Effective generalized equations of secure hyperelliptic curve digital signature algorithms
generalized estimating equations
generalized estimating equations广义估计方程（GeneralizedEstimatingEquations，GEE）是一种用于分析长期或重复测量数据的统计方法。
与传统的线性模型不同，GEE是一种非参数方法，它可以处理不同时间点或不同个体之间的相关性，并且可以处理非正态分布的数据。
作为一种广泛应用的分析方法，GEE已经在许多领域得到了应用，例如医学、社会科学、心理学等。
本文将介绍GEE的基本原理、模型构建、参数估计和模型诊断等方面，以此帮助读者更好地理解和应用GEE方法。
二、基本原理在分析长期或重复测量数据时，传统的线性模型通常不能很好地处理数据的相关性和非正态性。
GEE的基本思想是通过构建广义线性模型（Generalized Linear Model，GLM）来处理这些问题。
具体来说，GEE将目标变量与一组协变量进行回归，同时通过引入协方差结构来考虑数据的相关性。
因此，GEE的模型可以表示为：$$g(mu_{ij})=boldsymbol{x}_{ij}^{top}boldsymbol{beta}+bolds ymbol{z}_{ij}^{top}boldsymbol{alpha}$$其中，$g$是一个已知的连接函数，$mu_{ij}$是第$i$个个体在第$j$个时间点的目标变量的均值，$boldsymbol{x}_{ij}$和$boldsymbol{z}_{ij}$是分别与固定效应$boldsymbol{beta}$和随机效应$boldsymbol{alpha}$相关的协变量，$i=1,2,...,n$，$j=1,2,...,m_i$，$n$是个体数，$m_i$是第$i$个个体的测量次数。
在GEE中，我们假设数据的协方差矩阵可以表示为：$$boldsymbol{Sigma}_i=boldsymbol{A}(boldsymbol{phi})boldsym bol{R}_iboldsymbol{A}^{top}(boldsymbol{phi})$$其中，$boldsymbol{Sigma}_i$是第$i$个个体的协方差矩阵，$boldsymbol{R}_i$是一个已知的相关矩阵，$boldsymbol{A}(boldsymbol{phi})$是一个已知的函数，$boldsymbol{phi}$是一个未知的参数向量。
土木工程专业英语
1.荷载短期荷载short-time load 临界荷载critical load 持续荷载sustained loads恒载dead load 活载live load 峰值荷载peak load 冲击荷载impact load 2.专业名词力矩面等横截面cross section 隔离体 a free body 轴力axial forces 带肩梁ledger beam正应力the normal stress 剪应力the shear stress 固定铰支座 a pin support 可动铰支座 a roller support 平面内弯矩in-plane bending 平面外弯矩out-of-plane bending简支梁a simple beam 悬臂梁 a cantilever beam 分布力distributed load 均布力uniformly distributed load 静定结构statically determinate structure 超静定结构statically indeterminate structure 角焊缝fillet weld 对接焊缝groove weld外缘outer edges 中性轴the neutral axis 形心矩centroidal distance沙石混凝土sand-and-stone concrete 预应力混凝土pre stressed concrete复合应力combined stress 极限应变limiting tensile strain 平均正应力mean normal stress名义抗剪强度nominal shear strength 惯性力inertia force 地震作用seismic action广义位移generalized displacement 扭矩torsion 预加应力pre stress托梁corbel3.材料平面顶deck 屋面防水层water proof roofing 金属箔层压板foil-laminated钢筋steel 涂料paint 木条板lath 灰泥plaster 楔子wedge基础footing 横向钢筋transverse reinforcement 纵筋longitudinal reinforcement 弯起纵筋bent-up longitudinal steel 单向板one-way slabs 腹筋the web steel 楼梯踏步stair tread 顶棚抹灰plastered ceilings 承重墙bearing wall第 1 页/共 4 页轻质幕墙light weight curtain walls 桁架truss 构件member 谷仓grain elevator桥墩bridge pier 大型结构heavy structure 梯井stair shaft高层写字楼high-rise office 预埋构件metal insert 作业平台work plat form企口木板tongue-and-groove plank 施工架constructed yoke 走道脚手架 a walkway scaffold铅垂线the plumb line 喷雾器fog sprays 型钢structural steel 剪力墙shear wall平板flat slab 合成薄板synthetic film 防护墙板endosing wall panels人字起重机derrick crane 卫生间设施bathroom groups 服务竖井the service shaft隔气层vapor barriers 隔热层insulation 结露点dew point 空心板hollow plank竖向剪力墙shear-resistant vertical wall 预制构件pre cast member 隔板wall panel4.其他1应力等值线 a stress contour 数值分析numerical analysis 悬索基础cable structures实验研究experimental investigation 超静定次数degree of statical indeterminaly叠加法method of superposition 基本结构released structure高跨比span-depth ratio弯矩图bending moment diagram 附着deposit 弹性模量modulus of elasticity水化hydrate 硬化harden 变量variables 环境相对湿度ambient relative humidity蒸发evaporate 定向立方体单元oriented elementary cube初步结论tentative conclusion斜向拉力diagonal tension 微分长度单元 a differential length 应力迹线stress trajectory骨料咬合作用aggregate interlock 销栓作用dowel action 延性ductility扭转力偶twisting couple 力臂lever arm 分数fraction 取代in lieu of地震高发区zones of high earthquake probability 平立面in plan elevation平动translation 转动rotation 凹部depressions 凸起projection 凸口recess 在现场on the site 误差error 通用规范applicable codes滑模施工slip form operations 养护care 锚固be anchored in 挠度deflection5.其他2侧向支持sway bracing 先张法pre tensioning technique 后张法post tensioning technique安全系数safety factor 安全储备margin of safety 附属cust-in fittings防火等级fire ratings 不匀称沉降differential settlement 深基础deep foundation扩展式基础spread foundation 符合基础combined footings 条形基础strap footings垂直于at right angles to 类似于analogous to 单位力法unit-load method大小相等方向相反be equal in magnitude and opposite in direction静力平衡方程equations of static equilibrium 与……有关pertain to求合力from a summation of force 一组联立方程 a set of simultaneous equations协调方程equations of compatibility 经验方程empirical equation大一个数量级an order of magnitude longer 第二面积积分the second moment-area thorea·b dot product a*b cross product 位移互等定理reciprocal displacement theorem第 3 页/共 4 页液压控制系统hydraulic master control system 功的互等定理…………work ……与……成正比in direct proportion to 与……一致be geared to。
有限元分析中英文对照资料知识讲解
有限元分析中英文对照资料The finite element analysisFinite element method, the solving area is regarded as made up of many small in the node connected unit (a domain), the model gives the fundamental equation of sharding (sub-domain) approximation solution, due to the unit (a domain) can be divided into various shapes and sizes of different size, so it can well adapt to the complex geometry, complex material properties and complicated boundary conditions Finite element model: is it real system idealized mathematical abstractions. Is composed of some simple shapes of unit, unit connection through the node, and under a certain load.Finite element analysis: is the use of mathematical approximation method for real physical systems (geometry and loading conditions were simulated. And by using simple and interacting elements, namely unit, can use a limited number of unknown variables to approaching infinite unknown quantity of the real system.Linear elastic finite element method is a ideal elastic body as the research object, considering the deformation based on small deformation assumption of. In this kindof problem, the stress and strain of the material is linear relationship, meet the generalized hooke's law; Stress and strain is linear, linear elastic problem boils down to solving linear equations, so only need less computation time. If the efficient method of solving algebraic equations can also help reduce the duration of finite element analysis.Linear elastic finite element generally includes linear elastic statics analysis and linear elastic dynamics analysis from two aspects. The difference between the nonlinear problem and linear elastic problems:1) nonlinear equation is nonlinear, and iteratively solving of general;2) the nonlinear problem can't use superposition principle;3) nonlinear problem is not there is always solution, sometimes even no solution. Finite element to solve the nonlinear problem can be divided into the following three categories:1) material nonlinear problems of stress and strain is nonlinear, but the stress and strain is very small, a linear relationship between strain and displacement at this time, this kind of problem belongs to the material nonlinear problems. Due to theoretically also cannot provide the constitutive relation can be accepted, so, general nonlinear relations between stress and strain of the material based on the test data, sometimes, to simulate the nonlinear material properties available mathematical model though these models always have their limitations. More important material nonlinear problems in engineering practice are: nonlinear elastic (including piecewise linear elastic, elastic-plastic and viscoplastic, creep, etc.2) geometric nonlinear geometric nonlinear problems are caused due to the nonlinear relationship between displacement. When the object the displacement is larger, the strain and displacement relationship is nonlinear relationship. Research on this kind of problemIs assumes that the material of stress and strain is linear relationship. It consists of a large displacement problem of large strain and large displacement little strain. Such as the structure of the elastic buckling problem belongs to the large displacement little strain, rubber parts forming process for large strain.3) nonlinear boundary problem in the processing, problems such as sealing, the impact of the role of contact and friction can not be ignored, belongs to the highly nonlinear contact boundary. At ordinary times some contact problems, such as gear, stamping forming, rolling, rubber shock absorber, interference fit assembly, etc., when a structure and another structure or external boundary contact usually want to consider nonlinear boundary conditions. The actual nonlinear may appear at the same time these two or three kinds of nonlinear problems.Finite element theoretical basisFinite element method is based on variational principle and the weighted residual method, and the basic solving thought is the computational domain is divided into a finite number of non-overlapping unit, within each cell, select some appropriate nodes as solving the interpolation function, the differential equation of the variables in the rewritten by the variable or its derivative selected interpolation node value and the function of linear expression, with the aid of variational principle or weighted residual method, the discrete solution of differential equation. Using different forms of weight function and interpolation function, constitute different finite element methods. 1. The weighted residual method and the weighted residual method of weighted residual method of weighted residual method: refers to the weighted function is zero using make allowance for approximate solution of the differential equation method is called the weighted residual method. Is a kind of directly from the solution of differential equation and boundary conditions, to seek the approximate solution of boundary value problems of mathematical methods. Weighted residual method is to solve the differential equation of the approximate solution of a kind of effective method. Hybrid method for the trial function selected is the most convenient, but under the condition of the same precision, the workload is the largest. For internal method and the boundary method basis function must be made in advance to meet certain conditions, the analysis of complex structures tend to have certain difficulty, but the trial function is established, the workload is small. No matter what method is used, when set up trial function should be paid attention to are the following:(1) trial function should be composed of a subset of the complete function set. Have been using the trial function has the power series and trigonometric series, spline functions, beisaier, chebyshev, Legendre polynomial, and so on.(2) the trial function should have until than to eliminate surplus weighted integral expression of the highest derivative low first order derivative continuity.(3) the trial function should be special solution with analytical solution of the problem or problems associated with it. If computing problems with symmetry, should make full use of it. Obviously, any independent complete set of functions can be used as weight function. According to the weight function of the different options fordifferent weighted allowance calculation method, mainly include: collocation method, subdomain method, least square method, moment method and galerkin method. The galerkin method has the highest accuracy.Principle of virtual work: balance equations and geometric equations of the equivalent integral form of "weak" virtual work principles include principle of virtual displacement and virtual stress principle, is the floorboard of the principle of virtual displacement and virtual stress theory. They can be considered with some control equation of equivalent integral "weak" form. Principle of virtual work: get form any balanced force system in any state of deformation coordinate condition on the virtual work is equal to zero, namely the system of virtual work force and internal force of the sum of virtual work is equal to zero. The virtual displacement principle is the equilibrium equation and force boundary conditions of the equivalent integral form of "weak"; Virtual stress principle is geometric equation and displacement boundary condition of the equivalent integral form of "weak". Mechanical meaning of the virtual displacement principle: if the force system is balanced, they on the virtual displacement and virtual strain by the sum of the work is zero. On the other hand, if the force system in the virtual displacement (strain) and virtual and is equal to zero for the work, they must balance equation. Virtual displacement principle formulated the system of force balance, therefore, necessary and sufficient conditions. In general, the virtual displacement principle can not only suitable for linear elastic problems, and can be used in the nonlinear elastic and elastic-plastic nonlinear problem.Virtual mechanical meaning of stress principle: if the displacement is coordinated, the virtual stress and virtual boundary constraint counterforce in which they are the sumof the work is zero. On the other hand, if the virtual force system in which they are and is zero for the work, they must be meet the coordination. Virtual stress in principle, therefore, necessary and sufficient condition for the expression of displacement coordination. Virtual stress principle can be applied to different linear elastic and nonlinear elastic mechanics problem. But it must be pointed out that both principle of virtual displacement and virtual stress principle, rely on their geometric equation and equilibrium equation is based on the theory of small deformation, they cannot be directly applied to mechanical problems based on large deformation theory. 3,,,,, the minimum total potential energy method of minimum total potential energy method, the minimum strain energy method of minimum total potential energy method, the potential energy function in the object on the external load will cause deformation, the deformation force during the work done in the form of elastic energy stored in the object, is the strain energy.The convergence of the finite element method, the convergence of the finite element method refers to when the grid gradually encryption, the finite element solution sequence converges to the exact solution; Or when the cell size is fixed, the more freedom degree each unit, the finite element solutions tend to be more precise solution. Convergence condition of the convergence condition of the finite element finite element convergence condition of the convergence condition of the finite element finite element includes the following four aspects: 1) within the unit, the displacement function must be continuous. Polynomial is single-valued continuous function, sochoose polynomial as displacement function, to ensure continuity within the unit. 2) within the unit, the displacement function must include often strain. Total can be broken down into each unit of the state of strain does not depend on different locations within the cell strain and strain is decided by the point location of variables. When the size of the units is enough hours, unit of each point in the strain tend to be equal, unit deformation is uniform, so often strain becomes the main part of the strain. To reflect the state of strain unit, the unit must include the displacement functions often strain. 3) within the unit, the displacement function must include the rigid body displacement. Under normal circumstances, the cell for a bit of deformation displacement and displacement of rigid body displacement including two parts. Deformation displacement is associated with the changes in the object shape and volume, thus producing strain; The rigid body displacement changing the object position, don't change the shape and volume of the object, namely the rigid body displacement is not deformation displacement. Spatial displacement of an object includes three translational and three rotational displacement, a total of six rigid body displacements. Due to a unit involved in the other unit, other units do rigid body displacement deformation occurs will drive unit, thus, to simulate real displacement of a unit, assume that the element displacement function must include the rigid body displacement. 4) the displacement function must be coordinated in public boundary of the adjacent cell. For general unit of coordination is refers to the adjacent cell in public node have the same displacement, but also have the same displacement along the edge of the unit, that is to say, to ensure that the unit does not occur from cracking and invade the overlap each other. To do this requires the function on the common boundary can be determined by the public node function value only. For general unit and coordination to ensure the continuity of the displacement of adjacent cell boundaries. However, between the plate and shell of the adjacent cell, also requires a displacement of the first derivative continuous, only in this way, to guarantee the strain energy of the structure is bounded. On the whole, coordination refers to the public on the border between neighboring units satisfy the continuity conditions. The first three, also called completeness conditions, meet the conditions of complete unit is complete unit; Article 4 is coordination requirements, meet the coordination unit coordination unit; Otherwise known as the coordinating units. Completeness requirement is necessary for convergence, all four meet, constitutes a necessary and sufficient condition for convergence. In practical application, to make the selected displacement functions all meet the requirements of completeness and harmony, it is difficult in some cases can relax the requirement for coordination. It should be pointed out that, sometimes the coordination unit than its corresponding coordination unit, its reason lies in the nature of the approximate solution. Assumed displacement function is equivalent to put the unit under constraint conditions, the unit deformation subject to the constraints, this just some alternative structure compared to the real structure. But the approximate structure due to allow cell separation, overlap, become soft, the stiffness of the unit or formed (such as round degree between continuous plate unit in the unit, and corner is discontinuous, just to pin point) for the coordination unit, the error of these two effects have the possibility of cancellation, so sometimes use thecoordination unit will get very good results. In engineering practice, the coordination of yuan must pass to use "small pieces after test". Average units or nodes average processing method of stress stress average units or nodes average processing method of stress average units or nodes average processing method of stress of the unit average or node average treatment method is the simplest method is to take stress results adjacent cell or surrounding nodes, the average value of stress.1. Take an average of 2 adjacent unit stress. Take around nodes, the average value of stressThe basic steps of finite element method to solve the problemThe structural discretization structure discretization structure discretization structure discretization to discretization of the whole structure, will be divided into several units, through the node connected to each other between the units; 2. The stiffness matrix of each unit and each element stiffness matrix and the element stiffness matrix and the stiffness matrix of each unit (3) integrated global stiffness matrix integrated total stiffness matrix integrated overall stiffness matrix integrated total stiffness matrix and write out the general balance equations and write out the general balance equations and write out the general balance equations and write a general equation 4. Introduction of supporting conditions, the displacement of each node 5. Calculate the stress and strain in the unit to get the stress and strain of each cell and the cell of the stress and strain and the stress and strain of each cell.For the finite element method, the basic ideas and steps can be summarized as: (1) to establish integral equation, according to the principle of variational allowance and the weight function or equation principle of orthogonalization, establishment and integral expression of differential equations is equivalent to the initial-boundary value problem, this is the starting point of the finite element method. Unit (2) the area subdivision, according to the solution of the shape of the area and the physical characteristics of practical problems, cut area is divided into a number of mutual connection, overlap of unit. Regional unit is divided into finite element method of the preparation, this part of the workload is bigger, in addition to the cell and node number and determine the relationship between each other, also said the node coordinates, at the same time also need to list the natural boundary and essential boundary node number and the corresponding boundary value. (3) determine the unit basis function, according to the unit and the approximate solution of node number in precision requirement, choose meet certain interpolation condition basis function interpolation function as a unit. Basis function in the finite element method is selected in the unit, due to the geometry of each unit has a rule in the selection of basis function can follow certain rules. (4) the unit will be analysis: to solve the function of each unit with unit basis functions to approximate the linear combination of expression; Then approximate function generation into the integral equation, and the unit area integral, can be obtained with undetermined coefficient (i.e., cell parameter value) of each node in the algebraic equations, known as the finite element equation.(5) the overall synthesis: after the finite element equation, the area of all elements inthe finite element equation according to certain principles of accumulation, the formation of general finite element equations. (6) boundary condition processing: general boundary conditions there are three kinds of form, divided into the essential boundary conditions (dirichlet boundary condition) and natural boundary conditions (Riemann boundary conditions) and mixed boundary conditions (cauchy boundary conditions). Often in the integral expression for natural boundary conditions, can be automatically satisfied. For essential boundary conditions and mixed boundary conditions, should be in a certain method to modify general finite element equations satisfies. Solving finite element equations (7) : based on the general finite element equations of boundary conditions are fixed, are all closed equations of the unknown quantity, and adopt appropriate numerical calculation method, the function value of each node can be obtained.有限元分析有限元法求解区域是由许多小的节点连接单元（域），该模型给出了切分的基本方程（子域名）的近似解，由于单位（域）可以分为不同的形状和大小不同的尺寸，所以它能很好的适应复杂的几何形状、材料特性和边界条件复杂，复杂有限元模型：它是真实系统的理想化的数学抽象。
