Applications of the division theorem inH s

We prove a version of the division theorem in Sobolev spaces with an estimate of the constant ass tends to infinity. We then apply it to derive spatial decay estimates for time-periodic solutions of linear wave equations in one space dimension and to prove that the space of decaying solutions is finite-dimensional. The main point is to show that some of the arguments used to analyze embedded eigenvalues of Schrödinger operators can be extended to cases where positivity arguments are not available. This has implications for nonlinear Klein-Gordon equations. A different approach, based on the proof of the stable manifold theorem, is also worked out, under slightly different assumptions.     

多体动系统动力学方程在流形上的辛算法

多体系统动力学方程的数值方法一直是数学与力学家们的热门研 究课题.特别是多体系统动力学微分/代数方程组形式的数学模型，是所 谓的指标-3问题，它的求解是一个难题.目前流行的关于它的数值方法都 有不尽人意的地方，主要是对出现的所谓的违约问题和刚性问题未很好地 解决.多体系统动力学方程在流形上的辛算法是近几年出现的新的数值方 法，它将闭环型多体系统动力学的方程的约束部分和常微分方程部分利 用所谓的辛方法同时进行处理，其中的一些方法已证明是有效的，所以， 用它求解多体系统动力学方程前景看好.本文介绍了这些新的理论，并提 出了一些有待解决的问题.     

LAGRANGIAN MECHANICS ON KAHLER MANIFOLDS

IntroductionLagrangian mechanics is an important part of analytical mechanics.It was establishedby J.Lagrange by means of the Euclidean analytical geometry and calculus in his famouswork“Analytical Mechanics”in1788.Now,itwas established in Riemann manif…     

On a dynamical system on matrix algebra

LetM(n) be the algebra of alln×n complex matrices. We consider a dynamical system onM(n) defined by the vector fieldV(X)=[[X ^*,X],X], (X M(n)). It arises as the gradient flow for two kinds of variational problems onM(n). Given anyX ₀ M(n), letX(t) be the trajectory starting atX ₀. We study the global behavior ofX(t) ast . We show that, ifX ₀ is semisimple, thenX(t) converges exponentially to a normal matrix. IfX ₀ is not semisimple, then the behavior ofX(t) is completely different and difficult to analyze. We give some results also in this case. Furthermore, we discuss about a center manifold approach to our dynamical system.     

C^r随机中心流形定理

随机中心流形定理在非线性随机分叉理论的研究中具有关键作用。本文对Ｃ＾ｒ非线性随机系统给出Ｃ＾ｒ中心流形的存在性定理，并讨论了中心流形的稳定性及例子。     

HAMILTONIAN MECHANICS ON KAHLER MANIFOLDS

Using the mechanical principle, the theory of modern geometry and advanced calculus, Hamiltonian mechanics was generalized to Kahler manifolds, and the Hamiltonian mechanics on Kahler manifolds was established. Then the complex mathematical aspect of Hamiltonian vector field and Hamilton's equations was obtained, and so on.     

Bifurcation phenomena in viscoelastic flows through a symmetric 1:4 expansion

In this work we present an investigation of viscoelastic flow in a planar sudden expansion with expansion ratio D/d = 4. We apply the modified FENE–CR constitutive model based on the non-linear finite extensibility dumbbells (FENE) model. The governing equations were solved using a finite volume method with the high-resolution CUBISTA scheme utilised for the discretisation of the convective terms in the stress and momentum equations. Our interest here is to investigate two-dimensional steady-state solutions where, above a critical Reynolds number, stable asymmetric flow states are known to occur. We report a systematic parametric investigation, clarifying the roles of Reynolds number (0.01 < Re < 100), Weissenberg number (0 < We < 100) and the solvent viscosity ratio (0.3 < β < 1). For most simulations the extensibility parameter of the FENE model was kept constant, at a value L² = 100, but some exploration of its effect in the range 100–500 shows a rather minor influence. The results given comprise flow patterns, streamlines and vortex sizes and intensities, and pressure and velocity distributions along the centreline (i.e. y = 0). For the Newtonian case, in agreement with previous studies, a bifurcation to asymmetric flow was observed for Reynolds numbers greater than about 36. In contrast viscoelasticity was found to stabilise the flow; setting β = 0.5 and We = 2 as typical values, resulted in symmetric flow up to a Reynolds number of about 46. We analyse these two cases in particular detail.     

二元机翼颤振的分叉点类别的判定

应用中心流形理论将二元机翼颤振这一四维系统降为二维系统，用后继函数判别法对分叉点的真假中心及稳定性问题进行了分析．     

On Local Analysis of Oscillations of a Non-ideal and Non-linear Mechanical Model

It is of major importance to consider non-ideal energy sources in engineering problems. They act on an oscillating system and at the same time experience a reciprocal action from the system. Here, a non-ideal system is studied. In this system, the interaction between source energy and motion is accomplished through a special kind of friction. Results about the stability and instability of the equilibrium point of this system are obtained. Moreover, its bifurcation curves are determined. Hopf bifurcations are found in the set of parameters of the oscillating system.     

Liouville theorem for pseudoharmonic maps from Sasakian manifolds

In this paper, we derive a sub-gradient estimate for pseudoharmonic maps from noncompact complete Sasakian manifolds which satisfy the CR sub-Laplace comparison property, to simply-connected Riemannian manifolds with nonpositive sectional curvature. As its application, we obtain some Liouville theorems for pseudoharmonic maps. In the Appendix, we modify the method and apply it to harmonic maps from noncompact complete Sasakian manifolds.