NOTATION, TERMINOLOGY, AND PRELIMINARY RESULTS
高等数学教材翻译英文
高等数学教材翻译英文Mathematical Analysis: Advanced Calculus Textbook TranslationAbstract:In this paper, we will discuss the process of translating a high-level mathematics textbook from Chinese to English. The main focus will be on translating various mathematical concepts, equations, and proofs accurately while maintaining a clear and coherent writing style. Additionally, attention will be given to organizing the translated content in a logical and visually appealing manner.1. Introduction:Mathematics is a universal language that transcends cultural and geographical boundaries. As such, it is crucial to provide accurate translations of mathematics textbooks to facilitate the exchange of knowledge across different linguistic communities. This paper aims to outline the key considerations and strategies involved in translating a high-level mathematics textbook, with a focus on advanced calculus.2. Translating Mathematical Concepts:When translating mathematical concepts from Chinese to English, it is important to convey the precise meaning while considering the linguistic differences between the two languages. This involves selecting appropriate mathematical terms, symbols, and notations that are commonly understood in the English-speaking mathematical community. Additionally, effortsshould be made to ensure consistency in terminology usage throughout the translated textbook.3. Translating Equations and Formulas:Equations and formulas play a central role in mathematics, and translating them accurately is of utmost importance. In the translation process, it is necessary to pay attention to the syntax, grammar, and semantics of mathematical expressions. The translator should strive to maintain the original meaning, while adapting the structure and notation tofit the conventions of the English language. Clear and concise explanations should accompany each equation or formula to enhance understanding for English-speaking readers.4. Translating Proofs and Theorems:In advanced calculus textbooks, proofs and theorems are fundamental components that demonstrate mathematical concepts and promote logical reasoning. Translating proofs effectively requires not only an understanding of the mathematical content, but also the ability to convey the step-by-step reasoning in a clear and concise manner. Using appropriate mathematical terminology and logical connectors is crucial to ensure the coherence and effectiveness of the translated proofs.5. Organizing Translated Content:To enhance readability, the translated textbook should be well-organized. Each chapter should have a clear structure, with sections and subsections that logically present the content. Within each section, a coherent flow of ideas should be maintained. The integration of mathematical illustrationsand diagrams can further support comprehension for English-speaking readers. Tables, charts, and graphs should also be included whenever necessary, but care should be taken to ensure that all visual elements are properly labeled and explained.6. Conclusion:Translating a high-level mathematics textbook, particularly in the field of advanced calculus, requires thorough understanding of both the mathematical concepts and the linguistic nuances of Chinese and English. Accurate translation of mathematical content, equations, proofs, and theorems is essential in delivering a comprehensible textbook to English-speaking readers. Furthermore, organizing the translated content in a logical and visually appealing manner enhances the overall readability and accessibility of the textbook. By following these considerations and strategies, a successful translation of a high-quality mathematics textbook can be achieved.。
定积分的发展史
定积分的发展史起源定积分的概念起源于求平面图形的面积和其他一些实际问题。
定积分的思想在古代数学家的工作中,就已经有了萌芽。
比如古希腊时期阿基米德在公元前240年左右，就曾用求和的方法计算过抛物线弓形及其他图形的面积.公元 263 年我国刘徽提出的割圆术，也是同一思想。
在历史上,积分观念的形成比微分要早。
但是直到牛顿和莱布尼茨的工作出现之前(17世纪下半叶），有关定积分的种种结果还是孤立零散的，比较完整的定积分理论还未能形成，直到牛顿-—莱布尼茨公式建立以后,计算问题得以解决，定积分才迅速建立发展起来。
未来的重大进展，在微积分才开始出现，直到16世纪。
此时的卡瓦列利与他的indivisibles方法 ,并通过费尔马工作,开始卡瓦列利计算度N = 9 × N的积分奠定现代微积分的基础，卡瓦列利的正交公式。
17世纪初巴罗提供的第一个证明微积分基本定理。
牛顿和莱布尼茨在一体化的重大进展是在17世纪独立发现的牛顿和莱布尼茨的微积分基本定理. 定理演示了一个整合和分化之间的连接. 这方面，分化比较容易地结合起来，可以利用来计算积分。
特别是微积分基本定理,允许一个要解决的问题更广泛的类。
同等重要的是,牛顿和莱布尼茨开发全面的数学框架。
由于名称的微积分，它允许精确的分析在连续域的功能。
这个框架最终成为现代微积分符号积分是直接从莱布尼茨的工作.正式积分定积分概念的理论基础是极限。
人类得到比较明晰的极限概念，花了大约2000年的时间.在牛顿和莱布尼茨的时代，极限概念仍不明确。
因此牛顿和莱布尼茨建立的微积分的理论基础还不十分牢靠，有些概念还比较模糊,由此引起了数学界甚至哲学界长达一个半世纪的争论，并引发了“第二次数学危机”。
经过十八、十九世纪一大批数学家的努力，特别是柯西首先成功地建立了极限理论，魏尔斯特拉斯进一步给出了现在通用的极限的定义，极限概念才完全确立，微积分才有了坚实的基础,也才有了我们今天在教材中所见到的微积分。
数学证明 因为 所以 英文术语
数学证明因为所以英文术语Mathematics has always been a subject of great fascination and importance in our lives. It is the language of the universe, a tool that helps us understand the world around us and solve complex problems. In this essay, we will explore the concept of mathematical proof and its significance in the realm of English terminology.Proof is the cornerstone of mathematics, a means by which we can establish the validity of a statement or the truthfulness of a claim. It is a systematic and logical process that involves the use of well-defined rules and principles to arrive at a conclusion. Mathematical proofs are not merely a collection of calculations or formulas; they are a structured and rigorous way of thinking that allows us to build a solid foundation for our understanding of the world.One of the key aspects of mathematical proof is the use of logical reasoning. This involves the application of deductive logic, where we start with a set of premises or assumptions and then use a series of logical steps to arrive at a conclusion. This process is often referred to as the "if-then" statement, where we say, "If these conditions are true, then this conclusion must also be true."Another important aspect of mathematical proof is the use of mathematical language and terminology. In the realm of English, there are many specialized terms and concepts that are essential to the understanding and communication of mathematical ideas. These terms include concepts such as "axiom," "theorem," "lemma," "corollary," and "hypothesis," among others.Axioms are the fundamental building blocks of mathematical proof. They are statements that are accepted as true without the need for further proof. Theorems, on the other hand, are statements that can be proven to be true based on a set of axioms and previously established theorems. Lemmas are intermediate results that are used to prove a larger theorem, while corollaries are statements that follow directly from a theorem.The process of proving a mathematical statement often involves the use of logical connectives such as "if," "then," "and," "or," and "not." These connectives are used to construct complex logical statements and to establish the relationships between different parts of a proof.In addition to the use of logical reasoning and specialized