从等周不等式谈Green公式的一个应用

Green公式给出了平面图形面积与曲线积分的关系,利用这种关系给出了平面曲线等周不等式的一个简捷证明,并给出了平面图形面积的一个近似计算公式以及两个教学计算实例.     

反函数的原函数公式及其应用

在本文第一部分给出了反函数原函数公式一个简单的证明，第二部分列举了几个用反函的原函数公式求积的例子，最后用此公式给出了Ｙｏｕｎｇ不等式一个证明，此证明不必借助于定积分的几何意义和图形面积的几何直观，从而是数学意义上更为严格的证明。     

Winkler地基上固支薄板自由振动问题的准Green函数方法

将准Green函数方法应用于求解Winkler地基上固支薄板的自由振动问题.即利用问题的基本解和边界方程构造一个准Green函数,这个函数满足了问题的齐次边界条件.采用Green公式,将Winkler地基上固支薄板自由振动问题的振型控制微分方程化为第二类Fredholm积分方程.通过边界方程的适当选择,积分方程核的奇异性被克服了.数值算例表明,该方法具有较高的精度,是一种有效的数学方法.     

建立几何模型,巧记乘法公式

在乘法公式的学习中,对公式常常理解不到位,记忆不准确.如果从几何的角度出发,建立直观的、形象的几何模型,则便于理解和记忆.下面我们举例进行说明. 一、平方差公式 公式:(a b)(a-b)=a2-b2. 几何模型:长方形的面积. 构造过程:构造一个长为(a b)、宽为(a-b)的长方形(见图1),则其面积为S=(a b)(a-b).下面,我们将长方形沿虚线截取宽为b、长为(a-b)的一个小长方形,并将它补成如图2所示的图形.     

任意多边形面积公式的推导及其应用

通过对Green公式的离散变形,建立了重积分到曲线积分的公式,用多种方法推导出任意多形的面积公式.     

从阿基米德关于拋物線弓形面积的算法談到圓弓形面积近似公式

苏联吉西略夫所著几何敎科書中所引用的圓弓形面积近似公式,几年来已有不少人提出討論和証明。今年1月号数学通报上又同时發表了三篇这样的文章,其中宋三元先生一文中所用方法是用三角形逼近圓弓形。用三角形逼近的方法,古时阿基米德即曾用来計算拋物線弓形的面积。本文先从阿氏算法談起,再討論椭圓弓形,从而作为特別情形,得出圓弓形面积的第一个近似公式。感觉有趣的是三种弓形的面积公式之間的密切联系的存在,尤其是圓弓形面积的兩个近似公式的直覌意义。最后,附帶提到大于半圓的圓弓形。錯誤之处,希望大家給予指正。     

格林公式的几何意义

我们知道格林公式指出了二重积分和曲线积分的联系。本文的目的是要讨论格林公式的几何意义。格林公式:设P(x,y),Q(x,y)以及偏导数(?)P/(?)y和(?)Q/(?)x是闭单连通区域(D)上的连续函数,(D)的边界(L)是光滑的简单曲线,那末有从格林公式的证明知道,不需要增加条件,下式亦成立:和以下就(1)式来讨论几何意义,对于(2)式是类似的。我们假定读者已经知道什么是平面图形的面积。现在给出一个判别平面图形是否有面积的方法:     

找准几何概型解题突破口

几何概型有两个特点：一是无限性，即在一次试验中，基本事件的个数是无限的；二是等可能性，即每一个基本事件发生的可能性是均等的．几何概型问题求解概率的公式P（A）=d的测度/D的测度（分母不为0），其中“测度”的意义依几何区域D确定，当D分别是线段、平面图形和立体图形时，相应的“测度”分别是长度、面积和体积．在解题时，     

平面中Poisson方程的Dirichlet问题

基于向量旋转内积不变的特点,通过对Green积分公式的推广,得到与空间Green三个公式相似的平面Green公式.从而,得到平面中Poisson方程Robin问题的解和平面中Poisson方程Dirichlet问题的解.     

简支梯形底扁球壳自由振动问题的准Green函数方法

以简支梯形底扁球壳的自由振动问题为例,详细阐明了准Green函数方法的思想.即利用问题的基本解和边界方程构造一个准Green函数,此函数满足了问题的齐次边界条件,采用Green公式,将简支梯形底扁球壳自由振动问题的振形控制微分方程化为两个耦合的第二类Fredholm积分方程.边界方程有多种选择,在选定一种边界方程的基础上,可以通过建立一个新的边界方程来表示问题的边界,以克服积分核的奇异性.最后由积分方程的离散化方程组有非平凡解的条件,求得固有频率.数值结果表明,该方法具有较高的精度.     

Contour-ordered Green’s functions in stochastic field theory

We briefly review the functional formulation of the perturbation theory for various Green’s functions in quantum field theory. In particular, we discuss the contour-ordered representation of Green’s functions at a finite temperature. We show that the perturbation expansion of time-dependent Green’s functions at a finite temperature can be constructed using the standard Wick rules in the functional form without introducing complex time and evolution backward in time. We discuss the factorization problem for the corresponding functional integral. We construct the Green’s functions of the solution of stochastic differential equations in the Schwinger-Keldysh form with a functional-integral representation with explicitly intertwined physical and auxiliary fields.     