对分数阶扩散方程的紧致差分方法
Article history: Received 28 April 2009 Received in revised form 17 July 2009 Accepted 23 July 2009 Available online 3 August 2009 MSC: 65M06 65M12 65M15 35Q51 35Q53 78M20 Keywords: Fractional diffusion equation Finite difference Compact scheme Padé approximant Stability Convergence Fourier analysis
Journal of Computational Physics 228 (2009) 7792–7804
Contents lists available at ScienceDirect
Journal of Computational Physics
journal homepage: /locate/jcp
E-mail address: mrcui@ 0021-9991/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jcp.2009.07.021
M.R. Cui / Journal of Computational Physics 228 (2009) 7792–7804
1. Introduction Fractional differential equations (FDEs) have attracted increasing attention because they have applications in various fields of science and engineering. For example, they can describe many physical and chemical processes, biological systems, etc. The main physical purpose for investigating diffusion equations of fractional order is to describe phenomena of anomalous diffusion in transport processes through complex and/or disordered systems including fractal media, and fractional kinetic equations have proved particularly useful in the context of anomalous slow diffusion, see, for example, the excellent review paper [1]. An overview of the basic theory of fractional differentiation, fractional-order differential equations, methods of their solution and applications can be found in the book [2]. There have been several numerical methods proposed for solving the space and/or time FDEs up to now. Lynch et al. [3] developed two numerical schemes, one explicit and another one semi-implicit, for solving the transport problem with anomalous diffusion modeled by a partial differential equation of fractional order. Meerschaert and Tadjeran [4] studied the onedimensional radial flow model, they found that the fractional derivative describes more accurately the early arrival that cannot be explained by the classical advection–dispersion equations and they presented an implicit Euler method, based on a
generalizing equation 等式
generalizing equation 等式英文版Generalizing Equation: The Foundation of Mathematical UnderstandingMathematics, the language of the universe, is built upon a foundation of equations. These equations, in their simplest form, are statements of equality between two or more expressions. However, the power of equations lies not just in their direct representations of relationships, but also in their ability to generalize and abstract. Generalizing an equation is the process of transforming a specific instance of an equation into a more general form that can describe a wider range of situations.The process of generalizing an equation often involves identifying patterns and commonalities among different equations. For example, consider the equation y = mx + b, which represents a linear relationship between two variables. This equation can be generalized to describe any linearrelationship, regardless of the specific values of m and b. By identifying the underlying structure of the equation, we can abstract away the specific details and focus on the more general properties of the relationship.The importance of generalizing equations extends beyond the realm of pure mathematics. In applied mathematics and sciences, equations are often generalized to model real-world phenomena. By identifying patterns and trends in data, scientists and engineers can develop generalized equations that predict and explain observed behaviors. These generalized equations form the basis for many important technologies and theories, ranging from the laws of physics to economic models.In summary, generalizing equations is a crucial skill in mathematics. It allows us to identify patterns and abstract away specific details, focusing on the underlying structure and properties of relationships. By doing so, we can build more robust and flexible models that describe a wide range of situations. Generalizing equations is not just a mathematicaltool; it is a powerful framework for understanding and explaining the world.中文版等式：数学理解的基石数学，作为宇宙的语言，是建立在等式的基础之上的。
常微分方程中的英文单词和短语
常微分方程
Ordinary Differential Equtions
第一章 基本概念
Chapter 1 Basic Concepts
常微分方 程
阶 线性 非线性 偏微分方 程
通解 特解 初值条件 初值问题 柯西问题 几何解释 积分曲线 线素 线素场 方向场 等斜线 对称形式 奇异点 通积分
boundary finite closed interval open set finite semi-open interval
Lagrange formula
l ocal finite covering theorem
bar area monotone decreasing sectorial area comparability solution minimum solution maximum solution sl ope
topological dynamical system
differential dynamical system
stability of solution Liapunov stability
asymptotic stability asymptotic stability field attraction domain globe asymptotic stability
第六章 线 性 微 分 方 程 组
Chapter 6 Linear Differential Equations
存在和唯一 性定理 齐次线性微 分方程组 基本解组 朗斯基行列 式 刘维尔公 式 解矩阵 基解矩阵 常数矩阵 非齐次线性 微分方程 组 常数列向 量 常数变异公 式 计算公式 结构公式 常系数线性 微分方程 组 矩阵指数函 数 标准基解矩 阵 若尔当标准 型 矩阵函数 实值解 复值解 代数余子 式 特征方程 算子式 算子多项 式 拉普拉斯逆 变换
带狄利克雷边界条件的小初值耗散半线性波动方程外问题解的破裂及生命跨度估计
带狄利克雷边界条件的小初值耗散半线性波动方程外问题解的破裂及生命跨度估计徐根海;吴邦【摘要】研究在高维外区域上带狄利克雷边界条件的耗散半线性波动方程ut-Δu+ut=|u|p的初边值问题.证明了无论初值多么小,当1<p<1+2/n(n≥3)时,解会在有限时间内破裂;且当1<p<1+2/n时,得到了解的生命跨度上界估计.证明过程中运用了试探函数法.【期刊名称】《丽水学院学报》【年(卷),期】2018(040)002【总页数】9页(P1-9)【关键词】半线性波动方程;破裂;外问题;耗散;狄利克雷边界条件【作者】徐根海;吴邦【作者单位】丽水学院工学院,浙江丽水323000;浙江理工大学理学院,浙江杭州310018【正文语种】中文【中图分类】O1750 引言考虑带狄利克边界条件的小初值耗散半线性波动方程外问题可以用公式表示为：其中=Rn\B1表示在 Rn（n≥3)上单位球 B1的补集，ε＞0 表示初值的小性。
初值（u0，u1）满足：对于在Rn中的Cauchy问题，已有的研究结果表明其中存在一个临界指标=1+，该指标称为Fujita 指标[1]，见 Nakao 和 Ono[2]、Li和 Zhou[3]、Todorova 和Yordanov[4]、Zhang[5]等学者的文献。
破裂情形的生命跨度研究见Nishihara[6]、Ikeda和Ogawa[7]及Lai和Zhou[8]等学者的文献。
外区域上小初值耗散半线性波动方程的初边值问题（1），也引起了很多人的关注，研究成果可见Ikehata[9-11]，Nakao[12]，Racke[13-14]，Shibata[15]，Ikehata[16-18]，Lai和 Yin[19]，Lin、Jiang 和 Yin[20]及 Wu、Ma 和 Jin[21]等学者的文献。
本文主要研究问题（1）解的有限时间破裂及生命跨度估计。
Ogawa等[22]证明了当1＜p＜1+时，带狄利克雷边界条件的初边值外问题解会在有限时间内破裂，但是并没有给出生命跨度估计。
常微分方程 双语课程常用词汇表
《常微分方程》双语课程常用词汇表acceleration n. 加速度constant acceleration 常加速度downward acceleration 向下的加速度gravitational acceleration 重力加速度total acceleration 总加速度upward acceleration 向上的加速度account（for）v. 占去algebra 代数algebraic equation 代数方程linear algebra 线性代数the fundamental theorem of algebra 代数学基本定理amount v. 总计n. 总数amplitude n. 振幅application n. 应用by repeated application of 反复应用apply v. 应用approach v. 趋近于approach zero 趋近于零approach infinity 趋近于无穷area n. 面积cross-sectional area 横截面积the horizontal cross-sectional area 水平方向的截面积arrange v. 安排、整理、排列arrangement n. 安排、整理、排列rearrangement n. 重新安排、重新整理、重新排列associate v. 联系associated a.. 相应的associated with 对应于…的associated homogeneous linear equation 相应的齐线性方程associativity n. 结合律assume v. 假设assumption n. 假设asymptote 渐近线oblique asymptote 斜渐近线[əˈbli:k]axis 数轴negative x -axis 负x 轴positive y -axis 正y 轴x -axis x 轴y -axis y 轴base n. 基be present v. 出现body 天体boundary n. 边界bounded a. 有界的unbounded a. 无界的brine 盐水calculus 微积分elementary calculus 初等微积分capacitor 电容器case 情形exceptional case 例外情形chain rule （求导的）链式法则clockwise 顺时针clockwise direction 顺时针方向counterclockwise 逆时针counterclockwise direction 逆时针方向close v. 闭closed container 封闭的容器closed interval 闭区间coefficient 系数constant coefficient 常系数leading coefficient 首项系数undetermined coefficient 待定系数variable coefficient 变系数collect v. 整理collect coefficients 整理系数column 列commute v. 交换commutative a. 交换的commutativity 交换性property of commutativity 交换性质complete a. 完备的incomplete a. 不完备的complex a. 复的complex conjugate 复共轭的complex conjugate pairs 复共轭对complex conjugate roots 复共轭根component n. 分量componentwise 分量形式composition 复合compress v. 压缩compute v. 计算concentration n. 浓度condition 条件a given initial condition 一个给定的初始条件initial condition 初始条件necessary condition 必要条件sufficient condition 充分条件the given condition 给定的条件conjugate 共轭的constant 常数arbitrary constant 任意常数constant multiple 数乘constant of integration 积分常数constant speed 常速度damping constant 阻尼常数positive constant 正的常数continuity n. 连续性discontinuity 不连续性continuous 连续的continuous function 连续函数continuous partial derivative 连续偏导数discontinuous 不连续的piecewise continuous 分段连续的convention 惯例convergence n. 收敛absolute convergence 绝对收敛uniform convergence 一致收敛coordinate 坐标coordinate axis 坐标轴polar coordinates 极坐标corresponding a. 相应的cube 立方，立方体current 电流cylinder 柱，柱面dashpot 减震器decomposition 分解partial fraction decomposition 部分分式分解defect n. 亏量defective v. 亏损的define v. 定义definition n. 定义degenerate a. 退化的denominator 分母derive v. 导出derivation n. 求导（数）derivative n. 导数[diˈrivətiv] antiderivative 原函数first derivative 一阶导数second derivative 二阶导数the highest derivative 最高阶导数determine v. 确定determinant 行列式determinant of coefficients 系数行列式operational determinant 算子行列式diagonal 对角线principal diagonal 主对角线differ v. 不同difference n. 差differentiable 可微的differentiable function 可微函数differentiability 可微性differentiability condition 可微性条件differential n. 微分differential form 微分形式differentiate v. 微分differentiate term wise 逐项微分differentiation n. 微分（运算）term-by-term differentiation 逐项微分displacement 位移distance 距离distinct 不同的distinct real roots 不同的实根distributives 分配性diverge v. 发散divide v. 划分，除subdivide 细分domain 定义域double 重，二重，双double root 二重根duplicate v. 复制、重复duplication n. 复制、重复eliminate v. 消去elimination n. 消元法、消去the method of elimination 消元法、消去法eigenvalue n. 特征值complex conjugate eigenvalue 复共轭特征值defective eigenvalue 不完备的特征值multiple eigenvalue 多重特征值zero eigenvalue 零特征值eigenvector 特征向量generalized eigenvector 广义特征向量rank generalized eigenvector r 阶广义特征向量element 元素diagonal element 对角元off-diagonal element 非对角元element wise 逐个元素地ellipse 椭圆elliptical orbit 椭圆型轨道employ v. 利用employ the technique of 利用…技术enable v. 使能够entry n. 元素equate v. 使相等equation 方程Bernoulli equation 伯努利方程Bessel’s equation 贝赛尔方程characteristic equation 特征方程cubic equation 三次方程differential equation 微分方程eigenvector equation 特征向量方程exact differential equation 恰当微分方程higher-degree equation 高次方程homogeneous equation 齐次方程Legend re’s equation 勒让德方程linear first-order equation 一阶线性方程Logistic equation 逻辑斯蒂方程natural growth equation 自然增长方程ordinary differential equation 常微分方程partial differential equation 偏微分方程quadratic equation 二次方程reducible equation 可降阶方程second-degree equation 二次方程separable differential equation 可分离变量方程simultaneous equations 联立方程组equilibrium position 平衡位置equivalent 等价的be equivalent to 等价于equivalently 等价地error 误差average error 平均误差existence 存在existence-uniqueness theorem 存在唯一性定理exponent 指数negative exponent 负指数exponential 指数(的)exponential function 指数函数matrix exponential 矩阵指数factor n. 因式,因子v. 分解因式common factor 公因式,公因子integrating factor 积分因子linear factor 一次因式irreducible quadratic factor 二次不可约因式factorization n. 因子分解field 场、域direction field 方向场first 第一的the first two… 前两（个）……flow v. 流动n. 流量inflow n. 流入(量) outflow n. 流出(量)focus 焦点following 下面的force 力external force 外力external period ice force 周期性外力frictional force 摩擦力form 形式decimal form 小数形式explicit form 显式形式implicit form 隐式形式polar form 极坐标形式the standard form 标准形式upper triangular form 上三角形式former a. 以前的the former 前者formula 公式fraction 分式,分数frequency 频率function 函数analytic function 解析函数coefficient function 系数函数complementary function 补函数component function 分量函数constant-valued function 常数值函数continuous function 连续函数piecewise continuous function 分段连续函数decreasing function 单调减函数differentiable function 可微函数n times differentiable function n 阶可微函数twice differentiable function 二阶可微函数sufficiently differentiable function 足够阶可微函数discontinuous function 不连续函数elementary function 初等函数factorial function 分式函数increasing function 单调增函数matrix-valued function 矩阵值函数position function 位置函数rational function 有理函数real-valued function 实值函数trigonometric function 三角函数unknown function 未知函数vector-valued function 向量值函数generalize (to) v. 推广generalization n. 推广graph 图象hemispherical 半球形的hold v. 成立homogeneous 齐(次)的nonhomogenous 非齐(次)的hyperbolic 双曲型的hyperbolic cosine 双曲余弦hyperbolic sine 双曲正弦hypothesis n. 假设hypotheses n. 假设(复数形式) identity 恒等式identity principle 恒等原理trigonometric identity 三角恒等式illustrate v. 说明imaginary part 虚部immaterial a. 不重要的, 不相干的imply v. 意味着, 暗示impulse 脉冲independent a. 独立的, 不相关的independent of 独立于……inductor 电感器initial 开始的, 最初的initial condition 初始条件initial position 初始位置initial population 初始人口数initial velocity 初始速度integer 整数nonnegative integer 非负整数integral 积分definite integral 定积分improper integral 非正常积分indefinite integral 不定积分integral sign 积分号integrate v. 积分integrate by parts 分部积分integration n. 积分integration of both sides 两边积分interior n. 内部in terms of 根据interval 区间closed interval 闭区间interval of real number 实数区间open interval 开区间subinterval 子区间bounded subinterval 有界子区间the ends of the interval 区间的端点the whole interval 整个区间involve v. 包含，涉及Kepl er’s laws of planetary motion 开普勒行星运动定律latter a. 后期的，末期的the latter 后者left-hand side 左边like 类似，相似like powers 同次幂like term 同类项limit 极限take the limit 取极限upper limit 上极限line 线，线条line segment 线段real line 实数轴straight line 直线tangent line 切线the line tangent (to) 与…相切的直线the entire real line 整个实轴linear 线性的linear combination 线性组合linear dependence 线性相关linear independence 线性无关nonlinear 非线性的linearly 线性地linearly dependent 线性相关的linearly independent 线性无关的linearly independent solutions 线性无关解linearity 线性性liter 升logarithm 对数logarithmic term 含有对数的项long division 长除法major semi axis 长半轴mass (物体的)质量mathematical model 数学模型mathematical modeling 数学建模matrix 矩阵augmented matrix 增广矩阵coefficient matrix 系数矩阵diagonal matrix 对角矩阵exponential matrix 指数矩阵fundamental matrix 基解矩阵identity matrix 单位矩阵inverse matrix 逆矩阵matrix addition 矩阵加法matrix multiplication 矩阵乘法nonsingular matrix 非奇异矩阵singular matrix 奇异矩阵square matrix 方阵upper triangular matrix 上三角矩阵zero matrix 零矩阵mean value theorem for integral 积分中值定理method 方法straightforward method 直接的方法minimum 最小值minus prep. 减,减去;负的minus sign 负号motion 运动free undamped motion 无阻尼自由运动simple harmonic motion 简谐运动multiply v. 乘,倍增multiplication n. 乘法multiplicity n. 重数nature 自然, 本质nilpotent 幂零的number 数complex number 复数imaginary number 虚数negative number 负数nonnegative number 非负数positive number 正数real number 实数unknown number 未知数numerator (分数的)分子operate v. 运算,作用operation n. 运算,操作elementary row operation 初等行变换operator 算子polynomial operator 多项式算子orbit 轨道order 阶first-order equation 一阶方程fourth-order equation 四阶方程of exponential order 指数阶的second-order equation 二阶方程nth-order equation n 阶方程the mixed second-order partial derivative 二阶混合偏导数the order of a differential equation 微分方程的阶origin 原点original 原来的the original equation 原方程the original form 原来的形式oscillate v. 振动oscillation n. 振动forced oscillation 强迫振动free oscillation 自由振动parabola 抛物线 [pə'ræbələ]parameter 参数variation of parameters 常数变易法parameterize v. 参数化parameterization n. 参数化particle 粒子phase angle 相角phase portrait 相图plane 平面point 点end point 端点isolated point 孤立点ordinary point 常点singular point 奇点regular singular point 正则奇点irregular singular point 非正则奇点polynomial 多项式n th-degree polynomial n 次多项式a polynomial of degree n n 次多项式a polynomial of lower degree 次数较低的多项式Taylor polynomial 泰勒多项式possible 可能的possibility 可能性power 幂power function 幂函数in powers of x x 的幂in powers of x −a x −a 的幂presence 出现, 在场preceding 前面的prime 求导符号“撇”problem 问题mathematical problem 数学问题initial value problem 初始值问题proceed v. 继续进行, 继续下去product 乘, 积dot product 点积product rule 乘法法则scalar product 点积,数积,内积property 性质proposition 命题quotient 商radius 半径radius of convergence 收敛半径rate 速率at a rate of 以…的速率at a rate proportional to 以与…成正比的速率birth rate 出生率death rate 死亡率time rate of change of (something) …关于时间的变化率interest rate 利率reactant 反应物readily 容易地real part 实部recall v. 记起,回顾rectangle 长方形, 矩形open rectangle 开矩形recurrence relation 递推关系many-term recurrence relation 多项间的递推关系two-term recurrence relation 两项间的递推关系recursion formula 递推公式reduce v. 化简, 简化, 约简reduction n. 化简, 简化, 约简reduction of order 降阶resistor 电阻器result n. 结果v. 导致(in)revolution n. 旋转right-hand side 右边root 根characteristic root 特征根complex root 复根double root 二重根equal roots 相等的根k-fold root k 重根rational root 有理根real root 实根repeated root 重根the square root 平方根triple root 三重根rotation n. 旋转counterclockwise rotation 逆时针旋转routine 例行的; 平凡的a routine matter 平凡的情形row 行scalar 纯量(的), 数量(的), 标量(的)series 级数binomial series 二项式级数geometric series 几何级数harmonic series 调和级数infinite series 无穷级数power series 幂级数convergent power series 收敛的幂级数power series representation 幂级数表示power series in x x的幂级数power series in x −a x −a 的幂级数power series solution 幂级数解Taylor series 泰勒级数set 集合show v. 证明side 边left-hand side 左边right-hand side 右边simple 简单的simplify v. 简化, 化简simplification n. 简化, 化简singularity 奇异性slope 斜率slope field 斜率场smooth 光滑的piecewise smooth 逐段光滑的solute n. 溶质，溶解物solution n. 解explicit solution 显式解implicit solution 隐式解infinitely many solutions 无穷多解negative-valued solution 负值解period ice solution 周期解positive-valued solution 正值解singular solution 奇解solution curve 解曲线the (a) genera l solution 通解the particular solution 特解solve v. 解solvent n. 溶剂some 某个some open interval 某个开区间spring 弹簧spring constant 弹性系数step 步骤finitely many steps 有限多步stretch 拉伸subject(to) a. 易受…的ad.在…条件下subscript 下标even subscript 偶下标odd subscript 奇下标substitute v. 代入substitution n. 代入direct substitution 直接代入back substitution 回代subtract v. 减去subtraction n. 减去suffice v. 足够sufficient n. 足够的, 充分的sufficient condition 充分条件sum n. 和sum zero 总和为零summand 被加数summation 求和(法), 累加, 加法the index of summation 求和的指标the sum of…and … …与…的和superposition 叠加symmetry 对称symmetric form 对称形式system 方程组,系统first-order system 一阶方程组higher-order system 高阶方程组two-dimensional system 二维系统upper triangular system 上三角方程组take 取, 实施take the Laplace transform 取拉普拉斯变换take the limit 取极限tangent 正切(的),切线(的)be tangent to 与…相切time 时间per unit of time 单位时间time lag 时滞tank 箱, 柜, 罐water tank 水箱term 项constant term 常数项termwise addition 逐项相加term by term 逐项the first term 第一项the first few terms 前几项the genera l term 通项, 一般项the leading term 首项terminology 术语trajectory 轨道, 轨迹transform v. 转化n. 变换Laplace transform 拉普拉斯变换inverse Laplace transform 拉普拉斯反变换transformation n. 变换,转化transpose v. , n. 转置,移项triangle 三角(形)right triangle 直角三角形triple 三重的, 三次的, 三倍的triple eigenvalue 三重特征根trivial 平凡的, 不重要的trivial case 平凡情况nontrivial 非平凡的tuple 组n -tuple n 元组unique 唯一的uniqueness 唯一性unique solution 唯一解unknown 未知的the unknown 未知量value 值numerical value 数值absolute value 绝对值variable 变量dependent variable 因变量independent variable 自变量variable of integration 积分变量vector 向量acceleration vector 加速度向量column vector 列向量constant vector 常数向量position vector 位置向量radius vector 向径, 矢径row vector 行向量solution vector 解向量unit vector 单位向量verify v. 证明vanish 等于零vanish identically 恒等于零variable 变量dependent variable 因变量independent variable 自变量separation of variable 变量分离voltage 电压volume 列Wronskian 伏朗斯基行列式yield 产生zero 零nonzero 非零。
第十七章拉格朗日方程
ri
n
Fi
i 1
( jk1qrij
q
j
kn
) (F
j1 i1
i
ri q j
)q
j
kn
[( X i
j1 i1
xi q j
Yi
yi q j
Z i
zi q j
)]q j
k
Q j q j
j 1
15
ri jk1qrij q j (i1,2, n)
(c)
代入质点系动力学普遍方程，得：
n
n
n
(Fi miai )ri Fi ri miairi 0 (d )
QA Ma QB QBe QBr QBe ma , QBr mar
10
From the general equation of dynamics we have
(QA QBe QBr cos )xA (QBe cos Q sin QBr )sB 0.
