"图解"数学物理方法的教学实践

《数学物理方法》虽然是数学课，但是它是介绍在物理中应用的数学．物理课程多媒体教学的经验启发我们，合理地选择《数学物理方法》课程中有物理背景的内容，恰当地使用可视化以展现数学公式的物理图像，就能提高教学效果，对一些随时间演变的方程，还可以用动画来表现．我们用两种方法实现可视化，一是用解析解作图，另一个是通过数值求解方程，再将数值计算结果作图．本文介绍我们在《数学物理方法》课开展可视化教学的实践．     

数学物理方法在PN结分析中的应用

将数学思想方法应用于现代高科技各专业技术领域，并构建成典型的（物理）模型和解决问题的方法，从而形成科学研究中实用性很强的数学物理方法．     

Lie Symmetrical Perturbation and Adiabatic Invariants of Generalized Hojman Type for Disturbed Nonholonomic Systems

For a nonholonomic mechanics system with the action of small disturbance, the Lie symmetrical perturbation and adiabatic invariants of generalized Hojman type are studied under general infinitesimal transformations of groups in which the generalized coordinates and time are variable. On the basis of the invariance of disturbed nonholonomic dynamical equations under general infinitesimal transformations, the determining equations, the constrained restriction equations and the additional restriction equations of Lie symmetries of the system are constructed, which only depend on the variables t, qs and q^.s. Based on the definition of higher-order adiabatic invariants of a mechanical system, the perturbation of Lie symmetries for a nonholonomic system with the action of small disturbance is investigated, and the Lie symmetrical adiabatic invariants, the weakly Lie symmetrical adiabatic invariants and the strongly Lie symmetrical adiabatic invariants of generalized Hojman type of disturbed nonholonomic systems are obtained. An example is given to illustrate applications of the results.     

维格纳（Wigner，Eugene Paul，1902-1995）匈牙利-美国物理学家（Phgsicist，Hungary-America）

维格纳是一个大商人的儿子，1925年在柏林大学获得博士学位。1930年前他一直在柏林教书。1930年，他与诺伊曼一起被邀请至美国，任美国普林斯顿大学数学物理教授，1937年成为美国公民。1936年，他创立了中子吸收的理论，在建造核反应堆时显示了它的用途。维格纳还提出，核力不依赖于电荷，因此在原子核内，中子与质子在核力方面有相似的性质。这对汤川秀澍提出的介子理论是一个十分有用的概念。     

Modified Form of Wigner Functions for Non-Hamiltonian Systems

Quantization of non-Hamiltonian systems （such as damped systems） often gives rise to complex spectra and corresponding resonant states, therefore a standard form calculating Wigner functions cannot lead to static quasiprobability distribution functions. We show that a modified form of the Wigner functions satisfies a ＊-genvalue equation and can be derived from deformation quantization for such systems.     

Modular Epidemic Spreading in Small-World Networks

We study the epidemic spreading of the susceptible-infected-susceptible model on small-world networks with modular structure. It is found that the epidemic threshold increases linearly with the modular strength. Furthermore, the modular structure may influence the infected density in the steady state and the spreading velocity at the beginning of propagation. Practically, the propagation can be hindered by strengthening the modular structure in the view of network topology. In addition, to reduce the probability of reconnection between modules may also help to control the propagation.     

Initial-value Problems for Extended KdV-Burgers Equations via Generalized Conditional Symmetries

We classify initial-value problems for extended KdV-Burgers equations via generalized conditional symmetries. These equations can be reduced to Cauchy problems for some systems of first-order ordinary differential equations. The obtained reductions cannot be derived within the framework of the standard Lie approach.     

Algebraic Treatment of the MIC-Kepler System in Spherical Coordinates

The MIC-Kepler system is studied via the Milshtein Strakhovenko variant of the so（2,1） Lie algebra. Green＇s function is constructed in spherical coordinates, with the help of the Kustaanheimo Stiefel variables and the generators of the S0（2,1） group. The energy spectrum and the normalized wavefunctions of the bound states are obtained.     

Modification of the Clarkson-Kruskal Direct Method for a Coupled System

A new idea is put forward to modify the Clarkson-Kruskal （CK） direct method. Using the usual CK direct method to a coupled KdV system, two types of usual similarity reductions can be obtained. However, the application of the modified CK direct method leads to three types of new similarity reductions different from the usual ones.