-dimensional KAM theorem which allows for multiple no...In this paper, we establish an estimate for the solutions of small-divisor equation of higher order with large variable coefficient. Then by formulating an infinite-dimensional KAM theorem which allows for multiple normal frequencies and unbounded perturbations, we prove that there are many periodic solutions for the coupled KdV equation subject to small Hamiltonian perturbations.展开更多
A class of weaker nondegeneracy conditions is given and an existence theorem of invariant tori is prove n for small perturbations of degenerate integrable infinite dimensional Hamiltonian systems under the weaker nond...A class of weaker nondegeneracy conditions is given and an existence theorem of invariant tori is prove n for small perturbations of degenerate integrable infinite dimensional Hamiltonian systems under the weaker nondegeneracy conditions. The measure estimates of the parameter set are also given for which invariant tori exist. It is valuable to point out that by the motivation of finite dimensional situation the nondegeneracy conditions may be the weakest. Mainly KAM machine is used to prove the existence of invariant tori. The measure estimates for small divisor conditions, on which the measure estimates of the parameter set are based, will be given in the second paper.展开更多
In this paper, we prove that a Hamiltonian system possesses either a four-dimensional invaxiant disc or an invariant Cantor set with positive （n ＋ 2）-dimensional Lebesgue measure in the neighborhood of an elliptic ...In this paper, we prove that a Hamiltonian system possesses either a four-dimensional invaxiant disc or an invariant Cantor set with positive （n ＋ 2）-dimensional Lebesgue measure in the neighborhood of an elliptic equilibrium provided that its lineaxized system at the equilibrium satisfies some small divisor conditions. Both of the invariant sets are foliated by two-dimensional invaxiant tori carrying quasi-oeriodic solutions.展开更多
A semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. A bounded semiring is a semiring equipped with a compatible bounded partial order. I...A semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. A bounded semiring is a semiring equipped with a compatible bounded partial order. In this paper, properties of zero divisors and prime elements of a bounded semiring are studied. In particular, it is proved that under some mild assumption, the set Z（A） of nonzero zero divisors of A is A ／ {0, 1}, and each prime element of A is a maximal element. For a bounded semiring A with Z（A） = A ／ {0, 1}, it is proved that A has finitely many maximal elements if ACC holds either for elements of A or for principal annihilating ideals of A. As an application of prime elements, we show that the structure of a bounded semiring A is completely determined by the structure of integral bounded semirings if either ｜Z（A）｜ = 1 or ｜Z（A）｜ -- 2 and Z（A）2 ≠ 0. Applications to the ideal structure of commutative rings are also considered. In particular, when R has a finite number of ideals, it is shown that the chain complex of the poset I（R） is pure and shellable, where I（R） consists of all ideals of R.展开更多
基金Supported by National Natural Science Foundation of China （Grant No. 10725103）, 973 Program （Grant No 2010CB327900） and Research Foundation for Doctor Programme （Grant No. 20080246） The authors are indebted to X. Yuan for his helpful discussion and encouragements, and to the two reviewers for invaluable suggestions.
文摘In this paper, we establish an estimate for the solutions of small-divisor equation of higher order with large variable coefficient. Then by formulating an infinite-dimensional KAM theorem which allows for multiple normal frequencies and unbounded perturbations, we prove that there are many periodic solutions for the coupled KdV equation subject to small Hamiltonian perturbations.
基金the National Natural Science Foundation of China.
文摘A class of weaker nondegeneracy conditions is given and an existence theorem of invariant tori is prove n for small perturbations of degenerate integrable infinite dimensional Hamiltonian systems under the weaker nondegeneracy conditions. The measure estimates of the parameter set are also given for which invariant tori exist. It is valuable to point out that by the motivation of finite dimensional situation the nondegeneracy conditions may be the weakest. Mainly KAM machine is used to prove the existence of invariant tori. The measure estimates for small divisor conditions, on which the measure estimates of the parameter set are based, will be given in the second paper.
基金Supported by NNSF of China (Grant 10531050)the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20070284004)
文摘In this paper, we prove that a Hamiltonian system possesses either a four-dimensional invaxiant disc or an invariant Cantor set with positive （n ＋ 2）-dimensional Lebesgue measure in the neighborhood of an elliptic equilibrium provided that its lineaxized system at the equilibrium satisfies some small divisor conditions. Both of the invariant sets are foliated by two-dimensional invaxiant tori carrying quasi-oeriodic solutions.
基金Acknowledgements This work was supported by the National Natural Science Foundation of China （Grant No. 11271250）.
文摘A semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. A bounded semiring is a semiring equipped with a compatible bounded partial order. In this paper, properties of zero divisors and prime elements of a bounded semiring are studied. In particular, it is proved that under some mild assumption, the set Z（A） of nonzero zero divisors of A is A ／ {0, 1}, and each prime element of A is a maximal element. For a bounded semiring A with Z（A） = A ／ {0, 1}, it is proved that A has finitely many maximal elements if ACC holds either for elements of A or for principal annihilating ideals of A. As an application of prime elements, we show that the structure of a bounded semiring A is completely determined by the structure of integral bounded semirings if either ｜Z（A）｜ = 1 or ｜Z（A）｜ -- 2 and Z（A）2 ≠ 0. Applications to the ideal structure of commutative rings are also considered. In particular, when R has a finite number of ideals, it is shown that the chain complex of the poset I（R） is pure and shellable, where I（R） consists of all ideals of R.