The system has two degree of freedom, we can choose xA and sB as independent virtual displacement. Moreover,
generalized coordinates.
Suppose there are n particles in the system and that there are s
constraints which all are ideal ones. The number of degrees of
ri q j
0
( j 1,2 , k) .
(f)
generalized inertial forces
有限元分析法英文简介
有限元分析法英文简介The finite element analysisFinite element method, the solving area is regarded as made up of many small in the node connected unit (a domain), the model gives the fundamental equation of sharding (sub-domain) approximation solution, due to the unit (a domain) can be divided into various shapes and sizes of different size, so it can well adapt to the complex geometry, complex material properties and complicated boundary conditionsFinite element model: is it real system idealized mathematical abstractions. Is composed of some simple shapes of unit, unit connection through the node, and under a certain load.Finite element analysis: is the use of mathematical approximation method for real physical systems (geometry and loading conditions were simulated. And by using simple and interacting elements, namely unit, can use a limited number of unknown variables to approaching infinite unknown quantity of the real system.Linear elastic finite element method is a ideal elastic body as the research object, considering the deformation based on small deformation assumption of. In this kind of problem, the stress and strain of the material is linear relationship, meet the generalized hooke's law; Stress and strain is linear, linear elastic problem boils down to solving linear equations, so only need less computation time. If theefficient method of solving algebraic equations can also help reduce the duration of finite element analysis.Linear elastic finite element generally includes linear elasticstatics analysis and linear elastic dynamics analysis from two aspects. The difference between the nonlinear problem and linear elastic problems:1) nonlinear equation is nonlinear, and iteratively solving of general;2) the nonlinear problem can't use superposition principle;3) nonlinear problem is not there is always solution, sometimes evenno solution. Finite element to solve the nonlinear problem can bedivided into the following three categories: 1) material nonlinear problems of stress and strain is nonlinear, but the stress and strain is very small, a linear relationship between strain and displacement atthis time, this kind of problem belongs to the material nonlinear problems. Due to theoretically also cannot provide the constitutive relation can be accepted, so, general nonlinear relations between stress and strain of the material based on the test data, sometimes, tosimulate the nonlinear material properties available mathematical model though these models always have their limitations. More importantmaterial nonlinear problems in engineering practice are: nonlinearelastic (including piecewise linear elastic, elastic-plastic and viscoplastic, creep, etc.2) geometric nonlinear geometric nonlinear problems are caused dueto the nonlinear relationship between displacement. When the object thedisplacement is larger, the strain and displacement relationship is nonlinear relationship. Research on this kind of problem Is assumes that the material of stress and strain is linear relationship. It consists of a large displacement problem of large strain and large displacementlittle strain. Such as the structure of the elastic buckling problem belongs to the large displacement little strain, rubber parts forming process for large strain.3) nonlinear boundary problem in the processing, problems such as sealing, the impact of the role of contact and friction can not be ignored, belongs to the highly nonlinear contact boundary.At ordinary times some contact problems, such as gear, stamping forming, rolling, rubber shock absorber, interference fit assembly, etc., when a structure and another structure or external boundary contact usually want to consider nonlinear boundary conditions. The actual nonlinear may appear at the same time these two or three kinds of nonlinear problems.Finite element theoretical basisFinite element method is based on variational principle and the weighted residual method, and the basic solving thought is the computational domain is divided into a finite number of non-overlapping unit, within each cell, select some appropriate nodes as solving the interpolation function, the differential equation of the variables inthe rewritten by the variable or its derivative selected interpolation node value and the function of linear expression, with the aid ofvariational principle or weighted residual method, the discrete solution of differential equation. Using different forms of weight function and interpolation function, constitute different finite element methods. 1. The weighted residual method and the weighted residual method of weighted residual method of weighted residual method: refers to the weighted function is zero using make allowance for approximate solution of the differential equation method is called the weighted residual method. Is a kind of directly from the solution of differential equation and boundary conditions, to seek the approximate solution of boundary value problems of mathematical methods. Weighted residual method is to solve the differential equation of the approximate solution of a kind of effective method.Hybrid method for the trial function selected is the most convenient, but under the condition of the same precision, the workload is the largest. For internal method and the boundary method basis function must be made in advance to meet certain conditions, the analysis of complex structures tend to have certain difficulty, but the trial function is established, the workload is small. No matter what method is used, when set up trial function should be paid attention to are the following:(1) trial function should be composed of a subset of the complete function set. Have been using the trial function has the power seriesand trigonometric series, spline functions, beisaier, chebyshev, Legendre polynomial, and so on.(2) the trial function should have until than to eliminate surplus weighted integral expression of the highest derivative low first order derivative continuity.(3) the trial function should be special solution with analytical solution of the problem or problems associated with it. If computing problems with symmetry, should make full use of it. Obviously, any independent complete set of functions can be used as weight function. According to the weight function of the different options for different weighted allowance calculation method, mainly include: collocation method, subdomain method, least square method, moment method andgalerkin method. The galerkin method has the highest accuracy.Principle of virtual work: balance equations and geometric equations of the equivalent integral form of "weak" virtual work principles include principle of virtual displacement and virtual stress principle, is the floorboard of the principle of virtual displacement and virtual stress theory. They can be considered with some control equation of equivalent integral "weak" form. Principle of virtual work: get form any balanced force system in any state of deformation coordinate condition on the virtual work is equal to zero, namely the system of virtual work force and internal force ofthe sum of virtual work is equal to zero. The virtual displacement principle is the equilibrium equation and force boundary conditions of the equivalent integral form of "weak"; Virtual stress principle is geometric equation and displacement boundary condition of the equivalentintegral form of "weak". Mechanical meaning of the virtual displacement principle: if the force system is balanced, they on the virtual displacement and virtual strain by the sum of the work is zero. On the other hand, if the force system in the virtual displacement (strain) and virtual and is equal to zero for the work, they must balance equation. Virtual displacement principle formulated the system of force balance, therefore, necessary and sufficient conditions. In general, the virtual displacement principle can not only suitable for linear elastic problems, and can be used in the nonlinear elastic and elastic-plastic nonlinear problem.Virtual mechanical meaning of stress principle: if the displacementis coordinated, the virtual stress and virtual boundary constraint counterforce in which they are the sum of the work is zero. On the other hand, if the virtual force system in which they are and is zero for the work, they must be meet the coordination. Virtual stress in principle, therefore, necessary and sufficient condition for the expression of displacement coordination. Virtual stress principle can be applied to different linear elastic and nonlinear elastic mechanics problem. But it must be pointed out that both principle of virtual displacement and virtual stress principle, rely on their geometric equation andequilibrium equation is based on the theory of small deformation, they cannot be directly applied to mechanical problems based on large deformation theory. 3,,,,, the minimum total potential energy method of minimum total potential energy method, the minimum strain energy methodof minimum total potential energy method, the potential energy function in the object on the external load will cause deformation, the deformation force during the work done in the form of elastic energy stored in the object, is the strain energy.The convergence of the finite element method, the convergence of the finite element method refers to when the grid gradually encryption, the finite element solution sequence converges to the exact solution; Or when the cell size is fixed, the more freedom degree each unit, thefinite element solutions tend to be more precise solution. Convergence condition of the convergence condition of the finite element finite element convergence condition of the convergence condition of the finite element finite element includes the following four aspects: 1) within the unit, the displacement function must be continuous. Polynomial is single-valued continuous function, so choose polynomial as displacement function, to ensure continuity within the unit. 2) within the unit, the displacement function must include often strain. Total can be broken down into each unit of the state of strain does not depend on different locations within the cell strain and strain is decided by the point location of variables. When the size of the units is enough hours, unit of each point in the strain tend to be equal, unit deformation is uniform, so often strain becomes the main part of the strain. To reflect the state of strain unit, the unit must include the displacement functions often strain. 3) within the unit, the displacement function must include the rigid body displacement. Under normal circumstances,the cell for a bit of deformation displacement and displacement of rigid body displacement including two parts. Deformation displacement is associated with the changes in the object shape and volume, thus producing strain; The rigid body displacement changing the object position, don't change the shape and volume of the object, namely the rigid body displacement is not deformation displacement. Spatial displacement of an object includes three translational and three rotational displacement, a total of six rigid body displacements. Due to a unit involved in the other unit, other units do rigid body displacement deformation occurs willdrive unit, thus, to simulate real displacement of a unit, assume that the element displacement function must include the rigid body displacement. 4) the displacement function must be coordinated in public boundary of the adjacent cell. For general unit of coordination isrefers to the adjacent cell in public node have the same displacement, but also have the same displacement along the edge of the unit, that is to say, to ensure that the unit does not occur from cracking and invade the overlap each other. To do this requires the function on the common boundary can be determined by the public node function value only. For general unit and coordination to ensure the continuity of the displacement of adjacent cell boundaries. However, between the plate and shell of the adjacent cell, also requires a displacement of the first derivative continuous, only in this way, to guarantee the strain energy of the structure is bounded. On the whole, coordination refers to thepublic on the border between neighboring units satisfy the continuity conditions. The first three, also called completeness conditions, meet the conditions of complete unit is complete unit; Article 4 is coordination requirements, meet the coordination unit coordination unit; Otherwise known as the coordinating units. Completeness requirement is necessary for convergence, all four meet, constitutes a necessary and sufficient condition for convergence. In practical application, to make the selected displacement functions all meet the requirements of completeness and harmony, it is difficult in some cases can relax the requirement for coordination. It should be pointed out that, sometimes the coordination unit than its corresponding coordination unit, its reason lies in the nature of the approximate solution. Assumed displacement function is equivalent to put the unit under constraint conditions, the unit deformation subject to the constraints, this just some alternative structure compared to the real structure. But the approximate structure due to allow cell separation, overlap, become soft, the stiffness of the unit or formed (such as round degree between continuous plate unit in the unit, and corner is discontinuous, just to pin point) for the coordination unit, the error of these two effects have the possibility of cancellation, so sometimes use the coordination unit will get very good results. In engineering practice, the coordination of yuan must pass to use "small pieces after test". Average units or nodes average processing method of stress stress average units or nodes average processing method of stress average units or nodesaverage processing method of stress of the unit average or node average treatment method is the simplest method is to take stress results adjacent cell or surrounding nodes, the average value of stress. 1. Take an average of 2 adjacent unit stress. Take around nodes, the average value of stressThe basic steps of finite element method to solve the problemThe structural discretization structure discretization structure discretization structure discretization to discretization of the whole structure, will be divided into several units, through the node connected to each other between the units; 2. The stiffness matrix of each unit and each element stiffness matrix and the element stiffness matrix and the stiffness matrix of each unit (3) integrated global stiffness matrix integrated total stiffness matrix integrated overall stiffness matrix integrated total stiffness matrix and write out the general balance equations and write out the general balance equations and write out the general balance equations and write a general equation4. Introduction of supporting conditions, the displacement of each node5. Calculate the stress and strain in the unit to get the stress and strain of each cell and the cell of the stress and strain and the stress and strain of each cell.For the finite element method, the basic ideas and steps can be summarized as: (1) to establishintegral equation, according to the principle of variational allowance and the weight function or equation principle oforthogonalization, establishment and integral expression of differential equations is equivalent to the initial-boundary value problem, this is the starting point of the finite element method. Unit (2) the area subdivision, according to the solution of the shape of the area and the physical characteristics of practical problems, cut area is divided into a number of mutual connection, overlap of unit. Regional unit is divided into finite element method of the preparation, this part of the workload is bigger, in addition to the cell and node number and determine the relationship between each other, also said the node coordinates, at the same time also need to list the natural boundary and essential boundary node number and the corresponding boundary value. (3) determine the unit basis function, according to the unit and the approximate solution of node number in precision requirement, choose meet certain interpolation condition basis function interpolation function as a unit. Basisfunction in the finite element method is selected in the unit, due to the geometry of each unit has a rule in the selection of basis function can follow certain rules. (4) the unit will be analysis: to solve the function of each unit with unit basis functions to approximate thelinear combination of expression; Then approximate function generation into the integral equation, and the unit area integral, can be obtained with undetermined coefficient (i.e., cell parameter value) of each node in the algebraic equations, known as the finite element equation. (5) the overall synthesis: after the finite element equation, the area ofall elements in the finite element equation according to certainprinciples of accumulation, the formation of general finite element equations. (6) boundary condition processing: general boundaryconditions there are three kinds of form, divided into the essential boundary conditions (dirichlet boundary condition) and natural boundary conditions (Riemann boundary conditions) and mixed boundary conditions (cauchy boundary conditions). Often in the integral expression fornatural boundary conditions, can be automatically satisfied. Foressential boundary conditions and mixed boundary conditions, should bein a certain method to modify general finite element equations satisfies. Solving finite element equations (7) : based on the general finite element equations of boundary conditions are fixed, are all closed equations of the unknown quantity, and adopt appropriate numerical calculation method, the function value of each node can be obtained.。
