电磁场切向边值关系的严密数学理论

阐述了矢量场沿一阶无穷小线段的线积分虽然也是一阶无穷小量,但特殊情况下要求计算精度必须达到二阶无穷小量,为此目的需要把一阶无穷小线段分成无穷多个二阶无穷小线元来计算线积分,并且给出如此计算所得结果遵循的规律,即二阶无穷小精度下矢量场沿一阶无穷小线段的线积分定理.最后介绍了这个定理在旋度理论和电磁场切向边值关系理论中的应用.     

散度概念的从头构建法

不以高斯公式为前提,从头出发计算矢量场穿出无穷小闭合曲面的通量,以此建立散度概念并推导空间直角坐标系中散度的计算公式.给出一个优美的面积分公式,它能根据闭合曲面的形状和大小一般性地计算该曲面包围的空间区域的体积.指出通过计算矢量场穿出无穷小正六面体表面的通量来推导散度的计算公式的时候,人们通常会陷入误解.     

矢量场穿过运动高斯面的通量随时间变化率的严密理论

证明了当计算矢量场穿过运动高斯面的通量随时间的变化率时,积分符号与求导符号的顺序是可以交换的.导出了运动曲面上的面元矢量随时间变化率的公式.给出了矢量场穿过运动高斯面的通量随时间变化率的几种算法.     

多块对接网格技术在电磁场散射问题中的应用

采用多块对接网格技术和时域有限体积法(FVTD)研究了典型目标和多体的电磁场散射问题.控制方程采用三维一般曲线坐标系下的时变麦克斯韦方程组.时间方向采用四步Runge-Kutta方法,空间离散采用基于通量雅可比矩阵特征结构的通矢量差分分裂,依赖变量采用MUSCL插值.时间方向的计算精度为2阶,空间方向的计算精度可达3阶.典型算例的雷达散射截面(RCS)的计算结果与理论解吻合很好.对于多体问题计算与文献结果相当一致,说明该算法具有对复杂拓扑结构外形(包括多体问题)进行数值模拟的能力.     

一类正交增强的阶谱六面体矢量单元构造

利用阶谱正交多项式的优越性,从单元系统矩阵形成的角度分析阶谱六面体矢量单元的基函数,提出具有正交增强的阶谱六面体矢量单元(ORHHVFE)构造目的;并利用该目的构造的阶谱六面体矢量单元与其它阶谱六面体矢量单元(HHVFE)进行了金属腔本征模问题的数值计算对比实验.结果表明ORHHVFE具有与其它HHVFE同等的数值计算精度,且由ORHHVFE形成的有限元系统矩阵条件数获得了极大的改善.     

矢量场类比法在电磁学教学中的应用探讨

电磁学是物理学的重要分支,但是学生总觉得这部分知识很难学,对于高斯定理和环路定理的理解也比较模糊.本文推广矢量场类比法,将矢量场类比为流体中的流速场,用水的流量类比为通量,流速沿环路的分量类比为环量.通过研究流体中的"起点"和"终点",类比为矢量场中"源"和"汇",用流体里的"漩涡"类比矢量场的"涡旋",并引出高斯定理和环路定理,让学生理解这些定理背后的物理含义.这种方法对后续学习电场强度通量、磁感应强度通量、电场线和磁感应线都有帮助,也有助于理解电场"有源无旋"和磁场"有旋无源"的性质.     

光盘光学系统的矢量衍射理论分析

本文以完整的矢量衍射理论分析了光盘光学系统。首先将入射的聚焦光束分解为平面波谱 ,得到每个平面波的振幅矢量 ;然后计算光盘对每个平面波的衍射 ,得到衍射波的振幅矢量 ,从而得到了整个衍射波场的空间频谱 ;最后计算物镜光瞳上的能通量 ,得到光盘系统的读出信号。在衍射计算中 ,光盘被定义为二维金属光栅 ,根据信息符的不同模型 ,选用坐标变换方法、耦合波方法或模态方法。对于典型的 DVD光学系统来说 ,矢量理论的结论与标量理论相差甚远 ,要得到正确的结果就必须采用矢量衍射理论     

分数阶系统的一种稳定性判定定理及在分数阶统一混沌系统同步中的应用

根据Lyapunov稳定定理及其逆定理和分数阶系统稳定定理,提出了如果整数阶系统稳定,其对应的阶次小于1的分数阶形式的系统也稳定的分数阶系统稳定的判定定理,并给出了详细的证明过程.并将该理论运用于分数阶混沌系统的同步,实现了未知参数分数阶统一混沌系统的自适应同步,仿真结果证实了该理论的正确性. 关键词： 分数阶系统 混沌 Lyapunov稳定定理 Lyapunov稳定逆定理     

