5.4 Four fundamental subspaces

四阶微分方程组英文The Four-Dimensional Differential Equation SystemThe study of differential equations has long been a fundamental aspect of mathematics, with applications spanning a wide range of scientific and engineering disciplines. Among the most complex and intriguing forms of these equations are the four-dimensional differential equation systems, which possess a unique set of properties and challenges. In this essay, we will delve into the intricacies of these systems, exploring their mathematical foundations, their applications, and the insights they provide into the natural world.At the core of a four-dimensional differential equation system lies the concept of a system of coupled partial differential equations, where the variables depend on four independent variables, typically representing the three spatial dimensions and time. These systems are often used to model complex physical phenomena that involve the interplay of multiple interrelated processes, such as fluid dynamics, electromagnetic fields, and quantum mechanics.One of the defining characteristics of four-dimensional differentialequation systems is their inherent complexity. The addition of a fourth dimension to the problem domain introduces a level of complexity that is not present in lower-dimensional systems. This complexity manifests itself in the form of increased mathematical rigor, the need for advanced numerical methods, and the challenge of visualizing and interpreting the solutions.Despite the challenges, the study of four-dimensional differential equation systems has yielded numerous insights and advancements in various fields. In the realm of fluid dynamics, for example, these systems are used to model the behavior of fluids in three-dimensional space, taking into account the temporal evolution of the flow. This has led to a deeper understanding of phenomena such as turbulence, vortex formation, and the behavior of fluids in complex geometries.In the field of electromagnetism, four-dimensional differential equation systems are employed to describe the propagation of electromagnetic waves, including the behavior of light, radio waves, and other forms of radiation. These models have been instrumental in the development of advanced communication technologies, as well as in the study of the interactions between electromagnetic fields and matter.Furthermore, in the realm of quantum mechanics, four-dimensionaldifferential equation systems play a crucial role in the description of the behavior of subatomic particles and the evolution of quantum states. The Schrödinger equation, a fundamental equation in quantum mechanics, is a four-dimensional differential equation system that describes the wave function of a particle and its evolution over time.The mathematical complexity of four-dimensional differential equation systems has also led to the development of sophisticated numerical methods and computational tools. These tools enable researchers and engineers to simulate and analyze the behavior of these systems, allowing for the prediction of complex phenomena and the design of innovative technologies.One of the challenges in working with four-dimensional differential equation systems is the difficulty in visualizing and interpreting the solutions. The addition of a fourth dimension makes it challenging to represent the solutions in a way that is intuitive and easily understandable. However, advancements in computer graphics and visualization techniques have helped to overcome this obstacle, allowing researchers to gain deeper insights into the behavior of these systems.Despite the challenges, the study of four-dimensional differential equation systems continues to be an active and vibrant area ofresearch, with applications in a wide range of fields. As our understanding of these systems deepens, we can expect to see further advancements in fields such as fluid dynamics, electromagnetism, and quantum mechanics, leading to new discoveries and technological breakthroughs.In conclusion, the four-dimensional differential equation system is a fascinating and complex area of study that has far-reaching implications for our understanding of the natural world. Its mathematical intricacies, computational challenges, and diverse applications make it a rich and rewarding field of inquiry, with the potential to unlock new insights and drive innovation in a wide range of scientific and engineering disciplines.。

新视野大学英语第四册Unit4课文翻译新视野大学英语第四册Unit 4课文翻译新视野大学英语第四册第四单元的课文跟电信网络有关，下面是店铺分享的课文翻译，欢迎大家阅读!新视野大学英语第四册Unit 4课文翻译一个将会大大提高发展中国家生活水准的转变正方兴未艾。

