Negative Eigenvalue

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negative eigenvalues

Understanding negative eigenvaluesSometimes, negative eigenvalues are reported in an eigenvalue buckling analysis. In most cases such negative eigenvalues indicate that the structure would buckle if the load were applied in the opposite direction. A classical example is a plate under shear loading; the plate will buckle at the same value for positive and negative applied shear load. Buckling under reverse loading can also occur in situations where it may not be expected. For example, a pressure vessel under external pressure may exhibit a negative eigenvalue (buckling under internal pressure) due to local buckling of a stiffener. Such “physical” negative buckling modes are usually readily understood once they are displayed and can usually be avoided by applying a preload before the buckling analysis.Negative eigenvalues sometimes correspond to buckling modes that cannot be understood readily in terms of physical behavior, particularly if a preload is applied that causes significant geometric nonlinearity. In this case a geometrically nonlinear load-displacement analysis should be performed (“Unstable collapse and postbuckling analysis,”Section 6.2.4).Correcting an overconstrained modelA node set containing all the nodes in the chains of constraints associated with a particular zero pivot is generated automatically and can be displayed in the Visualization module of Abaqus/CAE.There is no unique way to remove the overconstraints in this model. For example, if one JOIN and REVOLUTE (five constraints) combination is replaced with a SLOT connector element, which enforces only the two translation constraints in the plane of the mechanism, there are no redundancies. Alternatively, you could remove the REVOLUTE from one of the connector elements and also use a SLOT connection instead of a JOIN in one of the other connector elements.Another alternative is to relax some of the constraints. In the example outlined here, an elastic body could replace one or more of the rigid bodies. You could also relax the Lagrange multiplier-based constraints (e.g., JOIN or REVOLUTE) by using CARTESIAN and CARDAN connection types with appropriate elastic stiffnesses (see “Connector behavior,”Section 28.2.1).After analyzing the chains of constraints, you have to decide which constraints have to be removed to render the model properly constrained and also best fit the modeling goals. For this example the three constraints cannot be removed randomly. Removing any three combinations of the six boundary conditions, for example, would make the problem worse: the model is still overconstrained, and three rigid body modes have been added to the model. Moreover, you should remove the constraints that do not affect the kinematics of the model. For example, you cannot completely remove a JOIN connection from any ofthe connector elements since the model would be different from that originally intended.。

非负不可约矩阵最大特征值的估计法

非负不可约矩阵最大特征值的估计法王芳芳;杨晋【摘要】To compute bounds for the greatest eigenvalue of nonnegative irreducible matrices,two matrices were construct-ed.The upper and lower bounds for the greatest eigenvalue of nonnegative irreducible matrices were estimated by using Perron-Frobenius theory and sums of rows and columns of constructed matrices.A limit estimation formula was deduced.The results show that the estimated value of the method of estimating the greatest eigenvalue of nonnegative irreducible matrices based on Perron-Frobenius theory is more accurate than the existing conclusion.%为了估计非负不可约矩阵最大特征值的界,构造2个新矩阵,利用Perron-Frobenius定理和新构造矩阵的行和与列和的性质,估计非负不可约矩阵最大特征值的上、下界,并推导极限估计式.结果表明,这种基于Perron-Frobenius定理的估计非负不可约矩阵最大特征值的方法的估计范围比已有结论更精确.【期刊名称】《济南大学学报（自然科学版）》【年(卷),期】2017(031)004【总页数】5页(P334-338)【关键词】非负不可约矩阵;最大特征值;估计【作者】王芳芳;杨晋【作者单位】太原理工大学数学学院, 山西太原030024;太原理工大学数学学院, 山西太原030024【正文语种】中文【中图分类】O151.21非负不可约矩阵理论一直是线性代数中最活跃的研究领域之一，在数值分析、计算机科学、动态规划以及人口统计学等领域中具有重要的应用价值。

数学专业英语词汇（N）

数学专业英语词汇(N)n ary relation n元关系n ball n维球n cell n维胞腔n chromatic graph n色图n coboundary n上边缘n cocycle n上循环n connected space n连通空间n dimensional column vector n维列向量n dimensional euclidean space n维欧几里得空间n dimensional rectangular parallelepiped n维长方体n dimensional row vector n维行向量n dimensional simplex n单形n dimensional skeleton n维骨架n disk n维圆盘n element set n元集n fold extension n重扩张n gon n角n graph n图n homogeneous variety n齐次簇n person game n人对策n simplex n单形n sphere bundle n球丛n th member 第n项n th partial quotient 第n偏商n th power operation n次幂运算n th root n次根n th term 第n项n times continuously differentiable n次连续可微的n times continuously differentiable function n次连续可微函数n tuple n组n tuply connected domain n重连通域n universal bundle n通用丛nabla 倒三角算子nabla calculation 倒三角算子计算nabla operator 倒三角算子napier's logarithm 讷代对数natural boundary 自然边界natural boundary condition 自然边界条件natural coordinates 自然坐标natural equation 自然方程natural equivalence 自然等价natural exponential function 自然指数函数natural frequency 固有频率natural geometry 自然几何natural injection 自然单射natural isomorphism 自然等价natural language 自然语言natural logarithm 自然对数natural mapping 自然映射natural number 自然数natural oscillation 固有振荡natural sine 正弦真数natural transformation 自然变换naught 零necessary and sufficient conditions 必要充分的条件necessary and sufficient statistic 必要充分统计量necessary condition 必要条件necessity 必然性negation 否定negation sign 否定符号negation symbol 否定符号negative 负数negative angle 负角negative binomial distribution 负二项分布negative complex 负复形negative correlation 负相关negative definite form 负定型negative definite hermitian form 负定埃尔米特形式negative definite quadratic form 负定二次形式negative function 负函数negative number 负数negative operator 负算子negative parity 负电阻negative part 负部分negative particular proposition 否定特称命题negative proposition 否定命题negative rotation 反时针方向旋转negative semidefinite 半负定的negative semidefinite eigenvalue problem 半负定特盏问题negative semidefinite form 半负定型negative semidefinite quadratic form 半负定二次形式negative sign 负号negative skewness 负偏斜性negative variation 负变差negligible quantity 可除量neighborhood 邻域neighborhood base 邻域基neighborhood basis 邻域基neighborhood filter 邻域滤子neighborhood retract 邻域收缩核neighborhood space 邻域空间neighborhood system 邻域系neighborhood topology 邻域拓扑neighboring vertex 邻近项点nephroid 肾脏线nerve 神经nested intervals 区间套net 网net function 网格函数net of curves 曲线网net of lines 直线网network 网络network analysis 网络分析network flow problem 网络潦题network matrix 网络矩阵neumann boundary condition 诺伊曼边界条件neumann function 诺伊曼函数neumann problem 诺伊曼问题neumann series 诺伊曼级数neutral element 零元素neutral line 中线neutral plane 中性平面neutral point 中性点newton diagram 牛顿多边形newton formula 牛顿公式newton identities 牛顿恒等式newton interpolation polynomial 牛顿插值多项式newton method 牛顿法newton potential 牛顿位势newtonian mechanics 牛顿力学nice function 佳函数nil ideal 零理想nil radical 幂零根基nilalgebra 幂零代数nilpotency 幂零nilpotent 幂零nilpotent algebra 幂零代数nilpotent element 幂零元素nilpotent group 幂零群nilpotent ideal 幂零理想nilpotent matrix 幂零矩阵nilpotent radical 幂零根基nine point circle 九点圆nine point finite difference scheme 九点有限差分格式niveau line 等位线niveau surface 等位面nodal curve 结点曲线nodal line 交点线nodal point 节点node 节点node locus 结点轨迹node of a curve 曲线的结点noetherian category 诺特范畴noetherian object 诺特对象nomogram 算图nomographic 列线图的nomographic chart 算图nomography 图算法non additivity 非加性non archimedean geometry 非阿基米德几何non archimedean valuation 非阿基米德赋值non countable set 不可数集non critical point 非奇点non denumerable 不可数的non denumerable set 不可数集non developable ruled surface 非可展直纹曲面non enumerable set 不可数集non euclidean geometry 非欧几里得几何学non euclidean motion 非欧几里得运动non euclidean space 非欧几里得空间non euclidean translation 非欧几里得平移non euclidean trigonometry 非欧几里得三角学non homogeneity 非齐non homogeneous chain 非齐次马尔可夫链non homogeneous markov chain 非齐次马尔可夫链non isotropic plane 非迷向平面non linear 非线性的non negative semidefinite matrix 非负半正定阵non oriented graph 无向图non parametric test 无分布检验non pascalian geometry 非拍斯卡几何non ramified extension 非分歧扩张non rational function 无理分数non relativistic approximation 非相对性近似non reversibility 不可逆性non singular 非奇的non stationary random process 不平稳随机过程non steady state 不稳定状态non symmetric 非对称的non symmetry 非对称non zero sum game 非零和对策nonabsolutely convergent series 非绝对收敛级数nonagon 九边形nonassociate 非结合的nonassociative ring 非结合环nonbasic variable 非基本变量noncentral chi squre distribution 非中心分布noncentral f distribution 非中心f分布noncentral t distribution 非中心t分布noncentrality parameter 非中心参数nonclosed group 非闭群noncommutative group 非交换群noncommutative ring 非交换环noncommutative valuation 非交换赋值noncommuting operators 非交换算子noncomparable elements 非可比元素nondegeneracy 非退化nondegenerate collineation 非退化直射变换nondegenerate conic 非退化二次曲线nondegenerate critical point 非退化临界点nondegenerate distribution 非退化分布nondegenerate set 非退化集nondense set 疏集nondenumerability 不可数性nondeterministic automaton 不确定性自动机nondiagonal element 非对角元nondiscrete space 非离散空间nonexistence 不存在性nonfinite set 非有限集nonholonomic constraint 不完全约束nonhomogeneity 非齐性nonhomogeneous 非齐次的nonhomogeneous linear boundary value problem 非齐次线性边值问题nonhomogeneous linear differential equation 非齐次线性微分方程nonhomogeneous linear system of differential equations 非齐次线性微分方程组nonisotropic line 非迷向线nonlimiting ordinal 非极限序数nonlinear equation 非线性方程nonlinear functional analysis 非线性泛函分析nonlinear lattice dynamics 非线性点阵力学nonlinear operator 非线性算子nonlinear optimization 非线性最优化nonlinear oscillations 非线性振动nonlinear problem 非线性问题nonlinear programming 非线性最优化nonlinear restriction 非线性限制nonlinear system 非线性系统nonlinear trend 非线性瞧nonlinear vibration 非线性振动nonlinearity 非线性nonlogical axiom 非逻辑公理nonlogical constant 非逻辑常数nonmeager set 非贫集nonmeasurable set 不可测集nonnegative divisor 非负除数nonnegative number 非负数nonnumeric algorithm 非数值的算法nonorientable contour 不可定向周线nonorientable surface 不可定向的曲面nonorthogonal factor 非正交因子nonparametric confidence region 非参数置信区域nonparametric estimation 非参数估计nonparametric method 非参数法nonparametric test 非参数检定nonperfect set 非完备集nonperiodic 非周期的nonperiodical function 非周期函数nonplanar graph 非平面图形nonprincipal character 非重贞nonrandom sample 非随机样本nonrandomized test 非随机化检验nonrational function 非有理函数nonremovable discontinuity 非可去不连续点nonrepresentative sampling 非代表抽样nonresidue 非剩余nonsense correlation 产生错觉相关nonsingular bilinear form 非奇双线性型nonsingular curve 非奇曲线nonsingular linear transformation 非退化线性变换nonsingular matrix 非退化阵nonspecial group 非特殊群nonstable 不稳定的nonstable homotopy group 非稳定的同伦群nonstandard analysis 非标准分析nonstandard model 非标准模型nonstandard numbers 非标准数nonsymmetric relation 非对称关系nonsymmetry 非对称nontangential 不相切的nontrivial element 非平凡元素nontrivial solution 非平凡解nonuniform convergence 非一致收敛nonvoid proper subset 非空真子集nonvoid set 非空集nonzero vector 非零向量norm 范数norm axioms 范数公理norm form 范形式norm of a matrix 阵的范数norm of vector 向量的模norm preserving mapping 保范映射norm residue 范数剩余norm residue symbol 范数剩余符号norm topology 范拓朴normability 可模性normal 法线normal algorithm 正规算法normal basis theorem 正规基定理normal bundle 法丛normal chain 正规链normal cone 法锥面normal congruence 法汇normal coordinates 正规坐标normal correlation 正态相关normal curvature 法曲率normal curvature vector 法曲率向量normal curve 正规曲线normal density 正规密度normal derivative 法向导数normal dispersion 正常色散normal distribution 正态分布normal distribution function 正态分布函数normal equations 正规方程normal error model 正规误差模型normal extension 正规开拓normal family 正规族normal force 法向力normal form 标准型normal form problem 标准形问题normal form theorem 正规形式定理normal function 正规函数normal homomorphism 正规同态normal integral 正规积分normal linear operator 正规线性算子normal mapping 正规映射normal matrix 正规矩阵normal number 正规数normal operator 正规算子normal order 良序normal plane 法面normal polygon 正规多角形normal polynomial 正规多项式normal population 正态总体normal probability paper 正态概率纸normal process 高斯过程normal sequence 正规序列normal series 正规列normal set 良序集normal simplicial mapping 正规单形映射normal solvable operator 正规可解算子normal space 正规空间normal surface 法曲面normal tensor 正规张量normal to the surface 曲面的法线normal valuation 正规赋值normal variate 正常变量normal variety 正规簇normal vector 法向量normality 正规性normalization 标准化normalization theorem 正规化定理normalize 正规化normalized basis 正规化基normalized function 规范化函数normalized variate 正规化变量normalized vector 正规化向量normalizer 正规化子normalizing factor 正则化因数normed algebra 赋范代数normed linear space 赋范线性空间normed space 赋范线性空间northwest corner rule 北午角规则notation 记法notation free from bracket 无括号记号notation of backus 巴科斯记号notion 概念nought 零nowhere convergent sequence 无处收敛序列nowhere convergent series 无处收敛级数nowhere dense 无处稠密的nowhere dense set 无处稠密点集nowhere dense subset 无处稠密子集nuclear operator 核算子nuclear space 核空间nucleus of an integral equation 积分方程的核null 零null class 零类null divisor 零因子null ellipse 零椭圆null function 零函数null hypothesis 虚假设null line 零线null matrix 零矩阵null method 衡消法null plane 零面null point 零点null ray 零射线null relation 零关系null representation 零表示null sequence 零序列null set 空集null solution 零解null system 零系null transformation 零变换null vector 零向量nullity 退化阶数nullring 零环nullspace 零空间number 数number defined by cut 切断数number defined by the dedekind cut 切断数number field 数域number interval 数区间number line 数值轴number notation 数记法number of partitions 划分数number of repetitions 重复数number of replications 重复数number of sheets 叶数number sequence 数列number set 数集number system 数系number theory 数论number variable 数变量numeration 计算numerator 分子numeric representation of information 信息的数值表示numerical 数值的numerical algorithm 数值算法numerical axis 数值轴numerical calculation 数值计算numerical coding 数值编码numerical coefficient 数字系数numerical computation 数值计算numerical constant 数值常数numerical data 数值数据numerical determinant 数字行列式numerical differentiation 数值微分numerical equality 数值等式numerical equation 数字方程numerical error 数值误差numerical example 数值例numerical function 数值函数numerical inequality 数值不等式numerical integration 数值积分法numerical invariant 不变数numerical mathematics 数值数学numerical method 数值法numerical model 数值模型numerical operator 数字算子numerical quadrature 数值积分法numerical series 数值级数numerical solution 数值解numerical solution of linear equations 线性方程组的数值解法numerical stability 数值稳定性numerical table 数表numerical value 数值numerical value equation 数值方程nutation 章动。

The algebraic eigenvalue problem代数特征值问题

CLARENDON PRESS • OXFORD 1965
Contents
1. THEORETICAL BACKGROUND Page
Introduction Definitions Eigenvalues and eigenvectors of the transposed matrix Distinct eigenvalues Similarity transformations Multiple eigenvalues and canonical forms for general matrices Defective system of eigenvectors The Jordan (classical) canonical form The elementary divisors Companion matrix of the characteristic polynomial of A Non-derogatory matrices The Frobenius (rational) canonical form Relationship between the Jordan and Frobenius canonical forms Equivalence transformations Lambda matrices Elementary operations Smith's canonical form The highest common factor offc-rowedminors of a A-matrix Invariant factors of (A —XI) The triangular canonical form Hermitian and symmetric matrices Elementary properties of Hermitian matrices Complex symmetric matrices Reduction to triangular form by unitary transformations Quadratic forms Necessary and sufficient conditions for positive definiteness Differential equations with constant coefficients Solutions corresponding to non-linear elementary divisors Differential equations of higher order Second-order equations of special form Explicit solution of By = —Ay Equations of the form (AB— XI)x — 0 The minimum polynomial of a vector The minimum polynomial of a matrix Cayley-Hamilton theorem Relation between minimum polynomial and canonical forms Principal vectors Elementary similarity transformations Properties of elementary matrices Reduction to triangular canonical form by elementary similarity transformations Elementary unitary transformations Elementary unitary Hermitian matrices Reduction to triangular form by elementary unitary transformations Normal matrices Commuting matrices