紧聚焦角向偏振分数阶涡旋光诱导磁化场特性

基于理查德-沃尔夫矢量衍射理论和逆法拉第效应, 首次推导出紧聚焦角向偏振分数阶涡旋光诱导磁化场表达式, 分析出角向偏振分数阶涡旋光诱导磁化场可近似等效为有限个相邻整数阶涡旋角向偏振光诱导磁化场与其交叉诱导磁化场的加权叠加. 数值模拟了紧聚焦条件下不同分数阶涡旋角向偏振光诱导磁化场的分布特性. 模拟结果表明角向偏振分数阶涡旋光诱导磁化场呈非对称分布. 当取分数阶涡旋拓扑荷α∈ [0.5, 1.5]时, 随着涡旋拓扑荷的增加, 磁化场横向平面分布出现分裂现象, 同时还引起磁斑垂直于光轴方向的自移效应. 当α = 0.5 或 1.5 时, 磁斑中心最大偏移量达0.24λ; 当分数阶涡旋拓扑荷α∈ (2, 3]时, 磁化场横向平面分布分裂出 2 个强度热点, 强度分布环径逐渐变大. 当分数阶涡旋拓扑荷趋于α = 3整数时, 磁化横向平面分布也趋于圆对称性分布. 特别有趣的是, 当分数阶涡旋拓扑荷取半整数, 尤其大于 3 时, 磁化场强度热点数与其包围的暗点数均与涡旋阶数存在正相关关系, 其中热点数为α - 0.5, 暗点数为α - 1.5. 这些研究结果将可应用于全光磁记录和磁性粒子的捕获与操控等.     

光子晶体传输特性的时域精细积分法分析

将精细积分法应用于时域有限差分法中,提出了一种求解光子晶体传输特性的时域精细积分法,并对其计算精度及稳定性进行了分析.从一阶麦克斯韦方程出发,在空间上采用Yee元胞进行差分离散,结合吸收边界条件及激励源表达式将方程整理为标准的一阶常微分方程组形式.通过时间步长的精细划分和指数矩阵的加法定理,在时间上利用精细积分法对齐次微分方程进行积分求解,并结合激励向量的特解得到空间离散的场分量,最终通过傅里叶变换求得方程的解.利用时域精细积分法对光子晶体进行了实例计算,并将其结果分别与时域有限差分法和四阶龙格库塔法在精度、稳定性等方面进行了比较,结果表明时域精细积分法具有更高的计算精度,并且克服了时域有限差分法以及四阶龙格库塔法在计算稳定性上对时间步长的限制.提出的方法具有精确、稳定的特点,为光子晶体传输特性的研究提供了一种新的有效的分析方法.     

Conformal invariance and Hojman conservedquantities of first order Lagrange systems

In this paper the conformal invariance by infinitesimal transformations of first order Lagrange systems is discussed in detail. The necessary and sufficient conditions of conformal invariance and Lie symmetry simultaneously by the action of infinitesimal transformations are given. Then it gets the Hojman conserved quantities of conformal invariance by the infinitesimal transformations. Finally an example is given to illustrate the application of the results.     

非傍轴矢量光束二阶矩的发散特性

基于非傍轴矢量光束横截面上光强的精确表示和矢量二阶矩理论,应用严格的角谱表示的矢量衍射理论,对任意非傍轴矢量衍射光束的传输进行了系统的研究.给出了光束的束宽、横向发散角的积分表达式,论证了非傍轴矢量光束二阶矩的发散特性.对非傍轴矢量光束的束宽、发散角的定义及一些相关的问题进行了分析和讨论. 关键词： 矢量衍射理论 光束传输 矢量二阶矩 发散特性     

三阶矢量有限元方法精确计算轴对称谐振腔高阶模

为了进一步提高数值求解谐振腔高阶模的精度，本文提出了三阶矢量有限元方法，并针对二阶矢量有限元轴对称谐振腔高阶模计算程序Cafe对曲线边界的计算能力较差和计算速度较慢的缺点做了改进. 在这些改进的基础上编制了三阶矢量有限元轴对称谐振腔高阶模计算程序meshmatrix3，得到了很好的结果.     