一些不久前还是信息闭塞的地方正在迅速获得最新的通信技术，这将促进他们吸纳国内外投资。

亚洲、拉丁美洲和东欧的许多国家也许需要10年时间来改善其交通、电力供应和其他公用设施。

但是只一根直径小于半毫米的光纤电缆就可以比由铜丝制成的粗电缆承载更多的信息。

由于安装了光纤电缆、数字转换器和最新的无线传输系统，从北京到布达佩斯的一系列城区和工业区正在直接步入信息时代。

一个蛛网般的数字和无线通信网络已经发展到亚洲的大部分地区和东欧的部分地区。

所有这些发展中地区都把先进的通信技术看作一种跨越经济发展诸阶段的途径。

例如，信息技术的广泛应用有望缩短劳动密集型的组装工业转向涉及工程、营销和设计的那些产业所需的时间。

现代通信技术“将使中国、越南这样的国家比那些困于旧技术的国家拥有巨大的优势”。

这些国家应以多快的速度向前发展是人们争论的一个问题。

许多专家认为，越南在目前急需电话的情况下，却要求所有的移动电话都必须是昂贵的数字型电话，这种做法太超前了。

一位专家说：“这些国家缺乏成本估算和选择技术的经验。

”然而毋庸争辩，通信技术将是区分输赢的关键因素。

看一看俄罗斯的情况吧。

由于其坚实的数学和科学教育基础，它应该在信息时代有繁荣的发展。

问题是，它的国内电话系统是一堆生锈的20世纪30年代的老古董。

为了解决这一问题，俄罗斯已经开始铺设光纤电缆，并制定了投入400亿美元建设多项通信工程的战略计划。

但是由于其经济陷于低迷，几乎没有资金来着手解决最基本的问题。

与俄罗斯相比，在未来10年中，中国大陆计划对通信设备投入1,000亿美元。

从某种意义上说，中国的落后成了一种有利因素，因为这一发展正好发生在新技术比铜线电缆系统更便宜的时候。

Dual problem对偶问题442,446EEconomics 经济学435,439Eigencourse 457,458Eigenvalue 特征值283,287,374,499 Eigenvalue changes 特征值变换439284,294,300 Eigenvalues of 的特征值297Eigenvalues of 的特征值362Eigenvalues of 的特征值Eigenvector basis 基底的特征向量399Eigenvectors 特征向量283,287,374Eigshow 290,368Elimination 消元法45-66,83,86,135Ellipse 椭圆290,346,366,382Energy 能量343,409Engineering 工程409,419Error 误差211,218,219,225,481,483 Error equation 误差方程477Euler angles 欧拉角474Euler’s formula 欧拉公式311,426,430,497Even 偶数113,246,258,452 Exponential 指数的314,319,327FFactorization 因式分解95,110,235,348,370,374 False proof 假证明305,338Fast Fourier Transform 快速傅立叶变换393,493,511,565Feasible set 可行集440,441FFT(see Fast见快速傅里叶变换509-514Fourier Transform)Fibonacci 菲波那契75,266,268,301,302,306,308 Finite difference 有限差分315-317,417Finite elements 有限元412,419First-order system 一阶方程组315,326Fixed-free 固定-自由410,414,417,419Force balance 平衡力412FORTRAN 16,38Forward difference 前向差分30Four Fundamental Subspaces 四个基本子空间184-199,368,424,507 Fourier series 傅立叶序列233,448,450,452Fourier Transform 傅立叶变换393,509-514Fredholm Alternative 弗雷德霍姆择一203Free 自由133,135,137,144,146,155 Full column rank 列满秩157,170,405Full row rank 行满秩159,405Function space 函数空间121,448,449 Fundamental Theorem of Linear线性代数基本定理188,198,368Algebra(see Four Fundamental Subspaces)GGaussian elimination 高斯消元法45,49,135Gaussian probability distribution 高斯概率分布455Gauss-Jordan 高斯-约当83,84,91,469Gauss-Seidel 高斯-塞德尔481,484,485,489Gene expression data 基因表达数据457Geometric series 等比数列436Gershgorin circles 格尔什戈林圆491Gibbs phenomenon 吉布斯效应451Givens rotation 吉文斯旋转471Google 谷歌368,369,434Gram-Schmidt 格拉姆-施密特223,234,236,241,370,469 Graph 图表74,143,307,311,420,422,423 Group 群119,354HHalf-plane 半平面7Heat equation 热方程式322,323Heisenberg 海森堡305,310Hilbert space 希尔伯特空间447,449Hook e’s Law 虎克定律410,412Householder reflections 镜像变换237,469,472Hyperplane 超平面30,42IIll-conditioned matrix 病态矩阵371,473,474Imaginary 虚数289Independent 独立的26,27,134,168,200,300 Initial value 初值313Inner product 内积11,56,108,448,502,506 Input and output basis 基底输入输出399Integral 积分24,385,386Interior point method 内点法445Intersection of spaces 交空间129,183Inverse matrix 逆矩阵24,81,27082Inverse of的逆Invertible 可逆的86,173,200,248Iteration 迭代481,482,484,489,492JJacobi 雅可比481,483,485,489Jordan form 约当型356,357,358,361,482 JPEG 364,373KKalman filter 卡尔曼滤波器93,214Kernel 核377,380Kirchhoff’s Laws 基尔霍夫定律143,189,420,424-427 Krylov 克雷洛夫491,492L225,480norm 和范数Lagrange multiplier 拉格朗日乘子445Lanczos method 兰索斯方法490,492LAPACK 线性代数软件包98,237,486Leapfrog method 跳步法317,329Least squares 最小平方218,219,236,405,408,453184,186,192,425Left nullspace左零空间Left-inverse 左逆的81,86,154,405Length 长度12,232,447,448,501Line 线34,40,221,474Line of springs 线弹簧411Linear combination 线性组合1,3Linear equation 线性方程23Linear programming 线性规划440Linear transformation 线性变换44,375-398Linearity 线性关系44,245,246Linearly independent 线性独立26,134,168,169,200 LINPACK 线性系统软件包465Loop 环路307,425,426Lower triangular 下三角9598,100,474Lucas numbers 卢卡斯数306MMaple 38,100Mathematica 38,100MATLAB 17,37,237,243,290,337,513 Matrix(see full page 570) 矩阵22,384,387Matrix exponential 矩阵指数314,319,327Matrix multiplication 矩阵乘法58,59,67,389----。

