Lecture Notes for A Mathematical Introduction to Robotic Manipulation

合集下载

mathematica教程第一章Mathematica基础知识

启动Mathematica后, 屏幕上出现称为Notebook 的Mathematica 系统集成界面:
Mathematica在用户区输入的内容被 Mathematica用一个具有扩展名为 “.ma” 的文件名来纪录，该文件名是退出 Mathematica时保存在用户区输入内容的默认文件名，一般是文件名：“Newnb-1.ma” 。
看磁盘中的安装文件Setup; 用鼠标双击安装文件Setup, 屏幕上出现一些选择对话框; 用鼠标点击所有选择对话框的OK按钮或键入字母y, 则系统就
在你的计算机上安装了Mathematica数学软件。 Mathematica 的安装成功后, 系统会在Windows【开始】菜
单的【程序】子菜单中加入启动Mathematica命令的图标, 用鼠标单击它就可以启动Mathematica系统,见下图：
为精确数参与计算和公式推导。
1.2.2Mathematica数的运算符
数的运算有：加、减、乘、除和乘方,它们在Mathematica 中的符号为：加（＋）、减（－）、乘（*）、除（/）和乘方（^）。不同类型的数参与运算，其结果的类型为：
如果运算数有复数，则计算结果为复数类型; 如果运算数没有复数，但有实数，则计算结果为实数类型
f(imin, jmin +m)}, {f(imin+1, jmin) ，f(imin+1, jmin +1), f(imin+1, jmin +2), ……, f(imin+1 , jmin +m)}, …… {f(imin+n, jmin) ，f(imin+n, jmin +1)，f(imin+n, jmin +2), ……，f(imin +n, jmin +m)} 其中： imax – 1 imin + n imax , jmax – 1 jmin +m jmax

Note Notes on Chvatal’s conjecture

Discrete Mathematics 247(2002)255–/locate/discNoteNotes on ChvÃa tal’s conjectureYi Wang 1Department of Applied Mathematics,Dalian University of Technology,Dalian 116024,People’s Republic of ChinaReceived 26May 1999;revised 5September 2000;accepted 22January 2001AbstractUsing Kleitman’s lemma and results of Sch o nheim and MiklÃo s it is shown that if w (D )=|D |=2,then every maximum-sized intersecting family in D contains all base elements of D .Then,the converse of this statement is conjectured and shown that this is equivalent to that of ChvÃa tal.c2002Elsevier Science B.V.All rights reserved.Keywords:Finite set;Downset;Intersecting familyLet X be a ÿnite set,P (X )the power set of X and A ⊆P (X ).A is called a downset on X if A ∈A and B ⊆A imply that B ∈A .F ⊆A is called an intersecting family in A if A ∩B =∅for all A;B ∈A .S ⊆A is called a star in A if there exists an element x ∈X such that x ∈A for all A ∈S .Let w (A )denote the maximum size of an intersecting family in A and let s (A )denote the maximum size of a star in A .Clearly w (A )¿s (A ).ChvÃa tal conjectured that there is equality if A is a downset.Conjecture 1(ChvÃa tal [3]).If A is a downset,then w (A )=s (A ).There are two particular interesting cases where the conjecture is true.If A =P (X ),then w (A )=s (A )=2n −1,which is the simplest result in intersection theory [1].If A ={A ⊆X :|A |6k }for some positive integer k 6n=2,then w (A )=s (A )= k i =1 n −1i −1 ,which is the corollary of the famous Erd o s–Ko–Rado Theorem [5].So far ChvÃa tal’s conjecture has been veriÿed for some further classes of downsets [2,4,7–10].For ex-ample,Sch o nheim [8]showed that if the intersection of all maximal members of a E-mail address:wang e@ (Y.Wang).1Present address:Department of Mathematics,Nanjing University,Nanjing 210093,People’s Republic of China.0012-365X/02/$-see front matter c 2002Elsevier Science B.V.All rights reserved.PII:S 0012-365X (01)00317-X256Y.Wang/Discrete Mathematics247(2002)255–259downset A is non-empty,then w(A)=|A|=2and ChvÃa tal’s conjecture is true for A. On the other hand,MiklÃo s[7]showed that if w(A)=|A|=2then the intersection of all maximal members of A is non-empty.In this note,we explore ChvÃa tal’s conjecture by means of a classical lemma of Kleitman.As an application,we give a simple proof of MiklÃo s’s result.Moreover,we formulate a conjecture and show that it is equivalent to ChvÃa tal’s conjecture.The following lemma is due to Kleitman.We include the proof of it for our purpose.Weÿrst need several notations.For A⊆P(X)and x∈X,let A(x)= {A\{x}:x∈A∈A};A( x)={A∈A:x∈A}.A is called an upset on X if A∈A and B⊇A imply that B∈A.Lemma1(Kleitman[6]).If U is an upset on X and D is a downset on X;where |X|=n;then|U∩D|6|U||D|=2n:(1)Proof.We proceed by induction on n,assuming that the result is true for n−1and considering the case n.Let x∈X.Then both U(x)and U( x)are upsets,and both D(x)and D( x)are downsets on the(n−1)-set X\{x}.Note that U(x)⊇U( x)and D(x)⊆D( x),we have|U∩D|=|U(x)∩D(x)|+|U( x)∩D( x)|6|U(x)||D(x)|=2n−1+|U( x)||D( x)|=2n−1=(|U(x)|+|U( x)|)(|D(x)|+|D( x)|)2n+(|U(x)|−|U( x)|)(|D(x)|−|D( x)|)2n6|U||D|=2n:Thus the proof is completed by induction.Remark1.It is not di cult to see that equality holds if and only if for all x∈X, |U( x)∩D( x)|=|U( x)||D( x)|=2n−1;|U(x)∩D(x)|=|U(x)||D(x)|=2n−1and either U( x)=U(x)or D( x)=D(x).In what follows,we always let X be a set of n elements and let D denote a downset on X.Let F be a maximum-sized intersecting family in D.The set T of minimal members of F is called a bottom of D.Denote∇T={A⊆X:A⊇T for some T∈T}.Clearly∇T is an upset on X and F=∇T∩D.Let Bot(D)denote the setof bottoms of D for all the maximum-sized intersecting families.Y.Wang/Discrete Mathematics247(2002)255–259257An immediate consequence of Lemma1is the following corollary,which has been formulated and proved in[8].Corollary1.If D is a downset;then w(D)6|D|=2.Proof.Let T∈Bot(D)and U=∇T.Then|U|62n−1since U is intersecting.Hence by Lemma1,we havew(D)=|U∩D|6|U||D|=2n6|D|=2:A maximal member of a downset D is called a base of D.Let B=B(D)denote the set of bases of D and let M=M(D)denote the intersection of bases of D.We say that D is a perfect downset if w(D)=|D|=2.Lemma2(Sch o nheim[8]).If M=∅;then D is perfect and ChvÃa tal’s conjecture is true for D.Now we present a characterization for perfect downsets by means of Lemma1.Theorem1.A downset D is perfect if and only if for any T∈Bot(D)and U=∇T;|U∩D|=|U||D|=2n:Proof.Suppose that D is perfect.Let T∈Bot(D)and U=∇T.Then|U∩D|=|D|=2. Thus by Lemma1,we have|U|¿2n|U∩D||D|=2n−1:(2)But|U|62n−1,hence equality in(2)holds,i.e.,|U∩D|=|U||D|=2n. Conversely,suppose that|U∩D|=|U||D|=2n where T∈Bot(D)and U=∇T. Then by Remark1,either U( x)=U(x)or D( x)=D(x)holds for any x∈X.But U( x)=U(x)cannot hold for all x∈X since U=P(X).Hence there exists an x∈X such that D( x)=D(x),which implies that D is perfect.Corollary2.If D is perfect;then Bot(D)⊆Bot(P(X)).In particular;if Y⊆X then Bot(P(Y))⊆Bot(P(X)).Lemma3.If T∈Bot(P(X));N={T:T∈T}and N⊆Y⊆X;then T∈Bot(P(Y)).Proof.Denote U=∇T.Let x∈Y.Then x∈N,which implies that U(x)=U( x).So|U( x)|=|U|=2=2n−2.Note that U( x)is still intersecting,hence U( x)is a maximum-sized intersecting family in P(X( x)),and therefore T∈Bot(P(X( x))).Continuingthis process,we canÿnally conclude that T∈Bot(P(Y))as required.258Y.Wang /Discrete Mathematics 247(2002)255–259As an application of the above discussion,we may give a simple proof of the following result of MiklÃo s.Theorem 2(MiklÃo s [7]).If D is perfect ;then M =∅and Bot(D )=Bot(P (M )).Proof.Let T ∈Bot(D )and U =∇T .Then |U ∩D |=|U ||D |=2n from Theorem 1.Denote N = {T :T ∈T }and let x ∈N .Then U ( x )=U (x ),which implies that D ( x )=D (x )from Remark 1.So x ∈M .Thus N ⊆M .It follows that T ∈Bot(P (M ))from Corollary 2and Lemma 3.Consequently M =∅and Bot(D )⊆Bot(P (M )).We next show that Bot(P (M ))⊆Bot(D )by induction on n .Let T ∈Bot(P (M ))and x ∈X \M .Denote D 1=D (x )and D 2=D ( x ).Then D j is a downset on the (n −1)-set X \{x }and M (D j )⊇M (j =1;2).Hence T ∈Bot(P (M (D j )))by Corollary 2.Applying the induction hypothesis to D j ,we obtain T ∈Bot(D j ).So|∇T ∩D |=|∇T ∩D 1|+|∇T ∩D 2|=|D 1|=2+|D 2|=2=|D |=2;which implies that T ∈Bot(D ).Consequently Bot(P (M ))⊆Bot(D ).Thus Bot(P (M ))=Bot(D )and the proof is complete.We say that D is a full downset if every maximum-sized intersecting family in D contains all base elements of D .Corollary 3.Every perfect downset is full .Proof.Suppose that D is a perfect downset and that F is a maximum-sized inter-secting family in D .Let T be the set of minimal members of F .Then T ∈Bot(D ),which follows that T ∈Bot(P (M ))from Theorem 2.Hence T ⊆M for any T ∈T ,and therefore T ⊆B for any B ∈B .Thus B ∈F ,and D is therefore full.Finally,we formulate a conjecture,which has been observed by MiklÃo s in [7],and show that this is equivalent to that of ChvÃa tal.Conjecture 2.Every full downset is perfect .Theorem 3.Conjecture 1holds if and only if Conjecture 2holds .Proof.Suppose that Conjecture 1is true and that D is a full downset.Then M =∅since there exists a star as a maximum-sized intersecting family in D .Thus D is perfect by Lemma 2and Conjecture 2is therefore true.Now suppose that Conjecture 1is false.Then we will ÿnd a downset which is full but not perfect.In fact,let D be a minimum-sized downset violating Conjecture 1.Then D is not perfect.On the other hand,assume that D is not full,then there must be a maximum-sized intersecting family in D which does not contain some base BY.Wang/Discrete Mathematics247(2002)255–259259 of D.Note that D =D\{B}is still a downset,hence Conjecture1is true for D by the minimality of D,i.e.,w(D )=s(D ).However,w(D)¿s(D);s(D)¿s(D )and w(D)=w(D ).So w(D)=s(D),which contradicts the hypothesis to D.Thus D is full and Conjecture2is therefore false.AcknowledgementsThe author would like to thank the anonymous referees for their careful reading and valuable suggestions.References[1]I.Anderson,Combinatorics of Finite Sets,Oxford University Press,London,1987.[2]C.Berge,A theorem related to the ChvÃa tal conjecture,Proceedings of the Fifth British CombinatorialConference,Aberdeen,1975,pp.35–40.[3]V.ChvÃa tal,Unsolved Problem No.7,in:C.Berge,D.K.Ray-Chaudhuri(Eds.),Hypergraph Seminar,Lecture Notes in Mathematics,Vol.411,Springer,Berlin,1974.[4]V.ChvÃa tal,Intersecting families of edges in hypergraphs having hereditary property,in:C.Berge,D.K.Ray-Chaudhuri(Eds.),Hypergraph Seminar,Lecture Notes in Mathematics,Vol.411,Springer,Berlin,1974,pp.61–66.[5]P.Erd˝o s,C.Ko,R.Rado,Intersection theorems for systems ofÿnite sets,Quart.J.Math.Oxford12(1961)313–320.[6]D.J.Kleitman,Families of non-disjoint subsets,bin.Theory1(1966)153–155.[7]D.MiklÃo s,Great intersecting families of edges in hereditary hypergraphs,Discrete Math.48(1984)95–99.[8]J.Sch o nheim,Hereditary systems and ChvÃa tal’s conjecture,Proceedings of the Fifth BritishCombinatorial Conference,Aberdeen,1975,pp.537–539.[9]H.Snevily,A new result on ChvÃa tal’s conjecture,bin.Theory A61(1992)137–141.[10]P.Stein,On ChvÃa tal’s conjecture related to hereditary systems,Discrete Math.43(1983)97–105.。