terminology, mathematical proofs also often involve the use of mathematical symbols and notation. These symbols, such as the equals sign (=), the greater than or less than signs (> and <), and the summation symbol (Σ), are used to represent mathematical conceptsand operations in a concise and unambiguous way.The importance of mathematical proof cannot be overstated. It is the foundation upon which our understanding of the world is built, and it is the means by which we can verify the accuracy and reliability of our mathematical models and theories. Without proof, mathematics would be nothing more than a collection of arbitrary rules and formulas, lacking the rigor and certainty that we have come to expect from this field of study.In conclusion, the concept of mathematical proof is a fundamental aspect of mathematics and is essential to the understanding and communication of mathematical ideas in the realm of English terminology. By mastering the use of logical reasoning, specialized terminology, and mathematical notation, we can deepen our understanding of the world around us and unlock the secrets of the universe.。
专业词汇&缩写
专业词汇A：abound——大量存在account——描述accuracy——精确性ad hoc ——尤其，特定地advent——出现adequate——足够的aerodynamic——空气动力学aircraft——飞行器alleviate——减轻，缓和altitude——海拔ambitious——有抱负的amplifier——放大器amplify——放大annual——每年的anode——阳极analog——模拟analog modulation——模拟调制antenna——天线applicability——适用性applied——适用的；外加的approach——方法；研究；途径approximate——近似arbitrary——任意的arrangement——排列；方案artificial——人工a set of variables ——一组变量as far as…be concerned 就……而言aspects of ——……方面assumed——假定的asynchronous transmission——异步传输atomic——原子的attain——达到，实现attenuation——衰减attitude——姿态anti-clockwise——反时钟方向autocode——自动编码autocontrol——自控autopilot——自动驾驶autograph——自动记录仪B：binary——二进制brilliancy——亮度boost——提升burst——脉冲bus network——总线网络back and forth——来回C：cathode——阴极carrier——电磁载波（电磁载子）carrier——载波，载流子category——分类；范畴charge——电荷，充电central——中心的；最重要的circuit——电路circumstances——状况，环境classify——分类classification——分类clockwise——时钟方向coefficient——系数commonplace——寻常的事物compensate——补偿component——组成context——环境；上下文codec——编码器coherent——相关的，一致的concern——涉及concentrated——集中的concept——概念conclusion——结论configuration——构造，结构consists——构成，包括consequently——因此constant——恒定的constitute——构造，组织constraints——约束contain——包含correction——修正correlation——相关,相互关联cost——成本crew——船员criteria——标准current——电流D：defined——下定义density——密度determinant——行列式determination——测定deterministic——确定的detractors——批评者development system——开发系统dictate——指示；要求differential equation——微分方程digital——数字式的dimension——维数diode ——二极管direct——指示discrete——离散的distributed parameter——分散参数distribution——分配disturbance——扰动domain——领域dominate——支配，使服从draft——制图dynamic analysis——动态分析dynamic response——动态响应E：effect——效应efficiency——效率elastic——有弹性的electrical-optical conversion 光电转换electromagnetic wave——电磁波electromagnetic carrier——电磁载波electron——电子emphasis——强调；重要性encounter——遇到equation——方程essentially——本质上evelope——包络，包迹exact——确切的、精确的existence——存在expect——期待extend——扩展extremely——极度的F：failure——失效field——电磁场field-effect transistor——场效应管filter——滤波器fixed telephone service 固定电话服务flip-flop——触发器formulation——公式化（表达）frequency hopped——跳频frequency spectrum——频谱G：gateway——关卡，门径general-purpose——通用的graphical——绘图的gross national product——国民生产总值H：handover——移交hard-to-pass(road)——难通行的(路)harden——硬化hardware——硬件heavy-type——重型的heading——艏向history——历史状态I：impedance——阻抗（全电阻）implementation——实现，履行independent——独立的instant——瞬间interaction——相互影响insofar as 到这样的程度，在……范围内integrated circuits——集成电路interval——间隔intuition——直觉inverse——相反的investigation——调查；研究J：jet plane——喷气式飞机L：leads——导线limitation——局限line——电线linear vector space——线性矢量空间likewise——同样的lumped parameter——集中参数M：matrices（pl.）——矩阵matrix algebra——矩阵代数massed——聚集的microprocessor——微（信息）处理机microcomputer——微型计算机minimum——最小量modulator——调制器modem——调制解调器modification——修正multi-access——多路存取multi-path fading——多径衰弱multivariable——多变量multifold——多倍multiply——加倍，倍增N：navigation——航行，导航n-dimensional ——n维的n-dimensional vector space n维状态空间Nyquist rate——奈奎斯特率Noise-figure——噪音系数notation——符号numerical——数字的O：obviously——显然地occupy——占有Ohm’s law——欧姆定律one-way(traffic)——单向(交通)optical carrier——光载波optimal control——最优控制optimization——最优化optimization theory——最优化理论ordinary differential equation常微分方程oscillator——振荡器oxidize——氧化P：package——包packet switching——分组交parametric——参量particular——特定的partial differential equation 偏微分方程performance criteria——性能指标phase-plane ——相平面phase-shift keying——移相键控换perform——执行pilot——驾驶员plant——机器，设备被控对象plus——正的；增加potential——电位power plant——电厂precise——精确的predictable——可断定的principally——主要prior——优先的probability theory——概率论process——过程progress——进步；发展prominently——显著地property——性质proponents——提倡者R：radio-relay transmission 无线电中继传输radio——收音机radio frequencies——无线电频率rationale——理论，原理的阐述Rayleigh fading——雷利衰落reaction——反应realistic——现实recognition——识别rectifier——整流器regulate——调节regulator——稳压器relatively——相对地relay——继电器，中继器reliability——可靠性remove——移除represent——表示，表现resonant frequency——谐振频率resistance——电阻（值）ring network——环形网络root mean square——均方根routine——例程（程序）；常规的rust-free——无锈的S：sample-data——采样数据scalar——数量的，标量的；数量，标量scanning——扫描self-cooling——自冷的separately——分离，独立sequence——顺序sequential——连续的；时序、顺序serial data——串行数据shift——偏移；转换simply——默认；简单的simultaneously——同时地；联立地situation——情况small-and-medium(size)——中小型spring——弹簧specialist——专家specifications——规范，说明书spectrum——频谱spectrum allocation——频谱分配spectrogram——(光)谱图spectrograph——摄谱仪state variable——状态变量stationary——静态的status——状态stimulus——刺激，鼓励stochastic——随机的strategy——方法subsequent——后序的subject——受……限制T：tedious——冗长的terminology——术语termed——称为thermodynamics——热力学time-invariant——时不变的time-variant——时变的to-and-to(motion)——往复(运动) trade deficit ——贸易赤字transistor——晶体管transceiver——无线电收发机transmit——传送transmission——变送器transition——转换trail-and-error——试凑法trend——趋势triangle——三角形tripod——三脚架,三角桌,凳子,三角鼎tutorial——指导性的U：ultrared——红外(线)的ultrasonic——超声的uniform——一致的union——联合；一致Union message ——联邦咨文up-and-down(traffic)——上下行(交通) V：verify——证实very large scale integrated circuits（VLSI）超大规模集成电路volume——体积W：white-hot——自热的,白炽的Y：yaw——偏航缩写词汇AC,ac——Alternating Current（交流电）/Adaptive Control（自适应控制）A/D,ADC ——Analog-to-Digital Converter（数模转换器）CA TV ——Cable Television（有线电视）CAD ——Computer-Aided Design（计算机辅助设计）CAI——Computer-Aided Instruction（计算机辅助教学）CPU——Central Processing Unit（中央处理器）DSP——Digital Signal Processing/Digital Signal Processor（数字信号处理）DC,dc ——Direct Current（直流电）FM ——Frequency Modulation（调频）HDTV——High Definition Television（高清晰度电视）IEE ——Institute of Electrical Engineers（电机工程师协会）IEEE——Institute of Electrical and Electronic Engineers（电气电子工程师协会）LED ——Light Emitting Diode（发光二极管）LANs ——Local Area Networks（局域网）SNR ——Signal to Noise Ratio（信噪比）WWW——World Wide Web（环球网）一些词汇和短语breakthrough——突破condition——限制，条件constraint——强制，约束necessitate——需要restriction——约束，限制因素recommendation、recommend——推荐，建议要求/建议：demand、requirement、suggestion overcharge——超越textbook——教科书undergo——经受,进行be suddenly connected or disconnected——突然接通或断开consist of、arise from、result from、serve as、abide by、find use in。
[14],[17],[4],[1],[13],[7],[2],[3],[18],[16],[10],[15],[11],[9],[8],[12],and[6].