第一单元一、英译汉亚里士多德是古希腊的哲学家和科学家。

他的作品涵盖了许多学科，包括物理学、生物学、动物学、逻辑学、伦理学、诗歌、戏剧、音乐、语言学、政治和政府，构成了第一个综合的西方哲学体系。

亚里士多德是第一个将人类的知识领域划分为不同学科的人，如数学，生物学和伦理学。

他相信人所有的观念和所有的知识在根本上都是基于感知能力。

他对自然科学的看法构成了他许多作品的基础。

他几乎对他所处时期的每一个人类知识领域都作出了贡献。

他的作品包含了人们所知的最早的关于逻辑的正式研究，即使在今天，亚里士多德哲学所涵盖的方方面面仍是学术研究的重要课题。

他的哲学对所有的西方哲学理论的发展有着经久不衰的影响。

在去世2,300多年后，亚里士多德仍是最有影响力的哲学家和科学家之一。

二、汉译英The Doctrine of the Mean is the core of Confucianism. The so-called "mean" by Confucius doesn't mean "compromise" but a "moderate" and "just-right" way when understanding and handling objective things. Confucius advocated that this thought should not only be treated as a way to understand and deal with things but also be integrated into one's daily conduct to make it a virtue through self-cultivation and training. The Doctrine of the Mean is not only the core of Confucianism but also an important component of traditional Chinese culture. From the time it came into being to the present, it has played an invaluable role in the construction of national spirit, the transmission of national wisdom, and the development of national culture.第二单元二、英译汉人们普遍认为，威廉•莎士比亚是最伟大的英语作家和世界杰出的戏剧家。

斯坦福大学开设的数学基础54集（Math 54）是一门专门为工程师和自然科学领域的学生设计的数学课程。

它不仅涵盖了微积分和线性代数等基础数学知识，还涉及了部分常微分方程和离散数学的内容。

而且，Math 54 注重理论与实践的结合，旨在培养学生解决实际问题的数学能力。

1. Math 54 的课程设置Math 54 为期一学期，主要包括以下几个部分的内容：微积分部分：介绍微积分的基本概念和技巧，包括导数、积分、微分方程等内容。

线性代数部分：介绍向量空间、矩阵、线性变换、特征值与特征向量等内容。

常微分方程部分：概述一阶和高阶微分方程的解法，包括分离变量法、特征方程法、变参数法等。

离散数学部分：介绍命题逻辑、集合论、图论等内容。

其课程设置合理，内容丰富，有助于学生全面掌握数学基础知识。

2. Math 54 的教学模式Math 54 采用了大学数学教学中常见的讲授和讨论相结合的教学模式。

教师在课堂上对数学知识进行讲解，并通过例题训练学生的解题能力。

课程还设置了小组讨论与作业等环节，学生通过交流与合作，提高解决数学问题的能力。

3. Math 54 的教学目标Math 54 旨在培养学生的数学思维和解决实际问题的能力。

通过学习此课程，学生将能够掌握微积分、线性代数、常微分方程和离散数学的基本理论和方法，并能够运用这些知识解决工程和自然科学领域的实际问题。

4. Math 54 的师资力量Math 54 的教师团队由斯坦福大学数学系的教授和讲师组成。

这些教师在数学领域拥有丰富的教学经验和深厚的学术造诣，能够为学生提供优质的数学教育。

斯坦福大学数学基础54集（Math 54）作为一门为工程师和自然科学领域的学生设计的数学课程，以其丰富的内容、合理的设置、高水平的教学与师资力量，为学生提供了扎实的数学学习基础，有助于他们在工程和科学领域取得成功。

