Non-genericity of infinitesimal variations of Hodge structures arising in some geometric co

A Classical InequalitiesTheorem 1.(AM-GM inequality )Let a 1,···,a n be positive real numbers.Then,we havea 1+···+a n n≥n √a 1···a n .Theorem 2.(Weighted AM-GM inequality )Let λ1,···,λn real positive numbers with λ1+···+λn =1.For all x 1,···,x n >0,we haveλ1·x 1+···+λn ·x n ≥x 1λ1···x n λn .Theorem 3.(GM-HM inequality )Let a 1,···,a n be positive real numbers.Then,we haven √a 1···a n ≥n 1a 1+1a 2+···+1a nTheorem 4.(QM-AM inequality )Let a 1,···,a n be positive real numbers.Then,we have a 1+a 2+···+a 2n n ≥a 1+···+a n nTheorem 5.(Power Mean inequality )Let x 1,···,x n >0.The power mean of order p is defined byM 0(x 1,x 2,...,x n )=n √x 1···x n ,M p (x 1,x 2,...,x n )= x p1+···+x n p n 1p (p =0).Then the function M p (x 1,x 2,...,x n ):R →R is continuous and monotone increasing.Theorem 6.(Rearrangement inequality )Let x 1≥···≥x n and y 1≥···≥y n be real numbers.For any permutation σof {1,...,n },we haveni =1x i y i ≥n i =1x i y σ(i )≥n i =1x i y n +1−i .149Theorem7.(The Cauchy3-Schwarz4-Bunyakovsky5inequality)Let a1,···,a n,b1,···,b n be real numbers.Then,(a12+···+a n2)(b12+···+b n2)≥(a1b1+···+a n b n)2. Remark.This inequality apparently wasfirstly mentioned in a work of A.L. Cauchy in1821.The integral form was obtained in1859by V.Y.Bunyakovsky. The corresponding version for inner-product spaces obtained by H.A.Schwartz in1885is mainly known as Schwarz’s inequality.In light of the clear historical precedence of Bunyakovsky’s work over that of Schwartz,the common practice of referring to this inequality as CS-inequality may seem unfair.Nevertheless in a lot of modern books the inequality is named CSB-inequality so that both Bunyakovsky and Schwartz appear in the name of this fundamental inequality.By setting a i=x i√y i and b i=√y i the CSB inequality takes the following formTheorem8.(Cauchy’s inequality in Engel’s form) Let x1,···,x n,y1,···,y n be positive real numbers.Then,x21 y1+x22y2+···+x2ny n≥(x1+x2+···+x n)2y1+y2+···+y nTheorem9.(Chebyshev’s inequality6)Let x1≥···≥x n and y1≥···≥y n be real numbers.We havex1y1+···+x n y nn ≥x1+···+x nny1+···+y nn.Theorem10.(H¨o lder’s inequality7)Let x1,···,x n,y1,···,y n be positive real numbers.Suppose that p>1andq>1satisfy1p +1q=1.Then,we haveni=1x i y i≤ni=1x i p1p ni=1y i q1q3Louis Augustin Cauchy(1789-1857),french mathematician4Hermann Amandus Schwarz(1843-1921),german mathematician5Viktor Yakovlevich Bunyakovsky(1804-1889),russian mathematician 6Pafnuty Lvovich Chebyshev(1821-1894),russian mathematician.7Otto Ludwig H¨o lder(1859-1937),german mathematician150Theorem11.(Minkowski’s inequality8) If x1,···,x n,y1,···,y n>0and p>1,thenni=1x i p1p+ni=1y i p1p≥ni=1(x i+y i)p1pDefinition1.(Convex functions.)We say that a function f(x)is convex on a segment[a,b]if for all x1,x2∈[a,b]fx1+x22≤f(x1)+f(x2)2Theorem12.(Jensen’s inequality9)Let n≥2andλ1,...,λn be nonnegative real numbers such thatλ1+···+λn=1. If f(x)is convex on[a,b]thenf(λ1x1+···+λn x n)≤λ1f(x1)+···+λn x nfor all x1,...,x n∈[a,b].Definition2.(Majorization relation forfinite sequences)Let a=(a1,a2,...,a n)and b=(b1,b2,...,b n)be two(finite)sequences of real numbers such that a1≥a2≥···≥a n and b1≥b2≥···≥b n.We say that the sequence a majorizes the sequence b and we writea b or b≺aif the following two conditions are satisfyied(i)a1+a2+···+a k≥b1+b2+···+b k,for all k,1≤k≤n−1;(ii)a1+a2+···+a n=b1+b2+···+b n.Theorem13.(Majorization inequality|Karamata’s inequality10)Let f:[a,b]−→R be a convex function.Suppose that(x1,···,x n)majorizes (y1,···,y n),where x1,···,x n,y1,···,y n∈[a,b].Then,we obtainf(x1)+···+f(x n)≥f(y1)+···+f(y n).8Hermann Minkowski(1864-1909),german mathematician.9Johan Ludwig William Valdemar Jensen(1859-1925),danish mathematician.10Jovan Karamata(1902-2967),serbian mathematician.151Theorem14.(Muirhead’s inequality11|Bunching Principle)If a=(a1,a2,...,a n)and b=(b1,b2,...,b n)are two nonincreasing sequences of nonnegative real numbers such that a majorizes b,then we havesym x a11···x a n n≥symx b11···x b n nwhere the sums are taken over all n!permutations of variables x1,x2,...,x n. Theorem15.