几类非负矩阵特征值反问题

The Inverse Eigenvalue Problem for Several Classes of Nonnegative MatricesA DissertationSubmitted for the Degree of MasterOn computational mathematicsby Tian YuUnder the Supervision ofProf. Wang Jinlin(College of Mathematics and Information Sciences)Nanchang Hangkong University, Nanchang, ChinaJune, 2011摘要非负矩阵理论一直是矩阵理论中最活跃的研究领域之一，在数学、自然科学的其他分支以及社会科学中都广泛涉及到,例如博弈论、Markov链（随机矩阵）、概率论、概率算法、数值分析、离散分布、群论、matrix scaling、小振荡弹性系统（振荡矩阵）和经济学等等.近年来，特征值反问题是矩阵理论研究的热点，本文将就非负矩阵特征值反问题（NIEP）这一问题进行研究.文章主要研究几类特殊形式的非负矩阵特征值反问题，得到了相关问题的充分必要条件和一些充分条件，进而给出了这几类特殊形式的非负矩阵特征值反问题数值算法，并通过数值算例来验证相关定理的正确性以及算法的准确性.主要工作如下：第一章是绪论部分，阐述了非负矩阵特征值反问题的重要意义和发展历程，介绍国内外研究现状.第二章，研究非负三对角矩阵特征值反问题.首先对三阶非负三对角矩阵特征值反问题，分几种情形进行讨论，解决了三阶非负三对角矩阵特征值反问题，得到了三阶非负三对角矩阵特征值反问题有解的充分必要条件.然后对n阶非负三对角矩阵特征值反问题，通过非负三对角矩阵截断矩阵特征多项式，并结合Jacobi 矩阵特征值的关系，得到了非负三对角矩阵的特征值的相关性质，并最终解决了非负三对角矩阵特征值反问题.第三章，研究非负五对角矩阵特征值反问题.三阶非负五对角矩阵，即是三阶非负矩阵，文中给出了其特征值反问题有解的充分必要条件，而对于n阶非负五对角矩阵特征值反问题，由于其复杂性，文中仅给出了它的一些充分条件.第四章，研究非负循环矩阵特征值反问题.首先总结了NIEP近些年来取得的研究成果，提出实循环矩阵特征值反问题，并成功解决了实循环矩阵特征值反问题，得到其充分必要条件.最后在实循环矩阵特征值反问题的基础上提出非负循环矩阵特征值反问题，得到了充分条件和相关推论.第五章，根据第二、三、四章的结论给出相关算法和实例.第六章，在总结全文的同时，提出了需要进一步研究的问题.关键词：特征值，反问题，非负三对角矩阵，非负五对角矩阵，非负循环矩阵AbstractThe theory of nonnegative matrices has always been one of the most active research areas in the matrix theory and has been widely applied in mathematics and other branches of natural and social sciences. There are, for example, game theory, Markov chains (stochastic martices), theory of probability, probabilistic algorithms, numerical analysis, discrete distribution, group theory, matrix scaling, theory of small osillations of elastic systems (oscillation marrices), economics and so on. In recent years, the inverse eigenvalue problem comes to be the focus of the matrix theory. This thesis will study the inverse eigenvalue problem for nonnegative matrices (NIEP). The major researches of this theisis focus on the inverse eigenvalue problem for several special classes of nonnegative matrices, the necessary and sufficient conditions and some sufficient conditions of which are derived. Moreover, the numerical algorithms of the inverse eigenvalue problem for these special classes of nonnegative matrices are given, the accuracy of which together with the correcteness of related theories is testified by several numerical examples. The main procedures of this theisis are as follows:In the first chapter, the significance and the development of the inverse eigenvalue problem for nonnegative matrices are addressed, and the research situation home and abroad is introduced.In the second chapter, the inverse eigenvalue problem for nonnegative tridiagonal matrices is studied. First, the inverse eigenvalue problem for 33⨯ nonnegative tridiagonal matrices is solved by discussion of a variety of situations. Moveover ，the necessary and sufficient conditions of the solutions of the inverse eigenvalue problem for 33⨯ nonnegative tridiagonal matrices are derived. Then, the properties of eigenvalue of n n ⨯ nonnegative tridiagonal matrices are derived by characteristic polynomial of truncated matrices of nonnegative tridiagonal matrices, with the combination of the relationship between eigenvalues of Jacobi matrix. Finally, the inverse eigenvalue problem for nonnegative tridiagonal matrices is solved.In the third chapter, the inverse eigenvalue problem for nonnegative five-diagonal matrices is studied. 33⨯ nonnegative five-diagonal matrices is also 33⨯ nonnegative matrices, the necessary and sufficient conditions of the solutions of the inverse eigenvalue problem for which are given in this thesis. For the inverseeigenvalue problem for n nnonnegative five-diagonal matrices, only some sufficient conditions are given because of its complexity.In the fourth chapter, the inverse eigenvalue problem for nonnegative circulant matrices is studied. First, some remarkable conclusions of the inverse eigenvalue problem for nonnegative matrices in recent years are summarized. Then, the inverse eigenvalue problem for real circulant matrices is advanced and successfully solved, the necessary and sufficient conditions of which are given also. Finally, the inverse eigenvalue problem for nonnegative circulant matrices is advanced based on the inverse eigenvalue problem for real circulant matrices, whose sufficient conditions and some relevant conclusions are given.In the fifth chapter, some algorithms and numerical examples are given based on the conclusions derived in the previous three chapters.In the sixth chapter, the summary of the paper is given and the future research work is put forward.Key words: eigenvalue, inverse problem, nonnegative tridiagonal matrices, nonnegative five-diagonal matrices, nonnegative circulant matrices目录第一章绪论 (1)1.1选题的依据与意义 (1)1.2非负矩阵特征值反问题的研究现状 (2)1.3研究的主要内容 (3)第二章非负三对角矩阵特征值反问题 (5)2.1引言 (5)2.2三阶非负三对角矩阵特征值反问题 (6)n阶非负三对角矩阵特征值反问题 (24)2.3第三章非负五对角矩阵特征值反问题 (33)3.1引言 (33)3.2非负五对角矩阵特征值反问题相关结论 (33)第四章非负循环矩阵特征值反问题 (38)4.1引言 (38)4.2一类特殊矩阵的特征值反问题 (40)4.3非负循环矩阵特征值反问题 (42)第五章算法设计及实例 (45)5.1非负三对角矩阵特征值反问题算法 (45)5.2非负五对角矩阵特征值反问题算法 (47)5.3实循环矩阵特征值反问题算法 (49)5.4非负循环矩阵特征值反问题算法 (50)第六章总结与展望 (53)6.1全文总结 (53)6.2工作展望 (53)参考文献 (54)攻读硕士学位期间发表的论文 (57)致谢 (58)IV第一章绪论1.1 选题的依据与意义反问题，顾名思义是相对于正问题而言的，它是根据事物的演化结果，由可观测的现象来探求事物的内部规律或所受的外部影响，由表及里，索隐探秘.在数学中有着许多反问题，例如已知两个自然数的乘积，如何求这两个自然数；已知导数，如何求原函数；已知一个角的三角函数值，如何求这个角的度数，等等.近些年来，人们在生活、工业生产、科学探索中经常遇到反问题，对反问题的研究也越来越受到重视.事实上，对于一般问题来说，反问题要比正问题复杂.如前面提到的求角度数问题，已知一个角，求其三角函数值是唯一的，但如果只知道一个角的三角函数值而不对这个角加以约束，这样的角将会有无穷多个，因而反问题的解一般来说不唯一.另外，反问题的解也极不稳定.因此，对反问题的研究主要包括以下几个方面：存在性、唯一性、稳定性、数值方法和实际应用.矩阵特征值反问题（又称代数特征值反问题或逆特征值问题），就是根据已给定的特征值和/或特征向量等信息来确定一个矩阵，使得该矩阵满足所给的条件.矩阵特征值反问题的来源非常广泛.它不仅来自于数学物理反问题的离散化，而且来自固体力学、粒子物理、量子物理、结构设计、系统参数识别、自动控制等许多领域.由于矩阵特征值反问题的应用广泛性，因而自从此类问题被提出来的几十年里，受到了大量学者的深入研究，得到了一系列优秀成果.本文研究的非负矩阵特征值反问题正是在此期间提出来的，它作为矩阵特征值反问题的一个重要分支，尤其是在概率统计、随机分布、系统分析方面有着重要应用.所谓非负矩阵特征值反问题就是根据已给定的特征值信息来确定一个非负矩阵，使得该非负矩阵满足所给的条件.例如在概率统计中提出一类随机矩阵（矩阵的元素行和为1），这类矩阵在Markov链中有着重要应用，假如对矩阵的特征值有某些特殊要求，能否构造和如何构造出此类矩阵？非负矩阵特征值反问题从提出到现在的几十年间，虽然受到了大量学者的研究，但由于其复杂性，目前仍存在大量的疑难问题尚未解决，这也是它吸引众多学者研究的魅力所在.因而从以上可以看出对非负矩阵特征值的研究无论是对数学本身的发展还是对其它科学的发展都有着重要的意义及广阔的前景.1.2 非负矩阵特征值反问题的研究现状非负矩阵特征值反问题的提出始于上个世纪50年代，它是由矩阵特征值反问题抽离出来的一个子问题.1937年，Kolmogorov [1]首先提出了给定一个复数z 何时为某个非负矩阵特征值的问题.1949年，Suleimanova [2]扩展了Kolmogorov 提出的问题，称为非负矩阵特征值反问题（简称NIEP），即寻找以一组复数12{,,,}n σλλλ= 为特征值的n 阶非负矩阵A ，并且假若能够找到这样一个矩阵A ，就说矩阵A 实现了σ.Kolmogorov 问题显然很容易回答，Minc [3]给出了解答，即对于33⨯阶正循环矩阵，总可以找到一个这样的矩阵使得给定的复数z 作为它的特征值.然而NIEP 从提出至今仍未得到很好地解决，为此一些学者首先从NIEP 的必要条件开始研究.Loewy 和Londow [4]、Johnson [5]给出文献[6]中NIEP 的四个必要条件，其中最后一个条件称为JLL 条件.1998年，Laffey 和Meehan [7]又对奇数阶非负矩阵进行了讨论，给出了奇数阶非负矩阵迹为零的JLL 条件.由于一般的n 阶NIEP 无法直接解答，一批学者考虑了低阶矩阵的情形.1978年，Loewy 和Londow [4]完全解决3n =时的NIEP，给出了四个充分必要条件.45n =、时的NIEP，目前只解决了迹为零的情形.1996年，Reams [8]解决了4n =时迹为零的情形，即：令1234{,,,}σλλλλ=为一组复数，假若120,0,S S =≥30S ≥和2244S S ≤(这里的41k k i i S λ==∑)，则必存在一个4阶非负矩阵能够实现σ.1999年，Laffey 和Meehan [9]解决了5n =时迹为零的情形.上面介绍了6n <的情形，然而当6n ≥时，NIEP 却是一个极大地挑战，到目前为止，未见任何形式的解答.虽然NIEP 未曾从正面给出很好的解答，但却吸引大批学者对12{,,}n σλλλ= ，的特殊形式作出深入探讨，这其中包括H.Suleimanova、H.Perfect、R.Kellogg、Salzman、Guo Wuwen 等等.Suleimanova [2]证明0(2,3,,)i i n λ≤= 的σ可被实现的充分必要条件是10ni i λ=≥∑.Kellogg [10]对σ的序列进行分块研究，给出了某些符合要求的分块可被实现.Guo Wuwen [11-12]对已可实现的σ修正做了研究，其中修正后可被实现与σ中最大的数有着密切关系，文献[12]定理3.1的结论尤为重要，它在研究扩展σ可被实现中被广泛引用.另外值得一提的是Ricardo.Soto、Alberto.Borobia、Julio.Moro 三位近十年来在非负矩阵特征值反问题上做了大量深入的研究，文献[13-18]集中反映了他们在这一块的研究成果.上面介绍了NIEP，如果把上面的非负矩阵换成非负对称矩阵，则称为非负对称矩阵特征值反问题（简称SNIEP）；如果把上面的一组复数12{,,}n σλλλ= ，换成一组实数，则称为非负矩阵实特征值反问题（简称RNIEP）.SNIEP和RNIEP都是NIEP的子问题，它们是研究NIEP的重要组成部分，虽然两者研究的都是实特征值，但它们并不完全等价.一般地，当5n≥时，这是两个完全不同的问题.目前，当4n≤时，SNIEP已被完全解决，当5n=时，R.Loewy和J.J.Mcdonald在文献[9]中做了详细的讨论.而当6n≥时，尚无人解决.文献[19-21]给出了SNIEP的相关结论.4n=时的RNIEP已被解决，事实上，Loewy和Londow在文献[4]中给出的NIEP四个必要条件也是4阶RNIEP的充分条件.当5n≥时，目前尚未有所突破.另外，文献[2,10,22,23,24,25]给出了RNIEP的相关结论.随机矩阵和双随机矩阵作为非负矩阵的两种特殊形式，在研究NIEP中有着极为重要的应用，这里把它们归为一类问题，即随机和双随机矩阵特征值反问题.Johnson[26]证明了如果一个非负矩阵A有正Perron根ρ，则存在一个随机矩阵与1Aρ同谱.1981年，Soules[27]给出了一种构造对称双随机矩阵的方法并得到构造对称双随机矩阵的充分条件.以上是NIEP的主要研究的方向，由于NIEP的复杂性和作者的水平限度，可能衍生出更多的小问题，本文没有一一涉及到，在此后面将不再叙述.此外，由于NIEP研究不够成熟，关于它的数值计算目前研究的不多.Robert.Orsi[28]利用交错射影的思想构造出一种迭代方法来计算非负矩阵特征值反问题，但需指出的是这种迭代并不一定会能得出好的结果，仍需要找到好的判定条件.O.Rojo等在文献[29-30]中通过快速Fourier变化巧妙地得到一种构造对称非负矩阵的方法，大大节省计算时间，这种方法通过在Matlab上实现，证明效率是非常高的.目前，国内尚无对此方面的研究的相关文献.从以上可以看出，虽然非负矩阵特征值反问题的研究得到了一定的成果，但仍有大量的问题需要解决，本文将从几类特殊矩阵来探讨此类问题，进一步促进此方向的研究.例如：能否给出非负（对称）三对角矩阵的特征值反问题的充要条件以及如何实现？如何实现非负循环矩阵的特征值反问题？等等.1.3 研究的主要内容本文研究几类特殊形式的非负矩阵特征值反问题，得到了相关问题的充分必要条件和一些充分条件，进而给出这几种特殊形式的非负矩阵特征值反问题算法，并通过数值算例来验证相关定理的正确性以及算法的准确性.主要工作如下：第一章是绪论部分，阐述了非负矩阵特征值反问题的重要意义和发展历程，介绍国内外研究现状.第二章，研究非负三对角矩阵特征值反问题.首先对三阶非负三对角矩阵特征值反问题，分几种情形进行讨论，解决了三阶非负三对角矩阵特征值反问题，得到了三阶非负三对角矩阵特征值反问题有解的充分必要条件.然后对n阶非负三对角矩阵特征值反问题，通过非负三对角矩阵截断矩阵特征多项式，并结合Jacobi矩阵特征值的关系，得到了非负三对角矩阵的特征值的相关性质，并最终解决了非负三对角矩阵特征值反问题.第三章，研究非负五对角矩阵特征值反问题.三阶非负五对角矩阵，即是三阶非负矩阵，文中给出了其特征值反问题有解的充分必要条件，而对于n阶非负五对角矩阵特征值反问题，由于其复杂性，文中仅给出了它的一些充分条件.第四章，研究非负循环矩阵特征值反问题.首先总结了NIEP近些年来取得的研究成果，提出实循环矩阵特征值反问题，并成功解决了实循环矩阵特征值反问题，得到其充分必要条件.最后在实循环矩阵特征值反问题的基础上提出非负循环矩阵特征值反问题，得到了充分条件和相关推论.第五章，根据第二、三、四章的结论给出相关算法和实例.第六章，在总结全文的同时，提出了需要进一步研究的问题.南昌航空大学硕士学位论文第二章非负矩阵特征值反问题第二章非负三对角矩阵特征值反问题2.1 引言在控制论、振动理论、结构设计中经常要求根据已给的特征值/或特征向量来构造矩阵，即是特征值反问题（或特征值逆问题）.三对角矩阵作为一类特殊矩阵，在实际问题中常出现，是研究矩阵理论的一个重要方面，因而有必要对其特征值反问题进行研究.文章的引言部分已给出了非负矩阵特征值反问题的研究现状，可以看出对于非负三对角矩阵的特征值反问题一直缺乏研究，本章将对这一问题进行研究.首先给出如下定义.定义 2.1.1 设n 阶实三对角矩阵形式如下：11112211100n n n n n n x y z x y T z x y z x ----⎡⎤⎢⎥⎢⎥⎢⎥=⎢⎥⎢⎥⎢⎥⎣⎦. （1）若0(1,2,,)i i y z i n =>= ，则称n T 为Jacobi 矩阵；（2）若0,0,0i i i x y z ≥≥≥，则称n T 为非负三对角矩阵；（3）若0,0i i i x y z ≥=≥，则称n T 为非负对称三对角矩阵；若0,0i i i x y z ≥=>，则称n T 为非负Jacobi 矩阵.非负三对角矩阵特征值反问题：给定一组复数12{,,,}n σλλλ= ，寻找非负三对角矩阵A 以σ为特征值，并且假设能够找到这样一个矩阵，就说矩阵A 实现了σ.下面再给出两个引理.引理 2.1.1[31](广义Perron 定理) 设A 是一个n n ⨯阶非负矩阵.定义Perron 根如下：()max{:()}A A ρλλσ=∈.则()A ρ为A 的特征值，并且其相应的特征向量0x ≥（即向量x 的每个元素均大于等于零）.引理 2.1.2[4] 设123{,,}σλλλ=是一个由复数构成的序列，并且假设σ满足如下条件：（i）13max{:}i i i λλσσ≤≤∈∈；（ii）σσ=；（iii）11230s λλλ=++≥；（iv）2123s s ≤.则σ能被一个非负矩阵A 实现.2.2 三阶非负三对角矩阵特征值反问题设12{,,,}n σλλλ= 是一个由n 个复数构成的序列，文献[6]给出由Loewy 和Londow [4]、Johnson [5]得到的NIEP 四个必要条件，显然这四个条件对非负三对角矩阵特征值反问题也适用，即（i）Perron 根max{:}i i ρλλσσ=∈∈；（ii）σσ=；（iii）定义1(1,2,)nk k i i s k λ===∑ ，则有0k s ≥；（iv）（JLL 条件）1(,1,2,)m m kk m s n s k m -≤= .二阶非负矩阵特征值反问题有如下结论.引理2.2.1 给定两个数12,λλ，则12{,}σλλ=可以被非负矩阵实现的充分必要条件是12,λλ均为实数（不妨设12λλ≥）并且12λλ≥.证明首先可以证明这两个数是实数.实矩阵的特征值如果是复数（虚部不为零），则会以共轭对的形式出现，不妨将12,λλ设为,(x yi x yi i +-=.假设σ可以被实现，则存在一个非负矩阵A 以12,λλ为特征值.令非负矩阵a c A d b ⎡⎤=⎢⎥⎣⎦（,,,a b c d 均大于等于零），则2()acI A a b ab cd d bλλλλλ---==-++---. （2-1）由式（2-1）知，有1220a b x λλ+=+=≥，（2-2） 2212ab cd x y λλ=-=+. （2-3）由式（2-2）中20a b x +=≥，根据均值不等式的关系知ab 的最大值为2x .而由式（2-3）有222ab x y cd x =++≥，显然当12,λλ是复数时，20,y ab x ≠>，矛盾.故12,λλ不可能是复数.充分性.当12λλ≥时，可以分为两种情形讨论即20λ≥和20λ<.而120λλ==时，显然可以被零矩阵实现.当20λ≥时，σ可以被1200λλ⎡⎤⎢⎥⎣⎦实现.当20λ≤时，可以取定,a b 均大于等于零使得式（2-2）成立，这时120ab cd λλ=-≤，显然可以取无数个均大于等于零,c d 使得式（2-3）成立.这样就存在一个矩阵a c d b ⎡⎤⎢⎥⎣⎦实现σ. 必要性.由于12λλ≥，故只需证12λλ<时，σ不能被现实即可.当12λλ<时，由式（2-2）有120a b λλ+=+<，而,a b 均大于等于零，矛盾.证毕.引理 2.2.2 给定三个实数123,,λλλ，如果123(2,3),i i λλλλ≥=≥和1230λλλ++≥，则123{,,}σλλλ=可被非负矩阵A 实现.证明分三种情形讨论.当0(1,2,3)i i λ≥=时，令123000000A λλλ⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦，则A 可实现σ.当1230λλλ≥≥≥时，令13131313202202200A λλλλλλλλλ+-⎡⎤⎢⎥⎢⎥-+⎢⎥=⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦，则A 可实现σ. 当1230λλλ≥≥≥时，令1231231231230A λλλλλλλλλλλλ⎡+++-⎢=+-++⎢⎢⎢⎥⎣⎦，则A 可实现σ.证毕.定理 2.2.3 给定一组实数12{,,,}n σλλλ= 12()n λλλ≥≥≥ ，1n 表示其中0(1,2,,)i i n λ>= 的个数，2n 表示0(1,2,,)i i n λ<= 的个数.