schur补定理
schur补定理"Schur Complement Theorem"The Schur Complement Theorem is a fundamental result in linear algebra that provides a powerful tool for solving systems of linear d after the Swiss mathematician Issai Schur,the theorem allows us to simplify large matrices into smaller ones,making them easier to work with.The theorem states that for a given partitioned matrix,if the block in the lower-right corner is invertible,then the original matrix is invertible as well.Moreover,it gives us an explicit formula for computing the inverse of the original matrix in terms of the inverses of the smaller blocks.To understand the Schur Complement Theorem,let's consider a2x2block matrix:A BC DHere,A and D are square matrices,while B and C are rectangular matrices.If D is invertible,then we can define the Schur complement of D with respect to A as:S=A-BD^(-1)C.The Schur Complement Theorem then tells us that if D is invertible,the original matrix is invertible if and only if S is invertible.Moreover,the inverse of the original matrix can be expressed as:(A B)^(-1)=(S^(-1)S^(-1)C^T-S^(-1)B^T)D^(-1).This formula allows us to compute the inverse of the original matrix using the inverses of the smaller blocks.The Schur Complement Theorem has numerous applications in various fields of mathematics and engineering.It is particularly useful in solving large systems of linear equations arising in circuit analysis,control theory,and optimization problems.By applying the theorem,we can reduce the computational complexity and improve efficiency in solving these systems.In addition to its practical applications,the Schur Complement Theorem also has deep connections to other areas of mathematics,such as graph theory,combinatorial optimization,and convex analysis.It has been extensively studied and generalized in different contexts,leading to further developments in linear algebra and related fields.In conclusion,the Schur Complement Theorem is a powerful tool in linear algebra that allows us to simplify large matrices by exploiting the invertibility of smaller blocks. Its applications are widespread,and it plays a crucial role in solving systems of linear equations efficiently. The theorem's elegance and versatility make it an essential result in the field of mathematics.。
高等数学中定义定理的英文表达
高等数学中定义定理的英文表达Value of function ：函数值Variable ：变数Vector ：向量Velocity ：速度Vertical asymptote ：垂直渐近线Volume ：体积X-axis ：x轴x-coordinate ：x坐标x-intercept ：x截距Zero vector ：函数的零点Zeros of a polynomial ：多项式的零点TTangent function ：正切函数Tangent line ：切线Tangent plane ：切平面Tangent vector ：切向量Total differential ：全微分Trigonometric function ：三角函数Trigonometric integrals ：三角积分Trigonometric substitutions ：三角代换法Tripe integrals ：三重积分SSaddle point ：鞍点Scalar ：纯量Secant line ：割线Second derivative ：二阶导数Second Derivative Test ：二阶导数试验法Second partial derivative ：二阶偏导数Sector ：扇形Sequence ：数列Series ：级数Set ：集合Shell method ：剥壳法Sine function ：正弦函数Singularity ：奇点Slant asymptote ：斜渐近线Slope ：斜率Slope-intercept equation of a line ：直线的斜截式Smooth curve ：平滑曲线Smooth surface ：平滑曲面Solid of revolution ：旋转体Space ：空间Speed ：速率Spherical coordinates ：球面坐标Squeeze Theorem ：夹挤定理Step function ：阶梯函数Strictly decreasing ：严格递减Strictly increasing ：严格递增Sum ：和Surface ：曲面Surface integral ：面积分Surface of revolution ：旋转曲面Symmetry ：对称RRadius of convergence ：收敛半径Range of a function ：函数的值域Rate of change ：变化率Rational function ：有理函数Rationalizing substitution ：有理代换法Rational number ：有理数Real number ：实数Rectangular coordinates ：直角坐标Rectangular coordinate system ：直角坐标系Relative maximum and minimum ：相对极大值与极小值Revenue function ：收入函数Revolution , solid of ：旋转体Revolution , surface of ：旋转曲面Riemann Sum ：黎曼和Riemannian geometry ：黎曼几何Right-hand derivative ：右导数Right-hand limit ：右极限Root ：根P、QParabola ：拋物线Parabolic cylinder ：抛物柱面Paraboloid ：抛物面Parallelepiped ：平行六面体Parallel lines ：并行线Parameter ：参数Partial derivative ：偏导数Partial differential equation ：偏微分方程Partial fractions ：部分分式Partial integration ：部分积分Partiton ：分割Period ：周期Periodic function ：周期函数Perpendicular lines ：垂直线Piecewise defined function ：分段定义函数Plane ：平面Point of inflection ：反曲点Polar axis ：极轴Polar coordinate ：极坐标Polar equation ：极方程式Pole ：极点Polynomial ：多项式Positive angle ：正角Point-slope form ：点斜式Power function ：幂函数Product ：积Quadrant ：象限Quotient Law of limit ：极限的商定律Quotient Rule ：商定律M、N、OMaximum and minimum values ：极大与极小值Mean Value Theorem ：均值定理Multiple integrals ：重积分Multiplier ：乘子Natural exponential function ：自然指数函数Natural logarithm function ：自然对数函数Natural number ：自然数Normal line ：法线Normal vector ：法向量Number ：数Octant ：卦限Odd function ：奇函数One-sided limit ：单边极限Open interval ：开区间Optimization problems ：最佳化问题Order ：阶Ordinary differential equation ：常微分方程Origin ：原点Orthogonal ：正交的LLaplace transform ：Leplace 变换Law of Cosines ：余弦定理Least upper bound ：最小上界Left-hand derivative ：左导数Left-hand limit ：左极限Lemniscate ：双钮线Length ：长度Level curve ：等高线L'Hospital's rule ：洛必达法则Limacon ：蚶线Limit ：极限Linear approximation：线性近似Linear equation ：线性方程式Linear function ：线性函数Linearity ：线性Linearization ：线性化Line in the plane ：平面上之直线Line in space ：空间之直线Lobachevski geometry ：罗巴切夫斯基几何Local extremum ：局部极值Local maximum and minimum ：局部极大值与极小值Logarithm ：对数Logarithmic function ：对数函数IImplicit differentiation ：隐求导法Implicit function ：隐函数Improper integral ：瑕积分Increasing/Decreasing Test ：递增或递减试验法Increment ：增量Increasing Function ：增函数Indefinite integral ：不定积分Independent variable ：自变数Indeterminate from ：不定型Inequality ：不等式Infinite point ：无穷极限Infinite series ：无穷级数Inflection point ：反曲点Instantaneous velocity ：瞬时速度Integer ：整数Integral ：积分Integrand ：被积分式Integration ：积分Integration by part ：分部积分法Intercepts ：截距Intermediate value of Theorem ：中间值定理Interval ：区间Inverse function ：反函数Inverse trigonometric function ：反三角函数Iterated integral ：逐次积分HHigher mathematics 高等数学/高数E、F、G、HEllipse ：椭圆Ellipsoid ：椭圆体Epicycloid ：外摆线Equation ：方程式Even function ：偶函数Expected Valued ：期望值Exponential Function ：指数函数Exponents , laws of ：指数率Extreme value ：极值Extreme Value Theorem ：极值定理Factorial ：阶乘First Derivative Test ：一阶导数试验法First octant ：第一卦限Focus ：焦点Fractions ：分式Function ：函数Fundamental Theorem of Calculus ：微积分基本定理Geometric series ：几何级数Gradient ：梯度Graph ：图形Green Formula ：格林公式Half-angle formulas ：半角公式Harmonic series ：调和级数Helix ：螺旋线Higher Derivative ：高阶导数Horizontal asymptote ：水平渐近线Horizontal line ：水平线Hyperbola ：双曲线Hyper boloid ：双曲面DDecreasing function ：递减函数Decreasing sequence ：递减数列Definite integral ：定积分Degree of a polynomial ：多项式之次数Density ：密度Derivative ：导数of a composite function ：复合函数之导数of a constant function ：常数函数之导数directional ：方向导数domain of ：导数之定义域of exponential function ：指数函数之导数higher ：高阶导数partial ：偏导数of a power function ：幂函数之导数of a power series ：羃级数之导数of a product ：积之导数of a quotient ：商之导数as a rate of change ：导数当作变率right-hand ：右导数second ：二阶导数as the slope of a tangent ：导数看成切线之斜率Determinant ：行列式Differentiable function ：可导函数Differential ：微分Differential equation ：微分方程partial ：偏微分方程Differentiation ：求导法implicit ：隐求导法partial ：偏微分法term by term ：逐项求导法Directional derivatives ：方向导数Discontinuity ：不连续性Disk method ：圆盘法Distance ：距离Divergence ：发散Domain ：定义域Dot product ：点积Double integral ：二重积分change of variable in ：二重积分之变数变换in polar coordinates ：极坐标二重积分CCalculus ：微积分differential ：微分学integral ：积分学Cartesian coordinates ：笛卡儿坐标图片一般指直角坐标Cartesian coordinates system ：笛卡儿坐标系Cauch’s Mean Value Theorem ：柯西均值定理Chain Rule ：连锁律Change of variables ：变数变换Circle ：圆Circular cylinder ：圆柱Closed interval ：封闭区间Coefficient ：系数Composition of function ：函数之合成Compound interest ：复利Concavity ：凹性Conchoid ：蚌线Cone ：圆锥Constant function ：常数函数Constant of integration ：积分常数Continuity ：连续性at a point ：在一点处之连续性of a function ：函数之连续性on an interval ：在区间之连续性from the left ：左连续from the right ：右连续Continuous function ：连续函数Convergence ：收敛interval of ：收敛区间radius of ：收敛半径Convergent sequence ：收敛数列series ：收敛级数Coordinate：s：坐标Cartesian ：笛卡儿坐标cylindrical ：柱面坐标polar ：极坐标rectangular ：直角坐标spherical ：球面坐标Coordinate axes ：坐标轴Coordinate planes ：坐标平面Cosine function ：余弦函数Critical point ：临界点Cubic function ：三次函数Curve ：曲线Cylinder：圆柱Cylindrical Coordinates ：圆柱坐标A、BAbsolute convergence ：绝对收敛Absolute extreme values ：绝对极值Absolute maximum and minimum ：绝对极大与极小Absolute value ：绝对值Absolute value function ：绝对值函数Acceleration ：加速度Antiderivative ：反导数Approximate integration ：近似积分Approximation ：逼近法by differentials ：用微分逼近linear ：线性逼近法by Simpson’s Rule ：Simpson法则逼近法by the Trapezoidal Rule ：梯形法则逼近法Arbitrary constant ：任意常数Arc length ：弧长Area ：面积under a curve ：曲线下方之面积between curves ：曲线间之面积in polar coordinates ：极坐标表示之面积of a sector of a circle ：扇形之面积of a surface of a revolution ：旋转曲面之面积Asymptote ：渐近线horizontal ：水平渐近线slant ：斜渐近线vertical ：垂直渐近线Average speed ：平均速率Average velocity ：平均速度Axes, coordinate ：坐标轴Axes of ellipse ：椭圆之轴Binomial series ：二项级数。
中英文力学对准
一般力学类：分析力学 analytical mechanics拉格朗日乘子 Lagrange multiplier拉格朗日[量] Lagrangian拉格朗日括号 Lagrange bracket循环坐标 cyclic coordinate循环积分 cyclic integral哈密顿[量] Hamiltonian哈密顿函数 Hamiltonian function正则方程 canonical equation正则摄动 canonical perturbation正则变换 canonical transformation正则变量 canonical variable哈密顿原理 Hamilton principle作用量积分 action integral哈密顿-雅可比方程 Hamilton-Jacobi equation作用--角度变量 action-angle variables阿佩尔方程 Appell equation劳斯方程 Routh equation拉格朗日函数 Lagrangian function诺特定理 Noether theorem泊松括号 poisson bracket边界积分法 boundary integral method并矢 dyad运动稳定性 stability of motion轨道稳定性 orbital stability李雅普诺夫函数 Lyapunov function渐近稳定性 asymptotic stability结构稳定性 structural stability久期不稳定性 secular instability弗洛凯定理 Floquet theorem倾覆力矩 capsizing moment自由振动 free vibration固有振动 natural vibration暂态 transient state环境振动 ambient vibration反共振 anti-resonance衰减 attenuation库仑阻尼 Coulomb damping同相分量 in-phase component非同相分量 out-of -phase component超调量 overshoot 参量[激励]振动 parametric vibration模糊振动 fuzzy vibration临界转速 critical speed of rotation阻尼器 damper半峰宽度 half-peak width集总参量系统 lumped parameter system 相平面法 phase plane method相轨迹 phase trajectory等倾线法 isocline method跳跃现象 jump phenomenon负阻尼 negative damping达芬方程 Duffing equation希尔方程 Hill equationKBM方法 KBM method, Krylov-Bogoliu- bov-Mitropol'skii method马蒂厄方程 Mathieu equation平均法 averaging method组合音调 combination tone解谐 detuning耗散函数 dissipative function硬激励 hard excitation硬弹簧 hard spring, hardening spring谐波平衡法harmonic balance method久期项 secular term自激振动 self-excited vibration分界线 separatrix亚谐波 subharmonic软弹簧 soft spring ,softening spring软激励 soft excitation邓克利公式 Dunkerley formula瑞利定理 Rayleigh theorem分布参量系统 distributed parameter system优势频率 dominant frequency模态分析 modal analysis固有模态natural mode of vibration同步 synchronization超谐波 ultraharmonic范德波尔方程 van der pol equation频谱 frequency spectrum基频 fundamental frequencyWKB方法 WKB methodWKB方法Wentzel-Kramers-Brillouin method缓冲器 buffer风激振动 aeolian vibration嗡鸣 buzz倒谱cepstrum颤动 chatter蛇行 hunting阻抗匹配 impedance matching机械导纳 mechanical admittance机械效率 mechanical efficiency机械阻抗 mechanical impedance随机振动 stochastic vibration, random vibration隔振 vibration isolation减振 vibration reduction应力过冲 stress overshoot喘振surge摆振shimmy起伏运动 phugoid motion起伏振荡 phugoid oscillation驰振 galloping陀螺动力学 gyrodynamics陀螺摆 gyropendulum陀螺平台 gyroplatform陀螺力矩 gyroscoopic torque陀螺稳定器 gyrostabilizer陀螺体 gyrostat惯性导航 inertial guidance 姿态角 attitude angle方位角 azimuthal angle舒勒周期 Schuler period机器人动力学 robot dynamics多体系统 multibody system多刚体系统 multi-rigid-body system机动性 maneuverability凯恩方法Kane method转子[系统]动力学 rotor dynamics转子[一支承一基础]系统 rotor-support- foundation system静平衡 static balancing动平衡 dynamic balancing静不平衡 static unbalance动不平衡 dynamic unbalance现场平衡 field balancing不平衡 unbalance不平衡量 unbalance互耦力 cross force挠性转子 flexible rotor分频进动 fractional frequency precession半频进动half frequency precession油膜振荡 oil whip转子临界转速 rotor critical speed自动定心 self-alignment亚临界转速 subcritical speed涡动 whirl固体力学类：弹性力学 elasticity弹性理论 theory of elasticity均匀应力状态 homogeneous state of stress 应力不变量 stress invariant应变不变量 strain invariant应变椭球 strain ellipsoid均匀应变状态 homogeneous state of strain应变协调方程 equation of strain compatibility拉梅常量 Lame constants各向同性弹性 isotropic elasticity旋转圆盘 rotating circular disk 楔wedge开尔文问题 Kelvin problem布西内斯克问题 Boussinesq problem艾里应力函数 Airy stress function克罗索夫--穆斯赫利什维利法 Kolosoff- Muskhelishvili method基尔霍夫假设 Kirchhoff hypothesis板 Plate矩形板 Rectangular plate圆板 Circular plate环板 Annular plate波纹板 Corrugated plate加劲板 Stiffened plate,reinforcedPlate中厚板 Plate of moderate thickness弯[曲]应力函数 Stress function of bending 壳Shell扁壳 Shallow shell旋转壳 Revolutionary shell球壳 Spherical shell[圆]柱壳 Cylindrical shell锥壳Conical shell环壳 Toroidal shell封闭壳 Closed shell波纹壳 Corrugated shell扭[转]应力函数 Stress function of torsion 翘曲函数 Warping function半逆解法 semi-inverse method瑞利--里茨法 Rayleigh-Ritz method松弛法 Relaxation method莱维法 Levy method松弛 Relaxation量纲分析 Dimensional analysis自相似[性] self-similarity影响面 Influence surface接触应力 Contact stress赫兹理论 Hertz theory协调接触 Conforming contact滑动接触 Sliding contact滚动接触 Rolling contact压入 Indentation各向异性弹性 Anisotropic elasticity颗粒材料 Granular material散体力学 Mechanics of granular media热弹性 Thermoelasticity超弹性 Hyperelasticity粘弹性 Viscoelasticity对应原理 Correspondence principle褶皱Wrinkle塑性全量理论 Total theory of plasticity滑动 Sliding微滑Microslip粗糙度 Roughness非线性弹性 Nonlinear elasticity大挠度 Large deflection突弹跳变 snap-through有限变形 Finite deformation 格林应变 Green strain阿尔曼西应变 Almansi strain弹性动力学 Dynamic elasticity运动方程 Equation of motion准静态的Quasi-static气动弹性 Aeroelasticity水弹性 Hydroelasticity颤振Flutter弹性波Elastic wave简单波Simple wave柱面波 Cylindrical wave水平剪切波 Horizontal shear wave竖直剪切波Vertical shear wave体波 body wave无旋波 Irrotational wave畸变波 Distortion wave膨胀波 Dilatation wave瑞利波 Rayleigh wave等容波 Equivoluminal wave勒夫波Love wave界面波 Interfacial wave边缘效应 edge effect塑性力学 Plasticity可成形性 Formability金属成形 Metal forming耐撞性 Crashworthiness结构抗撞毁性 Structural crashworthiness 拉拔Drawing破坏机构 Collapse mechanism回弹 Springback挤压 Extrusion冲压 Stamping穿透Perforation层裂Spalling塑性理论 Theory of plasticity安定[性]理论 Shake-down theory运动安定定理 kinematic shake-down theorem静力安定定理 Static shake-down theorem 率相关理论 rate dependent theorem载荷因子load factor加载准则 Loading criterion加载函数 Loading function加载面 Loading surface塑性加载 Plastic loading塑性加载波 Plastic loading wave简单加载 Simple loading比例加载 Proportional loading卸载 Unloading卸载波 Unloading wave冲击载荷 Impulsive load阶跃载荷step load脉冲载荷 pulse load极限载荷 limit load中性变载 nentral loading拉抻失稳 instability in tension加速度波 acceleration wave本构方程 constitutive equation完全解 complete solution名义应力 nominal stress过应力 over-stress真应力 true stress等效应力 equivalent stress流动应力 flow stress应力间断 stress discontinuity应力空间 stress space主应力空间 principal stress space静水应力状态hydrostatic state of stress对数应变 logarithmic strain工程应变 engineering strain等效应变 equivalent strain应变局部化 strain localization应变率 strain rate应变率敏感性 strain rate sensitivity应变空间 strain space有限应变 finite strain塑性应变增量 plastic strain increment 累积塑性应变 accumulated plastic strain 永久变形 permanent deformation内变量 internal variable应变软化 strain-softening理想刚塑性材料 rigid-perfectly plastic Material刚塑性材料 rigid-plastic material理想塑性材料 perfectl plastic material 材料稳定性stability of material应变偏张量deviatoric tensor of strain应力偏张量deviatori tensor of stress 应变球张量spherical tensor of strain应力球张量spherical tensor of stress路径相关性 path-dependency线性强化 linear strain-hardening应变强化 strain-hardening随动强化 kinematic hardening各向同性强化 isotropic hardening强化模量 strain-hardening modulus幂强化 power hardening塑性极限弯矩 plastic limit bending Moment塑性极限扭矩 plastic limit torque弹塑性弯曲 elastic-plastic bending弹塑性交界面 elastic-plastic interface弹塑性扭转 elastic-plastic torsion粘塑性 Viscoplasticity非弹性 Inelasticity理想弹塑性材料 elastic-perfectly plastic Material极限分析 limit analysis极限设计 limit design极限面limit surface上限定理 upper bound theorem上屈服点upper yield point下限定理 lower bound theorem下屈服点 lower yield point界限定理 bound theorem初始屈服面initial yield surface后继屈服面 subsequent yield surface屈服面[的]外凸性 convexity of yield surface截面形状因子 shape factor of cross-section 沙堆比拟 sand heap analogy屈服Yield屈服条件 yield condition屈服准则 yield criterion屈服函数 yield function屈服面 yield surface塑性势 plastic potential能量吸收装置 energy absorbing device能量耗散率 energy absorbing device塑性动力学 dynamic plasticity塑性动力屈曲 dynamic plastic buckling塑性动力响应 dynamic plastic response塑性波 plastic wave运动容许场 kinematically admissible Field静力容许场 statically admissibleField流动法则 flow rule速度间断 velocity discontinuity滑移线 slip-lines滑移线场 slip-lines field移行塑性铰 travelling plastic hinge塑性增量理论 incremental theory ofPlasticity米泽斯屈服准则 Mises yield criterion普朗特--罗伊斯关系 prandtl- Reuss relation特雷斯卡屈服准则 Tresca yield criterion洛德应力参数 Lode stress parameter莱维--米泽斯关系 Levy-Mises relation亨基应力方程 Hencky stress equation赫艾--韦斯特加德应力空间Haigh-Westergaard stress space洛德应变参数 Lode strain parameter德鲁克公设 Drucker postulate盖林格速度方程Geiringer velocity Equation结构力学 structural mechanics结构分析 structural analysis结构动力学 structural dynamics拱 Arch三铰拱 three-hinged arch抛物线拱 parabolic arch圆拱 circular arch穹顶Dome空间结构 space structure空间桁架 space truss雪载[荷] snow load风载[荷] wind load土压力 earth pressure地震载荷 earthquake loading弹簧支座 spring support支座位移 support displacement支座沉降 support settlement超静定次数 degree of indeterminacy机动分析 kinematic analysis 结点法 method of joints截面法 method of sections结点力 joint forces共轭位移 conjugate displacement影响线 influence line三弯矩方程 