Integral Theorems Based on a New Gradient Operator Derived from Biomembranes （Part Ⅱ）： Applications

Based on the second gradient operator and corresponding integral theorems such as the second divergence theorem, the second gradient theorem, the second curl theorem, and the second circulation theorem on curved surfaces, a few new scalar differential operators are defined and a series of integral transformations are derived. Interesting transformations between the average curvature and the Gauss curvature are presented. Various conserved integrals related to the Gauss curvature and the second fundamental tensor are disclosed. The important applications of the results in disciplines such as the geometry, physics, mechanics, and biology are briefly discussed.     

Integral Theorems Based on a New Gradient Operator Derived from Biomembranes (Part II): Applications

Introduction Yin et al.[1, 2] described a gradient operator ? derived from biomembranes with “the second gradient operator” defined on a curved surface. Yin[2] then used the second gradient operator to develop a set of integral theorems named “the second category of integral theorems” on curved surfaces, including the second divergence theorem, the second gradient theorem, the second curl theorem, and the second circulation theorem: d d ?2 dA C A? A = ?K∫∫ ∫ ∫∫i v si Li v A iv (1) …     

Integral Theorems Based on a New Gradient Operator Derived from Biomembranes （Part I）： Fundamentals

A new gradient operator was derived in recent studies of topological structures and shape transitions in biomembranes. Because this operator has widespread potential uses in mechanics, physics, and biology, the operator‘s general mathematical characteristics should be investigated. This paper explores the integral characteristics of the operator. The second divergence and the differential properties of the operator are used to demonstrate new integral transformations for vector and scalar fields on curved surfaces, such as the second divergence theorem, the second gradient theorem, the second curl theorem, and the second circulation theorem. These new theorems provide a mathematical basis for the use of this operator in many disciplines.     

Symmetrical Fundamental Tensors, Differential Operators, and Integral Theorems in Differential Geometry

To make the geometrical basis for soft matters with curved surfaces such as biomembranes as simple as possible, a symmetrical analytical system was developed in conventional differential geometry. The conventional second fundamental tensor is replaced by the so-called conjugate fundamental tensor. Because the differential properties of the conjugate fundamental tensor and the first fundamental tensor are symmetrical, the symmetrical analytical system including the symmetrical differential operators, symmetrical differential characteristics, and symmetrical integral theorems for tensor fields defined on curved surfaces can be constructed. From the symmetrical analytical system, the symmetrical integral theorems for mean curvature and Gauss curvature, with which the symmetrical Minkowski integral formulas are easily deduced just as special cases, can be derived. The applications of this symmetrical analytical system to biology not only display its simplicity and beauty, but also show its powers in depicting the symmetrical patterns of networks of biomembrane nanotubes. All these symmetrical patterns in soft matters should be just the reasonable and natural results of the symmetrical analytical system.     

关于第二积分中值定理“中间点”的再注记

对于第二积分中值定理中的“中间点”,给出并证明了比文献[1]更一般的结论,文献[1]的有关定理可以看成此处定理的推论.     

3个隐式代数曲面的四次GC1拼接及其在三通管道工程上的应用研究

用构造性代数几何方法,研究了3个截平面中有2个平行,另1个与其垂直时,3个隐式代数曲面GC1光滑拼接曲面的构造问题。得到了:GC0控制曲面存在的充分必要条件定理和光滑拼接曲面存在的充分必要条件定理。在此基础上又进行了三通管道工程上的应用研究。     

三维Minkowski空间中二次曲面的分类

研究了二次曲面绕不同坐标轴旋转后再进行平移变换下的不变量,根据这些不变量来研究二次曲面的等价类,达到分类的目的.在三维Minkowski空间中,二次曲面的变换有旋转和平移,旋转又分为在正交标架下绕类空轴、类时轴的旋转和在伪正交标架下绕类光轴的旋转,在不同的旋转变换下有不同的不变量,分类结果也不同.     

