Geometry and Physics Today

“Geometry,” in the sense of the classical differential geometry of smooth manifolds (CDG), is put under scrutiny from the point of view of Abstract Differential Geometry (ADG). We explore potential physical implications of viewing things under the light of ADG, especially matters concerning the “gauge theories” of modern physics, when the latter are viewed (as they are actually regarded currently) as “physical theories of a geometrical character.” Thence, “physical geometry,” in connection with physical laws and the associated with them, within the background spacetime manifoldless context of ADG, “differential” equations, are also being discussed.     

太阳米粒结构相关哈特曼-夏克波前传感模拟研究

基于互相关因子和绝对差分算法以及相关哈特曼－夏克波前传感器的基本原理，根据室内模拟太阳米料结构研究了低对比度扩展目标情况下应用相关哈特曼－夏克波前传感器探测波前误差的可行性，结果表明，相关哈特曼－夏克波前传感器能够有效探测光学波前误差。     

Diffractive optics: Design, realization, and applications

This paper presents methods for designing and recording optimal computer-generated diffractive optical elements. The design method is based on an analytic ray-tracing procedure for minimizing aberrations. The recording involves computer-generated masks and multiple lithographic processes in order to form reflective and transmissive multilevel, surface relief-phase, diffractive elements. As a result, the elements can have high diffraction efficiencies over a broad range of incidence angles. Even generalized diffractive elements that operate with highly uniform diffraction efficiency and polychromatic radiation can be designed and recorded by optimizing the shape and height of the relief gratings. To illustrate the effectiveness of the diffractive optical elements, they have been incorporated into a number of applications, involving visible as well as infra-red radiation. Some that deal with coordinate transformation, beam shaping, and polarization control are briefly reviewed.     

Critical value sets and dynamical systems

Elementary applications of singularity theory to dynamical systems are outlined. The main idea is to consider the geometry of the wrinkled surface generated by the time development of a one variable solution as a function of initial conditions or system parameters. Applications of singularity theory to vibro-impacting systems are discussed in some detail and further applications to dynamics are briefly reviewed.     

Weak Density of Smooth Maps in W 1, 1(M,N) for Non-Abelian π1(N)

We prove that smooth maps are dense in the sense of biting convergence in W ^{1, 1}(M, N) when M and Nare compact Riemannian manifolds and N is closed.     

多孔径扫描波面恢复技术的精度评定及影响因素

多孔径扫描波面恢复技术是一项新的高精度大孔径面形检测手段.本文从建立波面恢复精度的评价标准出发,结合实测结果进行讨论,并对影响精度的两个重要因素即拼接模式及拼接区大小进行分析.     

Univalent Interpolation in Besov Spaces and Superposition into Bergman Spaces

We establish the best possible condition for point singularities to be removable for nonlinear elliptic equations in divergent form with lower order terms from the non-linear Kato classes.      

The hp‐version of the BEM with quasi‐uniform meshes for a three‐dimensional crack problem: The case of a smooth crack having smooth boundary curve

The article considers a three‐dimensional crack problem in linear elasticity with Dirichlet boundary conditions. The crack in this model problem is assumed to be a smooth open surface with smooth boundary curve. The hp‐version of the boundary element method with weakly singular operator is applied to approximate the unknown jump of the traction which is not L₂‐regular due to strong edge singularities. Assuming quasi‐uniform meshes and uniform distributions of polynomial degrees, we prove an a priori error estimate in the energy norm. The estimate gives an upper bound for the error in terms of the mesh size h and the polynomial degree p. It is optimal in h for any given data and quasi‐optimal in p for sufficiently smooth data. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Dynamics of coupled systems

A survey in the form of a review of the author’s research results in the area of dynamics of coupled rotations and coupled systems is presented. The theorem on the existence of a trigger of the coupled singularities and the separatrix in the form of the number eight is presented with a few examples of coupled rotations of the simple systems with debalances. Also, a survey of models and dynamics of coupled systems composed of a number of deformable bodies (plates, beams or belts) with different properties of materials and discrete layer properties is presented and mathematically described. The constitutive stress–strain relations for materials of the coupled sandwich structure elements are presented for different properties: elastic, viscoelastic and creeping. The characteristic modes of the coupled system vibrations are obtained and analyzed for different kinds of materials and structure composition. The visualization of the characteristic numbers and modes and eigenamplitude forms are presented. Structural analysis of sandwich structure vibrations is done.     

Decaying states of perturbed wave equations

We study the solutions of perturbed wave equations that represent free wave motion outside some ball. When there are no trapped rays, it is shown that every solution whose total energy decays to zero must be smooth. This extends results of Rauch to the even-dimensional case and to systems having more than one sound speed. In these results, obstacles are not considered. We show that, even allowing obstacles, waves with compact spatial support cannot decay, assuming a unique continuation hypothesis. An example with obstacle is given where nonsmooth, compactly supported, decaying waves exist.