Operator trigonometry of Hotelling correlation, Frobenius condition, Penrose twistor

Three new applications of the author’s operator trigonometry are presented. These take place in three diverse mathematical domains: matrix statistics, numerical analysis, and theoretical physics. The goal is to continue to develop and expand the scope of the operator trigonometry.     

One dimensional metrical geometry

One dimensional metrical geometry may be developed in either an affine or projective setting over a general field using only algebraic ideas and quadratic forms. Some basic results of universal geometry are already present in this situation, such as the Triple quad formula, the Triple spread formula and the Spread polynomials, which are universal analogs of the Chebyshev polynomials of the first kind. Chromogeometry appears here, and the related metrical and algebraic properties of the projective line are brought to the fore.      

Promoting uncertainty to support preservice teachers’ reasoning about the tangent relationship

Including opportunities for students to experience uncertainty in solving mathematical tasks can prompt learners to resolve the uncertainty, leading to mathematical understanding. In this article, we examine how preservice secondary mathematics teachers’ thinking about a trigonometric relationship was impacted by a series of tasks that prompted uncertainty. Using dynamic geometry software, we asked preservice teachers to compare angle measures of lines on a coordinate grid to their slope values, beginning by investigating lines whose angle measures were in a near-linear relationship to their slopes. After encountering and resolving the uncertainty of the exact relationship between the values, preservice teachers connected what they learned to the tangent relationship and demonstrated new ways of thinking that entail quantitative and covariational reasoning about this trigonometric relationship. We argue that strategically using uncertainty can be an effective way of promoting preservice teachers’ reasoning about the tangent relationship.     

细管道内表面光电检测方法研究

管道是气体和液体传输的重要手段,管道内表面的检测对于工业和国防中管道泄漏事故的预防,减少环境污染和经济损失非常重要。随着电子和半导体技术的发展,光电器件逐步趋于小型化。介绍了基于激光阵列、PSD光电检测、光环截面以及结构光检测等光电检测方法的测量原理和系统构成,并在此基础上对不同光电检测方法的优缺点进行了分析和比较。分析结果表明：光电检测技术适用于管道内表面检测;并朝着快速识别缺陷、管道内表面瑕疵的精确三维测量以及三维图像直观显示管壁缺陷的方向发展。     

Three-dimensional vertex model in statistical mechanics from Baxter-Bazhanov model

We find that the Boltzmann weight of the three-dimensional Baxter-Bazhanov model is dependent on four spin variables which are the linear combinations of the spins on the corner sites of the cube, and the Wu-Kadanoff-Wegner duality between the cube- and vertex-type tetrahedron equations is obtained explicitly for the Baxter-Bazhanov model. Then a three-dimensional vertex model is obtained by considering the symmetry property of the weight function, which corresponds to the three-dimensional Baxter-Bazhanov model. The vertex-type weight function is parametrized as the dihedral angles between the rapidity planes connected with the cube. We write down the symmetry relations of the weight functions under the actions of the symmetry groupG of the cube. The six angles with a constraint condition appearing in the tetrahedron equation can be regarded as the six spectra connected, with the six spaces in which the vertextype tetrahedron equation is defined.     

Travelling Randomly on the Poincaré Half-Plane with a Pythagorean Compass

A random motion on the Poincaré half-plane is studied. A particle runs on the geodesic lines changing direction at Poisson-paced times. The hyperbolic distance is analyzed, also in the case where returns to the starting point are admitted. The main results concern the mean hyperbolic distance (and also the conditional mean distance) in all versions of the motion envisaged. Also an analogous motion on orthogonal circles of the sphere is examined and the evolution of the mean distance from the starting point is investigated.     

战车用惯性定位定向系统的支架误差分析及补偿

战车用惯性定位定向系统由于采用双轴陀螺平台结构,不可避免地存在着支架误差,从而影响系统航向精度。应用球面三角法和方向余弦矩阵法分别推导出双轴陀螺平台的支架误差公式,发现由这两种方法推导出来的公式计算结果是一致的。最后通过数学仿真和分析,给出了支架误差与载体的姿态角之间的定量关系。根据此误差公式,对惯性定位定向系统的航向输出能够进行误差补偿,从而保证了动态条件下定位定向系统的航向输出精度。     

Constructing knowledge about the trigonometric functions and their geometric meaning on the unit circle

Processes of knowledge construction are investigated. A learner is constructing knowledge about the trigonometric functions and their geometric meaning on the unit circle. The analysis is based on the dynamically nested epistemic action model for abstraction in context. Different tasks are offered to the learner. In his effort to perform the different tasks, he has the opportunity to understand the process used to create unit circle representations of trigonometric expressions. The theoretical framework of abstraction in context is used to analyse the evolution of the learner's construction of knowledge in the transition from ‘triangle’ trigonometry to ‘circle’ trigonometry.     

Integrating mathematics,statistics, and technology in vocational and workplace education

In the workplace mathematics and statistics are essential for communication and decision-making. Process workers at lower classifications of skill levels are likely to be confronted with statistical charts and warnings about nonconformity. Mathematics, statistics, and technology education in and for the workplace must take account of the cultural diversity which exists within and between workplaces. The design of generic mathematics, and in some cases statistics, curricula rarely reflect actual workplace practice except at a superficial level. One way of overcoming these problems is for mathematics/statistics educators to work in cooperation with industry, particularly at the local level, in a way that will encourage and support lifelong learning yet remain critical of the uses to which mathematics, statistics, and technology are put. This paper outlines some ways in which to address the challenge of making mathematics, statistics, and technology education take on real meaning within the context of the workplace.     

Spherical f-Tilings by Two Noncongruent Classes of Isosceles Triangles-Ⅱ

In this work,we give a complete classification of spherical dihedral f-tilings when the prototiles are two noncongruent isosceles triangles with certain adjacency pattern.As it will be shown,this class is composed by two discrete families denoted by Em,m ≥ 2,m ∈ N,Fk,k ≥ 4,k ∈ N and two sporadic tilings denoted by G and H.