Lecture notes for ESSLLI-97

begin
let r be any rule in Q remove r from Q if c(r) 2 Cn(S ) then =
begin end
S := S fc(r)g for every rule s such that p(s) 2 Cn(S ) do add s to Q;
end
Figure 1: Algorithm to compute a basis for CnB (W )
Then, for every set of formulas W there is a least set T closed under PC + B and containing W .
Proposition 2.1 Let B be a set of inference rules (defaults without justi cations).
1
2 Introduction to Default Logic
Default logic is a knowledge representation mechanism allowing for reasoning in the presence of incomplete information. It handles the logical aspects of modalities such as \normally", \usually", etc. Syntactically, default logic extends the rst order logic (we will be treating propositional case almost exclusively) by introducing new entities called default rules or, simply, defaults. A default rule is a construct of the form : r = ' : M 1 ;# : : ; M m where '; 1; : : : ; k ; # are formulas of the language. The formula ' is called the premise or prerequisite of r and is denoted by p(r). The set f 1 ; : : : ; k g is called the set of justi cation of r and is denoted by j (r). The formula # is called the conclusion or consequent of r and is denoted c(r). Justi cations are used in default logic to explicitly represent conditions blocking applicability of defaults. That is, application of a rule of proof is quali ed by the absence of explicit information that would implying inconsistency of one of the justi cations of the rule. Put in yet another way, a default is applicable if its premise has been already established and all its justi cations are consistent, that is, their negations are not provable. It is precisely that presence of justi cations that allows us to model modalities such as \normally" and \usually" within default logic. In our format, a default rule has just one premise. This is an immaterial restriction since we will be assuming the usual rules of logic anyway. Default logic deals with default theories, that is, pairs (D; W ), where D is a collection of defaults and W is a collection of formulas. Default logic subsumes standard proof systems. It turns out that usual inference rules of the form ' # can simply be considered as defaults with empty set of justi cations. All major approaches to semantics of default logic are based on the natural semantics for a proof system in which propositional logic (PC) is extended by a collection of such standard inference rules (say B ). We will denote such systems by PC + B . We will now describe some properties of systems PC + B . A set of formulas T is closed under an inference rule r if the fact that ' 2 T implies that # 2 T . Similarly, T is closed under a set of rules B is it is closed under all rules in B . A set of formulas T is closed under inference in PC + B if T is closed under B and under propositional provability. We have the following fact. 2

zz美国数学本科生、研究生基础课程参考书目

zz美国数学本科生、研究生基础课程参考书目--------------------------------------------------------------------------------转一个港大数学系去哥大的师姐的东西。

那个师姐的最大传奇之处在于，她和他在中科大的男朋友分别是当年从大陆和香港去哥大读数学的唯一的学生……以下正文。

在网上找书的时候恰好看到这个，看着觉得的确是经典书目大全，贴在这里供学弟学妹们参考：）其中所谓第几学年云云，各校要求不同，像我所在的学校，一般学生第一年选三到四门基础课（代数、分析、几何三大类中至少各挑一门），学年末进行qualifying笔试。

第二年开始选自己喜爱方向的高级课程，并通过qualifying口试。

第三年开始做research，并通过第二语言考试（法语或德语或俄语，一般人都选法语，因为代数几何经典大作都是法语的）. 而Princeton就没有基础课，只有seminar类型的课……美国数学研究生基础课程参考书目第一学年秋季学期春季学期几何与拓扑I 几何与拓扑II1、James R. Munkres, Topology较新的拓扑学的教材适用于本科高年级或研究生一年级2、Basic Topology by Armstrong本科生拓扑学教材3、Kelley, General Topology一般拓扑学的经典教材，不过观点较老4、Willard, General Topology一般拓扑学新的经典教材5、Glen Bredon, Topology and geometry研究生一年级的拓扑、几何教材6、Introduction to Topological Manifolds by John M. Lee研究生一年级的拓扑、几何教材，是一本新书7、From calculus to cohomology by Madsen很好的本科生代数拓扑、微分流形教材代数I 代数II1、Abstract Algebra Dummit最好的本科代数学参考书，标准的研究生一年级代数教材2、Algebra Lang标准的研究生一、二年级代数教材，难度很高，适合作参考书3、Algebra Hungerford标准的研究生一年级代数教材，适合作参考书4、Algebra M,Artin标准的本科生代数教材5、Advanced Modern Algebra by Rotman较新的研究生代数教材，很全面6、Algebra：a graduate course by Isaacs较新的研究生代数教材7、Basic algebra V ol I&II by Jacobson经典的代数学全面参考书，适合研究生参考分析基础复分析I 实分析I1、Walter Rudin, Principles of mathematical analysis本科数学分析的标准参考书2、Walter Rudin, Real and complex analysis标准的研究生一年级分析教材3、Lars V. Ahlfors, Complex analysis本科高年级和研究生一年级经典的复分析教材4、Functions of One Complex V ariable I，J.B.Conway研究生级别的单变量复分析经典5、Lang, Complex analysis研究生级别的单变量复分析参考书6、Complex Analysis by Elias M. Stein较新的研究生级别的单变量复分析教材7、Lang, Real and Functional analysis研究生级别的分析参考书8、Royden, Real analysis标准的研究生一年级实分析教材9、Folland, Real analysis标准的研究生一年级实分析教材第二学年秋季学期春季学期代数III 代数IV1、Commutative ring theory, by H. Matsumura较新的研究生交换代数标准教材2、Commutative Algebra I&II by Oscar Zariski , Pierre Samuel 经典的交换代数参考书3、An introduction to Commutative Algebra by Atiyah标准的交换代数入门教材4、An introduction to homological algebra ,by weibel较新的研究生二年级同调代数教材5、A Course in Homological Algebra by P.J.Hilton,U.Stammbach经典全面的同调代数参考书6、Homological Algebra by Cartan经典的同调代数参考书7、Methods of Homological Algebra by Sergei I. Gelfand, Y uri I. Manin高级、经典的同调代数参考书8、Homology by Saunders Mac Lane经典的同调代数系统介绍9、Commutative Algebra with a view toward Algebraic Geometry by Eisenbud 高级的代数几何、交换代数的参考书，最新的交换代数全面参考代数拓扑I 代数拓扑II1、Algebraic Topology, A. Hatcher最新的研究生代数拓扑标准教材2、Spaniers "Algebraic Topology"经典的代数拓扑参考书3、Differential forms in algebraic topology, by Raoul Bott and Loring W. Tu 研究生代数拓扑标准教材4、Massey, A basic course in Algebraic topology经典的研究生代数拓扑教材5、Fulton , Algebraic topology：a first course很好本科生高年级和研究生一年级的代数拓扑参考书6、Glen Bredon, Topology and geometry标准的研究生代数拓扑教材，有相当篇幅讲述光滑流形7、Algebraic Topology Homology and Homotopy高级、经典的代数拓扑参考书8、A Concise Course in Algebraic Topology by J.P.May研究生代数拓扑的入门教材，覆盖范围较广9、Elements of Homotopy Theory by G.W. Whitehead高级、经典的代数拓扑参考书实分析II 泛函分析1、Royden, Real analysis标准研究生分析教材2、Walter Rudin, Real and complex analysis标准研究生分析教材3、Halmos，"Measure Theory"经典的研究生实分析教材，适合作参考书4、Walter Rudin, Functional analysis标准的研究生泛函分析教材5、Conway,A course of Functional analysis标准的研究生泛函分析教材6、Folland, Real analysis标准研究生实分析教材7、Functional Analysis by Lax高级的研究生泛函分析教材8、Functional Analysis by Y oshida高级的研究生泛函分析参考书9、Measure Theory, Donald L. Cohn经典的测度论参考书微分拓扑李群、李代数1、Hirsch, Differential topology标准的研究生微分拓扑教材，有相当难度2、Lang, Differential and Riemannian manifolds研究生微分流形的参考书，难度较高3、Warner,Foundations of Differentiable manifolds and Lie groups标准的研究生微分流形教材，有相当的篇幅讲述李群4、Representation theory: a first course, by W. Fulton and J. Harris李群及其表示论的标准教材5、Lie groups and algebraic groups, by A. L. Onishchik, E. B. V inberg李群的参考书6、Lectures on Lie Groups W.Y.Hsiang李群的参考书7、Introduction to Smooth Manifolds by John M. Lee较新的关于光滑流形的标准教材8、Lie Groups, Lie Algebras, and Their Representation by V.S. V aradarajan最重要的李群、李代数参考书9、Humphreys, Introduction to Lie Algebras and Representation Theory , Springer-V erlag, GTM-9标准的李代数入门教材第三学年秋季学期春季学期微分几何I 微分几何II1、Peter Petersen, Riemannian Geometry标准的黎曼几何教材2、Riemannian Manifolds: An Introduction to Curvature by John M. Lee最新的黎曼几何教材3、doCarmo, Riemannian Geometry.标准的黎曼几何教材4、M. Spivak, A Comprehensive Introduction to Differential Geometry I—V全面的微分几何经典，适合作参考书5、Helgason , Differential Geometry,Lie groups,and symmetric spaces标准的微分几何教材6、Lang, Fundamentals of Differential Geometry最新的微分几何教材，很适合作参考书7、kobayashi/nomizu, Foundations of Differential Geometry经典的微分几何参考书8、Boothby,Introduction to Differentiable manifolds and Riemannian Geometry标准的微分几何入门教材，主要讲述微分流形9、Riemannian Geometry I.Chavel经典的黎曼几何参考书10、Dubrovin, Fomenko, Novikov “Modern geometry-methods and applications”V ol 1—3经典的现代几何学参考书代数几何I 代数几何II1、Harris,Algebraic Geometry: a first course代数几何的入门教材2、Algebraic Geometry Robin Hartshorne经典的代数几何教材，难度很高3、Basic Algebraic Geometry 1&2 2nd ed. I.R.Shafarevich.非常好的代数几何入门教材4、Principles of Algebraic Geometry by giffiths/harris全面、经典的代数几何参考书，偏复代数几何5、Commutative Algebra with a view toward Algebraic Geometry by Eisenbud高级的代数几何、交换代数的参考书，最新的交换代数全面参考6、The Geometry of Schemes by Eisenbud很好的研究生代数几何入门教材7、The Red Book of V arieties and Schemes by Mumford标准的研究生代数几何入门教材8、Algebraic Geometry I : Complex Projective V arieties by David Mumford复代数几何的经典调和分析偏微分方程1、An Introduction to Harmonic Analysis,Third Edition Yitzhak Katznelson调和分析的标准教材，很经典2、Evans, Partial differential equations偏微分方程的经典教材3、Aleksei.A.Dezin，Partial differential equations，Springer-V erlag偏微分方程的参考书4、L. Hormander "Linear Partial Differential Operators, " I&II偏微分方程的经典参考书5、A Course in Abstract Harmonic Analysis by Folland高级的研究生调和分析教材6、Abstract Harmonic Analysis by Ross Hewitt抽象调和分析的经典参考书7、Harmonic Analysis by Elias M. Stein标准的研究生调和分析教材8、Elliptic Partial Differential Equations of Second Order by David Gilbarg 偏微分方程的经典参考书9、Partial Differential Equations ，by Jeffrey Rauch标准的研究生偏微分方程教材复分析II 多复分析导论1、Functions of One Complex V ariable II，J.B.Conway单复变的经典教材，第二卷较深入2、Lectures on Riemann Surfaces O.Forster黎曼曲面的参考书3、Compact riemann surfaces Jost黎曼曲面的参考书4、Compact riemann surfaces Narasimhan黎曼曲面的参考书5、Hormander " An introduction to Complex Analysis in Several V ariables" 多复变的标准入门教材6、Riemann surfaces , Lang黎曼曲面的参考书7、Riemann Surfaces by Hershel M. Farkas标准的研究生黎曼曲面教材8、Function Theory of Several Complex V ariables by Steven G. Krantz高级的研究生多复变参考书9、Complex Analysis: The Geometric V iewpoint by Steven G. Krantz高级的研究生复分析参考书专业方向选修课：1、多复分析2、复几何3、几何分析4、抽象调和分析5、代数几何6、代数数论7、微分几何8、代数群、李代数与量子群9、泛函分析与算子代数10、数学物理11、概率理论12、动力系统与遍历理论13、泛代数*数学基础：1、halmos ,native set theory2、fraenkel ,abstract set theory3、ebbinghaus ,mathematical logic4、enderton ,a mathematical introduction to logic5、landau, foundations of analysis6、maclane ,categories for working mathematican 应该在核心课程学习的过程中穿插选修假设本科应有的水平分析Walter Rudin, Principles of mathematical analysis Apostol , mathematical analysisM.spivak , calculus on manifoldsMunknes ,analysis on manifoldsKolmogorov/fomin , introductory real analysis Arnold ,ordinary differential equations代数：linear algebra by Stephen H. Friedberglinear algebra by hoffmanlinear algebra done right by Axleradvanced linear algebra by Romanalgebra ,artina first course in abstract algebra by rotman几何：do carmo, differential geometry of curves and surfaces Differential topology by PollackHilbert ,foundations of geometryJames R. Munkres, Topology。

