Hyperbolic geometry
英文论文写作中一些可能用到的词汇
英⽂论⽂写作中⼀些可能⽤到的词汇英⽂论⽂写作过程中总是被⾃⼰可怜的词汇量击败, 所以我打算在这⾥记录⼀些在阅读论⽂过程中见到的⼀些⾃⼰不曾见过的词句或⽤法。
这些词句查词典都很容易查到，但是只有带⼊论⽂原⽂中才能体会内涵。
毕竟原⽂和译⽂中间总是存在⼀条看不见的思想鸿沟。
形容词1. vanilla: adj. 普通的, 寻常的, 毫⽆特⾊的. ordinary; not special in any way.2. crucial: adj. ⾄关重要的, 关键性的.3. parsimonious：adj. 悭吝的, 吝啬的, ⼩⽓的.e.g. Due to the underlying hyperbolic geometry, this allows us to learn parsimonious representations of symbolic data by simultaneously capturing hierarchy and similarity.4. diverse: adj. 不同的, 相异的, 多种多样的, 形形⾊⾊的.5. intriguing: adj. ⾮常有趣的, 引⼈⼊胜的; 神秘的. *intrigue: v. 激起…的兴趣, 引发…的好奇⼼; 秘密策划(加害他⼈), 密谋.e.g. The results of this paper carry several intriguing implications.6. intimate: adj. 亲密的; 密切的. v.透露; (间接)表⽰, 暗⽰.e.g. The above problems are intimately linked to machine learning on graphs.7. akin: adj. 类似的, 同族的, 相似的.e.g. Akin to GNN, in LOCAL a graph plays a double role: ...8. abundant: adj. ⼤量的, 丰盛的, 充裕的.9. prone: adj. 有做(坏事)的倾向; 易于遭受…的; 俯卧的.e.g. It is thus prone to oversmoothing when convolutions are applied repeatedly.10.concrete: adj. 混凝⼟制的; 确实的, 具体的(⽽⾮想象或猜测的); 有形的; 实在的.e.g. ... as a concrete example ...e.g. More concretely, HGCN applies the Euclidean non-linear activation in...11. plausible: adj. 有道理的; 可信的; 巧⾔令⾊的, 花⾔巧语的.e.g. ... this interpretation may be a plausible explanation of the success of the recently introduced methods.12. ubiquitous: adj. 似乎⽆所不在的;⼗分普遍的.e.g. While these higher-order interac- tions are ubiquitous, an evaluation of the basic properties and organizational principles in such systems is missing.13. disparate: adj. 由不同的⼈(或事物)组成的;迥然不同的;⽆法⽐较的.e.g. These seemingly disparate types of data have something in common: ...14. profound: adj. 巨⼤的; 深切的, 深远的; 知识渊博的; 理解深刻的;深邃的, 艰深的; ⽞奥的.e.g. This has profound consequences for network models of relational data — a cornerstone in the interdisciplinary study of complex systems.15. blurry: adj. 模糊不清的.e.g. When applying these estimators to solve (2), the line between the critic and the encoders $g_1, g_2$ can be blurry.16. amenable: adj. 顺从的; 顺服的; 可⽤某种⽅式处理的.e.g. Ou et al. utilize sparse generalized SVD to generate a graph embedding, HOPE, from a similarity matrix amenableto de- composition into two sparse proximity matrices.17. elaborate: adj. 复杂的;详尽的;精⼼制作的 v.详尽阐述;详细描述;详细制订;精⼼制作e.g. Topic Modeling for Graphs also requires elaborate effort, as graphs are relational while documents are indepen- dent samples.18. pivotal: adj. 关键性的；核⼼的e.g. To ensure the stabilities of complex systems is of pivotal significance toward reliable and better service providing.19. eminent: adj. 卓越的，著名的，显赫的;⾮凡的;杰出的e.g. To circumvent those defects, theoretical studies eminently represented by percolation theories appeared.20. indispensable: adj. 不可或缺的;必不可少的 n. 不可缺少的⼈或物e.g. However, little attention is paid to multipartite networks, which are an indispensable part of complex networks.21. post-hoc: adj. 事后的e.g. Post-hoc explainability typically considers the question “Why the GNN predictor made certain prediction?”.22. prevalent: adj. 流⾏的;盛⾏的;普遍存在的e.g. A prevalent solution is building an explainer model to conduct feature attribution23. salient: adj. 最重要的;显著的;突出的. n. 凸⾓;[建]突出部;<军>进攻或防卫阵地的突出部分e.g. It decomposes the prediction into the contributions of the input features, which redistributes the probability of features according to their importance and sample the salient features as an explanatory subgraph.24. rigorous: adj. 严格缜密的;严格的;谨慎的;细致的;彻底的;严厉的e.g. To inspect the OOD effect rigorously, we take a causal look at the evaluation process with a Structural Causal Model.25. substantial: adj. ⼤量的;价值巨⼤的;重⼤的;⼤⽽坚固的;结实的;牢固的. substantially: adv. ⾮常;⼤⼤地;基本上;⼤体上;总的来说26. cogent: adj. 有说服⼒的;令⼈信服的e.g. The explanatory subgraph $G_s$ emphasizes tokens like “weak” and relations like “n’t→funny”, which is cogent according to human knowledge.27. succinct: adj. 简练的;简洁的 succinctly: adv. 简⽽⾔之，简明扼要地28. concrete: adj. 混凝⼟制的;确实的，具体的(⽽⾮想象或猜测的);有形的;实在的 concretely: adv. 具体地;具体;具体的;有形地29. predominant：adj. 主要的;主导的;显著的;明显的;盛⾏的;占优势的动词1. mitigate: v. 减轻, 缓和. (反 enforce)e.g. In this work, we focus on mitigating this problem for a certain class of symbolic data.2. corroborate: v. [VN] [often passive] (formal) 证实, 确证.e.g. This is corroborated by our experiments on real-world graph.3. endeavor: n./v. 努⼒, 尽⼒, 企图, 试图.e.g. It encourages us to continue the endeavor in applying principles mathematics and theory in successful deployment of deep learning.4. augment: v. 增加, 提⾼, 扩⼤. n. 增加, 补充物.e.g. We also augment the graph with geographic information (longitude, latitude and altitude), and GDP of the country where the airport belongs to.5. constitute: v. (被认为或看做)是, 被算作; 组成, 构成; (合法或正式地)成⽴, 设⽴.6. abide: v. 接受, 遵照(规则, 决定, 劝告); 逗留, 停留.e.g. Training a graph classifier entails identifying what constitutes a class, i.e., finding properties shared by graphs in one class but not the other, and then deciding whether new graphs abide to said learned properties.7. entail: v. 牵涉; 需要; 使必要. to involve sth that cannot be avoided.e.g. Due to the recursive definition of the Chebyshev polynomials, the computation of the filter $g_α(\Delta)f$ entails applying the Laplacian $r$ times, resulting cal operator affecting only 1-hop neighbors of a vertex and in $O(rn)$ operations.8. encompass: v. 包含, 包括, 涉及(⼤量事物); 包围, 围绕, 围住.e.g. This model is chosen as it is sufficiently general to encompass several state-of-the-art networks.e.g. The k-cycle detection problem entails determining if G contains a k-cycle.9. reveal: v. 揭⽰, 显⽰, 透露, 显出, 露出, 展⽰.10. bestow: v. 将(…)给予, 授予, 献给.e.g. Aiming to bestow GCNs with theoretical guarantees, one promising research direction is to study graph scattering transforms (GSTs).11. alleviate: v. 减轻, 缓和, 缓解.12. investigate: v. 侦查(某事), 调查(某⼈), 研究, 调查.e.g. The sensitivity of pGST to random and localized noise is also investigated.13. fuse: v. (使)融合, 熔接, 结合; (使)熔化, (使保险丝熔断⽽)停⽌⼯作.e.g. We then fuse the topological embeddings with the initial node features into the initial query representations using a query network$f_q$ implemented as a two-layer feed-forward neural network.14. magnify: v. 放⼤, 扩⼤; 增强; 夸⼤(重要性或严重性); 夸张.e.g. ..., adding more layers also leads to more parameters which magnify the potential of overfitting.15. circumvent: v. 设法回避, 规避; 绕过, 绕⾏.e.g. To circumvent the issue and fulfill both goals simultaneously, we can add a negative term...16. excel: v. 擅长, 善于; 突出; 胜过平时.e.g. Nevertheless, these methods have been repeatedly shown to excel in practice.17. exploit: v. 利⽤(…为⾃⼰谋利); 剥削, 压榨; 运⽤, 利⽤; 发挥.e.g. In time series and high-dimensional modeling, approaches that use next step prediction exploit the local smoothness of the signal.18. regulate: v. (⽤规则条例)约束, 控制, 管理; 调节, 控制(速度、压⼒、温度等).e.g. ... where $b >0$ is a parameter regulating the probability of this event.19. necessitate: v. 使成为必要.e.g. Combinatorial models reproduce many-body interactions, which appear in many systems and necessitate higher-order models that capture information beyond pairwise interactions.20. portray:描绘, 描画, 描写; 将…描写成; 给⼈以某种印象; 表现; 扮演(某⾓⾊).e.g. Considering pairwise interactions, a standard network model would portray the link topology of the underlying system as shown in Fig. 2b.21. warrant: v. 使有必要; 使正当; 使恰当. n. 执⾏令; 授权令; (接受款项、服务等的)凭单, 许可证; (做某事的)正当理由, 依据.e.g. Besides statistical methods that can be used to detect correlations that warrant higher-order models, ... (除了可以⽤来检测⽀持⾼阶模型的相关性的统计⽅法外, ...)22. justify: v. 证明…正确(或正当、有理); 对…作出解释; 为…辩解(或辩护); 调整使全⾏排满; 使每⾏排齐.e.g. ..., they also come with the assumption of transitive, Markovian paths, which is not justified in many real systems.23. hinder:v. 阻碍; 妨碍; 阻挡. (反 foster: v. 促进; 助长; 培养; ⿎励; 代养, 抚育, 照料(他⼈⼦⼥⼀段时间))e.g. The eigenvalues and eigenvectors of these matrix operators capture how the topology of a system influences the efficiency of diffusion and propagation processes, whether it enforces or mitigates the stability of dynamical systems, or if it hinders or fosters collective dynamics.24. instantiate:v. 例⽰；⽤具体例⼦说明.e.g. To learn the representation we instantiate (2) and split each input MNIST image into two parts ...25. favor:v. 赞同;喜爱, 偏爱; 有利于, 便于. n. 喜爱, 宠爱, 好感, 赞同; 偏袒, 偏爱; 善⾏, 恩惠.26. attenuate: v. 使减弱; 使降低效⼒.e.g. It therefore seems that the bounds we consider favor hard-to-invert encoders, which heavily attenuate part of the noise, over well conditioned encoders.27. elucidate:v. 阐明; 解释; 说明.e.g. Secondly, it elucidates the importance of appropriately choosing the negative samples, which is indeed a critical component in deep metric learning based on triplet losses.28. violate: v. 违反, 违犯, 违背(法律、协议等); 侵犯(隐私等); 使⼈不得安宁; 搅扰; 亵渎, 污损(神圣之地).e.g. Negative samples are obtained by patches from different images as well as patches from the same image, violating the independence assumption.29. compel:v. 强迫, 迫使; 使必须; 引起(反应).30. gauge: v. 判定, 判断(尤指⼈的感情或态度); (⽤仪器)测量, 估计, 估算. n. 测量仪器(或仪表);计量器;宽度;厚度;(枪管的)⼝径e.g. Yet this hyperparameter-tuned approach raises a cubic worst-case space complexity and compels the user to traverse several feature sets and gauge the one that attains the best performance in the downstream task.31. depict: v. 描绘, 描画; 描写, 描述; 刻画.e.g. As they depict different aspects of a node, it would take elaborate designs of graph convolutions such that each set of features would act as a complement to the other.32. sketch: n. 素描;速写;草图;幽默短剧;⼩品;简报;概述 v. 画素描;画速写;概述;简述e.g. Next we sketch how to apply these insights to learning topic models.33. underscore：v. 在…下⾯划线;强调;着重说明 n.下划线e.g. Moreover, the walk-topic distributions generated by Graph Anchor LDA are indeed sharper than those by ordinary LDA, underscoring the need for selecting anchors.34. disclose: v. 揭露;透露;泄露;使显露;使暴露e.g. Another drawback lies in their unexplainable nature, i.e., they cannot disclose the sciences beneath network dynamics.35. coincide: v. 同时发⽣;相同;相符;极为类似;相接;相交;同位;位置重合;重叠e.g. The simulation results coincide quite well with the theoretical results.36. inspect: v. 检查;查看;审视;视察 to look closely at sth/sb, especially to check that everything is as it should be名词1. capacity: n. 容量, 容积, 容纳能⼒; 领悟(或理解、办事)能⼒; 职位, 职责.e.g. This paper studies theoretically the computational capacity limits of graph neural networks (GNN) falling within the message-passing framework of Gilmer et al. (2017).2. implication: n. 可能的影响(或作⽤、结果); 含意, 暗指; (被)牵连, 牵涉.e.g. Section 4 analyses the implications of restricting the depth $d$ and width $w$ of GNN that do not use a readout function.3. trade-off:(在需要⽽⼜相互对⽴的两者间的)权衡, 协调.e.g. This reveals a direct trade-off between the depth and width of a graph neural network.4. cornerstone:n. 基⽯; 最重要部分; 基础; 柱⽯.5. umbrella: n. 伞; 综合体; 总体, 整体; 保护, 庇护(体系).e.g. Community detection is an umbrella term for a large number of algorithms that group nodes into distinct modules to simplify and highlight essential structures in the network topology.6. folklore:n. 民间传统, 民俗; 民间传说.e.g. It is folklore knowledge that maximizing MI does not necessarily lead to useful representations.7. impediment:n. 妨碍,阻碍,障碍; ⼝吃.e.g. While a recent approach overcomes this impediment, it results in poor quality in prediction tasks due to its linear nature.8. obstacle:n. 障碍;阻碍; 绊脚⽯; 障碍物; 障碍栅栏.e.g. However, several major obstacles stand in our path towards leveraging topic modeling of structural patterns to enhance GCNs.9. vicinity:n. 周围地区; 邻近地区; 附近.e.g. The traits with which they engage are those that are performed in their vicinity.10. demerit: n. 过失,缺点,短处; (学校给学⽣记的)过失分e.g. However, their principal demerit is that their implementations are time-consuming when the studied network is large in size. Another介/副/连词1. notwithstanding:prep. 虽然;尽管 adv. 尽管如此.e.g. Notwithstanding this fundamental problem, the negative sampling strategy is often treated as a design choice.2. albeit: conj. 尽管;虽然e.g. Such methods rely on an implicit, albeit rigid, notion of node neighborhood; yet this one-size-fits-all approach cannot grapple with the diversity of real-world networks and applications.3. Hitherto:adv. 迄今;直到某时e.g. Hitherto, tremendous endeavors have been made by researchers to gauge the robustness of complex networks in face of perturbations.短语1.in a nutshell: 概括地说, 简⾔之, ⼀⾔以蔽之.e.g. In a nutshell, GNN are shown to be universal if four strong conditions are met: ...2. counter-intuitively: 反直觉地.3. on-the-fly:动态的(地), 运⾏中的(地).4. shed light on/into:揭⽰, 揭露; 阐明; 解释; 将…弄明⽩; 照亮.e.g. These contemporary works shed light into the stability and generalization capabilities of GCNs.e.g. Discovering roles and communities in networks can shed light on numerous graph mining tasks such as ...5. boil down to: 重点是; 将…归结为.e.g. These aforementioned works usually boil down to a general classification task, where the model is learnt on a training set and selected by checking a validation set.6. for the sake of:为了.e.g. The local structures anchored around each node as well as the attributes of nodes therein are jointly encoded with graph convolution for the sake of high-level feature extraction.7. dates back to:追溯到.e.g. The usual problem setup dates back at least to Becker and Hinton (1992).8. carry out:实施, 执⾏, 实⾏.e.g. We carry out extensive ablation studies and sensi- tivity analysis to show the effectiveness of the proposed functional time encoding and TGAT-layer.9. lay beyond the reach of:...能⼒达不到e.g. They provide us with information on higher-order dependencies between the components of a system, which lay beyond the reach of models that exclusively capture pairwise links.10. account for: ( 数量或⽐例上)占; 导致, 解释(某种事实或情况); 解释, 说明(某事); (某⼈)对(⾏动、政策等)负有责任; 将(钱款)列⼊(预算).e.g. Multilayer models account for the fact that many real complex systems exhibit multiple types of interactions.11. along with: 除某物以外; 随同…⼀起, 跟…⼀起.e.g. Along with giving us the ability to reason about topological features including community structures or node centralities, network science enables us to understand how the topology of a system influences dynamical processes, and thus its function.12. dates back to:可追溯到.e.g. The usual problem setup dates back at least to Becker and Hinton (1992) and can conceptually be described as follows: ...13. to this end:为此⽬的;为此计;为了达到这个⽬标.e.g. To this end, we consider a simple setup of learning a representation of the top half of MNIST handwritten digit images.14. Unless stated otherwise:除⾮另有说明.e.g. Unless stated otherwise, we use a bilinear critic $f(x, y) = x^TWy$, set the batch size to $128$ and the learning rate to $10^{−4}$.15. As a reference point:作为参照.e.g. As a reference point, the linear classification accuracy from pixels drops to about 84% due to the added noise.16. through the lens of:透过镜头. (以...视⾓)e.g. There are (at least) two immediate benefits of viewing recent representation learning methods based on MI estimators through the lens of metric learning.17. in accordance with:符合；依照；和…⼀致.e.g. The metric learning view seems hence in better accordance with the observations from Section 3.2 than the MI view.It can be shown that the anchors selected by our Graph Anchor LDA are not only indicative of “topics” but are also in accordance with the actual graph structures.18. be akin to:近似, 类似, 类似于.e.g. Thus, our learning model is akin to complex contagion dynamics.19. to name a few:仅举⼏例;举⼏个来说.e.g. Multitasking, multidisciplinary work and multi-authored works, to name a few, are ingrained in the fabric of science culture and certainly multi-multi is expected in order to succeed and move up the scientific ranks.20. a handful of:⼀把;⼀⼩撮;少数e.g. A handful of empirical work has investigated the robustness of complex networks at the community level.21. wreak havoc: 破坏;肆虐;严重破坏;造成破坏;浩劫e.g. Failures on one network could elicit failures on its coupled networks, i.e., networks with which the focal network interacts, and eventually those failures would wreak havoc on the entire network.22. apart from: 除了e.g. We further posit that apart from node $a$ node $b$ has $k$ neighboring nodes.。
丘维声解析几何 英文引用
Geometry Analysis by Qiu WeishengGeometry analysis, a branch of mathematics that deals with the properties and relationships of geometric figures, plays a fundamental role in various fields such as physics, computer graphics, and engineering. In this article, we will delve into the key concepts and techniques of geometry analysis.Euclidean GeometryEuclidean geometry, named after the ancient Greek mathematician Euclid, forms the foundation of modern geometry. It is based on a set of axioms and rules, from which various theorems and propositions can be derived.One of the most famous theorems in Euclidean geometry is the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This theorem has numerous applications in fields such as trigonometry and physics.Another important concept in Euclidean geometry is congruence. Two geometric figures are said to be congruent if they have the same shape and size. Congruence can be established through various methods, such as side-angle-side (SAS), angle-angle-side (AAS), or side-side-side (SSS) congruence criteria.Analytic GeometryAnalytic geometry, also known as coordinate geometry, combines algebraic techniques with geometry to study geometric figures. It introduces a coordinate system, where points on a plane are represented by ordered pairs (x, y).The distance between two points in a coordinate system can be calculated using the distance formula:d = √((x₂-x₁)² + (y₂-y₁)²)Analytic geometry also provides a method to determine the equation of geometric figures. For example, the equation of a straight line can be written in the form y = mx + c, where m is the gradient (slope) and c is the y-intercept.Transformational GeometryTransformational geometry investigates the properties of geometric figures under transformations such as translation, rotation, reflection, and dilation. These transformations preserve the shape and size of the figures.Translations shift figures to a different location while preserving their orientation and size. Rotations involve rotating a figure around a fixed point by acertain angle. Reflections mirror a figure across a line called the line of reflection. Dilations scale a figure up or down by multiplying its coordinates by a scale factor.Transformational geometry is widely used in computer graphics and animation to create realistic visual effects. It provides a mathematical framework for manipulating and moving objects in a virtual 3D space.Non-Euclidean GeometryNon-Euclidean geometry explores geometries that do not adhere to Euclid’s axioms. One of the most well-known non-Euclidean geometries is spherical geometry, which deals with figures on the surface of a sphere. In spherical geometry, the sum of the angles in a triangle exceeds 180 degrees.Another important non-Euclidean geometry is hyperbolic geometry, where the sum of angles in a triangle is less than 180 degrees. Hyperbolic geometry finds applications in the theory of relativity and the study of curved spaces.ConclusionGeometry analysis encompasses a wide range of topics and techniques that are crucial in various disciplines. From the fundamental principles of Euclidean geometry to the applications of transformational geometry and the exploration of non-Euclidean geometries, the study of geometry offers powerful tools for understanding and modeling the physical and abstract world around us.Note: This document is an original work by Qiu Weisheng and does not contain any images, AI, artificial intelligence, machine learning, GPT, or explicit mentions of Markdown.。
