KP系统Lax算子及主对称的换位公式

本文讨论的是ＫＰ系统Ｌａｘ算子及主对称的换位公式．通过拓广速降函数空间及对 ＫＰ方程 Ｌａｘ算子的讨论,找到了 Ｌａｘ算子的表示向量;并通过对 Ｌａｘ算子、 Ｌａｘ流、 Ｌａｘ算子表示向量之间联系的讨论,得出了计算 Ｌａｘ算子李括号的表示向量的方法,从而解决了 ＫＰ方程主对称的换位公式问题．最后本文还利用伴随算子给出了从ＫＰ方程任一主对称得到其一个对称的公式．     

论多目标分式规划

论多目标分式规划徐增堃（浙江师范大学数学系，金华３２１００４）基金项目：浙江省教委自然科学基金资助课题．１９９０年１０月２９日收到，１９９１年１０月５日收到第一次修改稿．１９９２年９月７日收到第二次修改稿，１９９３年２月１３日收到第三次修改、压缩稿．...     

回归曲线的加权正交级数估计

回归曲线的加权正交级数估计杨瑛（北京大学概率统计系，北京１００８７１）国家自然科学基金资助项目．１９８９年１２月３日收到．１９９０年１１月２２日收到第一次修改稿．１９９２年８月７日收到第二次修改稿．一、引论自１９６２年Ｃｅｎｃｏｖ［１］提出了密度函数...     

凹(凸)算子的不动点定理及其应用

凹（凸）算子的不动点定理及其应用李福义，梁展东（山西大学数学系，太原０３０００６）１９９２年１０月６日收到．近二十年来，人们对α凹（－α凸）算子，凹（凸）算子，一般凹（凸）算子以及减算子这四类非线性算于，用Ｈｉｌｂｅｒｔ投影距离、单调迭代技巧、拓扑度理论等各种方法进行了广泛深入的研究，条件彼不断减弱、结论不断深化，新旧更替，分别参见［１—７］，［８－１２］，［１１—１３］，［１４］．本文引...     

具有Lenard递推结构之发展方程族的Lax表示

该文对具有Ｌｅｎａｒｄ递推结构的发展方程族，通过转换速推结构为一个算子方程给出了Ｌａｘ表示的一种理论描述．这种描述可应用于各种１＋１维可积系统族Ｌａｘ表示的寻求之中，本文仅就ＫｄＶ可积族和Ａｎｔｏｎｏｗｉｃｚ—Ｆｏｒｄｙ可积族详细阐述了Ｌａｘ表示的具体构造过程．     

谱问题、发展方程族及Lax表示的等价性

讨论了四对谱问题、相应的保谱方程族以及它们Ｌａｘ表示的等价住，并利用这种等价关系可导出一些方程的Ｌａｘ表示．     

鲁棒稳定多项式的摄动界

鲁棒稳定多项式的摄动界安森建,王恩平（中国科学院系统科学研究所,北京１０００８０）基金项目：国家自然科学基金资助项目．１）作者现在通讯地址：北京大学力学系,邮政编码：１００８７１．１９９２年１月００４日收到,１９９２年４月７日收到第一次修改稿,１９９...     

二次超曲面在切点的几何连续性

二次超曲面在切点的几何连续性姜献峰，梁友栋（浙江大学，杭州３１００２７）１９９０年５月１７日收到．１９９２年７月１１日收到修改、压缩稿．在计算机辅助几何设计（ＣＡＧＤ）中，复杂曲面通常是用一系列较为简单的曲面片．经一定的几何连续拼接而成．参数曲面几何...     

凹(凸)算子的不动点定理及其应用

凹（凸）算子的不动点定理及其应用李福义，梁展东（山西大学数学系，太原０３０００６）１９９２年１０月６日收到．近二十年来，人们对α凹（－α凸）算子，凹（凸）算子，一般凹（凸）算子以及减算子这四类非线性算于，用Ｈｉｌｂｅｒｔ投影距离、单调迭代技巧、拓扑...     

非线性跟踪─微分器

非线性跟踪一微分器韩京清，王伟（中国科学院系统科学研究所，北京１０００８０）国家自然科学基金资助课题．１９９２年７月２５日收到，１９９３年２月１７日收到修改稿．一、前言在实际工程问题中，常存在由不连续或带随机噪声的量测信号合理地提取连续信号及微分信号...     

Integrable three-dimensional coupled nonlinear dynamical systems related to centrally extended operator Lie algebras and their Lax type three-linearization

The Hamiltonian representation for a hierarchy of Lax type equations on a dual space to the Lie algebra of integro-differential operators with matrix coefficients, extended by evolutions for eigenfunctions and adjoint eigenfunctions of the corresponding spectral problems, is obtained via some special Bäcklund transformation. The connection of this hierarchy with integrable by Lax two-dimensional Davey-Stewartson type systems is studied.     