![[14],[17],[4],[1],[13],[7],[2],[3],[18],[16],[10],[15],[11],[9],[8],[12],and[6].](https://img.taocdn.com/s3/m/6ed93cd033d4b14e85246851.png)
FORMALIZED MATHEMATICSVolume11,Number4,2003University of BiałystokBanach Space of Absolute SummableReal SequencesYasumasa Suzuki Take,Yokosuka-shiJapanNoboru EndouGifu National College of Technology Yasunari ShidamaShinshu UniversityNaganoSummary.A continuation of[5].As the example of real norm spaces, we introduce the arithmetic addition and multiplication in the set of absolutesummable real sequences and also introduce the norm.This set has the structureof the Banach space.MML Identifier:RSSPACE3.The notation and terminology used here are introduced in the following papers:[14],[17],[4],[1],[13],[7],[2],[3],[18],[16],[10],[15],[11],[9],[8],[12],and[6].1.The Space of Absolute Summable Real SequencesThe subset the set of l1-real sequences of the linear space of real sequences is defined by the condition(Def.1).(Def.1)Let x be a set.Then x∈the set of l1-real sequences if and only if x∈the set of real sequences and id seq(x)is absolutely summable.Let us observe that the set of l1-real sequences is non empty.One can prove the following two propositions:(1)The set of l1-real sequences is linearly closed.(2) the set of l1-real sequences,Zero(the set of l1-real sequences,the linearspace of real sequences),Add(the set of l1-real sequences,the linear space377c 2003University of BiałystokISSN1426–2630378yasumasa suzuki et al.of real sequences),Mult(the set of l1-real sequences,the linear space ofreal sequences) is a subspace of the linear space of real sequences.One can check that the set of l1-real sequences,Zero(the set of l1-real sequences,the linear space of real sequences),Add(the set of l1-real sequences,the linear space of real sequences),Mult(the set of l1-real sequences,the linear space of real sequences) is Abelian,add-associative,ri-ght zeroed,right complementable,and real linear space-like.One can prove the following proposition(3) the set of l1-real sequences,Zero(the set of l1-real sequences,the linearspace of real sequences),Add(the set of l1-real sequences,the linear spaceof real sequences),Mult(the set of l1-real sequences,the linear space ofreal sequences) is a real linear space.The function norm seq from the set of l1-real sequences into R is defined by: (Def.2)For every set x such that x∈the set of l1-real sequences holds norm seq(x)= |id seq(x)|.Let X be a non empty set,let Z be an element of X,let A be a binary operation on X,let M be a function from[:R,X:]into X,and let N be a function from X into R.One can check that X,Z,A,M,N is non empty.Next we state four propositions:(4)Let l be a normed structure.Suppose the carrier of l,the zero of l,theaddition of l,the external multiplication of l is a real linear space.Thenl is a real linear space.(5)Let r1be a sequence of real numbers.Suppose that for every naturalnumber n holds r1(n)=0.Then r1is absolutely summable and |r1|=0.(6)Let r1be a sequence of real numbers.Suppose r1is absolutely summableand |r1|=0.Let n be a natural number.Then r1(n)=0.(7) the set of l1-real sequences,Zero(the set of l1-real sequences,the linearspace of real sequences),Add(the set of l1-real sequences,the linear spaceof real sequences),Mult(the set of l1-real sequences,the linear space ofreal sequences),norm seq is a real linear space.The non empty normed structure l1-Space is defined by the condition (Def.3).(Def.3)l1-Space= the set of l1-real sequences,Zero(the set of l1-real sequences,the linear space of real sequences),Add(the set of l1-realsequences,the linear space of real sequences),Mult(the set of l1-realsequences,the linear space of real sequences),norm seq .banach space of absolute summable (379)2.The Space is Banach SpaceOne can prove the following two propositions:(8)The carrier of l1-Space=the set of l1-real sequences and for every set xholds x is an element of l1-Space iffx is a sequence of real numbers andid seq(x)is absolutely summable and for every set x holds x is a vectorof l1-Space iffx is a sequence of real numbers and id seq(x)is absolutelysummable and0l1-Space=Zeroseq and for every vector u of l1-Space holdsu=id seq(u)and for all vectors u,v of l1-Space holds u+v=id seq(u)+id seq(v)and for every real number r and for every vector u of l1-Spaceholds r·u=r id seq(u)and for every vector u of l1-Space holds−u=−id seq(u)and id seq(−u)=−id seq(u)and for all vectors u,v of l1-Spaceholds u−v=id seq(u)−id seq(v)and for every vector v of l1-Space holdsid seq(v)is absolutely summable and for every vector v of l1-Space holdsv = |id seq(v)|.(9)Let x,y be points of l1-Space and a be a real number.Then x =0iffx=0l1-Space and0 x and x+y x + y and a·x =|a|· x .Let us observe that l1-Space is real normed space-like,real linear space-like, Abelian,add-associative,right zeroed,and right complementable.Let X be a non empty normed structure and let x,y be points of X.The functorρ(x,y)yields a real number and is defined by:(Def.4)ρ(x,y)= x−y .Let N1be a non empty normed structure and let s1be a sequence of N1.We say that s1is CCauchy if and only if the condition(Def.5)is satisfied. (Def.5)Let r2be a real number.Suppose r2>0.Then there exists a natural number k1such that for all natural numbers n1,m1if n1 k1and m1k1,thenρ(s1(n1),s1(m1))<r2.We introduce s1is Cauchy sequence by norm as a synonym of s1is CCauchy.In the sequel N1denotes a non empty real normed space and s2denotes a sequence of N1.We now state two propositions:(10)s2is Cauchy sequence by norm if and only if for every real number rsuch that r>0there exists a natural number k such that for all naturalnumbers n,m such that n k and m k holds s2(n)−s2(m) <r.