如果12n n ≥且120(1,2,,)i n i i n λλ+-+≥= ，则12{,,,}n σλλλ= 可以被非负三对角矩阵实现.证明由引理2.2.1知120(1,2,,)i n i i n λλ+-+≥= 时，1{,}(1,2,,i n i i σλλ+-==2)n 可以被一个二阶非负矩阵2(1,2,,)i A i n = 实现，而2210(1,2,,i i n n n λ≥=++)，则22112{,,,}n n n σλλλ++= 可以被非负三对角矩阵22112{,,,}n n n diag λλλ++ 实现.因而12{,,}n σλλλ= ，可以被非负三对角矩阵22211212{,,,,,,,,n n n n diag A A A λλλ++ 12}n n n --0实现，其中12n n n --0表示12n n n --阶零矩阵.证毕.推论 2.2.4 给定一组实数12{,,,}n σλλλ= 12()n λλλ≥≥≥ ，1n 表示其中0(1,2,,)i i n λ>= 的个数，2n 表示0(1,2,,)i i n λ<= 的个数，11{1,2,,}n Γ= 对应的正特征值112222,,,,{1,2,,}n n n n n n λλλΓ=-+-+ 对应的负特征值2212,,,n n n n n λλλ-+-+ .如果12n n ≥，对于2Γ中的每个数j 都能在1Γ中找到一个数i 使得1220(1,2,,,1,2,,)i j i n j n n n n n λλ+≥==-+-+ 且每个i 对应一个j ，则12{,,,}n σλλλ= 可以被非负三对角矩阵实现.推论 2.2.5 给定一组实数12{,,,}n σλλλ= 12()n λλλ≥≥≥ ，1n 表示其中0(1,2,,)i i n λ>= 的个数，2n 表示0(1,2,,)i i n λ<= 的个数，11{1,2,,}n Γ= 对应的正特征值112,,,n λλλ ，222{1,2,,}n n n n n Γ=-+-+ 对应的负特征值2212,,,n n n n n λλλ-+-+ .如果12n n ≥，对于2Γ中的每个数j 都能在1Γ中找到一个数i 使得1220(1,2,,,1,2,,)i j i n j n n n n n λλ+≥==-+-+ 且每个i 对应一个j ，则12{,,,}n σλλλ= 可以被非负对称三对角矩阵实现.定理2.2.6 给定一组实数123123{,,}()σλλλλλλ=≥≥，如果3(1,2)i i λλ≥=和1230λλλ++>，假若123{,,}σλλλ=能被非负三对角矩阵111222300a b A c a b c a ⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦实现，则A 中的13,a a 均不能为零.证明设非负三对角矩阵111222300a b A c a b c a ⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦能够实现123{,,}σλλλ=，即123,,λλλ是矩阵A 的三个特征值.首先给出矩阵A 的特征多项式.11122231232211133212312132312322111332123121323112212312231100()()()()()()()()()()()a b I A c a b c a a a a b c a b c a a a a a a a a a a a a a b c a b c a a a a a a a a a a b c b c a a a a b c a b c λλλλλλλλλλλλλλλλλ---=-----=-------=-+++++-----=-+++++---++.由根与系数的关系知，有下列成立，123123a a a λλλ++=++，（2-4） 1213231213231122a a a a a a b c b c λλλλλλ++=++--，（2-5）123123122311a a a a b c a b c λλλ=--. （2-6）令123112132321233111,,,d d d b c t λλλλλλλλλλλλ++=++===和222b c t =，显然由3(1,2)i i λλ≥=和1230λλλ++>知10,0(2,3),0(1,2)i i d d i t i ><=≥=，则式（2-4）、式（2-5）和式（2-6）可改写成如下：1123d a a a =++，（2-7） 212132312d a a a a a a t t =++--，（2-8） 31231231d a a a a t a t =--. （2-9）下面用反证法证明13,a a 均不能为零.显然13,a a 不能同时为零，否则式（2-9）不成立.由于式（2-7）、式（2-8）和式（2-9）中的13,a a 是一个对称的关系，故不妨假设10a =.当130,0a a =≠时，有12322312331d a a d a a t t d a t=+⎧⎪=--⎨⎪=-⎩.（2-10）由式（2-10）可得133223233//t d a t a a d d a =-⎧⎨=-+⎩. （2-11）再来分析22313,,,,a a d d λ之间的关系.3(1,2)i i λλ≥=和1230λλλ++>，由123λλλ≥≥可知20λ>且1232λλλλ++≤.由式（2-10）有23120,a a d λ≤≤<和332d λ>.将213a d a =-带入式（2-11），有2133233()/t d a a d d a =--+. （2-12）将式（2-12）可以看做成2t 关于3a 的函数，对2t 关于3a 进行求导，可得'2213332/t d a d a =--. （2-13）显然'2t 在31(0,]a d ∈上有'20t >，而2t 又是关于31(0,]a d ∈上的连续函数，故2t在31a d =时取得最大值，这时3221d t d d =-+. （2-14）将12311213232,d d λλλλλλλλλ++=++=和1233d λλλ=带入式（2-14）中，可得32211231213231231213231231231232221232133121231232212123123121232123312()()()()()()2()()()[(d t d d λλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλ=-+=-+++++-+++++=++-+-+-+-=++-+-+-+=++-+++=1212312332312123132312123)]()[()()]()()()().λλλλλλλλλλλλλλλλλλλλλλλλλ+++-++++=++-+++=++ （2-15）由式（2-15）可知，当130,0a a =≠时，20t <.因为123{,,}σλλλ=能被非负三对角矩阵111222300a b c a b c a ⎡⎤⎢⎥⎢⎥⎢⎥⎣⎦实现时，总有11122200t b c t b c =≥⎧⎨=≥⎩，矛盾.因而13,a a 均不能为零.证毕.定理 2.2.7 给定一组全不为零的实数123{,,}σλλλ=123()λλλ≥≥，如果3(1,2)i i λλ≥=和1230λλλ++=，则123{,,}σλλλ=不能被非负三对角矩阵111222300a b A c a b c a ⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦实现.证明设非负三对角矩阵111222300a b A c a b c a ⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦能够实现123{,,}σλλλ=，即123,,λλλ是矩阵A 的三个特征值.同定理2.2.7的证明类似，可以给出矩阵A 的特征多项式，并由根与系数的关系可以得到式（2-4）、式（2-5）和式（2-6）.对于式（2-4），当1230λλλ++=时，由于0,1,2,3i a i ≥=，可知123a a a ==0=.而123,,λλλ全不为零和3(1,2)i i λλ≥=可知0(1,2)i i λ>=和30λ<.对于式（2-6）左边=1230λλλ<，右边=1231223110a a a a b c a b c --=，左右不相等，矛盾.故123{,,}σλλλ=不能被非负三对角矩阵111222300a b A c a b c a ⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦实现.证毕. 定理2.2.8 给定一组实数123123{,,}()σλλλλλλ=≥≥，如果3(1,2)i i λλ≥=和1230λλλ++>，则123{,,}σλλλ=不能被非负三对角矩阵111222000a b A c b c a ⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦实现. 证明设非负三对角矩阵矩阵111222000a b A c b c a ⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦能够实现123{,,}σλλλ=，即123,,λλλ是矩阵A 的三个特征值.首先给出矩阵A 的特征多项式.11122212221112321212221112321212112212221100()()()()()()()()().a b I A c b c a a a b c a b c a a a a a b c a b c a a a a a b c b c a b c a b c λλλλλλλλλλλλλλλλλ---=----=------=-++----=-++--++由根与系数的关系知，有下列成立：12312a a λλλ++=+，（2-16） 121323121122a a b c b c λλλλλλ++=--，（2-17） 123122211a b c a b c λλλ=--. （2-18）令123112132321233111,,,d d d b c t λλλλλλλλλλλλ++=++===和222b c t =，显然由3(1,2)i i λλ≥=和1230λλλ++>知10,0(2,3),0(1,2)i i d d i t i ><=≥=，则式（2-16）、式（2-17）和式（2-18）可改写成如下：112d a a =+，（2-19） 21212d a a t t =--，（2-20） 31221d a t a t =--. （2-21）先讨论12a a =的情形.当12a a =时，由式（2-19）可知1122da a ==，则式（2-20）和式（2-21）可化为：21212/4t t d d +=-，（2-22）1123()2dt t d +=-. （2-23）这里式（2-22）和式（2-23）两式中的12t t +必须相等，因而有231212/4d d d d -=-. （2-24）将123112132321233,,d d d λλλλλλλλλλλλ++=++==代入式（2-24）中可得到只关于123,,λλλ的方程，即2123123121323123312312132312312333322222123123123123233222221231231232332()/4(),()4()()80,3()3()143()4(()(3)λλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλ-++-++=++++-+++++=+++++++++-++++++22333222212312312323322222211232322331232222112323123)0,()()()0,(())()()0,(())()(())0.λλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλ=++-+---+=--++-+--=---+--=上式最终可化为22123123()(())0λλλλλλ----=. （2-25）由式（2-25）知要使得式（2-22）和式（2-23）两式中的12t t +相等，就必须满足1230λλλ--=或22123()0λλλ--=，故可得123λλλ=+或123()λλλ=±-.已知1233,(1,2)i i λλλλλ≥≥≥=和1230λλλ++>，显然无论是123λλλ=+还是123()λλλ=±-均不满足已知条件，因而12a a ≠.下面讨论12a a ≠的情形.结合式（2-19）、式（2-20）和式（2-21）联解，可得 2111312111()2a d a d a d t a d -+-=-，（2-26）21113122111211()()2a d a d a d t a d a d a d -+-=----. （2-27）对于式（2-26）和式（2-27）可以看成12,t t 关于1111(0,)(,)22d da d ∈ 的函数，下面把式（2-26）和式（2-27）分在两个区间上讨论.（i）当11(0,2da ∈时，先讨论1t ，令21111312()H a d a d a d =-+-，实际上1H 就是1t 的分子部分.因为20d <，所以有21211113()2d dH a d a d <-+-.由式（2-14）3210d d d -+<知312d d d <，这样2121111()2d d H a d a <-+.令2122111()2d dH a d a =-+,则在11(0,]2d a ∈上有12H H <.对2H 关于1a 求导，可得'2211111123(23)H a d a a d a =-=-，显然在112(0,)3d a ∈上有'20H >，故在11(0,2d a ∈上有'20H >，因而2H 在11(0,2d a ∈上单调递增，又因2H 在112d a =处有定义，则当112da =时，2H 取得最大值，且22111212112(((4)2228d d d d dH d d d =-+=+. （2-28）由1233,(1,2)i i λλλλλ≥≥≥=和1230λλλ++>可知1232λλλλ++≤，则12d λ≤.将1231d λλλ++=和1213232d λλλλλλ++= 代入式（2-28）中，可得21212222121323222312312132321223121323(4)8[4()]8[()()3()]8[()()3()]80.d H d d λλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλλ=+≤+++=++++++=+++++< 这样，在11(0,)2d a ∈上就有20H <，故1H 在11(0,2da ∈上同样也有10H <.因为在11(0,)2d a ∈上1120a d -<，则有10t >. 下面再来讨论2t .将2t 通分化简，得3221111121232112()2a a d a d d d d d t a d -+-++-=-. （2-29）令32231111121232()H a a d a d d d d d =-+-++-，对3H 关于1a 进行求导，得到'2231111234H a a d d d =-+--.显然在11(0,2d a ∈上，'3H 单调递增，且当10a =时，'3H 取得最小值212d d --.将1231d λλλ++=和1213232d λλλλλλ++=代入212d d --中，前面已说明12d λ≤，因而可得到2212123121323221213232231231223()()()()()()()0.d d λλλλλλλλλλλλλλλλλλλλλλλλλλ--=-++-++>--++=-+-+=-++>由上面可以看到'3H 在11(0,)2d a ∈上有'30H >.因此3H 在11(0,)2da ∈上单调递增，又3H 在10a =处有定义，3123(0)0H d d d =->，故3H 在11(0,2da ∈上有30H >，则2t 在区间11(0,2da ∈有20t <.（ii）当111(,)2da d ∈时，同样先分析1t ，直接对1H 求导可得，'21111223H a d a d =--2111122()()a d a a d =--+. （2-30）对于式（2-30）中右边第二项有221222221213231223()()()()()0.a d d λλλλλλλλλλλλ-+>-+=-+++=-++>因而在111(,)2d a d ∈上有'10H >,故1H 在此区间上单调递增，又1H 在11a d =处有定义，则在11a d =处取得最大值，即11312()0H d d d d =-<.因此，在区间111(,)2d a d ∈上，有10H <，又1120a d ->，则10t <. 下面再来分析2t .对3H 求导，可得 '2231111234()H a a d d d =-+-+211111123()()a d a a d d d =-+-+. （2-31）对于式（2-31）中右边第三项有221222()()0d d d λ-+>-+>.因而在111(,)2d a d ∈上有'30H >，故3H 在此区间上单调递增，又3H 在112d a =处取得最小值，即322111*********1121232112123()(2(()2222()24()20.d d d dH d d d d d d d d d d d d da d d d d =-+-++-=-++->-+++-> 因此，在区间111(,)2d a d ∈上，有30H >，又1120a d ->，则20t >. 通过对式（2-26）和式（2-27）在1111(0,)(,)22d da d ∈ 上的分析，可以得出当11(0,)2d a ∈时，120,0t t ><；当111(,)2da d ∈时，120,0t t <>.因而当12a a ≠时，12,t t 无法满足同时大于等于零.这样，以上的推导就证明了不存在非负三对角矩阵111222000a b A c b c a ⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦能够实现123{,,}σλλλ=.证毕.定理2.2.9 给定一组实数123123{,,}()σλλλλλλ=≥≥，如果3(1,2)i i λλ≥=和1230λλλ++>，则123{,,}σλλλ=不能被非负三对角矩阵111222300a b A c a b c a ⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦实现，其中123,,a a a 全不为零.证明设非负三对角矩阵111222300a b A c a b c a ⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦能够实现123{,,}σλλλ=，其中123,,a a a 均不为零，即123,,λλλ是矩阵A 的三个特征值.首先给出矩阵A 的特征多项式.11122231232211133212312132312322111332123121323112212312231100()()()()()()()()()()()a b I A c a b c a a a a b c a b c a a a a a a a a a a a a a b c a b c a a a a a a a a a a b c b c a a a a b c a b c λλλλλλλλλλλλλλλλλ---=-----=-------=-+++++-----=-+++++---++.由根与系数的关系知，有式（2-4）、式（2-5）和式（2-6）成立.令123112132321233111,,,d d d b c t λλλλλλλλλλλλ++=++===和222b c t =，由3(1,2)i i λλ≥=和1230λλλ++>知10,0(2,3),0(1,2)i i d d i t i ><=≥=，则式（2-4）、式（2-5）和式（2-6）可改写成式（2-7）、式（2-8）和式（2-9）.下面分两种情形讨论：13a a =和13a a ≠.（i）当13a a =时.由式（2-8）和（2-9）式分别得到212121323212122t t a a a a a a d a a a d +=++-=+-，（2-32） 2123121a a d t t a -+=. （2-33）显然式（2-32）和式（2-33）都有120t t +>，下面证明两式不可能相等.令23221231111231121211()2a a d a a d a d d f a a a a d a a --+-=+--=-.对于上式中的分子部分，令3241111123()H a a a d a d d =-+-.对4H 求导可得'24111232H a a d d =-+.令'40H =得1a =4H 的两个极值点分别在(,0)-∞和12(,)3d+∞上，因而4H 在区间1(0,2d 单调.因为1212a a d +=，则显然112d a <.当10a =时，43(0)0H d =->.当112d a =时，333111211243(0)()024282d d d d d d d H d =-+->-->.因此在区间1(0,)2d 内无法找到1a 满足41()0H a =，即找不到1a 使得1()0f a =，则式（2-32）和式（2-33）不相等.故当13a a =时，无法找到非负三对角矩阵111222300a b A c a b c a ⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦满足条件. （ii）当13a a ≠时.由式（2-7）、式（2-8）和式（2-9）联解，得 21233231121323231()a a a d d a t a a a a a a d a a ++-=++---，（2-34） 2123323231()a a a d d a t a a ++-=-. （2-35）将式（2-34）通分可得：2121323*********131()()()a a a a a a d a a a a a d d a t a a ++---+-+=-212332131()a a a d d a a a -+-+=-. （2-36） ①当13a a <时.由式（2-36）可知21233211312121131321121123131()2(2)422.a a a d d a t a a d d a d a a d d d d d d a a a a -+-+=--->----+>=--因为221221213231223222()()()0d d d λλλλλλλλλλλ+<+++=+++<，则10t >. 由式（2-35）有212332323121113231321121123131()42(2)4240.a a a d d a t a a d d d d d a a d d d d d d a a a a ++-=-+-<-++<=<--②当13a a >时.对于式（2-36）分子部分有2221121123321112()(2)0424d d d da a a d d a d d d -+-+>--=-+>.因而10t <.对于式（2-33）分子部分有322112112332312()(2)0424d d d d a a a d d a d d ++-<+=+<.因而20t >.由以上的分析可以得出无论123,,a a a 如何取值均不能满足12,t t 均大于等于零.这样，就证明了找不到一个非负三对角矩阵能够实现123{,,}σλλλ=.证毕.由定理2.2.6、定理2.2.7、定理2.2.8和定理2.2.9可以得出下面的结论. 推论2.2.10 给定一组实数123123{,,}()σλλλλλλ=≥≥，如果3(1,2)i i λλ≥=和1230λλλ++≥，则123{,,}σλλλ=不能被非负三对角矩阵111222300a b A c a b c a ⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦实现. 推论2.2.11 推论2.2.3、定理2.2.6、定理2.2.7、定理2.2.8、定理2.2.9和推论2.2.10的结论中非负三对角矩阵均可改为非负对称三对角矩阵，结论依然成立.注：推论2.2.10和2.2.11实际上也是对广义Perron 定理[31]一种验证. 定理 2.2.12 给定三个实数123,,λλλ，如果132(2,3),0i i λλλλ≥=≤<，和1230λλλ++≥，则123{,,}σλλλ=不能被非负三对角矩阵111222300a b A c a b c a ⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦实现. 证明设非负三对角矩阵111222300a b A c a b c a ⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦能够实现123{,,}σλλλ=，即123,,λλλ是矩阵A 的三个特征值.首先给出矩阵A 的特征多项式.11122231232211133212312132312322111332123121323112212312231100()()()()()()()()()()()a b I A c a b c a a a a b c a b c a a a a a a a a a a a a a b c a b c a a a a a a a a a a b c b c a a a a b c a b c λλλλλλλλλλλλλλλλλ---=-----=-------=-+++++-----=-+++++---++.由根与系数的关系知，有式（2-4）、式（2-5）和式（2-6）成立.。