(Schur’s inequality12)Let x,y,z be nonnegative real numbers.For any r>0,we havecycx r(x−y)(x−z)≥0.Remark.The case r=1of Schur’s inequality issymx3−2x2y+xyz≥0By espanding both the sides and rearranging terms,each of following inequalities is equivalent to the r=1case of Schur’s inequality•x3+y3+z3+3xyz≥xy(x+y)+yz(y+z)+zx(z+x)•xyx≥(x+y−z)(y+z−x)(z+x−y)•(x+y+z)3+9xyz≥4(x+y+z)(xy+yz+zx)Theorem16.(Bernoulli’s inequality13)For all r≥1and x≥−1,we have(1+x)r≥1+rx.Definition3.(Symmetric Means)For given arbitrary real numbers x1,···,x n,the coefficient of t n−i in the poly-nomial(t+x1)···(t+x n)is called the i-th elementary symmetric functionσi. This means that(t+x1)···(t+x n)=σ0t n+σ1t n−1+···+σn−1t+σn.11Robert Muirhead(1860-1941),english matematician.12Issai Schur(1875-1941),was Jewish a mathematician who worked in Germany for most of his life.He considered himself German rather than Jewish,even though he had been born in the Russian Empire in what is now Belarus,and brought up partly in Latvia.13Jacob Bernouilli(1654-1705),swiss mathematician founded this inequality in1689.How-ever the same result was exploited in1670by the english mathematician Isaac Barrow.152For i∈{0,1,···,n},the i-th elementary symmetric mean S i is defined byS i=σi ni .Theorem17.(Newton’s inequality14)Let x1,...,x n>0.For i∈{1,···,n},we haveS2i≥S i−1·S i+1 Theorem18.(Maclaurin’s inequality15) Let x1,...,x n>0.For i∈{1,···,n},we haveS1≥S2≥3S3≥···≥nS n14Sir Isaac Newton(1643-1727),was the greatest English mathematician of his generation. He laid the foundation for differential and integral calculus.His work on optics and gravitation make him one of the greatest scientists the world has known.15Colin Maclaurin(1698-1746),Scottish mathematican.153。

banach不动点定理英文Banach Fixed-Point Theorem:The Banach Fixed-Point Theorem, also known as the Contractive Mapping Theorem, is a fundamental result in functional analysis and mathematics. It states the conditions under which a mapping (function) from a metric space to itself will have a unique fixed point. The theorem is named after the Polish mathematician Stefan Banach, who played a significant role in its development.Statement of the Theorem:Let (X,d) be a complete metric space, and T:X→X be a contraction mapping on X. In other words, there exists a constant0≤k<1 such that for all x,y in X, the following inequality holds:d(T(x),T(y))≤k⋅d(x,y)Then, the mapping T has a unique fixed point x∗in X, meaning T(x ∗)=x∗.Key Concepts:•Metric Space (X, d): A set X along with a distance function d that satisfies the properties of non-negativity,symmetry, and the triangle inequality.•Complete Metric Space: A metric space in which every Cauchy sequence converges to a limit that is also within thespace.•Contraction Mapping: A mapping �:�→�T:X→X for which there exists a constant �<1k<1 such that the distancebetween the images of any two points under the mapping isalways reduced by a factor of �k.•Fixed Point: A point �∗x∗in the space �X such that �(�∗)=�∗T(x∗)=x∗.The Banach Fixed-Point Theorem is widely used in various branches of mathematics, including analysis, topology, and applied mathematics, and it has important applications in proving the existence and uniqueness of solutions to certain types of equations and problems.。

Daniel D.LeeBell Laboratories Lucent Technologies Murray Hill,NJ07974H.Sebastian SeungDept.of Brain and Cog.Sci.Massachusetts Institute of TechnologyCambridge,MA02138 AbstractNon-negative matrix factorization(NMF)has previously been shown tobe a useful decomposition for multivariate data.Two different multi-plicative algorithms for NMF are analyzed.They differ only slightly inthe multiplicative factor used in the update rules.One algorithm can beshown to minimize the conventional least squares error while the otherminimizes the generalized Kullback-Leibler divergence.The monotonicconvergence of both algorithms can be proven using