织物三维着装系统中几何建模和纹理映射技术的研究

从几何建模和纹理映射方面介绍了实现高质量的织物三维着装效果的计算机仿真方法。通过B样条曲面构造人体与服装模型,利用基于曲线的纹理映射方法,较好地解决了人体与服装曲面的光滑感以及织物图案“贴”到服装曲面后的变形。     

CAD数据修复中寻找相邻曲面的空间包围盒算法

提出了基于ACIS平台寻找相邻曲面的算法,首先对读入的模型的所有曲面求出其空间包围盒,然后对每个曲面的每条边求出曲线的空间包围盒,利用空间包围盒是否相交确定是否可能相邻,最后运用边界匹配算法确定曲面的相邻关系.该算法思想简单,易实现.     

第二型曲面积分的定号法则

第二型曲面积分积分号前±号的选择一直是困扰教与学的疑难问题 ,本文给出了一种简洁有效的方法 :一点定号法则     

积分第二中值定理“中间值”的渐近性

根据函数的特点,讨论了当积分区间趋于无穷大时,积分第二中值定理"中间值"的渐近性问题,利用函数的Taylor展式,得出了中间值的渐近估计式.     

单位球中Moebius截曲率的内蕴性

设x:Mm→Sn为Sn中无脐点子流形,K为截曲率的下确界,本文给出截曲率两个拼挤定理。     

R2,1中曲面的伪球线汇

线汇通过线产生于曲面间变换的经典方法中.如果保留原始一些曲面的几何性质,这些转变是特别有趣的.线汇的两个参数族作为线空间的的曲面来研究.利用活动标架来研究线汇,给出了3维闵氏空间R2,1中常Gauss曲率曲面间统一的Backlund变换和Bianchi's置换定理的证明.最后,利用定理的结果构造了一些伪球曲面.     

Yoshizawa稳定性和有界性定理的注记

研究了Yoshizawa稳定性和有界性.利用常微分方程定性理论中公认的普遍且强有力的Liapunov第二方法,将某些稳定性和有界性定理的条件减弱,并给出证明,从而改进了全局一致渐近稳定的定义.     

积分第二中值定理逆问题及其渐近性

对积分第二中值定理作了进一步的研究,得到了积分第二中值定理的逆问题及其逆问题的渐进性.研究表明,本文定理对于探讨有关积分问题有着十分重要的作用.     

适应数控加工的空间曲面的建立

论述了适应数控加工B样条曲面建立的方法,指出了B样条曲面具有一、二阶连续的特性,可以用多个曲面片的拼接而组成任意形状的自由曲面,从而解决了数控加工中空间曲面建模的难题,为建立空间自由曲面提供了理论依据.     

一类具有连续偏差变元的二阶半线性微分方程的振动准则

考虑了一类具有连续偏差变元的二阶半线性微分方程,利用积分变换和广义Riccati变换,给出了此类方程的振动准则.     

Second Shape Finding Analysis of Membrane Structures

A second shape finding method was developed to improve the nonlinear finite element based shape finding method. The curved shape is obtained by raising the control points above the projection planeThe convergence was improved using pseudo material properties to get a preliminary shape, and then using the real properties to get the final shape. A large number of examples were analyzed to verify the validity and practicality of this method. The results show that the final curved surface after the second shape finding process is always quite similar to the first one. Moreover, the curved surface obtained after the second shape finding process is accurate and will be realized in real materials.     

曲梁剪应力的积分方程解

直接对曲梁剪应力的积分方程求解 ,导出了曲梁剪应力和径向应力的计算公式 .这些公式不仅满足平衡方程 ,而且满足曲梁上、下表面处力的边界条件 .将该理论用于研究悬臂曲梁在自由端受集中力作用的情况 ,计算结果表明 ,与其他采用附加假设的近似解相比 ,据此得到的应力解具有很高精度 ,同弹性理论解和有限元解非常接近 .     

以曲面为顶的CAD实体体积计算

根据多重积分理论，用Ｓｈｅｐａｒｄ曲面和加权最小二乘曲面来实现三维空间散乱数据点所围成的曲顶实体体积的计算．