three-moment equation单位虚力 unit virtual force刚度系数 stiffness coefficient柔度系数 flexibility coefficient力矩分配 moment distribution力矩分配法moment distribution method力矩再分配 moment redistribution分配系数 distribution factor矩阵位移法matri displacement method单元刚度矩阵 element stiffness matrix单元应变矩阵 element strain matrix总体坐标 global coordinates贝蒂定理 Betti theorem高斯--若尔当消去法 Gauss-Jordan elimination Method屈曲模态 buckling mode复合材料力学 mechanics of composites 复合材料composite material纤维复合材料 fibrous composite单向复合材料 unidirectional composite泡沫复合材料foamed composite颗粒复合材料 particulate composite层板Laminate夹层板 sandwich panel正交层板 cross-ply laminate斜交层板 angle-ply laminate层片Ply多胞固体 cellular solid膨胀 Expansion压实Debulk劣化 Degradation脱层 Delamination脱粘 Debond纤维应力 fiber stress层应力 ply stress层应变ply strain层间应力 interlaminar stress比强度 specific strength强度折减系数 strength reduction factor强度应力比 strength -stress ratio横向剪切模量 transverse shear modulus 横观各向同性 transverse isotropy正交各向异 Orthotropy剪滞分析 shear lag analysis短纤维 chopped fiber长纤维 continuous fiber纤维方向 fiber direction纤维断裂 fiber break纤维拔脱 fiber pull-out纤维增强 fiber reinforcement致密化 Densification最小重量设计 optimum weight design网格分析法 netting analysis混合律 rule of mixture失效准则 failure criterion蔡--吴失效准则 Tsai-W u failure criterion 达格代尔模型 Dugdale model断裂力学 fracture mechanics概率断裂力学 probabilistic fracture Mechanics格里菲思理论 Griffith theory线弹性断裂力学 linear elastic fracturemechanics, LEFM弹塑性断裂力学 elastic-plastic fracture mecha-nics, EPFM断裂 Fracture脆性断裂 brittle fracture解理断裂 cleavage fracture蠕变断裂 creep fracture延性断裂 ductile fracture晶间断裂 inter-granular fracture准解理断裂 quasi-cleavage fracture穿晶断裂 trans-granular fracture裂纹Crack裂缝Flaw缺陷Defect割缝Slit微裂纹Microcrack折裂Kink椭圆裂纹 elliptical crack深埋裂纹 embedded crack[钱]币状裂纹 penny-shape crack预制裂纹 Precrack 短裂纹 short crack表面裂纹 surface crack裂纹钝化 crack blunting裂纹分叉 crack branching裂纹闭合 crack closure裂纹前缘 crack front裂纹嘴 crack mouth裂纹张开角crack opening angle,COA裂纹张开位移 crack opening displacement, COD裂纹阻力 crack resistance裂纹面 crack surface裂纹尖端 crack tip裂尖张角 crack tip opening angle,CTOA裂尖张开位移 crack tip openingdisplacement, CTOD裂尖奇异场crack tip singularity Field裂纹扩展速率 crack growth rate稳定裂纹扩展 stable crack growth定常裂纹扩展 steady crack growth亚临界裂纹扩展 subcritical crack growth 裂纹[扩展]减速 crack retardation止裂crack arrest止裂韧度 arrest toughness断裂类型 fracture mode滑开型 sliding mode张开型 opening mode撕开型 tearing mode复合型 mixed mode撕裂 Tearing撕裂模量 tearing modulus断裂准则 fracture criterionJ积分 J-integralJ阻力曲线 J-resistance curve断裂韧度 fracture toughness应力强度因子 stress intensity factorHRR场 Hutchinson-Rice-Rosengren Field守恒积分 conservation integral有效应力张量 effective stress tensor应变能密度strain energy density能量释放率 energy release rate内聚区 cohesive zone塑性区 plastic zone张拉区 stretched zone热影响区heat affected zone, HAZ延脆转变温度 brittle-ductile transitiontemperature剪切带shear band剪切唇shear lip无损检测 non-destructive inspection双边缺口试件double edge notchedspecimen, DEN specimen单边缺口试件 single edge notchedspecimen, SEN specimen三点弯曲试件 three point bendingspecimen, TPB specimen中心裂纹拉伸试件 center cracked tension specimen, CCT specimen中心裂纹板试件 center cracked panelspecimen, CCP specimen紧凑拉伸试件 compact tension specimen, CT specimen大范围屈服large scale yielding小范围攻屈服 small scale yielding韦布尔分布 Weibull distribution帕里斯公式 paris formula空穴化 Cavitation应力腐蚀 stress corrosion概率风险判定 probabilistic riskassessment, PRA损伤力学 damage mechanics损伤Damage连续介质损伤力学 continuum damage mechanics细观损伤力学 microscopic damage mechanics累积损伤 accumulated damage脆性损伤 brittle damage延性损伤 ductile damage宏观损伤 macroscopic damage细观损伤 microscopic damage微观损伤 microscopic damage损伤准则 damage criterion损伤演化方程 damage evolution equation 损伤软化 damage softening损伤强化 damage strengthening 损伤张量 damage tensor损伤阈值 damage threshold损伤变量 damage variable损伤矢量 damage vector损伤区 damage zone疲劳Fatigue低周疲劳 low cycle fatigue应力疲劳 stress fatigue随机疲劳 random fatigue蠕变疲劳 creep fatigue腐蚀疲劳 corrosion fatigue疲劳损伤 fatigue damage疲劳失效 fatigue failure疲劳断裂 fatigue fracture疲劳裂纹 fatigue crack疲劳寿命 fatigue life疲劳破坏 fatigue rupture疲劳强度 fatigue strength疲劳辉纹 fatigue striations疲劳阈值 fatigue threshold交变载荷 alternating load交变应力 alternating stress应力幅值 stress amplitude应变疲劳 strain fatigue应力循环 stress cycle应力比 stress ratio安全寿命 safe life过载效应 overloading effect循环硬化 cyclic hardening循环软化 cyclic softening环境效应 environmental effect裂纹片crack gage裂纹扩展 crack growth, crack Propagation裂纹萌生 crack initiation循环比 cycle ratio实验应力分析 experimental stressAnalysis工作[应变]片 active[strain] gage基底材料 backing material应力计stress gage零[点]飘移zero shift, zero drift应变测量 strain measurement应变计strain gage应变指示器 strain indicator应变花 strain rosette应变灵敏度 strain sensitivity机械式应变仪 mechanical strain gage 直角应变花 rectangular rosette引伸仪 Extensometer应变遥测 telemetering of strain横向灵敏系数 transverse gage factor 横向灵敏度 transverse sensitivity焊接式应变计 weldable strain gage 平衡电桥 balanced bridge粘贴式应变计 bonded strain gage粘贴箔式应变计bonded foiled gage粘贴丝式应变计 bonded wire gage 桥路平衡 bridge balancing电容应变计 capacitance strain gage 补偿片 compensation technique补偿技术 compensation technique基准电桥 reference bridge电阻应变计 resistance strain gage温度自补偿应变计 self-temperature compensating gage半导体应变计 semiconductor strain Gage集流器slip ring应变放大镜 strain amplifier疲劳寿命计 fatigue life gage电感应变计 inductance [strain] gage 光[测]力学 Photomechanics光弹性 Photoelasticity光塑性 Photoplasticity杨氏条纹 Young fringe双折射效应 birefrigent effect等位移线 contour of equalDisplacement暗条纹 dark fringe条纹倍增 fringe multiplication干涉条纹 interference fringe等差线 Isochromatic等倾线 Isoclinic等和线 isopachic应力光学定律 stress- optic law主应力迹线 Isostatic亮条纹 light fringe 光程差optical path difference热光弹性 photo-thermo -elasticity光弹性贴片法 photoelastic coating Method光弹性夹片法 photoelastic sandwich Method动态光弹性 dynamic photo-elasticity空间滤波 spatial filtering空间频率 spatial frequency起偏镜 Polarizer反射式光弹性仪 reflection polariscope残余双折射效应 residual birefringent Effect应变条纹值 strain fringe value应变光学灵敏度 strain-optic sensitivity 应力冻结效应 stress freezing effect应力条纹值 stress fringe value应力光图 stress-optic pattern暂时双折射效应 temporary birefringent Effect脉冲全息法 pulsed holography透射式光弹性仪 transmission polariscope 实时全息干涉法 real-time holographicinterfero - metry网格法 grid method全息光弹性法 holo-photoelasticity全息图Hologram全息照相 Holograph全息干涉法 holographic interferometry 全息云纹法 holographic moire technique 全息术 Holography全场分析法 whole-field analysis散斑干涉法 speckle interferometry散斑Speckle错位散斑干涉法 speckle-shearinginterferometry, shearography散斑图Specklegram白光散斑法white-light speckle method云纹干涉法 moire interferometry[叠栅]云纹 moire fringe[叠栅]云纹法 moire method云纹图 moire pattern离面云纹法 off-plane moire method参考栅 reference grating试件栅 specimen grating分析栅 analyzer grating面内云纹法 in-plane moire method脆性涂层法 brittle-coating method条带法 strip coating method坐标变换 transformation ofCoordinates计算结构力学 computational structuralmecha-nics加权残量法weighted residual method有限差分法 finite difference method有限[单]元法 finite element method配点法 point collocation里茨法 Ritz method广义变分原理 generalized variational Principle最小二乘法 least square method胡[海昌]一鹫津原理 Hu-Washizu principle 赫林格-赖斯纳原理 Hellinger-Reissner Principle修正变分原理 modified variational Principle约束变分原理 constrained variational Principle混合法 mixed method杂交法 hybrid method边界解法boundary solution method有限条法 finite strip method半解析法 semi-analytical method协调元 conforming element非协调元 non-conforming element混合元 mixed element杂交元 hybrid element边界元 boundary element强迫边界条件 forced boundary condition 自然边界条件 natural boundary condition 离散化 Discretization离散系统 discrete system连续问题 continuous problem广义位移 generalized displacement广义载荷 generalized load广义应变 generalized strain广义应力 generalized stress界面变量 interface variable 节点 node, nodal point[单]元 Element角节点 corner node边节点 mid-side node内节点 internal node无节点变量 nodeless variable杆元 bar element桁架杆元 truss element梁元 beam element二维元 two-dimensional element一维元 one-dimensional element三维元 three-dimensional element轴对称元 axisymmetric element板元 plate element壳元 shell element厚板元 thick plate element三角形元 triangular element四边形元 quadrilateral element四面体元 tetrahedral element曲线元 curved element二次元 quadratic element线性元 linear element三次元 cubic element四次元 quartic element等参[数]元 isoparametric element超参数元 super-parametric element亚参数元 sub-parametric element节点数可变元 variable-number-node element拉格朗日元 Lagrange element拉格朗日族 Lagrange family巧凑边点元 serendipity element巧凑边点族 serendipity family无限元 infinite element单元分析 element analysis单元特性 element characteristics刚度矩阵 stiffness matrix几何矩阵 geometric matrix等效节点力 equivalent nodal force节点位移 nodal displacement节点载荷 nodal load位移矢量 displacement vector载荷矢量 load vector质量矩阵 mass matrix集总质量矩阵 lumped mass matrix相容质量矩阵 consistent mass matrix阻尼矩阵 damping matrix瑞利阻尼 Rayleigh damping刚度矩阵的组集 assembly of stiffnessMatrices载荷矢量的组集 consistent mass matrix质量矩阵的组集 assembly of mass matrices 单元的组集 assembly of elements局部坐标系 local coordinate system局部坐标 local coordinate面积坐标 area coordinates体积坐标 volume coordinates曲线坐标 curvilinear coordinates静凝聚 static condensation合同变换 contragradient transformation形状函数 shape function试探函数 trial function检验函数test function权函数 weight function样条函数 spline function代用函数 substitute function降阶积分 reduced integration零能模式 zero-energy modeP收敛 p-convergenceH收敛 h-convergence掺混插值 blended interpolation等参数映射 isoparametric mapping双线性插值 bilinear interpolation小块检验 patch test非协调模式 incompatible mode 节点号 node number单元号 element number带宽 band width带状矩阵 banded matrix变带状矩阵 profile matrix带宽最小化minimization of band width波前法 frontal method子空间迭代法 subspace iteration method 行列式搜索法determinant search method逐步法 step-by-step method纽马克法Newmark威尔逊法 Wilson拟牛顿法 quasi-Newton method牛顿-拉弗森法 Newton-Raphson method 增量法 incremental method初应变 initial strain初应力 initial stress切线刚度矩阵 tangent stiffness matrix割线刚度矩阵 secant stiffness matrix模态叠加法mode superposition method平衡迭代 equilibrium iteration子结构 Substructure子结构法 substructure technique超单元 super-element网格生成 mesh generation结构分析程序 structural analysis program 前处理 pre-processing后处理 post-processing网格细化 mesh refinement应力光顺 stress smoothing组合结构 composite structure流体动力学类：流体动力学 fluid dynamics连续介质力学 mechanics of continuous media介质medium流体质点 fluid particle无粘性流体 nonviscous fluid, inviscid fluid连续介质假设 continuous medium hypothesis流体运动学 fluid kinematics水静力学 hydrostatics 液体静力学 hydrostatics支配方程 governing equation伯努利方程 Bernoulli equation伯努利定理 Bernonlli theorem毕奥-萨伐尔定律 Biot-Savart law欧拉方程Euler equation亥姆霍兹定理 Helmholtz theorem开尔文定理 Kelvin theorem涡片 vortex sheet库塔-茹可夫斯基条件 Kutta-Zhoukowskicondition布拉休斯解 Blasius solution达朗贝尔佯廖 d'Alembert paradox 雷诺数 Reynolds number施特鲁哈尔数 Strouhal number随体导数 material derivative不可压缩流体 incompressible fluid 质量守恒 conservation of mass动量守恒 conservation of momentum 能量守恒 conservation of energy动量方程 momentum equation能量方程 energy equation控制体积 control volume液体静压 hydrostatic pressure涡量拟能 enstrophy压差 differential pressure流[动] flow流线stream line流面 stream surface流管stream tube迹线path, path line流场 flow field流态 flow regime流动参量 flow parameter流量 flow rate, flow discharge涡旋 vortex涡量 vorticity涡丝 vortex filament涡线 vortex line涡面 vortex surface涡层 vortex layer涡环 vortex ring涡对 vortex pair涡管 vortex tube涡街 vortex street卡门涡街 Karman vortex street马蹄涡 horseshoe vortex对流涡胞 convective cell卷筒涡胞 roll cell涡 eddy涡粘性 eddy viscosity环流 circulation环量 circulation速度环量 velocity circulation 偶极子 doublet, dipole驻点 stagnation point总压[力] total pressure总压头 total head静压头 static head总焓 total enthalpy能量输运 energy transport速度剖面 velocity profile库埃特流 Couette flow单相流 single phase flow单组份流 single-component flow均匀流 uniform flow非均匀流 nonuniform flow二维流 two-dimensional flow三维流 three-dimensional flow准定常流 quasi-steady flow非定常流unsteady flow, non-steady flow 暂态流transient flow周期流 periodic flow振荡流 oscillatory flow分层流 stratified flow无旋流 irrotational flow有旋流 rotational flow轴对称流 axisymmetric flow不可压缩性 incompressibility不可压缩流[动] incompressible flow 浮体 floating body定倾中心metacenter阻力 drag, resistance减阻 drag reduction表面力 surface force表面张力 surface tension毛细[管]作用 capillarity来流 incoming flow自由流 free stream自由流线 free stream line外流 external flow进口 entrance, inlet出口exit, outlet扰动 disturbance, perturbation分布 distribution传播 propagation色散 dispersion弥散 dispersion附加质量added mass ,associated mass收缩 contraction镜象法 image method无量纲参数 dimensionless parameter几何相似 geometric similarity运动相似 kinematic similarity动力相似[性] dynamic similarity平面流 plane flow势 potential势流 potential flow速度势 velocity potential复势 complex potential复速度 complex velocity流函数 stream function源source汇sink速度[水]头 velocity head拐角流 corner flow空泡流cavity flow超空泡 supercavity超空泡流 supercavity flow空气动力学 aerodynamics低速空气动力学 low-speed aerodynamics 高速空气动力学 high-speed aerodynamics 气动热力学 aerothermodynamics亚声速流[动] subsonic flow跨声速流[动] transonic flow超声速流[动] supersonic flow锥形流 conical flow楔流wedge flow叶栅流 cascade flow非平衡流[动] non-equilibrium flow细长体 slender body细长度 slenderness钝头体 bluff body钝体 blunt body翼型 airfoil翼弦 chord薄翼理论 thin-airfoil theory构型 configuration后缘 trailing edge迎角 angle of attack失速stall脱体激波detached shock wave 波阻wave drag诱导阻力 induced drag诱导速度 induced velocity临界雷诺数critical Reynolds number前缘涡 leading edge vortex附着涡 bound vortex约束涡 confined vortex气动中心 aerodynamic center气动力 aerodynamic force气动噪声 aerodynamic noise气动加热 aerodynamic heating离解 dissociation地面效应 ground effect气体动力学 gas dynamics稀疏波 rarefaction wave热状态方程thermal equation of state喷管Nozzle普朗特-迈耶流 Prandtl-Meyer flow瑞利流 Rayleigh flow可压缩流[动] compressible flow可压缩流体 compressible fluid绝热流 adiabatic flow非绝热流 diabatic flow未扰动流 undisturbed flow等熵流 isentropic flow匀熵流 homoentropic flow兰金-于戈尼奥条件 Rankine-Hugoniot condition状态方程 equation of state量热状态方程 caloric equation of state完全气体 perfect gas拉瓦尔喷管 Laval nozzle马赫角 Mach angle马赫锥 Mach cone马赫线Mach line马赫数Mach number马赫波Mach wave当地马赫数 local Mach number冲击波 shock wave激波 shock wave正激波normal shock wave斜激波oblique shock wave头波 bow wave附体激波 attached shock wave激波阵面 shock front激波层 shock layer压缩波 compression wave反射 reflection折射 refraction散射scattering衍射 diffraction绕射 diffraction出口压力 exit pressure超压[强] over pressure反压 back pressure爆炸 explosion爆轰 detonation缓燃 deflagration水动力学 hydrodynamics液体动力学 hydrodynamics泰勒不稳定性 Taylor instability 盖斯特纳波 Gerstner wave斯托克斯波 Stokes wave瑞利数 Rayleigh number自由面 free surface波速 wave speed, wave velocity 波高 wave height波列wave train波群 wave group波能wave energy表面波 surface wave表面张力波 capillary wave规则波 regular wave不规则波 irregular wave浅水波 shallow water wave深水波deep water wave重力波 gravity wave椭圆余弦波 cnoidal wave潮波tidal wave涌波surge wave破碎波 breaking wave船波ship wave非线性波 nonlinear wave孤立子 soliton水动[力]噪声 hydrodynamic noise 水击 water hammer空化 cavitation空化数 cavitation number 空蚀 cavitation damage超空化流 supercavitating flow水翼 hydrofoil水力学 hydraulics洪水波 flood wave涟漪ripple消能 energy dissipation海洋水动力学 marine hydrodynamics谢齐公式 Chezy formula欧拉数 Euler number弗劳德数 Froude number水力半径 hydraulic radius水力坡度 hvdraulic slope高度水头 elevating head水头损失 head loss水位 water level水跃 hydraulic jump含水层 aquifer排水 drainage排放量 discharge壅水曲线back water curve压[强水]头 pressure head过水断面 flow cross-section明槽流open channel flow孔流 orifice flow无压流 free surface flow有压流 pressure flow缓流 subcritical flow急流 supercritical flow渐变流gradually varied flow急变流 rapidly varied flow临界流 critical flow异重流density current, gravity flow堰流weir flow掺气流 aerated flow含沙流 sediment-laden stream降水曲线 dropdown curve沉积物 sediment, deposit沉[降堆]积 sedimentation, deposition沉降速度 settling velocity流动稳定性 flow stability不稳定性 instability奥尔-索末菲方程 Orr-Sommerfeld equation 涡量方程 vorticity equation泊肃叶流 Poiseuille flow奥辛流 Oseen flow剪切流 shear flow粘性流[动] viscous flow层流 laminar flow分离流 separated flow二次流 secondary flow近场流near field flow远场流 far field flow滞止流 stagnation flow尾流 wake [flow]回流 back flow反流 reverse flow射流 jet自由射流 free jet管流pipe flow, tube flow内流 internal flow拟序结构 coherent structure 猝发过程 bursting process表观粘度 apparent viscosity 运动粘性 kinematic viscosity 动力粘性 dynamic viscosity 泊 poise厘泊 centipoise厘沱 centistoke剪切层 shear layer次层 sublayer流动分离 flow separation层流分离 laminar separation 湍流分离 turbulent separation 分离点 separation point附着点 attachment point再附 reattachment再层流化 relaminarization起动涡starting vortex驻涡 standing vortex涡旋破碎 vortex breakdown 涡旋脱落 vortex shedding压[力]降 pressure drop压差阻力 pressure drag压力能 pressure energy型阻 profile drag滑移速度 slip velocity无滑移条件 non-slip condition 壁剪应力 skin friction, frictional drag壁剪切速度 friction velocity磨擦损失 friction loss磨擦因子 friction factor耗散 dissipation滞后lag相似性解 similar solution局域相似 local similarity气体润滑 gas lubrication液体动力润滑 hydrodynamic lubrication 浆体 slurry泰勒数 Taylor number纳维-斯托克斯方程 Navier-Stokes equation 牛顿流体 Newtonian fluid边界层理论boundary later theory边界层方程boundary layer equation边界层 boundary layer附面层 boundary layer层流边界层laminar boundary layer湍流边界层turbulent boundary layer温度边界层thermal boundary layer边界层转捩boundary layer transition边界层分离boundary layer separation边界层厚度boundary layer thickness位移厚度 displacement thickness动量厚度 momentum thickness能量厚度 energy thickness焓厚度 enthalpy thickness注入 injection吸出suction泰勒涡 Taylor vortex速度亏损律 velocity defect law形状因子 shape factor测速法 anemometry粘度测定法 visco[si] metry流动显示 flow visualization油烟显示 oil smoke visualization孔板流量计 orifice meter频率响应 frequency response油膜显示oil film visualization阴影法 shadow method纹影法 schlieren method烟丝法smoke wire method丝线法 tuft method。
q-分数阶微分方程研究现状
