We study a class of quartic polynomial Poincare equations by applying a recurrence formula of focal value. We give the necessary and sufficient conditions for the origin to be a center, and prove that the order of fin...We study a class of quartic polynomial Poincare equations by applying a recurrence formula of focal value. We give the necessary and sufficient conditions for the origin to be a center, and prove that the order of fine focus at the origin for this class of equations is at most 6. Key words quartic polynomial Poincare equation - center - fine focus - order CLC number O 175. 12 Foundation item: Supported by the National Natural Science Foundation of China (19531070)Biography: TIAN De-sheng (1966-), male, Ph. D candidate, research direction: qualitative theory of differential equation.展开更多
A new general equation of state is presented, which can be used to express not only common cubic equations of state, but also quartic equations of state and so on. Main advantage of the new equation over the previous ...A new general equation of state is presented, which can be used to express not only common cubic equations of state, but also quartic equations of state and so on. Main advantage of the new equation over the previous general equations is that it is in simple form, and is easy to manipulate mathematically.展开更多
In this paper, we discuss the positive definite problem of a binary quartic form and obtain a necessary and sufficient condition. In addition we give two examples to show that there are some errors in the paper [1].
The goal of the present paper is to investigate some new HUR-stability results by applying the alternative fixed point of generalized quartic functional equationin β-Banach modules on Banach algebras. The concept of ...The goal of the present paper is to investigate some new HUR-stability results by applying the alternative fixed point of generalized quartic functional equationin β-Banach modules on Banach algebras. The concept of Ulam-Hyers-Rassias stability (briefly, HUR-stability) originated from Th. M. Rassias stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.展开更多
We prove that diophantine equation in title has at most one positive integer solution for any positive integers A>1, B>1. It follows that Lucas problem is very simple to solve and a recent result of Bennett ...We prove that diophantine equation in title has at most one positive integer solution for any positive integers A>1, B>1. It follows that Lucas problem is very simple to solve and a recent result of Bennett is very simple to prove.展开更多
For any fixed odd prime p, let N(p) denote the number of positive integer solutions (x, y) of the equation y^2 = px(x^2 + 2). In this paper, using some properties of binary quartic Diophantine equations, we pro...For any fixed odd prime p, let N(p) denote the number of positive integer solutions (x, y) of the equation y^2 = px(x^2 + 2). In this paper, using some properties of binary quartic Diophantine equations, we prove that ifp ≡ 5 or 7(mod 8), then N(p) = 0; ifp ≡ 1(mod 8), then N(p) 〈 1; if p〉 3 andp ≡ 3(rood 8), then N(p) ≤ 2.展开更多
文摘We study a class of quartic polynomial Poincare equations by applying a recurrence formula of focal value. We give the necessary and sufficient conditions for the origin to be a center, and prove that the order of fine focus at the origin for this class of equations is at most 6. Key words quartic polynomial Poincare equation - center - fine focus - order CLC number O 175. 12 Foundation item: Supported by the National Natural Science Foundation of China (19531070)Biography: TIAN De-sheng (1966-), male, Ph. D candidate, research direction: qualitative theory of differential equation.
文摘A new general equation of state is presented, which can be used to express not only common cubic equations of state, but also quartic equations of state and so on. Main advantage of the new equation over the previous general equations is that it is in simple form, and is easy to manipulate mathematically.
文摘In this paper, we discuss the positive definite problem of a binary quartic form and obtain a necessary and sufficient condition. In addition we give two examples to show that there are some errors in the paper [1].
文摘The goal of the present paper is to investigate some new HUR-stability results by applying the alternative fixed point of generalized quartic functional equationin β-Banach modules on Banach algebras. The concept of Ulam-Hyers-Rassias stability (briefly, HUR-stability) originated from Th. M. Rassias stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.
文摘We prove that diophantine equation in title has at most one positive integer solution for any positive integers A>1, B>1. It follows that Lucas problem is very simple to solve and a recent result of Bennett is very simple to prove.
基金Foundation item: Supported by the Natural Science Foundation of Shaanxi Province(2009JM1006)
文摘For any fixed odd prime p, let N(p) denote the number of positive integer solutions (x, y) of the equation y^2 = px(x^2 + 2). In this paper, using some properties of binary quartic Diophantine equations, we prove that ifp ≡ 5 or 7(mod 8), then N(p) = 0; ifp ≡ 1(mod 8), then N(p) 〈 1; if p〉 3 andp ≡ 3(rood 8), then N(p) ≤ 2.