(11)For every sequence v1of l1-Space such that v1is Cauchy sequence bynorm holds v1is convergent.References[1]Grzegorz Bancerek.The ordinal numbers.Formalized Mathematics,1(1):91–96,1990.[2]Czesław Byliński.Functions and their basic properties.Formalized Mathematics,1(1):55–65,1990.380yasumasa suzuki et al.[3]Czesław Byliński.Functions from a set to a set.Formalized Mathematics,1(1):153–164,1990.[4]Czesław Byliński.Some basic properties of sets.Formalized Mathematics,1(1):47–53,1990.[5]Noboru Endou,Yasumasa Suzuki,and Yasunari Shidama.Hilbert space of real sequences.Formalized Mathematics,11(3):255–257,2003.[6]Noboru Endou,Yasumasa Suzuki,and Yasunari Shidama.Real linear space of real sequ-ences.Formalized Mathematics,11(3):249–253,2003.[7]Krzysztof Hryniewiecki.Basic properties of real numbers.Formalized Mathematics,1(1):35–40,1990.[8]Jarosław Kotowicz.Monotone real sequences.Subsequences.Formalized Mathematics,1(3):471–475,1990.[9]Jarosław Kotowicz.Real sequences and basic operations on them.Formalized Mathema-tics,1(2):269–272,1990.[10]Jan Popiołek.Some properties of functions modul and signum.Formalized Mathematics,1(2):263–264,1990.[11]Jan Popiołek.Real normed space.Formalized Mathematics,2(1):111–115,1991.[12]Konrad Raczkowski and Andrzej Nędzusiak.Series.Formalized Mathematics,2(4):449–452,1991.[13]Andrzej Trybulec.Subsets of complex numbers.To appear in Formalized Mathematics.[14]Andrzej Trybulec.Tarski Grothendieck set theory.Formalized Mathematics,1(1):9–11,1990.[15]Wojciech A.Trybulec.Subspaces and cosets of subspaces in real linear space.FormalizedMathematics,1(2):297–301,1990.[16]Wojciech A.Trybulec.Vectors in real linear space.Formalized Mathematics,1(2):291–296,1990.[17]Zinaida Trybulec.Properties of subsets.Formalized Mathematics,1(1):67–71,1990.[18]Edmund Woronowicz.Relations and their basic properties.Formalized Mathematics,1(1):73–83,1990.Received August8,2003。
数学对计算机重要吗英语作文
Mathematics: The Backbone of ComputerScienceIn the realm of computer science, mathematics plays a pivotal role, serving as both the foundation and thedriving force behind many technological advancements. The intertwining of these two disciplines is not merely coincidental; rather, it is a symbiotic relationship that has given birth to numerous groundbreaking inventions and theories.The fundamental concepts of computer science, such as algorithms, data structures, and programming languages, are deeply rooted in mathematical principles. Algorithms, for instance, are essentially mathematical models that outline a finite set of steps to solve a problem or compute a function. The efficiency and effectiveness of these algorithms are often measured using mathematical notation and terminology, such as time complexity and space complexity.Data structures, another crucial aspect of computer science, are also deeply influenced by mathematics. From arrays and lists to graphs and trees, these structures aredesigned to efficiently store and manipulate data,借鉴了数学中的集合论和图论等概念。
不同科技学科单词
colon n.冒号dash n.连接号comma n.逗号underscore n.下划线discipline n.学科fundamental s n.基本原理corollary n.推论assume v.假设assessment n.评估assignment n.任务access n.通道入口v.接近accurate adj.准确的精确的analysis n.分析elementary adj.基础的algebra n.代数calculus n.微积分probability n.概率论prescribe v.规定指示开药方protocol n. （数据传输）协议laboratory n. 实验课conduct v.组织实施指挥enforce v.实施强迫fundamental adj.基本的comprehensive adj.综合的comprehension n.理解力complement n.补充second half 下半场revise v.复习修正reinforce v.增强review v.复习notify v.通知公告notification n. 通知subsequent adj.随后的curriculum n.课程circuit n.电路v. 环行router n.路由器server n.服务器projector n.投影仪analogous adj.类似的analogue adj.模拟的n.类似物analogy n.类比propagate v.传播普及繁衍accelerate v.加速framework n.结构框架dispensable adj.非必需的indispensable adj.必不可少的terminology n.术语deploy v.部署有效利用module n.单元模块modulate v.调制调节modulation n.调制model n/v. 模型做模型建模modem 等于modulator-demodulator n.调制解调器cellular adj.蜂窝的细胞的interference n.干涉干扰slide n.幻灯片v.滑动降低slice v./n.薄片切precise adj.精确的heterogeneous adj.多种多样由不同组成的homogeneous adj.同类的由相同组成的(uniform)无线通信：spectrum n.谱频谱声谱spectral adj.domain n.范围领域定义域域time domain时域frequency domain频域transmit v.传输传播发射transmitter n.发射机receiver n.接收机binary adj.二进制的bipolar adj.两极的双极性的repeater n.中继器转发器electromagnetic noise电磁干扰electromagnetic wave 电磁波interfere v.干扰阻碍interference n.interval n.间隔conjugate adj.共轭的distort v.扭曲失真distortion n.失真variance n.方差变化不一致variation n.变化mean n.平均值standard deviation 标准差covariance n.协方差integration n.积分integrate v.使结合一体化求积分derivation n.导数derivate v.派生衍生求导derive v.导出源于获得corrupted adj.损坏的腐败的modulation n.调制demodulation n.解调amplitude n.振幅广阔丰富detection n.检测检波imaginary part 虚部real part 实部imaginary adj.虚构的想象的虚部的proximity n.接近临近proximate adj.近似的最接近的obstruct v.阻塞阻碍obstructed adj. unobstructed adj.decay v.腐朽衰败衰落attenuation n.衰减attenuate v.antenna gain 天线增益unitless 无单位的partition n./v.分裂隔板隔开random variable 随机变量conductor n.导体指挥检票员fluctuate v.波动carrier n.载波inter-symbol interference 符号间干扰ISIreplica n.复制副本overlap n./v.重叠convolution n.卷积convolute v.回旋卷绕discrete system 离散系统decompose v.分解腐烂sequence n.序列impulse n.脉冲一般指冲激信号velocity n.速率threshold n.阈值临界值coherent adj.相干的连贯的synchronize v.同步同时发生synchronization n.phase n.