数学软件mathematica讲义-简略版

表示双阶乘 n(n-2)(n-4) …
函数 ln x
Log[a,x]
表示以数 a 为底的对数函数 Binomial[n, m] log a x
表示二项式系数 C
m n
Sin[x]
表示正弦函数 sin x
Re[z]
取复数 z 的实部
Cos[x]
表示余弦函数 cos x
Im[z]
取复数 z 的虚部
Tan[x]
(2) 课堂练习 1、计算下列各式的数值，保留 10 位有效数字。
(1)、（2）、
（3）、（4）、
（5）、
（6）、
（7）、
（8）、
（9）、
（10）、
2、计算 861、1638、2415 的最大公约数。计算 48、105、120 的最小公倍数。
3、计算组合数、、。计算
、
、
。
4、对
和
分别计算
联的元素）放在一起组成的一个整体。在 Mathematica 中，任何用一对花括号括起来的一组元素都代一个表，其中的元素用逗号分隔且各元素可以具有不同的类型，特别其中的元素还可以是一个表。表的形式是： { 元素 1，元素 2，元素 3，……，元素 n }。建表命令有：命令形式 1: Table[ 通项公式 f(i)，{i ，imin，imax，h}] 功能：产生一个表{ f(imin) ，f(imin +h)，f(imin +2h)，……，f(imin +nh)} , imax – h ≤ imin + nh ≤ imax, h>0.注：省略 h，则默认 h=1. 命令形式 2: Table[ 通项公式 f，{循环次数 n}], f 为常数功能：产生 n 个 f 的一个表{ f ，f，f，……，f } 命令形式 3: Table[ 通项公式 f(i , j)，{{i ，imin，imax}, {j ，jmin，jmax}] 功能：产生一个二维表。例如：In[11]: = Table[i-j, {i, 1 , 6}, {j, 1 , 2}]

Mathematica讲义

数学与统计软件讲义§0 引论计算机与数学计算机是应数学计算要求而产生的，计算机技术水平又因计算能力的提高而提高。

数学计算的发展与计算机的发展是互相促进，交织在一起的。

§0.1数学对计算机发展的促进：§0.1.1巨大的计算需求刺激了计算机发展航海测量、大地测量对计算的需求（Pascal机械计算机）；美国人口调查；二战对计算的巨大需求：弹道计算；密码破译、英国防空、战争物资生产等；§0.1.2数学方法为计算机设计提供了理论基础门电路——布尔代数算法——程序数学模型——各种应用程序等§0.2计算机对数学发展的促进§0.2.1在经济、管理领域的大规模应用资源分配——数学规划、运筹学等经济预测——多元统计分析、时间序列分析等§0.2.2其它应用石油勘探；气象预报；汽车、飞机等工业产品设计；地震预报；核武器研制等。

§0.2.3对数学理论的促进机器证明——吴文俊的工作；（偏）微分方程求解——混沌问题的发现§0.3数学计算的类型§0.3.1形式演算对各种抽象的表达式进行变换，其结果仍然是一个抽象的表达式。