Green's function-based integral approaches to linear transient boundary-value problems and their stability characteristics (I)

Two Green's function-based formulations are applied to the governing differential equation which describes unsteady heat or mass transport in an isotropic homogeneous 1-D domain. In this first part of a two series of papers, the linear form of the differential equation is addressed. The first formulation, herein denoted the quasi-steady Green element (QSGE) formulation, uses the Laplace differential operator as auxiliary equation to obtain the singular integral representation of the governing equation, while the second, denoted the transient Green element (TGE), uses the transient heat equation as auxiliary equation. The mathematical simplicity of the Green's function of the first formulation enhances the ease of solution of the integral equations and the resultant discrete equations. From the point of computational convenience, therefore, the first formulation is preferred. The stability characteristics of the two formulations are evaluated by examining how they propagate various Fourier harmonics in speed and amplitude. We found that both formulations correctly reproduce the theoretical speed of the harmonics, but fail to propagate the amplitude of the small harmonics correctly for Courant value of about unity. The QSGE formulation with difference weighting values between 0.67 and 0.75, and the TGE formulation provide optimal performance in numerical stability.     

Boundary conditions in intrinsic nonlinear elasticity

The intrinsic formulation of the displacement-traction problem of nonlinear elasticity is a system of partial differential equations and boundary conditions whose unknown is the Cauchy–Green strain tensor field instead of the deformation as is customary. We explicitly identify here the boundary conditions satisfied by the Cauchy–Green strain tensor field appearing in such intrinsic formulations.     

Music algorithm for locating point-like scatterers contained in a sample on flat substrate

In this paper, we consider a MUSIC algorithm for locating point-like scatterers contained in a sample on flat substrate. Based on an asymptotic expansion of the scattering amplitude proposed by Ammari et al., the reconstruction problem can be reduced to a calculation of Green function corresponding to the background medium. In addition, we use an explicit formulation of Green function in the MUSIC algorithm to simplify the calculation when the cross-section of sample is a half-disc. Numerical experiments are included to demonstrate the feasibility of this method.     

Global L theory and regularity for the 3D nonlinear Wigner-Poisson-Fokker-Planck system

A global existence, uniqueness and regularity theorem is proved for the simplest Markovian Wigner-Poisson-Fokker-Planck model incorporating friction and dissipation mechanisms. The proof relies on Green function and energy estimates under mild formulation, making essential use of the Husimi function and the elliptic regularization of the Fokker-Planck operator.     

麦克斯韦方程和锥形折射

通过对麦克斯韦方程格林函数的计算 ,我们得到它的求解公式 ,然后解释为什么一束光线射到冰洲石表面上会产生两束折射光线 ,并描述了折射光线的路径     

FFT-based homogenization on periodic anisotropic translation invariant spaces

In this paper we derive a discretisation of the equation of quasi-static elasticity in homogenization in form of a variational formulation and the so-called Lippmann–Schwinger equation, in anisotropic spaces of translates of periodic functions. We unify and extend the truncated Fourier series approach, the constant finite element ansatz and the anisotropic lattice derivation. The resulting formulation of the Lippmann–Schwinger equation in anisotropic translation invariant spaces unifies and analyses for the first time both the Fourier methods and finite element approaches in a common mathematical framework. We further define and characterize the resulting periodised Green operator. This operator coincides in case of a Dirichlet kernel corresponding to a diagonal matrix with the operator derived for the Galerkin projection stemming from the truncated Fourier series approach and to the anisotropic lattice derivation for all other Dirichlet kernels. Additionally, we proof the boundedness of the periodised Green operator. The operator further constitutes a projection if and only if the space of translates is generated by a Dirichlet kernel. Numerical examples for both de la Vallée Poussin means and Box splines illustrate the flexibility of this framework.     

Self-adjoint realization of a class of third-order differential operators with an eigenparameter contained in the boundary conditions

The present paper deals with a class of third-order differential operators with eigenparameter dependent boundary conditions. Using operator theoretic formulation, the self-adjointness of this operator is proved, the properties of spectrum are investigated, its Green function and the resolvent operator are also obtained.     

Resolvent Operator and Self-Adjointness of Sturm–Liouville Operators with a Finite Number of Transmission Conditions

In this paper, we consider a Sturm–Liouville operator with eigenparameter-dependent boundary conditions and transmission conditions at a finite number of interior points. We introduce a Hilbert space formulation such that the problem under consideration can be interpreted as an eigenvalue problem for a suitable self-adjoint linear operator. We construct Green function of the problem and resolvent operator. We establish the self-adjointness of the discontinuous Sturm–Liouville operator.     

Green-Funktionen für Wärmeleiter mit beweglichen Rändern

Zusammenfassung Es werden Green-Funktionen für Randwertaufgaben der Wärmeleitungsgleichung zusammengestellt. Die Ränder sind zeitlich linear veränderlich. Die angegebenen Green-Funktionen haben die Form von gut konvergierenden Thetareihen. Für die Randbedingungen wird eine Matrix-Schreibweise benutzt, die es erlaubt, eine Vielzahl von Spezialfällen einheitlich zu behandeln.

---

Summary The present work contains a collection of Green functions for heat conductors whose boundaries vary linearly with time. The Green functions are represented in terms of theta series. A matrix formulation is used for the boundary conditions, which allows us to treat a number of special problems in a unified manner.