数学专业英语词汇（H）_数学物理英语词汇
h closed h闭的haar condition 哈尔条件haar measure 哈尔测度hadamard criterion 阿达玛判别准则hadamard gap condition 阿达玛间断条件hadamard matrix 阿达玛矩阵hadamard method of descent 阿达玛下降法hadamard multiplication theorem 阿达玛乘法定理hadamard three circles theorem 阿达玛三圆定理half 半half angle 半角half angle formulas 半角公式half axis 半轴half closed interval 半闭区间half exact 半正合的half line 半直线half neighborhood 半邻域half plane 半无限平面half plane of absolute convergence 绝对收敛半平面half plane of convergence 收敛半平面half round 半圆的half side formulas 半边公式half space 半空间halve 对分halving method 二等分法hamilton characteristic function 哈密顿特寨数hamilton formula 哈密顿公式hamilton function 哈密顿函数hamilton jacobi equation 哈密顿雅可比方程hamilton jacobi theory 哈密顿雅可比理论hamilton principle 哈密顿原理hamiltonian 哈密顿函数hamiltonian circuit 哈密顿回路hamiltonian group 哈密顿群hamiltonian operator 哈密顿算子hamiltonian path 合密顿道路hand 边handle 环柄handle of the second kind 交叉套handlebody 环柄体hankel transformation 汉克尔变换harmonic analysis 低分析harmonic analyzer 傅里叶分析仪harmonic conjugate 低共轭点harmonic constant 低常数harmonic curve 低曲线harmonic differential equation 低微分方程harmonic division 低分割harmonic function 低函数harmonic integral 低积分harmonic mapping 低映射harmonic mean 低平均harmonic measure 低测度harmonic motion 低运动harmonic oscillation 谐振动harmonic progression 低级数harmonic ratio 低比harmonic series 低级数harmonic synthesis 傅里叶综合法harmonicity 低性hasse diagram 哈塞图hausdorff group 豪斯道夫群hausdorff measure 豪斯道夫测度hausdorff metric 豪斯道夫度量hausdorff separation axiom 廉斯道夫分离公理hausdorff space 分离空间haversine 半正矢heat 热heat conduction 热传导hecke character 黑克特贞hecke operator 黑克算子hectoliter 百升hectometer 百米helicograph 螺旋规helicoid 螺旋面helicoidal surface 螺旋面helix 螺旋线hemi continuous 半连续的hemihedry 半对称hemipyramid 半棱锥体hemisphere 半球hemispherical 半球面的hemispherical shape 半球形hendecagon 十一边形henselization 享泽莱化heptagon 七边形heptahedron 七面体hereditarily enumerable set 遗传可数集hereditarily generating system 遗传的生成系hereditarily indecomposable continuum 遗传不可分解的连续统hereditarily normal space 遗传正规空间hereditary class 遗传类hereditary property 遗传性质hereditary set 遗传集hereditary system of sets 集的遗传系heredity 遗传性hermite function 埃尔米特函数hermite interpolation formula 埃尔米特插值公式hermite interpolation polynomial 埃尔米特插值多项式hermite normal form 埃尔米特正规形式hermite polynomial 埃尔米特多项式hermite reciprocity law 埃尔米待互反律hermitian bilinear functional 埃尔米特双线性泛函hermitian conjugate 埃尔米特共轭阵hermitian form 埃尔米特形式hermitian inner product module 埃尔米特内积模hermitian inner product space 埃尔米特空间hermitian kernel 埃尔米特核hermitian matrix 埃尔米特矩阵hermitian metric 埃尔米特度量hermitian operator 埃尔米特算子hermitian polynomiat 埃尔米特多项式hermitian transformation 埃尔米特变换hero formula 海伦公式hesse normal form 海赛正规形式hessian 海赛形式hessian group 海赛群hessian matrix 海赛矩阵hexagon 六边形hexagonal 六边形的hexagonal net 六边形网格hexagonal system 六角系hexahedral 六面体的hexahedron 六面体hexakistetrahedron 六四面体hierarchical classification 谱系分类hierarchy 分层high speed computer 高速计算机higher algebra 高等代数higher commutator 广义换位子higher derivative 高阶导数higher mathematics 高等数学higher order term 高阶项higher plane curve 高次平面曲线higher singularity 高次奇异性highest common divisor 最大公约highest common factor 最大公因子highest derivative 最高阶导数highest order 最高位highest weight 最高权hilbert basis theorem 希耳伯特基定理hilbert cube 希耳伯特超平行体hilbert inequality 希耳伯特不等式hilbert integral 希耳伯特积分hilbert matrix 希耳伯特矩阵hilbert modular form 希耳伯特模形式hilbert modular function 希耳伯特模函数hilbert modular group 希耳伯特模群hilbert nullstellensatz 希耳伯特零点定理hilbert parallelotope 希耳伯特超平行体hilbert problems 希耳伯特问题hilbert space 希耳伯特空间hill differential equation 希耳微分方程histogram 直方图history 履历hodograph 速端曲线hodograph transformation 速端曲线变换hodometer 路程表holding domain 解域holomorph convex manifold 全形凸廖holomorph separable manifold 全形可分廖holomorphic 正则的holomorphic completeness 全纯完全性holomorphic convexity 正则凸性holomorphic differential 全纯微分holomorphic differential form 全纯微分形式holomorphic divisor 全纯除子holomorphic function 全纯函数holomorphic manifold 复解析廖holomorphic mapping 全纯映射holomorphic part 全纯部分holomorphy 正则holonomic condition 完全性条件holonomic reference system 完整参考系holonomic system 完整系holonomy 完整holonomy group 完整群homeomorph 同胚象homeomorphic 同胚的homogeneity 齐性homogeneity formula 齐性公式homogeneity of variances 同方差性homogeneity relation 齐性关系homogeneous 均匀的homogeneous cartesian co ordinates 齐次笛卡儿坐标homogeneous coordinates 齐次笛卡儿坐标homogeneous distribution 均匀分布homogeneous element 齐次元素homogeneous equation 齐次方程homogeneous function 齐次函数homogeneous function of order k k阶齐次函数homogeneous ideal 齐次理想homogeneous integral equation 齐次积分方程homogeneous linear boundary value problem 齐次线性边值问题homogeneous linear differential equation 齐次线性微分方程homogeneous linear transformation 齐次线性变换homogeneous lineare transformation 齐次线性变换homogeneous markov chain 齐次马尔可夫链homogeneous markov process 齐次马尔可夫过程homogeneous operator 齐次算子homogeneous polynomial 齐次多项式homogeneous space 商空间homogeneous system of differential equations 齐次微分方程组homogeneous system of linear equations 齐次线性方程组homogeneous variational problem 齐次变分问题homographic function 单应函数homological algebra 同碟数homological dimension 同惮数homological invariant 同祷变量homologous mappings 同党射homologous simplicial map 同单形映射homologous to zero 同第零homology 同调homology algebra 同碟数homology class 同掂homology equivalence 同等价homology equivalent complex 同等价复形homology functor 同弹子homology group 同岛homology manifold 同滴homology module 同担homology operation 同邓算homology sequence 同凋列homology simplex 同单形homology spectral sequence 同底序列homology sphere 同凋homology theory 同帝homology type 同低homomorphic group 同态群homomorphic image 同态象homomorphism 同态homomorphism theorem 同态定理homoscedastic 同方差的homoscedasticity 同方差性homothetic transformation 相似扩大homothety 相似扩大homotopic 同伦的homotopic invariant 同伦不变量homotopic map 同伦映射homotopic path 同伦道路homotopically equivalent space 同伦等价空间homotopy associativity 同伦结合性homotopy category of topological spaces 拓扑空间同伦范畴homotopy chain 同伦链homotopy class 同伦类homotopy classification 同伦分类homotopy equivalence 同伦等价homotopy excision theorem 同伦分割定理homotopy extension 同伦扩张homotopy group 同伦群homotopy group functor 同伦群函子homotopy inverse 同伦逆的homotopy operator 同伦算子homotopy sequence 同伦序列homotopy set 同伦集homotopy sphere 同伦球面homotopy theorem 同伦定理homotopy theory 同伦论homotopy type 同伦型homotopyassociative 同伦结合的horizon 水平线horizontal axis 水平轴horizontal component 水平分量horizontal coordinates 水平坐标horizontal plane 水平面horizontal projection 水平射影horned sphere 角形球面horocycle 极限圆horosphere 极限球面horse power 马力hungarian method 匈牙利法hurewicz isomorphism theorem 胡列维茨同构定理hydrodynamics 铃动力学hydromechanics 铃力学hydrostatics 铃静力学hyper graeco latin square 超格勒科拉丁方格hyper octahedral group 超八面体群hyperabelian function 超阿贝耳函数hyperalgebraic manifold 超代数廖hyperarithmetical 超算术的hyperarithmetical relation 超算术关系hyperbola 双曲线hyperbolic 双曲线的hyperbolic automorphism 双曲代换hyperbolic catenary 双曲悬链线hyperbolic cosecant 双曲余割hyperbolic cosine 双曲余弦hyperbolic cotangent 双曲余切hyperbolic cylinder 双曲柱hyperbolic elliptic motion 双曲椭圆运动hyperbolic equation 双曲型方程hyperbolic function 双曲函数hyperbolic geometry 双曲几何学hyperbolic inverse point 双曲逆点hyperbolic involution 双曲对合hyperbolic line 双曲线hyperbolic motion 双曲运动hyperbolic orbit 双曲线轨道hyperbolic paraboloid 双曲抛物面hyperbolic plane 双曲平面hyperbolic point 双曲点hyperbolic riemann surface 双曲型黎曼曲面hyperbolic rotation 双曲旋转hyperbolic secant 双曲正割hyperbolic sine 双曲正弦hyperbolic space 双曲空间hyperbolic spiral 双曲螺线hyperbolic substitution 双曲代换hyperbolic system 双曲型组hyperbolic tangent 双曲正切hyperbolic tangent function 双曲正切hyperbolic type 双曲型hyperbolicity 双曲性hyperboloid 双曲面hyperboloid of one sheet 单叶双曲面hyperboloid of revolution 旋转双曲面hyperboloid of two sheets 双叶双曲面hypercohomology 超上同调hypercomplex 超复数hypercomplex number 超复数hypercone 超锥hyperconjugation 超共轭hypercyclic group 超循环群hypercyclide 超四次圆纹曲面hyperelliptic 超椭圆的hyperelliptic function 超椭圆函数hyperelliptic integral 超椭圆积分hyperelliptic theta function 超椭圆函数hyperfinite c* algebra 超有限c*代数hypergeometric differential equation 超几何微分方程hypergeometric distribution 超几何分布hypergeometric function 超几何函数hypergeometric function of the second kind 第二类超几何函数hypergeometric series 超几何级数hypergeometry 超几何学hypergroup 超群hypermatrix 超矩阵hypernormal dispersion 超正态方差hyperplane 超平面hyperplane coordinates 超平面坐标hyperplane of support 支撑超平面hyperplane section 超平面截面hyperquadric 超二次曲面hyperreal numbers 超实数hyperspace 超空间hypersphere 超球面hyperstonian space 超斯通空间hypersurface 超曲面hypocycloid 内摆线hypocycloidal 圆内旋轮线的hypoelliptic operator 次椭圆型算子hypoellipticity 次椭圆性hypotenuse 斜边hypothesis 假设hypothetical population 假言总体hypotrochoid 长短辐圆内旋轮线。
低维拓扑简介
考研论坛»数学»低维拓扑knight51发表于2005-7-28 08:34低维拓扑<P>下面说说低维拓扑的内容：低维拓扑是微分拓扑的一部分，主要研究3，4维流形与纽结理论。
又叫几何拓扑。
主要以代数拓扑与微分拓扑为工具。
它与微分几何和动力系统关系密切。
国外搞这个方向的也几乎都搞微分几何和动力系统。
我国这个方向北大最牛，美国是伯克利和普林斯顿最牛。
比起代数几何来，它比较好入门。
初学者只需要代数拓扑，微分拓扑，黎曼几何的知识就行了。
美国这方面比较牛，几乎每个搞基础数学研究的都会低维拓扑。
</P><DIV class=postcolor>纠正一下上面的错误，美国也不是每个搞基础数的都精通低维拓扑，而是懂一些低维拓扑的知识。
如果入门后还想更加深入了解它，那还需要读一些双曲几何和拓扑动力系统的书。
</DIV><!-- THE POST --><!-- THE POST --><DIV class=postcolor>下面介绍一下这方面的牛人：Bill Thurston studied at New College, Sarasota, Florida. He received his B.S. from there in 1967 and moved to the University of California at Berkeley to undertake research under Morris Hirsch's and Stephen Smale's supervision. He was awarded his doctorate in 1972 for a thesis entitled Foliations of 3-manifolds which are circle bundles. This work showed the existence of compact leaves in foliations of 3-dimensional manifolds.After completing his Ph.D., Thurston spent the academic year 1972-73 at the Institute for Advanced Study at Princeton. Then, in 1973, he was appointed an assistant professor of mathematics at Massachusetts Institute of Technology. In 1974 he was appointed professor of mathematics at Princeton University.Throughout this period Thurston worked on foliations. Lawson ([5]) sums up this work:-It is evident that Thurston's contributions to the field of foliations are of considerable depth. However, what sets them apart is their marvellous originality. This is also true of his subsequent work on Teichmüller space and the theory of 3-manifolds.In [8] Wall describes Thurston's contributions which led to him being awarded a Fields Medal in 1982. In fact the1982 Fields Medals were announced at a meeting of the General Assembly of the International Mathematical Union in Warsaw in early August 1982. They were not presented until the International Congress in Warsaw which could not be held in 1982 as scheduled and was delayed until the following year. Lectures on the work of Thurston which led to his receiving the Medal were made at the 1983 International Congress. Wall, giving that address, said:-Thurston has fantastic geometric insight and vision: his ideas have completely revolutionised the study of topology in 2 and 3 dimensions, and brought about a new and fruitful interplaybetween analysis, topology and geometry.Wall [8] goes on to describe Thurston's work in more detail:-The central new idea is that a very large class of closed 3-manifolds should carry a hyperbolic structure - be the quotient of hyperbolic space by a discrete group of isometries, or equivalently, carry a metric of constant negative curvature. Although this is a natural analogue of the situation for 2-manifolds, where such a result is given by Riemann's uniformisation theorem, it is much less plausible - even counter-intuitive - in the 3-dimensional situation.Kleinian groups, which are discrete isometry groups of hyperbolic 3-space, were first studied by Poincaré and a fundamental finiteness theorem was proved by Ahlfors. Thurston's work on Kleinian groups yielded many new results and established a well known conjecture. Sullivan describes this geometrical work in [6], giving the following summary:-Thurston's results are surprising and beautiful. The method is a new level of geometrical analysis - in the sense of powerful geometrical estimation on the one hand, and spatial visualisation and imagination on the other, which are truly remarkable.Thurston's work is summarised by Wall [8]:-Thurston's work has had an enormous influence on 3-dimensional topology. This area has a strong tradition of 'bare hands' techniques and relatively little interaction with other subjects. Direct arguments remain essential, but 3-dimensional topology has now firmly rejoined the main stream of mathematics.Thurston has received many honours in addition to the Fields Medal. He held a Alfred P Sloan Foundation Fellowship in 1974-75. In 1976 his work on foliations led to his being awarded the Oswald Veblen Geometry Prize of the American Mathematical Society. In 1979 he was awarded the Alan T Waterman Award, being the second mathematician to receive such an award (the first being Fefferman in 1976).</DIV><!-- THE POST -->第2个牛人：Michael Freedman entered the University of California at Berkeley in 1968 and continued his studies at Princeton University in 1969. He was awarded a doctorate by Princeton in 1973 for his doctoral dissertation entitled Codimension-Two Surgery. His thesis supervisor was William Browder.After graduating Freedman was appointed a lecturer in the Department of Mathematics at the University of California at Berkeley. He held this post from 1973 until 1975 when he became a member of the Institute for Advanced Study at Princeton. In 1976 he was appointed as assistant professor in the Department of Mathematics at the University of California at San Diego.Freedman was promoted to associate professor at San Diego in 1979. He spent the year 1980/81 at the Institute for Advanced Study at Princeton returning to the University of California at San Diego where he was promoted to professor on 1982. He holds this post in addition to the Charles Lee Powell Chair of Mathematics which he was appointed to in 1985.Freedman was awarded a Fields Medal in 1986 for his work on the Poincaré conjecture. The Poincaré conjecture, one of the famous problems of 20th-century mathematics, asserts that a simply connected closed 3-dimensional manifold is a 3-dimensional sphere. The higher dimensional Poincaréconjecture claims that any closed n-manifold which is homotopy equivalent to the n-sphere must be the n-sphere. When n = 3 this is equivalent to the Poincaré conjecture. Smale proved the higher dimensional Poincaré conjecture in 1961 for n at least 5. Freedman proved the conjecture for n = 4 in 1982 but the original conjecture remains open.Milnor, describing Freedman's work which led to the award of a Fields Medal at the International Congress of Mathematicians in Berkeley in 1986, said:-Michael Freedman has not only proved the Poincaré hypothesis for 4-dimensional topological manifolds, thus characterising the sphere S4, but has also given us classification theorems, easy to state and to use but difficult to prove, for much more general 4-manifolds. The simple nature of his results in the topological case must be contrasted with the extreme complications which are now known to occur in the study of differentiable and piecewise linear 4-manifolds. ... Freedman's 1982 proof of the 4-dimensional Poincaré hypothesis was an extraordinary tour de force. His methods were so sharp as to actually provide a complete classification of all compact simply connected topological 4-manifolds, yielding many previously unknown examples of such manifolds, and many previously unknown homeomorphisms between known manifolds.Freedman has received many honours for his work. He was California Scientist of the Year in 1984 and, in the same year, he was made a MacArthur Foundation Fellow and also was elected to the National Academy of Sciences. In 1985 he was elected to the American Academy of Arts and Science. In addition to being awarded the Fields Medal in 1986, he also received the Veblen Prize from the American Mathematical Society in that year. The citation for the Veblen Prize reads (see [3]):-After the discovery in the early 60s of a proof for the Poincaré conjecture and other properties of simply connected manifolds of dimension greater than four, one of the biggest open problems, besides the three dimensional Poincaré conjecture, was the classification of closed simply connected four manifolds. In his paper, The topology of four-dimensional manifolds, published in the Journal of Differential Geometry (1982), Freedman solved this problem, and in particular, the four-dimensional Poincaré conjecture. The major innovation was the solution of the simply connected surgery problem by proving a homotopy theoretic condition suggested by Casson for embedding a 2-handle, i.e. a thickened disc in a four manifold with boundary.Besides these results about closed simply connected four manifolds, Freedman also proved:(a) Any four manifold properly equivalent to R4 is homeomorphic to R4; a related result holds for S3 R.