数学专业英语词汇(部分)

数学专业英语词汇（部分）n ary relation n元关系n ball n维球n cell n维胞腔n chromatic graph n色图n coboundary n上边缘n cocycle n上循环n connected space n连通空间n dimensional column vector n维列向量n dimensional euclidean space n维欧几里得空间n dimensional rectangular parallelepiped n维长方体n dimensional row vector n维行向量n dimensional simplex n单形n dimensional skeleton n维骨架n disk n维圆盘n element set n元集n fold extension n重扩张n gon n角n graph n图n homogeneous variety n齐次簇n person game n人对策n simplex n单形n sphere bundle n球丛n th member 第n项n th partial quotient 第n偏商n th power operation n次幂运算n th root n次根n th term 第n项n times continuously differentiable n次连续可微的n times continuously differentiable function n次连续可微函数n tuple n组n tuply connected domain n重连通域n universal bundle n通用丛nabla 倒三角算子nabla calculation 倒三角算子计算nabla operator 倒三角算子napier's logarithm 讷代对数natural boundary 自然边界natural boundary condition 自然边界条件natural coordinates 自然坐标natural equation 自然方程natural equivalence 自然等价natural exponential function 自然指数函数natural frequency 固有频率natural geometry 自然几何natural injection 自然单射natural isomorphism 自然等价natural language 自然语言natural logarithm 自然对数natural mapping 自然映射natural number 自然数natural oscillation 固有振荡natural sine 正弦真数natural transformation 自然变换naught 零necessary and sufficient conditions 必要充分的条件necessary and sufficient statistic 必要充分统计量necessary condition 必要条件necessity 必然性negation 否定negation sign 否定符号negation symbol 否定符号negative 负数negative angle 负角negative binomial distribution 负二项分布negative complex 负复形negative correlation 负相关negative definite form 负定型negative definite hermitian form 负定埃尔米特形式negative definite quadratic form 负定二次形式negative function 负函数negative number 负数negative operator 负算子negative parity 负电阻negative part 负部分negative particular proposition 否定特称命题negative proposition 否定命题negative rotation 反时针方向旋转negative semidefinite 半负定的negative semidefinite eigenvalue problem 半负定特盏问题negative semidefinite form 半负定型negative semidefinite quadratic form 半负定二次形式negative sign 负号negative skewness 负偏斜性negative variation 负变差negligible quantity 可除量neighborhood 邻域neighborhood base 邻域基neighborhood basis 邻域基neighborhood filter 邻域滤子neighborhood retract 邻域收缩核neighborhood space 邻域空间neighborhood system 邻域系neighborhood topology 邻域拓扑neighboring vertex 邻近项点nephroid 肾脏线nerve 神经nested intervals 区间套net 网net function 网格函数net of curves 曲线网net of lines 直线网network 网络network analysis 网络分析network flow problem 网络潦题network matrix 网络矩阵neumann boundary condition 诺伊曼边界条件neumann function 诺伊曼函数neumann problem 诺伊曼问题neumann series 诺伊曼级数neutral element 零元素neutral line 中线neutral plane 中性平面neutral point 中性点newton diagram 牛顿多边形newton formula 牛顿公式newton identities 牛顿恒等式newton interpolation polynomial 牛顿插值多项式newton method 牛顿法newton potential 牛顿位势newtonian mechanics 牛顿力学nice function 佳函数nil ideal 零理想nil radical 幂零根基nilalgebra 幂零代数nilpotency 幂零nilpotent 幂零nilpotent algebra 幂零代数nilpotent element 幂零元素nilpotent group 幂零群nilpotent ideal 幂零理想nilpotent matrix 幂零矩阵nilpotent radical 幂零根基nine point circle 九点圆nine point finite difference scheme 九点有限差分格式niveau line 等位线niveau surface 等位面nodal curve 结点曲线nodal line 交点线nodal point 节点node 节点node locus 结点轨迹node of a curve 曲线的结点noetherian category 诺特范畴noetherian object 诺特对象nomogram 算图nomographic 列线图的nomographic chart 算图nomography 图算法non additivity 非加性non archimedean geometry 非阿基米德几何non archimedean valuation 非阿基米德赋值non countable set 不可数集non critical point 非奇点non denumerable 不可数的non denumerable set 不可数集non developable ruled surface 非可展直纹曲面non enumerable set 不可数集non euclidean geometry 非欧几里得几何学non euclidean motion 非欧几里得运动non euclidean space 非欧几里得空间non euclidean translation 非欧几里得平移non euclidean trigonometry 非欧几里得三角学non homogeneity 非齐non homogeneous chain 非齐次马尔可夫链non homogeneous markov chain 非齐次马尔可夫链non isotropic plane 非迷向平面non linear 非线性的non negative semidefinite matrix 非负半正定阵non oriented graph 无向图non parametric test 无分布检验non pascalian geometry 非拍斯卡几何non ramified extension 非分歧扩张non rational function 无理分数non relativistic approximation 非相对性近似non reversibility 不可逆性non singular 非奇的non stationary random process 不平稳随机过程non steady state 不稳定状态non symmetric 非对称的non symmetry 非对称non zero sum game 非零和对策nonabsolutely convergent series 非绝对收敛级数nonagon 九边形nonassociate 非结合的nonassociative ring 非结合环nonbasic variable 非基本变量noncentral chi squre distribution 非中心分布noncentral f distribution 非中心f分布noncentral t distribution 非中心t分布noncentrality parameter 非中心参数nonclosed group 非闭群noncommutative group 非交换群noncommutative ring 非交换环noncommutative valuation 非交换赋值noncommuting operators 非交换算子noncomparable elements 非可比元素nondegeneracy 非退化nondegenerate collineation 非退化直射变换nondegenerate conic 非退化二次曲线nondegenerate critical point 非退化临界点nondegenerate distribution 非退化分布nondegenerate set 非退化集nondense set 疏集nondenumerability 不可数性nondeterministic automaton 不确定性自动机nondiagonal element 非对角元nondiscrete space 非离散空间nonexistence 不存在性nonfinite set 非有限集nonholonomic constraint 不完全约束nonhomogeneity 非齐性nonhomogeneous 非齐次的nonhomogeneous linear boundary value problem 非齐次线性边值问题nonhomogeneous linear differential equation 非齐次线性微分方程nonhomogeneous linear system of differential equations 非齐次线性微分方程组nonisotropic line 非迷向线nonlimiting ordinal 非极限序数nonlinear equation 非线性方程nonlinear functional analysis 非线性泛函分析nonlinear lattice dynamics 非线性点阵力学nonlinear operator 非线性算子nonlinear optimization 非线性最优化nonlinear oscillations 非线性振动nonlinear problem 非线性问题nonlinear programming 非线性最优化nonlinear restriction 非线性限制nonlinear system 非线性系统nonlinear trend 非线性瞧nonlinear vibration 非线性振动nonlinearity 非线性nonlogical axiom 非逻辑公理nonlogical constant 非逻辑常数nonmeager set 非贫集nonmeasurable set 不可测集nonnegative divisor 非负除数nonnegative number 非负数nonnumeric algorithm 非数值的算法nonorientable contour 不可定向周线nonorientable surface 不可定向的曲面nonorthogonal factor 非正交因子nonparametric confidence region 非参数置信区域nonparametric estimation 非参数估计nonparametric method 非参数法nonparametric test 非参数检定nonperfect set 非完备集nonperiodic 非周期的nonperiodical function 非周期函数nonplanar graph 非平面图形nonprincipal character 非重贞nonrandom sample 非随机样本nonrandomized test 非随机化检验nonrational function 非有理函数nonremovable discontinuity 非可去不连续点nonrepresentative sampling 非代表抽样nonresidue 非剩余nonsense correlation 产生错觉相关nonsingular bilinear form 非奇双线性型nonsingular curve 非奇曲线nonsingular linear transformation 非退化线性变换nonsingular matrix 非退化阵nonspecial group 非特殊群nonstable 不稳定的nonstable homotopy group 非稳定的同伦群nonstandard analysis 非标准分析nonstandard model 非标准模型nonstandard numbers 非标准数nonsymmetric relation 非对称关系nonsymmetry 非对称nontangential 不相切的nontrivial element 非平凡元素nontrivial solution 非平凡解nonuniform convergence 非一致收敛nonvoid proper subset 非空真子集nonvoid set 非空集nonzero vector 非零向量norm 范数norm axioms 范数公理norm form 范形式norm of a matrix 阵的范数norm of vector 向量的模norm preserving mapping 保范映射norm residue 范数剩余norm residue symbol 范数剩余符号norm topology 范拓朴normability 可模性normal 法线normal algorithm 正规算法normal basis theorem 正规基定理normal bundle 法丛normal chain 正规链normal cone 法锥面normal congruence 法汇normal coordinates 正规坐标normal correlation 正态相关normal curvature 法曲率normal curvature vector 法曲率向量normal curve 正规曲线normal density 正规密度normal derivative 法向导数normal dispersion 正常色散normal distribution 正态分布normal distribution function 正态分布函数normal equations 正规方程normal error model 正规误差模型normal extension 正规开拓normal family 正规族normal force 法向力normal form 标准型normal form problem 标准形问题normal form theorem 正规形式定理normal function 正规函数normal homomorphism 正规同态normal integral 正规积分normal linear operator 正规线性算子normal mapping 正规映射normal matrix 正规矩阵normal number 正规数normal operator 正规算子normal order 良序normal plane 法面normal polygon 正规多角形normal polynomial 正规多项式normal population 正态总体normal probability paper 正态概率纸normal process 高斯过程normal sequence 正规序列normal series 正规列normal set 良序集normal simplicial mapping 正规单形映射normal solvable operator 正规可解算子normal space 正规空间normal surface 法曲面normal tensor 正规张量normal to the surface 曲面的法线normal valuation 正规赋值normal variate 正常变量normal variety 正规簇normal vector 法向量normality 正规性normalization 标准化normalization theorem 正规化定理normalize 正规化normalized basis 正规化基normalized function 规范化函数normalized variate 正规化变量normalized vector 正规化向量normalizer 正规化子normalizing factor 正则化因数normed algebra 赋范代数normed linear space 赋范线性空间normed space 赋范线性空间northwest corner rule 北午角规则notation 记法notation free from bracket 无括号记号notation of backus 巴科斯记号notion 概念nought 零nowhere convergent sequence 无处收敛序列nowhere convergent series 无处收敛级数nowhere dense 无处稠密的nowhere dense set 无处稠密点集nowhere dense subset 无处稠密子集nuclear operator 核算子nuclear space 核空间nucleus of an integral equation 积分方程的核null 零null class 零类null divisor 零因子null ellipse 零椭圆null function 零函数null hypothesis 虚假设null line 零线null matrix 零矩阵null method 衡消法null plane 零面null point 零点null ray 零射线null relation 零关系null representation 零表示null sequence 零序列null set 空集null solution 零解null system 零系null transformation 零变换null vector 零向量nullity 退化阶数nullring 零环nullspace 零空间number 数number defined by cut 切断数number defined by the dedekind cut 切断数number field 数域number interval 数区间number line 数值轴number notation 数记法number of partitions 划分数number of repetitions 重复数number of replications 重复数number of sheets 叶数number sequence 数列number set 数集number system 数系number theory 数论number variable 数变量numeration 计算numerator 分子numeric representation of information 信息的数值表示numerical 数值的numerical algorithm 数值算法numerical axis 数值轴numerical calculation 数值计算numerical coding 数值编码numerical coefficient 数字系数numerical computation 数值计算numerical constant 数值常数numerical data 数值数据numerical determinant 数字行列式numerical differentiation 数值微分numerical equality 数值等式numerical equation 数字方程numerical error 数值误差numerical example 数值例numerical function 数值函数numerical inequality 数值不等式numerical integration 数值积分法numerical invariant 不变数numerical mathematics 数值数学numerical method 数值法numerical model 数值模型numerical operator 数字算子numerical quadrature 数值积分法numerical series 数值级数numerical solution 数值解numerical solution of linear equations 线性方程组的数值解法numerical stability 数值稳定性numerical table 数表numerical value 数值numerical value equation 数值方程nutation 章动obelisk 方尖形object 对象object language 对象语言object variable 对象变数objective analysis 客观分析objective function 目标函数oblate ellipsoid 扁椭面oblate spheroid 扁椭面oblique angle 斜角oblique angled 斜角的oblique astroid 斜星形线oblique circular cone 斜圆锥oblique circular cylinder 斜圆柱oblique cone 斜圆锥oblique coordinate system 斜角坐标系oblique coordinates 斜角坐标oblique parallelepiped 斜六面体oblique prism 斜棱柱oblique pyramid 斜棱锥oblique strophoid 斜环诉oblique triangle 斜三角形observable 可观测的observable component 可观测分量observation 观测observation function 观测函数observational error 观测误差observer 观察器观测员obstruction 障碍obstruction cocycle 障碍闭上链obstruction theory 障碍理论obstruction to lifting f 提升的障碍obtuse 钝的obtuse angle 钝角obtuse triangle 钝角三角形octagon 八角形octahedral 八面体octahedral form 八面体形式octahedral group 八面体群octahedron 八面体octal digit 八进制数字octal notation 八进记数法octal system 八进制系octant 卦限octuple 八倍的odd 奇的odd dimensional space 奇维空间odd even check 奇偶校验odd function 奇函数odd number 奇数odd parity 负电阻odd permutation 奇置换oddness 奇数性omega consistent 相容性omega group 群one address 单地址one address instruction 单地址指令one digit 单位的one dimensional 一维的one dimensional boundary value problem 一维边值问题one dimensional differential equation 一维微分方程one dimensional integral 单积分one dimensional space 一维空间one figure 单位的one figure number 单位数one parameter 单参数的one parameter family of curves 曲线的单参数族one place number 单位数one place predicate calculus 一元谓词演算one point compactification 单点紧化one point set 退化集one point union 一点并one sample method 单样本法one sided 单侧的one sided continuity 单侧连续性one sided derivative 单边导数one sided differentiability 单侧可微性one sided limit 单侧极限one sided lower approximate limit 单侧下近似极限one sided surface 单侧曲面one sided test 单侧检验one sided upper approximate limit 单侧偏大近似极限one term 单项的one to many correspondence 一对多对应one to many mapping 一对多映射one to one 一对一的one to one correspondence 一一对应one to one mapping 一一映射one to one relation 一一关系one to one sequence 一一序列one valued 单值的one valued function 单值函数onevaluedness 单值性onto hpmpmprphism 满射同态onto mapping 满射open ball 开球open base 开底open cover 开覆盖open disk 开圆盘open interval 开区间open manifold 开廖open map 开映射open mapping theorem 开映射定理open neighborhood 开邻域open ordered set 开有序集open parallelepiped 开平行六面体open rectangular parallelepiped 开长方体open relation 开关系open segment 开线段open semicircle 开半圆open set 开集open sphere 开球open star 开星形open subprogram 开型子程序open subroutine 开型子程序open surface 开曲面operand 运算对象operating characteristic 运算特征operating system 控制系统operation 运算operation code 操纂operation of symmetric difference 对称差运算operational 运算的operational amplifler 运算放大器operational calculus 算子演算operational code 操纂operational research 运筹学operational unit 运算部件operations research 运筹学operator 算子operator algebra 算子代数operator automorphism 算子同构operator domain 算子域operator equation 算子方程operator function 算子函数operator homomorphism 算子同态operator isomorphism 算子同构operator method 符号法operator norm 算子范数operator of finite rank 有限秩算子operator set 算子域operator valued function 算子函数oppose 反对opposite 逆的opposite angles 对角opposite category 对偶范畴opposite sides 对边opposite sign 导号opposite vector 反向向量opposite vertex 对顶opposition 对立optimal 最佳的optimal control 最优控制optimal disjunctive normal form 最优析取范式optimal normal form 最优标准形optimal policy 最优策略optimal position 最优位置optimal process 最优过程optimal step size 最优步长optimal strategy 最优策略optimal trajectory 最优轨道optimal value function 最优值函数optimality 最优性optimality criterion 最优性判别optimality policy 最优策略optimality principle 最优性原理optimalization problem 最优化问题optimally useful direction 最优可用方向optimization 最优化optimization of scheduling 工序的最优化optimization problem 最优化问题optimum 最佳optimum conditions 最优条件optimum programming 最优规划optimum solution 最优解optimum strategy 最优策略optimum system 最佳系统optimum system control 最佳系统控制optimum value 最优值optional sampling 任意抽样optional selection 任意抽样orbit 轨道orbit curve 轨道曲线orbit determination 轨道计算orbit space 轨道空间orbital stability 轨道稳定性order 次数order bounded set 有序有界集order boundedness 有序有界性order code 指令码order complete set 有序完全集order continuous topology 有序连续的拓扑order convergence 有序收敛order function 序函数order homomorphic group 序同态群order homomorphic image 序同态象order homomorphism 序同态order homomorphism operator 序同态算子order interval 有序区间order isomorphic field 序同构域order isomorphic group 序同构群order of a differential equation 微分方程的阶order of a group 群的阶order of an infinitesimal 无穷小的阶order of approximation 逼近的阶order of branch point 支点的阶数order of contact 接触度order of convergence 收敛的阶order of infinities 无穷大的阶order of infinity 无穷大的阶order of magnitude 绝对值的阶order of polynomial 多项式的阶order of rational integral function 有理整函数的阶order of units 位数order of zero point 零点的阶order preserving correspondence 保序对应order preserving isomorphism 保序同构order register 指令寄存器order relation 序关系order simple group 序单群order statistic 顺序统计量order type 有序型ordered 有序的ordered chain complex 有序链复形ordered factor group 有序商群ordered field 有序域ordered group 有序群ordered linear space 有序线性空间ordered module 有序模ordered pair 序对ordered product 有序积ordered sample 有序样本ordered set 有序集ordered set bounded below 下有界有序集ordered simplex 有序单形ordered singular boundary 有序奇异边界ordered triplet 有序三个一组orderisomorphism 序同构orderpreserving relation 保序关系ordinal 序数ordinal number 序数ordinal number class 序数类ordinal number of the second class 第二类的序数ordinal product 序数积ordinal series 序数列ordinal type 有序型ordinary 导常的ordinary differential equation 常微分方程ordinary differential operator 常微分算子ordinary dirichlet series 狄利克雷级数ordinary integral element 寻常积分元素ordinary linear differential equation 线性常微分方程ordinary multiple point 寻常多重点ordinary point 单点ordinary singularity 寻常奇点ordinate 纵坐标ordinate axis 纵轴orient 定向orientability 可定向性orientable 可定向的orientable surface 可定向曲面orientation 定向orientation class 定向类orientation of surface 曲面的定向orientation of torsion 挠率的定向orientation preserving automorphism 保持定向自同构orientation preserving isomorphism 保持定向同构orientation preserving parameter transformation 保持定向参数变换orientation reversing 反转定向的oriented chain complex 有向链复形oriented circle 有向圆oriented contour 有向围道oriented element 有向元oriented homology group 有向同岛oriented intersection 有向交叉oriented line 有向直线oriented plane 有向平面oriented polygon 有向多角形oriented space 有向空间oriented sphere bundle 有向球丛oriented surface 有向曲面origin 坐标的原点origin of coordinates 坐标的原点original 初始的ortho pinacoid 正交平行双面orthocenter 垂心orthocentric 垂心的orthocentric quadrangle 垂心四边形orthocentric tetrahedron 垂心四面体orthocomplement 正交补orthogonal 直交的orthogonal axonometry 正轴测射影法orthogonal basis 正交基orthogonal circle 正交圆orthogonal complement 正交补orthogonal coordinate system 正交坐标系orthogonal coordinates 正交坐标orthogonal decomposition 正交分解orthogonal expansion 正交函数展开orthogonal functions 正交函数orthogonal groups 正交群orthogonal involution 正交对合orthogonal matrix 正交矩阵orthogonal polynomial expansion 正交多项式展开orthogonal polynomials 正交多项式orthogonal projection 正射影orthogonal projector 正交射影算子orthogonal sequence 正交序列orthogonal series 正交级数orthogonal space 正交空间orthogonal square 正交方格orthogonal stochastic process 正交随机过程orthogonal subspaces 正交子空间orthogonal substitution 正交代换orthogonal sum 正交和orthogonal system 正交系orthogonal tests 独立检验orthogonal trajectory 正交轨线orthogonal transformation 正交变换orthogonal vectors 正交向量orthogonality 正交性orthogonality of columns 列正交性orthogonality of lines 行正交性orthogonality relation 正交关系orthogonality rows 行正交性orthogonalization procedure 正交化过程orthogonalization process 正交化过程orthogonally irreducible representation 正交不可约表示orthography 正投影法orthoheliotropism 直向阳性orthohexagonal 正六方的orthomorphism 正交射orthonormal 规格化正交的orthonormal basis 标准正交基orthonormal sequence 标准正交序列orthonormality 标准正交性orthonormalization 标准正交化orthonormalization process 标准正交化过程orthonormalized basis 标准正交化基orthopole 正交极orthotomy 面正交性oscillate 振动oscillating divergent series 振动发散级数oscillating infinite determinant 振动无穷行列式oscillating series 振荡级数oscillation 振动oscillation equation 振动方程oscillation function 振动函数oscillation of a function 函数的振幅oscillation theorem 振动定理oscillatory 振动的osculating circle 密切圆osculating cone 密切锥面osculating conic 密切二次曲线osculating curve 密切曲线osculating figure 密切图形osculating helix 密切螺旋线osculating parabola 密切抛物线osculating plane 密切平面osculating quadric 密切二次曲面osculating sphere 密切球面osculation 密切osculatory 密切的outdegree 出度outer 外部的outer automorphism 外自同构outer boundary 外边界outer capacity 外容量outer diameter 外径outer direct product 外直积outer lebesgue measure 勒贝格外测度outer measure 外测度outer point 外点outer product 外积outer semidirect product 外半直积output 输出output alphabet 输出字母表output device 输出装置output function 输出函数output quantity 输出量output signal 输出信号output state 输出状态output unit 输出设备outside 外部outward normal 外法线oval 卵形线oval surface 卵形曲面overconvergence 过度收敛overcrossing point 上交叉点overdetermined problem 超定问题overdetermined system 超定组overdetermined system of partial differential equations 偏微分方程的超定组overfield 扩张域overflow 溢出overidentification 过分识别overlap 交叠overlapping domains 交叠域overrelaxation 超松弛p adic algebra p进代数p adic fraction p进分数p adic integer p进整数p adic number p进数p adic representation p进表示p adic system p进法p adic valuation p进赋值pack 束pair 对paired comparison 成对比较法paired samples 成双样本pairing 配对pairwise 两两的pairwise independent events 互相独立事件paleogeometry 古几何学pantograph 比例画器放大器parabola 抛物线parabola of order n n阶抛物线parabolic coordinates 抛物线坐标parabolic curve 抛物曲线parabolic cusp 抛物尖点parabolic cylinder 抛物柱面parabolic cylinder function 抛物柱面函数parabolic differential equation 抛物型微分方程parabolic folium 抛物叶形线parabolic orbit 抛物线轨道parabolic point 抛物点parabolic regression 抛物回归parabolic segment 抛物线段parabolic spiral 抛物螺线parabolic type 抛物型parabolograph 抛物图形paraboloid 抛物面paraboloid of revolution 回转抛物面paraboloidal coordinates 抛物面坐标paracompact space 仿紧空间paradox 悖论paradoxical 荒谬的paradoxical set 不合理集合parallactic displacement 视差位移parallel 平行线parallel algorithm 并联算法parallel angle 平行角parallel axiom 平行公理parallel body 平行体parallel circle 平行圆parallel coordinates 平行坐标parallel curve 平行曲线parallel displacement 平行位移parallel edge 平行棱parallel line 平行线parallel operation 并行运算parallel plane 平面平行的parallel projection 平行射影parallel rule 平行直尺parallel shift 并行进位parallel slit 平行缝隙parallel slit domain 平行裂纹域parallel strip 平行带parallel subspace 平行子空间parallel surface 平行曲面parallel translation 并行进位parallelepiped 平行六面体parallelepipedal neighborhood 平行六面体邻域parallelism 平行parallelizability 可平行性parallelizable 可平行的parallelizable manifold 可平行廖parallelizable variety 可平行廖parallelogram 平行四边形parallelogram axiom 平行四边形公理parallelogram identity 平行四边形恒等式parallelogram law 平行四边形定律parallelogram of forces 力的平行四边形parallelotope 超平行体parallels of latitude 纬度线parameter 参数parameter curve 参数曲线parameter group 参数群parameter integral 参数积分parameter representation 参数表示parameter space 参数空间parameter variety 参数簇parametric curvature 参数曲率parametric curve 参数曲线parametric equation 参数方程parametric estimation 参数估计parametric hypothesis 参数假设parametric optimal solution 参数最优解parametric optimization 参数最优化parametric programming 参数规划parametric representation 参数表示parametric surface 参数曲面parametric test 参数检验parametric variable 参变数parametrics 参数学parametrix 拟基本解parametrization 参数化法parametron 变参数元件parastatistics 仲统计法paraxial 旁轴的paraxial rays 旁轴光线parent population 母体parenthesis 圈括弧parenthesize 引入括弧内parity 奇偶性parity check 奇偶校验parity digit 奇偶校验位parseval equality 帕谢互尔等式parted hyperboloid 双叶双曲面partial 部分的partial carry 部分进位partial confounding 部分混区partial correlation 偏相关partial correlation coefficient 偏相关系数partial cross section 部分截面partial denominator 偏分母partial derivative 偏导数partial difference 偏差partial difference equation 偏差分方程partial difference quotient 偏差商partial differential 偏微分partial differential coefficient 偏导数partial differential equation 偏微分方程partial differential equation of elliptic type 椭圆型偏微分方程partial differential equation of first order 一阶偏微分方程partial differential equation of hyperbolic type 双曲型偏微分方程partial differential equation of mixed type 混合型偏微分方程partial differential equation of parabolic type 抛物型偏微分方程partial differential operator 偏微分算子partial differentiation 偏微分法partial fraction 部分分数partial fraction decomposition 部分分数展开partial fraction expansion 部分分数展开partial function 部分函数partial functor 部分函子partial graph 部分图partial linear differential equation 线性偏微分方程partial mapping 部分映射partial matrix 子阵partial numerator 偏分子partial ordering 偏序partial product 部分积partial rank correlation coefficient 偏等级相关系数partial recursive function 部分递归函数partial recursive scheme 部分递归格式partial regression 偏回归partial subgraph 部分子图partial sum 部分和partially adjoint operator 部分伴随算子partially computable function 部分可计算的函数partially differentiable function 可偏微函数partially isometric operator 部分等距算子partially ordered set 半序集partially ordered space 半序空间particular 特别的particular integral 特别积分particular solution 特解partition 分划partition function 分折函数partition of an integer 整数的分拆partition of unity 单位分解partition problem 划分问题partly well ordered set 半良序集pascal distribution 帕斯卡分布pascal hexagon 帕斯卡六边形pascal line 帕斯卡线pascal triangle 帕斯卡三角形passive 被动的path 道路path component 道路连通区path connected space 道路连通空间path matrix 道路矩阵path of a projectile 弹道path of integration 积分路线path space 道路空间pathological function 病态函数pattern 图形pauli matrices 泡利阵payoff 支付payoff function 支付函数payoff matrix 支付阵peak 峰pearson distribution 皮尔逊分布pedal 垂足线pedal circle 垂足圆pedal curve 垂足线pedal surface 垂足曲面pedal transformation 垂足变换pedal triangle 垂足三角形peirce decomposition 皮尔斯分解penalty function 罚函数penalty method 补偿法pencil 束pencil of circles 圆束pencil of conics 二次曲线束pencil of curves 曲线束pencil of lines 线束pencil of parallel lines 平行线束pencil of planes 面束pencil of spheres 球束pendulum 摆pentad 拼五小组pentadecagon 十五边形pentagon 五角形pentagonal number 五角数pentagonal pyramid 五角锥pentagram 五角星pentahedron 五面体pentaspherical coordinates 五球坐标percent 百分率percent error 百分误差percentage 百分率percentile 百分位数perfect cube 完全立方perfect field 完全域perfect form 完全型perfect group 完全群perfect information 完全信息perfect magic square 完全幻方perfect mapping 完全映射perfect matching 全匹配perfect number 完全数perfect number of the second kind 第二类完全数perfect set 完备集perfect space 完全空间perfect square 完全平方perfectly normal space 完全正规空间perigon 周角perimeter 周围period 周期period matrix 周期矩阵period of a function 函数的周期period of oscillation 振荡的周期period relations 周期关系式period strip 周期带periodic 周期的periodic distribution 周期广义函数periodic element 周期元素periodic error 周期误差periodic function 周期函数periodic group 周期群periodic law 周期律periodic motion 周期运动periodic orbit 周期轨道periodic solution 周期解periodic system 周期系periodical decimal fraction 循环十进小数periodicity 周期性periodicity interval 周期性区间periodicity of function 函数的周期性periodogram 周期图periodogram analysis 周期图分析periodograph 周期图periodparallelogram 周期平行四边形peripheral 周边的peripheral device 外围设备peripheral unit 外围设备periphery 周边permanently convergent series 永久收敛级数permissible error 容许误差permissible function 容许函数permissible value 容许值permutation 置换。

