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.     

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) …     

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.     

The comparison theorem for Bergman and Kobayashi metrics on Cartan-Hartogs domain of the second type

In this paper, the holomorphic sectional curvature under invariant metric on a Cartan-Hartogs domain of the second type YII(N,p,K) is presented and an invariant K?]lher metric which is complete and not less than the Bergman metric is constructed, such that its holomorphic sectional curvature is bounded above by a negative constant. Hence a comparison theorem for the Bergman and Kobayashi metrics on YII(N,p,K) is obtained.     

Einstein-K(a)hler metric on Cartan-Hartogs domain of the second type

The Einstein-Kahler metric for the Cartan-Hartogs domain of the second type is described. Firstly, the Monge-Ampère equation for the metric to an ordinary differential equation in the auxiliary function X=X(z,w) is reduced, by which an implicit function in X is obtained. Secondly, for some cases, the explicit forms of the complete Einstein-Kahler metrics on Cartan-Hartogs domains which are the non-homogeneous domains are obtained. Thirdly, the estimate of holomorphic sectional curvature under the Einstein-Kahler metric is given, and in some cases the comparison theorem for Kobayashi metric and Einstein-Kahler metric on Cartan-Hartogs domain of the second type is established.     

ON THE SOLUTION OF THE SINGULAR INTEGRAL EQUATIONS WITH BOTH CAUCHY AND CONVOLUTION KERNEL

The following equations are basic forms of C-K equation (which is simplified in the following as singu-lar integral equations with convolution, that is C-K equations):where a,b,a_j,b_j are known constants or known functions, and find its solution f L_P(R), {0} or {α,β}.There were rather complete investigations on the method of solution for equations of Cauchy type aswell as integral equations of convolution type. But there is not investigation to the C-K equations, nodoubt, such that is important.     

Invariant line strain theory and its application to the crystallography of solid-state phase transformations

The formation mechanisms of the crystallographic features of phase transformation, including orientation relationship, habit plane, growth direction and transformation strain, are described; the early theoretical studies on the invariant line strain model are summarized; and the details of a "three-dimensional invariant line strain model proposed by one of the authors and his colleague abroad are presented. The experimental results on the crystallographic features of needle-, rod- or lath-shaped precipitates formed in the FCC <=>BCC and HCP <=>BCC precipitation transformations were in excellent agreement with the predictions from the model, thus suggesting that the model could well serve as a phenomenological theory of crystallography for diffusion-controlled phase transformations.     

Protein adsorption on materials surfaces with nano-topography

Protein adsorption behavior on the surfaces of biomedical materials is highly related to the biocom- patibility of the materials. In the past, numerous research reports were mainly focused on the effect of chemical components of a material’s surface on protein adsorption. The effect of surface topography on protein adsorption, the topic of this review, has recently received keen interest. The influence of surface nano-topographic factors, including roughness, curvature and geometry, on protein adsorption as well as the protein adsorption behavior, such as the amount of protein adsorbed, the activity and morphology of adsorbed protein, is introduced.     

Similarity solution to a heat equation with convection in an infinite medium

A second order heat equation with convection in an infinite medium is studied. Suitable similarity transformations are used to reduce the parabolic heat equation to a class of singular nonlinear boundary value problems. Numerical solutions are presented for different representations of heat conduction, heat convection, heat flux, and power law parameters by utilizing the shooting technique. The results reveal the heat transfer characteristic and the effect of parameters on the solutions.     

一类L-函数的函数方程

L-functions and their functional equation s of the quadratic fields have played an important role in the study of class nu mber and related problems. In ［2］ the authors studied the L-functions on a kind of real quadratic fields and obtained some interesting resu lts on a kind of real quadratic fields with class number one. In this paper, as a further development of our study on such real quadratic fields, the authors ha ve proved the following theorem.     

常曲率黎曼流形中具平行平均曲率向量的二维浸入曲面

构造了常曲率黎曼流形中具平行平均曲率向量的二维浸入曲面上的全纯微分形式,在同胚球时,给出了其上的Frenet-Boruvka公式及维数定理;估计了浸入的高斯曲率及像的面积;并研究了该浸入的Pinching问题.     

四维欧氏空间中的曲面

研究四维欧氏空间中的一类曲面,给出了具有常平均曲率、常高斯曲率及相关平均曲率和高斯曲率的这类曲面的分类·具有常平均曲率的曲面有4种;具有常高斯曲率的曲面有3种;具有相关非常数平均曲率和非常数高斯曲率的曲面有3种     

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.     

基于高斯曲率改进的PM模型

在Perona和Malik的各项异性扩散方程的消噪模型中,一些小梯度的细节信息和噪声会被扩散掉,而在这些细节信息点往往具有零高斯曲率的特征,根据这一特点,对PM模型加以改进。新的模型不但可以保持零高斯曲率的图像特征,如:直线边缘、曲线边缘、角点、斜坡和小尺度特征,还可以增强尖锐的边缘。实验表明,改进的模型比PM保留了更多的图像信息,因此该模型可以大量地应用于图像处理和计算机视觉。     

Invariants for Parallel Mapping

This paper analyzes the geometric quantities that remain unchanged during parallel mapping (i.e., mapping from a reference curved surface to a parallel surface with identical normal direction). The second gradient operator, the second class of integral theorems, the Gauss-curvature-based integral theorems, and the core property of parallel mapping are used to derive a series of parallel mapping invariants or geometrically conserved quantities. These include not only local mapping invariants but also global mapping invariants found to exist both in a curved surface and along curves on the curved surface. The parallel mapping invariants are used to identify important transformations between the reference surface and parallel surfaces. These mapping invariants and transformations have potential applications in geometry, physics, biomechanics, and mechanics in which various dynamic processes occur along or between parallel surfaces.     

完整非保守力学系统的Noether逆定理与Lie对称性逆问题

研究完整非保守系统的Ｌｉｅ对称性逆问题：由已知积分寻求相应的Ｌｉｅ对称变换。主要方法是,按Ｎｏｅｔｈｅｒ逆定理由已知积分找到相应的Ｎｏｅｔｈｅｒ广义准对称变换,再将所得变换代入Ｌｉｅ对称性的确定方程来判断变换是否Ｌｉｅ的,举例说明方法的应用。     

三维Minkowski空间中具有逐点1型高斯映射的时间轴旋转曲面

讨论R31 中具有逐点1型高斯映射的第一类和第二类时间轴旋转曲面。证明了时间轴旋转曲面具有第一类逐点1型高斯映射,等价于该曲面的平均曲率为常数;非类光洛伦兹圆锥面是惟一具有第二类逐点1型高斯映射的有理类时间轴旋转曲面。