六方硫化钒晶体的空间群推演

晶体空间群推演是晶体结构描述和表达的关键环节,也是理论印证实验晶体结构的手段。因涉及数学方法、物理技术和图形软件,所以不仅是晶体结构教学的重点也是难点。本文基于六方硫化钒晶体结构测定的基本结构数据,用自编晶体学程序cryst 2015在图形软件上再现硫化钒的晶体结构。在点空间和三维实向量空间中,由线性映射表达平移和非平移对称操作,得到硫化钒的空间群和对应的矩阵表示群。由生成元的幂次连乘导出空间群群元素的生成顺序,以及由空间群导致的晶体学分类。这种借助实例的连贯推演方法,有助于建立晶体学知识结构框架,尤其能帮助初学者理解晶体学的科学思想体系。     

基于SLAM的连续运动测量坐标转换算法研究

本文基于同步定位与地图创建(Simultaneous Localization and Mapping,SLAM)原理,通过双目视觉系统的坐标转换方法,建立动态坐标系与静态基准点之间的关系,实现连续运动测量。通过实验,设计运动路线,验证坐标转换算法,并得到了建立在世界坐标系下的空间线形曲线。实验结果表明：转换后的坐标,x、y、z三个方向的最大误差不超过0.734mm。     

The Invariant and Quasi-invariant Transformations of the Stable Lévy Processes

Let ξ(t),t∈[0,1] be a strictly stable Lévy process with the index of stability α∈(0,2). By ℘ _ξ we denote the law of ξ in the Skorokhod space . For arbitrary ξ we construct ℘ _ξ -quasi-invariant semigroup of transformations of . Under some nondegeneracy condition on the spectral measure of the stable process we construct ℘ _ξ -quasi-invariant group of transformations of . In symmetric case this group is a group of the invariant transformations.      

Quantum Fractals on <Emphasis Type="Italic">n</Emphasis>–Spheres. Clifford Algebra Approach

Using the Clifford algebra formalism we extend the quantum jumps algorithm of the Event Enhanced Quantum Theory (EEQT) to convex state figures other than those stemming from convex hulls of complex projective spaces that form the basis for the standard quantum theory. We study quantum jumps on n-dimensional spheres, jumps that are induced by symmetric configurations of non-commuting state monitoring detectors. The detectors cause quantum jumps via geometrically induced conformal maps (M?bius transformations) and realize iterated function systems (IFS) with fractal attractors located on n-dimensional spheres. We also extend the formalism to mixed states, represented by “density matrices” in the standard formalism, (the n-balls), but such an extension does not lead to new results, as there is a natural mechanism of purification of states. As a numerical illustration we study quantum fractals on the circle (one-dimensional sphere and pentagon), two–sphere (octahedron), and on three-dimensional sphere (hypercubetesseract, 24 cell, 600 cell, and 120 cell). The attractor, and the invariant measure on the attractor, are approximated by the powers of the Markov operator. In the appendices we calculate the Radon-Nikodym derivative of the SO(n + 1) invariant measure on Sⁿ under SO(1, n + 1) transformations and discuss the Hamilton’s “icossian calculus” as well as its application to quaternionic realization of the binary icosahedral group that is at the basis of the 600 cell and its dual, the 120 cell. As a by-product of this work we obtain several Clifford algebraic results, such as a characterization of positive elements in a Clifford algebra as generalized Lorentz “spin–boosts”, and their action as M?bius transformation on n-sphere, and a decomposition of any element of Spin⁺(1, n + 1) into a spin–boost and a spin–rotation, including the explicit formula for the pullback of the SO(n + 1) invariant Riemannian metric with respect to the associated M?bius transformation.     

Preparation of highly (111)-oriented (Pb,La)(Zr,Sn,Ti)O<Subscript>3</Subscript> (PLZST) antiferroelectric thin films by modified sol-gel process using a novel tin source,dibutyloxide of tin

Highly (111)-oriented Pb_0.97La_0.02Zr_0.85Sn_0.13 Ti_0.02O₃(PLZST) antiferroelectric thin films were deposited on Pt(111)/Ti/SiO₂/Si(100) substrates through a modified sol-gel process technique. The electric field-induced antiferroelectric-to-ferroelectric (AFE‐FE) phase transformation behaviour was examined by C-V measurement. The results indicated that antiferroelectric (AFE) to ferrroelectric (FE) switching field , FE to AFE switching field were 315 kV/cm and 240 kV/cm respectively. The temperature dependence of the dielectric constant showed that the Curie temperature (T _c) of the PLZST antiferroelectric thin films was 171°C. The voltage dependent current density of the highly (111)-oriented PLZST film was less than 1.3 × 10⁻⁶ A/cm² over electric field range from 0 to ± 427 kV/cm.     

Oscillation Theorems for Quasilinear Elliptic Differential Equations

By using vector Riccati transformation and averaging technique, some oscillation criteria for the quasilinear elliptic differential equation of second order, ΣNi,j=1Di[Ψ（y）Aij（x）｜Dy｜^p-2Djy]＋p（x）f（y）=0, are obtained. These theorems extend and include earlier results for the semilinear elliptic equation and PDE with p-Laplacian.     

用超对称幺正变换解双光子过程的Jaynes-Cummings模型

本文发现了描写双光子过程的Jaynes-Cummings哈密顿量的超对称结构.根据超对称量子力学理论,引入了超对称幺正变换使其哈密顿量对角化.从而,得到了它的本征值、本征态,同时也计算了态的演化和跃迁几率.     

四元数方法及其应用

本文全面地评述了四元数方法及其在现代科学技术中的应用,指出四元数具有其他定位参数和变换算子的综合优点和综合功能,不仅在刚体动力学中有广泛的用途,而且在一般多体系统的力学分析和控制中也有极其广阔的应用前景。      

正则变换的一种群表示

经典力学中的哈密顿正则变换所涉及的4个母函数F₁(q,Q),F₂(q,P),F₃(p,P),F₄(p,Q)和4种正则变量q,p,Q,P之间所有的关系,可以由7个基本关系式经线性变换而得到,这些变换是勒让德变换,变换是由32个8×8的变换矩阵来实现的,而这32个矩阵以4:1的关系与具有8个群元的D₄点群同态。热力学中的4个状态函数G(P,T),H(P,S),U(V,S),F(V,T)和4个热力学变量P,V,T,S之间的变换关系恰好与正则变换关系相同。热力学状态方程是源于宏观测量的实验结果的概括,而哈密顿正则变换是经典力学的理论性总结,它们的群表示是相同的,即它们的数学结构是相同的, 这种共性表明热力学变换是一维哈密顿正则变换的实例。