数学问题解答

１９９９年７月号问题解答（解答由问题提供人给出）１２０１．已知ｘ,ｙ,ｚ都是正实数,且满足ｘ·ｙ·ｚ＝１;求证ｘ２ｙ＋ｚ＋ｙ２ｚ＋ｘ＋ｚ２ｘ＋ｙ≥３２;证明：设ｓ＝ｘ＋ｙ＋ｚ,则ｘ２ｙ＋ｚ＋ｙ２ｚ＋ｘ＋ｚ２ｘ＋ｙ＝ｘ２ｓ－ｘ＋ｙ２ｓ－ｙ＋ｚ２ｓ－ｚ＝ｘ２ｓ－ｘ＋ｘ＋ｙ２ｓ－ｙ＋ｙ＋ｚ２ｓ－ｚ＋ｚ－ｓ＝ｓ·ｘｓ－ｘ＋ｙｓ－ｙ＋ｚｓ－ｚ－ｓ＝ｓ·ｘｓ－ｘ＋１＋ｙｓ－ｙ＋１＋ｚｓ－ｚ＋１－３－ｓ＝ｓ２·１ｓ－ｘ＋１ｓ－ｙ＋１ｓ－ｚ－４ｓ≥ｓ２·３２（ｓ－ｘ）＋（ｓ－ｙ）＋（ｓ－ｚ）－４ｓ＝９２ｓ－４ｓ…     

关于点可数序列邻域网

Ｉｔｉｓｋｎｏｗｎｔｏａｌｌｔｈａｔｔｈｅｔｈｅｏｒｙｏｆｔｈｅｐｏｉｎｔ－ｃｏｕｎｔａｂｌｅｃｏｖｅｒｓｂｅｏｎｅｏｆｔｈｅｍｏｓｔｉｍｐｏｒ－ｔａｎｔｓｕｂｊｅｃｔｓｉｎＧｅｎｅｒａｌＴｏｐｏｌｏｇｙ．Ｔｈｅｐｏｉｎｔ－ｃｏｕｎｔａｂｌｅｃｏｖｅｒｓｗｉｔｈｖａｒｉｏｕｓｃｈａｒａｃｔｅｒｈａｖｅｂｅｅｎｄｉｓｃｕｓｓｅｄｂｙｍａｎｙｔｏｐｏｌｏｇｉｓｔｓ,ｓｕｃｈａｓＬｉｎａｎｄＴａｎａｋａ’ｓｐａｐｅｒ［１］ｏｎｐｏｉｎｔ－ｃｏｕｎｔａｂｌｅＫ－ｎｅｔｗｏｒｋ．Ｉｎｔｈｉｓｐａｐｅｒ,ｗ…     

The Degree Theory of a New Operator

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＩｔｉｓｗｅｌｌｋｎｏｗｎｔｈａｔｔｈｅｄｅｇｒｅｅｔｈｅｏｒｙｆｏｒａｋ－ｓｅｔ－ｃｏｎｔｒａｃｔｏｐｅｒａｔｏｒｈａｓｍａｎｙａｐｐｌｉｃａｔｉｏｎｓｏｎｔｈｅｅｘｉｓｔｅｎｃｅｏｆｔｈｅｓｏｌｕｔｉｏｎｓｏｆｓｏｍｅｅｑｕａｔｉｏｎｓ（ｓｅｅ［１］，［２］，［３］）．Ｈｏｗｅｖｅｒ，ｓｏｍｅｏｐｅｒａ－ｔｏｒｓａｒｅｎｏｔｋ－ｓｅｔ－ｃｏｎｔｒａｃｔｏｐｅｒａｔｏｒｓ．Ｉｔｉｓａｎｅｗｐｒｏｂｌｅｍｔｈａｔｗｅｅｓｔａｂｌｉｓｈａｄｅｇｒｅｅｔｈｅｏｒｙｆｏ…     

On the Finite-dimensional Optimal Approximations to the Infinite dimensional Linear Operators

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＨａｎｄＫｂｅｒｅａｌｏｒｃｏｍｐｌｅｘｉｎｆｉｎｉｔｅｄｉｍｅｎｓｉｏｎａｌＨｉｌｂｅｒｓｐａｃｅ．Ａ（ｂｏｕｎｄｅｄｌｉｎｅａｒ）ｏｐ－ｅｒａｔｏｒＡ：Ｈ→Ｋｉｓｓａｉｄｔｏｂｅｉｎｆｉｎｉｔｅ－ｄｉｍｅｎｓｉｏｎａｌ（ｏｒ,ｆｉｎｉｔｅ－ｄｉｍｅｎｓｉｏｎａｌ）ｉｆｔｈｅｒａｎｇｅｏｆＡｉｓｉｎｆｉｎｉｔｅｄｉｍｅｎｓｉｏｎａｌ（ｏｒ,ｆｉｎｉｔｅｄｉｍｅｎｓｉｏｎａｌ）．Ａｉｓｓａｉｄｔｏｂｅｎ－ｄｉｍｅｎｓｉｏｎａｌｉｆｔｈｅｒａｎｇｅｏｆＡｉｓｎ…     