an auxiliary func-tion analogous to that used for proving convergence of the Expectation-Maximization algorithm.The algorithms can also be interpreted as diag-onally rescaled gradient descent,where the rescaling factor is optimallychosen to ensure convergence.1IntroductionUnsupervised learning algorithms such as principal components analysis and vector quan-tization can be understood as factorizing a data matrix subject to different constraints.De-pending upon the constraints utilized,the resulting factors can be shown to have very dif-ferent representational properties.Principal components analysis enforces only a weak or-thogonality constraint,resulting in a very distributed representation that uses cancellations to generate variability[1,2].On the other hand,vector quantization uses a hard winner-take-all constraint that results in clustering the data into mutually exclusive prototypes[3]. We have previously shown that nonnegativity is a useful constraint for matrix factorization that can learn a parts representation of the data[4,5].The nonnegative basis vectors that are learned are used in distributed,yet still sparse combinations to generate expressiveness in the reconstructions[6,7].In this submission,we analyze in detail two numerical algorithms for learning the optimal nonnegative factors from data.2Non-negative matrix factorizationWe formally consider algorithms for solving the following problem:Non-negative matrix factorization(NMF)Given a non-negative matrix,find non-negative matrix factors and such that:(1)NMF can be applied to the statistical analysis of multivariate data in the following manner. Given a set of of multivariate-dimensional data vectors,the vectors are placed in the columns of an matrix where is the number of examples in the data set.This matrix is then approximately factorized into an matrix and an matrix. Usually is chosen to be smaller than or,so that and are smaller than the original matrix.This results in a compressed version of the original data matrix.What is the significance of the approximation in Eq.(1)?It can be rewritten column by column as,where and are the corresponding columns of and.In other words,each data vector is approximated by a linear combination of the columns of, weighted by the components of.Therefore can be regarded as containing a basis that is optimized for the linear approximation of the data in.Since relatively few basis vectors are used to represent many data vectors,good approximation can only be achieved if the basis vectors discover structure that is latent in the data.The present submission is not about applications of NMF,but focuses instead on the tech-nical aspects offinding non-negative matrix factorizations.Of course,other types of ma-trix factorizations have been extensively studied in numerical linear algebra,but the non-negativity constraint makes much of this previous work inapplicable to the present case [8].Here we discuss two algorithms for NMF based on iterative updates of and.Because these algorithms are easy to implement and their convergence properties are guaranteed, we have found them very useful in practical applications.Other algorithms may possibly be more efficient in overall computation time,but are more difficult to implement and may not generalize to different cost functions.Algorithms similar to ours where only one of the factors is adapted have previously been used for the deconvolution of emission tomography and astronomical images[9,10,11,12].At each iteration of our algorithms,the new value of or is found by multiplying the current value by some factor that depends on the quality of the approximation in Eq.