abaqus个人学习笔记小结1【范本模板】

1、abaqus中的力载荷集中力concentrated force、压强pressure（垂直于表面)、表面分布力surface traction （设定沿着某方向）pressure只能施加在面上（几何的面，单元的面）,为垂直于表面的分布力；surface traction只能施加在面上（几何的面，单元的面）,为沿着某一方向的分布力；concentrated force只能施加在点上(几何的点,节点)，要使得集中力产生的效果等同于分布力，则需要将集中力施加在参考点上，然后将参考点与作用面上的节点进行耦合约束coupling(distributed coupling）,而不要直接施加在节点上.一般，如果不要求等效均布力，则集中力最好施加在几何的点上。

确实需要施加节点力，则施加在节点上.对于有限元软件，所有的力载荷本质上都由程序处理成节点力。

2、abaqus计算热电耦合出现Too many attempts made for this increment(1）调整一下计算载荷施加的速度或者调整载荷大小，要么把计算步长设置的小一点,尝试次数设的多一点.这个提示是说计算的过程中直到设定的尝试次数极限仍然求解失败。

（2) 分析步主要有初始分析步和后续分析步，每个分析步可以用来描述一个分析过程，例如在后续分析步中施加不同荷载，在初始分析步中施加边界条件等。

增量步是在分析步里面根据模型计算收敛情况设置的，简单模型可以设置较少的增量步,并可使初始增量为1；复杂模型设置多一点增量步，并减少初始增量值。

超过设置的允许增量步数，则计算停止。

（3)检查模型，是否存在刚体位移，过约束,接触定义不当等问题（4)分别建立四个边界条件，BC—1，BC—2，BC-3，BC-4，每一个边界条件定义板的一边固结的支承条件就行了。

之前是建立了一个BC—1,四边的约束都定义在BC-1里面，就算不下去了,不清楚原因。

仅供参考学习。

（5）1。

计量经济学中英文词汇对照

Common variance Common variation Communality variance Comparability Comparison of bathes Comparison value Compartment model Compassion Complement of an event Complete association Complete dissociation Complete statistics Completely randomized design Composite event Composite events Concavity Conditional expectation Conditional likelihood Conditional probability Conditionally linear Confidence interval Confidence limit Confidence lower limit Confidence upper limit Confirmatory Factor Analysis Confirmatory research Confounding factor Conjoint Consistency Consistency check Consistent asymptotically normal estimate Consistent estimate Constrained nonlinear regression Constraint Contaminated distribution Contaminated Gausssian Contaminated normal distribution Contamination Contamination model Contingency table Contour Contribution rate Control

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。

Abaqus错误信息和警告信息汇总

Fixed time is too largeToo many attamps have been madeTHE SOLUTION APPEARS TO BE DIVERGING.CONVERGENCE ISJUDGED UNLIKELY.Time increment required is less than the minimum specified这样的信息几乎是无用信息（除了告诉你的模型分析失败以外，没有告诉你任何有用的东西）。

宜再查找别的信息来考察。

根据经验，改小增量步也不一定能收敛，虽然也有人报告过改好的先例，我是从来没有遇到过，也从来没有那个奢望。

所以我一般从模型的设置入手。

---------------------------------------------------------------------------Error意味着运算失败，but出现warning可能还能算，而且有些运算必定会出现warning（比如接触分析必定出“负特征值”，下有详述）。

很多警告只是通知性质的，或者只是说明一下而已，不一定都是模型有问题。

比如以下warning完全可以忽略：xxxxx will （not）printed，这种只是通知你一声，某些玩意儿不输出了。

还有：The parameter frequency cannot be used wit h the parameter field. It will be ignored（都说某某被ignored了）.---------------------------------------------------------------------------A系列如果模型能算，且结果合理，那么大部分警告信息可以不管。

但是以下除外：1 numerical sigularity(数值奇异)：刚体位移（欠约束）solver problem. numerical sigularity when processing node105 instance pile D.O.F. 1 ratio=1.735e132 Zero pivot（零主元）：过约束或者欠约束。

秩1修正矩阵特征值问题的推广及其应用（英文）

秩1修正矩阵特征值问题的推广及其应用（英文）吕海玲;明清河【摘要】本文给出了秩1修正矩阵特征值问题推广的新证明,证明过程主要应用了一个行列恒等式.在此基础上,把秩1修正矩阵的特征值问题推广到块特征值问题.最后给出一个应用说明结论的重要性.%We prove a spectral perturbation theorem for an extension eigenvalues of rank-one update matrix of special structure.The main idea behind our proof is from the simple relation between the determinants for a matrix and this result.Furthermore,we extent this theorem to the block eigenvalues problem.At last one application of the result is given to illustrate the usefulness of the theorem.【期刊名称】《枣庄学院学报》【年(卷),期】2011(000)005【总页数】4页(P29-32)【关键词】秩1更新;行列式;谱【作者】吕海玲;明清河【作者单位】枣庄学院信息科学与工程学院,山东枣庄277160;枣庄学院数学与统计学院,山东枣庄277160【正文语种】中文【中图分类】O151.211 IntroductionIn this paper we prove a spectral perturbation theorem for an extension eigenvalues of rank－ one update matrix of special structure，which shows how to modify r eigenvalues of a matrix of order n，(r≤n)，ia a rank－k updated matrix，without changing any of the n－rremaining eigenvalues．This theorem plays a relevan t role in the study of the nonnegative inverse eigenvalue problem(NIEP)．The main idea behind our proof is from the simple relation between the determinants of a matrix and this result，using a well known determinant identity．Furthermore，we extent this theorem to the block eigenvalues problem．By using this extension，we give a Application on eigenvalues problem of matrix perturbation of special structure．Because we apply a classic determinant equality to our spectral analysis，we are able to find explicit expression of the characteristic polynomial of the rank－r update matrix．All eigenvalues of the matrix are immediately available．Lemma1 If A is an invertible n×n matrix，and u and v are two n－dimensional column vectors，thenProof．We may assume A=I，the n × n identity matrix，sincethen(1)follows fromin the general case．In this special case，the result comes from theequalityso(2)becomesRemark1 If A is an invertible n×n matrix，B is a n×r matrix，C is a r×n matrix，thenIn the next section we present the main result．2 Main resultLet A be an n×n matrix．The eigenvalues of A are all the complex zeros of the characteristic polynomial pA(λ)=det(λI－ A)of A．Letσ(A)= {λ1，λ2，…，λn }be the set of the eigenvalues of A，counting algebraic multiplicity，that is spectrum of A．Theorem 1［1］ Let u and v are two n － dimensional column vectors such that u is an eigenvector of A associated with eigenvalue λ1． Then，the eigenvalues of A + uvT are {λ1+vT u，λ2，…，λn}，counting algebraic multiplicity．The following result is an extension of the theorem 1．This extension shows how to change r eigenvalues λ1，λ2，…，λr，r≤ n，of a matrix A of order n，via a rank － k updated matrix，without changing any of the n －rremaining eigenvaluesλr+1，λr+2，…，λn．Theorem 2 Let A be an n × n matrix with eigenvalues λ1，λ2，…，λn．LetX =[x1 x2 …xr ]be such that rank(X)=r and AX=Xdiag [λ1，λ2，…，λr]，r≤n．Let C be a r × n matrix．Then the matrix A+XC has eigenvalues γ1，γ2，…，γr，λr+1，λr+2，…，λn．where γ1，γ2，…，γr are eigenvalues of the matrix K+CX with K=diag [λ1，λ2，…，λr]．Proof Letλ ∉ σ(A)be any comple x number．Then，by applying remark 1 to the equalityW e haveThe condition AX=Xdiag [λ1，λ2，…，λr]implies thatso(7)becomesSince the above equality is true for allλ ∉ σ(A)，the theorem is p roved．Rem ark2．1 By Theorem 2．1，the characteristic polynomial of A+XC isRemark2．2 Since A and AT have the same eigenvalues counting algebraic multiplicity，the conclusion of Theorem 2．1 also holds for A+XC，whereX= [x1 x2 … xr ]be such that rank(X)=r and AX=Xdiag [λ1，λ2，…，λr]．Furthermore，we extent this theorem to the block eigenvalues problem Definition 1［4］．A matrix X of order n is a block eigenvalue of a matrix A of order mn，if there exists a block vector V of full rank，such that AV=VX，X is a block eigenvector of A．The matrix A is partitioned into m ×m blocks of or der n，and the block vector V．Definition 2［4］．A set of block eigenvalues of a block matrix is acomplete set if the set of all the eigenvalues of these block eigenvalues is the set of the matrix．Let us suppose now that we have computed mn scalar eigenvalues of a partitioned matrix A．We can construct a complete set of block eigenvalues by taking m matrix of order n in Jordan form where the diagonal elements are those scalar eigenvalues．Furthermore，if the scalar eigenvalues of A are distinct，these m matrix are diagonal matrix as is shown in the following construction:where theλi，i=1，…，mn，are the eigenvaluesof A．The proof that the matrix Xj，j=1，…，m，are a complete set of block eigenvalues of A is in ［1，p．74］．Theorem 2．If the scalar eigenvalues of A are distinct，let V and C be the block vectors such that V is a block eigenvector of A associated with block eigenvalues X1，Then，the eigenvalues of A + VCT are μ1，…，μn，λn+1，…，λ2n，…λ(m－1)n+1，…，λmn where μ1，…，μn are eigenvalues of the matrix K+CT V with K=diag［λ1，…，λn］．Proof．The same to theorem 1．3 Application of the theoremA direct consequence of Theorem 2．1 is the following．One Application of the result is given to illustrate the eigenvalues problem with the perturbation matrix．Proposition 3．1Let A，B，C，D ∈ Cn×n，D=A+B，where B is the perturbation of A．If B=XC，where X= ［x1，x2，…，xn］，xi is aneigenvector of A dissociate with eigenvalue xi，i=1，2，…，n．So thatthen，the eigenvalues of A+B are the eigenvalues of the matrix diag ［λ1，λ2，…，λn］+CX．References［1］Jiu D，Zhou A H．Eigenvalues of rank －one updated matrix with some applications［J］．Applied Mathematics Letters，2007，20:1223－1226．［2］Ricrdo L S，Oscar R．Applications of a Brauer theorem in the nonnegative inverse eigenvalue problem［J］．Linear Algebra and its Applications，2006，416:844 －856．［3］Bapat R B，Raghavan E S．Nonnegative Matrices and Applications，Cambridge University press，1997．［4］Dennis J E，Traub J F and Weber R．P．On the matrix polynomial，lambda－ matrix and block eigenvalue problem，Tech．Rep．71 － 109，Computer Science Department，Cornell Univ，Ithaca，NY and Carnegie －Mellon Univ．，Pitsburgh，PA，(1971)．。

数学专业词汇(N)

