Consider the differential equationii-λp（t）u=0, （1）where the parameter γ∈R<sup>+</sup>, and t∈C u（t）∈C, and ρ（t） is the elliptic function ofWeierstrass with periods ω<sub>1</sub>...Consider the differential equationii-λp（t）u=0, （1）where the parameter γ∈R<sup>+</sup>, and t∈C u（t）∈C, and ρ（t） is the elliptic function ofWeierstrass with periods ω<sub>1</sub>,=2α and ω<sub>2</sub>=2αi （α∈R）. It is shown that ρ（t） has the following properties: （i） ρ（0） =0, （ii） ρ（it）=-ρ（t）,t∈C （iii） for any t∈R, ρ（t）∈R and ρ（t）≤0, and for t∈[-α, 0], ρ（t） increases from展开更多
We consider in this paper the boundary value problems of nonlinear systems the form εY″=F(t,Y,Y′,ε), -1<t<1, Y(-1,ε)=A(ε), Y(1,ε)=B(ε). Supoosing some or all of the components of F , that is, ...We consider in this paper the boundary value problems of nonlinear systems the form εY″=F(t,Y,Y′,ε), -1<t<1, Y(-1,ε)=A(ε), Y(1,ε)=B(ε). Supoosing some or all of the components of F , that is, f i satisfy 2 f y′ 2 i t =0 =0, we say that F possesses a generalized turning point at t =0. Our goal is to give sufficient conditions for the existence of solution of the problems and to study the asymptotic behavior of the solution when F possesses a generalized turning point at t =0. We mainly discuss regular singular crossings.展开更多
As we know that the power series method is a very effective method for solving the Ordinary differential equations (ODEs) which have variable coefficient, so in this paper we have studied how to solve second-order ord...As we know that the power series method is a very effective method for solving the Ordinary differential equations (ODEs) which have variable coefficient, so in this paper we have studied how to solve second-order ordinary differential equation with variable coefficient at a singular point <em>t</em> = 0 and determined the form of second linearly independent solution. Based on the roots of initial equation there are real and complex cases. When the roots of initial equation are real then there are three kinds of second linearly independent solutions. If the roots of the initial equation are distinct complex numbers, then the solution is complex-valued.展开更多
Lee等人(LEE C F,CHEN H L.A novel data hiding scheme based on modulus function.The Journal of Systems and Software,2010,83(5):832-843)提出了一种基于模函数的数据隐写方法,在图像质量可接受的情况下,每个像素的最大嵌入容量为...Lee等人(LEE C F,CHEN H L.A novel data hiding scheme based on modulus function.The Journal of Systems and Software,2010,83(5):832-843)提出了一种基于模函数的数据隐写方法,在图像质量可接受的情况下,每个像素的最大嵌入容量为4位二进制数。但是当每个像素的嵌入量为4位时,隐写图像的质量较差,容易引起攻击者的注意。对该方法进行了改进,缩小了像素改变的范围。理论分析及模拟实验表明,改进方法不但保留了原方法的各种优点,而且使PSNR值增加1.5～3.5 dB,提高了隐写图像的视觉不可见性及抵御RS攻击的能力。展开更多
In this paper, we consider the general ordinary quasi-differential expression τ of order n with complex coefficients and its formal adjoint τ<sup>+</sup> on the interval [a,b). We shall show in the case ...In this paper, we consider the general ordinary quasi-differential expression τ of order n with complex coefficients and its formal adjoint τ<sup>+</sup> on the interval [a,b). We shall show in the case of one singular end-point and under suitable conditions that all solutions of a general ordinary quasi-differential equation are in the weighted Hilbert space provided that all solutions of the equations and its adjoint are in . Also, a number of results concerning the location of the point spectra and regularity fields of the operators generated by such expressions may be obtained. Some of these results are extensions or generalizations of those in the symmetric case, while the others are new.展开更多
In this paper, we have considered the general ordinary quasi-differential operators generated by a general quasi-differential expression τ<sub>p,q</sub> in L<sup>p</sup>w</sub>-spaces of...In this paper, we have considered the general ordinary quasi-differential operators generated by a general quasi-differential expression τ<sub>p,q</sub> in L<sup>p</sup>w</sub>-spaces of order n with complex coefficients and its formal adjoint τ<sup>+</sup><sub>q',p' </sub>in L<sup>p</sup>w</sub>-spaces for arbitrary p,q∈[1,∞). We have proved in the case of one singular end-point that all well-posed extensions of the minimal operator T<sub>0</sub> (τ<sub>p,q</sub>) generated by such expression τ<sub>p,q</sub> and their formal adjoint on the interval [a,b) with maximal deficiency indices have resolvents which are Hilbert-Schmidt integral operators and consequently have a wholly discrete spectrum. This implies that all the regularly solvable operators have all the standard essential spectra to be empty. Also, a number of results concerning the location of the point spectra and regularity fields of the operators generated by such expressions can be obtained. Some of these results are extensions or generalizations of those in the symmetric case, while others are new.展开更多
In this paper, we consider the general quasi-differential expressions each of order n with complex coefficients and their formal adjoints on the interval (a,b). It is shown in direct sum spaces of functions defined on...In this paper, we consider the general quasi-differential expressions each of order n with complex coefficients and their formal adjoints on the interval (a,b). It is shown in direct sum spaces of functions defined on each of the separate intervals with the cases of one and two singular end-points and when all solutions of the equation and its adjoint are in (the limit circle case) that all well-posed extensions of the minimal operator have resolvents which are HilbertSchmidt integral operators and consequently have a wholly discrete spectrum. This implies that all the regularly solvable operators have all the standard essential spectra to be empty. These results extend those of formally symmetric expression studied in [1-10] and those of general quasi-differential expressions in [11-19].展开更多