(1).We prove that the quality of the approximation improves monotonically with the application of these multiplicative update rules.In practice,this means that repeated iteration of the update rules is guaranteed to converge to a locally optimal matrix factorization.3Cost functionsTofind an approximate factorization,wefirst need to define cost functions that quantify the quality of the approximation.Such a cost function can be constructed using some measure of distance between two non-negative matrices and.One useful measure is simply the square of the Euclidean distance between and[13],(2)This is lower bounded by zero,and clearly vanishes if and only if.Another useful measure isWe now consider two alternative formulations of NMF as optimization problems: Problem1Minimize with respect to and,subject to the constraints .Problem2Minimize with respect to and,subject to the constraints .Although the functions and are convex in only or only,they are not convex in both variables together.Therefore it is unrealistic to expect an algorithm to solve Problems1and2in the sense offinding global minima.However,there are many techniques from numerical optimization that can be applied tofind local minima. Gradient descent is perhaps the simplest technique to implement,but convergence can be slow.Other methods such as conjugate gradient have faster convergence,at least in the vicinity of local minima,but are more complicated to implement than gradient descent [8].The convergence of gradient based methods also have the disadvantage of being very sensitive to the choice of step size,which can be very inconvenient for large applications.4Multiplicative update rulesWe have found that the following“multiplicative update rules”are a good compromise between speed and ease of implementation for solving Problems1and2.Theorem1The Euclidean distance is nonincreasing under the update rules(4)The Euclidean distance is invariant under these updates if and only if and are at a stationary point of the distance.Theorem2The divergence is nonincreasing under the update rules(5)The divergence is invariant under these updates if and only if and are at a stationary point of the divergence.Proofs of these theorems are given in a later section.For now,we note that each update consists of multiplication by a factor.In particular,it is straightforward to see that this multiplicative factor is unity when,so that perfect reconstruction is necessarily afixed point of the update rules.5Multiplicative versus additive update rulesIt is useful to contrast these multiplicative updates with those arising from gradient descent [14].In particular,a simple additive update for that reduces the squared distance can be written as(6) If are all set equal to some small positive number,this is equivalent to conventional gradient descent.As long as this number is sufficiently small,the update should reduce .Now if we diagonally rescale the variables and set(8) Again,if the are small and positive,this update should reduce.If we now setminFigure1:Minimizing the auxiliary function guarantees that for.Lemma2If is the diagonal matrix(13) then(15) Proof:Since is obvious,we need only show that.To do this,we compare(22)(23)is a positive eigenvector of with unity eigenvalue,and application of the Frobenius-Perron theorem shows that Eq.17holds.We can now demonstrate the convergence of Theorem1:Proof of Theorem1Replacing in Eq.(11)by Eq.(14)results in the update rule:(24) Since Eq.(14)is an auxiliary function,is nonincreasing under this update rule,accordingto Lemma1.Writing the components of this equation explicitly,we obtain(28)Proof:It is straightforward to verify that.To show that, we use convexity of the log function to derive the inequality(30) we obtain(31) From this inequality it follows that.Theorem2then follows from the application of Lemma1:Proof of Theorem2:The minimum of with respect to is determined by setting the gradient to zero:7DiscussionWe have shown that application of the update rules in Eqs.(4)and(5)are guaranteed to find at least locally optimal solutions of Problems1and2,respectively.The convergence proofs rely upon defining an appropriate auxiliary function.We are currently working to generalize these theorems to more complex constraints.The update rules themselves are extremely easy to implement computationally,and will hopefully be utilized by others for a wide variety of applications.We acknowledge the support of Bell Laboratories.We would also like to thank Carlos Brody,Ken Clarkson,Corinna Cortes,Roland Freund,Linda Kaufman,Yann Le Cun,Sam Roweis,Larry Saul,and Margaret Wright for helpful discussions.References[1]Jolliffe,IT(1986).Principal Component Analysis.New York:Springer-Verlag.