(b) There is a nonsmoothable closed four manifold.© The four-dimensional Hauptvermutung is false; i.e. there are four manifolds with inequivalent combinatorial triangulations.Finally, we note that the results of the above mentioned paper, together with Donaldson's work, produced the startling example of an exotic smoothing of R4.In his reply Freedman thanked his teachers (who he said included his students) and also gave some fascinating views on mathematics [3]:-My primary interest in geometry is for the light it sheds on the topology of manifolds. Here it seems important to be open to the entire spectrum of geometry, from formal to concrete. By spectrum, I mean the variety of ways in which we can think about mathematical structures. At one extreme the intuition for problems arises almost entirely from mental pictures. At the other extreme the geometric burden is shifted to symbolic and algebraic thinking. Of course this extreme is only a middle ground from the viewpoint of algebra, which is prepared to go much further in the direction of formal operations and abandon geometric intuition altogether.In the same reply Freedman also talks about the influence mathematics can have on the world and the way that mathematicians should express their ideas:-In the nineteenth century there was a movement, of which Steiner was a principal exponent, to keep geometry pure and ward off the depredations of algebra. Today I think we feel that much of the power of mathematics comes from combining insights from seemingly distant branches of the discipline. Mathematics is not so much a collection of different subjects as a way of thinking. As such, it may be applied to any branch of knowledge. I want to applaud the efforts now being made by mathematicians to publish ideas on education, energy, economics, defence, and world peace. Experience inside mathematics shows that it isn't necessary to be an old hand in an area to make a contribution. Outside mathematics the situation is less clear, but I cannot help feeling that there, too, it is a mistake to leave important issues entirely to experts.In June 1987 Freedman was presented with the National Medal of Science at the White House by President Ronald Reagan. The following year he received the Humboldt Award and, in 1994, he received the Guggenheim Fellowship Award.<DIV class=postcolor>介绍第3个牛人：Simon Donaldson's secondary school education was at Sevenoaks School in Kent which he attended from 1970 to 1975. He then entered Pembroke College, Cambridge where he studied until 1980, receiving his B.A. in 1979. One of his tutors at Cambridge described him as a very good student but certainly not the top student in his year. Apparently he would always come to his tutorials carrying a violin case.In 1980 Donaldson began postgraduate work at Worcester College, Oxford, first under Nigel Hitchen's supervision and later under Atiyah's supervision. Atiyah writes in [2]:-In 1982, when he was a second-year graduate student, Simon Donaldson proved a result that stunned the mathematical world.This result was published by Donaldson in a paper Self-dual connections and the topology of smooth 4-manifolds which appeared in the Bulletin of the American Mathematical Society in 1983. Atiyah continues his description of Donaldson's work [2]:-Together with the important work of Michael Freedman ..., Donaldson's result implied that there are "exotic" 4-spaces, i.e. 4-dimensional differentiable manifolds which are topologically but not differentiably equivalent to the standard Euclidean 4-space R4. What makes this result so surprising is that n = 4 is the only value for which such exotic n-spaces exist. These exotic 4-spaces have the remarkable property that (unlike R4) they contain compact sets which cannot be contained inside any differentiably embedded 3-sphere !After being awarded his doctorate from Oxford in 1983, Donaldson was appointed a Junior Research Fellow at All Souls College, Oxford. He spent the academic year 1983-84 at the Institute for Advanced Study at Princeton, After returning to Oxford he was appointed Wallis Professor of Mathematics in 1985, a position he continues to hold.Donaldson has received many honours for his work. He received the Junior Whitehead Prize from the London Mathematical Society in 1985. In the following year he was elected a Fellow of the Royal Society and, also in 1986, he received a Fields Medal at the International Congress at Berkeley. In 1991 Donaldson received the Sir William Hopkins Prize from the Cambridge Philosophical Society. Then, the following year, he received the Royal Medal from the Royal Society. He also received the Crafoord Prize from the Royal Swedish Academy of Sciences in 1994:-... for his fundamental investigations in four-dimensional geometry through application of instantons, in particular his discovery of new differential invariants ...Atiyah describes the contribution which led to Donaldson's award of a Fields Medal in [2]. He sums up Donaldson's contribution:-When Donaldson produced his first few results on 4-manifolds, the ideas were so new and foreign to geometers and topologists that they merely gazed in bewildered admiration.Slowly the message has gotten across and now Donaldson's ideas are beginning to be used by others in a variety of ways. ... Donaldson has opened up an entirely new area; unexpected and mysterious phenomena about the geometry of 4-dimensions have been discovered. Moreover the methods are new and extremely subtle, using difficult nonlinear partial differential equations. On the other hand, this theory is firmly in the mainstream of mathematics, having intimate links with the past, incorporating ideas from theoretical physics, and tying in beautifully with algebraic geometry.The article [3] is very interesting and provides both a collection of reminiscences by Donaldson on how he came to make his major discoveries while a graduate student at Oxford and also a survey of areas which he has worked on in recent years. Donaldson writes in [3] that nearly all his work has all come under the headings:-(1) Differential geometry of holomorphic vector bundles.(2) Applications of gauge theory to 4-manifold topology.and he relates his contribution to that of many others in the field.Donaldson's work in summed up by R Stern in [6]:-In 1982 Simon Donaldson began a rich geometrical journey that is leading us to an exciting conclusion to this century. He has created an entirely new and exciting area of research through which much of mathematics passes and which continues to yield mysterious and unexpected phenomena about the topology and geometry of smooth 4-manifolds</DIV><DIV class=postcolor>下面continue介绍第4个牛人：Robion Kirby。
漫谈微分几何、多复变函数与代数几何(Differential geometry, functions
漫谈微分几何、多复变函数与代数几何（Differential geometry, functions of complex variable and algebraic geometry）Differential geometry and tensor analysis, developed with the development of differential geometry, are the basic tools for mastering general relativity. Because general relativity's success, to always obscure differential geometry has become one of the central discipline of mathematics.Since the invention of differential calculus, the birth of differential geometry was born. But the work of Euler, Clairaut and Monge really made differential geometry an independent discipline. In the work of geodesy, Euler has gradually obtained important research, and obtained the famous Euler formula for the calculation of normal curvature. The Clairaut curve of the curvature and torsion, Monge published "analysis is applied to the geometry of the loose leaf paper", the important properties of curves and surfaces are represented by differential equations, which makes the development of classical differential geometry to reach a peak. Gauss in the study of geodesic, through complicated calculation, in 1827 found two main curvature surfaces and its product in the periphery of the Euclidean shape of the space not only depends on its first fundamental form, the result is Gauss proudly called the wonderful theorem, created from the intrinsic geometry. The free surface of space from the periphery, the surface itself as a space to study. In 1854, Riemann made the hypothesis about geometric foundation, and extended the intrinsic geometry of Gauss in 2 dimensional curved surface, thus developing n-dimensional Riemann geometry, with the development of complex functions. A group of excellentmathematicians extended the research objects of differential geometry to complex manifolds and extended them to the complex analytic space theory including singularities. Each step of differential geometry faces not only the deepening of knowledge, but also the continuous expansion of the field of knowledge. Here, differential geometry and complex functions, Lie group theory, algebraic geometry, and PDE all interact profoundly with one another. Mathematics is constantly dividing and blending with each other.By shining the charming glory and the differential geometric function theory of several complex variables, unit circle and the upper half plane (the two conformal mapping establishment) defined on Poincare metric, complex function theory and the differential geometric relationships can be seen distinctly. Poincare metric is conformal invariant. The famous Schwarz theorem can be explained as follows: the Poincare metric on the unit circle does not increase under analytic mapping; if and only if the mapping is a fractional linear transformation, the Poincare metric does not change Poincare. Applying the hyperbolic geometry of Poincare metric, we can easily prove the famous Picard theorem. The proof of Picard theorem to modular function theory is hard to use, if using the differential geometric point of view, can also be in a very simple way to prove. Differential geometry permeates deep into the theory of complex functions. In the theory of multiple complex functions, the curvature of the real differential geometry and other series of calculations are followed by the analysis of the region definition metric of the complex affine space. In complex situations, all of the singular discrete distribution, and in more complex situations, because of the famous Hartogsdevelopment phenomenon, all isolated singularities are engulfed by a continuous region even in singularity formation is often destroyed, only the formation of real codimension 1 manifold can avoid the bad luck. But even this situation requires other restrictions to ensure safety". The singular properties of singularities in the theory of functions of complex functions make them destined to be manifolds. In 1922, Bergman introduced the famous Bergman kernel function, the more complex function or Weyl said its era, in addition to the famous Hartogs, Poincare, Levi of Cousin and several predecessors almost no substantive progress, injected a dynamic Bergman work will undoubtedly give this dead area. In many complex function domains in the Bergman metric metric in the one-dimensional case is the unit circle and Poincare on the upper half plane of the Poincare, which doomed the importance of the work of Bergman.The basic object of algebraic geometry is the properties of the common zeros (algebraic families) of any dimension, affine space, or algebraic equations of a projective space (defined equations),The definitions of algebraic clusters, the coefficients of equations, and the domains in which the points of an algebraic cluster are located are called base domains. An irreducible algebraic variety is a finite sub extension of its base domain. In our numerical domain, the linear space is the extension of the base field in the number field, and the dimension of the linear space is the number of the expansion. From this point of view, algebraic geometry can be viewed as a study of finite extension fields. The properties of algebraic clusters areclosely related to their base domains. The algebraic domain of complex affine space or complex projective space, the research process is not only a large number of concepts and differential geometry and complex function theory and applied to a large number of coincidence, the similar tools in the process of research. Every step of the complex manifold and the complex analytic space has the same influence on these subjects. Many masters in related fields, although they seem to study only one field, have consequences for other areas. For example: the Lerey study of algebraic topology that it has little effect on layer, in algebraic topology, but because of Serre, Weil and H? Cartan (E? Cartan, eldest son) introduction, has a profound impact on algebraic geometry and complex function theory. Chern studies the categories of Hermite spaces, but it also affects algebraic geometry, differential geometry and complex functions. Hironaka studies the singular point resolution in algebraic geometry, but the modification of complex manifold to complex analytic space and blow up affect the theory of complex analytic space. Yau proves that the Calabi conjecture not only affects algebraic geometry and differential geometry, but also affects classical general relativity. At the same time, we can see the important position of nonlinear ordinary differential equations and partial differential equations in differential geometry. Cartan study of symmetric Riemann space, the classification theorem is important, given 1, 2 and 3 dimensional space of a Homogeneous Bounded Domain complete classification, prove that they are all homogeneous symmetric domains at the same time, he guessed: This is also true in the n-dimensional equivalent relation. In 1959, Piatetski-Shapiro has two counterexample and find the domain theory of automorphic function study in symmetry, in the 4 and 5dimensional cases each find a homogeneous bounded domain, which is not a homogeneous symmetric domain, the domain he named Siegel domain, to commemorate the profound work on Siegel in 1943 of automorphic function. The results of Piatetski-Shapiro has profound impact on the theory of complex variable functions and automorphic function theory, and have a profound impact on the symmetry space theory and a series of topics. As we know, Cartan transforms the study of symmetric spaces into the study of Lie groups and Lie algebras, which is directly influenced by Klein and greatly develops the initial idea of Klein. Then it is Cartan developed the concept of Levi-Civita connection, the development of differential geometry in general contact theory, isomorphic mapping through tangent space at each point on the manifold, realize the dream of Klein and greatly promote the development of differential geometry. Cartan is the same, and concluded that the importance of the research in the holonomy manifold twists and turns, finally after his death in thirty years has proved to be correct. Here, we see the vast beauty of differential geometry.As we know, geodesic ties are associated with ODE (ordinary differential equations), minimal surfaces and high dimensional submanifolds are associated with PDE (partial differential equations). These equations are nonlinear equations, so they have high requirements for analysis. Complex PDE and complex analysis the relationship between Cauchy-Riemann equations coupling the famous function theory, in the complex case, the Cauchy- Riemann equations not only deepen the unprecedented contact and the qualitative super Cauchy-Riemann equations (the number of variables is greater than the number of equations) led to a strange phenomenon. This makes PDE and the theory ofmultiple complex functions closely integrated with differential geometry.Most of the scholars have been studying the differential geometry of the intrinsic geometry of the Gauss and Riemann extremely deep stun, by Cartan's method of moving frames is beautiful and concise dumping, by Chern's theory of characteristic classes of the broad and profound admiration, Yau deep exquisite geometric analysis skills to deter.When the young Chern faced the whole differentiation, he said he was like a mountain facing the shining golden light, but he couldn't reach the summit at one time. But then he was cast as a master in this field before Hopf and Weil.If the differential geometry Cartan development to gradually change the general relativistic geometric model, then the differential geometry of Chern et al not only affect the continuation of Cartan and to promote the development of fiber bundle in the form of gauge field theory. Differential geometry is still closely bound up with physics as in the age of Einstein and continues to acquire research topics from physicsWhy does the three-dimensional sphere not give flatness gauge, but can give conformal flatness gauge? Because 3D balls and other dimension as the ball to establish flat space isometric mapping, so it is impossible to establish a flatness gauge; and n-dimensional balls are usually single curvature space, thus can establish a conformal flat metric. In differential geometry, isometry means that the distance between the points on the manifold before and after the mapping remains the same. Whena manifold is equidistant from a flat space, the curvature of its Riemann cross section is always zero. Since the curvature of all spheres is positive constant, the n-dimensional sphere and other manifolds whose sectional curvature is nonzero can not be assigned to local flatness gauge.But there are locally conformally flat manifolds for this concept, two gauge G and G, if G=exp{is called G, P}? G between a and G transform is a conformal transformation. Weyl conformal curvature tensor remains unchanged under conformal transformation. It is