相位phasor n.矢量相量（包括幅度和角度）amplify v.放大增强dashed line虚线millisecond n.毫秒hexagon n.六角形prone adj.有...的倾向的易于...的deteriorate v.恶化退化sectorial adj.扇形的sectorization n.分段化扇形化sector n.扇形align v.校准对其一致alignment n.电信网络：topology n.拓扑学memory n.内存buffer n./v.缓存缓冲allocate v.分配指定拨出slot n./v.空挡位置idle adj./v.空闲的闲置的懒惰的congestion n.拥挤充血占线音congest v. busy tone 忙音processor n.处理器nodal adj.节点的encapsulation n.封装encapsulate v.压缩封入内部decapsulate v.解封装intermittent adj.间歇性的retrieve v.检索，恢复，取回identifer n.标识符proprietary adj.专有的专利的specification n.说明规范cleartext password 明文密码encrypt v.加密encryption n.intact adj.完好无损的integrate v.整合adj.完全的hierarchical adj.分层次的分等级的iterate v.迭代iteration n.recursive adj.递归的congest v.充血拥塞congestion n.idle adj./v.空闲的懒惰的encapsulate v.封装constrain v.约束constraint n.约束条件consent n./v.同意准许collision n.冲突collide v.reassemble v.重新装配fragment v./n.碎片片段payload n.有效负荷净负荷enumerate v.枚举列举数学相关：diagram n.图表v.用图表法histogram n.直方图柱状图exponential adj.指数的exponent n.指数orthogonal adj.正交的proportional adj.成比例的statistical adj.统计的integer n.整数constant n.常数variable n.变量symmetrical adj.对称的biased adj.偏的有偏见的bias n.偏见偏差arithmetic n./adj.算数算数的computation n.计算formula n.公式megabyte n.1MB mega百万10^6multi前缀多…multiple adj.多重的复杂的n.倍数multiply v.乘繁殖domain定义域range 值域slope n.斜率parameter n.参数加and/plus/added to 减minus/taken from/subtracted from subtraction减法4 plus 6 equals to 10. 8 minus 3 is 5.4 added to 6 is 10. 3 taken(subtracted) from 8 leaves 5.乘time/multiplied by 除divided to/into3 times 5 is 15. 8 into 24 equals 3.3 multiplies by 5 is 15. 24 divided by 8 is 3.3/4 three fourth s24/25 twenty-four twenty-fifth s10^7 the 7th power of 10 10 to the 7th powerx分之y y over xtranspose v.移项变换顺序转置multiplication n.相乘power n.幂4 to the power of 3. 4的3次方exponential adj./n.指数的logarithm n.对数logarithmic adj.initialization n.初始化probability n.概率proportional adj.成比例的linear adj.线性的nolinear adj.非线性的equation n.方程式fraction n.分数比例率coefficient n.系数denote v.表示指代(记为)threshold n.临界值阈值门槛converge v.趋于收敛于diameter n.直径weighted adj.加权的directed adj.有向的time-varying adj.时变的bifurcate v./adj. 分叉分支chaotic adj.无序的混沌的mutual adj.相互的共有的internal adj.内部的国内的external adj.外部的国外的fractal n.不规则形分形adjacent adj.相邻的shift v.移动替换transition n.转变过渡discrete adj.离散的分离的column n.列圆柱row n.行cylinder n.圆柱体identical adj.同一的identity n.同一性身份identify v.识别等同于geometric n.几何的denominator n.分母numerator n.分子2D continuous function 二维连续函数origin n.原点开端起源coordinate system 坐标系diagonal n./adj.对角线斜线density n.密度dense adj.稠密的密度大的normalized adj.标准化归一化dimensionless n.无穷小量dimension n.方面维3Dinverse adj./n.相反的倒转的倒数reciprocal n./adj.倒数的相互的交互的product n.乘积normal distribution 正态分布square n./adj.平方正方形monotonic adj.单调的argument n.自变量binomial n./adj. 二项式binomial distribution 二项分布inequality n.不等式不平等intersection n.交集交叉点intersect v.相交交叉横穿union n.并集编程相关：syntax n.语法句法tuple n. 数组set n.集合array n.数组package n.软件包extract n./v.提取摘录matrix n.矩阵loop n.循环iterate v.迭代execute v.执行function n.函数parameter n.参数segment n.段部分v.分割划分algorithm n.算法conjecture n.猜想theory n.理论theorem n.定理colon n.冒号interface n.界面v.连接交流图像处理：format n.格式v.格式化formation n.形成组成extract v.提取convert v.转变转换merge v.合并split v.分离划分slice v.切割RGB和BGR蓝绿红ROI: region of interestpixel n.像素photosite n.像素attribute n.属性variable n.变量element n.元素要素degrade v.退化降低degradation n.compress v.压缩compression n.压缩deburr v.清理毛刺blur n./v.模糊motion n./v.运动移动Spatial quantization 空间的量化intensity n.强度intense adj.强烈的紧张的激烈的inpainting n.图像修复texture n.质地实质纹理segment v.分割划分segmentation n.分割partition v.分割partitioning n.extract v.提取storage n.储存transmit v.传输发送transfer v.转移translate v.翻译shift v.转移换convert v.转变换算transition n.过渡转变transit v.运送经过lossy adj.有损耗的lossy compression 有损压缩acquire v.获得acquisition n.获得采集capture v.采集捕捉引起sensor n.传感器individual n.个人adj.单独的charge v./n.收费充电电荷accumulate v.累积horizontal adj./n.水平的水平面horizon n.地平线vertical adj./n.垂直的垂直线vertex n.顶点voltage n.电压volume n.体积amplify v.放大增强register v./n.注册登记显示shift register n.移位寄存器interpolate v.篡改插入mosaic n./v.马赛克demosaic v.去马赛克illuminate v.照亮阐明illumination n.光源灯饰启示illustrate v.举例说明digital adj.数字的digitize v.数字化analog n./adj.模拟analog image 模拟图像index n.索引（复数indexes/indices）denote v. 表示指示标志imply v.表示暗示donate v.捐赠sample n./v. 采样取样quantization n.量化quantize v.scan v./n.扫描discrete adj.分离的离散的continuous adj.连续的coordinate v.协作搭配n.坐标coordinate system 坐标系resolution n.解决决心分辨率interpolation n.插入插值填写interpolate v.resize v.调整大小real-time 实时的ultrasound n.超声波capacitor n.电容器circuit n.电路回路环线circuitry n.电路电路图prism n.棱镜棱柱contour n./v.轮廓边界线abrupt adj.突然的不连贯的tuple n.元组hue n.色调saturation n.饱和度offset n.偏移量gain n.增益clip v.剪掉裁剪template n.样板模板intermediate adj./n.中间的中间事物intermediate image 中间过渡图像eliminate v.消除淘汰flip v.反转打开overlap v.重叠superimpose v.叠加magnitude n.大小数值periodic adj.周期的series n.级数系列mitigate v.减轻缓和ramp n.斜坡坡道渐变notate v.以符号标记notation n.记法符号conjugate v./adj./n.共轭complex conjugate 复共轭harmonics n.谐波impulse train冲击序列translation v.平移翻译replica n.复制副本dilated adj.扩大膨胀dilate v. dilation n.sinusoid 正弦曲线aliasing n.混叠wrap v./n.包裹环绕围巾卷饼wraparound error 环绕错误suppress v.抑制阻止mitigate v.使减轻使缓和transition n.过渡转变transit v.运输经过agenda n.议程表待议日程arithmetic n.算数运算arithmetical adj. algorithm n.算法geometry n.几何geometric adj. medium adj.中间的五分熟的平均的median adj./n.中间值中位数prior adj.先前的正式的prior probability先验概率matrix n.矩阵metric n.度量标准adj.十进制的trim v.修剪调整emit v.发出发射beam n.光线波束project v.投射预测projection n.投影diffuse v./adj.扩散弥漫漫反射传播synchronous adj.同步的synchrony n.intercept n./v.截距拦截截断slope n./v.斜率倾斜斜坡vector n.矢量向量unit vector单位向量prime n.上标撇slash n.斜杠arbitrary adj.