On Torsion Theories of Regulars and V—Rings

ＬｅｔＲｂｅａｃｏｍｍｕｔａｔｉｖｅｒｉｎｇｗｉｔｈｉｄｅｎｔｉｔｙ，τｂｅａｈｅｒｅｄｉｔａｒｙｔｏｒｓｉｏｎｔｈｅｏｒｙｏｎＲ－ｍｏｄ．Ｔｈｅｃｏｎｃｅｐｔｉｏｎｓｗｈｉｃｈｗｅｈａｖｅｕｓｅｄｉｎｔｈｅｐａｐｅｒｃａｎｂｅｓｅｅｎｉｎ［１］．Ｄｅｆｉｎｉｔｉｏｎ１　ＴｈｅｒｉｎｇＲｉｓτυ－ｒｉｎｇｉｆｅｖｅｒｙｓｉｍｐｌｅｍｏｄｕｌｅｉｓτ－ｉｒｊｅｃｔｉｖｅ，ｔｈｅｒｉｎｇＲｉｓτ－ｒｅｇｕｌａｒｒｉｎｇｉｆｅｖｅｒｙＲ－ｍｏｄｕｌｅｉｓτ－ｆｌａｔ．Ｚｈａｎｇ［２］ｓｔｕｄｉｅｄτυ…     

Minimum Ellipsiod and Optim alDesign

§０．　ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｅｐａｐｅｒ［１］，ＫｉｅｆｅｒａｎｄＷｏｌｆｏｗｉｔｚｓｏｌｖｅｄｔｈｅｅｑｕｉｖａｌｅｎｃｅｏｆＧ－ｏｐｔｉｍａｌｄｅｓｉｇｎａｎｄＤ－ｏｐｔｉｍａｌｄｅｓｉｇｎ．Ｉｎｔｈｅｐａｐｅｒ［２］，ＫａｉｌｉｎａｎｄＳｔｕｄｄｅｎｓｔｕｄｉｅｄｔｈｅａｌｇｅｂｒａｉｃｓｔｒｕｃｔｕｏｅｏｆｃｏｎｃｅｓ－ｓｉｏｎａｌｄｅｓｉｇｎ，Ｄ－ｏｐｔｉｍａｌａｎｄＣ－ｏｐｔｉｍａｌｄｅｓｉｇｎ．Ｉｎｔｈｅｐａｐｅｒ［３］，ＳｐｒｕｉｌｌｓｔｕｄｉｅｄｔｈｅＣ－ｏｐｔｉｍ…     

一道错题与一个有意义的结论

题目：若ｔｇＡ、ｔｇ３Ａ是方程ｘ２＋６ｘ＋７＝０的根．求２ｓｉｎ２Ａ－ｃｏｓ４Ａ－１２ｃｏｓ２Ａ－ｃｏｓ４Ａ－１的值．这是某地区的一道期末统考题．命题者提供的解答是：∵２ｓｉｎ２Ａ－ｃｏｓ４Ａ－１２ｃｏｓ２Ａ－ｃｏｓ４Ａ－１＝－ｃｏｓ２Ａ－ｃｏｓ４Ａ...     

Normal TL-subgroups of TL-subgroup

ＮｏｒｍａｌＴＬ－ｓｕｂｇｒｏｕｐｓｏｆＴＬ－ｓｕｂｇｒｏｕｐＮｏｒｍａｌＴＬ－ｓｕｂｇｒｏｕｐｓｏｆＴＬ－ｓｕｂｇｒｏｕｐＹｕＹａｎｄｏｎｇ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＹａｎｃｈｅｎｇＴｅａｃｈｅｒｓＣｏｌｅｇｅ，Ｙａｎｃｈ...     

CO—s闭拓扑

本文证明，当空间赋予ＣＯ－ｓ－闭石扑ｓ（τ）时，ｓ－连续函数概念与连续函数概念一致，研究了ｓ－ｘ闭空间；ｓ－闭空间的某些基本性质，且用其刻画几类ＣＯ－ｓ闭拓扑。     

Moore-Penrose Inverses and Group Inverses of Block k-Circulant Matrices

Ｍｏｏｒｅ－Ｐｅｎｒｏｓｅ　Ｉｎｖｅｒｓｅｓ　ａｎｄ　Ｇｒｏｕｐ　Ｉｎｖｅｒｓｅｓ　ｏｆ　Ｂｌｏｃｋ　ｋ－Ｃｉｒｃｕｌａｎｔ　ＭａｔｒｉｃｅｓＭｏｏｒｅ－ＰｅｎｒｏｓｅＩｎｖｅｒｓｅｓａｎｄＧｒｏｕｐＩｎｖｅｒｓｅｓｏｆＢｌｏｃｋｋ－Ｃｉｒｃｕｌａｎ...     