Constraints in Noncommutative Chern-Simons Mechanics and Hall Effect

We study noncommutative Chern-Simons mechanics and noncommutative Hall effect by Dirac theory in this paper. The magnetic field is introduced by means of minimal coupling. We show that the constraint set will enlarge when a dimensionless parameter takes zero value. In order to illustrate our idea, we study two specific models. One is noncommutative Chern-Simons mechanics which describes a charged particle on a noncommutative plane interacting with a perpendicular uniform magnetic field. The other is a charged particle on a noncommutative plane with a background uniform electromagnetic field. We show that when the dimensionless parameter tends to zero, the particle will live in a lower dimensional space in both models. Noncommutative Chern-Simons mechanics will reduce to a harmonic oscillator and the classical equations of motion of a charged particle in the background of a uniform electromagnetic field are governed by classical Hall law when the dimensionless parameter tends to zero.     

Synthetic Vector Analysis II

Physicists prefer approximate calculations. This is natural, since physics is an empirical science. What is surprising judicious mathematicians is that many physicists use their favorite approximate reasoning to establish such theorems of pure mathematics as Gauss's divergence theorem and Stokes' theorem. What is more surprising is that their discussions impressively appeal to our geometric and physical intuitions, so that the discussions appear cryptically convincing, though mathematicians feel forced to contend offhand that such discussions are mathematically untenable and flimsy. In our previous paper (Nishimura, H. (2002). International Journal of Theoretical Physics 41, 1165–1190) we have shown that once we realize that their discussions in establishing Gauss's divergence theorem and Stokes' theorem are not approximate (with errors) but infinitesimal (without errors), the discussions are bona fide authentic. What we should do is only transfer between the standard universe of sets and mappings whose set of real numbers contains no infinitesimals but zero and an intuitionistic universe of sets and mappings whose set of real numbers contains nilpotent infinitesimals in abundance and in coherence. The principal objective in this paper is to show that the same finesse can establish the celebrated Gauss–Bonnet theorem relating the topology and the Gaussian curvature of a surface, opening the way to the geometric theory of characteristic classes.     

级联二阶非线性法布里-珀罗谐振腔特性分析

在小信号近似下,得出了级联二阶过程的表达式。利用该式的有效三阶极化率的表达式,讨论了存在二阶非线性光学晶体的法－珀谐振腔的非线性光学特性。分析表明,由于谐振腔的存在,通过非线性晶体的功率密度超过入射功率密度几十倍以上,因而产生较大的、随相位失配变化的非线性相位漂移。当相位失配接近２π时,获得基叔光的最大相位漂移。该最大相位漂移对应的相位失配随着入射功率密度、非线性晶体的地阶极化率和晶体长度而变化。     

变质量力学系统的一般形式的非Noether守恒量

研究一般的无限小变换下变质量力学系统Lie对称性的非Noether守恒量, 进一步推广Hojma n定理. 给出变质量力学系统的一般形式的非Noether守恒量,并举例说明结果的应用. 关键词： 变质量系统 一般的无限小变换 非Noether守恒量     

Non-local theory solution of two collinear mode-I permeable cracks in a magnetoelectroelastic composite material plane

The non-local theory solution of two collinear mode-I permeable cracks in a magnetoelectroelastic composite material plane was investigated using the generalized Almansi's theorem and the Schmidt method. The problem was formulated through Fourier transform into two pairs of dual integral equations, in which the unknown variables are the jumps in displacements across the crack surfaces. To solve the dual integral equations, the displacement jumps across the crack surfaces were directly expanded as a series of Jacobi polynomials. Numerical examples were provided to show the effects of crack length, the distance between two collinear cracks and the lattice parameter on the stress field, the electric displacement field and the magnetic flux field near the crack tips. Unlike the classical elasticity solutions, it is found that no stress, electric displacement or magnetic flux singularities are present at the crack tips in a magnetoelectroelastic composite material plane. The non-local elastic solutions yield a finite hoop stress at the crack tip, thus allowing us to use the maximum stress as a fracture criterion.     

The infinitesimal operator for the semigroup of the Frobenius-Perron operator from image sequence data: vector fields and transport barriers from movies

In this paper, we present an approach to approximate the Frobenius-Perron transfer operator from a sequence of time-ordered images, that is, a movie dataset. Unlike time-series data, successive images do not provide a direct access to a trajectory of a point in a phase space; more precisely, a pixel in an image plane. Therefore, we reconstruct the velocity field from image sequences based on the infinitesimal generator of the Frobenius-Perron operator. Moreover, we relate this problem to the well-known optical flow problem from the computer vision community and we validate the continuity equation derived from the infinitesimal operator as a constraint equation for the optical flow problem. Once the vector field and then a discrete transfer operator are found, then, in addition, we present a graph modularity method as a tool to discover basin structure in the phase space. Together with a tool to reconstruct a velocity field, this graph-based partition method provides us with a way to study transport behavior and other ergodic properties of measurable dynamical systems captured only through image sequences.