q-分数阶微分方程研究现状English Answer:Fractional calculus, as a branch of mathematical analysis, has gained increasing attention in recent decades due to its various applications in diverse fields,including physics, engineering, and finance. Within fractional calculus, q-calculus, which involves q-derivatives and q-integrals, has emerged as a significant area of study.Q-fractional differential equations, which incorporate q-derivatives and q-integrals, have become a fascinating subject of investigation. These equations offer a more generalized framework compared to their classical counterparts, providing researchers with a powerful tool to model complex phenomena in various disciplines.The study of q-fractional differential equations has witnessed substantial progress in recent years, leading tosignificant advancements in both theoretical foundations and practical applications. Here are some key aspects related to the current research landscape in q-fractional differential equations:1. Theoretical Development:Establishing new mathematical tools and techniques for analyzing q-fractional differential equations.Exploring the existence, uniqueness, and stability of solutions to q-fractional differential equations.Developing numerical methods for solving q-fractional differential equations efficiently and accurately.2. Applications:Modeling anomalous diffusion and transport phenomena in physics.Analyzing fractional-order control systems in engineering.Studying financial models with long-range dependencies and memory effects.3. Interdisciplinary Collaborations:Combining q-fractional differential equations with other mathematical disciplines, such as probability theory and numerical analysis.Exploring applications of q-fractional differential equations in fields such as biology, medicine, and social sciences.4. Open Challenges:Further development of analytical and numerical techniques for solving q-fractional differential equations.Investigating the connections between q-fractionaldifferential equations and other areas of mathematics, including partial differential equations and integral equations.Exploring new applications of q-fractionaldifferential equations in emerging fields, such asartificial intelligence and machine learning.The research on q-fractional differential equations continues to expand rapidly, with numerous researchers worldwide contributing to its development. It isanticipated that this field will continue to yieldsignificant theoretical and practical advancements in the years to come.中文回答：q-分数阶微分方程是分数阶微积分的一个分支，由于其在物理、工程和金融等领域的广泛应用，近年来受到越来越多的关注。
多因素广义估计方程调整基线的方法
多因素广义估计方程调整基线的方法In the field of statistics, the generalized estimating equations (GEE) approach is commonly used to analyze correlated data. It is a useful method for estimating population-averaged effects in longitudinal and clustered data. 多因素广义估计方程（GEE）方法是统计学领域中常用的方法，用于分析相关数据。
它是一种在纵向和集群数据中估计总体平均效应的有用方法。
When it comes to adjusting the baseline in a GEE model, there are several methods that can be employed. One common approach is to use the empirical method, where the baseline is adjusted based on the empirical mean or median of the response variable at the baseline time point. 另一种调整GEE模型基线的方法是使用经验方法，其中基线根据基线时间点的响应变量的经验均值或中位数进行调整。
Another method for adjusting the baseline in a GEE model is through the use of the working correlation structure. By specifying a working correlation structure, it is possible to account for the correlation between repeated measures and adjust the baseline accordingly. 调整GEE模型基线的另一种方法是通过使用工作相关结构。
广义相加模型 广义估计方程
广义相加模型广义估计方程英文回答：Generalized Additive Model (GAM)。
Generalized additive models (GAMs) are a type of semi-parametric regression model that allows for non-linear relationships between the response variable and the predictor variables. GAMs are an extension of generalized linear models (GLMs), which are themselves a generalization of linear regression models. GAMs are more flexible than GLMs because they allow for non-linear relationships between the response variable and the predictor variables, while GLMs assume that the relationships are linear.GAMs are fitted using a process called backfitting, which involves fitting a series of simpler models to the data. The first model is fitted to the response variable and the predictor variables, and then the residuals from this model are used to fit a second model. This process isrepeated until a final model is fitted that adequately describes the data.GAMs are a powerful tool for modeling complex relationships between the response variable and the predictor variables. However, they can be computationally intensive to fit, and they can be difficult to interpret if the relationships between the response variable and the predictor variables are complex.Generalized Estimating Equations (GEE)。
A more general effective-viscosity hypothesis
(where Ul,zis the only non-zero (mean) velocity gradient and peff is the effective viscosity). This formula has been used with considerable success by, among others, Ng (1971) for wall boundary layers and by Rodi (1972) for free shear flows. However, Bradshaw (1 973) has observed that the Boussinesq hypothesis fails for boundary layers over curved surfaces and inferred that this failure is due to the form of the stress-strain relation rather than to the inapplicability of an effective-viscosityapproach. For flows in which more than one Reynolds stress is needed to close the mean momentum equations, the Boussinesq hypothesis may be generalized to give the isotropic-viscosity assumption
By S. B . POPE
A discussion of the applicability of an effective-viscosityapproach to turbulent flow suggests that there are flow situations where the approach is valid andyet present hypotheses fail. The general form of an effective-viscosity formulation is shown to be a finite tensor polynomial. For two-dimensional flows, the coefficients of this polynomial are evaluated from the modelled Reynolds-stress equations of Launder, Reece & Rodi (1975). The advantage of the proposed effective-viscosityformulation, equation (4.3), over isotropic-viscosity hypotheses is that the whole velocity-gradient tensor affects the predicted Reynolds stresses. Two notable consequences of this are that (i) the complete Reynoldsstress tensor is realistically modelled and (ii)the influence of streamline curvature on the Reynolds stresses is incorporated.
分红模型中的极限定理
Stochastic Processes and their Applications119(2009)2401–2435/locate/spaLimit theorems for individual-based models ineconomics andfinanceDaniel Remenik∗Center for Applied Mathematics,Cornell University,657Rhodes Hall,14853Ithaca,NY,United States Received13June2008;received in revised form15October2008;accepted9December2008Available online16December2008AbstractThere is a widespread recent interest in using ideas from statistical physics to model certain types of problems in economics andfinance.The main idea is to derive the macroscopic behavior of the market from the random local interactions between agents.Our purpose is to present a general framework that encompasses a broad range of models,by proving a law of large numbers and a central limit theorem for certain interacting particle systems with very general state spaces.To do this we draw inspiration from some work done in mathematical ecology and mathematical physics.Thefirst result is proved for the system seen as a measure-valued process,while to prove the second one we will need to introduce a chain of embeddings of some abstract Banach and Hilbert spaces of test functions and prove that thefluctuations converge to the solution of a certain generalized Gaussian stochastic differential equation taking values in the dual of one of these spaces.c 2008Elsevier B.V.All rights reserved.MSC:60K35;60B12;46N30;62P50;91B70Keywords:Individual-based model;Interacting particle system;Law of large numbers;Central limit theorem;Fluctuation process;Measure-valued process;Finance;Economics1.IntroductionWe consider interacting particle systems of the following form.There is afixed number N of particles,each one having a type w∈W.The particles change their types via two mechanisms. Thefirst one corresponds simply to transitions from one type to another at some given rate.The∗Tel.:+16072553399;fax:+16072559860.E-mail address:dir4@.0304-4149/$-see front matter c 2008Elsevier B.V.All rights reserved.doi:10.1016/j.spa.2008.12.0012402 D.Remenik /Stochastic Processes and their Applications 119(2009)2401–2435second one involves a direct interaction between particles:pairs of particles interact at a certain rate and acquire new types according to some given (random)rule.We will allow these rates to depend directly on the types of the particles involved and on the distribution of the whole population on the type space.Our purpose is to prove limit theorems,as the number of particles N goes to infinity,for the empirical random measures νN t associated to these systems.νN t is defined as follows:if ηN t (i )∈W denotes the type of the i th particle at time t ,thenνN t =1N N i =1δηN t (i ),where δw is the probability measure on W assigning mass 1to w .Our first result,Theorem 1,provides a law of large numbers for νN t on a finite time interval[0,T ]:the empirical measures converge in distribution to a deterministic continuous path νt in the space of probability measures on W ,whose evolution is described by a certain system of integro-differential equations.Theorem 2analyzes the fluctuations of the finite system νN t around νt ,and provides an appropriate central limit result:the fluctuations are of order 1/√N ,and the asymptotic behavior of the process √N νN t −νt has a Gaussian nature.This second result is,as could be expected,much more delicate than the first one.In recent years there has been an increasing interest in the use of interacting particle systems to model phenomena outside their original application to statistical physics,with special attention given to models in ecology,economics,and finance.Our model is specially suited for the last two types of problems,in particular because we have assumed a constant number of particles,which may represent agents in the economy or financial market (ecological problems,on the other hand,usually require including birth and death of particles).Particle systems were first used in this context in [1],and they have been used recently by many authors to analyze a variety of problems in economics and finance.The techniques that have been used are diverse,including,for instance,ideas taken from the Ising model in [1],the voter model in [2],the contact process in [3],the theory of large deviations in [4],and the theory of queuing networks in [5,6].Our original motivation for this work comes precisely from financial modeling.It is related to some problems studied by Darrell Duffie and coauthors (see Examples 2.1and 3.3)in which they derive some models from the random local interactions between the financial agents involved,based on the ideas of Duffie and Sun [7].Our initial goal was to provide a general framework in which this type of problems could be rigorously analyzed,and in particular prove a law of large numbers for them.In our general setting,W will be allowed to be any locally compact complete separable metric space.Considering type spaces of this generality is one of the main features of our model,and it allows us to provide a unified framework to deal with models of different nature (for instance,the model in Example 2.1has a finite type space and the limit solves a finite system of ordinary differential equations,while in Example 3.3the type space is R and the limit solves a system of uncountably many integro-differential equations).To achieve this first goal,we based our model and techniques on ideas taken from the mathematical biology literature,and in particular on [8],where the authors study a model that describes a spatial ecological system where plants disperse seeds and die at rates that depend on the local population density,and obtain a deterministic limit similar to ours.We remark that,following their ideas,our results could be extended to systems with a non-constant population by adding assumptions which allow to control the growth of the population,but we have preferred to keep this part of the problem simple.D.Remenik /Stochastic Processes and their Applications 119(2009)2401–24352403The central limit result arose as a natural extension of this original question,but,as we already mentioned,it is much more delicate.The extra technical difficulties are related with the fact that the fluctuations of the process are signed measures (as opposed to the process νN t which takes values in a space of probability measures),and the space of signed measures is not well suited for the study of convergence in distribution.The natural topology to consider for this space in our setting,that of weak convergence,is in general not metrizable.One could try to regard this space as the Banach space dual of the space of continuous bounded functions on W and endow it with its operator norm,but this topology is too strong in general to obtain tightness for the fluctuations (observe that,in particular,the total mass of the fluctuations √N νN t −νt is not a priori bounded uniformly in N ).To overcome this difficulty we will show convergence of the fluctuations as a process taking values in the dual of a suitable abstract Hilbert space of test functions.We will actually have to consider a sequence of embeddings of Banach and Hilbert spaces,which will help us in controlling the norm of the fluctuations.This approach is inspired by ideas introduced in [9]to study weak convergence of some measure-valued processes using sequences of Sobolev embeddings.Our proof is based on [10],where the author proves a similar central limit result for a system of interacting diffusions associated with Boltzmann equations.The rest of the paper is organized as follows.Section 2contains the description of the general model,Section 3presents the law of large numbers for our system,and Section 4presents the central limit theorem,together with the description of the extra assumptions and the functional analytical setting we will use to obtain it.All the proofs are contained in Section 5.2.Description of the model2.1.Introductory exampleTo introduce the basic features of our model and fix some ideas,we begin by presenting one of the basic examples we have in mind.Example 2.1.We consider the model for over-the-counter markets introduced in [11].There is a “consol”,which is an asset paying dividends at a constant rate of 1,and there are N investors that can hold up to one unit of the asset.The total number of units of the asset remains constant in time,and the asset can be traded when the investors contact each other and when they are contacted by marketmakers.Each investor is characterized by whether he or she owns the asset or not,and by an intrinsic type that is “high”or “low”.Low-type investors have a holding cost when owning the asset,while high-type investors do not.These characteristics will be represented by the set of types W ={ho ,hn ,lo ,ln },where h and l designate the high-and low-type of an investor while o and n designate whether an investor owns or not the asset.At some fixed rate λd ,high-type investors change their type to low.This means that each investor runs a Poisson process with rate λd (independent from the others),and at each event of this process the investor changes his or her intrinsic type to low (nothing happens if the investor is already of low-type).Analogously,low-type investors change to high-type at some rate λu .The meetings between agents are defined as follows:each investor decides to look for another investor at rate β(understood as before,i.e.,at the times of the events of a Poisson process with rate β),chooses the investor uniformly among the set of N investors,and tries to trade.Additionally,each investor contacts a marketmaker at rate ρ.The marketmakers pair potential buyers and sellers,and the model assumes that this pairing happens instantly.At equilibrium,the rate at which investors trade through marketmakers is ρtimes the minimum between the fraction2404 D.Remenik/Stochastic Processes and their Applications119(2009)2401–2435of investors willing to buy and the fraction of investors willing to sell(see[11]for more details). In this model,the only encounters leading to a trade are those between hn-and lo-agents,since high-type investors not owning the asset are the only ones willing to buy,while low-type investors owning the asset are the only ones willing to sell.Theorem1will imply the following for this model:as N goes to infinity,the(random) evolution of the fraction of agents of each type converges to a deterministic limit which is the unique solution of the following system of ordinary differential equations:˙u ho(t)=2βu hn(t)u lo(t)+ρmin{u hn(t),u lo(t)}+λu u lo(t)−λd u ho(t),˙u hn(t)=−2βu hn(t)u lo(t)−ρmin{u hn(t),u lo(t)}+λu u ln(t)−λd u hn(t),˙u lo(t)=−2βu hn(t)u lo(t)−ρmin{u hn(t),u lo(t)}−λu u lo(t)+λd u ho(t),˙u ln(t)=2βu hn(t)u lo(t)+ρmin{u hn(t),u lo(t)}−λu u ln(t)+λd u hn(t).(2.1)Here u w(t)denotes the fraction of type-w investors at time t.This deterministic limit corresponds to the one proposed in[11]for this model(see the referred paper for the interpretation of this equations and more on this model).2.2.Description of the general modelWe will denote by I N={1,...,N}the set of particles in the system.In line with our original financial motivation,we will refer to these particles as the“agents”in the system(like the investors of the aforementioned example).The possible types for the agents will be represented by a locally compact Polish(i.e.,separable,complete,metrizable)space W.Given a metric space E,P(E)will denote the collection of probability measures on E,which will be endowed with the topology of weak convergence.When E=W,we will simply write P=P(W).We will denote by P a the subset of P consisting of purely atomic measures.The Markov processνN t we are interested in takes values in P a and describes the evolution of the distribution of the agents over the set of types.We recall that it is defined asνN t=1NNi=1δηNt(i),whereδw is the probability measure on W assigning mass1to w∈W andηN t(i)corresponds to the type of the agent i at time t.In other words,the vectorηN t∈W I N gives the configuration of the set of agents at time t,while for any Borel subset A of W,νN t(A)is the fraction of agents whose type is in A at time t.The dynamics of the process is defined by the following rates:•Each agent decides to change its type at a certain rateγ(w,νNt )that depends on its currenttype w and the current distributionνN t.The new type is chosen according to a probability measure a(w,νN t,d w )on W.