[2]Turk,M&Pentland,A(1991).Eigenfaces for recognition.J.Cogn.Neurosci.3,71–86.[3]Gersho,A&Gray,RM(1992).Vector Quantization and Signal Compression.Kluwer Acad.Press.[4]Lee,DD&Seung,HS.Unsupervised learning by convex and conic coding(1997).Proceedingsof the Conference on Neural Information Processing Systems9,515–521.[5]Lee,DD&Seung,HS(1999).Learning the parts of objects by non-negative matrix factoriza-tion.Nature401,788–791.[6]Field,DJ(1994).What is the goal of sensory coding?Neural Comput.6,559–601.[7]Foldiak,P&Young,M(1995).Sparse coding in the primate cortex.The Handbook of BrainTheory and Neural Networks,895–898.(MIT Press,Cambridge,MA).[8]Press,WH,Teukolsky,SA,Vetterling,WT&Flannery,BP(1993).Numerical recipes:the artof scientific computing.(Cambridge University Press,Cambridge,England).[9]Shepp,LA&Vardi,Y(1982).Maximum likelihood reconstruction for emission tomography.IEEE Trans.MI-2,113–122.[10]Richardson,WH(1972).Bayesian-based iterative method of image restoration.J.Opt.Soc.Am.62,55–59.[11]Lucy,LB(1974).An iterative technique for the rectification of observed distributions.Astron.J.74,745–754.[12]Bouman,CA&Sauer,K(1996).A unified approach to statistical tomography using coordinatedescent optimization.IEEE Trans.Image Proc.5,480–492.[13]Paatero,P&Tapper,U(1997).Least squares formulation of robust non-negative factor analy-b.37,23–35.[14]Kivinen,J&Warmuth,M(1997).Additive versus exponentiated gradient updates for linearprediction.Journal of Information and Computation132,1–64.[15]Dempster,AP,Laird,NM&Rubin,DB(1977).Maximum likelihood from incomplete data viathe EM algorithm.J.Royal Stat.Soc.39,1–38.[16]Saul,L&Pereira,F(1997).Aggregate and mixed-order Markov models for statistical languageprocessing.In C.Cardie and R.Weischedel(eds).Proceedings of the Second Conference on Empirical Methods in Natural Language Processing,81–89.ACL Press.。

极限思想外文翻译pdfBSHM Bulletin, 2014Did Weierstrass’s differential calculus have a limit-avoiding character? His,,,definition of a limit in styleMICHIYO NAKANENihon University Research Institute of Science & Technology, Japan In the 1820s, Cauchy founded his calculus on his original limit concept and,,,developed his the-ory by using inequalities, but he did not apply theseinequalities consistently to all parts of his theory. In contrast, Weierstrass consistently developed his 1861 lectures on differential calculus in terms of epsilonics. His lectures were not based on Cauchy’s limit and are distin-guished bytheir limit-avoiding character. Dugac’s partial publication of the 1861 lecturesmakes these differences clear. But in the unpublished portions of the lectures,,,,Weierstrass actu-ally defined his limit in terms ofinequalities. Weierstrass’slimit was a prototype of the modern limit but did not serve as a foundation of his calculus theory. For this reason, he did not providethe basic structure for the modern e d style analysis. Thus it was Dini’s 1878 text-book that introduced the,,,definition of a limit in terms of inequalities.IntroductionAugustin Louis Cauchy and Karl Weierstrass were two of the most important mathematicians associated with the formalization of analysis on the basis of the e d doctrine. In the 1820s, Cauchy was the first to give comprehensive statements of mathematical analysis that were based from the outset on a reasonably clear definition of the limit concept (Edwards 1979, 310). He introduced various definitions and theories that involved his limit concept. His expressions were mainly verbal, but they could be understood in terms of inequalities: given an e, find n or d (Grabiner 1981, 7). As we show later, Cauchy actually paraphrased his limit concept in terms of e, d, and n0 inequalities, in his more complicated proofs. But it was Weierstrass’s 1861 lectures which used the technique in all proofs and also in his defi-nition (Lutzen? 