§0.3.2数值计算对指定变量的值，计算表达式的值，其结果是一个数值。

§0.4常用数学与统计软件在计算机发展初期，通常是针对某一具体的目的，特定的计算编制特定的程序进行计算。

随着计算的需求增加，人们为了提高效率，将大的计算过程分解一系列小的计算过程，针对这些小计算过程编制所谓的子程序。

这些子程序具有相当的通用性，即对选定的子程序进行适当的组合，就可以完成许多大的、特定的计算目的，从而形成许多庞大的科学计算程序库。

但是，调用这些子程序的接口较为复杂，而且还需要具有相当的程序语言知识才能把它们组合成完成特定目的的计算程序，使用比较麻烦。

随着计算机技术的发展，人们进一步把这些子程序集成为统一的软件包。

Mata手册说明书

Contents[M-0]Introduction to the Mata manualintro.................................................Introduction to the Mata manual[M-1]Introduction and adviceintro........................................................Introduction and advice ing Mata with ado-ﬁles ﬁrst....................................................Introduction andﬁrst session help........................................................Obtaining help in Stata how.............................................................How Mata works ing Mata interactively LAPACK.........................................The LAPACK linear-algebra routines limits..................................................Limits and memory utilization naming......................................Advice on naming functions and variables permutation................................An aside on permutation matrices and vectors returnedargs..................................Function arguments used to return results source.....................................................Viewing the source code e and speciﬁcation of tolerances[M-2]Language deﬁnitionnguage deﬁnition break..............................................Break out of for,while,or do loop class...........................................Object-oriented programming(classes) ments continue...........................Continue with next iteration of for,while,or do loop declarations...................................................Declarations and types do..............................................................do...while(exp) errors.................................................................Error codes exp..................................................................Expressions for......................................................for(exp1;exp2;exp3)stmt ftof...................................................Passing functions to functions goto...................................................................goto label if.............................................................if(exp)...else... op arith........................................................Arithmetic operators op assignment..................................................Assignment operator op colon...........................................................Colon operators op conditional..................................................Conditional operator op increment.......................................Increment and decrement operators op join..............................................Row-and column-join operators op kronecker........................................Kronecker direct-product operator op logical........................................................Logical operators op range..........................................................Range operators op transpose.............................................Conjugate transpose operator12Contents optargs.........................................................Optional arguments pointers..................................................................Pointers pragma...............................................Suppressing warning messages reswords...........................................................Reserved words return........................................................return and return(exp) e of semicolons struct..................................................................Structures e of subscripts syntax............................................Mata language grammar and syntax version............................................................Version control void................................................................V oid matrices while.............................................................while(exp)stmt[M-3]Commands for controlling Matamands for controlling Mata end...................................................Exit Mata and return to Stata mata.....................................................Mata invocation command mata clear.....................................................Clear Mata’s memory mata describe.....................................Describe contents of Mata’s memory mata drop...................................................Drop matrix or function mata help......................................................Obtain help in Stata mata matsave..............................................Save and restore matrices mata memory.........................................Report on Mata’s memory usage mata mlib....................................................Create function library mata mosave................................Save function’s compiled code in objectﬁle mata rename..............................................Rename matrix or function mata set......................................Set and display Mata system parameters mata stata...................................................Execute Stata command mata which........................................................Identify function namelists........................................Specifying matrix and function names [M-4]Index and guide to functions intro....................................................Index and guide to functions io...................................................................I/O functions manipulation....................................................Matrix manipulation mathematical.........................................Important mathematical functions matrix............................................................Matrix functions programming................................................Programming functions scalar..................................................Scalar mathematical functions solvers................................Functions to solve AX=B and to obtain A inverse standard..........................................Functions to create standard matrices stata........................................................Stata interface functions statistical........................................................Statistical functions string..................................................String manipulation functions utility.......................................................Matrix utility functions [M-5]Mata functions intro...............................................................Mata functionsContents3 abbrev().........................................................Abbreviate strings abs().......................................................Absolute value(length) adosubdir().........................................Determine ado-subdirectory forﬁle all()..........................................................Element comparisons args()........................................................Number of arguments asarray().........................................................Associative arrays ascii().....................................................Manipulate ASCII codes assert().....................................................Abort execution if false blockdiag()...................................................Block-diagonal matrix buﬁo()........................................................Buffered(binary)I/O byteorder().............................................Byte order used by computer C()...............................................................Make complex c()...............................................................Access c()value callersversion()........................................Obtain version number of caller cat()....................................................Loadﬁle into string matrix chdir().......................................................Manipulate directories cholesky().........................................Cholesky square-root decomposition cholinv()...................................Symmetric,positive-deﬁnite matrix inversion cholsolve()............................Solve AX=B for X using Cholesky decomposition comb()binatorial function cond()...........................................................Condition number conj()plex conjugate corr()....................................Make correlation matrix from variance matrix cross().............................................................Cross products crossdev()..................................................Deviation cross products cvpermute().................................................Obtain all permutations date()...................................................Date and time manipulation deriv()........................................................Numerical derivatives designmatrix().....................................................Design matrices det()........................................................Determinant of matrix diag().................................................Replace diagonal of a matrix diag().......................................................Create diagonal matrix diag0cnt()..................................................Count zeros on diagonal diagonal().........................................Extract diagonal into column vector dir()....................................................................File list direxists()..................................................Whether directory exists direxternal().....................................Obtain list of existing external globals display().............................................Display text interpreting SMCL displayas()........................................................Set display level displayﬂush()............................................Flush terminal-output buffer Dmatrix().......................................................Duplication matrix docx*().......................................Generate Ofﬁce Open XML(.docx)ﬁle dsign()............................................FORTRAN-like DSIGN()function e()...................................................................Unit vectors editmissing()..........................................Edit matrix for missing values edittoint()......................................Edit matrix for roundoff error(integers) edittozero().......................................Edit matrix for roundoff error(zeros) editvalue()............................................Edit(change)values in matrix eigensystem()............................................Eigenvectors and eigenvalues4Contentseigensystemselect()pute selected eigenvectors and eigenvalues eltype()..................................Element type and organizational type of object epsilon().......................................Unit roundoff error(machine precision) equilrc()..............................................Row and column equilibration error().........................................................Issue error message errprintf()..................................Format output and display as error message exit()..........................................................Terminate execution exp().................................................Exponentiation and logarithms factorial()..............................................Factorial and gamma function favorspeed()...................................Whether speed or space is to be favored ferrortext()......................................Text and return code ofﬁle error code fft().............................................................Fourier transform ﬁleexists().......................................................Whetherﬁle exists ﬁllmissing()..........................................Fill matrix with missing values ﬁndexternal().................................Find,create,and remove external globals ﬁndﬁle().................................................................Findﬁle ﬂoatround().................................................Round toﬂoat precision fmtwidth().........................................................Width of%fmt fopen()..................................................................File I/O fullsvd()............................................Full singular value decomposition geigensystem().................................Generalized eigenvectors and eigenvalues ghessenbergd()..................................Generalized Hessenberg decomposition ghk()...................Geweke–Hajivassiliou–Keane(GHK)multivariate normal simulator ghkfast().....................GHK multivariate normal simulator using pregenerated points gschurd()...........................................Generalized Schur decomposition halton().........................................Generate a Halton or Hammersley set hash1()...........................................Jenkins’one-at-a-time hash function hessenbergd()..............................................Hessenberg decomposition Hilbert()..........................................................Hilbert matrices I()................................................................Identity matrix inbase()...........................................................Base conversion indexnot()..................................................Find character not in list invorder()............................................Permutation vector manipulation invsym().............................................Symmetric real matrix inversion invtokens()...............................Concatenate string rowvector into string scalar isdiagonal()..............................................Whether matrix is diagonal isﬂeeting()...........................................Whether argument is temporary isreal()......................................................Storage type of matrix isrealvalues()..................................Whether matrix contains only real values issymmetric().................................Whether matrix is symmetric(Hermitian) isview()....................................................Whether matrix is view J().............................................................Matrix of constants Kmatrix()mutation matrix lapack()PACK linear-algebra functions liststruct()...................................................List structure’s contents Lmatrix().......................................................Elimination matrix logit()...........................................Log odds and complementary log-logContents5 lowertriangle().........................................Extract lower or upper triangle lud()...........................................................LU decomposition luinv()......................................................Square matrix inversion lusolve()...................................Solve AX=B for X using LU decomposition makesymmetric().............................Make square matrix symmetric(Hermitian) matexpsym().......................Exponentiation and logarithms of symmetric matrices matpowersym().........................................Powers of a symmetric matrix mean()............................................Means,variances,and correlations mindouble().................................Minimum and maximum nonmissing value minindex().......................................Indices of minimums and maximums minmax().................................................Minimums and maximums missing().......................................Count missing and nonmissing values missingof()................................................Appropriate missing value mod()..................................................................Modulus moptimize().....................................................Model optimization more().....................................................Create–more–condition negate().......................................................Negate real matrix norm()....................................................Matrix and vector norms normal()................................Cumulatives,reverse cumulatives,and densities optimize().....................................................Function optimization panelsetup()...................................................Panel-data processing pathjoin()....................................................File path manipulation pinv().................................................Moore–Penrose pseudoinverse polyeval()........................................Manipulate and evaluate polynomials printf().............................................................Format output qrd()...........................................................QR decomposition qrinv().............................Generalized inverse of matrix via QR decomposition qrsolve()...................................Solve AX=B for X using QR decomposition quadcross().............................................Quad-precision cross products range()..................................................Vector over speciﬁed range rank().............................................................Rank of matrix Re()..................................................Extract real or imaginary part reldif()..................................................Relative/absolute difference rows()........................................Number of rows and number of columns rowshape().........................................................Reshape matrix runiform()..............................Uniform and nonuniform pseudorandom variates runningsum().................................................Running sum of vector schurd()......................................................Schur decomposition select()..............................................Select rows,columns,or indices setbreakintr()..................................................Break-key processing sign()...........................................Sign and complex quadrant functions sin()..........................................Trigonometric and hyperbolic functions sizeof().........................................Number of bytes consumed by object solve tol().....................................Tolerance used by solvers and inverters solvelower()..........................................Solve AX=B for X,A triangular solvenl().........................................Solve systems of nonlinear equations sort().......................................................Reorder rows of matrix6Contentssoundex().............................................Convert string to soundex code spline3()..................................................Cubic spline interpolation sqrt().................................................................Square root st addobs()....................................Add observations to current Stata dataset st addvar().......................................Add variable to current Stata dataset st data()...........................................Load copy of current Stata dataset st dir()..................................................Obtain list of Stata objects st dropvar()...........................................Drop variables or observations st global()........................Obtain strings from and put strings into global macros st isfmt()......................................................Whether valid%fmt st isname()................................................Whether valid Stata name st local()..........................Obtain strings from and put strings into Stata macros st macroexpand().......................................Expand Stata macros in string st matrix().............................................Obtain and put Stata matrices st numscalar().......................Obtain values from and put values into Stata scalars st nvar()........................................Numbers of variables and observations st rclear().....................................................Clear r(),e(),or s() st store().................................Modify values stored in current Stata dataset st subview()..................................................Make view from view st tempname()...............................................Temporary Stata names st tsrevar().....................................Create time-series op.varname variables st updata()....................................Determine or set data-have-changedﬂag st varformat().................................Obtain/set format,etc.,of Stata variable st varindex()...............................Obtain variable indices from variable names st varname()...............................Obtain variable names from variable indices st varrename()................................................Rename Stata variable st vartype()............................................Storage type of Stata variable st view()..........................Make matrix that is a view onto current Stata dataset st viewvars().......................................Variables and observations of view st vlexists()e and manipulate value labels stata()......................................................Execute Stata command stataversion().............................................Version of Stata being used strdup()..........................................................String duplication strlen()...........................................................Length of string strmatch()....................................Determine whether string matches pattern strofreal().....................................................Convert real to string strpos().....................................................Find substring in string strreverse()..........................................................Reverse string strtoname()...........................................Convert a string to a Stata name strtoreal().....................................................Convert string to real strtrim()...........................................................Remove blanks strupper()......................................Convert string to uppercase(lowercase) subinstr()...........................................................Substitute text sublowertriangle()...........................Return a matrix with zeros above a diagonal substr()......................................................Substitute into string substr()...........................................................Extract substring sum().....................................................................Sums svd()..................................................Singular value decomposition svsolve()..........................Solve AX=B for X using singular value decomposition swap()..............................................Interchange contents of variablesContents7 Toeplitz().........................................................Toeplitz matrices tokenget()........................................................Advanced parsing tokens()..................................................Obtain tokens from string trace().......................................................Trace of square matrix transpose()..................................................Transposition in place transposeonly().......................................Transposition without conjugation trunc()............................................................Round to integeruniqrows()..............................................Obtain sorted,unique values unitcircle()plex vector containing unit circle unlink().................................................................Eraseﬁle valofexternal().........................................Obtain value of external global Vandermonde()................................................Vandermonde matrices vec().........................................................Stack matrix columns xl()............................................................Excelﬁle I/O class[M-6]Mata glossary of common terms Glossary........................................................................ Subject and author index...........................................................。

新GRE数学全部知识点汇总讲解

GRE数学解题大全目录GRE数学解题大全 (1)代数与几何部分 (2)概率论部分 (5)1.排列(permutation): (5)2．组合(combination): (5)3．概率 (5)统计学部分 (8)1.mode（众数） (8)2.range（值域） (8)3.mean（平均数） (8)4.median（中数） (8)5.standard error（标准偏差） (9)6.standard variation (9)7.standard deviation (9)8.the calculation of quartile(四分位数的计算) (9)9．The calculation of Percentile (10)10.To find median using Stem-and-Leaf (茎叶法计算中位数) (11)11．To find the median of data given by percentage(按比例求中位数) (12)12：比较,当n<1时，n，1，2 和1，2,3的标准方差谁大 (13)13.算数平均值和加权平均值 (13)14.正态分布题. (13)15．正态分布 (13)GRE数学符号与概念 (16)常用数学公式 (19)精讲20题 (20)GRE数学考试词汇分类汇总 (26)代数-数论 (26)代数-基本数学概念 (27)代数-基本运算, 小数,分数 (27)代数-方程,集合,数列等 (28)几何-三角 (29)几何-平面, 立体 (29)几何-图形概念 (30)几何-坐标 (31)商业术语,计量单位 (31)GRE数学考试词汇首字母查询 (32)代数与几何部分1.正整数n有奇数个因子,则n为完全平方数2.因子个数求解公式:将整数n分解为质因子乘积形式,然后将每个质因子的幂分别加一相乘.n=a*a*a*b*b*c则因子个数=(3+1)(2+1)(1+1)eg. 200=2*2*2 * 5*5 因子个数=(3+1)(2+1)=12个3.能被8整除的数后三位的和能被8整除;能被9整除的数各位数的和能被9整除.能被3整除的数,各位的和能被3整除.4.多边形内角和=(n-2)x1805.菱形面积=1/2 x 对角线乘积6.欧拉公式：边数=面数+顶点数-28.三角形余玄定理C2=A2+B2-2ABCOSβ,β为AB两条线间的夹角9.正弦定理:A/SinA=B/SinB=C/SinC=2R(A,B,C是各边及所对应的角,R是三角形外接圆的半径)10.Y=k1X+B1,Y=k2X+B2，两线垂直的条件为K1K2=-111.N的阶乘公式:N!=1*2*3*....(N-2)*(N-1)*N 且规定0!=1 1!=1Eg:8!=1*2*3*4*5*6*7*812. 熟悉一下根号2、3、5的值sqrt(2)=1.414 sqrt(3)=1.732 sqrt(5)=2.23613. ...2/3 as many A as B: A=2/3*B...twice as many... A as B: A=2*B14. 华氏温度与摄氏温度的换算换算公式:(F-32)*5/9=CPS.常用计量单位的换算:（自己查查牛津大字典的附录吧）练习题:1：还有数列题：a1=2，a2=6，a n=a n-1/a n-2，求a150.解答: a n=a n-1/a n-2,所以a n-1=a n-2/a n-3,带入前式得a n=1/a n-3,然后再拆一遍得到a n=a n-6,也就是说,这个数列是以6为周期的,则a150=a144=...=a6,利用a1,a2可以计算出a6=1/3.如果实在想不到这个方法,可以写几项看看很快就会发现a150=a144,大胆推测该数列是以6为周期得,然后写出a1-a13(也就是写到你能看出来规律),不难发现a6=a12,a7=a13,然后那,稍微数数,就可以知道a150=a6了,同样计算得1/3.2：问摄氏升高30度华氏升高的度数与62比大小.key:F=30*9/5=54<623：那道费波拉契数列的题:已知,a1=1 a2=1 a n=a n-1+a n-2，问a1,a2,a3,a6四项的平均数和a1,a3,a4,a5四项的平均数大小比较。