a tensor field of (1,3) type on a manifold. When the Weyl conformal curvature tensor is zero, the curvature tensor of the manifold can be represented by the Ricci curvature tensor and the scalar curvature, so Penrose always emphasizes the curvature =Ricci+Weyl.The metric tensor g of an n-dimensional Riemann manifold is conformally equivalent to the flatness gauge locally, and is called conformally flat manifold. All Manifolds (constant curvature manifolds) whose curvature is constant are conformally flat, so they can be given conformal conformal metric. And all dimensions of the sphere (including thethree-dimensional sphere) are manifold of constant curvature, so they must be given conformal conformal metric. Conversely, conformally flat manifolds are not necessarily manifolds of constant curvature. But a wonderful result related to Einstein manifolds can make up for this regret: conformally conformally Einstein manifolds over 3 dimensions must be manifolds of constant curvature. That is to say, if we want conformally conformally flat manifolds to be manifolds of constant curvature, we must call Ric= lambda g, and this is thedefinition of Einstein manifolds. In the formula, Ric is the Ricci curvature tensor, G is the metric tensor, and lambda is constant. The scalar curvature S=m of Einstein manifolds is constant. Moreover, if S is nonzero, there is no nonzero parallel tangent vector field over it. Einstein introduction of the cosmological constant. So he missed the great achievements that the expansion of the universe, so Hubble is successful in the official career; but the vacuum gravitational field equation of cosmological term with had a Einstein manifold, which provides a new stage for mathematicians wit.For the 3 dimensional connected Einstein manifold, even if does not require the conformal flat, it is also the automatic constant curvature manifolds, other dimensions do not set up this wonderful nature, I only know that this is the tensor analysis summer learning, the feeling is a kind of enjoyment. The sectional curvature in the real manifold is different from the curvature of the Holomorphic cross section in the Kahler manifold, and thus produces different results. If the curvature of holomorphic section is constant, the Ricci curvature of the manifold must be constant, so it must be Einstein manifold, called Kahler- Einstein manifold, Kahler. Kahler manifolds are Kahler- Einstein manifolds, if and only if they are Riemann manifolds, Einstein manifolds. N dimensional complex vector space, complex projective space, complex torus and complex hyperbolic space are Kahler- and Einstein manifolds. The study of Kahler-Einstein manifolds becomes the intellectual enjoyment of geometer.Let's go back to an important result of isometric mapping.In this paper, we consider the isometric mapping between M and N and the mapping of the cut space between the two Riemann manifolds, take P at any point on M, and select two non tangent tangent vectors in its tangent space, and obtain its sectional curvature. In the mapping, the two tangent vectors on the P point and its tangent space are transformed into two other tangent vectors under the mapping, and the sectional curvature of the vector is also obtained. If the mapping is isometric mapping, then the curvature of the two cross sections is equal. Or, to be vague, isometric mapping does not change the curvature of the section.Conversely, if the arbitrary points are set, the curvature of the section does not change in nature, then the mapping is not isometric mapping The answer was No. Even in thethree-dimensional Euclidean space on the surface can not set up this property. In some cases, the limit of the geodesic line must be added, and the properties of the Jacobi field can be used to do so. This is the famous Cartan isometry theorem. This theorem is a wonderful application of the Jacobi field. Its wide range of promotion is made by Ambrose and Hicks, known as the Cartan-Ambrose-Hicks theorem.Differential geometry is full of infinite charm. We classify pseudo-Riemannian spaces by using Weyl conformal curvature tensor, which can be classified by Ricci curvature tensor, or classified into 9 types by Bianchi. And these things are all can be attributed to the study of differential geometry, this distant view Riemann and slightly closer to the Klein point of the perfect combination, it can be seen that the great wisdom Cartan, here you can see the profound influence of Einstein.From the Hermite symmetry space to the Kahler-Hodge manifold, differential geometry is not only closely linked with the Lie group, but also connected with algebra, geometry and topologyThink of the great 1895 Poicare wrote the great "position analysis" was founded combination topology unabashedly said differential geometry in high dimensional space is of little importance to this subject, he said: "the home has beautiful scenery, where Xuyuan for." (Chern) topology is the beauty of the home. Why do you have to work hard to compute the curvature of surfaces or even manifolds of high dimensions? But this versatile mathematician is wrong, but we can not say that the mathematical genius no major contribution to differential geometry? Can not. Let's see today's close relation between differential geometry and topology, we'll see. When is a closed form the proper form? The inverse of the Poicare lemma in the region of the homotopy point (the single connected region) tells us that it is automatically established. In the non simply connected region is de famous Rham theorem tells us how to set up, that is the integral differential form in all closed on zero.Even in the field of differential geometry ignored by Poicare, he is still in a casual way deeply affected by the subject, or rather is affecting the whole mathematics.The nature of any discipline that seeks to be generalized after its creation, as is differential geometry. From the curvature, Euclidean curvature of space straight to zero, geometry extended to normal curvature number (narrow Riemann space) andnegative constant space (Lobachevskii space), we know that the greatness of non Euclidean geometry is that it not only independent of the fifth postulate and other alternative to the new geometry. It can be the founder of triangle analysis on it. But the famous mathematician Milnor said that before differential geometry went into non Euclidean geometry, non Euclidean geometry was only the torso with no hands and no feet. The non Euclidean geometry is born only when the curvature is computed uniformly after the metric is defined. In his speech in 1854, Riemann wrote only one formula: that is, this formula unifies the positive curvature, negative curvature and zero curvature geometry. Most people think that the formula for "Riemann" is based on intuition. In fact, later people found the draft paper that he used to calculate the formula. Only then did he realize that talent should be diligent. Riemann has explored the curvature of manifolds of arbitrary curvature of any dimension, but the quantitative calculations go beyond the mathematical tools of that time, and he can only write the unified formula for manifolds of constant curvature. But we know,Even today, this result is still important, differential geometry "comparison theorem" a multitude of names are in constant curvature manifolds for comparison model.When Riemann had considered two differential forms the root of two, this is what we are familiar with the Riemann metric Riemannnian, derived from geometry, he specifically mentioned another case, is the root of four four differential forms (equivalent to four yuan product and four times square). This is the contact and the difference between the two. But he saidthat for this situation and the previous case, the study does not require substantially different methods. It also says that such studies are time consuming and that new insights cannot be added to space, and the results of calculations lack geometric meaning. So Riemann studied only what is now called Riemann metric. Why are future generations of Finsler interested in promoting the Riemann's not wanting to study? It may be that mathematicians are so good that they become a hobby. Cartan in Finsler geometry made efforts, but the effect was little, Chern on the geometric really high hopes also developed some achievements. But I still and general view on the international consensus, that is the Finsler geometry bleak. This is also the essential reason of Finsler geometry has been unable to enter the mainstream of differential geometry, it no beautiful properties really worth geometers to struggle, also do not have what big application value. Later K- exhibition space, Cartan space will not become mainstream, although they are the extension of Riemannnian geometry, but did not get what the big development.In fact, sometimes the promotion of things to get new content is not much, differential geometry is the same, not the object of study, the more ordinary the better, but should be appropriate to the special good. For example, in Riemann manifold, homogeneous Riemann manifold is more special, beautiful nature, homogeneous Riemann manifolds, symmetric Riemann manifold is more special, so nature more beautiful. This is from the analysis of manifold Lie group action angle.From the point of view of metric, the complex structure is given on the even dimensional Riemann manifold, and the complexmanifold is very elegant. Near complex manifolds are complex manifolds only when the near complex structure is integrable. The complex manifold must be orientable, because it is easy to find that its Jacobian must be nonnegative, whereas the real manifold does not have this property in general. To narrow the scope of the Kahler manifold has more good properties, all complex Submanifolds of Kahler manifolds are Kahler manifolds, and minimal submanifolds (Wirtinger theorem), the beautiful results captured the hearts of many differential geometry and algebraic geometry, because other more general manifolds do not set up this beautiful results. If the first Chern number of a three-dimensional Kahler manifold is zero, the Calabi-Yau manifold can be obtained, which is a very interesting manifold for theoretical physicists. The manifold of mirrors of Calabi-Yau manifolds is also a common subject of differential geometry in algebraic geometry. The popular Hodge structure is a subject of endless appeal.Differential geometry, an endless topic. Just as algebraic geometry requires double - rational equivalence as a luxury, differential geometry requires isometric transformations to be difficult. Taxonomy is an eternal subject of mathematics. In group theory, there are single group classification, multi complex function theory, regional classification, algebraic geometry in the classification of algebraic clusters, differential geometry is also classified.The hard question has led to a dash of young geometry and old scholars, and the prospect of differential geometry is very bright.。
高等数学专业名词中英文对照(全面)
微积分英文词汇,高数名词中英文对照，高等数学术语英语翻译一览关键词：微积分英文，高等数学英文翻译,高数英语词汇来源:上海外教网| 发布日期：2008—05-16 17:12V、X、Z:Value of function ：函数值Variable ：变数Vector ：向量Velocity ：速度Vertical asymptote ：垂直渐近线Volume ：体积X—axis ：x轴x—coordinate ：x坐标x—intercept ：x截距Zero vector ：函数的零点Zeros of a polynomial ：多项式的零点T:Tangent function ：正切函数Tangent line ：切线Tangent plane ：切平面Tangent vector :切向量Total differential :全微分Trigonometric function :三角函数Trigonometric integrals ：三角积分Trigonometric substitutions ：三角代换法Tripe integrals ：三重积分S：Saddle point ：鞍点Scalar ：纯量Secant line :割线Second derivative :二阶导数Second Derivative Test ：二阶导数试验法Second partial derivative :二阶偏导数Sector :扇形Sequence ：数列Series ：级数Set ：集合Shell method ：剥壳法Sine function ：正弦函数Singularity ：奇点Slant asymptote :斜渐近线Slope ：斜率Slope—intercept equation of a line ：直线的斜截式Smooth curve ：平滑曲线Smooth surface ：平滑曲面Solid of revolution :旋转体Space ：空间Speed ：速率Spherical coordinates :球面坐标Squeeze Theorem ：夹挤定理Step function ：阶梯函数Strictly decreasing :严格递减Strictly increasing :严格递增Sum :和Surface :曲面Surface integral ：面积分Surface of revolution :旋转曲面Symmetry ：对称R：Radius of convergence ：收敛半径Range of a function ：函数的值域Rate of change :变化率Rational function ：有理函数Rationalizing substitution ：有理代换法Rational number ：有理数Real number :实数Rectangular coordinates ：直角坐标Rectangular coordinate system ：直角坐标系Relative maximum and minimum ：相对极大值与极小值Revenue function ：收入函数Revolution ，solid of ：旋转体Revolution ，surface of ：旋转曲面Riemann Sum ：黎曼和Riemannian geometry ：黎曼几何Right-hand derivative ：右导数Right—hand limit ：右极限Root ：根P、Q:Parabola ：拋物线Parabolic cylinder ：抛物柱面Paraboloid ：抛物面Parallelepiped ：平行六面体Parallel lines ：并行线Parameter ：参数Partial derivative ：偏导数Partial differential equation :偏微分方程Partial fractions ：部分分式Partial integration ：部分积分Partiton :分割Period ：周期Periodic function ：周期函数Perpendicular lines :垂直线Piecewise defined function ：分段定义函数Plane ：平面Point of inflection ：反曲点Polar axis ：极轴Polar coordinate :极坐标Polar equation :极方程式Pole ：极点Polynomial ：多项式Positive angle :正角Point—slope form ：点斜式Power function :幂函数Product ：积Quadrant ：象限Quotient Law of limit ：极限的商定律Quotient Rule ：商定律M、N、O：Maximum and minimum values :极大与极小值Mean Value Theorem ：均值定理Multiple integrals :重积分Multiplier :乘子Natural exponential function ：自然指数函数Natural logarithm function :自然对数函数Natural number ：自然数Normal line ：法线Normal vector ：法向量Number ：数Octant ：卦限Odd function ：奇函数One-sided limit ：单边极限Open interval ：开区间Optimization problems ：最佳化问题Order ：阶Ordinary differential equation ：常微分方程Origin ：原点Orthogonal ：正交的L:Laplace transform ：Leplace 变换Law of Cosines ：余弦定理Least upper bound ：最小上界Left—hand derivative :左导数Left—hand limit :左极限Lemniscate :双钮线Length ：长度Level curve ：等高线L’Hospital's rule ：洛必达法则Limacon :蚶线Limit ：极限Linear approximation：线性近似Linear equation :线性方程式Linear function ：线性函数Linearity ：线性Linearization ：线性化Line in the plane ：平面上之直线Line in space :空间之直线Lobachevski geometry ：罗巴切夫斯基几何Local extremum ：局部极值Local maximum and minimum ：局部极大值与极小值Logarithm ：对数Logarithmic function ：对数函数I:Implicit differentiation ：隐求导法Implicit function ：隐函数Improper integral ：瑕积分Increasing/Decreasing Test ：递增或递减试验法Increment ：增量Increasing Function ：增函数Indefinite integral ：不定积分Independent variable ：自变数Indeterminate from ：不定型Inequality :不等式Infinite point ：无穷极限Infinite series :无穷级数Inflection point ：反曲点Instantaneous velocity ：瞬时速度Integer ：整数Integral ：积分Integrand ：被积分式Integration ：积分Integration by part :分部积分法Intercepts ：截距Intermediate value of Theorem ：中间值定理Interval ：区间Inverse function :反函数Inverse trigonometric function :反三角函数Iterated integral :逐次积分H:Higher mathematics 高等数学/高数E、F、G、H:Ellipse ：椭圆Ellipsoid ：椭圆体Epicycloid ：外摆线Equation :方程式Even function ：偶函数Expected Valued ：期望值Exponential Function ：指数函数Exponents ，laws of ：指数率Extreme value :极值Extreme Value Theorem :极值定理Factorial ：阶乘First Derivative Test :一阶导数试验法First octant ：第一卦限Focus ：焦点Fractions ：分式Function :函数Fundamental Theorem of Calculus ：微积分基本定理Geometric series ：几何级数Gradient ：梯度Graph ：图形Green Formula :格林公式Half-angle formulas ：半角公式Harmonic series ：调和级数Helix :螺旋线Higher Derivative :高阶导数Horizontal asymptote :水平渐近线Horizontal line :水平线Hyperbola ：双曲线Hyper boloid ：双曲面D：Decreasing function ：递减函数Decreasing sequence ：递减数列Definite integral :定积分Degree of a polynomial :多项式之次数Density ：密度Derivative ：导数of a composite function ：复合函数之导数of a constant function ：常数函数之导数directional ：方向导数domain of ：导数之定义域of exponential function ：指数函数之导数higher :高阶导数partial :偏导数of a power function :幂函数之导数of a power series ：羃级数之导数of a product :积之导数of a quotient ：商之导数as a rate of change :导数当作变率right-hand ：右导数second :二阶导数as the slope of a tangent :导数看成切线之斜率Determinant :行列式Differentiable function :可导函数Differential ：微分Differential equation ：微分方程partial :偏微分方程Differentiation ：求导法implicit ：隐求导法partial :偏微分法term by term ：逐项求导法Directional derivatives ：方向导数Discontinuity :不连续性Disk method ：圆盘法Distance ：距离Divergence :发散Domain ：定义域Dot product ：点积Double integral ：二重积分change of variable in ：二重积分之变数变换in polar coordinates ：极坐标二重积分C：Calculus ：微积分differential ：微分学integral ：积分学Cartesian coordinates :笛卡儿坐标一般指直角坐标Cartesian coordinates system ：笛卡儿坐标系Cauch’s Mean Value Theorem ：柯西均值定理Chain Rule ：连锁律Change of variables ：变数变换Circle ：圆Circular cylinder ：圆柱Closed interval :封闭区间Coefficient ：系数Composition of function ：函数之合成Compound interest ：复利Concavity :凹性Conchoid :蚌线Cone :圆锥Constant function :常数函数Constant of integration :积分常数Continuity ：连续性at a point ：在一点处之连续性of a function ：函数之连续性on an interval ：在区间之连续性from the left ：左连续from the right ：右连续Continuous function :连续函数Convergence ：收敛interval of ：收敛区间radius of :收敛半径Convergent sequence ：收敛数列series ：收敛级数Coordinate：s：坐标Cartesian ：笛卡儿坐标cylindrical ：柱面坐标polar ：极坐标rectangular ：直角坐标spherical ：球面坐标Coordinate axes ：坐标轴Coordinate planes :坐标平面Cosine function :余弦函数Critical point ：临界点Cubic function ：三次函数Curve ：曲线Cylinder:圆柱Cylindrical Coordinates ：圆柱坐标A、B：Absolute convergence ：绝对收敛Absolute extreme values ：绝对极值Absolute maximum and minimum ：绝对极大与极小Absolute value ：绝对值Absolute value function ：绝对值函数Acceleration ：加速度Antiderivative :反导数Approximate integration ：近似积分Approximation :逼近法by differentials ：用微分逼近linear ：线性逼近法by Simpson's Rule ：Simpson法则逼近法by the Trapezoidal Rule ：梯形法则逼近法Arbitrary constant ：任意常数Arc length :弧长Area ：面积under a curve :曲线下方之面积between curves ：曲线间之面积in polar coordinates ：极坐标表示之面积of a sector of a circle ：扇形之面积of a surface of a revolution ：旋转曲面之面积Asymptote ：渐近线horizontal :水平渐近线slant ：斜渐近线vertical ：垂直渐近线Average speed ：平均速率Average velocity ：平均速度Axes, coordinate ：坐标轴Axes of ellipse ：椭圆之轴Binomial series ：二项级数微积分词汇第一章函数与极限Chapter1 Function and Limit集合set元素element子集subset空集empty set并集union交集intersection差集difference of set基本集basic set补集complement set直积direct product笛卡儿积Cartesian product开区间open interval闭区间closed interval半开区间half open interval有限区间finite interval区间的长度length of an interval无限区间infinite interval领域neighborhood领域的中心centre of a neighborhood 领域的半径radius of a neighborhood 左领域left