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
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
本文研究一类低秩矩阵优化问题,其中惩罚项为目标矩阵奇异值的l_(p)(0<p<1)正则函数.基于半阈值函数在稀疏/低秩恢复问题中的良好性能,本文提出奇异值半阈值(singular value half thresholding,SVHT)算法来求解l_(p)正则矩阵优化...本文研究一类低秩矩阵优化问题,其中惩罚项为目标矩阵奇异值的l_(p)(0<p<1)正则函数.基于半阈值函数在稀疏/低秩恢复问题中的良好性能,本文提出奇异值半阈值(singular value half thresholding,SVHT)算法来求解l_(p)正则矩阵优化问题.SVHT算法的主要迭代利用了子问题的闭式解,但与现有算法不同,其本质上是对目标函数在当前点进行局部1/2近似,而不是局部线性或局部二次近似.通过构造目标函数的Lipschitz和非Lipschitz近似函数,本文证明了SVHT算法生成序列的任意聚点都是问题的一阶稳定点.在数值实验中,利用模拟数据和实际图像数据的低秩矩阵补全问题对SVHT算法进行测试.大量的数值结果表明,SVHT算法对低秩矩阵优化问题在速度、精度和鲁棒性等方面都表现优异.展开更多
In this paper,we consider the system of Sturm-Liouville singular BVP and present a sufficient and necessary condition for the existence of positive solutions by means of the fixed point theorem for regular cones.
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.
文摘本文研究一类低秩矩阵优化问题,其中惩罚项为目标矩阵奇异值的l_(p)(0<p<1)正则函数.基于半阈值函数在稀疏/低秩恢复问题中的良好性能,本文提出奇异值半阈值(singular value half thresholding,SVHT)算法来求解l_(p)正则矩阵优化问题.SVHT算法的主要迭代利用了子问题的闭式解,但与现有算法不同,其本质上是对目标函数在当前点进行局部1/2近似,而不是局部线性或局部二次近似.通过构造目标函数的Lipschitz和非Lipschitz近似函数,本文证明了SVHT算法生成序列的任意聚点都是问题的一阶稳定点.在数值实验中,利用模拟数据和实际图像数据的低秩矩阵补全问题对SVHT算法进行测试.大量的数值结果表明,SVHT算法对低秩矩阵优化问题在速度、精度和鲁棒性等方面都表现优异.
基金supported by the NSF of Shandong Province (No.2010AL013)
文摘In this paper,we consider the system of Sturm-Liouville singular BVP and present a sufficient and necessary condition for the existence of positive solutions by means of the fixed point theorem for regular cones.