Math 54作为一门为工程师和自然科学领域的学生设计的数学课程，一直以来都备受青睐。

2021版高考英语一轮复习Unit4Astronomythescienceofthestars词汇训练新人教版必修3基础知识默写篇一、分层单词写作词汇1. prep.与……不同;不像2. adv. 因此,因此3. adj.温顺的;文雅的4. n.气候阅读词汇1.astronomy n.2.system n.3.globe n.4.atmosphere n.5.fundamental adj.6.chain n.7.multiply vt.& vi.8.gravity n.9.satellite n.10.physicist n.11.crash vi.&vt.12.spaceship n.13.pull n.&vt.14.float vi.&vt.& n.15.oxygen n.拓展词汇1. adj.猛烈的;猛烈的;强暴的→ n.暴力→adv. 暴力地;猛烈地2. adj.有害的→ n.& v.损害→adj.无害的3. vi.存在;生存→n.存在;生存4. vi.&vt.(使)困惑;(使)为难 n.谜;难题→adj.困惑不解的→adj.令人困惑的5. n.生物学→ n.生物学家二、高频短语1. 及时;终于2. 熟悉;把握;明白得3. 产生;分娩4. 轮到某人5. 阻止;禁止6. 感到快乐;感到振奋7. 既然8. 突发;爆发9. 紧密注意;当心;提防10. 挡住(光线)三、经典句型1.This produced a chain reaction,which (使生命的进展成为可能).2. (专门多科学家所相信的是)the continued presence of water allowed the earth to dissolve harmful gases and acids into the oceans and seas.st month I (我足够有幸有机会)to make a trip into space with my friend Li Yanping, an astronomer.4.When we get closer to the moon,we shall feel its gravity pulling us,but it will(不像地球的引力那么大).5.But when I tried to step forward,I found I was carried (地球上的两倍远)and fell over.6....walking does need a bit of practice (既然重力改变了).基础知识运用篇一、语境词汇运用(一)词汇拼写(注意单词或短语的形式变化)1.Worse still, being addicted to cellphones might also lead to teena gers’ poor performance at school, (因此)adding to their elders’ worries.2.The final of the match was going on in an exciting (气氛) and won hearty applause at times.3.She (拉;拖) him gently towards her just now.4.Some (差不多的) mistakes are not allowed to make when you are doing the work.5.As we know, the earth is becoming warmer and warmer due to the global (气候)change.6. (不像) most people in the office who come to work by car, I usually come to work by bus.7.When the Chinese women volleyball team turned up, all the fans (感到振奋).8.Please tell me when Mr. Smith is arriving at the train station so that I can pick him up (及时).9.Once he has made up his mind, no one can him (阻止)carrying out the pla n.10.At last the firefighters put out the fire in the supermarket, which (突发;爆发)at ten this morning.11.You’ve made me suffer in the past, so now you, (轮到你), suffer again!12.She married at 17 and (分娩)her first child shortly after.13.She bought a heavy curtain to (挡住)the light.(二)单句填空1.He often says that he doesn’t believe in any (religious) and only believes in himself.2.The factory gives off gases. They may do to our health.(harm)3.Do you believe the of ghosts? Perhaps they only in the minds of some people.(exist)4.The look on her face suggested she was about the math problem.(puzzle)5.The robbers hit them . They couldn’t bear it any more so they decided to answer with and began their fightback. (violent)二、经典句型仿写1.了解唐朝的历史会使你更容易明白得唐诗。