neighborhood右领域right neighborhood映射mappingX到Y的映射mapping of X ontoY 满射surjection单射injection一一映射one-to-one mapping双射bijection算子operator变化transformation函数function逆映射inverse mapping复合映射composite mapping自变量independent variable因变量dependent variable定义域domain函数值value of function函数关系function relation值域range自然定义域natural domain单值函数single valued function多值函数multiple valued function单值分支one-valued branch函数图形graph of a function绝对值函数absolute value符号函数sigh function整数部分integral part阶梯曲线step curve当且仅当if and only if（iff)分段函数piecewise function上界upper bound下界lower bound有界boundedness无界unbounded函数的单调性monotonicity of a function 单调增加的increasing单调减少的decreasing单调函数monotone function函数的奇偶性parity(odevity) of a function 对称symmetry偶函数even function奇函数odd function函数的周期性periodicity of a function周期period反函数inverse function直接函数direct function复合函数composite function中间变量intermediate variable函数的运算operation of function基本初等函数basic elementary function 初等函数elementary function幂函数power function指数函数exponential function对数函数logarithmic function三角函数trigonometric function反三角函数inverse trigonometric function 常数函数constant function双曲函数hyperbolic function双曲正弦hyperbolic sine双曲余弦hyperbolic cosine双曲正切hyperbolic tangent反双曲正弦inverse hyperbolic sine反双曲余弦inverse hyperbolic cosine反双曲正切inverse hyperbolic tangent极限limit数列sequence of number收敛convergence收敛于a converge to a发散divergent极限的唯一性uniqueness of limits收敛数列的有界性boundedness of a convergent sequence子列subsequence函数的极限limits of functions函数当x趋于x0时的极限limit of functions as x approaches x0左极限left limit右极限right limit单侧极限one-sided limits水平渐近线horizontal asymptote无穷小infinitesimal无穷大infinity铅直渐近线vertical asymptote夹逼准则squeeze rule单调数列monotonic sequence高阶无穷小infinitesimal of higher order低阶无穷小infinitesimal of lower order同阶无穷小infinitesimal of the same order作者：新少年特工2007-10-8 18：37 回复此发言-—--——-—---——————-—----—--———----—---——-—-——-————--————-—-—————-——————————--———-2 高等数学-翻译等阶无穷小equivalent infinitesimal函数的连续性continuity of a function增量increment函数在x0连续the function is continuous at x0左连续left continuous右连续right continuous区间上的连续函数continuous function函数在该区间上连续function is continuous on an interval不连续点discontinuity point第一类间断点discontinuity point of the first kind第二类间断点discontinuity point of the second kind初等函数的连续性continuity of the elementary functions定义区间defined interval最大值global maximum value (absolute maximum)最小值global minimum value (absolute minimum）零点定理the zero point theorem介值定理intermediate value theorem第二章导数与微分Chapter2 Derivative and Differential速度velocity匀速运动uniform motion平均速度average velocity瞬时速度instantaneous velocity圆的切线tangent line of a circle切线tangent line切线的斜率slope of the tangent line位置函数position function导数derivative可导derivable函数的变化率问题problem of the change rate of a function导函数derived function左导数left—hand derivative右导数right—hand derivative单侧导数one—sided derivatives在闭区间【a,b】上可导is derivable on the closed interval ［a，b］切线方程tangent equation角速度angular velocity成本函数cost function边际成本marginal cost链式法则chain rule隐函数implicit function显函数explicit function二阶函数second derivative三阶导数third derivative高阶导数nth derivative莱布尼茨公式Leibniz formula对数求导法log— derivative参数方程parametric equation相关变化率correlative change rata微分differential可微的differentiable函数的微分differential of function自变量的微分differential of independent variable微商differential quotient间接测量误差indirect measurement error绝对误差absolute error相对误差relative error第三章微分中值定理与导数的应用Chapter3 MeanValue Theorem of Differentials and the Application of Derivatives 罗马定理Rolle’s theorem费马引理Fermat’s lemma拉格朗日中值定理Lagrange's mean value theorem驻点stationary point稳定点stable point临界点critical point拉格朗日中值公式Lagrange’s mean value formula柯西中值定理Cauchy’s mean value theorem洛必达法则L’Hospital's Rule0/0型不定式indeterminate form of type 0/0不定式indeterminate form泰勒中值定理Taylor’s mean value theorem泰勒公式Taylor formula余项remainder term拉格朗日余项Lagrange remainder term麦克劳林公式Maclaurin’s formula佩亚诺公式Peano remainder term凹凸性concavity凹向上的concave upward，cancave up凹向下的，向上凸的concave downward’ concave down 拐点inflection point函数的极值extremum of function极大值local（relative) maximum最大值global（absolute) mximum极小值local（relative）minimum最小值global（absolute）minimum目标函数objective function曲率curvature弧微分arc differential平均曲率average curvature曲率园circle of curvature曲率中心center of curvature曲率半径radius of curvature渐屈线evolute渐伸线involute根的隔离isolation of root隔离区间isolation interval切线法tangent line method第四章不定积分Chapter4 Indefinite Integrals原函数primitive function（antiderivative)积分号sign of integration被积函数integrand积分变量integral variable积分曲线integral curve积分表table of integrals换元积分法integration by substitution分部积分法integration by parts分部积分公式formula of integration by parts真分式proper fraction假分式improper fraction第五章定积分Chapter5 Definite Integrals曲边梯形trapezoid with曲边curve edge窄矩形narrow rectangle曲边梯形的面积area of trapezoid with curved edge积分下限lower limit of integral积分上限upper limit of integral积分区间integral interval分割partition积分和integral sum可积integrable矩形法rectangle method积分中值定理mean value theorem of integrals函数在区间上的平均值average value of a function on an integvals 牛顿－莱布尼茨公式Newton—Leibniz formula微积分基本公式fundamental formula of calculus换元公式formula for integration by substitution递推公式recurrence formula反常积分improper integral反常积分发散the improper integral is divergent反常积分收敛the improper integral is convergent无穷限的反常积分improper integral on an infinite interval无界函数的反常积分improper integral of unbounded functions绝对收敛absolutely convergent第六章定积分的应用Chapter6 Applications of the Definite Integrals元素法the element method面积元素element of area平面图形的面积area of a luane figure直角坐标又称“笛卡儿坐标（Cartesian coordinates）”极坐标polar coordinates抛物线parabola椭圆ellipse旋转体的面积volume of a solid of rotation旋转椭球体ellipsoid of revolution, ellipsoid of rotation曲线的弧长arc length of acurve可求长的rectifiable光滑smooth功work水压力water pressure引力gravitation变力variable force第七章空间解析几何与向量代数Chapter7 Space Analytic Geometry and Vector Algebra向量vector自由向量free vector单位向量unit vector零向量zero vector相等equal平行parallel向量的线性运算linear poeration of vector三角法则triangle rule平行四边形法则parallelogram rule交换律commutative law结合律associative law负向量negative vector差difference分配律distributive law空间直角坐标系space rectangular coordinates坐标面coordinate plane卦限octant向量的模modulus of vector向量a与b的夹角angle between vector a and b方向余弦direction cosine方向角direction angle向量在轴上的投影projection of a vector onto an axis数量积，外积，叉积scalar product，dot product,inner product 曲面方程equation for a surface球面sphere旋转曲面surface of revolution母线generating line轴axis圆锥面cone顶点vertex旋转单叶双曲面revolution hyperboloids of one sheet旋转双叶双曲面revolution hyperboloids of two sheets柱面cylindrical surface ，cylinder圆柱面cylindrical surface准线directrix抛物柱面parabolic cylinder二次曲面quadric surface椭圆锥面dlliptic cone椭球面ellipsoid单叶双曲面hyperboloid of one sheet双叶双曲面hyperboloid of two sheets旋转椭球面ellipsoid of revolution椭圆抛物面elliptic paraboloid旋转抛物面paraboloid of revolution双曲抛物面hyperbolic paraboloid马鞍面saddle surface椭圆柱面elliptic cylinder双曲柱面hyperbolic cylinder抛物柱面parabolic cylinder空间曲线space curve空间曲线的一般方程general form equations of a space curve空间曲线的参数方程parametric equations of a space curve螺转线spiral螺矩pitch投影柱面projecting cylinder投影projection平面的点法式方程pointnorm form eqyation of a plane法向量normal vector平面的一般方程general form equation of a plane两平面的夹角angle between two planes点到平面的距离distance from a point to a plane空间直线的一般方程general equation of a line in space方向向量direction vector直线的点向式方程pointdirection form equations of a line方向数direction number直线的参数方程parametric equations of a line两直线的夹角angle between two lines垂直perpendicular直线与平面的夹角angle between a line and a planes平面束pencil of planes平面束的方程equation of a pencil of planes行列式determinant系数行列式coefficient determinant第八章多元函数微分法及其应用Chapter8 Differentiation of Functions of Several Variables and Its Application 一元函数function of one variable多元函数function of several variables内点interior point外点exterior point边界点frontier point，boundary point聚点point of accumulation开集openset闭集closed set连通集connected set开区域open region闭区域closed region有界集bounded set无界集unbounded setn维空间n—dimentional space二重极限double limit多元函数的连续性continuity of function of seveal连续函数continuous function不连续点discontinuity point一致连续uniformly continuous偏导数partial derivative对自变量x的偏导数partial derivative with respect to independent variable x 高阶偏导数partial derivative of higher order二阶偏导数second order partial derivative混合偏导数hybrid partial derivative全微分total differential偏增量oartial increment偏微分partial differential全增量total increment可微分differentiable必要条件necessary condition充分条件sufficient condition叠加原理superpostition principle全导数total derivative中间变量intermediate variable隐函数存在定理theorem of the existence of implicit function曲线的切向量tangent vector of a curve法平面normal plane向量方程vector equation向量值函数vector-valued function切平面tangent plane法线normal line方向导数directional derivative梯度gradient数量场scalar field梯度场gradient field向量场vector field势场potential field引力场gravitational field引力势gravitational potential曲面在一点的切平面tangent plane to a surface at a point曲线在一点的法线normal line to a surface at a point无条件极值unconditional extreme values条件极值conditional extreme values拉格朗日乘数法Lagrange multiplier method拉格朗日乘子Lagrange multiplier经验公式empirical formula最小二乘法method of least squares均方误差mean square error第九章重积分Chapter9 Multiple Integrals二重积分double integral可加性additivity累次积分iterated integral体积元素volume element三重积分triple integral直角坐标系中的体积元素volume element in rectangular coordinate system 柱面坐标cylindrical coordinates柱面坐标系中的体积元素volume element in cylindrical coordinate system 球面坐标spherical coordinates球面坐标系中的体积元素volume element in spherical coordinate system 反常二重积分improper double integral曲面的面积area of a surface质心centre of mass静矩static moment密度density形心centroid转动惯量moment of inertia参变量parametric variable第十章曲线积分与曲面积分Chapter10 Line（Curve)Integrals and Surface Integrals对弧长的曲线积分line integrals with respect to arc hength第一类曲线积分line integrals of the first type对坐标的曲线积分line integrals with respect to x,y，and z第二类曲线积分line integrals of the second type有向曲线弧directed arc单连通区域simple connected region复连通区域complex connected region格林公式Green formula第一类曲面积分surface integrals of the first type对面的曲面积分surface integrals with respect to area有向曲面directed surface对坐标的曲面积分surface integrals with respect to coordinate elements第二类曲面积分surface integrals of the second type有向曲面元element of directed surface高斯公式gauss formula拉普拉斯算子Laplace operator格林第一公式Green’s first formula通量flux散度divergence斯托克斯公式Stokes formula环流量circulation旋度rotation，curl第十一章无穷级数Chapter11 Infinite Series一般项general term部分和partial sum余项remainder term等比级数geometric series几何级数geometric series公比common ratio调和级数harmonic series柯西收敛准则Cauchy convergence criteria，Cauchy criteria for convergence 正项级数series of positive terms达朗贝尔判别法D’Alembert test柯西判别法Cauchy test交错级数alternating series绝对收敛absolutely convergent条件收敛conditionally convergent柯西乘积Cauchy product函数项级数series of functions发散点point of divergence收敛点point of convergence收敛域convergence domain和函数sum function幂级数power series幂级数的系数coeffcients of power series阿贝尔定理Abel Theorem收敛半径radius of convergence收敛区间interval of convergence泰勒级数Taylor series麦克劳林级数Maclaurin series二项展开式binomial expansion近似计算approximate calculation舍入误差round—off error，rounding error欧拉公式Euler’s formula魏尔斯特拉丝判别法Weierstrass test三角级数trigonometric series振幅amplitude角频率angular frequency初相initial phase矩形波square wave谐波分析harmonic analysis直流分量direct component基波fundamental wave二次谐波second harmonic三角函数系trigonometric function system傅立叶系数Fourier coefficient傅立叶级数Forrier series周期延拓periodic prolongation正弦级数sine series余弦级数cosine series奇延拓odd prolongation偶延拓even prolongation傅立叶级数的复数形式complex form of Fourier series第十二章微分方程Chapter12 Differential Equation解微分方程solve a dirrerential equation常微分方程ordinary differential equation偏微分方程partial differential equation，PDE微分方程的阶order of a differential equation微分方程的解solution of a differential equation微分方程的通解general solution of a differential equation初始条件initial condition微分方程的特解particular solution of a differential equation初值问题initial value problem微分方程的积分曲线integral curve of a differential equation 可分离变量的微分方程variable separable differential equation 隐式解implicit solution隐式通解inplicit general solution衰变系数decay coefficient衰变decay齐次方程homogeneous equation一阶线性方程linear differential equation of first order非齐次non-homogeneous齐次线性方程homogeneous linear equation非齐次线性方程non—homogeneous linear equation常数变易法method of variation of constant暂态电流transient stata current稳态电流steady state current伯努利方程Bernoulli equation全微分方程total differential equation积分因子integrating factor高阶微分方程differential equation of higher order悬链线catenary高阶线性微分方程linera differential equation of higher order自由振动的微分方程differential equation of free vibration强迫振动的微分方程differential equation of forced oscillation串联电路的振荡方程oscillation equation of series circuit二阶线性微分方程second order linera differential equation线性相关linearly dependence线性无关linearly independce二阶常系数齐次线性微分方程second order homogeneour linear differential equation with constant coefficient二阶变系数齐次线性微分方程second order homogeneous linear differential equation with variable coefficient特征方程characteristic equation无阻尼自由振动的微分方程differential equation of free vibration with zero damping固有频率natural frequency简谐振动simple harmonic oscillation,simple harmonic vibration微分算子differential operator待定系数法method of undetermined coefficient共振现象resonance phenomenon欧拉方程Euler equation幂级数解法power series solution数值解法numerial solution勒让德方程Legendre equation微分方程组system of differential equations常系数线性微分方程组system of linera differential equations with constant coefficientV、X、Z：Value of function：函数值Variable:变数Vector：向量Velocity：速度Vertical asymptote：垂直渐近线Volume:体积X—axis：x轴x—coordinate：x坐标x-intercept：x截距Zero vector：函数的零点Zeros of a polynomial：多项式的零点T:Tangent function：正切函数Tangent line:切线Tangent plane:切平面Tangent vector：切向量Total differential：全微分Trigonometric function：三角函数Trigonometric integrals：三角积分Trigonometric substitutions：三角代换法Tripe integrals：三重积分S：Saddle point：鞍点Scalar：纯量Secant line：割线Second derivative：二阶导数Second Derivative Test：二阶导数试验法Second partial derivative：二阶偏导数Sector：扇形Sequence：数列Series:级数Set：集合Shell method:剥壳法Sine function：正弦函数Singularity：奇点Slant asymptote：斜渐近线Slope：斜率Slope—intercept equation of a line：直线的斜截式Smooth curve：平滑曲线Smooth surface：平滑曲面Solid of revolution：旋转体Space:空间Speed：速率Spherical coordinates：球面坐标Squeeze Theorem：夹挤定理Step function：阶梯函数Strictly decreasing：严格递减Strictly increasing:严格递增Sum：和Surface：曲面Surface integral:面积分Surface of revolution:旋转曲面Symmetry:对称R:Radius of convergence：收敛半径Range of a function:函数的值域Rate of change：变化率Rational function：有理函数Rationalizing substitution：有理代换法Rational number：有理数Real number：实数Rectangular coordinates：直角坐标Rectangular coordinate system：直角坐标系Relative maximum and minimum:相对极大值与极小值Revenue function：收入函数Revolution，solid of：旋转体Revolution，surface of：旋转曲面Riemann Sum:黎曼和Riemannian geometry：黎曼几何Right-hand derivative：右导数Right—hand limit:右极限Root：根P、Q：Parabola：拋物线Parabolic cylinder:抛物柱面Paraboloid：抛物面Parallelepiped：平行六面体Parallel lines：并行线Parameter：参数Partial derivative：偏导数Partial differential equation：偏微分方程Partial fractions：部分分式Partial integration：部分积分Partiton:分割Period：周期Periodic function：周期函数Perpendicular lines：垂直线Piecewise defined function：分段定义函数Plane：平面Point of inflection：反曲点Polar axis：极轴Polar coordinate:极坐标Polar equation：极方程式Pole：极点Polynomial：多项式Positive angle：正角Point-slope form:点斜式Power function：幂函数Product：积Quadrant：象限Quotient Law of limit：极限的商定律Quotient Rule：商定律M、N、O：Maximum and minimum values：极大与极小值Mean Value Theorem：均值定理Multiple integrals:重积分Multiplier：乘子Natural exponential function：自然指数函数Natural logarithm function：自然对数函数Natural number：自然数Normal line：法线Normal vector:法向量Number:数Octant:卦限Odd function：奇函数One-sided limit：单边极限Open interval:开区间Optimization problems：最佳化问题Order：阶Ordinary differential equation：常微分方程Origin：原点Orthogonal：正交的L:Laplace transform：Leplace变换Law of Cosines：余弦定理Least upper bound：最小上界Left—hand derivative：左导数Left—hand limit：左极限Lemniscate：双钮线Length：长度Level curve：等高线L'Hospital’s rule：洛必达法则Limacon：蚶线Limit：极限Linear approximation：线性近似Linear equation:线性方程式Linear function：线性函数Linearity：线性Linearization：线性化Line in the plane：平面上之直线Line in space:空间之直线Lobachevski geometry：罗巴切夫斯基几何Local extremum:局部极值Local maximum and minimum：局部极大值与极小值Logarithm:对数Logarithmic function:对数函数I:Implicit differentiation：隐求导法Implicit function：隐函数Improper integral：瑕积分Increasing/Decreasing Test：递增或递减试验法Increment：增量Increasing Function：增函数Indefinite integral：不定积分Independent variable：自变数Indeterminate from：不定型Inequality：不等式Infinite point：无穷极限Infinite series：无穷级数Inflection point：反曲点Instantaneous velocity：瞬时速度Integer:整数Integral:积分Integrand：被积分式Integration:积分Integration by part:分部积分法Intercepts：截距Intermediate value of Theorem:中间值定理Interval：区间Inverse function：反函数Inverse trigonometric function：反三角函数Iterated integral：逐次积分H：Higher mathematics高等数学/高数E、F、G、H:Ellipse：椭圆Ellipsoid：椭圆体Epicycloid:外摆线Equation：方程式Even function:偶函数Expected Valued：期望值Exponential Function:指数函数Exponents,laws of：指数率Extreme value:极值Extreme Value Theorem：极值定理Factorial：阶乘First Derivative Test:一阶导数试验法First octant:第一卦限Focus：焦点Fractions：分式Function:函数Fundamental Theorem of Calculus：微积分基本定理Geometric series：几何级数Gradient：梯度Graph:图形Green Formula:格林公式Half-angle formulas：半角公式Harmonic series：调和级数Helix:螺旋线Higher Derivative：高阶导数Horizontal asymptote：水平渐近线Horizontal line：水平线Hyperbola:双曲线Hyper boloid：双曲面D：Decreasing function:递减函数Decreasing sequence：递减数列Definite integral:定积分Degree of a polynomial：多项式之次数Density:密度Derivative：导数of a composite function：复合函数之导数of a constant function:常数函数之导数directional：方向导数domain of：导数之定义域of exponential function:指数函数之导数higher：高阶导数partial：偏导数of a power function：幂函数之导数of a power series：羃级数之导数of a product:积之导数of a quotient：商之导数as a rate of change：导数当作变率right-hand：右导数second：二阶导数as the slope of a tangent：导数看成切线之斜率Determinant:行列式Differentiable function：可导函数Differential：微分Differential equation：微分方程partial：偏微分方程Differentiation:求导法implicit：隐求导法partial:偏微分法term by term：逐项求导法Directional derivatives：方向导数Discontinuity：不连续性Disk method:圆盘法Distance：距离Divergence：发散Domain:定义域Dot product:点积Double integral:二重积分。
几何群论课程
3. Presentation of groups. Introduction of word problems, conjugacy problems and isomorphism problems.
4. Growth of groups. Cayley graph; word metric; growth function and growth rates; growth of abelian and nilpotent groups.