n ary relation n元关系n ball n维球n cell n维胞腔n chromatic graph n色图n coboundary n上边缘n cocycle n上循环n connected space n连通空间n dimensional column vector n维列向量n dimensional euclidean space n维欧几里得空间n dimensional rectangular parallelepiped n维长方体n dimensional row vector n维行向量n dimensional simplex n单形n dimensional skeleton n维骨架n disk n维圆盘n element set n元集n fold extension n重扩张n gon n角n graph n图n homogeneous variety n齐次簇n person game n人对策n simplex n单形n sphere bundle n球丛n th member 第n项n th partial quotient 第n偏商n th power operation n次幂运算n th root n次根n th term 第n项n times continuously differentiable n次连续可微的n times continuously differentiable function n次连续可微函数n tuple n组n tuply connected domain n重连通域n universal bundle n通用丛nabla 倒三角算子nabla calculation 倒三角算子计算nabla operator 倒三角算子napier's logarithm 讷代对数natural boundary 自然边界natural boundary condition 自然边界条件natural coordinates 自然坐标natural equation 自然方程natural equivalence 自然等价natural exponential function 自然指数函数natural frequency 固有频率natural geometry 自然几何natural injection 自然单射natural isomorphism 自然等价natural language 自然语言natural logarithm 自然对数natural mapping 自然映射natural number 自然数natural oscillation 固有振荡natural sine 正弦真数natural transformation 自然变换naught 零necessary and sufficient conditions 必要充分的条件necessary and sufficient statistic 必要充分统计量necessary condition 必要条件necessity 必然性negation 否定negation sign 否定符号negation symbol 否定符号negative 负数negative angle 负角negative binomial distribution 负二项分布negative complex 负复形negative correlation 负相关negative definite form 负定型negative definite hermitian form 负定埃尔米特形式negative definite quadratic form 负定二次形式negative function 负函数negative number 负数negative operator 负算子negative parity 负电阻negative part 负部分negative particular proposition 否定特称命题negative proposition 否定命题negative rotation 反时针方向旋转negative semidefinite 半负定的negative semidefinite eigenvalue problem 半负定特盏问题negative semidefinite form 半负定型negative semidefinite quadratic form 半负定二次形式negative sign 负号negative skewness 负偏斜性negative variation 负变差negligible quantity 可除量neighborhood 邻域neighborhood base 邻域基neighborhood basis 邻域基neighborhood filter 邻域滤子neighborhood retract 邻域收缩核neighborhood space 邻域空间neighborhood system 邻域系neighborhood topology 邻域拓扑neighboring vertex 邻近项点nephroid 肾脏线nerve 神经nested intervals 区间套net 网net function 网格函数net of curves 曲线网net of lines 直线网network 网络network analysis 网络分析network flow problem 网络潦题network matrix 网络矩阵neumann boundary condition 诺伊曼边界条件neumann function 诺伊曼函数neumann problem 诺伊曼问题neumann series 诺伊曼级数neutral element 零元素neutral line 中线neutral plane 中性平面neutral point 中性点newton diagram 牛顿多边形newton formula 牛顿公式newton identities 牛顿恒等式newton interpolation polynomial 牛顿插值多项式newton method 牛顿法newton potential 牛顿位势newtonian mechanics 牛顿力学nice function 佳函数nil ideal 零理想nil radical 幂零根基nilalgebra 幂零代数nilpotency 幂零nilpotent 幂零nilpotent algebra 幂零代数nilpotent element 幂零元素nilpotent group 幂零群nilpotent ideal 幂零理想nilpotent matrix 幂零矩阵nilpotent radical 幂零根基nine point circle 九点圆nine point finite difference scheme 九点有限差分格式niveau line 等位线niveau surface 等位面nodal curve 结点曲线nodal line 交点线nodal point 节点node 节点node locus 结点轨迹node of a curve 曲线的结点noetherian category 诺特范畴noetherian object 诺特对象nomogram 算图nomographic 列线图的nomographic chart 算图nomography 图算法non additivity 非加性non archimedean geometry 非阿基米德几何non archimedean valuation 非阿基米德赋值non countable set 不可数集non critical point 非奇点non denumerable 不可数的non denumerable set 不可数集non developable ruled surface 非可展直纹曲面non enumerable set 不可数集non euclidean geometry 非欧几里得几何学non euclidean motion 非欧几里得运动non euclidean space 非欧几里得空间non euclidean translation 非欧几里得平移non euclidean trigonometry 非欧几里得三角学non homogeneity 非齐non homogeneous chain 非齐次马尔可夫链non homogeneous markov chain 非齐次马尔可夫链non isotropic plane 非迷向平面non linear 非线性的non negative semidefinite matrix 非负半正定阵non oriented graph 无向图non parametric test 无分布检验non pascalian geometry 非拍斯卡几何non ramified extension 非分歧扩张non rational function 无理分数non relativistic approximation 非相对性近似non reversibility 不可逆性non singular 非奇的non stationary random process 不平稳随机过程non steady state 不稳定状态non symmetric 非对称的non symmetry 非对称non zero sum game 非零和对策nonabsolutely convergent series 非绝对收敛级数nonagon 九边形nonassociate 非结合的nonassociative ring 非结合环nonbasic variable 非基本变量noncentral chi squre distribution 非中心分布noncentral f distribution 非中心f分布noncentral t distribution 非中心t分布noncentrality parameter 非中心参数nonclosed group 非闭群noncommutative group 非交换群noncommutative ring 非交换环noncommutative valuation 非交换赋值noncommuting operators 非交换算子noncomparable elements 非可比元素nondegeneracy 非退化nondegenerate collineation 非退化直射变换nondegenerate conic 非退化二次曲线nondegenerate critical point 非退化临界点nondegenerate distribution 非退化分布nondegenerate set 非退化集nondense set 疏集nondenumerability 不可数性nondeterministic automaton 不确定性自动机nondiagonal element 非对角元nondiscrete space 非离散空间nonexistence 不存在性nonfinite set 非有限集nonholonomic constraint 不完全约束nonhomogeneity 非齐性nonhomogeneous 非齐次的nonhomogeneous linear boundary value problem 非齐次线性边值问题nonhomogeneous linear differential equation 非齐次线性微分方程nonhomogeneous linear system of differential equations 非齐次线性微分方程组nonisotropic line 非迷向线nonlimiting ordinal 非极限序数nonlinear equation 非线性方程nonlinear functional analysis 非线性泛函分析nonlinear lattice dynamics 非线性点阵力学nonlinear operator 非线性算子nonlinear optimization 非线性最优化nonlinear oscillations 非线性振动nonlinear problem 非线性问题nonlinear programming 非线性最优化nonlinear restriction 非线性限制nonlinear system 非线性系统nonlinear trend 非线性瞧nonlinear vibration 非线性振动nonlinearity 非线性nonlogical axiom 非逻辑公理nonlogical constant 非逻辑常数nonmeager set 非贫集nonmeasurable set 不可测集nonnegative divisor 非负除数nonnegative number 非负数nonnumeric algorithm 非数值的算法nonorientable contour 不可定向周线nonorientable surface 不可定向的曲面nonorthogonal factor 非正交因子nonparametric confidence region 非参数置信区域nonparametric estimation 非参数估计nonparametric method 非参数法nonparametric test 非参数检定nonperfect set 非完备集nonperiodic 非周期的nonperiodical function 非周期函数nonplanar graph 非平面图形nonprincipal character 非重贞nonrandom sample 非随机样本nonrandomized test 非随机化检验nonrational function 非有理函数nonremovable discontinuity 非可去不连续点nonrepresentative sampling 非代表抽样nonresidue 非剩余nonsense correlation 产生错觉相关nonsingular bilinear form 非奇双线性型nonsingular curve 非奇曲线nonsingular linear transformation 非退化线性变换nonsingular matrix 非退化阵nonspecial group 非特殊群nonstable 不稳定的nonstable homotopy group 非稳定的同伦群nonstandard analysis 非标准分析nonstandard model 非标准模型nonstandard numbers 非标准数nonsymmetric relation 非对称关系nonsymmetry 非对称nontangential 不相切的nontrivial element 非平凡元素nontrivial solution 非平凡解nonuniform convergence 非一致收敛nonvoid proper subset 非空真子集nonvoid set 非空集nonzero vector 非零向量norm 范数norm axioms 范数公理norm form 范形式norm of a matrix 阵的范数norm of vector 向量的模norm preserving mapping 保范映射norm residue 范数剩余norm residue symbol 范数剩余符号norm topology 范拓朴normability 可模性normal 法线normal algorithm 正规算法normal basis theorem 正规基定理normal bundle 法丛normal chain 正规链normal cone 法锥面normal congruence 法汇normal coordinates 正规坐标normal correlation 正态相关normal curvature 法曲率normal curvature vector 法曲率向量normal curve 正规曲线normal density 正规密度normal derivative 法向导数normal dispersion 正常色散normal distribution 正态分布normal distribution function 正态分布函数normal equations 正规方程normal error model 正规误差模型normal extension 正规开拓normal family 正规族normal force 法向力normal form 标准型normal form problem 标准形问题normal form theorem 正规形式定理normal function 正规函数normal homomorphism 正规同态normal integral 正规积分normal linear operator 正规线性算子normal mapping 正规映射normal matrix 正规矩阵normal number 正规数normal operator 正规算子normal order 良序normal plane 法面normal polygon 正规多角形normal polynomial 正规多项式normal population 正态总体normal probability paper 正态概率纸normal process 高斯过程normal sequence 正规序列normal series 正规列normal set 良序集normal simplicial mapping 正规单形映射normal solvable operator 正规可解算子normal space 正规空间normal surface 法曲面normal tensor 正规张量normal to the surface 曲面的法线normal valuation 正规赋值normal variate 正常变量normal variety 正规簇normal vector 法向量normality 正规性normalization 标准化normalization theorem 正规化定理normalize 正规化normalized basis 正规化基normalized function 规范化函数normalized variate 正规化变量normalized vector 正规化向量normalizer 正规化子normalizing factor 正则化因数normed algebra 赋范代数normed linear space 赋范线性空间normed space 赋范线性空间northwest corner rule 北午角规则notation 记法notation free from bracket 无括号记号notation of backus 巴科斯记号notion 概念nought 零nowhere convergent sequence 无处收敛序列nowhere convergent series 无处收敛级数nowhere dense 无处稠密的nowhere dense set 无处稠密点集nowhere dense subset 无处稠密子集nuclear operator 核算子nuclear space 核空间nucleus of an integral equation 积分方程的核null 零null class 零类null divisor 零因子null ellipse 零椭圆null function 零函数null hypothesis 虚假设null line 零线null matrix 零矩阵null method 衡消法null plane 零面null point 零点null ray 零射线null relation 零关系null representation 零表示null sequence 零序列null set 空集null solution 零解null system 零系null transformation 零变换null vector 零向量nullity 退化阶数nullring 零环nullspace 零空间number 数number defined by cut 切断数number defined by the dedekind cut 切断数number field 数域number interval 数区间number line 数值轴number notation 数记法number of partitions 划分数number of repetitions 重复数number of replications 重复数number of sheets 叶数number sequence 数列number set 数集number system 数系number theory 数论number variable 数变量numeration 计算numerator 分子numeric representation of information 信息的数值表示numerical 数值的numerical algorithm 数值算法numerical axis 数值轴numerical calculation 数值计算numerical coding 数值编码numerical coefficient 数字系数numerical computation 数值计算numerical constant 数值常数numerical data 数值数据numerical determinant 数字行列式numerical differentiation 数值微分numerical equality 数值等式numerical equation 数字方程numerical error 数值误差numerical example 数值例numerical function 数值函数numerical inequality 数值不等式numerical integration 数值积分法numerical invariant 不变数numerical mathematics 数值数学numerical method 数值法numerical model 数值模型numerical operator 数字算子numerical quadrature 数值积分法numerical series 数值级数numerical solution 数值解numerical solution of linear equations 线性方程组的数值解法numerical stability 数值稳定性numerical table 数表numerical value 数值numerical value equation 数值方程nutation 章动。

Singular values and eigenvalues of tensors a variational approach

ABSTRACT We propose a theory of eigenvalues, eigenvectors, singular values, and singular vectors for tensors based on a constrained variational approach much like the Rayleigh quotient for symmetric matrix eigenvalues. These notions are particularly useful in generalizing certain areas where the spectral theory of matrices has traditionally played an important role. For illustration, we will discuss a multilinear generalization of the Perron-Frobenius theorem. 1. INTRODUCTION It is well known that the eigenvalues and eigenvectors of a symmetric matrix A are the critical values and critical points of its Rayleigh quotient, x Ax/ x 2 2 , or equivalently, the critical values and points of the quadratic form x Ax constrained to vectors with unit l2 -norm, {x | x 2 = 1}. If L : Rn × R → R is the associated Lagrangian with Lagrange multiplier λ, L(x, λ) = x Ax − λ( x

mathematic求解薛定谔方程程序

5.编程求解薛定谔方程：BeginPackage["QuantumWell`"]Clear[PsiSym,PsiASym,Spectrum]PsiSym::usage="PsiSym[x_,k,a]determines the symmetric eigenfunctionfor a potential well of depth-V0.The input parameter k fixes the energy and2a the width of the well.Psisym is useful for a numerical representationof eigenfunctions."PsiASym::usage="PsiASym[x_,k_,a_]determines the antisymmetric eigenfunction for a potential well of depth-V0.The input parameter k fixes the energy and2a the width of the well. PsiASym is useful for a numerical representation of eigenfunctions."Spectrum::usage="Spectrum[V0_,a_]calculates the negative eigenvalues in a potential well.V0is the potential depth and2a the width of the well.The eigenvalues are returend as a list and are available in the variables lsymand lasym as replacement rules.The corresponding plots of eigenfunctions are stored in the variables Plsym and Plasym.The determining equation forthe eieenvalues is plotted."(*-一define global variables-一*)Plsym::usage="Variables containing the symmetric plots of the eigenfunctions."Plasym::usage="Variables containing the antisymmetric plots of the eigenfunctions."lsym::usage="List of symmetric eigenvalues."lasym::usage="List of antisymmetric eigenvalues."k::usage="Eigenvalue."Begin["`Private`"](*一symmetric eigenfunctions一*)Psisym[x,k,a]:=Module[{kapa,Al]},Kapa=k Tan[k a];(*一normalization constant一*)A1=1/Sqrt[a Exp[-2a kapa](1+1/(kapa a)+kapa/(k^2a)+Kapa^2/k^2)];(*一define the three domains of solution一一*)Which[-Infinity<x&&x<-a,A1Exp[kapa x],-a<=x&&x<=a,A1Exp[-kapa a]cos[kx]/Cos[k a],a<x&&x(Infinity,A1Exp[-kapa x]]](*---antisymmetric eigenfunctions---*)Pa iASym[x_,k_,a_]:=Module[{kapa,Al},Kapa=一k Cut[k a];(*一normalization constant一*)A1=1/Sqrt[a Exp[-2a kapa](1+1/(kapa a)+kapa/(k^2a)+Kapa^2/k^2)];(*一define the three domains of solution一*)Which[-Infinity<x&&x<-a，-Al Exp[kapa x],-a<=x&&x‘=a,A1Exp[-kapa a]Sin[k x]/Sin[k a],a<x&&x<Infinity,A1Exp[-kapa x]]](*一determination of the eigenvalues;plot of the eigenfunction:一*)Spectrum[V0_,a];=Module[{hbar=1，m=1,ymax,C2,rhs,lhssym,lhsasym,equatsym,equatasym,kmax,nsym,nasym, resultsym,resultasym},(*一define constants and the eigenvalue equation一*)C2=2m V0a^2/(hbar^2);Rhs=Tan[k a];Lhssym=Sqrt[C2-(k a)^2]/(k a);Lhsasym=-k a/Sqrt[C2-(k a)^2];equatsym=Sqrt[C2-(k a)^2]/(k a)一Tan[k a];equatasym=-k a/Sqrt[C2-(k a)^2]-Tan[k a];(*一location of the singularity in k一*)Kmax=Sqrt[C2/a^2];(*一number of symmetric eigenvalues一*)Nsym=Floor[N[kmax/(Pi/a)]]十1;(*一number of antisymmetric eigenvalues一*)nasym=Floor[N[(kmax一Pi/(2a))/(Pi/a)]]+1;(*-一initialize the lists for the eigenvalues一一*)Lsym={};lasym={};(*一calculate the symetric eigenvalue:一*)Do[resultsym=Chop[FindRoot[equatsym==0,{k,Pi/(2a)+0.1+(Pi/a)(i-1)}]];AppendTo[lsym,resultsym],{i,1,nsym}];(*一Chop[]replaces small numbers(<10^<(-10))by0一*)(*-一calculate he antisymmetric eigenvalues一*)Do[resultasym=Chop[FindRoot[equatasym=0,[k,Pi/(2a)+0.l+(Pi/a)(i-1)}]];AppendTo[lasym,resultasym],{i,1,nasym)];(*一plot the eigenvalue equation一*)Ymax=lhssym 1.5/.lsym[[1]];Off[Plot::plnr];Plot[{rhs,lhssym,lhsasym},{k,0.01,3kmax/2},P1otRange->{-ymax,ymax},Prolog->Thickness[0.001],PlotStyle->(RGBColor[1,0,0],Da shing[{}],Dashing[{1/60}]},AxesLabel->{"k",""}];On[Plot::plnr];(*一plot the symmetric eigenfunctions一*)Do[k1=k/.lsym[[i]];Plsym[i]=Plot[PsiSym[z,kl,a],{x,-2a,2a},AxesLabel->{"x","Psi_s"},P1otRange->All,Prolog->Thickness[0.001],P1otStyle->{Dashing[{1/(i20)}]}],(i,1,nsym)];(*一plot the antisymmetric eigenfunctions一*)Do[k1=k/.lasym[[i]];Plasym[i]=Plot[PsiASym[x,kl,a],[x,-2a,2a},AxesLabel->{"x","Psia"),P1otRange->All,Prolog->Thickness[0.001],Plotstyle->{Dashing[{1/(i 20)1]}],{i,1,nasym)];(*一print the eigenvalues一*)Print[""]:Print["---eigenvalues---’];Print["”];Do[k1=k/.lsym[[i]];If[i<=nasym,k2=k/.lasym[[i]],,k2="---"];Print["sym eigenvalue k",i,"=",k1,"asym eigenvalue k",i,"=",k2],{i,1,nsym}]]End[]EndPackage[]利用上面程序可以求解不同势阱高度和宽度的一维势阱薛定谔方程波函数和能量谱的程序包。

