金融衍生产品的力学方法分析（Ⅰ）——期指价格基本方程

类似连续介质力学方法,将期指价格变化看成是连续、有规律可寻的。根据期指特点,建立期指价格变化的基本方程。这是一个微分方程,其解显示时间与价格呈对数圆形关系。若将时间理解为相应价格的概率,则这一关系与基于统计理论分析的、著名的诺贝尔经济学奖（1997）获得者的期权定价Black-Scholes公式中主要假设-基础资产（在此为期指）价格呈对数正态分布-完全一致。表明了依据完全不同的两种分析方法,也会得到相同的结果。只是Black-Scholes是用假设给出,而作者则从微分方程的解推出。     

TORSION OF ELASTIC SHAFT OF REVOLUTION EMBEDDED IN AN ELASTIC HALF SPACE

The problem of torsion of elastic shaft of revolution embedded in an elastic half spaceis studied by the Line-Loaded Integral Equation Method(LLIEM).The problem isreduced to pair of one-dimensional Fredholm integral equations of the first kind due to thedistributions of the fictitious loads“Point Ring Couple(PRC)”and“Point Ring Couple inHalf Space(PRCHS”on the axis of symmetry in the interior and external ranges of theshaft occuied respectively.The direct discrete solution of this integral equations may beunstable,i.e.an ill-posed case occurs.In this paper,such an ill-posed Fredholm integralequation of first kind is replaced by a Fredholm integral equation of the second kind withsmall parameter,which provides a stable solution.This method is simpler and easier tocarry out on a computer than the Tikhonov’s regularization method for ill-posed problems.Numerical examples for conical,cylindrical,conical-cylindrical,and parabolic shafts aregiven.     

The Solution of Elastostatic Problems and the Principles of Minimum Potential Energy,Minimum Complementary Work,under Fuzzy Boundary Conditions

A Solution of elastostatic problem is defined on the basis of set theory andextended to the cases with fuzzy boundary conditions.Extension is alsogiven for the principles of minimum potential energy and minimum comple-mentary work with fuzzy boundary conditions.A quasisolution of an elasto-static problem is defined as an approximate solution with boundary conditionsmost close to the original.And the existance of quasisolution of an elasto-static problem can be proved on the basis of certain assumptions and thetheorem of minimum elementary potential energy.     

Primary research on Subjectivity Geometry

The so-called Subjectivity Geometry herein is a course for studying the relationship between the abstracted object configuration and its observed record by means of mathematical methods and described by mathematical language. There are many features differing from the common geometry when the effects of the subjectivity (such as the position of the observer, the functions of the visual system of human) are taken into account during the observing-recording process. In this paper, some basic assumptions are made; spherical observing record is suggested; the fundamental relationship between the abstracted object configuration and its corresponding observed record is studied and an example of application of the above theory is presented. We anticipate that the study of subjectivity geometry will influence, or will be associated with the study of physiology of the visual system, applied optics etc., and will be useful in surveying, pilotage and imitative biology, etc. Supported by Scientific Foundation of South China University of Technology.     

ASYMMETRIC DYNAMIC INSTABILITY OF AXISYMMETRIC POLAR DIMPLING OF THIN SHALLOW SPHERICAL SHELLS

If the parameterε~2 ,which measures the thickness-to-rise of the shell,is small,theaxisymmetric polar dimpling of shallow spherical shell due to quadratic pressuredistribution is dynamic instability,i.e.,a small perturbation can change it to an asymmetricpolar dimple mode.In two cases,the problem can he reduced to an eigenvalue problemTw_n=c_nw_n,where T can approximately be reduced to a Sturm-Liouville operator ifε~2《1.The existence of at least one real eigenvalue of T,which means that the axisymmetricpolar dimpling is dynamically unstable,is proved by spectral theorem or Hilbert theorem.Furthermore,an eigenfunction,which represents one of the asymmetric modes of theunstable dimple shell,belonging to an eigenvalue of T,is found.     

PRIMARY RESEARCH ON SUBJECTIVITY GEOMETRY

The so-called“Subjectivity Geometry ”herein is a course for studying the relationshipbetween the abstracted object configuration and its observed record by means ofmathematical methods and described by mathematical language.There are many featuresdiffering from the common geometry when the effects of the subjectivity (such as theposition of the observer,the functions of the visual system of human) are taken into accountduring the observing-recording process.In this paper ,some basic assumptions are made;spherical observing record is suggested:the fundamental relationship between theabstracted object configuration and its corresponding observed record is studied and anexample of application of the above theory is presented.We anticipate that the study ofsubjectivity geometry will influence,or will be associated with the study of physiology of thevisual system,applied opties etc.,and will be useful in surveying,pilotage and imitativebiology,etc.     

ANALYSIS OF FINANCIAL DERIVATIVES BY MECHANICAL METHOD (Ⅱ)-BASIC EQUATION OF MARKET PRICE OF OPTION

The basic equation of market price of option is formulated by taking assumptions based on the characteristics of option and similar method for formulating basic equations in solid mechanics: hv ₀(t) = m ₁ v ₀ ^–1(t) – n ₁ v ₀(t) + F, where h, m ₁, n ₁, F are constants. The main assumptions are: the ups and downs of market price v ₀(t) are determined by supply and demand of the market; the factors, such as the strike price, tenor, volatility, etc. that affect on v ₀(t) are demonstrated by using proportion or inverse proportion relation; opposite rules are used for purchasing and selling respectively. The solutions of the basic equation under various conditions are found and are compared with the solution v _f (t) of the basic equation of market price of futures. Furthermore the one-one correspondence between v _f and v ₀(t) is proved by implicit function theorem, which forms the theoretic base for study of v _f affecting on the market price of option v ₀(t).     

Hertz接触问题的精确的积分方程

本文给出Hertz接触问题的精确积分方程,也就是将压力作用下接触面中各点的水平位移也考虑在内而得到的积分方程.     

竖井的近似分析

本文用线载荷积分方程法对坚井在地压作用下作近似分析。竖井被考虑为嵌于均匀的各向同性的弹性半空间中。问题归结为Fredholm第一种积分方程,并给出数值结果。