三维Plumbed流形的不变量

本文给出了三维plumbed流形的由Blanchet，Habegger，Masbaum和Vogel给出的Witten型不变量的公式．作为特例对所有透镜空间和Poincare同调球给出了这些不变量的更具体的公式．     

一类链环的Kauffman多项式和三维流形不变量

研究了所谓广义树状链环的Kauffman括号多项式,给出了更一般三维流形的由Blanchet等得到的Witten型不变量的计算方法.     

线性常微分方程的不变量及其应用

在这篇文章里,我们证明了ｎ阶线性常微分方程在变换Ｔ＝（ｔ）与交换ｘ＝ｕ（ｔ）ｙ下分别有ｎ—１个不变量,并对后一情形给出了不变量的具体表述式．最后,我们还利用这些不变量研究了（１）的解的非振动性等问题．     

一种构造可积Hamilton系统的新方法

建议了一种新的构造可积Hamilton系统的方法。对于给定的Poisson流形,本文利用Dirac-Poisson结构构造其上的新Poisson括号[1],进而获得了新的可积Hamilton系统。构造的Poisson括号一般是非线的,并且这种方法也不同于通常的方法[2～4]。本文还给出了两个实例。     

广义量子动力学理论中通常复数量子力学的Hamilton量表述

讨论广义量子动力学理论中,通常复数量子力学的Hamilton量表述。在引入总迹Lagrange量、总迹Hamilton量和两种类型的Poison括号运算之后,不仅能得到用Poisson括号表达的算子的总迹泛函的运动方程,还能得到用Poisson括号表达的部分算子的运动方程。由此得到广义量子动力学理论中用Poisson括号和总迹Hamilton量表达的通常复数量子力学的一组基本运动方程。并且这组方程和Heisenberg方程是相溶的。     

可积的Riccati微分方程的不变量变换讨论

对于可积的Riccati微分方程:L[y]=-y′+p(x)yn+Q(x)y+R(x)(p(x)R(x)≠0,n≠0,1)(0)L[y]=-y′+p(x)y2+Q(x)y+R(x)(p(x)R(x)≠0)(1)利用其不变量变换,给出方程(0)和(1)的可积充分条件,并对方程(1)的特解形式L[y0]=0,讨论其不变量变换的等效性;同时,对方程(1)的非特解形式L[y0]≠0,讨论其可积性.     

某些三维流形不变量的计算

Ｔｈｅｃａｌｃｕａｌｔｉｏｎｏｆ〈ｚｋ１,ｚｋ２,…,ｚｋｎ〉ＬｏｆｔｈｉｓｐａｐｅｒｗｉｌｌｂｅｉｎｔｅｒｍｏｆｔｈｅＫａｕｆｆｍａｎｂｒａｃｋｅｔ－ｐｏｌｙｎｏｍｉａｌ［１,２,３］．ＴｈｅＫａｕｆｆｍａｎｂｒａｃｋｅｔｐｏｌｙｎｏｍｉａｌｏｆａｐｌａｎａｒｄｉａｇｒａｍｏｆａｎｕｎｏｒｉｅｎｔｅｄｌｉｎｋｉｓａｎｅｌｅｍｅｎｔ〈Ｄ〉∈Ｚ［Ａ,Ａ－１］ｄｅｆｉｎｅｄｂｙｔｈｅｆｏｌｌｏｗｉｎｇｐｒｏｃｅｓｓ．ＡｓｔａｔｅｏｆＤｉｓｄｅｆｉｎｅｄｔｏｂｅａｍａｐｓｆｒｏｍｔｈｅｃｒｏｓｓｉｎｇ…     

一类具有逐段常变量微分方程的渐近概周期解

利用差分方程的渐近概周期序列解,讨论了一类具有逐段常变量微分方程的渐近概周期解的存在性．     

常系数无穷维Hamilton系统的二阶循环算子结构

该文借助于有限和形式的常系数Hamilton算子,将一般体系循环算子的获得方法应用到无穷维形式的Hamilton正则系统.在结果方面,获得了约束条件下一阶常系数Hamilton算子所允许的循环算子的一般结构及其系数的具体形式.又通过算例验证了结论的正确性与便捷性.     