MATHEMATICS（PUREANDAPPLIED）：（纯粹数学和应用）

Professor & Head of DepartmentNT Bishop, MA(Cambridge), PhD(Southampton), FRASSenior LecturersJ Larena, MSc(Paris), PhD(Paris)D Pollney, PhD(Southampton)CC Remsing, MSc(Timisoara), PhD(Rhodes)Vacant LecturersEOD Andriantiana, PhD(Stellenbosch)V Naicker, MSc(KwaZulu-Natal)AL Pinchuck, MSc(Rhodes), PhD(Wits)Lecturer, Academic Development M Lubczonok, Masters(Jagiellonian)Mathematics (MA T) is a six-semester subject and Applied Mathematics (MAP) is a four-semester subject. These subjects may be taken as major subjects for the degrees of BSc, BA, BJourn, BCom, BBusSci, BEcon and BSocSc, and for the diploma HDE(SEC).To major in Mathematics, a candidate is required to obtain credit in the following courses: MAT1C; MAM2; MAT3. See Rule S.23.To major in Applied Mathematics, a candidate is required to obtain credit in the following courses: MAT1C, MAM2; MAP3. See Rule S.23.The attention of students who hope to pursue careers in the field of Bioinformatics is drawn to the recommended curriculum that leads to postgraduate study in this area, in which Mathematics is a recommended co-major with Biochemistry, and for which two years of Computer Science and either Mathematics or Mathematical Statistics are prerequisites. Details of this curriculum can be foundin the entry for the Department of Biochemistry, Microbiology and Biotechnology.See the Departmental Web Page http://www.ru.ac.za/departments/mathematics/ for further details, particularly on the content of courses.First-year level courses in MathematicsMathematics 1 (MAT1C) is given as a year-long semesterized two-credit course. Credit in MAT1C must be obtained by students who wish to major in certain subjects (such as Applied Mathematics, MATHEMATICS (PURE AND APPLIED)Physics and Mathematical Statistics) and by students registered for the BBusSci degree.Introductory Mathematics (MAT1S) is recommended for Pharmacy students and for Science students who do not need MAT1C or MAT1C1.Supplementary examinations may be recommended for any of these courses, provided that a candidate achieves a minimum standard specified by the Department.Mathematics 1L (MA T1L) is a full year course for students who do not qualify for entry into any of the first courses mentioned above. This is particularly suitable for students in the Social Sciences and Biological Sciences who need to become numerate or achieve a level of mathematical literacy. A successful pass in this course will give admission to MA T1C.First yearMAT1CThere are two first-year courses in Mathematics for candidates planning to major in Mathematics or Applied Mathematics. MAT1C1 is held in thefirst semester and MAT1C2 in the second semester. Credit may be obtained in each course separately and, in addition, an aggregate mark of at least 50%will be deemed to be equivalent to a two-creditcourse MAT1C, provided that a candidate obtains the required sub-minimum (40%) in each component. Supplementary examinations may be recommended in either course, provided that a candidate achieves a minimum standard specified by the department. Candidates obtaining less than 40% for MAT1C1 are not permitted to continue with MAT1C2.MAT1C1 (First semester course): Basic concepts (number systems, functions), calculus (limits,continuity, differentiation, optimisation, curvesketching, introduction to integration), propositional calculus, mathematical induction, permutations, combinations, binomial theorem, vectors, lines andplanes, matrices and systems of linear equations.MAT1C2 (Second semester course): Calculus (integration, applications of integration, improper integrals), complex numbers, differential equations, partial differentiation, sequences and series.MAT1S (Semester course: Introductory Mathematics) (about 65 lectures)Estimation, ratios, scales (log scales), change of units, measurements; Vectors, systems of equations, matrices, in 2-dimensions; Functions: Review of coordinate geometry, absolute values (including graphs); Inequalities; Power functions, trig functions, exponential functions, the number e (including graphs); Inverse functions: roots, logs, ln (including graphs); Graphs and working with graphs; Interpretation of graphs, modeling; Descriptive statistics (mean, standard deviation, variance) with examples including normally distributed data; Introduction to differentiation and basic derivatives; Differentiation techniques (product, quotient and chain rules); Introduction to integration and basic integrals; Modeling, translation of real-world problems into mathematics.MAT 1L: Mathematics Literacy This course helps students develop appropriatemathematical tools necessary to represent and interpret information quantitatively. It also develops skills and meaningful ways of thinking, reasoning and arguing with quantitative ideas in order to solve problems in any given context.Arithmetic: Units of scientific measurement, scales, dimensions; Error and uncertainty in measure values.Fractions and percentages - usages in basic science and commerce; use of calculators and spreadsheets. Algebra: Polynomial, exponential, logarithmic and trigonometric functions and their graphs; modelling with functions; fitting curves to data; setting up and solving equations. Sequences and series, presentation of statistical data.Differential Calculus: Limits and continuity; Rules of differentiation; Applications of Calculus in curvesketching and optimisation.Second Year Mathematics 2 comprises two semesterized courses,MAM201 and MAM202, each comprising of 65 lectures. Credit may be obtained in each course seperately. An aggregate mark of 50% will grant the two-credit course MAM2, provided a sub-minimumof 40% is achieved in both semesters. Each semester consists of a primary and secondary stream which are run concurrently at 3 and 2 lectures per week, respectively. Additionally, a problem-based course in Mathematical Programming contributes to the class record and runs throughout the academic year.MAM201 (First semester):Advanced Calculus (39 lectures): Partial differentiation: directional derivatives and the gradient vector; maxima and minima of surfaces; Lagrange multipliers. Multiple integrals: surface and volume integrals in general coordinate systems. Vector calculus: vector fields, line integrals, fundamental theorem of line integrals, Green’s theorem, curl and divergence, parametric curves and surfaces.Ordinary Differential Equations (20 lectures): First order ordinary differential equations, linear differential equations of second order, Laplace transforms, systems of equations, series solutions, Green’s functions.Mathematical Programming 1 (6 lectures): Introduction to the MATLAB language, basic syntax, tools, programming principles. Applicationstaken from MAM2 modules. Course runs over twosemesters.MAM202 (Second semester):Linear Algebra (39 lectures): Linear spaces, inner products, norms. Vector spaces, spans, linear independence, basis and dimension. Linear transformations, change of basis, eigenvalues, diagonalization and its applications.Groups and Geometry (20 lectures): Number theory and counting. Groups, permutation groups, homomorphisms, symmetry groups in 2 and 3 dimensions. The Euclidean plane, transformations and isometries. Complex numbers, roots of unity and introduction to the geometry of the complex plane.Mathematical Programming 2 (6 lectures): Problem-based continuation of Semester 1.Third-year level courses inMathematics and Applied Mathematics Mathematics and Applied Mathematics are offered at the third year level. Each consists of four modules as listed below. Code TopicSemester Subject AM3.1 Numerical analysis 1 Applied MathematicsAM3.2 Dynamical systems 2 Applied Mathematics AM3.4 Partial differentialequations 1 Applied Mathematics AM3.5 Advanced differentialequations 2 Applied MathematicsM3.1 Algebra 2 MathematicsM3.2 Complex analysis 1 MathematicsM3.3 Real analysis 1 MathematicsM3.4 Differential geometry 2 Mathematics Students who obtain at least 40% in all of the above modules will be granted credit for both MAT3 and MAP3, provided that the average of the Applied Mathematics modules is at least 50% AND the average of the Mathematics modules is at least 50%. Students who obtain at least 40% for any FOUR of the above modules and with an average mark over the four modules of at least 50%, will be granted credit in either MAT3 or MAP3. If three or four of the modules are from Applied Mathematics then the credit will be in MAP3, otherwise it will be in MAT3.Module credits may be carried forward from year to year.Changes to the modules offered may be made from time-to-time depending on the interests of the academic staff.Credit for MAM 2 is required before admission to the third year courses.M3.1 (about 39 lectures) AlgebraAlgebra is one of the main areas of mathematics with a rich history. Algebraic structures pervade all modern mathematics. This course introduces students to the algebraic structure of groups, rings and fields. Algebra is a required course for any further study in mathematics.Syllabus: Sets, equivalence relations, groups, rings, fields, integral domains, homorphisms, isomorphisms, and their elementary properties.M3.2 (about 39 lectures) Complex Analysis Building on the first year introduction to complex numbers, this course provides a rigorous introduction to the theory of functions of a complex variable. It introduces and examines complex-valued functions of a complex variable, such as notions of elementary functions, their limits, derivatives and integrals. Syllabus: Revision of complex numbers, Cauchy- Riemann equations, analytic and harmonic functions, elementary functions and their properties, branches of logarithmic functions, complex differentiation, integration in the complex plane, Cauchy’s Theorem and integral formula, Taylor and Laurent series, Residue theory and applications. Fourier Integrals.M3.3 (about 39 lectures) Real AnalysisReal Analysis is the field of mathematics that studies properties of real numbers and functions on them. The course places great emphasis on careful reasoning and proof. This course is an essential basis for any further study in mathematics.Syllabus: Topology of the real line, continuity and uniform continuity, Heine-Borel, Bolzano-Weierstrass, uniform convergence, introduction to metric spaces.M3.4 (about 39 lectures) Differential Geometry Roughly speaking, differential geometry is concerned with understanding shapes and their properties in terms of calculus. This elementary course on differential geometry provides a perfect transition to higher mathematics and its applications. It is a subject which allows students to see mathematics for what it is - a unified whole mixing together geometry, calculus, linear algebra, differential equations, complex variables, calculus of variations and topology.Syllabus: Curves (in the plane and in the space), curvature, global properties of curves, surfaces, the first fundamental form, isometries, the second fundamental form, the normal and principal curvatures, the Gaussian and mean curvatures, the Gauss map, geodesics.AM3.1 (about 39 lectures) Numerical Analysis Many mathematical problems cannot be solved exactly and require numerical techniques. These techniques usually consist of an algorithm which performs a numerical calculation iteratively until certain tolerances are met. These algorithms can be expressed as a program which is executed by a computer. The collection of such techniques is called “numerical analysis”.Syllabus: Systems of non-linear equations, polynomial interpolation, cubic splines, numerical linear algebra, numerical computation of eigenvalues, numerical differentiation and integration, numerical solution of ordinary and partial differential equations, finite differences,, approximation theory, discrete Fourier transform.AM3.2 (about 39 lectures) Dynamical Systems This module is about the dynamical aspects of ordinary differential equations and the relations between dynamical systems and certain fields of applied mathematics (like control theory and the Lagrangian and Hamiltonian formalisms of classical mechanics). The emphasis is on the mathematical aspects of various constructions and structures rather than on the specific physical/mechanical models. Syllabus Linear systems; Linear control systems; Nonlinear systems (local theory); Nonlinear control systems; Nonlinear systems (global theory); Applications : elements of optimal control and/or geometric mechanics.AM3.4 (about 39 lectures) Partial Differential EquationsThis course deals with the basic theory of partial differential equations (elliptic, parabolic and hyperbolic) and dynamical systems. It presents both the qualitative properties of solutions of partial differential equations and methods of solution. Syllabus: First-order partial equations, classification of second-order equations, derivation of the classical equations of mathematical physics (wave equation, Laplace equation, and heat equation), method of characteristics, construction and behaviour of solutions, maximum principles, energy integrals. Fourier and Laplace transforms, introduction to dynamical systems.AM3.5 (about 39 lectures)Advanced differential equationsThis course is an introduction to the study of nonlinearity and chaos. Many natural phenomena can be modeled as nonlinear ordinary differential equations, the majority of which are impossible to solve analytically. Examples of nonlinear behaviour are drawn from across the sciences including physics, biology and engineering.Syllabus:Integrability theory and qualitative techniques for deducing underlying behaviour such as phase plane analysis, linearisations and pertubations. The study of flows, bifurcations, the Poincare-Bendixson theorem, and the Lorenz equations.Mathematics and Applied Mathematics Honours Each of the two courses consists of either eight topics and one project or six topics and two projects.A Mathematics Honours course usually requires the candidate to have majored in Mathematics, whilst Applied Mathematics Honours usually requires the candidate to have majored in Applied Mathematics. The topics are selected from the following general areas covering a wide spectrum of contemporary Mathematics and Applied Mathematics: Algebra; Combinatorics; Complex Analysis; Cosmology; Functional Analysis; General Relativity; Geometric Control Theory; Geometry; Logic and Set Theory; Measure Theory; Number Theory; Numerical Modelling; Topology.Two or three topics from those offered at the third-year level in either Mathematics or Applied Mathematics may also be taken in the case of a student who has not done such topics before. With the approval of the Heads of Department concerned, the course may also contain topics from Education, and from those offered by other departments in the Science Faculty such as Physics, Computer Science, and Statistics. On the other hand, the topics above may also be considered by such Departments as possible components of their postgraduate courses.Master’s and Doctoral degrees in Mathematics or Applied MathematicsSuitably qualified students are encouraged to proceed to these degrees under the direction of the staff of the Department. Requirements for these degrees are given in the General Rules.A Master’s degree in either Mathematics or Applied Mathematics may be taken by thesis only, or by a combination of course work and a thesis. Normally four examination papers and/or essays are required apart from the thesis. The whole course of study must be approved by the Head of Department.。