任意的recap n.扼要重述preliminary adj./n.初步的预赛transpose n./v.转置调换erode v.腐蚀削弱erosion n.腐蚀dilate v.膨胀扩大dilation n.膨胀shrink v.缩水缩小减少fracture v.断裂破碎duality n.二元性对偶性optimize v.优化optimal adj.最佳的合适的speckled adj.有斑点的speckle n./v.斑点色斑做标记多维数据：regress v./n..倒退回归regression n.回归set v.放n.集合anomaly n.异常asymmetric adj.不对称的sparse adj.稀疏的sparsity n.稀疏性symmetry n.对称相似induce v.引起诱使inductive adj.归纳的诱导的induction n.归纳感应deduct v.演绎扣除deduction n.推论split v./n.分割划分sort v./n.分类排序permute v.改变顺序permutation n.categorize v.分类category n. categorical adj.分类的绝对的numeric adj.数值的quantitative adj.定量的数量的qualitative adj.定性的质量的autocorrelation 自相关temporal autocorrelation 时间自相关temporal adj.时间的暂时的redundant adj.多余的冗余converge v.收敛聚集convergence n.复杂网络：homogeneous adj.单一的均匀的同质的isomorphic adj.同构的同形的isomorphism n.bipartite adj.双向的二分的invertible adj.可逆的traverse v.穿过横越yield v./n.产生出产产量skeleton n.骨骼框架梗概backbone n.脊柱骨干initial adj.最初的首字母outlier n.离群值异常值局外人consensus n.共识一致看法counterclockwise 逆时针clockwise顺时针equilibrate v.使平衡equilibrium n.均衡平衡rational adj.理智的hierarchies n.层级分层分类sparse adj.稀疏的稀少的benchmark n.比较基准synchrony n.同步synchronization n.同时性同步性consensus n.共识检测估计：hypothesis n.假设猜想scenario n.场景terminology n.术语interpolate v.插入篡改defect n.缺点v.背叛crude adj.粗略的粗糙的abrupt adj.突然的陡峭的唐突的asymptotic adj.渐进的渐近线的marginal adj.边缘的微不足道的marginal probability边缘概率prior probability先验概率conditional probability条件概率posterior probability后验概率hypothesis testing假设检验ground truth真实值decision region决策域disjoint adj.不相交的overlap adj.重叠的curvature n.弧线曲率concave adj.凹的power spectral density能量谱密度inner product 内积点积norm 模transpose v.调换n.转置矩阵determinant n./adj.决定因素行列式subscript n./adj.下标注脚Wide Sense Stationary 广义平稳time invariant时不变discretize v.离散discrete adj.离散的distort v.失真扭曲superimpose v.叠加compose v.组成构成移动APP：tablet n.药片平板电脑agenda n.待议事项议事日程kernel n.核内核compiler n.编译器compile v.汇编编译搜集manifest adj./n.表明显示清单configurate v.配置使形成obfuscate v.使模糊混淆refine v.改进精炼virtual adj.虚拟的实际上的device driver 设备驱动程序accelerate v.加速促进template n.模板样板compact adj./n.紧凑的简洁的deploy v.部署调动配置refactor v./n.重构configure v.配置config=configuration n.配置install v.安装icon n.图标palette n.调色板选项板compatible adj.兼容的可共处的execute v.执行实施prototype v./n.原型雏形widget n.小工具窗口小部件toggle n./v.切换转换(键)embed v.嵌入内置portrait adj.纵向的n.肖像描绘纵向打印格式vertical adj.垂直的纵向的landscape adj.横向打印格式n.风景横向打印格式v.景观美化horizontal adj.水平的chaos n.混乱无序状态chaotic adj.混乱的enclose v.封装围住封上nest v./n.鸟巢嵌套一套物件align v.对齐使一致margin v./n.边缘利润余地custom n.顾客惯例adj.定做的自定义的current adj.当前的流行的n.水流concurrency n.并发性并行性thread n.线程v.穿起来asynchronous adj.异步的instantiate v.举例说明invoke v.调用提及引起sequential adj.按顺序的相继发生的dispatch n./v.派遣发送迅速处理parse v.分析解析immutable adj.不可改变的mutable adj.可变的易变的explicit adj.明确的详述的implicit adj.含蓄的facilitate v.促进使便利synchronize v.同步grant v./n.承认给予授予engage v.从事雇佣pane n.玻璃窗格repository n.数据库储藏室资源库retrieve n.找回检索refactor v./n.重构resume v.恢复重新开始customize v.定制排队论：stochastic adj.随机的stochastic process 随机过程swap v.交换替换constraint n.约束条件constrain v.约束限制强迫moderate adj.适中的温和的中等的regime n.制度状态机制ethical adj.道德的plagiarize v.抄袭剽窃plagiarism n.mutually exclusive events 互斥事件mutual adj.相互的exclusive adj.昂贵的独有的排斥的rigorous adj.严格的严密的novel n.小说novelty adj./n.新颖的新奇的事物intersection n.交集相交交点在B的条件下A的概率the probability of A given Bposterior probability后验概率posterior adj.其次的较后的prior probability先验概率prior adj.先前的优先的axiom n.公理定理binomial n./adj.二项分布geometric n./adj.几何分布uniform distribution均匀分布recursion n.递归循环recur v.再发生反复出现arbitrary adj.任意的estimator n.估计量估计函数revise v./n.修正复习denominator n.分母numerator n.分子numerous adj.许多的fraction n.分数decimal n.小数integer n.整数expected value 期望standard deviation 标准差covariance n.协方差preliminary n./adj.初步的预备的准备工作identity n.身份特征同一性identical adj.同一的完全相同的stationary adj.不动的平稳的strictly stationary严格平稳ergodicity n. 遍历性各态历经性ergodic adj.multivariate adj.多元的多变量的superpose v.叠加叠放superposition n.叠加lag v./n.掉队延迟滞后consecutive adj.连续的derive v.起源于获得导出derivation n.求导导数validate v.正式确认生效物联网：mindset n.思维模式trend n.趋势风尚curriculum n.课程combat v.战斗争论block diagram 狂徒converge v.集中汇集convergence n.汇集融合immersive adj.沉浸式immerse v.沉浸于浸没resistor n.电阻resistance n.电阻(值)accelerate v.加速促进acceleration n.加速度accelerometer n.加速度计acoustic adj./n.声音的声学appliance n.家用电器actuator n.执行器驱动器actuate v.驱动促使velocity n.速度displacement n.位移gravity n.重力coordinate v./n.协调坐标magnetometer n.磁力计magnetic adj.磁性的axis n.轴对称轴calibrate v.校准calibration n.校准刻度vibrate v.震动vibration n.震动oscillate v.振荡oscillation n.振荡gearbox n.变速箱utilize v.利用使用日常：obsess v.痴迷blackout v.断电propagate v.传播繁殖grid n.格子lattice n.格子pervade v.遍及弥漫prominent adj.突出的杰出的tremendous adj.极大的极棒的intrinsic adj.内在的disordered adj.杂乱的错乱的convention n.约定协定condense v.压缩浓缩eliminate v.消除淘汰consensus n.一致同意initial adj.最初的字首的initialization n.初始化interrelated adj.相互关联的interpersonal adj.人与人之间的split v.分离skip v.跳过conform v.遵守符合顺应cluster n.群formation n.形成formative adj.形成的有重大影响的format v.格式化n.格式subtle adj.细微的敏感的verify v.核实查证vary v.变化variation n.变化variant adj.变化的vertical adj.垂直的duplicate n./v./adj.完全一样的复制implement v.实施penalty n.惩罚penalize v.惩罚capital n./adj.大写lowercase n./adj.小写specify v.具体指出specific adj.明确的特定的respective adj.各自的分别的estimate v.估计assign v.赋值分配assignment n.任务作业execute v.执行valid adj.有效的合法的vivid adj.生动的鲜艳的dynamic adj.动态的有活力的compose v. 组成撰写排版decompose v.分解腐烂spectrum n.范围频谱光谱perceptual adj.感知的有知觉的perception n.认知知觉洞察力absorb v.吸收掌握scope n.范围视野dissipate v.驱散挥霍mandatory adj.强制的tedious adj.单调乏味的allude v.略微提到暗指discern v辨别了解recognize v.识别承认decompose v.分解腐烂衰变deficient adj. deficiency n.不足缺陷interval n.间隔desirable adj.令人满意的满足需要的complement n./v.