逐段常变量微分方程组概周期型解存在唯一性

通过差分方程指数二分法,我们研究了一类具有变系数逐段常变量微分方程组概周期与伪概周期解的存在唯一性与渐近稳定性,改进了已有文献的结果,并且得到了差分方程满足指数二分性的一个充分条件．     

A compressible Hamiltonian electromagnetic gyrofluid model

A Lie-Poisson bracket is presented for a four-field gyrofluid model with magnetic field curvature and compressible ions, thereby showing the model to be Hamiltonian. The corresponding Casimir invariants are presented, and shown to be associated to four Lagrangian invariants advected by distinct velocity fields. This differs from a cold ion limit, in which the Lie-Poisson bracket transforms into the sum of direct and semidirect products, leading to only three Lagrangian invariants.     

On the Linearization of Hamiltonian Systems on Poisson Manifolds

The linearization of a Hamiltonian system on a Poisson manifold at a given (singular) symplectic leaf gives a dynamical system on the normal bundle of the leaf, which is called the first variation system. We show that the first variation system admits a compatible Hamiltonian structure if there exists a transversal to the leaf which is invariant with respect to the flow of the original system. In the case where the transverse Lie algebra of the symplectic leaf is semisimple, this condition is also necessary.     

Compatible Poisson Brackets on Lie Algebras

We discuss the relationship between the representation of an integrable system as an L-A-pair with a spectral parameter and the existence of two compatible Hamiltonian representations of this system. We consider examples of compatible Poisson brackets on Lie algebras, as well as the corresponding integrable Hamiltonian systems and Lax representations.     

Hamiltonian formulation of generalized quantum dynamics

The Hamiltonian formulation of the usual complex quantum mechanics in the theory of generalized quantum dynamics is discussed. After the total trace Lagrangian, total trace Hamiltonian and two kinds of Poisson brackets are introduced, both the equations of motion of some total trace functionals which are expressed by total trace Poisson brackets and the equations of motion of some operators which are expressed by the without-total-trace Poisson brackets are obtained. Then a set of basic equations of motion of the usual complex quantum mechanics are obtained, which are also expressed by the Poisson brackets and total trace Hamiltonian in the generalized quantum dynamics. The set of equations of motion are consistent with the corresponding Heisenberg equations. Project supported by Prof. T.D. Lee’s NNSC Grant, the National Natural Science Foundation of China, the Foundation of Ph. D. Directing Programme of Chinese University, and the Chinese Academy of Sciences.     

The Poisson bracket compatible with the classical reflection equation algebra

We introduce a family of compatible Poisson brackets on the space of 2 × 2 polynomial matrices, which contains the reflection equation algebra bracket. Then we use it to derive a multi-Hamiltonian structure for a set of integrable systems that includes the XXX Heisenberg magnet with boundary conditions, the generalized Toda lattices and the Kowalevski top.      

HOMOCLINIC ORBITS IN PERTURBED GENERALIZED HAMILTONIAN SYSTEMS

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The Liouville Canonical Form for Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Hierarchies

We reduce an arbitrary pair of compatible nonlocal Poisson brackets of hydrodynamic type generated by metrics of constant Riemannian curvature (compatible Mokhov–Ferapontov brackets) to a canonical form, find an integrable system describing all such pairs, and, for an arbitrary solution of this integrable system, i.e., for any pair of compatible Poisson brackets in question, construct (in closed form) integrable bi-Hamiltonian systems of hydrodynamic type possessing this pair of compatible Poisson brackets of hydrodynamic type. The corresponding special canonical forms of metrics of constant Riemannian curvature are considered. A theory of special Liouville coordinates for Poisson brackets is developed. We prove that the classification of these compatible Poisson brackets is equivalent to the classification of special Liouville coordinates for Mokhov–Ferapontov brackets.     

Irreducible Representations of Quantum 3 × 3 Matrices

The irreducible representations of quantum 2×2 and 3×3 matrices at the roots of unity are classified.     

On Linearized Poisson Structures

It is shown that, up to a diffeomorphism, the linearized Poisson structure does not depend on the choice of the transversal near the zero section.