学习Mathematica资料集合

学习Mathematica 资料集合19 Answersactiveoldestvotesup vote438 down vote accepted+100Here's a collection of resources that I started on Mathgroup (a collection of Mathematica learning resources) and updated here at Stack Overflow. As this site is dedicated to Mathematica it makes more sense to maintain it here. This represents a huge amount of information; of course it's not exhaustive so feel free to improve it! Also, don't hesitate to share it and suggest other interesting links! Remember, you can always search the online Documentation Center of Mathematica, that is identical to the built-in help of the latest software version.Links to more advanced aspects of the program that you can start to appreciate once you understand the basics are provided in separate answers (below) as this post became too large.Tips and TricksAdvanced evaluation, patterns and neat algorithmsIntroductionIf you're just beginning try to have a look at these videos.Mathematica Basics, Elementary Programming in MathematicaHands-on Start to MathematicaSeveral introductory videos by Jon McLooneand many other video introductions and tutorials from the official Wolfram websiteAn elementary introduction to the Wolfram languageFast introduction for programmersIs it necessary to have a prior computational background or is it possible to learn Mathematica as a first programming language? What are the most common pitfalls awaiting new users?How To-s: full solutions for particular tasks from the onlinedocumentationEasy-to-understand animations explaining common Mathematica functionsSal Mangano's videos for using pure functions, Part and patterns Introductory videos of various applications of Mathematica What is the best Mathematica tutorial for young people?Basic advices for people new to MathematicaFunctional styleAvoid iterative programming using loops like For or Do, use instead functional programming functions Map, Scan, MapThread, Fold, FoldList, ... and pure functions. This makes the code cleaner and faster.Functional Programming, Functional Programming: Quick Start Pure functionsWhat does # mean in Mathematica?Alternatives to procedural loops and iterating over lists in MathematicaAn example: Programming a numerical method in the functionalstyleHow to understand the usage of Inner and Outer figuratively?Transpose and dimensionsSomething not easy to guess alone at the beginning: if you havex={1,2} and y={3,4},doing Transpose[{x,y}] or {x,y}ESC tr ESC in the front end will produce {{1,3},{2,4}} (format compatible with ListPlot).This animation helps understand why.You can also use the second argument of Transpose to reorder the indices of a multidimensional list.Don't forget to regularly control the output of the lists you generate using Dimensions.Get familiar with shorthand syntax (@, &, ##, /@, /., etc.)Operator Input Formswhen is f@g not the same as f[g]?Programming easily Getting help: Execute ?Map for example for a short description of a function, or press F1 on a function name for more details and examples about it. You can solve many problems by adapting examples to your needs.Auto-completion: Start typing the name of a function and (inMathematica 9+) select from the pop-up auto-completion menu, or press Ctrl+k to get a list of functions which names start with what has already been entered. Once the name of the function is written completely press Ctrl+Shift+k (on Mac, Cmd+k) to get a list of its arguments.Function templates: In Mathematica 9, after typing a function name, press Ctrl+Shift+k (on Mac, Cmd+Shift+k) and click on the desired form from the pop-up menu to insert a template with named placeholders for the arguments.Other useful shortcuts are described in the post Using the Mathematica front-end efficiently for editing notebooks. Use palettes in the Palettes menu especially when you're beginning.In Mathematica 8, use the natural input capability of Wolfram Alpha, for example type "= graph 2 x + 1 between 0 and 3" without thequotes and see the command associated with the result.TutorialsAn elementary introduction to the Wolfram language, byStephen WolframFast introduction for programmersFundamentals of Mathematica Programming (by Richard Gaylord, great tutorial for an overview of the logic behind Mathematica: patterns)Video tutorial also availableIntroduction to Mathematica (by Thomas Hahn, another succinct overview of Mathematica)Tutorial Collection by WRI (lots of extra documentation and examples, available as free PDFs, also available and up-to-date in Help > Virtual Book in Mathematica).Programming Paradigms via Mathematica (A First Course) Mathematica Tutorial: A New Resource for Developers Wolfram's Mathematica 101http://bmia.bmt.tue.nl/Software/Downloads/Campus/TrainingM athematicaEnglish.ziphttp://bmia.bmt.tue.nl/Software/Mathematica/Tutorials/index.ht ml A problem centered approachA beginner's guide to Mathematica/MathematiClub/tutorials.htmhttp://www.austi /mmcguff/mathematica//courses/hnichols/phys303//courses/ap1601y/ (Introduction to Computational Mathematics and Physics).au/pub/MATH2200/2012/Lectures/ (Applied Mathematics).au/pub/MATH2200/2009/Lectures (path for some lectures in pdf) /wiki/Mathematica /homes/ayg/CS590C/www/mathemati ca/math.html (Basic tutorial)https:///questions/4430998/mathematica-what -is-symbolic-programming (What is symbolic programming) http://www.cer.ethz.ch/resec/people/tsteger/Econ_Model_Math_ 1.pdf/enp/jjkelly (An introduction to Mathematica as well as some physics courses)Do you know of any web-based university course that is entirely Mathematica based?http://homepage.cem.itesm.mx/jose.luis.gomez/data/mathematic a (Tutorials in Spanish)Mathematica programming (some examples of the various programming paradigms that can be used in Mathematica)FAQ/my_notes/faq/mma_notes/MMA.htm (FAQ) https:///questions/tagged/mathematica?sort=faq&pagesize=15 (FAQ on Stack Overflow)https:///questions?sort=faq (FAQ on this site)/conferences/conference98/Lichtblau/ SymbolicFAQ.nb (Symbolic FAQ)BooksStephen Wolfram's The Mathematica Book (online, version 5.2), available for freeMathematica programming: an advanced introduction (online) by Leonid Shifrin, available for freeTutorial Collection by WRI (lots of extra documentation and examples, available as free pdfs, also available and up-to-date in Help > Virtual Book in Mathematica).Mathematica Cookbook by Sal Mangano (O'Reilly, 2010) Mathematica in Action by Stan Wagon (Springer, 2010) Mathematica: AProblem-Centered Approach by Roozbeh Hazrat (Springer, 2010)Mathematica Navigator by Heikki Ruskeepaa (Academic Press, 2009) The Mathematica GuideBooks (for Programming, Numerics, Graphics, Symbolics) by Michael Trott (Springer, 2004-2005) An introduction to programming with Mathematica by Paul R. Wellin, Richard J. Gaylord and Samuel N. Kamin (Cambridge University Press, 2005); contains an example of Domain Specific Language (DSL) creation.Mastering Mathematica by John W. Gray (Academic Press, 1997) Programming in Mathematica by Roman Mae。

艾特亚（Mata）教程与参考手册说明书

Title Intro—Introduction to the Mata manualContents Description Remarks and examples ReferencesAlso seeContentsSection Description[M-1]Introduction and advice[M-2]Language deﬁnition[M-3]Commands for controlling Mata[M-4]Categorical guide to Mata functions[M-5]Alphabetical index to Mata functions[M-6]Mata glossary of common termsDescriptionMata is a matrix programming language that can be used by those who want to perform matrix calculations interactively and by those who want to add new features to Stata.The Mata Reference Manual is comprehensive.If it seems overly comprehensive and too short on explanation as to why things work the way they do and how they could be used,we have a suggestion.See The Mata Book by William Gould(428pages)or An Introduction to Stata Programming by Christopher Baum(412pages).Baum’s book assumes that you are familiar with Stata but new to programming.Gould’s book assumes that you have some familiarity with programming and goes on from there.The books go well together.Remarks and examples This manual is divided into six sections.Each section is organized alphabetically,but there is an introduction in front that will help you get around.If you are new to Mata,here is a helpful reading list.Start by reading[M-1]First Introduction andﬁrst session[M-1]Interactive Using Mata interactively[M-1]How How Mata works12Intro—Introduction to the Mata manualYou mayﬁnd other things in section[M-1]that interest you.For a table of contents,see[M-1]Intro Introduction and adviceWhenever you see a term that you are unfamiliar with,see[M-6]Glossary Mata glossary of common termsNow that you know the basics,if you are interested,you can look deeper into Mata’s programming features:[M-2]Syntax Mata language grammar and syntax[M-2]Syntax is pretty dense reading,but it summarizes nearly everything.The other entries in[M-2] repeat what is said there but with more explanation;see[M-2]Intro Language deﬁnitionbecause other entries in[M-2]will interest you.If you are interested in object-oriented programming, be sure to see[M-2]class.Along the way,you will eventually be guided to sections[M-4]and[M-5].[M-5]documents Mata’s functions;the alphabetical order makes it easy toﬁnd a function if you know its name but makes learning what functions there are hopeless.That is the purpose of[M-4]—to present the functions in logical order.See[M-4]Intro Categorical guide to Mata functionsMathematical[M-4]Matrix Matrix functions[M-4]Solvers Matrix solvers and inverters[M-4]Scalar Scalar functions[M-4]Statistical Statistical functions[M-4]Mathematical Other important functionsUtility and manipulation[M-4]Standard Functions to create standard matrices[M-4]Utility Matrix utility functions[M-4]Manipulation Matrix manipulation functionsStata interface[M-4]Stata Stata interface functionsDate and time[M-4]Dates Date and time functionsString,I/O,and programming[M-4]String String manipulation functions[M-4]IO I/O functions[M-4]Programming Programming functionsIntro—Introduction to the Mata manual3 ReferencesBaum,C.F.2016.An Introduction to Stata Programming.2nd ed.College Station,TX:Stata Press.Gould,W.W.2018.The Mata Book:A Book for Serious Programmers and Those Who Want to Be.College Station, TX:Stata Press.Also see[M-1]First—Introduction andﬁrst session[M-6]GlossaryStata,Stata Press,and Mata are registered trademarks of StataCorp LLC.Stata andStata Press are registered trademarks with the World Intellectual Property Organization®of the United Nations.Other brand and product names are registered trademarks ortrademarks of their respective companies.Copyright c 1985–2023StataCorp LLC,College Station,TX,USA.All rights reserved.。

