Bipartite graphs are not universal fixers

For any permutation π of the vertex set of a graph G, the graph πG is obtained from two copies G^′ and G^″ of G by joining u∈V(G^′) and v∈V(G^″) if and only if v=π(u). Denote the domination number of G by γ(G). For all permutations π of V(G), γ(G)≤γ(πG)≤2γ(G). If γ(πG)=γ(G) for all π, then G is called a universal fixer. We prove that graphs without 5-cycles are not universal fixers, from which it follows that bipartite graphs are not universal fixers.     

Claw-free graphs are not universal fixers

For any permutation π of the vertex set of a graph G, the generalized prism πG is obtained by joining two copies of G by the matching {uπ(u):u∈V(G)}. Denote the domination number of G by γ(G). If γ(πG)=γ(G) for all π, then G is called a universal fixer. The edgeless graphs are the only known universal fixers, and are conjectured to be the only universal fixers. We prove that claw-free graphs are not universal fixers.     

全反射棱镜式环形激光器耦合输出的理论分析

对采用棱镜耦合器的全反射棱镜式环形激光器的耦合输出特性进行了理论研究, 导出了有“楔形”间隙的棱镜耦合器的反射系数和透射系数表达式, 给出了反射率和透过率随“楔形”间隙高度变化的理论计算曲线, 进而研究了有关材料选取、腔内振荡分量的选取及减小棱镜耦合器对环形激光腔内模式的影响等问题. 这些研究对于全反射棱镜式环形激光器在激光陀螺、激光角速度传感器等实际应用中有重要意义.     

Geometrical optics analysis for reduction of trapezoidal image distortion in a single prism-based optical fingerprint scanner

A single prism-based optical fingerprint scanner structure is mathematically analyzed by using geometrical optics approach. Important parameters including a tilting angle of the fingerprint accepting plane, a geometrical-optic distant difference, a trapezoidal image distortion, an object compression ratio, and an orientation of a two-dimensional image sensor are formulated for the first time in terms of all three angles of the prism and the prism material. In addition, based on our mathematical model, we propose to design all three angles of the prism in such a way that the plane accepting the fingertip observed from one side of the prism is normal to the optical axis of the system. In this way, the imaging plane is perpendicular to the optical axis, eliminating the trapezoidal image distortion and leading to ease of implementation. Experimental verification using three commercially available dispersion prisms made from three different glass materials and one prism based on our design concept shows that a prism with higher refractive index provides lower trapezoidal image distortion. The experimental data generally obeys our theoretical analysis.