In this article, we will derive an equality, where the Taylor series expansion around ε= 0 for any asymptotical analytical solution of the perturbed partial differential equation （PDE） with perturbing parameter e m...In this article, we will derive an equality, where the Taylor series expansion around ε= 0 for any asymptotical analytical solution of the perturbed partial differential equation （PDE） with perturbing parameter e must be admitted. By making use of the equality, we may obtain a transformation, which directly map the analytical solutions of a given unperturbed PDE to the asymptotical analytical solutions of the corresponding perturbed one. The notion of Lie-Bgcklund symmetries is introduced in order to obtain more transformations. Hence, we can directly create more transformations in virtue of known Lie-Bgcklund symmetries and recursion operators of corresponding unperturbed equation. The perturbed Burgers equation and the perturbed Korteweg-de Vries （KdV） equation are used as examples.展开更多
In [l], a property of roots of polynomials is considered, which involves the existence of local analytic solutions of polynomial-like functional iterative equations. In this paper we discuss this property and obtain a...In [l], a property of roots of polynomials is considered, which involves the existence of local analytic solutions of polynomial-like functional iterative equations. In this paper we discuss this property and obtain a succinct condition to decide whether this property holds. Our main result is: A polynomialλnzn+''' + λ2z2 + λlz + λ0 of degree n has a root or such that inf{|λnanm +... + λ2a2m + λ1am+ λ0|: m = 2, 3,.. .} > 0 if and only if at least one of the following two conditions holds: (i) the polynomial has a root β satisfying ｜β｜ > 1; (ii) the polynomial has a root β satisfying ｜β｜ < 1, and λ0≠0展开更多
In this paper we use an alternative method to study analytically and numerically for a nonlocal elastic bar in tension.The equilibrium equation of the model is a Fredholm integral equation of the second kind.With the ...In this paper we use an alternative method to study analytically and numerically for a nonlocal elastic bar in tension.The equilibrium equation of the model is a Fredholm integral equation of the second kind.With the aid of an efficient iterative method,we are able to get the approximate analytical solution.For the purpose of comparisons,numerical solutions are also obtained for two types of nonlocal kernels,which show the validity of the analytical solution.The effects of some related parameters are also investigated.展开更多
We present in the work two intriguing results in the entanglement classification of a pure and true tripartite entangled state of 2 × M × N under stochastic local operation and classical communication: (i) t...We present in the work two intriguing results in the entanglement classification of a pure and true tripartite entangled state of 2 × M × N under stochastic local operation and classical communication: (i) the internal symmetric properties of the nonlocal parameters in the continuous entangled class; (ii) the analytic expression for the total numbers of the true and pure entangled class 2 × M × N states. These properties help better understand the nature of the 2 × M × N entangled system.展开更多
基金0ne of the authors （H.Z. Liu） would like to express his sincere thanks to Dr. Shou-Feng Shen for his continuous encouragement and warm-hearted help.
文摘In this article, we will derive an equality, where the Taylor series expansion around ε= 0 for any asymptotical analytical solution of the perturbed partial differential equation （PDE） with perturbing parameter e must be admitted. By making use of the equality, we may obtain a transformation, which directly map the analytical solutions of a given unperturbed PDE to the asymptotical analytical solutions of the corresponding perturbed one. The notion of Lie-Bgcklund symmetries is introduced in order to obtain more transformations. Hence, we can directly create more transformations in virtue of known Lie-Bgcklund symmetries and recursion operators of corresponding unperturbed equation. The perturbed Burgers equation and the perturbed Korteweg-de Vries （KdV） equation are used as examples.
基金Supported by the National Natural Science Foundation of China.
文摘In [l], a property of roots of polynomials is considered, which involves the existence of local analytic solutions of polynomial-like functional iterative equations. In this paper we discuss this property and obtain a succinct condition to decide whether this property holds. Our main result is: A polynomialλnzn+''' + λ2z2 + λlz + λ0 of degree n has a root or such that inf{|λnanm +... + λ2a2m + λ1am+ λ0|: m = 2, 3,.. .} > 0 if and only if at least one of the following two conditions holds: (i) the polynomial has a root β satisfying ｜β｜ > 1; (ii) the polynomial has a root β satisfying ｜β｜ < 1, and λ0≠0
基金supported by the City University of Hong Kong (Grant No. 7008111)
文摘In this paper we use an alternative method to study analytically and numerically for a nonlocal elastic bar in tension.The equilibrium equation of the model is a Fredholm integral equation of the second kind.With the aid of an efficient iterative method,we are able to get the approximate analytical solution.For the purpose of comparisons,numerical solutions are also obtained for two types of nonlocal kernels,which show the validity of the analytical solution.The effects of some related parameters are also investigated.
基金supported by the National Natural Science Foundation of China (Grant Nos. 10935012, 10928510, 10821063 and 10775179)the Chinese Academy of Sciences Key Projects (Grant Nos. KJCX2-yw-N29 andH92A0200S2)the Scientific Research Fund of Graduate University, the Chinese Academy of Sciences
文摘We present in the work two intriguing results in the entanglement classification of a pure and true tripartite entangled state of 2 × M × N under stochastic local operation and classical communication: (i) the internal symmetric properties of the nonlocal parameters in the continuous entangled class; (ii) the analytic expression for the total numbers of the true and pure entangled class 2 × M × N states. These properties help better understand the nature of the 2 × M × N entangled system.