2003, 185–186).Weierstrass’s adoption of full epsilonic arguments, however, didnot mean that he attained a prototype of the modern theory. Modern analysis theory is founded on limits defined in terms of e d inequalities. His lectures were not founded on Cauchy’s limit or his own original definition of limit (Dugac 1973). Therefore, in order to clarify the formation of the modern theory, it will be necessary toidentify where the e d definition of limit was introduced and used as a foundation.We do not find the word ‘limit’ in the pu blished part of the 1861 lectures.Accord-ingly, Grattan-Guinness (1986, 228) characterizesWeierstrass’s analysis aslimit-avoid-ing. However, Weierstrass actually defined his limit in terms of epsilonics in the unpublished portion of his lectures. Histheory involved his limit concept, although the concept did not function as the foundation of his theory. Based on this discovery, this paper re-examines the formation of e d calculus theory, noting mathematicians’ treat-ments of their limits. We restrict ourattention to the process of defining continuity and derivatives. Nonetheless, this focus provides sufficient information for our purposes.First, we confirm that epsilonics arguments cannot representCauchy’s limit,though they can describe relationships that involved his limit concept. Next, we examine how Weierstrass constructed a novel analysis theory which was not based2013 British Society for the History of Mathematics52 BSHM Bulletinon Cauchy’s limits but could have involved Cauchy’s resu lts. Thenwe confirmWeierstrass’s definition of limit. Finally, we note that Dini organized his analysis textbook in 1878 based on analysis performed inthe e d style.Cauchy’s limit and epsilonic argumentsCauchy’s series of textbooks on calculus, Cours d’analyse (1821), Resume deslecons? donnees a l’Ecole royale polytechnique sur le calcul infinitesimal tomepremier (1823), and Lecons? sur le calcul differentiel (1829), are often considered as the main referen-ces for modern analysis theory, the rigour of which is rooted more in the nineteenth than the twentieth century.At the beginning of his Cours d’analyse, Cauchy defined the limit concept as fol-lows: ‘When the successively attributed values of the same variable indefinitely approach a fixed value, so that finally they differ from it by as little as desired, the last is called the limit of all theothers’ (1821, 19; English translation fromGrabiner 1981, 80). Starting from this concept, Cauchy developed a theory of continuous func-tions, infinite series, derivatives, and integrals, constructing an analysis based on lim-its (Grabiner 1981, 77).When discussing the evolution of the limit concept, Grabiner writes:‘This con-cept, translated into the algebra of inequalities, was exactly what Ca uchy needed for his calculus’ (1981, 80). From the present-day point of view, Cauchy described rather than defined his kinetic concept of limits. According to his ‘definition’—which has the quality of a translation or description—he could develop any aspectof the theory by reducing it to the algebra of inequalities.Next, Cauchy introduced infinitely small quantities into his theory. ‘When the suc-cessive absolute values of a variable decrease indefinitely, in such a way as to become less than any given quantity, that variable becomes what is called an infinitesimal. Such a variable has zero for its limit’ (1821, 19; English translationfrom Birkhoff and Merzbach 1973, 2). That is to say, in Cauchy’s framework ‘thelimit of variable x is c’ is intuitively understood as ‘x indefinitely approaches c’,and is represented as ‘jx cj is as little as desired’ or ‘jx cj is infinitesimal’.Cauchy’s idea of defining infinitesimals as variables of a special kind was original, because Leibniz and Euler, for example, had treated them as constants (Boyer 1989, 575; Lutzen? 