•Each agent contacts each other agent at a certain rate that depends on their current types w1 and w2and the current distributionνN t:the total rate at which a given type-w1agent contacts type-w2agents is given by Nλ(w1,w2,νN t)νN t({w2}).After a pair of agents meet,they choose together a new pair of types according to a probability measure b(w1,w2,νN t,d w 1⊗d w 2) (not necessarily symmetric in w1,w2)on W×W.For afixedµ∈P a,a(w,µ,d w )and b(w1,w2,µ,d w 1⊗d w 2)can be interpreted, respectively,as the transition kernels of Markov chains in W and W×W.D.Remenik /Stochastic Processes and their Applications 119(2009)2401–24352405Let B (W )be the collection of bounded measurable functions on W and C b (W )be the collection of bounded continuous functions on W .For ν∈P and ϕ∈B (W )(or,more generally,any measurable function ϕ)we write ν,ϕ = Wϕd ν.Observe thatνN t ,ϕ =1N N i =1ϕ(ηN t (i )).We make the following assumption:Assumption A.(A1)The rate functions γ(w,ν)and λ(w,w ,ν)are defined for all ν∈P .They are non-negative,measurable in w and w ,bounded respectively by constants γand λ,andcontinuous in ν.(A2)a (w,ν,·)and b (w,w ,ν,·)are measurable in w and w .(A3)The mappings ν−→ W γ(w,ν)a (w,ν,·)ν(d w)and ν−→ W Wλ(w 1,w 2,ν)b (w 1,w 2,ν,·)ν(d w 2)ν(d w 1),which assign to each ν∈P a a finite measure on W and W ×W ,respectively,are continuous with respect to the topology of weak convergence and Lipschitz with respect to the total variation norm:there are constants C a ,C b >0such that W γ(w,ν1)a (w,ν1,·)ν1(d w)− W γ(w,ν2)a (w,ν2,·)ν2(d w) TV≤C a ν1−ν2 TVand W W λ(w 1,w 2,ν1)b (w 1,w 2,ν1,·)ν1(d w 2)ν1(d w 1)− W W λ(w 1,w 2,ν2)b (w 1,w 2,ν2,·)ν2(d w 2)ν2(d w 1) TV ≤C b ν1−ν2 TV .We recall that the total variation norm of a signed measure µis defined byµ TV =supϕ: ϕ ∞≤1| µ,ϕ |.(A3)is satisfied,in particular,whenever the rates do not depend on ν.w of large numbers for νNt Our first result shows that the process νN t converges in distribution,as the number of agents N goes to infinity,to a deterministic limit that is characterized by a measure-valued system of differential equations (written in its weak form).2406 D.Remenik /Stochastic Processes and their Applications 119(2009)2401–2435Given a metric space S ,we will denote by D ([0,T ],S )the space of c`a dl`a g functions ν:[0,T ]−→S ,and we endow these spaces with the Skorohod topology (see [12]or [13]for a reference on this topology and weak convergence in general).Observe that our processes νN t have paths on D ([0,T ],P )(recall that we are endowing P with the topology of weak convergence,which is metrizable).We will also denote by C ([0,T ],S )the space of continuous functions ν:[0,T ]−→S .Theorem 1.Suppose that Assumption A holds.For any given T >0,consider the sequence of P -valued processes νN t on [0,T ],and assume that the sequence of initial distributions νN 0converges in distribution to some fixed ν0∈P .Then the sequence νN t converges in distribution in D ([0,T ],P )to a deterministic νt in C ([0,T ],P ),which is the unique solution of the following system of integro-differential equations:for every ϕ∈B (W )and t ∈[0,T ], νt ,ϕ = ν0,ϕ + t 0 W γ(w,νs ) W (ϕ(w )−ϕ(w))a (w,νs ,d w )νs (d w)d s + t 0 W W λ(w 1,w 2,νs ) W ×W(ϕ(w 1)+ϕ(w 2)−ϕ(w 1)−ϕ(w 2))×b (w 1,w 2,νs ,d w 1⊗d w 2)νs (d w 2)νs (d w 1)d s .(S1)Observe that,in particular,(S1)implies that for every Borel set A ⊆W and almost every t ∈[0,T ],d νt (A )d t =− A γ(w,νt )+ W λ(w,w ,νt )+λ(w ,w,νt ) νt (d w ) νt (d w)+ W γ(w,νt )a (w,νt ,A )νt (d w)+ W Wλ(w,w ,νt )× b (w,w ,νt ,A ×W )+b (w,w ,νt ,W ×A ) νt (d w )νt (d w).(S1 )Furthermore,standard measure theory arguments allow to show that the system (S1 )actually characterizes the solution of (S1)(by approximating the test functions ϕin (S1)by simple functions).(S1 )has an intuitive interpretation:the first term on the right side is the total rate at which agents leave the set of types A ,the second term is the rate at which agents decide to change their types to a type in A ,and the third term is the rate at which agents acquire types in A due to interactions between them.The following corollary of the previous result is useful when writing and analyzing the limiting equations (S1)or (S1 )(see,for instance,Example 3.3).Corollary 3.1.In the context of Theorem 1,assume that ν0is absolutely continuous with respect to some measure µon W and that the measures W γ(w,ν0)a (w,ν0,·)ν0(d w)andW Wλ(w 1,w 2,ν0)b (w 1,w 2,ν0,·)ν0(d w 1)ν0(d w 2)are absolutely continuous with respect to µand µ⊗µ,respectively.Then the limit νt is absolutely continuous with respect to µfor all t ∈[0,T ].D.Remenik/Stochastic Processes and their Applications119(2009)2401–24352407The following two examples show two different kinds of models:one with afinite type space and the other with W=R.Thefirst model is the one given in Example2.1.Example3.2(Continuation of Example2.1).To translate into our framework the model for over-the-counter markets of[11],we take W={ho,hn,lo,ln}and consider a set of parametersγ,a,λ,and b with all butλbeing independent ofνN t.Letγ(ho)=γ(hn)=λd,a(ho,·)=δlo,a(hn,·)=δln,γ(lo)=γ(ln)=λu,a(lo,·)=δho,a(ln,·)=δhn.Observe that with this definition,high-type investors become low-type at rateλd and low-type investors become high-type at rateλu,just as required.For the encounters between agents wetakeλ(hn,lo,ν)=λ(lo,hn,ν)=β+ρ2ν({hn})∧ν({lo})ν({hn})ν({lo})ifν({hn})ν({lo})>0,βifν({hn})ν({lo})=0,b(hn,lo,ν,·)=δ(ho,ln),and b(lo,hn,ν,·)=δ(ln,ho)(where a∧b=min{a,b}),and for all other pairs w1,w2∈W,λ(w1,w2,ν)=0(recall that the only encounters leading to a trade are those between hn-and lo-agents and vice versa,in which case trade always occurs).The ratesλ(hn,lo,ν)andλ(lo,hn,ν)have two terms:the rate βcorresponding to the rate at which hn-agents contact lo-agents,plus a second rate reflecting trades carried out via a marketmaker.The form of this second rate assures that hn-and lo-agents meet through marketmakers at the right rate ofρν({hn})∧ν({lo}).It is not difficult to check that these parameters satisfy Assumption A,using the fact that x∧y=(x+y−|x−y|)/2for x,y∈R.Now let u w(t)=νt({w}),whereνt is the limit ofνN t given by Theorem1.We need to compute the right side of(S1 )with A={w}for each w∈W.Take,for example,w=ho.We get˙u ho(t)=λu u lo(t)−λd u ho(t)+βu hn(t)u lo(t)+ρ2u hn(t)∧u lo(t)+βu lo(t)u hn(t)+ρ2u hn(t)∧u lo(t),which corresponds exactly to thefirst equation in(2.1).The other three equations follow similarly.Example3.3.Our second example is based on the model for information percolation in large markets introduced in[14].We will only describe the basic features of the model,for more details see the cited paper.There is a random variable X of concern to all agents which has two possible values,“high”or“low”.Each agent holds some information about the outcome of X,and this information is summarized in a real number x which is a sufficient statistic for the posterior probability assigned by the agent(given his or her information)to the outcome of X being high.We take these statistics as the types of the agents(so W=R).The model is set up so that these statistics satisfy the following:after a type-x1agent and a type-x2agent meet and share their information,x1+x2becomes a sufficient statistic for the posterior distribution of X assigned by both agents given now their shared information.In this model the agents change types only after contacting other agents,so we takeγ≡0, and encounters between agents occur at a constant rateλ>0.The transition kernel for the types2408 D.Remenik /Stochastic Processes and their Applications 119(2009)2401–2435of the agents after encounters is independent of νN t and is given byb (x 1,x 2,·)=b (x 2,x 1,·)=δ(x 1+x 2,x 1+x 2)for every x 1,x 2∈R .This choice for the parameters trivially satisfies Assumption A .To compute the limit of the process,let A be a Borel subset of R .Then,since γ≡0and λis constant,(S1 )gives ˙νt (A )=−2λνt (A )+λ R 2 δ(x +y ,x +y )(R ×A )+δ(x +y ,x +y )(A ×R ) νt (d y )νt (d x )=−2λνt (A )+2λ R 2δx +y (A )νt (d y )νt (d x )=−2λνt (A )+2λ ∞−∞νt (A −x )νt (d x ),where A −x ={y ∈R :y +x ∈A }.Therefore,˙νt (A )=−2λνt (A )+2λ(νt ∗νt )(A ).(3.1)Using Corollary 3.1we can write the last equation in a nicer form:if we assume that the initial condition ν0is absolutely continuous with respect to the Lebesgue measure,then the measures νt have a density g t with respect to the Lebesgue measure,and we obtain ˙g t (x )=−2λg t (x )+2λ ∞−∞g t (z −x )g t (z )d z =−2λg t (x )+2λ(g t ∗g t )(x ).This is the system of integro-differential equations proposed in [14]for this model (except for the factor of 2,which is omitted in that paper).4.Central limit theorem for νNt Theorem 1gives the law of large numbers for νN t ,in the sense that it obtains a deterministic limit for the process as the size of the market goes to infinity.We will see now that,under some additional hypotheses,we can also obtain a central limit result for our process:the fluctuations of νN t around the limit νt are of order 1/√N ,and they have,asymptotically,a Gaussian nature.As we mentioned in the Section 1,this result is much more delicate than Theorem 1,and we will need to work hard to find the right setting for it.The fluctuation process is defined as follows:σN t =√N νN t −νt .σN t is a sequence of finite signed measures,and our goal is to prove that it converges to the solution of a system of stochastic differential equations driven by a Gaussian process.As we explained in Section 1,regarding the fluctuation process as taking values in the space of signed measures,and endowing this space with the topology of weak convergence (which corresponds to seeing the process as taking values in the Banach space dual of C b (W )topologized with the weak ∗convergence)is not the right approach for this problem.The idea will be to replace the test function space C b (W )by an appropriate Hilbert space H 1and regard σN t as a linear functional acting on this space via the mapping ϕ−→ σN t ,ϕ .In other words,we will regard σN t as a process taking values in the dual H 1 of a Hilbert space H 1.D.Remenik /Stochastic Processes and their Applications 119(2009)2401–24352409The space H 1that we choose will depend on the type space W .Actually,whenever W is not finite we will not need a single space,but a chain of seven spaces embedded in a certain structure.Our goal is to handle (at least)the following four possibilities for W :a finite set,Z d ,a “sufficiently smooth”compact subset of R d ,and all of R d .We wish to handle these cases under a unified framework,and this will require us to abstract the necessary assumptions on our seven spaces and the parameters of the model.We will do this in Sections 4.1and 4.2,and then in Section 4.3we will explain how to apply this abstract setting to the four type spaces W that we just mentioned.4.1.General settingIn this and the next subsection we will assume as given the collection of spaces in which our problem will be embedded,and then we will make some assumptions on the parameters of our process that will assure that they are compatible with the structure of these spaces.The idea of this part is that we will try to impose as little as possible on these spaces,leaving their definition to be specified for the different cases of type space W .The elements we will use are the following:•Four separable Hilbert spaces of measurable functions on W ,H 1,H 2,H 3,and H 4.•Three Banach spaces of continuous functions on W ,C 0,C 2,and C 3.•Five continuous functions ρ0,ρ1,ρ2,ρ3,ρ4:W −→[1,∞)such that ρi ≤ρi +1for i =0,1,2,3,ρi ∈C i for i =0,2,3,and for all w ∈W ,ρp 1(w)≤C ρ4(w)for some C >0and p >1(this last requirement is very mild,as we will see in the examples below,but will be necessary in the proof of Theorem 2).The seven spaces and the five functions introduced above must be related in a specific way.First,we assume that the following sequence of continuous embeddings holds:C 0H 1c H 2C 2H 3C 3H 4,(B1)where the c under the second arrow means that the embedding is compact.We recall that a continuous embedding E 1 →E 2between two normed spaces E 1,E 2implies,in particular,that · E 2≤C · E 1for some C >0,while saying that the embedding is compact means that every bounded set in E 1is compact in E 2.Second,we assume that for i =1,2,3,4,if ϕ∈H i then|ϕ(w)|≤C ϕ H i ρi (w)(B2)for all w ∈W ,for some C >0which does not depend on ϕ.The same holds for the spaces C i :for i =0,2,3and ϕ∈C i ,|ϕ(w)|≤C ϕ C i ρi (w).(B3)The functions ρi will typically appear as weighting functions in the definition of the norms of the spaces H i and C i .They will dictate the maximum growth rate allowed for functions in these spaces.We will denote by H i and C i the topological duals of the spaces H i and C i ,respectively,endowed with their operator norms (in particular,the spaces H i and C i are Hilbert and Banach spaces themselves).Observe that (B1)implies the following dual continuous embeddings:H 4 C 3 H 3 C 2 H 2c H 1 C 0 .(B1 )2410 D.Remenik /Stochastic Processes and their Applications 119(2009)2401–2435Before continuing,let us describe briefly the main ideas behind the proof of our central limit theorem,which will help explain why this is a good setting for proving convergence of the fluctuation process.What we want to prove is that σN t converges in distribution,as a process taking values in H 1 ,to the solution σt of a certain stochastic differential equation (see (S2)).The approach we will take to prove this (the proof of Theorem 1follows an analogous line)is standard:we first prove that the sequence σN t is tight,then we show that any limit point of this sequence satisfies the desired stochastic differential equation,and finally we prove that this equation has a unique solution (in distribution).To achieve this we will follow the line of proof of [10].Our sequence of embeddings (B1 )corresponds there to a sequence of embeddings of weighted Sobolev spaces (see (3.11)in the cited paper);in particular,we will use a very similar sequence of spaces to deal with the case W =R d in Section 4.3.4.One important difficulty with this approach is the following:the operator J s associated with the drift term of our limiting equation (see (4.1)),as well as the corresponding operators J N s for σN t (see (5.9)),cannot in general be taken to be bounded as operators acting on any of the spaces H i .This forces us to introduce the spaces C i ,on which (B3)plus some assumptions on the rates of the process will assure that J s and J N s are bounded.The scheme of proof will be roughly as follows.We will consider the semimartingale decomposition of the real-valued process σN t ,ϕ ,for ϕ∈H 4,and then show that the martingale part defines a martingale in H 4 .This,together with a moment estimate on the norm of the martingale part in H 4 and the boundedness of the operators J N s in C 3 ,will allow us to deduce that σN t can be seen as a semimartingale in H 3 ,and moreover give its semimartingale decomposition.Next,we will give a uniform estimate (in N )of the norm of σN t in C 2 .This implies the same type of estimate in H 2 ,and this will allow us to obtain the tightness of σN t in H 1 .The fact that the embedding H 2 →H 1 is compact is crucial in this step.Then we will show that all limit points of σN t have continuous paths in H 1 and they all satisfy the desired stochastic differential equation (S2).Unfortunately,it will not be possible to achieve this last part in H 1 ,due to the unboundedness of J s in this space.Consequently,we are forced to embed the equation in the (bigger)space C 0 .The boundedness of J s in C 0 will also allow us to obtain uniqueness for the solutions of this equation in this space,thus finishing the proof.Our last Assumption D will assure that our abstract setting is compatible with the rates defining our process.Before that,we need to replace Assumptions (A1)and (A2)by stronger versions:Assumption C.(C1)There is a family of finite measures {Γ(w,z ,·)}w,z ∈W on W ,whose total masses arebounded by γ,such that for every w ∈W and every ν∈P we have γ(w,ν)a (w,ν,d w )= WΓ(w,z ,d w )ν(d z ).Γ(w,z ,·)is measurable in w and continuous in z .(C2)There is a family of measures {Λ(w 1,w 2,z ,·)}w 1,w 2,z ∈W on W ×W ,whose total massesare bounded by λ,such that for every w 1,w 2∈W and every ν∈P we have λ(w 1,w 2,ν)b (w 1,w 2,ν,d w 1⊗d w 2)= W ×WΛ(w 1,w 2,z ,d w 1⊗d w 2)ν(d z ).Λ(w 1,w 2,z ,·)is measurable in w 1and w 2and continuous in z .。
各向同性湍流速度分布的相似解