Mathematica软件说明书

Model More Sophisticated SystemsAs the size of engineering systems increase linearly, the size of the equations that describe those engineering systems increases exponentially.A key example is the modeling of multiple degree of freedom (DOF) robotic systems. As the number of joints increases, the transformation matrices requiredto describe joint motion exponentially increase in size. At some point, equation manipulation by hand is impractical; software support is hence needed.A corollary is that computer algebra systems can be used to model more sophisticated engineering systems than is possible by hand.Computationally Faster than Numeric Computation Numerical computation refers to the iterative solution of equations using software; this is computationally time-consuming.In many cases, computer algebra systems can be used to rearrange equations to an explicit formulation; this eliminates the need for time-consumingiterative approaches.Preserve Information about Model StructureBy delaying numeric evaluation until only strictly necessary, computer algebra systems preserve information about model structure and parameter relationships. This information can be used for code generation, parameter-based optimization, model simplification and more.How Do Engineers Use Computer Algebra Systems?IntroductionWe will now, through an exploratory application-based approach, illustrate how computer algebra systems are typically used by engineers.Maple, a math tool with a hybrid symbolic-numeric math engine, is used to illustrate each example. Maple’s broader calculation management features are used to document each example with natural math notation, images and text.Transformation Matrices for a Multi-DOF RobotThe modeling of robotic arm manipulators involves the derivation of transformation matrices; these matrices grow in size as the degrees of freedom increase. Deriving these matrices by hand would take hours, with a high probability of introducing errors. However, computer algebra systems will derive transformation matrices for an arbitrarily complex robot in a matter of seconds, while eliminating the risk of error.Figure 2 demonstrates how Maple can be used to generate the Denevit & Hartenberg matrices, which assist in determining the coordinate transformations between joints in a robotic manipulator.Figure 1.Maple provides a complete calculation management environment in which engineers can record and reuse the underlying reasoning, data and derivationsInverse KinematicsInverse kinematics involves finding the joint parametersto move a robot arm to a desired position. Figure 3demonstrates how the position constraints of a doublependulum can be rearranged to give the joint angles.Code Translation of Joint Angles to CThis equation in Figure 3 is now converted to C code.Computer algebra systems can identify and factor outcommon subexpressions; this makes the code far morenumerically efficient than hand-written code.Figure 2. Denavit and Hartenberg formulation forrobotic manipulatorsFigure 3. Rearranging inverse kinematics equationsFigure 5. Photovoltaic diode equationHowever, the equation can be rearranged using special functions. These functions are normally only encountered in advanced mathematical analysis; a few examples are LambertW, Fresnel, Bessel, and Appell. Special functions are increasingly implemented in computer algebra tools, allowing engineers to use these functions without any specialized training. This isRearranging Heat Balance EquationsThe thermal efficiency of a thermodynamic cycle is a function of the heat and mass flows around a system. Deriving the heat and mass balances require a lgebraic manipulation.Consider the Rankine cycle with two-stage regeneration in Figure 11.A heat balance on the two pre-heaters H1and H 2 gives these equations,where X 1and X 2 are the mass fractions of the working fluid extracted in the high and low pressure turbines, and h n is the specific enthalpy at pointsn = 1 .. 6.Figure 12 illustrates how Maple is employed to rearrange these equations to give X 1and X 2. If two states at points 1-6 are known (e.g. temperature and pressure) then• e nthalpy values can be determined,• a nd X 1 and X 2 can then be calculated.Integrating an Empirical Equation for Heat Capacity The specific heat capacity of many chemicals is often described by empirical polynomials in temperature.These polynomials can be integrated (as illustrated in Figure 13) to give an expression that can be used to calculate changes in enthalpy.Beam Deflection with Distributed and Point LoadStudents of structural and civil engineering encounter the Euler-Bernoulli beam deflection equation early in their education:This equation connects the deflection of the beam w(x) to the applied load q(x). With the appropriate initial and boundary conditions, the equation can be solved to give explicit expressions describing the deflection.Figure 11. Rankine Cycle with Two-Stage RegenerationFigure 12. Heat Balance on Preheaters and Symbolic Manipulation of EquationsFigure 13. Integrating Empirical Equation for Specific Heat CapacityFigure 14. Beam with distributed and point loadFor simple load cases, such as a simply supported beam with a distributed load, the Euler-Bernoulli equation can be solved manually.However, more complex load cases would take significant amounts of time to solve by hand. For example, consider the simply supported beam in Figure 14; the beam has a uniform load (across part of the beam) and a point load. Mathematically, the distributed load can be described by a Heaviside step function, and the point load described by a Dirac function. q(x) hence becomesFigure 15 describes how Maple is used to solve the Euler-Bernouilli equation with this loading.Closed Loop Transfer FunctionConsider the closed loop control system in Figure 16.Typically, an engineer may want to calculate the closedloop transfer function for such a system.Figure 17 illustrates how the closed loop transferfunction is derived using Maple. The steps, whilemathematically straightforward, are tedious to do byhand for non-trivial systems.Controllability Matrix of a DC MotorIn Figure 18, we extract the symbolic controllabilitymatrix of a DC motor described by differential equations.Figure 15. Solving the Euler-Bernoulli Beam Bending Equationwith a Distributed and Point LoadFigure 16. Closed Loop Control SystemFigure 17. Closed Loop Transfer Function for a Control LoopT erminal Settling VelocityConsider a spherical particle falling in a fluid. Figure 19 gives the equations for the drag force and the buoyancy force; the terminal settling velocity is reached when both are equal.The equations are rearranged to give an explicit expression for the settling velocity.Balancing Chemical EquationsMonomethylhydrazine (CH 6N 2) and Dinitrogen T etroxide (N 2O 4) are typically used in rocket propulsion as a fuel and oxidizer. In determining the theoretical rocket performance, the adiabatic flame temperature of the combustion products needs to be calculated; this partly involves balancing the carbon, hydrogen, oxygen and nitrogen atoms in the feed and combustion products.Assuming the combustion products contain CO , HNO ,H 2O , NO 2, O , CO 2, HO 2, H 2O 2, N 2, OH , H , H 2, NO , N 2O and O 2, the overall balance equation for the combustion of CH 6N 2 and N 2O 4 can be written thus.a CH 6N 2 +b N 2O 4 = n 1 CO + n 2 HNO + n 3 H 2O + n 4 NO 2 + n 5 O + n 6 CO 2 + n 7 HO 2 +n 8 H 2O 2 + n 9 N 2 + n 10 OH + n 11 H + n 12 H 2 + n 13 NO + n 14 N 2O n 15 O 2Generating the individual atom balances from the overall balance is painstaking because of the sheer number of chemical species. As illustrated in Figure 20, the process can be mechanized with computer algebra.Figure 18. Symbolic Controllability MatrixFigure 19. Terminal Settling Velocity of a Settling ParticleFigure 20.Balancing the Combustion Reaction of Monomethylhydrazine (CH 6N 2) and Dinitrogen Tetroxide (N 2O 4)Combustion can produce many more species than used in this example; generating the atom balanceequations by hand would be laborious, but is easy with computer algebra.Battery Modeling and Model ReductionElectrochemical battery models derived from porous electrode theory are described by partial differentialequations. While being physically accurate, these models are computationally intensive to simulation.Several techniques are used to simplify these models while retaining physical accuracy. These techniques include collocation and Galerkin’s method; bothtransform non-linear PDEs into a set of ODEs and are described elsewhere (Dao et al, 2012).ConclusionThe 1950s and 1960s saw the birth of the first computer algebra systems. Due to the skills and training of their creators, these innovative tools were first designed for the needs of mathematicians and physicists.Initially, a few forward-thinking engineers exploited symbolic math for advanced research applications. The benefits, however, remained out of reach for the vast majority of engineers.This started to change in the early eighties with the advent of cheap computing power. The next 30 years also saw the evolution of the human-centered design principles that radically improved the usability of computer algebra systems. | info@T oll-free: (US & Canada) 1-800-267-6583 | Direct:1-519-747-2373© Maplesoft, a division of Waterloo Maple Inc., 2017. Maplesoft and Maple are trademarks of Waterloo Maple Inc. All other trademarks are the property of their respective owners.Moreover, a maturing feature set, including tools formanaging calculations as well as doing calculations, made integrating mechanized algebra into the engineering design process much simplerComputer algebra systems thus gradually entered the mainstream consciousness of engineers, and have grown in popularity year-on-year. Moreover, the applications discussed in this white paper clearly demonstrate the benefits of mechanized algebra across the entire breadth of engineering.References[1] Simplification and order reduction of lithium-ionbattery model based on porous-electrode theory, Thanh-Son Dao et al., Journal of Power Sources, Volume 198, 15 January 2012, Pages 329-337。

高等数学英文版教材推荐

高等数学英文版教材推荐Higher Mathematics: Recommended English Edition TextbooksIntroductionHigher Mathematics, also known as advanced calculus, is a fundamental course for students pursuing degrees in science, engineering, and mathematics. It covers various topics that develop students' analytical, problem-solving, and critical thinking skills. In this article, we will recommend several English edition textbooks for studying higher mathematics.1. "Calculus: Early Transcendentals" by James Stewart"Calculus: Early Transcendentals" by James Stewart is a widely used textbook that provides thorough coverage of calculus topics. It offers a comprehensive approach to differential and integral calculus, including concepts such as limits, derivatives, and integrals. The book incorporates real-world applications and examples, allowing students to understand the practical significance of mathematical concepts. With clear explanations and a wealth of exercises, this textbook is suitable for both beginners and advanced learners.2. "Advanced Engineering Mathematics" by Erwin Kreyszig"Advanced Engineering Mathematics" by Erwin Kreyszig is a comprehensive textbook that covers various topics in higher mathematics. It encompasses differential equations, vector calculus, complex analysis, and numerical methods, among others. The book emphasizes the application ofmathematical principles to engineering problems, making it an excellent resource for engineering students. Its extensive collection of examples, exercises, and solutions aids in developing problem-solving skills.3. "Mathematical Methods for Physics and Engineering" by K.F. Riley, M.P. Hobson, and S.J. Bence"Mathematical Methods for Physics and Engineering" by K.F. Riley, M.P. Hobson, and S.J. Bence is a renowned textbook that combines mathematical theory with practical applications in physics and engineering. It covers topics like vector analysis, complex variables, Fourier series, and partial differential equations. The book explains mathematical concepts in a clear and concise manner, with numerous illustrations and worked examples to enhance understanding. It is an invaluable resource for students studying mathematical methods in the context of physical sciences and engineering.4. "Linear Algebra and Its Applications" by David C. Lay"Linear Algebra and Its Applications" by David C. Lay focuses on the study of vector spaces and linear transformations. The book presents fundamental concepts such as matrix algebra, determinants, eigenvalues, and eigenvectors. It demonstrates the applications of linear algebra in diverse fields, including computer science, economics, and physics. The text's accessible language, logical organization, and abundant exercises contribute to a comprehensive understanding of the subject.5. "Complex Variables and Applications" by James Ward Brown and Ruel V. Churchill"Complex Variables and Applications" by James Ward Brown and Ruel V. Churchill introduces complex analysis, which is a vital branch of mathematics for engineering, physics, and applied sciences. The book covers topics like complex functions, contour integration, and series expansions, along with their applications. It emphasizes the connections between complex analysis and real-world problems, promoting a deeper understanding of both the theory and its practical relevance.ConclusionThese recommended English edition textbooks provide comprehensive coverage of various topics in higher mathematics. Whether you are a beginner or an advanced learner, these textbooks offer clear explanations, practical examples, and extensive exercises to enhance your understanding and problem-solving skills. Choose the textbook that aligns with your course requirements and learning style to excel in the study of higher mathematics.。