线性代数课程专业词汇表英文单词或词组中文翻译书中出现页码Linear equation 线性方程 1Linear system(s) 线性方程组 1Consistent 有解 2Inconsistent 无解 2Solution set of linear system 线性方程组的解集合 2Equivalent systems 等价的线性方程组 3Row operations 行变换 5Strict triangular form 严格三角形式 5Back substitution 回代法 6Equivalent systems 等价的线性方程组 6Coefficient matrix 系数矩阵 7Coefficient matrix 系数矩阵 7Augmented matrix 增广矩阵 8Pivot 主元 8Free variables 自由未知量 14Lead variables 前变量 14Gaussian elimination 高斯消元法 15Overdetermined linear system 方程个数超过未知数个数的方程组 15 Row echelon form 行阶梯形 15Underdetermined linear system 方程个数低于未知数个数的方程组 17 Gauss-Jordan reduction 高斯-若当归纳法 18Reduced row echelon form 减少的行阶梯形 18Homogeneous linear system 齐次线性方程组 22Homogeneous system 齐次线性方程组 22nontrivial solution 非零解 22Trivial solution 平凡解，全零解 22Matrix algebra 矩阵代数 30Scalars 常数 30Column vector(s) 列向量 31Euclidean n-space 欧几里得空间 31Row vector(s) 行向量 31Vector(s）向量 31Addition of matrices 矩阵加法 32Addition of matrices 矩阵加法 32Equality of matrices 矩阵相等 32Scalar multiplication for matrices 矩阵的数乘 32Scalar multiplication of matrices 矩阵的数乘 32Zero matrix 零矩阵 33Scalar product 内积 34Linear combination 线性组合 36Consistency Theorem 解的存在性定理 37Multiplication of matrices 矩阵乘法 38Identity matrix 单位矩阵 47Inverse matrix 逆矩阵 48Invertible matrix 可逆矩阵 48Nonsingular matrix 非奇异矩阵 48Singular matrix 奇异矩阵 49Transpose of a matrix 矩阵的转置 49Transpose of matrix 矩阵的转置 49Symmetric matrix 对称矩阵 51Symmetric matrix 对称矩阵 51Adjacency matrix 邻接矩阵 52Graph(s) 图 52Angle between vectors 向量的夹角 56Markov chain(s） Markov链 57Elementary matrix 初等矩阵 62Row equivalent 行等价 64Row equivalent matrices 行等价矩阵 64Diagonal matrix 对角矩阵 67Lower triangular 下三角 67Triangular factorization 三角分解 67Triangular matrix 三角形矩阵 67Triangular matrix 三角形矩阵 67Upper triangular 上三角 67Upper triangular matrix 上三角矩阵 67LU factorization LU分解 68Matrix factorizations 矩阵分解 68Partitioned matrices 分块矩阵 72Vandermonde matrix 范德蒙矩阵 72Block multiplication 分块乘法 74Inner product 内积 78Determinant(s) 行列式 90Cofactor 代数余子式 93Minor 余子式 93Cofactor expansion 代数余子式展开 94Determinant of matrix 矩阵的行列式 95Skew symmetric 反对称 105Adjoint of a matrix 伴随矩阵 106Cramer’s rule 克莱姆法则 107Cryptography 密码学 108Addition of vectors 向量的加法 119Closure properties 封闭性 119Vector space 向量空间 119Zero vector 零向量 119C[a,b] 区间[a,b]上的连续函数 120Isomorphism between vector spaces 向量空间的同构 123Subspace(s) 子空间 123Zero subsapce 零空间 125Nullspace 零化空间 127Nullspace of matrix 矩阵的零化空间 127Span 张成 128Spanning set 生成集 129Linearly dependent 线性相关 136Linearly independent 线性无关 136Basis 基 145Dimension 维数 147Finite dimensional 有限维 147Infinite dimensional 无限维 147Standard basis 标准基 150change of basis 基的变换 151Coordinate vector 坐标向量 152Transition matrix 过渡矩阵 155Coordinates 坐标 157Column space 列空间 162Column space of matrix 矩阵的列空间 162Rank of a matrix 矩阵的秩 162Rank of matrix 矩阵的秩 162Row space 行空间 162Row space of matrix 矩阵的行空间 162Nullity 零化度 164Rank-Nullity Theorem Rank-Nullity定理 164Left inverse 左可逆 170Right inverse 右逆 170Full rank 满秩 171Linear transformation(s) 线性变换 175Linear operator 线性算子 176Image 象 181Kernel 核 181Contraction 收缩 192Dilation 扩张 192Similarity 相似性 199Similar matrices 相似矩阵 202Trace 迹 206Angle between vectors 向量的夹角 211Euclidean length 欧几里得长度 211Distance in 2-space 2维空间的距离 212Cauchy-Schwarz inequality 柯西-施瓦兹不等式 213 Orthogonality 正交性 213Scalar projection 数量投影 214equation of plane 平面方程 215Nonmal vector 正规向量 215Angle between vectors 向量的夹角 216Pythagorean Law Pythagorean 定理 216Correlations 相关 219Correlation matrix 相关矩阵 221Covariance 协方差 222Covariance matrix 协方差矩阵 222Factor analysis 因子分析 222Fundamental subspaces 基本子空间 227Range of a matrix 矩阵的值域 227Direct sum 直和 229Least squares problem(s) 最小二乘法问题 234Projection onto column space 列空间上的投射 236Normal equations 正规方程 237Inner product space 内积空间 245Length in inner product spaces 内积空间中的长度 246 Orthogonal set(s) 正交集合 255Orthonormal set(s) 标准正交集 255orthonormal basis 标准正交基 256Orthonormal basis 标准正交基 256Orthogonal matrices 正交矩阵 258Orthogonal matrix 正交矩阵 258Approximation of functions 函数的逼近 264Fourier coefficients 傅里叶系数 266Fourier matrix 傅里叶矩阵 269Gram-Schmidt process Gram-Schmidt过程 274Dimension Theorem 维数定理 283Orthogonal polynomials 正交多项式 283Hermite polynomials Hermite 多项式 287Jacobi polynomials Jacobi多项式 287Lagrange’s interpolating formula Lagrange 插值公式 288 Gaussian quadrature 高斯求积 289Characteristic value(s) 特征值 301Characteristic vector 特征向量 301Eigenvalue 特征值 301Eigenvector 特征向量 301Characteristic equation 特征方程 302Characteristic polynomial 特征多项式 302Eigenspace 特征空间 302Nilpotent 幂零的 311Companion matrix 友矩阵 313Linear differential equations 线性微分方程 313Initial value problems 初值问题 314Diagonalizable matrix 可对角化的矩阵 326Distance in n-space n维空间的距离 332Complex matrix 复矩阵 346Hermite matrix Hermite 矩阵 346Unitary matrix 酉矩阵 347Unitary matrix 酉矩阵 347Normal matrices 正规矩阵 351Singular values 奇异值 356Conic sections 二次曲线部分 371Quadratic equation in n variables n个变量的二次方程 376 Quadratic form in n variables n个变量的二次型 376 Definite quadratic form 定二次型 378Indefinite quadratic form 不定二次型 378Negative definite matrix 负定矩阵 378Negative definite quadratic form 负定二次型 378 Negative semidefinite matrix 半负定矩阵 378Negative semidefinite quadratic form 半负定二次型 378 Positive definite matrix 正定矩阵 378Positive definite quadratic form 正定二次型 378Positive semidefinite matrix 半正定矩阵 378Positive semidefinite quadratic form 半正定二次型 378 Local maximum 极大值 382Local minimum 极小值 382Positive definite matrix 正定矩阵 384Leading principal submatrix 顺序主子矩阵 385 Nonnegative matrix 非负矩阵 392Nonnegative vector 非负向量 392Positive matrix 正矩阵 392Positive matrix 正矩阵 392Reducible matrix 可约矩阵 394Frobenius theorem Frobenius 定理 395Absolute error 绝对误差 411Relative error 相对误差 411Back substitution 回代法 419QR factorization QR分解 448。