补充pros and cons(拉丁语) 优缺点nuts and bolts(螺母和螺栓) 基本组成部分aka (also known as)又叫做ultimate adj.最终的最重要的configuration n.布局配置configure v.安装形成hybrid n./adj.混合杂种immune adj.免疫的不受影响的layman n.外行门外汉exceed v.超越超过incoming adj.进来的进入的income n.收入alter v.改变修改alternate v./adj.交替轮流alternative adj./n.可选择的二选一alternatively adv.要不二择一on-demand adj.按需的emphasize v.强调negligible adj.可以忽略的precede v. 在之前的领先experiment v./n.实验尝试arbitrary adj.任意的suffice v.足够有能力sufficient adj.elastic adj.弹性的灵活的essentially adv.本质上实质上essential adj.必须的基本的excess adj./n.超额的过多的violate v.违反侵犯compose v.撰写组成排版compromise n./v.折中妥协intuitive adj.直觉的intuition n.直觉lengthy adj.冗长的过于详尽的grasp v./n.握紧领会characterize v.描述刻画convention n.习俗公约conventional adj.依照惯例的criterion n.标准规范criteria 复数interpret v.解释口译interpretation n.解释notation n.符号标记法mutual adj.相互的sketch v./n.绘制草图outage n.断电中断in all walks of life 在各行各业frontier n./adj.边境前沿embark v.开始着手上船possess v.具备拥有insight n.了解洞察力encounter v.遭遇碰到inherent adj.固有的内在的与生俱来的inherently adv. commonality n.共性sophisticated adj.复杂精妙的水平高的stepping stone 跳板垫脚石bracket n.括号圆括号square bracket 方括号superscript n.上角标hypothesis n.猜想假设submarine v./n./adj.海底的潜水艇。
佩雷尔曼关于庞加莱猜想的论文0303109
Proof. A gradient shrinking soliton gij(t), −∞ < t < 0, satisfies the equation
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NOTATION,TERMINOLOGY,AND PRELIMINARY RESULTS For the most part,our notation and terminology is taken from West[20].Let G=(V,E)be afinite simple graph with vertex set V and edge set E,where each e∈E is an unordered pair of distinct elements in V.If e=uv,then we say that u is adjacent to v or u and v are neighbors,and that e is incident to u and v.We will use the notation u∼v to denote that u is adjacent to v,and u v to denote that u is not adjacent to v.A subgraph H of G is a graph such that V(H)⊆V(G)and E(H)⊆E(G).An induced subgraph H of G is a subgraph with the added property that if u,v∈V(H),then uv∈E(H)if and only if uv∈E(G).A neighbor w of v is called a private neighbor of v with respect to a set S⊆V if no other vertex of S is adjacent to w.The degree of a vertex v,denoted d(v),is equal to the number of edges that are incident with v.A graph is said to be k-regular if each vertex has degree k.The open neighborhood of a vertex v,denoted N(v),is the set of vertices of G that are adjacent to v.The closed neighborhood of a vertex v is N[v]=N(v)∪{v}. An independent set I⊆V of G is a set of vertices such that no two vertices of I are adjacent.The independence number of G,denotedα(G),is defined as the maximum cardinality of an independent set of G.We say that a graph is well-covered if every maximal independent set of G has the same cardinality.Determining whether or not a graph is well-covered has been shown to be co-NP-complete[3][18].A path is an alternating sequence of vertices and edges such that every edge joins the vertex preceding it with the vertex succeeding it and no vertex is repeated.If there is a path from every vertex in a graph G to every other vertex in G,then G is said to be connected;otherwise G is disconnected.A graph G is said to be k-connectedif you must delete at least k vertices to disconnect G.If G−v is a disconnected graph,then v is called a cut-vertex.A bipartite graph is one whose vertex set can be partitioned into two independent sets.A complete bipartite graph,denoted K m,n where one independent set A contains m vertices and the other independent set B contains n vertices,is a bipartite graph containing all possible edges joining a vertex from A to a vertex in B.Note that(A,B)is called a bipartition of the graph.A graph is said to be claw-free if it contains no induced K1,3.A set of pairwise disjoint edges of a graph is called a matching.Planar graphs are those that can be drawn in the plane without any edges crossing.A dominating set D⊆V of G is a set such that each vertex v∈V is either in the set or adjacent to a vertex in the set.The domination number of G,denotedγ(G),is defined as the minimum cardinality of a dominating set of G.Note thatγ(G)≤α(G) for all graphs G,since any maximal independent set is a minimal dominating set.We say that a graph is well-dominated if every minimal dominating set of G has the same cardinality.Research in the area of well-dominated graphs was begun by Finbow, Hartnell and Nowakowski[5].They were interested in characterizing different classes of well-covered graphs and were able to prove the following lemma.Lemma1.1[5]:Every well-dominated graph is well-covered.Hence they showed that the well-dominated graphs are a subclass of the well-covered graphs and hoped that perhaps they would be able to characterize this subclass.Unfortunately,this task has proved difficult.In Chapter II,we will char-acterize planar,claw-free,3-connected,well-dominated graphs.Below are some lem-mas that both describe some properties of well-dominated graphs and will assist us in characterizing these graphs.Lemma 1.2:If G is a well-dominated graph and v is a vertex of G ,then there exists a minimum dominating set containing v and a minimum dominating set not containing v .Proof :To obtain a dominating set containing v ,place v in the set D ,delete N [v ]from G and continue in this greedy fashion until there are no vertices left.Then D is minimal and since G is well-dominated it is therefore minimum.To obtain a minimum dominating set not containing v ,we use the same greedy method except we use a neighbor of v as our initial vertex