Let a_1,..., a_9 be nonzero integers not of the same sign, and let b be an integer. Suppose that a_1,..., a_9 are pairwise coprime and a_1 + + a_9 ≡ b(mod 2). We apply the p-adic method of Davenport to find an explic...Let a_1,..., a_9 be nonzero integers not of the same sign, and let b be an integer. Suppose that a_1,..., a_9 are pairwise coprime and a_1 + + a_9 ≡ b(mod 2). We apply the p-adic method of Davenport to find an explicit P = P(a_1,..., a_9, n) such that the cubic equation a_1p_1~3+ + a9p_9~3= b is solvable with p_j 《 P for all 1 ≤ j ≤ 9. It is proved that one can take P = max{|a_1|,..., |a_9|}~c+ |b|^(1/3) with c = 2. This improves upon the earlier result with c = 14 due to Liu(2013).展开更多
The paper reports the dynamical study of a three-dimensional quadratic autonomous chaotic system with only two quadratic nonlinearities, which is a special case of the so-called conjugate Lue system. Basic properties ...The paper reports the dynamical study of a three-dimensional quadratic autonomous chaotic system with only two quadratic nonlinearities, which is a special case of the so-called conjugate Lue system. Basic properties of this system are analyzed by means of Lyapunov exponent spectrum and bifurcation diagram. The analysis shows that the system has complex dynamics with some interesting characteristics in which there are several periodic regions, but each of them has quite different periodic orbits.展开更多
基金supported by National Natural Science Foundation of China (Grant No. 11401154)
文摘Let a_1,..., a_9 be nonzero integers not of the same sign, and let b be an integer. Suppose that a_1,..., a_9 are pairwise coprime and a_1 + + a_9 ≡ b(mod 2). We apply the p-adic method of Davenport to find an explicit P = P(a_1,..., a_9, n) such that the cubic equation a_1p_1~3+ + a9p_9~3= b is solvable with p_j 《 P for all 1 ≤ j ≤ 9. It is proved that one can take P = max{|a_1|,..., |a_9|}~c+ |b|^(1/3) with c = 2. This improves upon the earlier result with c = 14 due to Liu(2013).
文摘The paper reports the dynamical study of a three-dimensional quadratic autonomous chaotic system with only two quadratic nonlinearities, which is a special case of the so-called conjugate Lue system. Basic properties of this system are analyzed by means of Lyapunov exponent spectrum and bifurcation diagram. The analysis shows that the system has complex dynamics with some interesting characteristics in which there are several periodic regions, but each of them has quite different periodic orbits.