以课堂教授为主，辅以一定量的习题课和讨论班的形式
期中期末考试成绩占60%，平时作业占40%
教学评估
杨文元：
8. Hyperbolic groups and subgroups. Automatic structures; Tits alternative in hyperbolic groups; Quasi-convex subgroups.
9. Boundary of hyperbolic groups. Limit sets; Visual metrics; Convergence group action on boundary.
2. Free groups and their subgroups. Definitions of free groups and their construction by words; Ping-Pong Lemma; review of basics of covering space theory; subgroups in free groups.
庞加莱猜想前言
庞加莱猜想-前言Wir m\"ussen wissen! Wir werden wissen!(我们必须知道！我们必将知道！)—— David Hilbert两年前科学版举行过一次版聚，我报告了低维拓扑里面的一些问题和进展，其中有一半篇幅是关于Poincar\'e 猜想。
版聚后，flyleaf 要求大家回去后把自己所讲的内容发在版上。
当时我甚至已经开始写了一两段，但后来又搁置了。
主要是因为自己对于低维拓扑还是一个门外汉，写出来的东西难免有疏漏之处，不敢妄下笔。
两年过去，我对低维拓扑这门学科的了解比原先多了，说话的底气也就比原先足了。
另外，由于Clay 研究所的百万巨赏，近年来Poincar\'e 猜想频频在媒体上曝光；而且Perelman 最近的工作使数学家们有理由相信我们已经充分接近于这一猜想的最后解决。
所以大概会有很多人对Poincar\'e 猜想的来龙去脉感兴趣，我也好借机一偿两年来的宿愿。
现代科学的高速发展使各学科之间的鸿沟加大，不同学科之间难以互相理解，所以非数学专业的读者在阅读本文时可能会遇到一些困难。
但限于篇幅和文章的形式，我也不可能对很多东西详细解释。
一些最基本的拓扑概念如“流形”，我将在本文的附录中解释。
还有一些“同调群”、“基本群”之类的名词，读者见到时大可不去理会它们的确切含义。
我将尽量避免使用这一类的专业术语。
作者并非拓扑方面的专家，对下面要说的很多内容都是道听途说，只知其然而不知其所以然；作者更不善于写作，写出来的东东总会枯燥无味，难登大雅之堂。
凡此种种，还请读者诸君海涵。
问题的由来Consid\'erons maintenant une vari\'et\'e [ferm\'ee] $V$ \`a trois dimensions ... Est-il possible que le groupe fondamental de $V$ ser\'eduise \`a la substitution identique, et que pourtant $V$ ne soit pas simplement connexe?—— Henri Poincar\'e在拓扑学家的眼里，篮球、排球和乒乓球并没有什么不同，它们都同胚于三维空间中的球面S^2. (我们把n+1维欧氏空间中到原点距离为1的点的集合记作S^n，称为n维球面(sphere)。
双曲线焦点三角形面积公式正切
双曲线焦点三角形面积公式正切The formula for the area of a triangle formed by the foci of a hyperbola is an interesting mathematical concept that can be explored from various perspectives. This formula, known as the tangent of the eccentricity of the hyperbola, provides a way to calculate the area of a triangle by using the distances between the foci and the vertices of the hyperbola.双曲线焦点三角形面积公式是一个有趣的数学概念，可以从各种角度进行探讨。
这个公式，即双曲线的偏心率切线，提供了一种计算三角形面积的方法，即利用双曲线焦点与顶点之间的距离。
From a geometric perspective, understanding the formation of a triangle using the foci of a hyperbola can be visually captivating. The unique shape of a hyperbola and its foci create an intricate relationship that allows for the calculation of the triangle's area. By visualizing the geometric properties of the hyperbola, one can appreciate the beauty of how the foci and vertices come together to form a triangle with a specific area based on the hyperbola's eccentricity.从几何学角度来看，了解使用双曲线焦点形成三角形的过程是令人着迷的。
双曲线中点弦公式推导过程
双曲线中点弦公式推导过程The midpoint chord formula for a hyperbola is a fundamental concept in analytic geometry and has numerous applications in various fields. This formula allows us to find the midpoint of a chord that lies on a hyperbola, given the coordinates of the endpoints of the chord. The derivation of this formula involves some advanced algebra and geometry techniques but is an essential tool for solving problems related to hyperbolas.双曲线中点弦公式是解析几何中的一个基本概念，在各个领域都有着广泛的应用。
这个公式使我们能够找到一个位于双曲线上的弦的中点，只要给出了该弦的端点的坐标。
推导这个公式涉及一些高级代数和几何技巧，但它是解决与双曲线相关的问题的重要工具。
To derive the midpoint chord formula for a hyperbola, we first need to understand the general equation of a hyperbola. A hyperbola is defined as the set of all points in a plane such that the absolute difference of the distances from two fixed points, called the foci, is constant. The equation of a hyperbola can be written in standard form as (x-h)^2/a^2 - (y-k)^2/b^2 = 1 or (y-k)^2/b^2 - (x-h)^2/a^2= 1, where (h, k) is the center of the hyperbola, and a and b are the lengths of the major and minor axes, respectively.要推导双曲线的中点弦公式，我们首先需要了解双曲线的一般方程。
ComplexGeometry
Complex GeometryA Conference in Honor ofDomingo Toledo’s60th BirthdayUniversity of UtahMarch24and25,2006AbstractsDaniel Allcock:A monstrous proposalA series of coincidences suggests an appearance of the monster simple group in the deck group of a branched cover of a particular arithmetic quotient of complex hyperbolic13-space,possibly with a moduli-space interpretation.Jim Carlson:New Hodge theory for cubic threefoldsThe moduli space of smooth cubic threefolds,like that of cubic surfaces,has a complex hyperbolic structure.That is,it can be realized as the quotient of the unit ball in a complex Euclidean space,minus a set of totally geodesic complex hyperplanes,modulo an arithmetic group.We discuss this construction and a natural partial compactification of it which is isomorphic to the ball modulo the group.Geometrically,the components added are(a)nodal cubics(b)the secant variety of a rational normal curve of degree four with a set of twelve points marked on it.The identification with a ball quotient involves two pieces of Hodge theory.First is the cyclic cover trick,which goes back to Picard and which was exploited by Deligne and Mostow in their study of moduli of points on the projective line and their relation to ball quotients.Second is a way of relating certain complex Hodge structures of weight four on a fourfold to other complex Hodge structures of weight one on an algebraic curve.(Joint work with Domingo Toledo and Daniel Allcock).Bill Goldman:Toledo’s invariant of surface group representations (tentative abstract)First,a historical account of early work of the author and of Toledo’s Math Scandinavica harmonic maps paper and how it led to the local rigidity of surface1groups in U(1,1)in U(n,1).This then led to Toledo’s global rigidity on surface group representations in U(n,1).I will also discuss connections with my work with Millson on higher dimensional local rigidity,which led into Corlette’s thesis on global rigidity.I may also try to survey some of the work in complex hyperbolic geometry(Parker,Falbel, Gusevskii)on the Toledo invariant and(maybe if time allows)work of Bradlow/Garcia-Prada/Gothen/Mundet/Xia on Higgs bundles.Luis Hern´a ndez:Almost-hermitian structures of minimal energyLet(M,g)be a compact Riemannian manifold and J an orthogonal almost-complex struc-ture on M.The energy of J is defined asE(J)=∇ω,Mwhereωis the K¨a hler form associated to J and g.Reminiscent of what happens in Yang-Mills theory,we show this energy decomposes into pieces according to a certain U n-representation(exactly two pieces when dim M=4), and a certain linear combination of such pieces(the difference of the two,in dimension4) turns out to be,not a topological invariant as in Yang-Mills theory,but a multiple of the total scalar curvature(and thus depending only on g)when the metric g happens to be conformallyflat(of ASD in dimension4).We’ll use this to give examples of minimal energy J,e.g.the almost-complex J given by Cayley multiplication on the round S6,the usual hermitian structure on S3×S1,etc.Misha Kapovich:Generalized triangle inequalities and their applications Abstract.Everybody knows how to construct triangles in the Euclidean plane given their side-lengths which satisfy the familiar triangle inequalities.In this talk I will explain how to generalize this in the setting of nonpositively curved symmetric spaces and buildings,where the real-valued distance function is replaced by an appropriate vector-valued function.If the time permits I will explain the relation of this problem to the geometric invariant theory and theoretical computer science.J´a nos Koll´a r:Holonomy groups of stable vector bundlesAbstract:We define the notion of holonomy group for a stable vector bundle F on a variety in terms of the Narasimhan–Seshadri unitary representation of its restriction to curves.Next we relate the holonomy group to the minimal structure group and to the decomposition of tensor powers of F.Finally we illustrate the principle that either the2holonomy is large or there is a clear geometric reason why it should be small.(Joint work with Balaji.)Bruno Klingler:On the Andre-Oort conjectureAbstract:The Andre-Oort conjecture describes the geometry of collection of special points on Shimura varieties:any irreducible component of the Zariski closure of a set of special points on a Shimura variety is conjectured to be a subvariety of Hodge type.I will explain a proof of this conjecture under the Generalized Riemann Hypothesis(joint work with Andrei Yafaev).Yum-Tong Siu:Multiplier ideal sheaves and thefinite generation of canon-ical ringstbaDennis Sullivan:Operations in the string spaces of a smooth manifold and compactified spaces of Riemann surfacesThere are canonical minimal energy area preservingflows on Riemann surfaces with input punctures where thefluid enters at given rates and output punctures where theflow exits at given rates.Theseflows are pictorially easy to analyze as the surface developes nodes. There are also versions on surfaces with boundary...The combinatorics of the orbits touching rest points of theflows leads to two discussions.Thefirst is a cell decomposition of the open moduli space of Riemann surfaces(discussed first by Giddings and Wolpert in a physics oriented paper and independently by C.F. Bodigheimer in a more precise mathematical treatment).The open moduli space cell discussion extends naturally to the nodal compactification of moduli space using the above mentioned pictures.The second discussion makes use of the combinatorics,in particular how theflow moves, splits and reconnects the orthogonal one manifolds to theflow,and regularized transversal-ity in families of poly curves in a manifold to construct operations in the algebraic topology of the free loop space as well as in that of path spaces with boundary conditions.The reg-ularization is a technical device which diffuses the objects acted upon using a measure on afinite dimensional space of diffeomorphisms constructed from a coordinate cover.3Anna Weinhard:Why you should maximize the Toledo invariantThe Toledo invariant associates a real number to every homomorphism of the fundamental group of a(closed)Riemann surface into a semisimple Lie group of Hermitian type.The Toledo invariant is locally constant and bounded.Homomorphisms which realize the max-imal possible value of the Toledo invariant were studied around15years ago by Goldman, Toledo and Hernandez.They showed that in many cases homomorphisms with maximal Toledo invariant have nice geometric properties.I will present recent work with Burger and Iozzi on homomorphisms of fundamental groups of(not necessarily closed)Riemann surfaces with maximal Toledo invariants extending their results.Our approach owes a lot to Toledo’s treatment of the problem when the Lie group of Hermitian type is the isometry group of a complex hyperbolic space.4。
切触几何与非线性微分方程导论
theorem for distributions. Finite type systems.
3. Finite-dimensional dynamics of evolutionary differential equations
fluid dynamics (Rapoport-Leas model). Dynamics of evolutionary systems.
4. Contact geometry
Contact structures on manifolds, contact distributions, contact forms.
equations. Symmetries. Geometric structures. Characteristic distributions.