in D .Lemma 1.3:Suppose G is a well-dominated graph with a cut-vertex v .Let H 1,...,H k be the components of G −v .Then each H i for i =1,...,k is well-dominated.Proof :By way of contradiction,suppose there exists an H i of G −v that is not well-dominated.Then there exist minimal dominating sets D i and D k +1of H i such that |D i |<|D k +1|.Let u be a vertex of H 1that is adjacent to v .Greed-ily choose a minimal dominating set,D 1,of H 1containing u .Choose arbitrary minimal dominating sets D j for each H j where j =1,i .Then D =k j =1D j and D ∗= i −1j =1D j ∪ k +1 j =i +1D j are both minimal dominating sets of G ,but |D |<|D ∗|.This contradicts the fact that G is well-dominated.Hence each H i is well-dominated.It is important to note that Lemma 1.3is not an if and only if statement.For(b)v(a)Figure 1:Although (b)is well-dominated,the graph shown in (a)is not well-dominated.example,let G be the graph shown in (a)of Figure 1with cut-vertex v .Then the component of G −v shown in (b)of Figure 1is well-dominated,but G is not.Two minimal dominating sets of G of different sizes are denoted by the white and boxed vertices respectively in Figure 1(a).The following lemma,shown by Campbell and Plummer [2],is an important tool in proving the characterization theorems in Chapter II.Lemma 1.4[2]:Let G be a well-covered graph and I be an independent set of G .If C is a component of G −N [I ],then C is well-covered.It is valuable to note that this technique for looking at well-covered graphs extends to well-dominated graphs as well.Lemma 1.5:Let G be a graph and I be an independent set of G .Let C be a component of G −N [I ].If C is not well-dominated,then G is not well-dominated.Proof :Let G be a graph,I be an independent set of G and C be a component of G −N [I ]that is not well dominated.Then there exist two minimal dominating setsof C,D1and D2,such that|D1|<|D2|.Let D be a minimal dominating set of G−N[I]−C.Then D∪I∪D1and D∪I∪D2are minimal dominating sets of G and|D∪I∪D1|<|D∪I∪D2|.Therefore G is not well-dominated.Often we have information that tells us how part of a graph,G,is defined,but not the entire graph.An induced subgraph of the graph on the vertices for which we have complete adjacency information at any given time in an argument we shall call a partial,P,of G.We define a vertex,x,of G to be a link vertex of P if x is contained in G−P and is a neighbor in G of a vertex of P.Suppose u and v are two vertices of a partial P of G and we know both u and v have neighbors in G outside of P.Consider a link vertex y adjacent to u and a link vertex z adjacent to v.Since we do not have complete information about y and z it is possible that these two vertices could be identical and therefore that u and v share a neighbor outside of P.For this reason,we never assume that vertices we have labeled as link vertices are distinct.Let L be the set of link vertices associated with a partial P.We can use these subgraphs of a graph,which we call partials,to determine whether or not the whole graph is well-dominated.Additional terminology will be introduced when needed.BIBLIOGRAPHY1.S.R.Campbell,M.N.Ellingham and G.F.Royle.A characterisation of well-covered cubic put.,13(1993)193-212.2.S.R.Campbell and M.D.Plummer.On well-covered3-polytopes.Ars Com-bin.,25A(1988)215-242.3.V.Chv´a tal and P.J.Slater.A note on well-covered graphs.Quo Vadis,GraphTheory?Ann.Discrete Math.,55(1993)179-182.4.G.A.Dirac.In abstrakten Graphen vorhande vollst¨a ndigene4-Graphen undihre Unterteilungen.Math.Nachr.,22(1960)61-85.5.A.Finbow,B.Hartnell,and R.J.Nowakowski.Well-dominated graphs:acollection of well-covered ones.Ars Combin.,25A(1988)5-10.6.J.F.Fink,M.S.Jacobson,L.F.Kinch and J.Roberts.On graphs havingdomination number half their order.Period.Math.Hungar.,16(1985)287-293.7.T.W.Haynes,S.T.Hedetniemi,and P.J.Slater.Fundamentals of Dominationin Graphs.New York:Marcel Dekker,Inc.(1998).8.T.W.Haynes,S.T.Hedetniemi,and P.J.Slater,ed.Domination in Graphs:Advanced Topics.New York:Marcel Dekker,Inc.(1998).9.B.L.Hartnell and M.D.Plummer.On4-connected claw-free well-coveredgraphs.Discrete Appl.Math.,64(1996)57-65.10.E.L.C.King.Some open problems in domination theory.Qualifying Paper,Department of Mathematics,Vanderbilt University(1999).11.C.Payan and N.H.Xuong.Domination-balanced graphs.J.Graph Theory,6(1982)23-32.12.M.R.Pinter.A construction which yields strongly well-covered graphs.Congr.Numer.,(Proceedings of the Twenty-eighth Southeastern International Confer-ence on Combinatorics,Graph Theory and Computing),128(1997)33-43. 13.M.R.Pinter.Planar strongly well-covered graphs.Graph Theory,Combina-torics,and Algorithms,Vol.1,2.Wiley-Intersci.Publ.(1995)869-879.14.M.R.Pinter.Strongly well-covered graphs.Discrete Math.,132(1994)231-246.15.M.R.Pinter.W2graphs and strongly well-covered graphs:two well-coveredgraph subclasses.Ph.D.Dissertation,Department of Mathematics,Vanderbilt University(1991).16.M.D.Plummer.Claw-free maximal planar graphs.Congr.Numer.,72(1990)9-23.17.B.Randerath and P.D.Vestergaard.A note on well-covered graphs with oddgirth.Aalborg University Report R-99-2019,(November1999)1-8.18.R.S.Sankaranarayana and plexity results for well-coveredworks,22(1992)247-262.19.J.Topp and L.Volkmann.On domination and independence numbers ofgraphs.Resultate Math.,17(1990)333-341.20.D.B.West.Introduction to Graph Theory,second edition.Prentice-Hall,Inc.(2001).。