第四单元课文AIn Search of DavosMan Peter GumbelGlobalization is sweeping aside national borders and changing relations between nations. What impact does this have on national identities and loyalties? Are they strengthened or weakened? The author investigates.全球化正在扫除国界、改变国与国之间的关系。

这对国家的认同和对国家的忠诚会带来什么影响呢？它们会得到加强还是削弱？作者对这些问题进行了探讨。

In Search of Davos ManPeter Gumbel1. William Browder was born in Princeton, New Jersey, grew up in Chicago, and studied at Stanford University in California. But don't call him an American. For the past 16 of his 40 years he has lived outside the U.S., first in London and then, from 1996, in Moscow, where he runs his own investment firm. Browder now manages $1.6 billion in assets. In 1998 he gave up his American passport to become a British citizen, since his life is now centered in Europe. "National identity makes no difference for me," he says. "I feel completely international. If you have four good friends and you like what you are doing, it doesn't matter where you are. That's globalization."寻找达沃斯人彼得·甘贝尔威廉·布劳德出生于新泽西州的普林斯顿，在芝加哥长大，就读于加利福尼亚州的斯坦福大学。

IndexEEconomics 经济学435,439 Eigencourse 457,458Eigenvalue 特征值283,287,374,499 Eigenvalue changes 特征值变换439Eigenvalues of 的特征值284,294,300 Eigenvalues of 的特征值297Eigenvalues of 的特征值362Eigenvector basis 基底的特征向量399Eigenvectors 特征向量283,287,374Eigshow 290,368Elimination 消元法45-66,83,86,135 Ellipse 椭圆290,346,366,382 Energy 能量343,409Engineering 工程409,419Error 误差211,218,219,225,481,483 Error equation 误差方程477Euler angles 欧拉角474Euler’s formula 欧拉公式311,426,430,497Even 偶数113,246,258,452 Exponential 指数的314,319,327FHilbert space 希尔伯特空间447,449Hooke’s Law 虎克定律410,412 Householder reflections 镜像变换237,469,472 Hyperplane 超平面30,42IIll-conditioned matrix 病态矩阵371,473,474 Imaginary 虚数289Independent 独立的26,27,134,168,200,300 Initial value 初值313Inner product 内积11,56,108,448,502,506 Input and output basis 基底输入输出399Integral 积分24,385,386Interior point method 内点法445Intersection of spaces 交空间129,183Inverse matrix 逆矩阵24,81,270Inverse of的逆82Invertible 可逆的86,173,200,248 Iteration 迭代481,482,484,489,492 JJacobi 雅可比481,483,485,489 Jordan form 约当型356,357,358,361,482 JPEG 364,373KKalman filter 卡尔曼滤波器93,214Kernel 核377,380Kirchhoff’s Laws 基尔霍夫定律143,189,420,424-427 Krylov 克雷洛夫491,492Lnorm 和范数225,480Lagrange multiplier 拉格朗日乘子445Lanczos method 兰索斯方法490,492LAPACK 线性代数软件包98,237,486Leapfrog method 跳步法317,329Least squares 最小平方218,219,236,405,408,453 Left nullspace左零空间184,186,192,425Left-inverse 左逆的81,86,154,405Length 长度12,232,447,448,501 Line 线34,40,221,474Line of springs 线弹簧411Linear combination 线性组合1,3Linear equation 线性方程23Linear programming 线性规划440Linear transformation 线性变换44,375-398Linearity 线性关系44,245,246Linearly independent 线性独立26,134,168,169,200 LINPACK 线性系统软件包465Loop 环路307,425,426Lower triangular 下三角9598,100,474Lucas numbers 卢卡斯数306MMaple 38,100Mathematica 38,100MATLAB 17,37,237,243,290,337,513 Matrix(see full page 570) 矩阵22,384,387Matrix exponential 矩阵指数314,319,327Matrix multiplication 矩阵乘法58,59,67,389Matrix notation 矩阵记号37Matrix space 矩阵空间121,122,175,181,311 Matrix 矩阵-1,2,-1 matrix -1，2，-1矩阵106,167,261,265,349,374,410,480Adjacency, 邻接矩阵74,80,311,369All-ones 全1矩阵251,262,307,348 Augmented 增广矩阵60,84,155Band 带状矩阵99,468,469。

Unit 4从科罗拉多州利特尔顿的科伦拜中学大规模的谋杀惨案到南联盟科索沃的种族净化和战争可以看出，我们今天生活的世界极度令人忧虑，但同时对未来它也充满了非凡的希望。