微分代数系统的标准型和分支

我们首先把微分代数系统归结为受限微分方程,且讨论了受限微分系统的参数化问题,即系统的状态空间形式。然后,借助于受限系统的状态空间形式,我们得到了受限微分系统平行于常微分方程的线性化问题和Ｈｏｐｆ分支理论问题的有关结果。     

LINEAR SYSTEMS ASSOCIATED WITH NUMERICAL METHODS FOR CONSTRAINED OPITMIZATION

Linear systems associated with numerical methods for constrained optimization are discussed in this paper. It is shown that the corresponding subproblems arise in most well-known methods, no matter line search methods or trust region methods for constrained optimization can be expressed as similar systems of linear equations. All these linear systems can be viewed as some kinds of approximation to the linear system derived by the Lagrange-Newton method. Some properties of these linear systems are analyzed.     

约束Hamilton系统的Lie对称性及其在场论中的应用

研究了约束Hamilton系统的Lie对称性,得到了场论系统的守恒量.首先给出约束Hamilton系统的正则运动方程和固有约束方程;其次构建了约束Hamilton 系统的Lie对称性确定方程和结构方程;然后给出了约束Hamilton系统的Lie守恒定理和守恒量;最后研究了复标量场与Chern-Simons项耦合系统的Lie对称性和另外一个例子以说明此方法在场论中的应用.     

A singular value decomposition algorithm based on solving hyperplane constrained nonlinear systems

A new algorithm for singular value decomposition (SVD) is presented through relating SVD problem to nonlinear systems whose solutions are constrained on hyperplanes. The hyperplane constrained nonlinear systems are solved with the help of Newton’s iterative method. It is proved that our SVD algorithm has the quadratic convergence substantially and all singular pairs are computable. These facts are also confirmed by some numerical examples.     

求解Hamilton约束的新算法

利用吴方法对多项式类型带约束的Hamilton系统作了研究.给出了判断系统是否正则的一个新算法.对于正则系统,可以得到Hamilton函数和运动方程,而对退化的系统给出了两个求解约束的新算法,得到带约束的Hamilton函数和运动方程.利用符号计算软件,这几个算法都可以在计算机上实现.     

Binary nonlinearization of AKNS spectral problem under higher-order symmetry constraints

Binary nonlinearization of AKNS spectral problem is extended to the cases of higher-order symmetry constraints. The Hamiltonian structures, Lax representations, r-matrices and integrals of motion in involution are explicitly proposed for the resulting constrained systems in the cases of the first four orders.The obtained integrals of motion are proved to be functionally independent and thus the constrained systems are completely integrable in the Liouville sense.     

Binary nonlinearization of spectral problems of the perturbation AKNS systems

The method of nonlinearization of spectral problems is extended to the perturbation AKNS systems, and a new kind of finite-dimensional Hamiltonian systems is obtained. It is shown that the obtained Hamiltonian systems are just the perturbation systems of the well-known constrained AKNS flows and thus their Liouville integrability is established by restoring from the Liouville integrability of the constrained AKNS flows. As a byproduct, the process of binary nonlinearization of spectral problems and the process of perturbation of soliton equations commute in the case of the AKNS hierarchy.     

Probability-one homotopies in computational science

Probability-one homotopy algorithms are a class of methods for solving nonlinear systems of equations that, under mild assumptions, are globally convergent for a wide range of problems in science and engineering. Convergence theory, robust numerical algorithms, and production quality mathematical software exist for general nonlinear systems of equations, and special cases such as Brouwer fixed point problems, polynomial systems, and nonlinear constrained optimization. Using a sample of challenging scientific problems as motivation, some pertinent homotopy theory and algorithms are presented. The problems considered are analog circuit simulation (for nonlinear systems), reconfigurable space trusses (for polynomial systems), and fuel-optimal orbital rendezvous (for nonlinear constrained optimization). The mathematical software packages HOMPACK90 and POLSYS_PLP are also briefly described.     

Noether’s Theorem for Constrained Hamiltonian System Under Generalized Operators北大核心CSCD

Noether’s symmetry and conserved quantity of singular systems under generalized operators were studied. Firstly, the Lagrangian equation of singular systems under generalized operators was established, and the primary constraints on the system were derived. Then the Lagrangian multiplier was introduced to establish the constrained Hamilton equation and the compatibility condition under generalized operators. Secondly, based on the invariance of the Hamilton action under the infinitesimal transformation, Noether’s theorem for constrained Hamiltonian systems under generalized operators was established, and the symmetry and corresponding conserved quantity of the system were given. Under certain conditions, Noether’s conservation of constrained Hamiltonian systems under generalized operators can be reduced to Noether’s conservation of integer-order constrained Hamiltonian systems. Finally, an example illustrates the application of the results. © 2022 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

Constrained controllability of nonlinear systems

In the paper, infinite-dimensional dynamical systems describedby nonlinear abstract differential equations are considered.Using a generalized open-mapping theorem sufficient conditionsfor constrained exact local controllability are formulated andproved. It is generally assumed that the values of controlsare in a convex and closed cone with the vertex at zero. Asan illustrative example. the constrained exact local controllabilityproblem for nonlinear delayed dynamical system is solved indetail. Some remarks and comments on the existing results forcontrollability of nonlinear dynamical systems are also presented.