Given general quasi-differential expressions , each of order n with complex coefficients and their formal adjoint are on the interval [a,b) respectively, we give a characterization of all regularly solvable operators ...Given general quasi-differential expressions , each of order n with complex coefficients and their formal adjoint are on the interval [a,b) respectively, we give a characterization of all regularly solvable operators and their adjoints generated by a general ordinary quasi-differential expression in the direct sum Hilbert spaces . The domains of these operators are described in terms of boundary conditions involving -solutions of the equations and their adjoint on the intervals [a<sub>p</sub>,b<sub>p</sub>). This characterization is an extension of those obtained in the case of one interval with one and two singular end-points of the interval (a,b), and is a generalization of those proved in the case of self-adjoint and J-self-adjoint differential operators as a special case, where J denotes complex conjugation.展开更多
文摘Consider the differential equationii-λp（t）u=0, （1）where the parameter γ∈R<sup>+</sup>, and t∈C u（t）∈C, and ρ（t） is the elliptic function ofWeierstrass with periods ω<sub>1</sub>,=2α and ω<sub>2</sub>=2αi （α∈R）. It is shown that ρ（t） has the following properties: （i） ρ（0） =0, （ii） ρ（it）=-ρ（t）,t∈C （iii） for any t∈R, ρ（t）∈R and ρ（t）≤0, and for t∈[-α, 0], ρ（t） increases from
文摘We consider in this paper the boundary value problems of nonlinear systems the form εY″=F(t,Y,Y′,ε), -1<t<1, Y(-1,ε)=A(ε), Y(1,ε)=B(ε). Supoosing some or all of the components of F , that is, f i satisfy 2 f y′ 2 i t =0 =0, we say that F possesses a generalized turning point at t =0. Our goal is to give sufficient conditions for the existence of solution of the problems and to study the asymptotic behavior of the solution when F possesses a generalized turning point at t =0. We mainly discuss regular singular crossings.
文摘As we know that the power series method is a very effective method for solving the Ordinary differential equations (ODEs) which have variable coefficient, so in this paper we have studied how to solve second-order ordinary differential equation with variable coefficient at a singular point <em>t</em> = 0 and determined the form of second linearly independent solution. Based on the roots of initial equation there are real and complex cases. When the roots of initial equation are real then there are three kinds of second linearly independent solutions. If the roots of the initial equation are distinct complex numbers, then the solution is complex-valued.
文摘Lee等人(LEE C F,CHEN H L.A novel data hiding scheme based on modulus function.The Journal of Systems and Software,2010,83(5):832-843)提出了一种基于模函数的数据隐写方法,在图像质量可接受的情况下,每个像素的最大嵌入容量为4位二进制数。但是当每个像素的嵌入量为4位时,隐写图像的质量较差,容易引起攻击者的注意。对该方法进行了改进,缩小了像素改变的范围。理论分析及模拟实验表明,改进方法不但保留了原方法的各种优点,而且使PSNR值增加1.5～3.5 dB,提高了隐写图像的视觉不可见性及抵御RS攻击的能力。
文摘In this paper, we consider the general ordinary quasi-differential expression τ of order n with complex coefficients and its formal adjoint τ<sup>+</sup> on the interval [a,b). We shall show in the case of one singular end-point and under suitable conditions that all solutions of a general ordinary quasi-differential equation are in the weighted Hilbert space provided that all solutions of the equations and its adjoint are in . Also, a number of results concerning the location of the point spectra and regularity fields of the operators generated by such expressions may be obtained. Some of these results are extensions or generalizations of those in the symmetric case, while the others are new.
文摘In this paper, we have considered the general ordinary quasi-differential operators generated by a general quasi-differential expression τ<sub>p,q</sub> in L<sup>p</sup>w</sub>-spaces of order n with complex coefficients and its formal adjoint τ<sup>+</sup><sub>q',p' </sub>in L<sup>p</sup>w</sub>-spaces for arbitrary p,q∈[1,∞). We have proved in the case of one singular end-point that all well-posed extensions of the minimal operator T<sub>0</sub> (τ<sub>p,q</sub>) generated by such expression τ<sub>p,q</sub> and their formal adjoint on the interval [a,b) with maximal deficiency indices have resolvents which are Hilbert-Schmidt integral operators and consequently have a wholly discrete spectrum. This implies that all the regularly solvable operators have all the standard essential spectra to be empty. Also, a number of results concerning the location of the point spectra and regularity fields of the operators generated by such expressions can be obtained. Some of these results are extensions or generalizations of those in the symmetric case, while others are new.
文摘In this paper, we consider the general quasi-differential expressions each of order n with complex coefficients and their formal adjoints on the interval (a,b). It is shown in direct sum spaces of functions defined on each of the separate intervals with the cases of one and two singular end-points and when all solutions of the equation and its adjoint are in (the limit circle case) that all well-posed extensions of the minimal operator have resolvents which are HilbertSchmidt integral operators and consequently have a wholly discrete spectrum. This implies that all the regularly solvable operators have all the standard essential spectra to be empty. These results extend those of formally symmetric expression studied in [1-10] and those of general quasi-differential expressions in [11-19].
文摘Given general quasi-differential expressions , each of order n with complex coefficients and their formal adjoint are on the interval [a,b) respectively, we give a characterization of all regularly solvable operators and their adjoints generated by a general ordinary quasi-differential expression in the direct sum Hilbert spaces . The domains of these operators are described in terms of boundary conditions involving -solutions of the equations and their adjoint on the intervals [a<sub>p</sub>,b<sub>p</sub>). This characterization is an extension of those obtained in the case of one interval with one and two singular end-points of the interval (a,b), and is a generalization of those proved in the case of self-adjoint and J-self-adjoint differential operators as a special case, where J denotes complex conjugation.