《统计词汇中英文对照》

《统计词汇中英文对照》Effect, 实验效应Eigenvalue, 特征值Eigenvector, 特征向量Ellipse, 椭圆Empirical distribution, 经验分布Empirical probability, 经验概率单位Enumeration data, 计数资料Equal sun-class number, 相等次级组含量Equally likely, 等可能Equivariance, 同变性Error, 误差/错误Error of estimate, 估计误差Error type I, 第一类错误Error type II, 第二类错误Estimand, 被估量Estimated error mean squares, 估计误差均方Estimated error sum of squares, 估计误差平方与Euclidean distance, 欧式距离Event, 事件Event, 事件Exceptional data point, 特殊数据点Expectation plane, 期望平面Expectation surface, 期望曲面Expected values, 期望值Experiment, 实验Experimental sampling, 试验抽样Experimental unit, 试验单位Explanatory variable, 说明变量Exploratory data analysis, 探索性数据分析Explore Summarize, 探索-摘要Exponential curve, 指数曲线Exponential growth, 指数式增长EXSMOOTH, 指数平滑方法Extended fit, 扩充拟合Extra parameter, 附加参数Extrapolation, 外推法Extreme observation, 末端观测值Extremes, 极端值/极值F distribution, F分布F test, F检验Factor, 因素/因子Factor analysis, 因子分析Factor Analysis, 因子分析Factor score, 因子得分Factorial, 阶乘Factorial design, 析因试验设计False negative, 假阴性False negative error, 假阴性错误Family of distributions, 分布族Family of estimators, 估计量族Fanning, 扇面Fatality rate, 病死率Field investigation, 现场调查Field survey, 现场调查Finite population, 有限总体Finite-sample, 有限样本First derivative, 一阶导数First principal component, 第一主成分First quartile, 第一四分位数Fisher information, 费雪信息量Fitted value, 拟合值Fitting a curve, 曲线拟合Fixed base, 定基Fluctuation, 随机起伏Forecast, 预测Four fold table, 四格表Fourth, 四分点Fraction blow, 左侧比率Fractional error, 相对误差Frequency, 频率Frequency polygon, 频数多边图Frontier point, 界限点Function relationship, 泛函关系Gamma distribution, 伽玛分布Gauss increment, 高斯增量Gaussian distribution, 高斯分布/正态分布Gauss-Newton increment, 高斯-牛顿增量General census, 全面普查GENLOG (Generalized liner models), 广义线性模型Geometric mean, 几何平均数Gini's mean difference, 基尼均差GLM (General liner models), 通用线性模型Goodness of fit, 拟与优度/配合度Gradient of determinant, 行列式的梯度Graeco-Latin square, 希腊拉丁方Grand mean, 总均值Gross errors, 重大错误Gross-error sensitivity, 大错敏感度Group averages, 分组平均Grouped data, 分组资料Guessed mean, 假定平均数Half-life, 半衰期Hampel M-estimators, 汉佩尔M估计量Happenstance, 偶然事件Harmonic mean, 调与均数Hazard function, 风险均数Hazard rate, 风险率Heading, 标目Heavy-tailed distribution, 重尾分布Hessian array, 海森立体阵Heterogeneity, 不一致质Heterogeneity of variance, 方差不齐Hierarchical classification, 组内分组Hierarchical clustering method, 系统聚类法High-leverage point, 高杠杆率点HILOGLINEAR, 多维列联表的层次对数线性模型Hinge, 折叶点Histogram, 直方图Historical cohort study, 历史性队列研究Holes, 空洞HOMALS, 多重响应分析Homogeneity of variance, 方差齐性Homogeneity test, 齐性检验Huber M-estimators, 休伯M估计量Hyperbola, 双曲线Hypothesis testing, 假设检验Hypothetical universe, 假设总体Impossible event, 不可能事件Independence, 独立性Independent variable, 自变量Index, 指标/指数Indirect standardization, 间接标准化法Individual, 个体Inference band, 推断带Infinite population, 无限总体Infinitely great, 无穷大Infinitely small, 无穷小Influence curve, 影响曲线Information capacity, 信息容量Initial condition, 初始条件Initial estimate, 初始估计值Initial level, 最初水平Interaction, 交互作用Interaction terms, 交互作用项Intercept, 截距Interpolation, 内插法Interquartile range, 四分位距Interval estimation, 区间估计Intervals of equal probability, 等概率区间Intrinsic curvature, 固有曲率Invariance, 不变性Inverse matrix, 逆矩阵Inverse probability, 逆概率Inverse sine transformation, 反正弦变换Iteration, 迭代Jacobian determinant, 雅可比行列式Joint distribution function, 分布函数Joint probability, 联合概率Joint probability distribution, 联合概率分布K means method, 逐步聚类法Kaplan-Meier, 评估事件的时间长度Kaplan-Merier chart, Kaplan-Merier图Kendall's rank correlation, Kendall等级有关Kinetic, 动力学Kolmogorov-Smirnove test, 柯尔莫哥洛夫-斯米尔诺夫检验Kruskal and Wallis test, Kruskal及Wallis检验/多样本的秩与检验/H 检验Kurtosis, 峰度Lack of fit, 失拟Ladder of powers, 幂阶梯Lag, 滞后Large sample, 大样本Large sample test, 大样本检验Latin square, 拉丁方Latin square design, 拉丁方设计Leakage, 泄漏Least favorable configuration, 最不利构形Least favorable distribution, 最不利分布Least significant difference, 最小显著差法Least square method, 最小二乘法Least-absolute-residuals estimates, 最小绝对残差估计Least-absolute-residuals fit, 最小绝对残差拟合Least-absolute-residuals line, 最小绝对残差线Legend, 图例L-estimator, L估计量L-estimator of location, 位置L估计量L-estimator of scale, 尺度L估计量Level, 水平Life expectance, 预期期望寿命Life table, 寿命表Life table method, 生命表法Light-tailed distribution, 轻尾分布Likelihood function, 似然函数Likelihood ratio, 似然比line graph, 线图Linear correlation, 直线有关Linear equation, 线性方程Linear programming, 线性规划Linear regression, 直线回归Linear Regression, 线性回归Linear trend, 线性趋势Loading, 载荷Location and scale equivariance, 位置尺度同变性Location equivariance, 位置同变性Location invariance, 位置不变性Location scale family, 位置尺度族Log rank test, 时序检验Logarithmic curve, 对数曲线Logarithmic normal distribution, 对数正态分布Logarithmic scale, 对数尺度Logarithmic transformation, 对数变换Logic check, 逻辑检查Logistic distribution, 逻辑斯特分布Logit transformation, Logit转换LOGLINEAR, 多维列联表通用模型Lognormal distribution, 对数正态分布Lost function, 缺失函数Low correlation, 低度有关Lower limit, 下限Lowest-attained variance, 最小可达方差LSD, 最小显著差法的简称Lurking variable, 潜在变量Main effect, 主效应Major heading, 主辞标目Marginal density function, 边缘密度函数Marginal probability, 边缘概率Marginal probability distribution, 边缘概率分布Matched data, 配对资料Matched distribution, 匹配过分布Matching of distribution, 分布的匹配Matching of transformation, 变换的匹配Mathematical expectation, 数学期望Mathematical model, 数学模型Maximum L-estimator, 极大极小L 估计量Maximum likelihood method, 最大似然法Mean, 均数Mean squares between groups, 组间均方Mean squares within group, 组内均方Means (Compare means), 均值-均值比较Median, 中位数Median effective dose, 半数效量Median lethal dose, 半数致死量Median polish, 中位数平滑Median test, 中位数检验Minimal sufficient statistic, 最小充分统计量Minimum distance estimation, 最小距离估计Minimum effective dose, 最小有效量Minimum lethal dose, 最小致死量Minimum variance estimator, 最小方差估计量MINITAB, 统计软件包Minor heading, 宾词标目Missing data, 缺失值Model specification, 模型的确定Modeling Statistics , 模型统计Models for outliers, 离群值模型Modifying the model, 模型的修正Modulus of continuity, 连续性模Morbidity, 发病率Most favorable configuration, 最有利构形Multidimensional Scaling (ASCAL), 多维尺度/多维标度Multinomial Logistic Regression , 多项逻辑斯蒂回归Multiple comparison, 多重比较Multiple correlation , 复有关Multiple covariance, 多元协方差Multiple linear regression, 多元线性回归Multiple response , 多重选项Multiple solutions, 多解Multiplication theorem, 乘法定理Multiresponse, 多元响应Multi-stage sampling, 多阶段抽样Multivariate T distribution, 多元T分布Mutual exclusive, 互不相容Mutual independence, 互相独立Natural boundary, 自然边界Natural dead, 自然死亡Natural zero, 自然零Negative correlation, 负有关Negative linear correlation, 负线性有关Negatively skewed, 负偏Newman-Keuls method, q检验NK method, q检验No statistical significance, 无统计意义Nominal variable, 名义变量Nonconstancy of variability, 变异的非定常性Nonlinear regression, 非线性有关Nonparametric statistics, 非参数统计Nonparametric test, 非参数检验Nonparametric tests, 非参数检验Normal deviate, 正态离差Normal distribution, 正态分布Normal equation, 正规方程组Normal ranges, 正常范围Normal value, 正常值Nuisance parameter, 多余参数/讨厌参数Null hypothesis, 无效假设Numerical variable, 数值变量Objective function, 目标函数Observation unit, 观察单位Observed value, 观察值One sided test, 单侧检验One-way analysis of variance, 单因素方差分析Oneway ANOVA , 单因素方差分析Open sequential trial, 开放型序贯设计Optrim, 优切尾Optrim efficiency, 优切尾效率Order statistics, 顺序统计量Ordered categories, 有序分类Ordinal logistic regression , 序数逻辑斯蒂回归Ordinal variable, 有序变量Orthogonal basis, 正交基Orthogonal design, 正交试验设计Orthogonality conditions, 正交条件ORTHOPLAN, 正交设计Outlier cutoffs, 离群值截断点Outliers, 极端值OVERALS , 多组变量的非线性正规有关Overshoot, 迭代过度Paired design, 配对设计Paired sample, 配对样本Pairwise slopes, 成对斜率Parabola, 抛物线Parallel tests, 平行试验Parameter, 参数Parametric statistics, 参数统计Parametric test, 参数检验Partial correlation, 偏有关Partial regression, 偏回归Partial sorting, 偏排序Partials residuals, 偏残差Pattern, 模式Pearson curves, 皮尔逊曲线Peeling, 退层Percent bar graph, 百分条形图Percentage, 百分比Percentile, 百分位数Percentile curves, 百分位曲线Periodicity, 周期性Permutation, 排列P-estimator, P估计量Pie graph, 饼图Pitman estimator, 皮特曼估计量Pivot, 枢轴量Planar, 平坦Planar assumption, 平面的假设PLANCARDS, 生成试验的计划卡Point estimation, 点估计Poisson distribution, 泊松分布Polishing, 平滑Polled standard deviation, 合并标准差Polled variance, 合并方差Polygon, 多边图Polynomial, 多项式Polynomial curve, 多项式曲线Population, 总体Population attributable risk, 人群归因危险度Positive correlation, 正有关Positively skewed, 正偏Posterior distribution, 后验分布Power of a test, 检验效能Precision, 精密度Predicted value, 预测值Preliminary analysis, 预备性分析Principal component analysis, 主成分分析Prior distribution, 先验分布Prior probability, 先验概率Probabilistic model, 概率模型probability, 概率Probability density, 概率密度Product moment, 乘积矩/协方差Profile trace, 截面迹图Proportion, 比/构成比Proportion allocation in stratified random sampling, 按比例分层随机抽样Proportionate, 成比例Proportionate sub-class numbers, 成比例次级组含量Prospective study, 前瞻性调查Proximities, 亲近性Pseudo F test, 近似F检验Pseudo model, 近似模型Pseudosigma, 伪标准差Purposive sampling, 有目的抽样QR decomposition, QR分解Quadratic approximation, 二次近似Qualitative classification, 属性分类Qualitative method, 定性方法Quantile-quantile plot, 分位数-分位数图/Q-Q图Quantitative analysis, 定量分析Quartile, 四分位数Quick Cluster, 快速聚类Radix sort, 基数排序Random allocation, 随机化分组Random blocks design, 随机区组设计Random event, 随机事件Randomization, 随机化Range, 极差/全距Rank correlation, 等级有关Rank sum test, 秩与检验Rank test, 秩检验Ranked data, 等级资料Rate, 比率Ratio, 比例Raw data, 原始资料Raw residual, 原始残差Rayleigh's test, 雷氏检验Rayleigh's Z, 雷氏Z值Reciprocal, 倒数Reciprocal transformation, 倒数变换Recording, 记录Redescending estimators, 回降估计量Reducing dimensions, 降维Re-expression, 重新表达Reference set, 标准组Region of acceptance, 同意域Regression coefficient, 回归系数Regression sum of square, 回归平方与Rejection point, 拒绝点Relative dispersion, 相对离散度Relative number, 相对数Reliability, 可靠性Reparametrization, 重新设置参数Replication, 重复Report Summaries, 报告摘要Residual sum of square, 剩余平方与Resistance, 耐抗性Resistant line, 耐抗线Resistant technique, 耐抗技术R-estimator of location, 位置R估计量R-estimator of scale, 尺度R估计量Retrospective study, 回顾性调查Ridge trace, 岭迹Ridit analysis, Ridit分析Rotation, 旋转Rounding, 舍入Row, 行Row effects, 行效应Row factor, 行因素RXC table, RXC表Sample, 样本Sample regression coefficient, 样本回归系数Sample size, 样本量Sample standard deviation, 样本标准差Sampling error, 抽样误差SAS(Statistical analysis system ), SAS统计软件包Scale, 尺度/量表Scatter diagram, 散点图Schematic plot, 示意图/简图Score test, 计分检验Screening, 筛检SEASON, 季节分析Second derivative, 二阶导数Second principal component, 第二主成分SEM (Structural equation modeling), 结构化方程模型Semi-logarithmic graph, 半对数图Semi-logarithmic paper, 半对数格纸Sensitivity curve, 敏感度曲线Sequential analysis, 贯序分析Sequential data set, 顺序数据集Sequential design, 贯序设计Sequential method, 贯序法Sequential test, 贯序检验法Serial tests, 系列试验Short-cut method, 简捷法Sigmoid curve, S形曲线Sign function, 正负号函数Sign test, 符号检验Signed rank, 符号秩Significance test, 显著性检验Significant figure, 有效数字Simple cluster sampling, 简单整群抽样Simple correlation, 简单有关Simple random sampling, 简单随机抽样Simple regression, 简单回归simple table, 简单表Sine estimator, 正弦估计量Single-valued estimate, 单值估计Singular matrix, 奇异矩阵Skewed distribution, 偏斜分布Skewness, 偏度Slash distribution, 斜线分布Slope, 斜率Smirnov test, 斯米尔诺夫检验Source of variation, 变异来源Spearman rank correlation, 斯皮尔曼等级有关Specific factor, 特殊因子Specific factor variance, 特殊因子方差Spectra , 频谱Spherical distribution, 球型正态分布Spread, 展布SPSS(Statistical package for the social science), SPSS统计软件包Spurious correlation, 假性有关Square root transformation, 平方根变换Stabilizing variance, 稳固方差Standard deviation, 标准差Standard error, 标准误Standard error of difference, 差别的标准误Standard error of estimate, 标准估计误差Standard error of rate, 率的标准误Standard normal distribution, 标准正态分布Standardization, 标准化Starting value, 起始值Statistic, 统计量Statistical control, 统计操纵Statistical graph, 统计图Statistical inference, 统计推断Statistical table, 统计表Steepest descent, 最速下降法Stem and leaf display, 茎叶图Step factor, 步长因子Stepwise regression, 逐步回归Storage, 存Strata, 层（复数）Stratified sampling, 分层抽样Stratified sampling, 分层抽样Strength, 强度Stringency, 严密性Structural relationship, 结构关系Studentized residual, 学生化残差/t化残差Sub-class numbers, 次级组含量Subdividing, 分割Sufficient statistic, 充分统计量Sum of products, 积与Sum of squares, 离差平方与Sum of squares about regression, 回归平方与Sum of squares between groups, 组间平方与Sum of squares of partial regression, 偏回归平方与Sure event, 必定事件Survey, 调查Survival, 生存分析Survival rate, 生存率Suspended root gram, 悬吊根图Symmetry, 对称Systematic error, 系统误差Systematic sampling, 系统抽样Tags, 标签Tail area, 尾部面积Tail length, 尾长Tail weight, 尾重Tangent line, 切线Target distribution, 目标分布Taylor series, 泰勒级数Tendency of dispersion, 离散趋势Testing of hypotheses, 假设检验Theoretical frequency, 理论频数Time series, 时间序列Tolerance interval, 容忍区间Tolerance lower limit, 容忍下限Tolerance upper limit, 容忍上限Torsion, 扰率Total sum of square, 总平方与Total variation, 总变异Transformation, 转换Treatment, 处理Trend, 趋势Trend of percentage, 百分比趋势Trial, 试验Trial and error method, 试错法Tuning constant, 细调常数Two sided test, 双向检验Two-stage least squares, 二阶最小平方Two-stage sampling, 二阶段抽样Two-tailed test, 双侧检验Two-way analysis of variance, 双因素方差分析Two-way table, 双向表Type I error, 一类错误/α错误Type II error, 二类错误/β错误UMVU, 方差一致最小无偏估计简称Unbiased estimate, 无偏估计Unconstrained nonlinear regression , 无约束非线性回归Unequal subclass number, 不等次级组含量Ungrouped data, 不分组资料Uniform coordinate, 均匀坐标Uniform distribution, 均匀分布Uniformly minimum variance unbiased estimate, 方差一致最小无偏估计Unit, 单元Unordered categories, 无序分类Upper limit, 上限Upward rank, 升秩Vague concept, 模糊概念Validity, 有效性VARCOMP (Variance component estimation), 方差元素估计Variability, 变异性Variable, 变量Variance, 方差Variation, 变异Varimax orthogonal rotation, 方差最大正交旋转Volume of distribution, 容积W test, W检验Weibull distribution, 威布尔分布Weight, 权数Weighted Chi-square test, 加权卡方检验/Cochran检验Weighted linear regression method, 加权直线回归Weighted mean, 加权平均数Weighted mean square, 加权平均方差Weighted sum of square, 加权平方与Weighting coefficient, 权重系数Weighting method, 加权法W-estimation, W估计量W-estimation of location, 位置W估计量Width, 宽度Wilcoxon paired test, 威斯康星配对法/配对符号秩与检验Wild point, 野点/狂点Wild value, 野值/狂值Winsorized mean, 缩尾均值Withdraw, 失访Youden's index, 尤登指数Z test, Z检验Zero correlation, 零有关Z-transformation, Z变换。

Eigenvalues of a real supersymmetric tensor

Abstract In this paper, we deﬁne the symmetric hyperdeterminant, eigenvalues and E-eigenvalues of a real supersymmetric tensor. We show that eigenvalues are roots of a one-dimensional polynomial, and when the order of the tensor is even, E-eigenvalues are roots of another one-dimensional polynomial. These two one-dimensional polynomials are associated with the symmetric hyperdeterminant. We call them the characteristic polynomial and the E-characteristic polynomial of that supersymmetric tensor. Real eigenvalues (E-eigenvalues) with real eigenvectors (E-eigenvectors) are called H-eigenvalues (Z-eigenvalues). When the order of the supersymmetric tensor is even, H-eigenvalues (Z-eigenvalues) exist and the supersymmetric tensor is positive deﬁnite if and only if all of its H-eigenvalues (Z-eigenvalues) are positive. An m th-order n -dimensional supersymmetric tensor where m is even has exactly n (m − 1)n −1 eigenvalues, and the number of its E-eigenvalues is strictly less than n (m − 1)n −1 when m ≥ 4. We show that the product of all the eigenvalues is equal to the value of the symmetric hyperdeterminant, while the sum of all the eigenvalues is equal to the sum of the diagonal elements of that supersymmetric tensor, multiplied by (m − 1)n −1 . The n (m − 1)n −1 eigenvalues are distributed in n disks in C. The centers and radii of these n disks are the diagonal elements, and the sums of the absolute values of the corresponding off-diagonal elements, of that supersymmetric tensor. On the other hand, E-eigenvalues are invariant under orthogonal transformations. © 2005 Elsevier Ltd. All rights reserved.

Perron–Frobenius theorem

Perron–Frobenius theorem:证明方阵，如果其行列的元素为正值，则存在最大的特征值，并且特征向量的每个元素是正的。

这个原理应用在统计推断，经济，人口统计学，搜索引擎的基础。

PF原理：说明了最大特征值的作用和实对称矩阵的作用。

如果临界矩阵A是个实对称矩阵，则具有下列性质：1. There is a positive real number r, called the Perron root or the Perron–Frobenius eigenvalue, such that r is an eigenvalue of A and any other eigenvalue λ (possibly, complex) is strictly smaller than r in absolute value, |λ| < r. Thus, the spectral radiusρ(A) is equal to r.2. The Perron–Frobenius eigenvalue is simple: r is a simple root of the characteristic polynomial of A. Consequently, eigenspace associated to r is one-dimensional. (Same is true for the left eigenspace i.e. eigenspace for A T.)3. There exists an eigenvector v =(v1,…,v n) of A with eigenvalue r such that all components of v are positive: A v = r v, v i > 0 for 1 ≤i≤n. (Respectively exists positive left eigenvector w : w T A = r w T, w i > 0.)转置就可以了。