一类非线性演化方程族的可积耦合

Starting from a Tu Guizhang‘s isospectral‘problem, a Lax pair is obtained by means of Tu scheme ( we call it Tu Lax pair ). By applying a gauge transformation between matrices, the Tu Lax pair is changed to its equivalent Lax pair with the traces of spectral matrices being zero, whose compatibility gives rise to a type of Tu hierarchy of equations. By making use of a high order loop algebra constructed by us, an integrable coupling system of the Tu hierarchy of equations are presented. Especially, as reduction cases, the integrable couplings of the celebrated AKNS hierarchy, TD hierarchy and Levi hierarchy are given at the same time.     

An integrable coupling hierarchy of the Mkdv_integrable systems, its Hamiltonian structure and corresponding nonisospectral integrable hierarchy

A four-by-four matrix spectral problem is introduced, locality of solution of the related stationary zero curvature equation is proved. An integrable coupling hierarchy of the Mkdv_integrable systems is presented. The Hamiltonian structure of the resulting integrable coupling hierarchy is established by means of the variational identity. It is shown that the resulting integrable couplings are all Liouville integrable Hamiltonian systems. Ultimately, through the nonisospectral zero curvature representation, a nonisospectral integrable hierarchy associated with the resulting integrable couplings is constructed.     

一族Liouville可积系及其3-Hamilton结构

构造了loop代数A↑～1的一个高阶子代数,设计了一个新的Lax对,利用屠格式获得了含8个位势的孤立子方程族;利用Gauteax导数直接验证了所得3个辛算子的线性组合仍为辛算子．因此该孤立族具有3-Hamilton结构,具有无穷多个对合的公共守恒密度,故Liouville可积．作为约化情形,得到了2个可积系,其中之一是著名的AKNS方程族．     

一个新的Lie代数和它的应用

通过构造一个新的Lie代数,利用它相应的Loop代数设计等谱Lax对,根据其相容性条件,得到了一族Lax可积方程族,其一种约化形式为著名的AKNS族．根据迹恒等式得到该方程族的Hamilton结构．利用该可积方程族可以进一步研究它的达布变换、对称、代数几何解等相关性质．     

New algebraic structures from Hermitian one-matrix model

Virasoro constraint is the operator algebra version of one-loop equation for a Hermitian one-matrix model, and it plays an important role in solving the model. We construct the realization of the Virasoro constraint from the Conformal Field Theory (CFT) method. From multi-loop equations of the one-matrix model, we get a more general constraint. It can be expressed in terms of the operator algebras, which is the Virasoro subalgebra with extra parameters. In this sense, we named as generalized Virasoro constraint. We enlarge this algebra with central extension, this is a new kind of algebra, and the usual Virasoro algebra is its subalgebra. And we give a bosonic realization of its subalgebra.     

Two expanding integrable systems and quasi-Hamiltonian function associated with an equation hierarchy

A Lie algebra sl(2) which is isomorphic to the known Lie algebra A₁ is introduced for which an isospectral Lax pair is presented, whose compatibility condition leads to a soliton-equation hierarchy. By using the trace identity, its Hamiltonian structure is obtained. Especially, as its reduction cases, a Sine equation and a complex modified KdV(cmKdV) equation are obtained,respectively. Then we enlarge the sl(2) into a bigger Lie algebra sl(4) so that a type of expanding integrable model of the hierarchy is worked out. However, the soliton-equation hierarchy is not integrable couplings. In order to generate the integrable couplings, an isospectral Lax pair is introduced. Under the frame of the zero curvature equation, we generate an integrable coupling whose quasi-Hamiltonian function is derived by employing the variational identity. Finally, two types of computing formulas of the constant γ are obtained, respectively.     

Symmetries of Quantum Lax Equations for the Painlevé Equations

Based on the fact that the Painlevé equations can be written as Hamiltonian systems with affine Weyl group symmetries, a canonical quantization of the Painlevé equations preserving such symmetries has been studied recently. On the other hand, since the Painlevé equations can also be described as isomonodromic deformations of certain second-order linear differential equations, a quantization of such Lax formalism is also a natural problem. In this paper, we introduce a canonical quantization of Lax equations for the Painlevé equations and study their symmetries. We also show that our quantum Lax equations are derived from Virasoro conformal field theory.     

一个新的Loop代数及其对Levi谱系的应用

本文基于loop代数A₁的一个子代数,构造了一个新的loop代数G;通过作一个适宜的Lax对变换,成功地将G应用于Levi等谱问题上,求得了Levi谱系的可积耦合,这种方法可以普遍地应用。