高等数学英文版引入教材

高等数学英文版引入教材IntroductionThe Importance of Introducing Advanced Mathematics in English TextbooksIn recent years, with the increasing globalization and internationalization of education, the need for English language skills has become more prominent. This extends to various academic disciplines, including mathematics. As a result, there has been a growing demand for English versions of textbooks, particularly in the field of advanced mathematics. This article aims to explore the significance and benefits of introducing an English version of a high-level mathematics textbook.Enhancing English Language ProficiencyThe introduction of an English version of a high-level mathematics textbook can greatly contribute to the improvement of students' English language proficiency. By engaging with mathematical concepts and theories in English, students are exposed to a wide range of mathematical vocabulary and terminology, as well as the grammatical structures used in mathematical contexts. This exposure enables students to strengthen their English language skills, particularly in reading and comprehending academic texts, thereby equipping them with valuable language tools for future academic pursuits.Facilitating International Academic ExchangeMany students aspire to pursue higher education abroad, where English is often the predominant language of instruction. By familiarizing students with advanced mathematical concepts in English from an early stage, the English version of a high-level mathematics textbook can help prepare students for their future academic endeavors. It removes the language barrier that students may face when studying mathematics in a foreign language, enabling them to seamlessly integrate into international academic environments and participate in academic exchanges with students from diverse linguistic backgrounds.Accessing Global Resources and PerspectivesThe availability of an English version of a high-level mathematics textbook opens up a vast repertoire of global resources and perspectives to students. It allows them to expand their knowledge by studying mathematical theories and practices from different cultural and academic contexts. Students can access research papers, articles, and online course materials in English, which offer diverse insights and approaches to problem-solving. By incorporating these global perspectives into their mathematical studies, students can develop a broader understanding of the subject and its applications in a global context.Enhancing Cross-Disciplinary SkillsMathematics is an interdisciplinary subject that intersects with various academic disciplines. By introducing an English version of a high-level mathematics textbook, students not only gain proficiency in mathematics but also develop cross-disciplinary skills. The use of English as a medium of instruction allows students to integrate mathematical concepts with othersubjects, such as physics, economics, and engineering, which are often taught in English. This integration fosters a holistic understanding of how mathematics intersects with other disciplines, enabling students to apply their mathematical knowledge in real-world scenarios.Preparing for International AssessmentsIn an increasingly competitive academic environment, many students participate in international assessments such as the International Baccalaureate (IB) or the Advanced Placement (AP) exams. These exams often require a solid understanding of advanced mathematical concepts and a high level of English language proficiency. By utilizing an English version of a high-level mathematics textbook, students can simultaneously develop their mathematical and English language skills, thereby enhancing their readiness for these international assessments. It provides an opportunity for students to excel in both academic areas, increasing their chances of gaining admission to prestigious universities.ConclusionThe introduction of an English version of a high-level mathematics textbook is a valuable educational resource that offers numerous advantages to students. It enhances their English language proficiency, facilitates international academic exchange, exposes them to global resources and perspectives, promotes cross-disciplinary skills, and prepares them for international assessments. By embracing this expansion of the academic curriculum, educational institutions can better equip students to thrive in an increasingly interconnected and multilingual world.。

《数学专业英语》课件

The course provides learners with feedback on their performance and guidance on areas where improvement is needed to help them achieve their learning goals effectively
Basic mathematical terms
This includes words like "equation," "function," "integral," etc., which are essential for understanding mathematical concepts and e combinations of symbols that represent relationships between mathematical entities They are used to express mathematical ideas conceptually and precisely
感谢观看
THANKS
Application of Mathematics Professional English
Use English to write research articles for international journals or conferences, enabling more people to learn about and cite your article.
Summary and Outlook
01
Deepen the understanding of mathematical English vocabulary and grammar

Mathematica教程(中科大教案)

学习必备欢迎下载绪论0.1 符号计算系统简介# 数值计算与符号计算1946 年世界上第一台计算机ENIAC (The Electronic Numerical Integrator and Computer) 是为数值积分服务的。

一提起计算机求解人们立刻想到的是数值求解，这是因为计算机的早期应用范围主要是数值求解。

其实数值求解是计算机求解的一个方面，计算机进行计算的另一方面即对数学表示式的处理已形成一门新的科学分支，称为符号计算或计算机代数，它是一门研究使用计算机进行数学公式推导的理论和方法，演算数学公式的理论和算法是它研究的中心课题。

数值计算：常量、变量、函数、运算符--〉数值、字符、逻辑量表达式€一个值多€一近似计算例：计算y=sin10+ln10。

其结果是1.75856。

在高级语言中，算术表达式由常量、变量、函数和运算符等组成，算术表达式的值为某一精度范围内的数值。

计算各类表达式的值是高级语言的主要工作。

符号计算(计算机代数):常量、变量值、函数值--〉数值、字符、逻辑量表达式€表达式多€多准确计算x 2 sin xdx ＝-(-2 + x 2 )cos x + 2 x sin x与数值计算相比，符号计算对计算机硬件和软件提出了更高的要求。

# 符号计算系统符号计算系统是一个表示数学知识和数学工具的系统，一个集成化的计算机数学软件系统。

# 数值计算、# 符号计算、# 图形演示# 程序设计公式推导、数值计算和图形可视化操作一致性和连贯性。

符号计算系统的对象从初等数学到高等数学，几乎涉及所有数学学科。

包括各种数学表达式的化简、多项式的四则运算、求最大公因式、因式分解(factor)、常微分方程和偏微分方程的解函数。

各种特殊函数的推导、函数的级数展开、矩阵和行列式的各种运算和线性方程组的符号解等。

和数值计算一样，算法也是符号计算的核心。

就算法而言，符号计算比数值计算能继承更多的更丰富的数学遗产，古典数学家许多算法仍然是核心算法的成员，近代数学的算法成果也在不断地充实到符号计算中。

1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。

Lecture Notes for A Mathematical Introduction to Robotic Manipulation
By Z.X. Li∗ and Y.Q. Wu
Dept. of ECE, Hong Kong University of Science & Technology School of ME, Shanghai Jiaotong University
Chapter 4 Robot Dynamics and Control
4.2 Intertial Properties of Rigid Body ◻ Kinetic energy of a rigid body:
Chapter Robot Dynamics and Control Lagrangian Equations Inertial Properties of Rigid Body Dynamics of an Open-chain Manipulator Newton-Euler Equations Coordinateinvariant algorithms for robot dynamics Lagrange’s Equations with Constraints
July
Chapter 4 Robot Dynamics and Control
Summer School-Math. Methods in Robotics@TU-BS.DE
-
July
Chapter Robot Dynamics and Control Lagrangian Equations Inertial Properties of Rigid Body Dynamics of an Open-chain Manipulator Newton-Euler Equations Coordinateinvariant algorithms for robot dynamics Lagrange’s Equations with Constraints
⇔
=
Py = m˙ y d Fx ,pter 4 Robot Dynamics and Control
4.1 Lagrangian Equations ◻ Generalization to multibody systems:
Chapter Robot Dynamics and Control Lagrangian Equations Inertial Properties of Rigid Body Dynamics of an Open-chain Manipulator Newton-Euler Equations Coordinateinvariant algorithms for robot dynamics Lagrange’s Equations with Constraints
y x θ
Chapter 4 Robot Dynamics and Control
4.1 Lagrangian Equations Example: Dynamics of a Spherical Pendulum
Chapter Robot Dynamics and Control Lagrangian Equations Inertial Properties of Rigid Body Dynamics of an Open-chain Manipulator Newton-Euler Equations Coordinateinvariant algorithms for robot dynamics Lagrange’s Equations with Constraints
4.2 Intertial Properties of Rigid Body
In A-frame
Chapter Robot Dynamics and Control Lagrangian Equations Inertial Properties of Rigid Body Dynamics of an Open-chain Manipulator Newton-Euler Equations Coordinateinvariant algorithms for robot dynamics Lagrange’s Equations with Constraints
⎡ l sin θ cos ϕ ⎢ r(θ, ϕ) = ⎢ l sin θ sin ϕ ⎢ −l cos θ ⎣ T= m ˙ r V = −mgl cos θ
˙ ˙ = ml (θ + ( − cos θ)ϕ )
⎤ ⎥ ⎥ ⎥ ⎦
θ
˙ ˙ ˙ L(q, q) = ml (θ + ( − cosθ )ϕ ) + mgl cos θ ϕ mg
Chapter Robot Dynamics and Control
Lagrangian Equations Inertial Properties of Rigid Body Dynamics of an Open-chain Manipulator Newton-Euler Equations Coordinate-invariant algorithms for robot dynamics Lagrange’s Equations with Constraints
Potential energy: mg V = mgl( − cos θ) Lagrangian function: ∂L ˙ ˙ ∂L = −mgl sin θ L = T − V = ml θ − mgl( − cos θ), ⇒ = ml θ, ˙ ∂θ ∂θ Equation of motion: d ∂L ∂L ¨ − = τ ⇒ ml θ + mgl sin θ = τ dt ˙ ∂θ
T=
∫
V
˙ ˙ ρ(r) p + Rr dV = ˙ ˙ + pT R
∫
˙ = m p
∫
V
ρ(r)rdV +
V
˙ ρ(r)( p
˙ ˙ ˙ + pT Rr + Rr )dV dV
˙ ∫ ρ(r) Rr
=
∫
y m q
m q m q
qi , i = , . . . , n: generalized coordinates Kinetic energy: ˙ T = T(q, q) Potential energy: V = V(q) Lagrangian: ˙ ˙ L(q, q) = T(q, q) − V(q) τi , i = , . . . , n: external force on qi Lagrangian Equation:
Chapter 4 Robot Dynamics and Control
4.1 Lagrangian Equations
Chapter Robot Dynamics and Control Lagrangian Equations Inertial Properties of Rigid Body Dynamics of an Open-chain Manipulator Newton-Euler Equations Coordinateinvariant algorithms for robot dynamics Lagrange’s Equations with Constraints
Generalized coordinate: θ∈S Kinematics: x = l sin θ, y = −l cos θ ˙ ˙ ˙ ˙ ⇒ x = l cos θ ⋅ θ, y = l sin θ ⋅ θ Kinetic energy: ˙ ˙ T(θ, θ) = m(˙ + y ) = ml θ x ˙
Chapter 4 Robot Dynamics and Control
Summer School-Math. Methods in Robotics@TU-BS.DE
-
July
Chapter Robot Dynamics and Control Lagrangian Equations Inertial Properties of Rigid Body Dynamics of an Open-chain Manipulator Newton-Euler Equations Coordinateinvariant algorithms for robot dynamics Lagrange’s Equations with Constraints ∗
r ra A gab
B
Volume occupied by the body: Mass density: Mass: Mass center: Relative to frame at the mass center:
V ρ(r) m=
r≜ m r=
∫
V
ρ(r)dV
V
∫
ρ(r)rdV
Chapter 4 Robot Dynamics and Control
x
d ∂L ∂L − = τi , i = , . . . , n dt ∂˙ i ∂qi q
Chapter 4 Robot Dynamics and Control
4.1 Lagrangian Equations Example: Pendulum equation
Chapter Robot Dynamics and Control Lagrangian Equations Inertial Properties of Rigid Body Dynamics of an Open-chain Manipulator Newton-Euler Equations Coordinateinvariant algorithms for robot dynamics Lagrange’s Equations with Constraints