这是一个充满变化的世界，其特点是严重的冲突、紧张、社会分工以及现代技术对自然环境造成的毁灭性冲击。

然而我们仍然有可能掌控自己的命运,并且把我们的生活塑造得更好, 大大超出前辈的想象。

这个世界是如何变成这样的？为什么我们的生活状况与我们的父辈们及祖父辈们的如此不同？未来，变化会朝着什么方向发展？这些问题是社会学主要关注的对象。

正因如此，社会学这个研究领域在现代知识生活中发挥着重要作用。

社会学是对人类社会生活群体和社会的科学研究。

这是一项炫目而迷人的事业，因为它的研究对象是我们这些社会人的行为。

社会学研究的范围相当广泛，包括对小到人和人在街头的偶遇，大到全球社会进程的调查。

我们将通过一个简短的例子来初步了解社会学的本质和研究目地。

你曾经恋爱过吗？很可能如此。

大多数人在青少年时期或者年龄稍大一些便懂了恋爱的感觉。

对我们大部分人来说，爱情与浪漫史是我们所经历过的最为强烈的感受。

为什么人们会坠入爱河呢？乍一看,答案似乎是显而易见的。

爱情表达了两个个体彼此间身体和情感的依附关系。

近来，我们或许会怀疑爱情到底是不是“永恒”的，但我们依然倾向于认为，恋爱体验源于人类的普遍情感。

对于两个相爱的人来说，他们会通过双方关系来满足自己的情感和性需要，可能通过结婚的形式吧，这看来理所当然。

然而，如今在我们看来似乎是不言自明的这种情况，事实上非常少见。

世上不是所有的人都会有恋爱这种经历，在它确实发生时，我们也很少将它和婚姻联系起来。

在我们的社会中，浪漫爱情这一观念直到最近才开始变得普遍，而在其他大部分文化中从未出现过。

只有现代社会才将爱情和性视做是密切相关之事。

在中世纪以及随后的几个世纪中，男人和女人结婚的目的就是为了保住家业, 或者养育孩子来经营家庭农场。

一旦结婚，他们可能会变成亲密的伴侣；然而，这种情况发生在婚后，而不是婚前。

学术词汇大全四级中常见学科词汇一览学术领域中的各个学科都有其独特的词汇，这些词汇是理解和交流学术知识的基础。

对于准备考取四级英语的学生来说，掌握常见的学科词汇是非常重要的。

本文将为大家提供学术词汇大全四级中常见学科词汇一览，希望能帮助大家在考试中取得好成绩。

1. 数学（Mathematics）- algebra 代数- calculus 微积分- geometry 几何学- trigonometry 三角学- statistics 统计学- probability 概率论- equation 方程- derivative 导数- integral 积分- matrix 矩阵2. 物理学（Physics）- mechanics 力学- thermodynamics 热力学- electromagnetism 电磁学- optics 光学- quantum mechanics 量子力学- relativity 相对论- nuclear physics 核物理学- particle physics 粒子物理学- astrophysics 天体物理学- acoustics 声学3. 化学（Chemistry）- organic chemistry 有机化学- inorganic chemistry 无机化学- physical chemistry 物理化学- analytical chemistry 分析化学- biochemistry 生物化学- polymer chemistry 高分子化学- environmental chemistry 环境化学- pharmaceutical chemistry 制药化学 - theoretical chemistry 理论化学- industrial chemistry 工业化学4. 生物学（Biology）- molecular biology 分子生物学- cell biology 细胞生物学- genetics 遗传学- ecology 生态学- evolution 进化论- microbiology 微生物学- zoology 动物学- botany 植物学- anatomy 解剖学- physiology 生理学5. 地球科学（Earth Science）- geology 地质学- meteorology 气象学- oceanography 海洋学- climatology 气候学- seismology 地震学- hydrology 水文学- geomorphology 地貌学- paleontology 古生物学- geophysics 地球物理学- planetary science 行星科学6. 计算机科学（Computer Science） - programming 编程- algorithm 算法- data structure 数据结构- artificial intelligence 人工智能- computer network 计算机网络- operating system 操作系统- database 数据库- software engineering 软件工程- computer graphics 计算机图形学- computer security 计算机安全7. 经济学（Economics）- microeconomics 微观经济学- macroeconomics 宏观经济学- econometrics 计量经济学- international economics 国际经济学 - finance 金融学- labor economics 劳动经济学- economic development 经济发展学- game theory 博弈论- econometric models 计量经济模型- economic policy 经济政策8. 心理学（Psychology）- cognitive psychology 认知心理学- developmental psychology 发展心理学 - social psychology 社会心理学- personality psychology 人格心理学- abnormal psychology 异常心理学- educational psychology 教育心理学- industrial-organizational psychology 工业组织心理学- clinical psychology 临床心理学- counseling psychology 咨询心理学- positive psychology 积极心理学通过掌握以上学科词汇，同学们可以在英语四级考试中更好地理解并回答与学术知识相关的问题。

