Characteristic Numbers of Matrix Lie Algebras

A notion of characteristic number of matrix Lie algebras is defined, which is devoted to distinguishing various Lie algebras that are used to generate integrable couplings of soliton equations. That is, the exact classification of the matrix Lie algebras by using computational formulas is given. Here the characteristic numbers also describe the relations between soliton solutions of the stationary zero curvature equations expressed by various Lie algebras.     

Hamiltonian Forms for a Hierarchy of Discrete Integrable Coupling Systems

A semi-direct sum of two Lie algebras of four-by-four matrices is presented, and a discrete four-by-four matrix spectral problem is introduced. A hierarchy of discrete integrable coupling systems is derived. The obtained integrable coupling systems are all written in their Hamiltonian forms by the discrete variational identity. Finally, we prove that the lattice equations in the obtained integrable coupling systems are all Liouville integrable discrete Hamiltonian systems.     

双侧下颌支矢状劈开截骨术后患者可施加合适咬合力的数值研究

双侧下颌支矢状劈开截骨术在临床手术中广泛用于治疗缩颌、凸颌和颌面不对称患者的颌面畸形症状。本文用数值方法研究双侧下颌支矢状劈开截骨术并前移下颌骨后患者可施加的合适咬合力。首先,用缩颌患者的CT断层扫面数据建立下颌骨、关节盘、关节窝和相关软组织的高度几何相似性的几何模型;然后,用有限元模型模拟双侧下颌支矢状劈开截骨术后患者的咬合过程。通过分析发现,术后关节盘上的位移处于正常的范围,而应力却远远超过正常范围,因此,将应力因素作为确定术后患者可施加的合适咬合力的标准。当施加正常咬合力的25%时,关节盘上的应力分布和术前没有显著性差异,同时下颌骨上的应力水平也处于正常范围。     

卡氏积图的Laplacian谱半径的上界

对近年来图的Laplacian谱半径上界的研究成果进行了简单梳理.利用2个图的卡氏积图的特征值,讨论了2个循环图的卡氏积图的Laplacian谱半径的上界问题,得到了几个上界,推广了已有文献的结论.     

W6*Sn的交叉数

早在20世纪50年代,Zarankiewicz 猜想完全2-部图K_{m,n}(m\leq n)的交叉数为\lfloor\frac{m}{2}\rfloor\times \lfloor\frac{m-1}{2}\rfloor\times\lfloor\frac{n}{2}\rfloor\times\lfloor\frac{n-1}{2}\rfloor (对任意实数x,\lfloor x\rfloor表示不超过x的最大整数). 目前这一猜想的正确性只证明了当m\leq6时成立. 假定著名的Zarankiewicz的猜想对m=7的情形成立,确定了6-轮W_{6}与星S_{n}的笛卡尔积图的交叉是 cr(W_{6}\times S_{n})=9\lfloor\frac{n}{2}\rfloor\times\lfloor\frac{n-1}{2}\rfloor+2n+5\lfloor\frac{n}{2}\rfloor.     

Generation of Nonlinear Evolution Equations by Reductions of the Self-Dual Yang–Mills Equations

With the help of some reductions of the self-dual Yang Mills （briefly written as sdYM） equations, we introduce a Lax pair whose compatibility condition leads to a set of （2 ＋ 1）-dimensional equations. Its first reduction gives rise to a generalized variable-coefficient Burgers equation with a forced term. Furthermore, the Burgers equation again reduces to a forced Burgers equation with constant coefficients, the standard Burgers equation, the heat equation, the Fisher equation, and the Huxley equation, respectively. The second reduction generates a few new （2 ＋ 1）-dimensional nonlinear integrable systems, in particular, obtains a kind of （2 ＋ 1）-dimensional integrable couplings of a new （2 ＋ 1）- dimensional integrable nonlinear equation.     

受均布速度作用的有限板面导纳研究

针对接触面积上典型的激励形式,均布速度激励,推导有限薄板上面导纳的理论计算方法。在推导过程中,将接触面积离散为较小的子面积,连续分布的激励和响应简化为作用在子面积中心的激励和响应,再利用有效点导纳的定义,获得有限薄板受均布速度激励的面导纳的理论计算公式;根据接触面积上复功率与面导纳的关系,推导出由传递点导纳获得有限板面导纳的测量方法。通过比较理论计算结果和实验测量结果,证明两种方法可行。对计算结果进行的分析还表明,子面积数的增加可以提高理论计算的精度,而测量精度在激励频率较高时受测量装置产生的弯矩影响较大。      

Two new discrete integrable systems

In this paper,we focus on the construction of new(1+1)-dimensional discrete integrable systems according to a subalgebra of loop algebra A 1.By designing two new(1+1)-dimensional discrete spectral problems,two new discrete integrable systems are obtained,namely,a 2-field lattice hierarchy and a 3-field lattice hierarchy.When deriving the two new discrete integrable systems,we find the generalized relativistic Toda lattice hierarchy and the generalized modified Toda lattice hierarchy.Moreover,we also obtain the Hamiltonian structures of the two lattice hierarchies by means of the discrete trace identity.     

基于平行玻璃板的半导体激光器阵列光束整形

为得到质量均衡的大功率激光束,分别以条形和面阵半导体激光器为模型,基于平行玻璃板对光束的偏移作用,对准直后激光束进行分割和重排,并在Zemax软件中进行了仿真。条形半导体激光器初始发散角为40和10,整形前已准直光束快慢轴方向的光参数积分别为0.455 2 mmmrad和20.484 mmmrad,光束质量相差较大,整形后快慢轴方向光参数积分别为2.731 2 mmmrad和3.414 0 mmmrad,实现了快慢轴方向光束质量均衡。利用平行玻璃板消除面阵半导体激光器中存在的发光死区,整形后快慢轴方向光参数积分别为7.002 mmmrad和10.242 mmmrad ,整形系统耦合效率为90.13%。     

L_p-对偶几何表面积

Lutwak结合Lp-混合体积给出了Lp-几何表面积的概念,本文结合Lp-对偶混合体积,给出了Lp-对偶几何表面积的概念,并且证明了相关不等式.