Hyperbolic, elliptic and parabolic MA equations. Decomposition of de
Rham complex. Tensor invariants. The Laplace invariants for linear
distributions and the Frobenius theorem. Characteristic and shuffling
symmetries of distributions. Maurer-Cartan equations. Lie-Bianchi
theorem for distributions. Finite type systems. Finite-dimensional dynamics of evolutionary differential equations
时空几何｜欧几里德（平面）几何非欧几里德（双曲、椭圆）几何
时空几何｜欧几里德（平面）几何非欧几里德（双曲、椭圆）几何数学研究的对象是“数”与“形”，形的数学就是几何学．它是以直观为主导，以培养人的空间洞察力与思维为目的．从数学发展的历史来看几何学的第一个最重要著作就是欧几里得(Euclid，约公元前330一275年)的《几何原本》．它被世界各国翻译成各种文字．它的印刷量仅次于“圣经”，所以不少人称《几何原本》为数学工作者的“圣经”。
《几何原本》在数学史乃至人类思想史上有着无比崇高的地位。
1 欧几里德几何（Euclid Geometry）－平面欧氏几何源于公元前3世纪。
古希腊数学家欧几里德把人们公认的一些几何知识作为定义和公理(公设)，在此基础上研究图形的性质，推导出一系列定理，组成演绎体系，写出《几何原本》，形成了欧氏几何。
按所讨论的图形在平面上或空间中，又分别称为“平面几何”与“立体几何”(欧几里得空间)。
Euclid(约公元前330一275) ↑在欧几里德以前，古希腊人已经积累了大量的几何知识，并开始用逻辑推理的方法去证明一些几何命题的结论。
欧几里德将早期许多没有联系和未予严谨证明的定理加以整理，写下《几何原本》一书，标志着欧氏几何学的建立。
这部划时代的著作共分13卷，465个命题。
其中有八卷讲述几何学，包含了现今中学所学的平面几何和立体几何的内容。
但《几何原本》的意义却绝不限于其内容的重要，或者其对诸定理的出色证明。
真正重要的是欧几里德在书中创造的公理化方法。
在证明几何命题时，每一个命题总是从再前一个命题推导出来的，而前一个命题又是从再前一个命题推导出来的。
我们不能这样无限地推导下去，应有一些命题作为起点。
这些作为论证起点，具有自明性并被公认下来的命题称为公理，如“两点确定一条直线”即是一例。
同样对于概念来讲也有些不加定义的原始概念，如点、线等。
在一个数学理论系统中，我们尽可能少地先取原始概念和不加证明的若干公理，以此为出发点，利用纯逻辑推理的方法，把该系统建立成一个演绎系统，这样的方法就是公理化方法。
高等数学专业英语名词
高等数学术语英语翻译V、X、Z:Value of function：函数值Vector：函数值Volume：体积X-axis：x轴x-coordinate：x坐标x-intercept：x截距Zero vector：函数的零点T:Tangent function：正切函数Tangent line：切线Total differential：全微分Trigonometric function：三角函数Tripe integrals：三重积分S:Second derivative：二阶导数Second partial derivative：二阶偏导数Sequence：数列Set：集合Slope：斜率Smooth curve：平滑曲线Smooth surface：平滑曲面Solid of revolution：旋转体Space：空间Speed：速率Spherical coordinates：球面坐标Sum：和Surface：曲面Surface integral：面积分Surface of revolution：旋转曲面Symmetry：对称Sine function：正弦函数Slant asymptote：斜渐近线R:Range of a function：函数的值域Rate of change：变化率Rational function：有理函数Rational number：有理数Real number：实数Rectangular coordinates：直角坐标Revolution,solid of：旋转体Revolution,surface of：旋转曲面Root：根P、Q:Parabola：拋物线Parabolic cylinder：抛物柱面Paraboloid：抛物面Parallelepiped：平行六面体Parallel lines：并行线Parameter：参数Partial derivative：偏导数Partial differential equation：偏微分方程Partial fractions：部分分式Partial integration：部分积分Partiton：分割Period：周期Periodic function：周期函数Perpendicular lines：垂直线Plane：平面Polar coordinate：极坐标Pole：极点Polynomial：多项式Positive angle：正角Power function：幂函数Product：积M、N、O:Maximum and minimum values：极大与极小值Multiple integrals：重积分Natural num ber：自然数Normal line：法线Number：数Odd function：奇函数One-sided li mit：单边极限Open interval：开区间Ordinary differential equation：常微分方程Orthogonal：正交的Origin：原点L:Law of Cosines：余弦定理Left-hand derivative：左导数Left-hand limit：左极限Length：长度Limit：极限Linear approximation：线性近似Linear equation：线性方程式Linear function：线性函数Linearity：线性Logarithm：对数Logarithmic function：对数函数I:Implicit function：隐函数Increment：增量Indefinite integral：不定积分Independent variable：自变数Indeterminate from：不定型Inequality：不等式Infinite point：无穷极限Infinite series：无穷级数Integer：整数Integral：积分Integrand：被积分式Integration：积分Intercepts：截距Interval：区间Inverse function：反函数Inverse trigonometric function：反三角函数Iterated integral：逐次积分Intermediate value of Theorem：中间值定理H:Higher mathematics高等数学/高数E、F、G、H:Ellipse：椭圆Ellipsoid：椭圆体Equation：方程式Even function：偶函数Expected Valued：期望值Exponential Function：指数函数Extreme value：极值Focus：焦点Fractions：分式Function：函数Gradient：梯度Graph：图形Higher Derivative：高阶导数Horizontal asymptote：水平渐近线Horizontal line：水平线Hyperbola：双曲线Hyper boloid：双曲面D:Decreasing function：递减函数Decreasing sequence：递减数列Definite integral：定积分Density：密度Derivative：导数higher：高阶导数partial：偏导数Determinant：行列式Differentiable function：可导函数Differential：微分Differential equation：微分方程partial：偏微分方程Differentiation：求导法implicit：隐求导法partial：偏微分法Discontinuity：不连续性Distance：距离Divergence：发散Domain：定义域Double integral：二重积分C:Calculus：微积分differential：微分学integral：积分学Circle：圆Circular cylinder：圆柱Closed interval：封闭区间Coefficient：系数Cone：圆锥Constant function：常数函数Constant of integration：积分常数Continuity：连续性Continuous function：连续函数Convergence：收敛Convergent sequence：收敛数列Coordinate：s：坐标polar：极坐标rectangular：直角坐标spherical：球面坐标Coordinate axes：坐标轴Cosine function：余弦函数Critical point：临界点Cubic function：三次函数Curve：曲线Cylinder：圆柱A、B:Absolute convergence：绝对收敛Absolute extreme values：绝对极值Absolute maximum and minimum：绝对极大与极小Absolute value：绝对值Absolute value function：绝对值函数Acceleration：加速度Antiderivative：反导Arbitrary constant：任意常数Arc length：弧长Area：面积Asymptote：渐近线horizontal：水平渐近线slant：斜渐近线vertical：垂直渐近线Average speed：平均速率Average velocity：平均速度微积分词汇第一章函数与极限Chapter1Function and Limit集合set元素element子集subset空集empty set并集union交集intersection 差集difference of set基本集basic set补集complement set直积direct product开区间open interval闭区间closed interval映射mapping一一映射one-to-one mapping变化transformation函数function自变量independent variable因变量dependent variable函数关系function relation值域range函数图形graph of a function绝对值函数absolute value符号函数sigh function整数部分integral part分段函数piecewise function函数的单调性monotonicity of a function单调增加的increasing单调减少的decreasing单调函数monotone function对称symmetry偶函数even function奇函数odd function周期period反函数inverse function直接函数direct function复合函数composite function中间变量intermediate variable函数的运算operation of function基本初等函数basic elementary function初等函数elementary function幂函数power function指数函数exponential function对数函数logarithmic function三角函数trigonometric function反三角函数inversetrigonometric function常数函数constant function极限limit数列sequence of number 收敛convergence发散divergent子列subsequence函数的极限limits of functions左极限left limit右极限right limit单侧极限one-sided limits无穷小infinitesimal无穷大infinity单调数列monotonic sequence高阶无穷小infinitesimal of higher order 低阶无穷小infinitesimal of lower order同阶无穷小infinitesimal of the same order2高等数学-翻译等阶无穷小equivalent infinitesimal函数的连续性continuity of a function增量increment不连续点discontinuity point第一类间断点discontinuity point of the first kind第二类间断点discontinuity point of the second kind定义区间defined interval最大值global maximum value(absolute maximum)最小值global minimum value (absolute minimum)零点定理the zero point theorem介值定理intermediate value theorem 第二章导数与微分Chapter2Derivative and Differential匀速运动uniform motion平均速度average velocity瞬时速度instantaneous velocity圆的切线tangent line of a circle切线tangent line切线的斜率slope of the tangent line 位置函数position function导数derivative可导derivable导函数derived function切线方程tangent equation隐函数implicit function显函数explicit function高阶导数nth derivative相关变化率correlative change rata微分differential可微的differentiable函数的微分differential of function自变量的微分differential of independent variable绝对误差absolute error相对误差relative error第三章微分中值定理与导数的应用Chapter3MeanValue Theorem of Differentials and the Application of Derivatives临界点critical point辅助函数auxiliary function不定式indeterminate form泰勒公式Taylor formula余项remainder term拐点inflection point函数的极值extremum of function极大值local(relative)maximum极小值local(relative)minimum曲率curvature平均曲率average curvature曲率中心center of curvature第四章不定积分Chapter4Indefinite Integrals原函数primitive function(antiderivative)积分号sign of integration被积函数integrand积分变量integral variable积分曲线integral curve积分表table of integrals换元积分法integration by substitution分部积分法integration by parts分部积分公式formula of integration by parts有理函数rational function第五章定积分Chapter5Definite Integrals曲边梯形trapezoid with曲边curve edge积分下限lower limit of integral积分上限upper limit of integral积分区间integral interval分割partition积分和integral sum可积integrable反常积分improper integral第六章定积分的应用面积元素element of area极坐标polar coordinates抛物线parabola椭圆ellipse旋转体的面积volume of a solid of rotation曲线的弧长arc length of acurve光滑smooth功work水压力water pressure引力gravitation变力variable force第七章空间解析几何与向量代数Chapter7Space Analytic Geometry and Vector Algebra相等equal平行parallel三角法则triangle rule平行四边形法则parallelogram rule交换律commutative law结合律associative law差difference分配律distributive law球面sphere轴axis顶点vertex抛物柱面parabolic cylinder二次曲面quadric surface椭圆锥面dlliptic cone椭球面ellipsoid椭圆柱面elliptic cylinder双曲柱面hyperbolic cylinder抛物柱面parabolic cylinder空间曲线space curve投影projection垂直perpendicular第八章多元函数微分法及其应用Chapter8Differentiation of Functions of Several Variables and Its Application一元函数function of one variable多元函数function of several variables边界点frontier point,boundary point开集openset闭集closed set有界集bounded set 无界集unbounded set二重极限double limit连续函数continuous function不连续点discontinuity point偏导数partial derivative高阶偏导数partial derivative of higher order二阶偏导数second order partial derivative全微分total differential偏增量oartial increment偏微分partial differential全增量total increment可微分differentiable必要条件necessary condition充分条件sufficient condition全导数total derivative法线normal line梯度gradient 无条件极值unconditional extreme values条件极值conditional extreme values最小二乘法method of least squares第九章重积分Chapter9Multiple Integrals二重积分double integral可加性additivity三重积分triple integral反常二重积分improper double integral曲面的面积area of a surface质心centre of mass密度density第十二章微分方程Chapter12Differential Equation常微分方程ordinary differential equation偏微分方程partial differential equation,PDE初始条件initial condition衰变decay齐次方程homogeneous equation一阶线性方程linear differential equation of first order非齐次non-homogeneous齐次线性方程homogeneous linear equation非齐次线性方程non-homogeneous linear equation全微分方程total differential equation高阶微分方程differential equation of higher order二阶线性微分方程second order linera differential equation线性相关linearly dependence 线性无关linearly independce variable coefficient微分方程组system of differential equations。
数学专业英语词汇(H)
数学专业英语词汇(H)数学专业英语词汇(H)数学专业英语词汇(H)h closed h闭的haar condition 哈尔条件haar measure 哈尔测度hadamard criterion 阿达玛判别准则hadamard gap condition 阿达玛间断条件hadamard matrix 阿达玛矩阵hadamard method of descent 阿达玛下降法hadamard multiplication theorem 阿达玛乘法定理hadamard three circles theorem 阿达玛三圆定理half 半half angle 半角half angle formulas 半角公式half axis 半轴half closed interval 半闭区间half exact 半正合的half line 半直线half neighborhood 半邻域half plane 半无限平面half plane of absolute convergence 绝对收敛半平面half plane of convergence 收敛半平面half round 半圆的half side formulas 半边公式half space 半空间halve 对分halving method 二等分法hamilton characteristic function 哈密顿特寨数hamilton formula 哈密顿公式hamilton function 哈密顿函数hamilton jacobi equation 哈密顿雅可比方程hamilton jacobi theory 哈密顿雅可比理论hamilton principle 哈密顿原理hamiltonian 哈密顿函数hamiltonian circuit 哈密顿回路hamiltonian group 哈密顿群hamiltonian operator 哈密顿算子hamiltonian path 合密顿道路hand 边handle 环柄handle of the second kind 交叉套handlebody 环柄体hankel transformation 汉克尔变换harmonic analysis 低分析harmonic analyzer 傅里叶分析仪harmonic conjugate 低共轭点harmonic constant 低常数harmonic curve 低曲线harmonic differential equation 低微分方程harmonic division 低分割harmonic function 低函数harmonic integral 低积分harmonic mapping 低映射harmonic mean 低平均harmonic measure 低测度harmonic motion 低运动harmonic oscillation 谐振动harmonic progression 低级数harmonic ratio 低比harmonic series 低级数harmonic synthesis 傅里叶综合法harmonicity 低性hasse diagram 哈塞图hausdorff group 豪斯道夫群hausdorff measure 豪斯道夫测度hausdorff metric 豪斯道夫度量hausdorff separation axiom 廉斯道夫分离公理hausdorff space 分离空间haversine 半正矢heat 热heat conduction 热传导hecke character 黑克特贞hecke operator 黑克算子hectoliter 百升hectometer 百米helicograph 螺旋规helicoid 螺旋面helicoidal surface 螺旋面helix 螺旋线hemi continuous 半连续的hemihedry 半对称hemipyramid 半棱锥体hemisphere 半球hemispherical 半球面的hemispherical shape 半球形hendecagon 十一边形henselization 享泽莱化heptagon 七边形heptahedron 七面体hereditarily enumerable set 遗传可数集hereditarily generating system 遗传的生成系hereditarily indecomposable continuum 遗传不可分解的连续统hereditarily normal space 遗传正规空间hereditary class 遗传类hereditary property 遗传性质hereditary set 遗传集hereditary system of sets 集的遗传系heredity 遗传性hermite function 埃尔米特函数hermite interpolation formula 埃尔米特插值公式hermite interpolation polynomial 埃尔米特插值多项式hermite normal form 埃尔米特正规形式hermite polynomial 埃尔米特多项式hermite reciprocity law 埃尔米待互反律hermitian bilinear functional 埃尔米特双线性泛函hermitian conjugate 埃尔米特共轭阵hermitian form 埃尔米特形式hermitian inner product module 埃尔米特内积模hermitian inner product space 埃尔米特空间hermitian kernel 埃尔米特核hermitian matrix 埃尔米特矩阵hermitian metric 埃尔米特度量hermitian operator 埃尔米特算子hermitian polynomiat 埃尔米特多项式hermitian transformation 埃尔米特变换hero formula 海伦公式hesse normal form 海赛正规形式hessian 海赛形式hessian group 海赛群hessian matrix 海赛矩阵hexagon 六边形hexagonal 六边形的hexagonal net 六边形网格hexagonal system 六角系hexahedral 六面体的hexahedron 六面体hexakistetrahedron 六四面体hierarchical classification 谱系分类hierarchy 分层high speed computer 高速计算机higher algebra 高等代数higher commutator 广义换位子higher derivative 高阶导数higher mathematics 高等数学higher order term 高阶项higher plane curve 高次平面曲线higher singularity 高次奇异性highest common divisor 最大公约highest common factor 最大公因子highest derivative 最高阶导数highest order 最高位highest weight 最高权hilbert basis theorem 希耳伯特基定理hilbert cube 希耳伯特超平行体hilbert inequality 希耳伯特不等式hilbert integral 希耳伯特积分hilbert matrix 希耳伯特矩阵hilbert modular form 希耳伯特模形式hilbert modular function 希耳伯特模函数hilbert modular group 希耳伯特模群hilbert nullstellensatz 希耳伯特零点定理hilbert parallelotope 希耳伯特超平行体hilbert problems 希耳伯特问题hilbert space 希耳伯特空间hill differential equation 希耳微分方程histogram 直方图history 履历hodograph 速端曲线hodograph transformation 速端曲线变换hodometer 路程表holding domain 解域holomorph convex manifold 全形凸廖holomorph separable manifold 全形可分廖holomorphic 正则的holomorphic completeness 全纯完全性holomorphic convexity 正则凸性holomorphic differential 全纯微分holomorphic differential form 全纯微分形式holomorphic divisor 全纯除子holomorphic function 全纯函数holomorphic manifold 复解析廖holomorphic mapping 全纯映射holomorphic part 全纯部分holomorphy 正则holonomic condition 完全性条件holonomic reference system 完整参考系holonomic system 完整系holonomy 完整holonomy group 完整群homeomorph 同胚象homeomorphic 同胚的homogeneity 齐性homogeneity formula 齐性公式homogeneity of variances 同方差性homogeneity relation 齐性关系homogeneous 均匀的homogeneous cartesian co ordinates 齐次笛卡儿坐标homogeneous coordinates 齐次笛卡儿坐标homogeneous distribution 均匀分布homogeneous element 齐次元素homogeneous equation 齐次方程homogeneous function 齐次函数homogeneous function of order k k阶齐次函数homogeneous ideal 齐次理想homogeneous integral equation 齐次积分方程homogeneous linear boundary value problem 齐次线性边值问题homogeneous linear differential equation 齐次线性微分方程homogeneous linear transformation 齐次线性变换homogeneous lineare transformation 齐次线性变换homogeneous markov chain 齐次马尔可夫链homogeneous markov process 齐次马尔可夫过程homogeneous operator 齐次算子homogeneous polynomial 齐次多项式homogeneous space 商空间homogeneous system of differential equations 齐次微分方程组homogeneous system of linear equations 齐次线性方程组homogeneous variational problem 齐次变分问题homographic function 单应函数homological algebra 同碟数homological dimension 同惮数homological invariant 同祷变量homologous mappings 同党射homologous simplicial map 同单形映射homologous to zero 同第零homology 同调homology algebra 同碟数homology class 同掂homology equivalence 同等价homology equivalent complex 同等价复形homology functor 同弹子homology group 同岛homology manifold 同滴homology module 同担homology operation 同邓算homology sequence 同凋列homology simplex 同单形homology spectral sequence 同底序列homology sphere 同凋homology theory 同帝homology type 同低homomorphic group 同态群homomorphic image 同态象homomorphism 同态homomorphism theorem 同态定理homoscedastic 同方差的homoscedasticity 同方差性homothetic transformation 相似扩大homothety 相似扩大homotopic 同伦的homotopic invariant 同伦不变量homotopic map 同伦映射homotopic path 同伦道路homotopically equivalent space 同伦等价空间homotopy associativity 同伦结合性homotopy category of topological spaces 拓扑空间同伦范畴homotopy chain 同伦链homotopy class 同伦类homotopy classification 同伦分类homotopy equivalence 同伦等价homotopy excision theorem 同伦分割定理homotopy extension 同伦扩张homotopy group 同伦群homotopy group functor 同伦群函子homotopy inverse 同伦逆的homotopy operator 同伦算子homotopy sequence 同伦序列homotopy set 同伦集homotopy sphere 同伦球面homotopy theorem 同伦定理homotopy theory 同伦论homotopy type 同伦型homotopyassociative 同伦结合的horizon 水平线horizontal axis 水平轴horizontal component 水平分量horizontal coordinates 水平坐标horizontal plane 水平面horizontal projection 水平射影horned sphere 角形球面horocycle 极限圆horosphere 极限球面horse power 马力hungarian method 匈牙利法hurewicz isomorphism theorem 胡列维茨同构定理hydrodynamics 铃动力学hydromechanics 铃力学hydrostatics 铃静力学hyper graeco latin square 超格勒科拉丁方格hyper octahedral group 超八面体群hyperabelian function 超阿贝耳函数hyperalgebraic manifold 超代数廖hyperarithmetical 超算术的hyperarithmetical relation 超算术关系hyperbola 双曲线hyperbolic 双曲线的hyperbolic automorphism 双曲代换hyperbolic catenary 双曲悬链线hyperbolic cosecant 双曲余割hyperbolic cosine 双曲余弦hyperbolic cotangent 双曲余切hyperbolic cylinder 双曲柱hyperbolic elliptic motion 双曲椭圆运动hyperbolic equation 双曲型方程hyperbolic function 双曲函数hyperbolic geometry 双曲几何学hyperbolic inverse point 双曲逆点hyperbolic involution 双曲对合hyperbolic line 双曲线hyperbolic motion 双曲运动hyperbolic orbit 双曲线轨道hyperbolic paraboloid 双曲抛物面hyperbolic plane 双曲平面hyperbolic point 双曲点hyperbolic riemann surface 双曲型黎曼曲面hyperbolic rotation 双曲旋转hyperbolic secant 双曲正割hyperbolic sine 双曲正弦hyperbolic space 双曲空间hyperbolic spiral 双曲螺线hyperbolic substitution 双曲代换hyperbolic system 双曲型组hyperbolic tangent 双曲正切hyperbolic tangent function 双曲正切hyperbolic type 双曲型hyperbolicity 双曲性hyperboloid 双曲面hyperboloid of one sheet 单叶双曲面hyperboloid of revolution 旋转双曲面hyperboloid of two sheets 双叶双曲面hypercohomology 超上同调hypercomplex 超复数hypercomplex number 超复数hypercone 超锥hyperconjugation 超共轭hypercyclic group 超循环群hypercyclide 超四次圆纹曲面hyperelliptic 超椭圆的hyperelliptic function 超椭圆函数hyperelliptic integral 超椭圆积分hyperelliptic theta function 超椭圆函数hyperfinite c* algebra 超有限c*代数hypergeometric differential equation 超几何微分方程hypergeometric distribution 超几何分布hypergeometric function 超几何函数hypergeometric function of the second kind 第二类超几何函数hypergeometric series 超几何级数hypergeometry 超几何学hypergroup 超群hypermatrix 超矩阵hypernormal dispersion 超正态方差hyperplane 超平面hyperplane coordinates 超平面坐标hyperplane of support 支撑超平面hyperplane section 超平面截面hyperquadric 超二次曲面hyperreal numbers 超实数hyperspace 超空间hypersphere 超球面hyperstonian space 超斯通空间hypersurface 超曲面hypocycloid 内摆线hypocycloidal 圆内旋轮线的hypoelliptic operator 次椭圆型算子hypoellipticity 次椭圆性hypotenuse 斜边hypothesis 假设hypothetical population 假言总体hypotrochoid 长短辐圆内旋轮线数学专业英语词汇(H) 相关内容:。