奇异值分解

SINGULAR V ALUE DECOMPOSITION OF A MATRIX-V ALUED IMPULSE RESPONSEThomas L.Marzetta and Jack SalzMathematical Sciences Research CenterBell Laboratories,Lucent Technologies600Mountain Avenue,Murray Hill,NJ07974(908)582-3090tlm@ABSTRACTThe singular value decomposition(svd),a standard tool for theoretical and computational matrix analysis,represents a matrix as an ordered product of a unitary matrix,a non-negative,real diagonal matrix,and a second unitary ma-trix.Our contribution is to extend the svd representation to a matrix-valued impulse response.The representation in-volves a countably inﬁnite number of vector-valued eigen-functions and scalar singular values,and it provides the most concise and elegant description of the action of a matrix-valuedﬁlter(for example,a space-time communication chan-nel)when driven by aﬁnite-duration input signal.1.INTRODUCTIONThe singular value decomposition(svd)represents a complex matrix as a product,,whereis a diagonal square matrix having row and column dimen-sions whose diagonal elements,the sin-gular values,are real and nonnegative,and and are unitary matrices having row-dimensions of and re-spectively,and column dimensions equal to[6]. The superscript“”denotes“conjugate transpose”.The svd provides the simplest way to visualize the action of matrix multiplication,and it is a natural theoretical and compu-tational tool for formulating problems of over-and under-determined systems of linear equations.Our contribution is to extend the svd representation to the matrix-valued impulse response of a space-time linear channel.The mathematical basis for the svd is the repre-sentation by Kelly and Root[1]of a nonnegative-deﬁnite matrix-valued kernel in terms of scalar eigenvalues and vec-tor eigenfunctions.We discuss only the continuous-time case,but analogous discrete-time results hold also.2.NOTATIONA complex matrix-valued impulse response is easily moti-vated by a multiple-antenna wireless link having trans-mit antennas and receive antennas.Over a time interval ,an input signal,denoted,an vector function of input time,drives the transmit antennas.The resulting output at the receive antennas,denoted ,is an vector function of output time,where(1)where is an matrix whose-th ele-ment is the impulse response,,that connects the-th transmit antenna to the-th receive antenna,where the impulse is applied at time.Any linear time-varying, delay-spread channel is described by such an impulse re-sponse.For a time-invariant channel the dependence of on its second variable vanishes.3.SVD OF MATRIX IMPULSE RESPONSELet be a time-varying impulse response. Then for any speciﬁedﬁnite input interval and any speciﬁed output interval,eitherﬁnite or inﬁnite, the impulse response has the following exact representation over the indicated ranges of input time,,and output time, ,(2)where,the singular values,are nonnegative real scalars, ,the output eigenfunctions,are complex func-tions of that are orthonormal over the output interval,and ,the input eigenfunctions,are complex func-tions of that are orthonormal over the input interval,(3)(4) Note that the input and the output eigenfunctions are not eigenfunctions of the impulse response,but rather eigen-functions of certain covariance kernels.In general a spec-iﬁcation of different input or output intervals results in a different svd representation,in the same way that removing rows or columns from a matrix changes its svd.The svd simultaneously diagonalizes the channel over both space and time.For a given input,,the output of the channel(1)takes the form(5) After multiplying both sides of the expression by(on the left),and integrating with respect to,we obtain(6) where the and are the scalar coefﬁcients that rep-resent the input signal and the output signal with respect to the orthonormal and respectively,(7)The attainable values of the output are linear combina-tions of those whose associated singular values are signiﬁcantly greater than zero.The power of the svd is immediately appreciated by considering the problem of choosing aﬁnite duration in-put function,subject to an energy constraint,to ap-proximate a desired output function,over a speciﬁed interval,in the mean-square sense.To do so we expand the input and output functions in terms of the input and out-put eigenfunctions respectively,and formulate the following minimization problem,(8)where is a Lagrange multiplier.The optimum solution for the coefﬁcients of the input waveform is(9)where is adjusted to satisfy the speciﬁed energy constraint. In contrast,this problem could not be tackled directly with Fourier methods because of theﬁnite-time support of the input waveform.4.DERIV ATION OF SVDWe begin with the matrix version of Mercer’s theorem[1, 5].Starting with the channel impulse response,we deﬁne a Hermitian nonnegative-deﬁnite covariance kernel,,as follows,(10)Under reasonable mathematical conditions,Kelly and Root [1]proved that the kernel can be expanded in terms of vector eigenfunctions and scalar eigenvalues as follows,(11) where(12) the are nonnegative real eigenvalues,and theare orthonormal eigenfunctions that satisfy(4).Next we use the vector-valued eigenfunctionsas inputs to the channel to obtain vector-valued scaled output functions.First,consider the case where the eigenvalue,,is strictly positive.Then we deﬁne the cor-responding output function as follows,(13)A successive application of the deﬁnition(13),the deﬁnition of the covariance kernel(10),the eigenfunction/eigenvalue relation(12),and the orthogonality of the eigenfunctions (4)proves that the that are associated with strictly positive eigenvalues are mutually orthonormal,(14) The output of the channel resulting from an application of an eigenfunction to the input of the channel whose eigen-value is zero is equal to zero,because otherwise,a“squar-ing”and integration of the expression with respect to wouldimply that the eigenvalue is greater than zero-a contradic-tion.For strictly positive eigenvalues we rewrite(13)as(15) Now we multiply both sides of(15)(on the right)by,and sum over such that is positive,(16) We claim that(17) To show this we apply an input to the channel which can be expressed as,where is spanned by the eigenfunctions having positive eigenvalues, and where is orthogonal to those same eigenfunc-tions,(18)and.Of the two additive components of,the channel responds only to.(It can be shown that the energy at the output of the channel due to is equal to zero.)The channel response to is identical to the response to,soThis proves(17)which,when substituted into(16)yields the desired svd representation.5.SPECIAL CASES OF SVDWe now consider some special cases of the svd:the diag-onal kernel,an impulse response that is separable in space and time,the time-invariant channel for large time-bandwidth product,andﬁnally the ideal bandlimited scalar channel. For three of these cases the svd exhibits distinct space-time structure.In general the svd makes no distinction between spatial and temporal degrees of freedom,and one cannot as-sociate individual singular values with either spatial or tem-poral diversity.5.1.Diagonal kernelHere the kernel(10)is diagonal,and the-th di-agonal element is itself a scalar kernel,.In turn, each scalar kernel has an expansion in terms of eigenvalues and scalar eigenfunctions.The corresponding vector-valued eigenfunctions of the matrix kernel have all but one of their components equal to zero.Thus the svd has a natural double-index form,(19) where the-index denotes space,the-index denotes time, and only the-th component of is nonzero,where(20)5.2.Separable impulse responseA separable impulse response,characteristic of theﬂat-fading multiple-antenna channel,is equal to the product of a con-stant matrix,,and a scalar impulse response,,(21) In turn,each of the two factors has its own svd,Again we have a double-index(e.g.,space-time)singular value decomposition of the form(19),where5.3.Time-invariant channel,large time-bandwidth prod-uctHere the channel impulse response is,whose matrix-valued frequency response isWe specify the same input and output intervals of, and we assume that the frequency response is smooth overfrequency intervals of duration comparable to.Then asymptotically,for large time-bandwidth products,,where is the channel bandwidth,the channel impulse response again has a double-index svd,(22) wherewhere the,,and constitute the svd of the matrix-valued frequency response,evaluated at the fre-quency,(23)Multiple-antenna OFDM(orthogonal frequency division multiplex)techniques can be interpreted in light of the svd (22).The wideband complex pulse that is fed to each trans-mit antenna is a linear combination of harmonically related sinewaves,that are extended periodically(the cyclic preﬁx) beyond the interval by an amount that exceeds the duration of the channel impulse response.This ensures that all of the output sinewaves exist in a transient-free state over an interval of duration greater than.5.4.Ideal bandlimited scalar channelThe ideal bandlimited scalar channel has an impulse re-sponse(24)The svd of the channel,for the input interval, and the output interval is(25) The are called prolate spheroidal wave functions.The covariance kernel for which the are eigen-functions is equal to the channel impulse response itself,a property unique to this channel.This property implies that the output eigenfunctions,in addition to being orthonormal over the inﬁnite time interval,are also orthogonal over the ﬁnite interval.To see this,we proceed as fol-lows(26) As a result of equating theﬁrst and last terms of the above equations,we haveThe double-orthogonality of the implies that the svd representation(25),where the output interval is, yields a second svd representation where the output interval is,(27)6.CONCLUSIONSThe svd of the matrix-valued impulse response is a powerful tool that enables us to analyze waveform channels with the same facility that we analyze ordinary matrix operations.7.REFERENCES[1]E.J.Kelly and W.L.Root,“A representation of vector-valued random processes”,MIT,Lincoln LaboratoryReport,1960.[2]D.Slepian and H.O.Pollak,“Prolate Spheroidal WaveFunctions,Fourier Analysis and Uncertainty–I”,BellSystem Tech.J.,vol.40,pp.43–63,1961.[3]ndau and H.O.Pollak,“Prolate SpheroidalWave Functions,Fourier Analysis and Uncertainty–II”,Bell System Tech.J.,vol.40,pp.65–84,1961. [4]ndau and H.O.Pollak,“Prolate SpheroidalWave Functions,Fourier Analysis and Uncertainty–III:The Dimension of the Space of Essentially Time-and Band-Limited Signals”,Bell System Tech.J.,vol.41,p.1295-1336,1962.[5]H.L.van Trees,Detection,Estimation,and Modula-tion Theory,Part I,pp.220–224,Wiley,New York,1968.[6]G.Strang,Linear Algebra and its Applications,thirdedition,Harcourt Brace Jovanovich,San Diego,1988.。

AbaqusCAE-CONT-W04-Q-DiskForging

Workshop 4Disk Forging AnalysisInteractive VersionNote: This workshop provides instructions on using contact interactions in terms of the Abaqus GUI interface. If you wish to use the Abaqus Keywords interface instead, please see the “Keywords” version of these instructions. Please complete either the Keywords or Interactive version of thisworkshop. IntroductionWhen you complete this workshop you will be able to: • Modify the shape of a rigid part. • Understand negative eigenvalue warning messages.• Identify regions in a model in need of mesh refinement.• Create X –Y plots in Visualization module.Compression of an elastic-plastic steel blank into a diskFigure W4–1 Model geometry. This workshop models the forging of a steel disk. The steel is defined to have elastic-plastic material behavior. The die is assumed to be rigid compared to the steel. The steel blank is held fixed along the bottom and along the axis of symmetry. The modelgeometry is shown in Figure W4–1. The blank is 8.0 cm ⨯ 3.175 cm with a 0.5 cm round at the top-right corner. The die is translated -2.223 cm in the 2-direction during the forging process.Steel blank Rigid die Axis ofsymmetryU2W4.2 PreliminariesEnter the working directory for this workshop:../contact/interactive/forgingRun the script ws_contact_forging.py using the following command:abaqus cae startup=ws_contact_forging.pyThe above command creates an Abaqus/CAE database named forging.cae in the current directory. The model geometry, material properties, and loading history have already been created in the database.Frictionless disk forging analysisNode-to-surface methodIn this section you will complete the rigid surface model by smoothing the rigid part geometry and defining the contact interaction. Once the model definition is complete, you will run the analysis and visualize the results.1.Smooth the part named rigidDie, by rounding the two corners with a filletradius of 0.005. Detailed instructions are below:a.Underneath the Parts container, expand the part named rigidDie. In thelist that appears, expand the Features container and double-click 2DAnalytic rigid shell-1. In the Edit Feature dialog box, click Edit SectionSketch.b.From the main menu bar, select Add→Fillet.c.Enter a fillet radius of 0.005.d.Select the two edges near the intersection to be filleted. Repeat theselection to fillet both corners.The final sketch of the rigid die is shown in Figure W4–2.e.Click Done in the prompt area when the sketch modification is complete.f.Click OK in the Edit Feature dialog box.Figure W4–2 Rigid Die.W4.3 Next, you will define the contact interaction.2.In the Model Tree, expand the Assembly container and double-click Surfaces. the surface rigidDie. In the viewport, select the rigid die and choose theside of the die facing the blank (Magenta). Click Done in the prompt area.4.Create a geometry-based surface named blank including the contact surface onthe blank. In the prompt area, select by angle as the selection method and click the right edge of the blank.This automatically selects all the edges of the blank that may that may come into contact with the die.5.Create a frictionless interaction property named NoFric.6.Create a contact interaction in Step-1 to define the interaction between thesurfaces rigidDie and blank. Use the Node to surface discretization method.Question W4–1:Which slave type should be used for this interaction, surfaceor node region?Question W4–2:Which contact formulation, small or finite sliding, do youthink is appropriate for this analysis? Why?7.Submit the job rigidDie for analysis. Use the job monitor to check for warningsand errors. Are there any negative eigenvalues? Open the message file (.msg) and search for negative eigenvalues.Negative eigenvalues indicate that the stiffness matrix is not positive definite. The stiffness matrix may become nonpositive definite for a number of reasons:•Lagrange multipliers are used (e.g., contact or hybrid elements). Abaqus automatically suppresses most negative eigenvalue warning messagesassociated with Lagrange multipliers, however. (The exception occurswhen three-dimensional, quadratic elements are used to define contactsurfaces.)•The stiffness matrix is assembled about a non-equilibrium state. The warnings in this case generally disappear with repeated iterations and canbe safely ignored.•The structure has become unstable (i.e., it has buckled or an unstable material response is occurring). Negative eigenvalue warnings that occurdue to real instabilities should be investigated very carefully.Question W4–3:What is causing the negative eigenvalues in this analysis—theLagrange multipliers associated with the contact constraints oran instability in the model?8.Look at the deformation of the model.Question W4–4:Where does it appear that refinement of the model is needed?9.Contour some results, such as CPRESS and PEEQ, in the Visualization module.W4.4 Question W4–5:Is the location of the largest plastic strains—that is, the peakvalue of PEEQ—where you would have expected it to be?Question W4–6:Are the contours of CPRESS difficult to visualize? What canbe done to make the CPRESS contours easier to visualize?10.Create an X–Y plot of the vertical reaction force at the rigid body reference node.The history output request to generate the necessary history output was created by the workshop setup script.a.In the Results Tree, expand the History Output container for the outputdatabase file named rigidDie.odb.b.Double-click the curve named Reaction force: RF2 PI: RIGIDDIE-1Node 1 in NSET REFPT.c.Click mouse-button 3 on the curve and select Save As from the menu thatappears. Name the curve RF2-rigid-n2s.The curve appears in the XYData container of the Results Tree.Question W4–7:Why is there a sudden increase in the force required to movethe die late in the analysis?paring the deformed shapes from earlier points in the analysis to later pointsin the analysis may give you some hints. The procedure is described below.a.In the toolbox, click, .b.From the toolbar, click and drag the slider to display differentincrements (alternatively, select Result→Step/Frame and select anincrement to be displayed).Surface-to-surface method1.In the Model Tree, copy the model named rigidDie to one named rigidDie-s2s.2.Collapse the model named rigidDie.3.In the new model, do the following:a.Edit the contact interaction and change the discretization method toSurface to surface.b.Edit the step and select the Unsymmetric equation solver.4.Create a job named rigidDie-s2s and submit it for analysis.5.From the main menu bar, select File→Save to save your model database.6.Examine the results obtained with the surface to surface method with thoseobtained previously. For example, compare the reaction force histories and thedeformed shapes. You will find that the reaction force histories are very similar;the level of penetration into the blank, however, is significantly reduced using the surface-to-surface method.W4.5 Disk forging analysis with frictionDisk forging analysis with penalty friction (μ = 0.1)In this section you will add penalty friction to the forging model and view the results.1.Copy the model rigidDie to a model named rigidDie-fric.2.Create a new contact property with penalty friction.a.In the Model Tree, double-click Interaction Properties. the property Friction and accept contact as the type.c.In the Edit Contact Property dialog box, select Mechanical→TangentialBehavior.d.Select Penalty from the list of available friction formulations. Enter afriction coefficient of 0.1 and accept the default settings for this frictionmodel.e.Click OK.3.Edit the contact interaction and select Friction as the interaction property.4.Create a job for this model and submit it for analysis.5.Look at the results in the Visualization module.Question W4–8:Where would you expect the peak plastic strains to be located?6.Contour the PEEQ variable, and find out where they are.Question W4–9:Can you explain why the region of highest plastic strain hasshifted?7.There is a contact surface variable that can be plotted when friction is in themodel: the frictional shear stress at the interface. Contour this variable asdescribed next.a.From the main menu bar, select Plot→Contours→On UndeformedShape (or click in the toolbox).b.From the main menu bar, select Result→Field Output. In the PrimaryVariable tabbed page, select CSHEAR1and click OK.As you saw earlier, it is difficult to view surface-based results using the defaultcontour plot settings when the model is two dimensional. As before, sweep theaxisymmetric model to better view the surface contours.Question W4–10: In terms of the global coordinates in the plane of definition, inwhich direction does a positive value of CSHEAR1 act alongthe top edge of the blank? In which direction along the side ofthe blank?W4.6 Disk forging analysis with Lagrange friction (μ = 0.1)Next, you will perform the forging analysis using the Lagrange multiplier formulation for friction.1.Copy the model rigidDie-fric to a model named rigidDie-lagr.2.Edit the contact property Friction so that the Lagrange multiplier frictionformulation is used.Continue to use a friction coefficient of 0.1.With Lagrange friction this analysis encounters convergence difficulties, whichcan be avoided by including contact stabilization.3.Create contact controls with a contact stabilization factor of 1.a.In the Model Tree, double-click Contact Controls. the controls stabilize and accept Abaqus/Standard contactcontrols as the type.c.In the Edit Contact Controls dialog box, switch to the Stabilizationtabbed page and choose Automatic stabilization with the default factorof 1.d.Click OK.4.Edit the contact interaction and choose stabilize as the contact controls.5.Create a job for this model and submit it for analysis.6.Study results in the Visualization module. How do they compare to the resultsusing the penalty method? Compare X–Y plots of the reaction forces at the rigidbody reference node for the two analyses.Question W4–11: How much more expensive was the analysis with Lagrangefriction compared to the analysis with the penalty methodfriction in terms of run time? In terms of required memory?Question W4–12: Was the additional computational expense justified for thismodel?Disk forging analysis with penalty friction (μ = 0.35)In this section you will increase the friction coefficient to 0.35.1.Copy the model rigidDie-fric to a model named rigidDie-mu35.2.Edit the contact property Friction so that penalty friction is used with a frictioncoefficient of 0.35.Question W4–13: Are there any additional changes to the model that aresuggested when high coefficients of friction are used(μ > 0.2)?3.Create a job for this model and submit it for analysis. Compare the computationalexpense for this analysis to the cost of the analysis where μ = 0.1. If you look atthe summary of the analysis given at the bottom of the message file, you will find that the analysis with the higher coefficient of friction was more expensive.4.Run the analysis without the unsymmetric solver.W4.7a.Copy the model rigidDie-mu35 to a model named rigidDie-mu35-SYMM.b.In the Model Tree, expand the Steps container and double-click Step-1.c.In the Edit Step dialog box, click the Other tab and choose theSymmetric matrix solver.d.Click OK.e.Create a job for this model and submit it for analysis.Question W4–14: How much CPU time was saved by using the unsymmetricsolver? More memory is needed with this solver. How muchmore memory was needed for the analysis with theunsymmetric solver?5.In the Visualization module, compare contour plots of pertinent variables for thetwo analyses with = 0.35. It is possible to have several windows inAbaqus/CAE so that multiple plots can be displayed simultaneously. Use thefollowing procedure to display contours for the two analyses side by side.a.If your current viewport is maximized, click the restore button on the rightside of the main menu bar.b.From the main menu bar, select Viewport→Create.The new viewport appears. The title bar of the new viewport ishighlighted to indicate that it is the current viewport; all work takes placein the current viewport. To make a viewport current, simply click on theviewport title bar.c.To arrange the viewports so that both are clearly visible, selectViewport→Tile Horizontally from the main menu bar.d.In one viewport, open the output database for the model rigidDie-mu35.In the other viewport, open the output database for the model rigidDie-mu35-SYMM.e. Create contour plots for various field output variables for comparison.pare X–Y plots of the reaction forces at the rigid body reference point fromthe two models.Question W4–15: Did you expect the reaction forces for the two analyses to besimilar or different? Why?7.Save your Abaqus/CAE model database.W4.8Optional exercisesSurface-to-surface formulation1.Repeat the friction analyses using the surface-to-surface finite-sliding formulation.Only consider the penalty friction methods since the Lagrange multiplier methodfor friction is not recommended for surface-to-surface contact.2.View at the deformed shape and contour plots in the Visualization module.Disk forging analysis with rough friction1.Copy one of the node-to-surface rigid die models and modify it so that the roughfrictional contact formulation is applied along the interface between the blank andthe die. This condition will prevent any sliding of the two surfaces when they arein contact.Question W4–16: Is this interface condition realistic for this type of problem?Question W4–17: What types of problems would you expect in this analysis?Why? Do you think the analysis will be able to complete thefull loading?2.Submit the model for analysis.3.View at the deformed shape and contour plots in the Visualization module.Question W4–18: Are the results consistent with your answers to the earlierquestions?Note: A script that creates the complete model described in these instructions is available for your convenience. Run this script if you encounter difficulties following the instructions outlined here or if you wish to check your work. The script is namedws_contact_forging_answer.pyand is available using the Abaqus fetch utility.。

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No doubt many people benifited from the moderator's answers but thatdoesnt give him any right to discard some people.It is not possible in a public forum to meet the "tastes"of someselective people.If you are an expert on some matter ,try to help the other fellow byanswering his "silly"queries which incidentlyy are not silly to theposter.If you have nothing to offer,just dont waste time writingrrealms about how you get frustrated by reading other posts not toyour taste.
(b) Watch out for beams that are free to spin about theiraxes.
PS: I am re-posting it coz previous one didn,t appear in the forum.
It is not an intellectual act to block informative messages like this.
Everybody out there is not an expert on ABAQUS.Experts are always encouraged to give expert advice.But at the sametime they should not feel on cloud no.9 and look down uponinexperienced users of ABAQUS as some kind of menace to them.This is a public forum every one has an equal place in it.And everything was so peaceful until there was a change in moderatorrecently,who started so called "quality"control of the forum.What is quality?It is a relative term.
Re
It has NOT been moderated, you're a little bit paranoiac not ?? Seemessage 9935 !!!! And please stop to be agressive, we are not herefor that, thx.Btw, I do not agree on point 3. Rigid body motions create numericalsingularities, not negative eigenvalues, or they are indirectconsequences.
We do need "quality" moderation or the forum will be full of "Trash"messages. In fact, a lot of efforts are exerted by the moderator who Iassume has to read every message before posting. The moderator also doesa lot of work to answer every question he can while keeping a highquality for the forum.I, personally, have benefited a lot from many of the answers that themoderator posts. I also get frustrated by a lot of messages that I wastetime reading, like for example those messages from people who want tolearn how to use ABAQUS.. Or questions like: what is finite elementmodeling.Personally, I am thankful that there is someone who wastes his time inreading and moderating these messages so that I don't waste my own timereading them.Again, these are just personal opinions
Simply stated it is fitness for the purpose.This forum,like many other famous forums on net,is supposed to caterto all people and not to show bias towards selected few.Like I said earlier there is no need to restrict any techinical postas long as it is not some kind of spam.
How come members' techinical postings come under "quality "scan and moderator'smeaningless messages,like thanks givings to his good ol' friends,being forced onus, pass the stringent"quality"criteria?The best government is that whaich is least governed.So is with moderation.Let this message appear in the forum and let people decide for themselves.
(1) Non-physica源自 material properties.
(a) Negative Young's modulus, negative slope ofstress-strain curve, etc.
(b) Check input file to be sure abaqus read the inputdeck properly.
There is no place for personal whims and fancies in a public place.The common law should prevail.All are equal here.Elite people,kindly bear with us,ignorant morans.
Well if you check the posts before the owner change, you will see thatbenz's responses amount to about 30% or more of the "expert's advice",as we might call it. Other historical memebers, such as Fern A.Tomassy or Dave Lindeman have slowed down significantly lately. Thereason? There are many, for sure, but you must agree that repeatinganswers every 2 or 3 weeks stops being interesting pretty quickly.So instead of complaining about some message being moderated andothers not, try to think about WHY it was blocked in the first place.You might find out that your same question (or a very similar one forthat matter) was answered less than 6 months ago.Or maybe all you have to do is take tha manual and read a little,remember that ABAQUS comes with online, pdf and printed manuals. Iagree that *some* sections could see an improvement, but after all thepreface of all manuals says that familiarity with the finite elementmethod is assumed (ABAQUS is a code that assumes graduate level inengineering). So in my opinion some questions actually don't belong inthe ABAQUS group but in a basic finite element or numerical methods(undergraduate) lesson.

Negative Eigenvalue