Similarity Solutions of Velocity Distributions inHomogeneous Isotropic TurbulenceZheng RanShanghai Institute of Applied Mathematics and Mechanics,Shanghai University, Shanghai 200072,P.R.ChinaAbstractThe starting point for this paper lies in the results obtained by Tatsumi (2004) for isotropic turbulence with the self-preserving hypothesis. A careful consideration of the mathematical structure of the one-point velocity distribution function equation obtained by Tatsumi (2004) leads to an exact analysis of all possible cases and to all admissible solutions of the problem. This paper revisits this interesting problem from a new point of view. Firstly, a new complete set of solutions are obtained. Based on these exact solutions, some physically significant consequences of recent advances in the theory of homogenous statistical solution of the Navier-Stokes equations are presented. The comparison with former theory was also made. The origin of non-gaussian character could be deduced from the above exact solutions.Keywords: isotropic turbulence, PDF, similarity solutionThe complete statistical information of turbulence is provided by the probability distribution functional of the velocity field in space and time. The evolution equation for this distribution functional was given by Hopf (1952) in terms of the characteristic functional equation. So far, however, no mathematical method has been known for dealing with such a functional equation generally, and only a few particular solutions given by Hopf himself have been available. A statistically equivalent formalism to distribution functional is given by an infinite set of the joint-probability distribution of the velocities at arbitrary number of points in space-time. The system of equations governing such joint-probability distributions was given by Lundgren (1967) and Monin (1967) independently. The set of equations, however, is indeterminate since each equation involves a higher-order distribution as new unknown. Such indeterminacy is known as the common difficulty of nonlinear physics and the closure of the set of equations is the central problem of turbulence. Recently, the hypothesis of cross-independence of two turbulent velocities has been proposed by Tatsumi (2001) for closing the equations for the velocity distributions given by Lundgren (1967) and Monin (1967). Using this hypothesis, the equations for one-and two-point velocity distributions in homogeneous isotropic turbulence are derived in closed form, and the velocity distributions are obtained from these equations, One-point velocity distribution is obtained as an inertial normal distribution (N1) including the energy dissipation rate as only parameter. The inertial similarity of homogeneous isotropic turbulence associated with the normal one-and two-point velocity distributions seems to give good prospects for the extension of the further approach to inhomogeneous turbulent flows and more complex turbulent phenomena.The start point lives in the work of Tatsumi (2001) with the hypothesis of cross-independence of two turbulent velocities. New similarity exact solutions could be found. This would be change subtlety the results presented by Tatsumi (2001).Let us consider turbulence in an incompressible viscous fluid, in particular, homogeneous isotropic turbulence whose probability distributions are uniform and isotropic in space. Take thecoordinates , the time t and denote the fluid velocity at point ( byand the pressure by . We denote the velocities at the two space pointsand at time t by (321,,x x x x =))))t x ,(t x u ,(t x p ,21,x x ()()t x u t x u ,,,21, the one-and two-point velocity distributions are defined, respectively, as()()()1111,,,v t x u t x v f −=δ (1)()()()()()221121212,,;,;,v t x u v t x u t x x v v f −−=δδ (2)where denotes the probability variable corresponding to the velocities ,2,1,=i v i ()t x u i,δthe three-dimensional delta function, andthe probability mean with respect to a certaininitial distribution.Tatisumi[5,6] introduced the sum and the difference of two velocities as21,u u (2121u u u +=+) (3) (1221u u u −=−) (4)and call them the cross-velocities of two velocities . The distributions of the cross-velocities are defined in accordance with the former.21,u u ()()()±±±±−=v t x u t x v g ,,,11δ (5)()()()()()vt x u v t x u t x x v v g −−=±±±,,;,;,211212δδ (6)where(2121v v v +=+) (7) (1221v v v −=−) (8)Tatsumi (2001) introduced a new independence relation between the cross-velocities by()()()t r v g t r v g t r v v g ,,,,,;,2−−++−+= (9)that may be called the cross-independence in contrast to the ordinary independence. Under this consideration, one can obtain the closed form equation for one-point velocity distribution (Ref.6, Eq.50)()()0,2=⋅⎥⎥⎦⎤⎢⎢⎣⎡∂∂+∂∂t v f v t tα (10)where()()t t εα31= (11)ε represents the mean energy dissipation rate.Let us consider homogeneous turbulence without external energy supply and obtain the velocity distribution as the solution of the governing equation (10), and we introduce the self-preserving hypothesis as()()()⎟⎟⎠⎞⎜⎜⎝⎛≡t l v Y t b t v f , (12) ()t l νξ=(13)On substitution from (12), Eq. (10) reduces to the following form02222122=+⎟⎟⎠⎞⎜⎜⎝⎛++Y a d dY a d Y d ξξξξ (14) ()().0,10=∞=Y Y (15)where()()211adt dl t l t =⋅−α (16)()()()222a dt db t b t t l =⋅α (17) We call these the scaling equations. Now, let us confine ourselves to the self-similar solution intime. Under this consideration=i a constant, (18)For .2,1=i The turbulence energy equation reads asε−=dtEd (19) For decaying turbulence, we have()10−=t E t E (20)()20−=t t εε (21)Substitution these into the scaling equations, we have()2010126l t a t l +=−ε (22)()()σω−+=t b t b 10(23) where1206εωa l =12a a =σ The complete solution are given in this paper, these are(the details could be seen in Appendix)： When3=σ(denotes type SI)，()2141ξξa eY −= (24)When23−=σκ(denotes type SI)，()⎟⎠⎞⎜⎝⎛−=−21424,23,4921ξσξξa F eY a (25)Whenσκ−=23(denotes type SIII), ()⎟⎠⎞⎜⎝⎛−=−21434,23,4321ξσξξa F eY a (26)When23=σ(denotes SVI) ()⎟⎠⎞⎜⎝⎛=−21444,21,4121ξξξa F eY a (27) We could deduce the corresponding asymptotic expansions of the above solutions, and give theexistence conditions for different type of solutions. (1) Existence condition for SIIn order to satisfy the boundary condition at infinity, we must have01>a (28)This is the only existence condition for SI, and also one of existence conditions for other three solutions.(2) Existence condition for SII()⎟⎠⎞⎜⎝⎛−∝σξξ4322F (29)The boundary condition leads to43>σ (30) which is one of the existence conditions for SII.(3) Existence condition for SIII()⎟⎠⎞⎜⎝⎛−∝4923σξξF (31)So49<σ (32) (4) Existence condition for SIV()214−∝ξξF (33)So the only existence condition for SIV is.01>a We can draw the following conclusion on the existence conditions for the complete set of solutions:(1)For all kind of solution: .01>a (2)For the second kind of solution ：43>σ；23≠σ; (3)For the third kind of solution ：490<<σ；23≠σ;A simple comparison shows that the special solution found by Tatsumi (2001) belongs to onespecial kind of our new set of solution. Other differences between Tatsumi’s solution and our new set of solutions would be discussed below.In order to give a complete description of turbulence features, we must introduce a probability structure based on these solutions. Let 4,3,2,1,=j p j denotes the probability of the j ’th class solution ，and the set P is called the state vector of this process defined by(4321,,,p p p p P ≡). (34)The state vector is a function of the parameter σ, we could be easy to write as follows (1)If 430≤<σ，the probability vector is ()()0,1,0,0,,,4321=≡p p p p P (35)(2)If2343<<σ，we have ()⎟⎠⎞⎜⎝⎛=≡0,21,21,0,,,4321p p p p P (36)(3)If23=σ, we have ()()1,0,0,0,,,4321=≡p p p p P (37)(5)If4923<<σ,()⎟⎠⎞⎜⎝⎛=≡0,21,21,0,,,4321p p p p P (38)(6)If349<≤σ，we have ()()0,0,1,0,,,4321=≡p p p p P (39)(7)If3=σ，()⎟⎠⎞⎜⎝⎛=≡0,0,21,21,,,4321p p p p P (40)(8)If3>σ，we have()()0,0,1,0,,,4321=≡p p p p P (41)Based on the above deduced probability structure, the average of turbulence could be given, whichrepresents the statistical distributions of turbulence. In the following discuss, we will deal with the one-point velocity distribution function, in principle, we could give the average of these physical quantities.∑==41j j j f p f (42)(1)If 430≤<σ，the probability vector is 3f f = (43)(2)If2343<<σ，we have (3221f f f +=) (44) (3)If23=σ, we have 4f f = (45)(5)If4923<<σ,(3221f f f +=) (46) (6)If349<≤σ，we have 2f f = (47)(7)If3=σ，(1221f f f +=) (48) (8)If3>σ，we have2f = (49)Actually, in a turbulent viscid fluid, the statistics of the velocity field at a fixed point is non-gaussian. For example, let us set 0.1=σ, and it belongs to the secondary case:(2)If2343<<σ，we have (3221f f f +=) (44) So, we could easily write the expression of the one-point velocity distribution as : (only presentedthe part in terms of the ()ξj Y )(3221Y Y Y +=) (45a) ⎭⎬⎫⎩⎨⎧+⋅+≈⎭⎬⎫⎩⎨⎧⎟⎠⎞⎜⎝⎛+⎟⎠⎞⎜⎝⎛=−−...43214,23,414,23,4521214212142121ξξξξξa ea F a F e Y a a(45b)We can see that Y is non-guassian type distribution. The comparison is presented in Figure 1.Figure.1 Comparison of the one-point velocity distribution solution[1]Hopf,E., Statistical hydromechanics and functional calculus. J. Rat. Mech. Anal. 1,87-123.(1952).[2]Kida, S., Murakami,Y ., J.Phys. Soc. Japan, 57,3657-3660(1988). [3]Lundgren,T.S., Phys. Fluids. 10,969-975(1967).[4]Monin, A.S., PMM J.Appl.Mech.31,1057-1068(1967).[5]Tatsumi,T., Mathematical physics of turbulence. In: Kambe, T., et al. (Ed.), Geometry and Statistics of Turbulence. Kluwer Academic Publishers, Dordrecht, pp,3-12. [6]Tatsumi, T. & Yoshimura, T., Fluid Dynamics Research, 35,123-158(2004).Appendix : Solutions of the PDF equationThe ideas of similarity and self-preservation were firstly introduced by von Karman (1938). Following the methods adopted by von Karman (1938), the one-pointl velocity distribution function satisfied02222122=+⎟⎟⎠⎞⎜⎜⎝⎛++f a d df a d f d ξξξξ （1） with the boundary condition()10=f ()0=∞fThe complete solution are given in this paper, these are ： When3=σ，()214ξξa ef −=When 23−=σκ，()⎟⎠⎞⎜⎝⎛−=−2144,23,4921ξσξξa F e f aWhen σκ−=23，()⎟⎠⎞⎜⎝⎛−=−2144,23,4321ξσξξa F e f aWhen 23=σ，()⎟⎠⎞⎜⎝⎛=−2144,21,4121ξξξa F e f aThe detailed calculation is given as following:A lot of useful partial differential equations can be reduced to confluent hypergeometric equations,()ςm k P ,is the solution of Whittaker equation as that defined by Whittaker and Waston041412222=⎥⎦⎤⎢⎣⎡−++−+W m d W d ςςκς （2） where()()()()z h P e z z y m z f ,κβ= （3）After some reduction, the equation of ()z y reads()()02222=⋅+⎥⎦⎤⎢⎣⎡′++′′′−z y g dzdy z f z h h dz y d β （4） where()()12212g f z h h z z f f f g +⎟⎠⎞⎜⎝⎛′+′′′+++′+′′−′=ββββ⎟⎟⎠⎞⎜⎜⎝⎛−+−⎟⎠⎞⎜⎝⎛′=4412221h h m h h g κ The solutions of above equation could be deduced in terms of Whittaker function. We discussed this equation in following special case:()λaz z f = （5） ()λAz z h = （6）The equation under this condition reads as()0221122=+⎥⎦⎤⎢⎣⎡−−−+−z qy dz dy z z dz y d λλαβλ （7） where()()()2222222224124z m z A z A q −+++++⎟⎟⎠⎞⎜⎜⎝⎛−=−−λλββκλαβλαλλλ The solution of this equation is()()λκαβλAz P ez z y m z ,= （8）For isotropic turbulence, the corresponding parameters satisfied221=−−βλ （9）11=−λ （10）221a =−λα （11）04222=⎟⎟⎠⎞⎜⎜⎝⎛−A αλ （12） ()04122=⎟⎠⎞⎜⎝⎛−++m λλββ （13）()222a A =+κλαβλ （14） Hence, we have2=λ （15） 81a −=α （16） 23−=β （17）41±=m （18）41a A ±= （19） ⎭⎬⎫⎩⎨⎧−±=2312a a κ （20） From above analysis, we can introduce two parameters to classification turbulence, they are:121,a a a =σ. According to Whittaker and Waston, if isn’t an integral,m 2()⎟⎠⎞⎜⎝⎛+−+=+−z m m F ze z P m z m k ,21,21212,κ （21）()⎟⎠⎞⎜⎝⎛−−−=−−−z m m F ze z P m zm k ,21,21212,κ （22）For the case0=κ，we must use the secondary Kummer formula,()⎟⎟⎠⎞⎜⎜⎝⎛+=+16;121021,0z m F zz P m m （23） By making use of the boundary condition, we could chose the rational parameters for isotropic turbulence. The solution of equation could be rewritten in()()()⎟⎠⎞⎜⎝⎛+−+⋅⋅=⎟⎠⎞⎜⎝⎛+−+⋅==++⎟⎠⎞⎜⎝⎛−++−λλλβαλλαβλκαβκκλλAz m m F z e A Az m m F Az e e z Az P e z z y m z A m m z A z m z ,21,21,21,212221212,22 （24） Let ，this resulted in the definition of exponent.0>A If we chose 41−=m ，in the above solution, the exponent of reads as z 021412232<−=+⎟⎠⎞⎜⎝⎛−×+−=++λλβm （25） The boundary condition would be broken under this condition. So we only chose()0y 41=m （26） Another condition must be satisfied 02=+A α （27） The solution is ()⎟⎠⎞⎜⎝⎛−⋅=−2,23,432Az F e z y Az κ （28） There is an important parameter in the above solution, the multiple values could be existed ： k When ⎭⎬⎫⎩⎨⎧−=2312a a κ， ()⎟⎠⎞⎜⎝⎛−⋅=−2,23,492Az F e z y Az σ （29） when ⎭⎬⎫⎩⎨⎧−−=2312a a κ， ()⎟⎠⎞⎜⎝⎛−⋅=−2,23,432Az F e z y Az σ （30） We must treat the other special case 0=k ，by using the secondary Kummer formula()⎟⎟⎠⎞⎜⎜⎝⎛+=+16;121021,0z m F z z P m m （31）where()z m m F e z m F z ,2,16;12210−=⎟⎟⎠⎞⎜⎜⎝⎛+ For this case, the solution of equation is()⎟⎠⎞⎜⎝⎛⋅=−2,21,412Az F ez y Az （32） Another reduced case for 3=σ，the solution is ()214ξξa e f −= （33）At last, we have already obtained a complete set solution of isotropic turbulence , depending on two parameters, these are ：When 3=σ，()214ξξa e f −= When 23−=σκ，()⎟⎠⎞⎜⎝⎛−=−2144,23,4921ξσξξa F e f a When σκ−=23，()⎟⎠⎞⎜⎝⎛−=−2144,23,4321ξσξξa F e f a When 23=σ，()⎟⎠⎞⎜⎝⎛=−2144,21,4121ξξξa F e f aReferences[1]Whittaker,E.T. and Waston, G .N., A course of modern analysis. Cambridge University Press, 1935[2]M.Abramowitz and I.A.Stegun, Handbook of mathematical functions. Dover, New York,1965[3]Wang, Z.X. and Guo,D. R., Special functions. The series of advanced physics of Peking University. Peking University Press,2000 (In Chinese)。
有限点集共超球的充分必要条件
有限点集共超球的充分必要条件杨定华【摘要】Using the theory and method of distance geometry, we proved that a necessary and sufficient condition for ∑(A,N ＋ 1 ) being on an (n - 1 )-dimensional hyper-sphere is that the matrix M(∑(A ,N ＋ 1 ) )=(a2kl) (k,l=0, 1, …, N) has rank n ＋1, where M ( ∑ ( A, N ＋ 1 ) ) is the distance square matrix of ∑(A,N＋1), ∑(A, N ＋1 )={A0,A1, …,An is an n-dimensional finite set of points in n-dimensional Euclidean space. A necessary and sufficient condition for 5 points on a 2-dimensional hyper-sphere was given.%运用距离几何的理论与方法,证明n维欧氏空间 En 中的n维有限点集Σ(A,N+1)={A0,A1,…,AΝ}在同一个n-1维超球面上的充要条件是:Σ(A,N+1)的距离平方矩阵M(Σ(A,N+1))=(a2kl)(k,l=0,1,…,N)的秩等于n+1.并给出了三维空间中5点共球的充分必要条件.【期刊名称】《吉林大学学报（理学版）》【年(卷),期】2011(049)003【总页数】6页(P465-470)【关键词】n维欧氏空间;n-1维超球面;距离平方矩阵;秩【作者】杨定华【作者单位】四川师范大学,数学与软件科学学院,成都,610066【正文语种】中文【中图分类】O184在平面几何中, Ptolemy定理可以表述为：在同一平面上不共线的4点{A0,A1,A2,A3}共圆的充分必要条件是(a01a23-a02a13-a03a12)(a02a13-a03a12-a01a23)(a03a12-a01a23-a02a13)=0,(1)这里akl为Ak与Al的距离, k,l=0,1,2,3. 如果在A0,A1,A2,A3同一平面上, 并且已知A0A2和A1A3为凸四边形A0A1A2A3对角线, 则Ptolemy定理可以简单地表述为： 4点A0,A1,A2,A3共圆的充分必要条件是a02a13=a03a12+a01a23.(2)本文考虑从距离方面推广Ptolemy定理, 运用距离几何的理论和方法以及矩阵代数工具, 证明了n维欧氏空间En中的n维有限点集Σ(A,N+1)={A0,A1,…,AN}在同一个n-1维超球面上的充分必要条件是：Σ(A,N+1)的距离平方矩阵的秩等于n+1. 定理1 设Σ(A,N+1)={A0,A1,…,AN}(N≥n+1)是n维欧氏空间En中的n维有限点集, 记akl为Ak与Al的距离, k,l=0,1,…,N, 则Σ(A,N+1)在同一个n-1维超球面上的充分必要条件是：Σ(A,N+1)的距离平方矩阵(3)的秩rank(M(Σ(A,N+1)))=n+1.(4)证明: 必要性. 因为Σ(A,N+1)在同一个n-1维超球面上, 设该超球S的球心为O, 由于Ak与点Al的距离akl(k,l=0,1,…,N)在刚体运动群作用下是不变量, 所以可以以超球S的球心O为原点建立直角坐标系, 并设Ak所对应的向量于是(5)这里r为超球S的半径, 并且(6)表示列向量的转置. 对Σ(A,N+1)的距离平方矩阵M(Σ(A,N+1))镶边得(7)考虑到式(5), 对式(7)中的矩阵做初等变换：将第1行分别加到第t(t=2,3,…,N+2)行, 得(8)对式(8)中的矩阵做初等变换：将第t(t=2,3,…,N+2)列乘以-1/2后, 得(9)由于有限点集Σ(A,N+1)是n维的, 易知向量组的秩为n, 于是, 向量组的Gram矩阵的秩(10)由于r>0, 并且矩阵的初等变换不改变矩阵的秩, 易知(11)从而得(12)因此, 易知rank(M(Σ(A,N+1)))≤n+1.(13)由于有限点集Σ(A,N+1)是n维的, 所以存在一个非退化的n维单形, 由对称性不妨假设Σ(A,n+1)={A0,A1,…,An}是非退化的n维单形, 记(14)为n维单形Σ(A,n+1)的距离平方矩阵, v(Σ(A,n+1))和r(Σ(A,n+1))分别是n维单形Σ(A,n+1)的体积和外接超球半径, 于是, 由文献[1-2]知,det(M(Σ(A,n+1)))=(-1)n2n+1(n!)2v2(Σ(A,n+1))r2(Σ(A,n+1)),(15)从而Σ(A,n+1)的距离平方矩阵M(Σ(A,n+1))的秩rank(M(Σ(A,n+1)))=n+1.(16)由于M(Σ(A,n+1))为M(Σ(A,N+1))的主子矩阵, 所以rank(M(Σ(A,N+1)))≥n+1,(17)于是得rank(M(Σ(A,N+1)))=n+1.(18)证毕.充分性. 为简便, 约定m=n+1, 并交替使用m和n+1.由于有限点集Σ(A,N+1)是n维的, 由对称性, 不妨假设Σ(A,n+1)={A0,A1,…,An}是非退化的n维单形, 设Σ(A,n+1)外接超球S的球心为O, 于是可以超球S的球心O 为原点建立直角坐标系, 设点Ak所对应的向量为点An+1所对应的向量为从而(19)这里r(Σ(A,n+1))为超球S的半径. 考虑Σ(A,n+2)={A0,A1,…,An+1}的距离平方矩阵(20)由于rank(M(Σ(A,N+1)))=n+1, 因此易知rank(M(Σ(A,n+2)))≤n+1,(21)于是得(22)从而(23)故(24)对式(24)中的行列式做不改变值变换：将第1行加到第t(t=2,3,…,n+3)行, 并整理得(25)这里: 将式(25)按第1列展开得(26)对式(26)中的第1个行列式做不改变值变换：将第1行的-1倍分别加到第t行(t=2,3,…,n+2), 整理后根据文献[3-8]的结果得(27)于是这里: 又易知(29)故易得(30)从而(31)将式(31)中的第2个行列式按第n+2列展开得根据式(30), 可得(33)考虑对式(33)中的第2个行列式做不改变其值的变换：将第n+2行的-r2倍加到第t(t=1,2,…,n+1)行, 整理得(34)由于Σ(A,n+1)={A0,A1,…,An}是非退化的n维单形, 因此由单形的体积公式[2]知(35)所以可得(36)即表明点An+1在非退化n维单形Σ(A,n+1)={A0,A1,…,An}的外接超球面上, 同理可以依次证明点An+2,An+3,…,AN在非退化的n维单形Σ(A,n+1)的外接超球面上, 从而证明了n维有限点集Σ(A,N+1)={A0,A1,…,AN}在同一个n-1维超球面上. 证毕.下面应用定理1推导Ptolemy定理. 根据定理1, 4点A0,A1,A2,A3共圆的充分必要条件是其距离平方矩阵的秩(37)所以经计算得到又因为(a01a23+a02a13+a03a12)>0, 于是得到式(1).反之, 只要注意得到任意不互相重合的4点A0,A1,A2,A3, 必有(40)由式(38)即可得到式(37), 再应用定理1即可得到4点A0,A1,A2,A3共圆.直接应用定理1可得:推论1 在同一平面上不共线的N+1点A0,A1,…,AN共圆的充要条件是其距离平方矩阵的秩(41)推论2 在三维空间中, 不共面的5点A0,A1,A2,A3,A4共球的充要条件是其距离平方矩阵的秩(42)推论3 在三维空间中, 不共面的N+1(N≥4)点A0,A1,…,AN共球的充要条件是其距离平方矩阵的秩(43)参考文献研究简报【相关文献】[1] Alexander R. The Geometry of Metric and Linear Space [M]. Berlin: Springer-Verlag, 1977.[2] Blumenthal L M. Theory and Applications on Distance Geometry [M]. 2nd ed. New York: Chelsea House Publishing Company, 1970.[3] YANG Lu, ZHANG Jing-zhong. A Class of Geometric Inequalities on Finite Set of Points [J]. Acta Math Sinica, 1980, 23(5)： 740-749. (杨路, 张景中. 关于有限点集的一类几何不等式 [J]. 数学学报, 1980, 23(5)： 740-749.)[4] ZHANG Jing-zhong, YANG Lu. A Class of Geometric Inequalities Concerning a Mass-Point System [J]. Jour China Univ Sci Technol, 1981, 11(2)： 1-8. (张景中, 杨路. 关于质点组的一类几何不等式 [J]. 中国科学技术大学学报, 1981, 11(2)： 1-8.)[5] YANG Lu, ZHANG Jing-zhong. Metric Equations in Geometry and Their Applications [R]. Trieste: Internat Centre for Theory Physics, 1989: 217-219.[6] YANG Ding-hua. Metric Equalities of a Generalized Abstract Distance Space [J]. Journal Zhejiang University: Science Edition, 2010, 37(3): 263-268. (杨定华. 广义抽象距离空间的度量方程 [J]. 浙江大学学报: 理学版, 2010, 37(3): 263-268.)[7] YANG Lu, ZHANG Jing-zhong. The Concept of the Rank of an Abstract Distance Space [J]. Jour China Univ Sci Technol, 1980, 10(4)： 52-65. (杨路, 张景中. 抽象距离空间的秩的概念[J]. 中国科学技术大学学报, 1980, 10(4)： 52-65.)[8] YANG Ding-hua. The Improvement of the Generalized Metric Equations and Applications (Ⅰ) [J]. Jour Sys Sci and Math, 2009, 29(3)： 425-432. (杨定华. 广义度量方程的改进及其应用(Ⅰ) [J]. 系统科学与数学, 2009, 29(3): 425-432.)。