环面上近三角剖分地图的计数

本文提供了环面上带边数和根面次这两个参数的有根近三角剖分的函数方程及其参数表达式，并给出了根面次为1以边数为参数的有根近三角剖分地图的精确解．     

f-Vectors of Polyhedra

The f-vector of a triangulation of a polyhedron X is the numbers of simplices at various dimensions. We prove that the affine span of f-vectors of X has dimension (n+s+1)/2, where n is the dimension of X, and s is the dimension of the part of X that is singular with respect to the local Euler characteristic.     

Cuts, matrix completions and graph rigidity

This paper brings together several topics arising in distinct areas: polyhedral combinatorics, in particular, cut and metric polyhedra; matrix theory and semidefinite programming, in particular, completion problems for positive semidefinite matrices and Euclidean distance matrices; distance geometry and structural topology, in particular, graph realization and rigidity problems. Cuts and metrics provide the unifying theme. Indeed, cuts can be encoded as positive semidefinite matrices (this fact underlies the approximative algorithm for max-cut of Goemans and Williamson) and both positive semidefinite and Euclidean distance matrices yield points of the cut polytope or cone, after applying the functions 1/π arccos(.) or √. When fixing the dimension in the Euclidean distance matrix completion problem, we find the graph realization problem and the related question of unicity of realization, which leads to the question of graph rigidity. Our main objective here is to present in a unified setting a number of results and questions concerning matrix completion, graph realization and rigidity problems. These problems contain indeed very interesting questions relevant to mathematical programming and we believe that research in this area could yield to cross-fertilization between the various fields involved.     

Light power density distribution of image spot of laser triangulation measuring

HaiMing Wang [Wang HaiMing, Long range optical triangulation utilising collimated probe beam. Optics and Lasers in Engineering, 23(11), 1995, 41–52] studied the influences of the size of the object spot on the image spot in laser triangulation measuring. This paper points out that the study method of HaiMing Wang is inappropriate from two points of view. The size and relative power density distribution of the image spot can be calculated either when assuming the object spot is an ideal point, which emits a uniform spherical wave over the object, or when assuming that the object surface is a light scattering surface. Both the conditions show that the size of the image spot is much bigger than that of HaiMing Wang’s.     

一种标定三角测量法激光位移计的方法

三角测量的原理公式中因变量、自变量及系统结构参数之间具有非线性关系，并且这些系统结构参数在工程上难以精确测量，因此激光位移计的标定变得很困难。本文通过设置辅助变量及组合参数，使变量之间关系线性化，然后采用最小二乘法拟合出组合参数的准确值，实现激光位移计的准确标定。在此过程中对自变量的测量值做适量修正，可部分消除光学系统畸变对组合参数拟合结果的影响，减小拟合残差。本文给出了用该方法标定激光位移计的实验结果     

Triangulated n-manifolds are determined by their [n2] + 1-skeletons

We will reconstruct compact, triangulated n-manifolds—without-boundaries from just their $\text{[}\text{n}\text{2}\text{] + 1}$ skeletons. Therefore, two such manifolds are isomorphic if they have simplicially isomorphic $\text{[}\text{n}\text{2}\text{+ 1}$-skeletons. Furthermore, when n is even, and the $\text{n}\text{2}$-homology group is zero, then the $\text{n}\text{2}$-skeleton is sufficient.     

Hypoelliptic systems connected with Newton's polyhedron

We introduce a general class of hypoelliptic systems of partial differential operators with variable coefficients generalizing the so‐called multi‐quasi‐elliptic differential operators. We study the regularity of solutions of these systems. More complete results are obtained in the case of two independent variables. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

运动目标空间位置坐标激光跟踪测量的数学模型

运动目标空间位置激光跟踪测量是国际计量测试领域的前沿课题，该测量系统集激光干涉测距、精密机械、计算机及控制系统和现代数值计算于一体，对空间运动目标进行跟踪并实时测量其坐标和姿态。它的测量范围为１～５ｍ，测量精度为５０μｍ／ｍ。它的测量原理主要是干涉法和三角法。在干涉法中，根据跟踪测量机构数量的多少又可分为一站法和多站法。一站法采用球坐标测量法，多站法通过解最小二乘方程组计算出被测点的坐标。文中对各种测量方法的数学理论模型进行了较为详细地论述。     

Delaunay triangulation and 3D adaptive mesh generation

Delaunay triangulation and its complementary structure the Voronoi polyhedra form two of the most fundamental constructs of computational geometry. Delaunay triangulation offers an efficient method for generating high-quality triangulations. However, the generation of Delaunay triangulations in 3D with Watson's algorithm, leads to the appearance of silver tetrahedra, in a relatively large percentage. A different method for generating high-quality tetrahedralizations, based on Delaunay triangulation and not presenting the problem of sliver tetrahedra, is presented. The method consists in a tetrahedra division procedure and an efficient method for optimizing tetrahedral meshes, based on the application of a set of topological Delaunay transformations for tetrahedra and a technique for node repositioning. The method is robust and can be applied to arbitrary unstructured tetrahedral meshes, having as a result the generation of high-quality adaptive meshes with varying density, totally eliminating the appearance of sliver elements. In this way it offers a convenient and highly flexible algorithm for implementation in a general purpose 3D adaptive finite element analysis system. Applications to various engineering problems are presented