Birkhoff系统的一类新型绝热不变量

研究Birkhoff系统对称性的摄动与绝热不变量.给出了未受扰Birkhoff系统的Lie对称性导致的Hojman型精确不变量.基于力学系统的高阶绝热不变量的定义，研究在小扰动作用下Birkhoff系统Lie对称性的摄动，得到了系统的一类新型绝热不变量.举例说明结果的应用. 关键词： Birkhoff系统 Lie对称性 摄动 绝热不变量     

Stationary turbulent closure via the Hopf functional equation

Exact closed-form solutions are exhibited for the Hopf equation for stationary incompressible 3D Navier-Stokes flow, for the cases of homogeneous forced flow (including a solution with depleted nonlinearity) and inhomogeneous flow with arbitrary boundary conditions. This provides an exact method for computing two- and higher-point moments, given the mean flow.     

Dynamical Invariant and Exact Mechanical Analyses for the Caldirola–Kanai Model of Dissipative Three Coupled Oscillators

We study the dynamical invariant for dissipative three coupled oscillators mainly from the quantum mechanical point of view. It is known that there are many advantages of the invariant quantity in elucidating mechanical properties of the system. We use such a property of the invariant operator in quantizing the system in this work. To this end, we first transform the invariant operator to a simple one by using a unitary operator in order that we can easily manage it. The invariant operator is further simplified through its diagonalization via three-dimensional rotations parameterized by three Euler angles. The coupling terms in the quantum invariant are eventually eliminated thanks to such a diagonalization. As a consequence, transformed quantum invariant is represented in terms of three independent simple harmonic oscillators which have unit masses. Starting from the wave functions in the transformed system, we have derived the full wave functions in the original system with the help of the unitary operators.     

Transition to chaos in lid-driven square cavity flow

To date, there are very few studies on the transition beyond second Hopf bifurcation in a lid-driven square cavity, due to the difficulties in theoretical analysis and numerical simulations. In this paper, we study the characteristics of the third Hopf bifurcation in a driven square cavity by applying a consistent fourth-order compact finite difference scheme rectently developed by us. We numerically identify the critical Reynolds number of the third Hopf bifurcation located in the interval of (13944.7021,13946.5333) by the method of bisection. Through Fourier analysis, it is discovered that the flow becomes chaotic with a characteristic of period-doubling bifurcation when the Reynolds number is beyond the third bifurcation critical interval. Nonlinear time series analysis further ascertains the flow chaotic behaviors via the phase diagram, Kolmogorov entropy and maximal Lyapunov exponent. The phase diagram changes interestingly from a closed curve with self-intersection to an unclosed curve and the attractor eventually becomes strange when the flow becomes chaotic.     

On α-Limit Sets in Lorenz Maps

The aim of this paper is to show that α-limit sets in Lorenz maps do not have to be completely invariant. This highlights unexpected dynamical behavior in these maps, showing gaps existing in the literature. Similar result is obtained for unimodal maps on [0,1]. On the basis of provided examples, we also present how the performed study on the structure of α-limit sets is closely connected with the calculation of the topological entropy.     

The solution of orthogonal Procrustes problems for a family of orthogonally invariant norms

The orthogonal Procrustes problem involves finding an orthogonal matrix which transforms one given matrix into another in the least-squares sense, and thus it requires the minimization of the Frobenius matrix norm. We consider, the solution of this problem for a family of orthogonally invariant norms which includes the Frobenius norm as a special case.     

关于自同构作用下的不变元

文章主要讨论了代数（环）、（双）Ｈｏｐｆ代数中自同构作用下不变元素的性质，得到一些新的结果：设Ａ是有限维半单代数，则ＡｕｔＡ作用下的不变元属于中心，并且举出反例说明这个性质不能一般化到有限维代数、有限维半单（双）Ｈｏｐｆ代数．     

变分方法与反向上下解

本文在非线性方程的下解不小于上解这一条件（即反向上下解条件）下研究了其解的存在性．我们证明了，如果非线性算子方程有变分结构，有一对反向上下解，对应的算子映某个锥入锥并且满足一定的辅助条件，那么这一方程在锥中至少有两个解．另外，本文还研究了反向上下解条件下非线性椭圆边值问题正解的存在性．     

Spontaneous translation symmetry breakdown for quantum lattice systems

We study the set of equilibrium states for quantum lattice states in the presence of a translation symmetry of the model. We derive a characterization of the spontaneous breaking of this symmetry, i.e., the decomposition of an invariant equilibrium state into a mixture of noninvariant equilibrium states, in terms of the separability in mean energy of these states for a class of perturbed dynamics.