Introduction to Linear Algebra,Fourth Edition by Gilbert StrangGood Treatment To Gain IntuitionGilbert Strangs textbooks have changed the entire approach to learning linear algebra -- away from abstract vector spaces to specific examples of the four fundamental subspaces: the column space and nullspace of A and A. Introduction to Linear Algebra, Fourth Edition includes challenge problems to complement the review problems that have been highlypraised in previous editions. The basic course is followed by seven applications: differential equations, engineering, graph theory, statistics, fourier methods and the FFT, linear programming, and computer graphics. Thousands of teachers in colleges and universities and now high schools are using this book, which truly explains this crucial subject. Chapter 1: Introduction to Vectors; Chapter 2: Solving Linear Equations; Chapter 3: Vector Spaces and Subspaces; Chapter 4: Orthogonality; Chapter 5: Determinants; Chapter 6: Eigenvalues and Eigenvectors; Chapter 7: Linear Transformations; Chapter 8: Applications; Chapter 9: Numerical Linear Algebra; Chapter 10: Complex Vectors and Matrices; Solutions to Selected Exercises; Final Exam. Matrix Factorizations. Conceptual Questions for Review. Glossary: A Dictionary for Linear Algebra Index Teaching Codes Linear Algebra in a Nutshell.Personal Review: Introduction to Linear Algebra, Fourth Edition by Gilbert StrangI had a very poor background in Linear algebra and I had a really advance optimization course. This book along with MIT opencourseware 18.06 (the course Prof Strang teaches at MIT) saved my semester. OCW is free. Iwent through some important video lectures and almost the whole book. Very simple english, covers all the basics, tells a bit about Matlab, has a chapter on engineering application examples(didn't go through that part). If you get confused regarding something in the book, which in very fewcases you might if your running though it like I did, watch the relatedlecture and then go back to the book. You'll get the BIG PICTURE... I did. Thank you Prof. Strang.For More 5 Star Customer Reviews and Lowest Price: Introduction to Linear Algebra, Fourth Edition by Gilbert Strang 5 Star CustomerReviews and Lowest Price!。

新视野大学英语读写教程第四册第四单元B篇原文和翻译篇一：新视野大学英语读写教程第四册10课答案及课文翻译新视野大学英语第四册学习辅导材料New Horizon College English(Book 4)2012年3月Unit 1Part One : Intensive Reading:Section A: The Tail of FameIII.1.idle2.justify3.discount4.distinct5.minute6.accused7.object8.contaminate9.sustain10.worshipIV.1.accusing...of2.end up3.came upon4.at her worst5.pay for6.run a risk of7.participate in8.other than9. object to/objected10. at best V.1.K2.G3.C4.E5.N6.O7.I8. L9.A10. D VI.1.delay2.pain3.hardship4.suffering5.fever6.defeat7.poverty8.treatment9.noise10.agonyVII.1.justify2.glorify3.exemplifies4.classified5.purified6.intensify7.identify8.terrifiedVIII.1.bravery2.jewelry3.delivery4.machinery5.robbery6 nursery 7.scenery 8.discoveryIX.1. other than for funerals and weddings2. other than to live an independent life3. other than that they appealed to his eye4. but other than that, he'll eat just about everything5. other than that it's somewhere in the town center.X.1. shouldn't have been to the cinema last night2. would have; told him the answer3. they needn't have gone at all4. must have had too much work to do5. might have been injured seriouslyXI.1. The plant does not grow well in soils other than the one in which it has been developed.2. Research findings show that we spend about two hours dreaming every night, no matter whatwe may have done during the day.3. Some people tend to justify their failure by blaming others for not trying their best.4. We remain true to our commitment: Whatever we promised to do; we would do it.5. Even Beethoven's father discounted the possibility that his son would one day become thegreatest musician in the world. The same is true of Edison, who seemed to his teacher to bequite dull.6. They were accused by authorities of threatening the state security.Ⅻ.l.出入除自己家以外的任何场所时，如果你带有宠物，一定要了解有关宠物的规定。

four dimensions 翻译中文四维空间，也称为四维度，是指具有四个独立方向的空间。

在数学和物理学中，四维空间被广泛研究和应用。

下面是一些四维空间的用法和中英文对照例句：1. 理论物理学家们研究了四维时空的性质。

Theoretical physicists study the properties of four-dimensional spacetime.2. 在四维空间中，物体可以在四个独立的方向上移动。

In four-dimensional space, objects can move in four independent directions.3. 通过引入四维度，我们可以更全面地描述物体的运动。

By introducing the fourth dimension, we can describe the motion of objects more comprehensively.4. 四维空间在相对论中发挥了重要的作用。

Four-dimensional space plays an important role in relativity theory.5. 相比于三维空间，四维空间具有更多的自由度。

Four-dimensional space has more degrees of freedom compared to three-dimensional space.6. 研究四维空间需要使用高级数学工具，如张量分析和微分几何。

Studying four-dimensional space requires the use of advanced mathematical tools, such as tensor analysis and differential geometry.7. 在四维空间中，我们可以通过超立方体来可视化对象。

In four-dimensional space, we can visualize objects using hypercubes.8. 人类的感知能力限制了我们对四维空间的直观理解。