数学组课程大纲
數學組課程大綱93.3.1101 [024002] 微積分(一) [Calculus (I)] , 4學分. 大一必修先修科目：無極限及連續性, 微分及其應用, 不定積分, Riemann 積分.102 [024001] 線性代數(一) [Linear Algebra (I)] , 3學分. 大一必修先修科目：無Gaussian 消去法, 矩陣計算, 行列式, 矩陣運算, 基底內積及垂直性.103 [024021] 數學導論(一) [Introduction to Mathematics (I)] , 3學分. 大一必修先修科目：無敘述及量化邏輯，基數, 真值表, 證明法, 謬論, 集合運算, 等價關係, 函數.111 [024003] 微積分(二) [Calculus (II)] , 4學分. 大一必修先修科目：微積分(一)瑕積分, 超越函數, 數列及級數, Taylor's 定理, 偏微分, 重積分及其應用.112 [024004] 線性代數(二) [Linear Algebra (II)] , 3學分. 大一必修先修科目：線性代數(一)線性變換, 固有值, 固有向量, 對角化, 二次型, 及正定矩陣.113 [024028] 數學導論(二) [Introduction to Mathematics (II)], 3學分. 大一選修先修科目：數學導論(一)實數, Schrőder-Bernstein定理, 次序, Zorn's 引理, 選擇公理.201 [024007] 高等微積分(一) [Advanced Calculus (I)] , 4學分. 大二必修先修科目：微積分(一) , 微積分(二)實數性質, 均勻連續, 函數序列與級數, 均勻收斂.203 [024018] 離散數學(一) [Discrete Mathematics (I)] , 3學分. 大二必修先修科目：微積分(一) , 微積分(二)排列, 組合方式, 排演原理, 圖的表示法, 圖的結構, 二分圖, 樣本, 最小生成樣本, 最短路, 歐拉迴路, 組合數學和基本圖論等.204 [024020] 拓樸學（一）[Topology I ] , 3學分. 大二選修先修科目：無賦距空間, 子空間, 積空間, 商空間, 收歛及連續, 分離公設, 緊緻性及連通性, 拓樸不變性.211 [024008] 高等微積分(二) [Advanced Calculus (II)] , 4學分. 大二必修先修科目：高等微積分(一)反函數及隱函數定理, Ｒn之拓樸性, 連續映射, 重積分. [024008]213 [024019] 離散數學(二) [Discrete Mathematics (II)] , 3學分. 大二選修先修科目：離散數學(一)Recurrence relation ,生成函數(generating function),圖的連通性,漢來爾頓路徑,圖的著色, matching.214[024022] 代數學(一) [Algebra (I)] , 3學分. 大二必修先修科目：無群, 子群, 商群, 對稱群, 置換群, 同態群的應用.215 [024009] 微分方程(一) [Differential Equations (I)] , 3學分. 大二必修先修科目：微積分(一) ,微積分(二)一階微分方程, ｎ階線性方程, 冪級數解, 線性微分方程組.301 [024029] 複變函數論(一) [Complex Analysis (I)] , 3學分. 大三必修先修科目：高等微積分(一) ,高等微積分(二)解析函數, 基本複變函數, Cauchy定理及積分公式, 極大模原理, Taylor級數, Laurent 級數, 零,點留數論.304[024031] 代數學(二) [Algebra (II)]大三選修先修科目：代數學(一)環, 商環理想, 多項式環, 體, 有限體, 尺規作圖.305 [024016] 微分方程(二) [Differential Equations (II)] , 3學分. 大三選修先修科目：微分方程(一)存在唯一性定理, Laplace轉換, 邊值問題, 動態系統, 基礎偏微分方程.306 [024403] 高等線性代數（一）[Advanced Linear Algebra I ] , 3學分. 大三選修先修科目：線性代數(一) ,線性代數(二)線性泛函, 偶空間, 伴隨算子, 對角化, 不變子空間, Jordan型, 重線性代數等.311 複變函數論（二）[Complex Analysis (II) ] , 3學分. 大三選修先修科目：無Conformal mapping, Mobius transformation, harmonic functions, Reflection principle, Riemann mapping theorem.314 拓樸學（二）[Topology 2 ] , 3學分. 大三選修先修科目：無緊空間(compact spaces)，分離性公理T0,T1,T2,T3,T3½及T4，Uryshon lemma and Tietze extension theorem及應用，基本群及其應用315 向量分析[Vector Analysis] , 3學分. 大三選修先修科目：無向量代數，向量函數，純量場與向量場，線，面與體積分，散度定理與旋度定理，格林定理與史多克定理，行列與線性正交變換，力學與電磁之應用。
stewart定理
Online Geometry Problem 805: Stewart's Theorem, Triangle, Sides,Cevian, Metric Relations, Measurement. Level: High School, SAT Prep,College, Mathematics EducationLet be given a triangle ABC and a point D on AB such that m = AD, n = BD, and x =CD. Then Stewart's theorem, also called Apollonius' theorem, statesthat: .Median length, Apollonius' TheoremThe following theorem has been attributed to Apollonius: In any triangle, the sum of the squares on any two sides is the square on half the third side together with twice the square on the median which bisects the third side. In other wa median of a triangle ABC and if AB = c, BC = a, AC = b, and BM = m, prove that .stewart's theorem in hyperbolic geometryIn euclidean geometry, the length of a cevian of a triangle is related tothe length of the sides byStewart's TheoremIn triangle ABC, if D lies on the line BC and the segments have lengths|AB|=c, |BC|=a, |CA|=b, |AD|=d, |BD|=m, |DC|=n, then(1) if D lies between B and C, then a(d2+mn) = mb2+nc2,(2) if D lies beyond C, then a(d2-mn) = mb2-nc2,(3) if D lies beyond B, then a(d2-mn) =-mb2+nc2.The proof in each case consists of an application of the Cosine Rule.Note that, if we regard a, m and n as signed lengths, with BC as thepositive direction, then the formula in (1) covers all cases. If we usesigned ratios along the line BC, then we can give a unified version.In the notation of Stewart's Theorem, for any point D on the line BC,d2 = (BD/BC)b2+(CD/CB)c2-(BD/BC)(CD/CB)a2.For the last term on the right, we have written mn as (m/a)(n/a)a2.When AD is a median, the formula simplifies since |BD|=|DC|=|BC|/2.Then we have an easy proof of part (1) of the followingCorollary(1) If AD is a median of ΔABC, then |AD|2 = (2|AB|2+2|AC|2-|BC|2)/4.(2) If the other medians are BE and CF, then|AD|2+|BE|2+|CF|2 = 3(|AB|2+|BC|2+|CA|2)/4.(3) If ΔABC is equilateral of side a, then each median has length (/3/2)a.Part (2) is obtained by applying part (1) to each median, and adding the results. Part (3) also follows from part (1) since, for an equilateral triangle, a = b = c.This gives d2= (3/4)a2, and the result follows. Of course, there are easier waysto prove this last part, but our method adapts easily to hyperbolic geometry.Most of these euclidean results have hyperbolic counterparts, and the proofs of these run in parallel to the euclidean proofs, as we now show.Stewart's Theorem in Hyperbolic GeomatryIn hyperbolic triangle ABC, if D lies on the hyperbolic line BC and thesegments have hyperbolic lengths d(A,B)=c, d(B,C)=a, d(C,A)=b,d(A,D)=d, d(B,D)=m, d(D,C)=n, then(1) if D lies between B and C, then sinh(a)cosh(d) = sinh(m)cosh(b)+sinh(n)cosh(c),(2) if D lies beyond C, then sinh(a)cosh(d) = sinh(m)cosh(b)-sinh(n)cosh(c),(3) if D lies beyond B, then sinh(a)cosh(d) =-sinh(m)cosh(b)+sinh(n)cosh(c).The proof is an analogue of the euclidean proof, with the hyperbolic Cosine Rulein place of the euclidean version.Once again, if we look at signed (hyperbolic) ratios, we get a unified result.In the notation of Stewart's Theorem, for any point D on the hyperbolic line BC, cosh(d) = -h(D,B,C)cosh(b)-h(D,C,B)cosh((c)Recall that h(X,Y,Z) = ±sinh(d(X,Y))/sinh(d(Y,Z)) with the positive sign if andonly if Y lies between X and Z. Thus, if D lies between B and C, then we haveh(D,B,C) =-sinh(d(D,B))/sinh(d(B,C) =-sinh(m)/sinh(a). The other cases are similar.As in the euclidean case, we get a simple result for the hyperbolic length of a median of a triangle.Corollary(1) If AD is a median of the hyperbolic triangle ABC, thencosh(d) = (cosh(b)+cosh(c))/2cosh(½a).(3) If ΔABC is a equilateral hyperbolic triangle of side a, theneach median has length d where cosh(d) = cosh(a)/cosh(½a).This follows from case (1) since for the median AD, we have m = n = ½a,and, for any a, sinh(a) =2 sinh(½a)cosh(½a).Part (3) folows at once since here a = b = c.There is no obvious (neat) analogue of part (2) of the euclidean corollary.It is appropriate at this point to mention another result on eq uilateral hyperbolic triangles. We know that the angles of such a triangle must all be equal. In the euclidean case, each angle is π/3, so all equilateral eucl idean triangles are similar.In hyperbolic geometry, triangles with eqaul angles are congruent, so we cannot expect an analogous result. In fact, the angles of an equilateral hyperbolic triangle depend on the (common) hyperbolic length of the sides.TheoremIf ΔABC is an equilateral hyperbolic triangle with sides of hyperbolic length a, then each of its angles is equal to θ, where cos(θ) = cosh(a)/(cosh(a)+1).ProofApplying the hyperbolic Cosine Rule to the traingle, we getcos(θ) = (cosh(a)cosh(a)-cosh(a))/sinh(a)sinh(a) = cosh(a)(cosh(a)-1)/(cosh2(a)-1). Cancelling (cosh(a)-1) from top and bottom, we get the result.Note that this gives a proof that all the angles are equal, but it also finds the angle θ.Also, as cosh(a) > 1, cosh(a)/(cosh(a)+1) > ½, so θ < π/3. Of course, this can also be deduced from the fact that the sum of the angles is less than π, so that 3θ < π.hyperbolic trianglesproof of stewart's theorem in euclidean geometryStewart's TheoremIn triangle ABC, if D lies on the line BC and the segments have lengths|AB|=c, |BC|=a, |CA|=b, |AD|=d, |BD|=m, |DC|=n, then(1) if D lies between B and C, then a(d2+mn) = mb2+nc2,(2) if D lies beyond C, then a(d2-mn) = mb2-nc2,(3) if D lies beyond B, then a(d2-mn) =-mb2+nc2.ProofBy the Cosine Rule applied to triangles ABD, ADC, we have cos(<ADB) = (d2+m2-c2)/2dm, cos(<ADC) = (d2+n2-b2)/2dn.(1) Here <ADC = π-<ADB, so cos(<ADC)=-cos(<ADB), so that (d2+n2-b2)/2dn = -(d2+m2-c2)/2dm. Simplifying this, we get (m+n)(d2+mn) = mb2+nc2. As m+n=a, we have (1).(2) Now <ADC = <ADB, so we get(d2+n2-b2)/2dn = (d2+m2-c2)/2dm. Simplifying this, we get (m+n)(d2-mn) = mb2-nc2. As m-n=a, we have (2).(3) The algebra is identical to that in (2), but here n-m=a, so we get (3).case (1) case (2)stewart's theorem。
双曲几何
（第一个资料）Poincare圆盘模型一个神奇的双曲世界今年恰逢PKU数学文化节十周年其间开办的很多讲座我都去了。
去听讲座的人好像都是数院的我恐怕是唯一一个中文系的。
考虑到我和中文系的MM没有共同话题因此每一次听讲座时我都会顺便四处打望看看有没有数院的美女下来可以和她“交谈”一下。
有趣的是我的做法与常人所想的恰好相反据说数院的已经盯上中文系的MM了而我一个中文系的竟然反过来去找数院的MM。
昨天有一个关于非欧几何的讲座这是目前所有的讲座中最为精彩的一次。
讲座里提到了Poincaré的一个双曲几何模型感觉非常有意思在这里和大家分享一下。
在所有的双曲几何模型中Poincaré的圆盘模型可能是最有趣的一个。
这个双曲世界存在于一个有限的平面区域里整个世界限制在一个单位圆的范围内。
这个世界中有两个最重要的物理定律一假如某物体X离原点O距离为d那么该物体的温度为1-d^2二物体的大小与温度成正比。
这样假如某个人从这个世界的中心走向边缘那么他的温度会从1慢慢变成0同时整个人慢慢变小。
他自身大小改变的同时周围的物体也等比例地放大或缩小而这个世界里的人视野有限看不见远处的东西因此他不会觉得自己变小了或者变大了。
因此在这个世界里物理学家们能够很轻易地发现第一定律但要发现第二定律则非常具有挑战性探索第二定律的过程必然很曲折并且很可能出现哥白尼时代的故事。
对于我们来说这个世界是有界的但对于这个世界中的人来说这个世界是无穷大的。
因为离原点越远人就越小于是相对来说他们所看到的空间也就越大。
当人的位置趋于边界时物体大小趋于0此时的空间将变得无穷大因此这个世界中的物体永远无法到达边界。
同时离原点越远的话越接近“绝对零度”这将非常不适宜生物的生存因此人们大多居住在原点离原点越远城市规模越小更远的地方则完全没有开发过只适合于疯狂的冒险家进行极限运动。
于是这个世界中的物理学家很自然地得到这个结论世界是无穷大的。
下面就神奇了。
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1This work was supported in part by The Geometry Center, University of Minnesota, an STC funded by NSF, DOE, and Minnesota Technology, Inc., by the Mathematical Sciences Research Institute, and by NSF research grants.
3. There is exactly one circle of any given radius with any given center.
4. All right angles are congruent to one another.
5. If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles.
Of course, volumes have been written on non-Euclidean geometry which the reader must consult for more exhaustive information.
2 The origins of hyperbolic geometry
Hyperbolic Geometry
J. W. Cannon, W. J. Floyd, R. Kenyon, and W. R. Parry 1
1 Introduction
Non-Euclidean, or hyperbolic, geometry was created in the first half of the nineteenth century in the midst of attempts to understand Euclid’s axiomatic basis for geometry. Einstein and Minkowski found in non-Euclidean geometry a geometric basis for the understanding of physical time and space. In the earlyery serious student of mathematics and physics studied nonEuclidean geometry. This has not been true of the mathematicians and physicists of our generation. Nevertheless with the passage of time it has become more and more apparent that the negatively curved geometries, of which hyperbolic non-Euclidean geometry is the prototype, are the generic forms of geometry. They have profound applications to the study of complex variables, to the topology of two- and threedimensional manifolds, to the study of finitely presented infinite groups, to physics, and to other disparate fields of mathematics. A working knowledge of hyperbolic geometry has become a prerequisite for workers in these fields.
Except for Euclid’s five fundamental postulates of plane geometry, which we paraphrase from Kline [5], most of the following historical material is taken from Felix Klein’s book [4]. Other historical references appear in the bibliography. Here are Euclid’s postulates:
These notes are intended as a relatively quick introduction to hyperbolic geometry. They review the wonderful history of non-Euclidean geometry. They give five different analytic models for and several combinatorial approximations to nonEuclidean geometry by means of which the reader can develop an intuition for the behavior of this geometry. They develop a number of the properties of this geometry which are particularly important in topology and group theory. They indicate some of the fundamental problems being approached by means of non-Euclidean geometry in topology and group theory.
Englishman Wallis (1616-1703) used the assumption that to every triangle there is a similar triangle of each given size.
Italian Saccheri (1667-1733) considered quadrilaterals with two base angles equal to a right angle and with vertical sides having equal length and deduced consequences from the (non-Euclidean) possibility that the remaining two angles were not right angles.
Proclus (ca. 400 AD) used as additional postulate the assumption that the points at constant distance from a given line on one side form a straight line.
2
1. Each pair of points can be joined by one and only one straight line segment.
2. Any straight line segment can be indefinitely extended in either direction.
Decisive progress came in the 19th century when mathematicians abandoned the effort to find a contradiction in the denial of postulate (5) and instead worked out carefully and completely the consequences of such a denial. Unusual consequences of that denial came to be recognized as fundamental and surprising properties of non-Euclidean geometry: equidistant curves on either side of a straight line were in fact not straight but curved; similar triangles were congruent; angle sums in a triangle were not equal to π, and so forth.
Of these five postulates, the fifth is by far the most complicated and unnatural. For two thousand years mathematicians attempted to deduce postulate (5) from the four simpler postulates. In each case one reduced the proof of postulate (5) to the conjunction of the first four postulates with an additional natural postulate which, in fact, proved to be equivalent to postulate (5):