2003, 164).In Cours d’analyse Cauchy at first gave a verbal definition of a continuous func-tion. Then, he rewrote it in terms of infinitesimals:[In other words,] the function f ðxÞ will rema in continuous relative to x in a given interval if (in this interval) an infinitesimalincrement in the variable always pro-duces an infinitesimal increment in the function itself. (1821, 43; English transla-tion from Birkhoff and Merzbach 1973, 2).He introduced the infinitesimal-involving definition and adopted a modified version of it in Resume (1823, 19–20) and Lecons? (1829, 278).Following Cauchy’s definition of infinitesimals, a continuous function can be defined as a function f ðxÞ in which ‘the variable f ðx þ aÞ f ðxÞ is an infinitelysmall quantity (as previously defined) whenever the variable a is, that is, that f ðx þ aÞ f ðxÞ approaches to zero as a does’, as notedby Edwards (1979, 311). Thus,the definition can be translated into the language of e dinequalities from a modern viewpoint. Cauchy’s infinitesimals are variables, and we can also takesuch an interpretation.Volume 29 (2014) 53Cauchy himself translated his limit concept in terms of e d inequalities. He changed ‘If the difference f ðx þ 1Þ f ðxÞ converges towards a certain limit k, for increasing values of x, (. . .)’ to‘First suppose that the quantity k has a finitevalue, and denote by e a number as small as we wish. . . . we cangive the number h a value large enough that, when x is equal to orgreater than h, the difference in question is always contained between the limits k e; k þ e’ (1821, 54; Englishtranslation from Bradley and Sandifer 2009, 35).In Resume , Cauchy gave a definition of a derivative: ‘if f ðxÞ is continuous, thenits derivative is the limit of the difference quotient,,yf(x，i),f(x), ,xias i tends to 0’ (1823, 22–23). He also translated the concept of derivative asfollows: ‘Designate by d and e two very small numbers; the first being chosen in such a way that, for numerical values of i less than d, [. . .], the ratio f ðx þ iÞ fðxÞ=i always remains greater than f ’ðx Þ e and less than f ’ðxÞ þ e’ (1823,44–45; English transla-tion from Grabiner 1981, 115).These examples show that Cauchy noted that relationships involving limits or infinitesimals could be rewritten in term of inequalities. Cauchy’s argumentsabout infinite series in Cours d’analyse, which dealt with the relationship betweenincreasing numbers and infinitesimals, had such a character. Laugwitz (1987, 264; 1999, 58) and Lutzen? (2003, 167) have noted Cauchy’s strict use of the e Ncharacterization of convergence in several of his proofs. Borovick and Katz (2012) indicate that there is room to question whether or not our representation using e d inequalities conveys messages different from Cauchy’s original intention. Butthis paper accepts the inter-pretations of Edwards, Laugwitz, and Lutzen?.Cauchy’s lectures mainly discussed properties of series and functions in the limit process, which were represented as relationships between his limits or his infinitesi-mals, or between increasing numbers and infinitesimals. His contemporaries presum-ably recognized the possibility of developing analysis theory in terms of only e, d, and n0 inequalities. With a few notable exceptions, all of Cauchy’s lectures could be rewrit-ten in terms of e d inequalities. Cauchy’s limits and hisinfinitesimals were not func-tional relationships,1 so they were not representable in terms of e d inequalities.Cauchy’s limit concept was the foundation of his theory. Thus, Weierstrass’s fullepsilonic analysis theory has a different foundation from that of Cauchy.Weierstrass’s 1861 lecturesWeierstrass’s consistent use of e d argumentsWeierstrass delivered his lectures ‘On the differential calculus’ at the GewerbeInsti-tut Berlin2 in the summer semester of 1861. Notes of these lectures were taken by1Edwards (1979, 310), Laugwitz (1987, 260–261, 271–272), andFisher (1978, 16–318) point out tha t Cauchy’s infinitesimals equate to a dependent variablefunction or aðhÞ that approaches zero as h ! 0. Cauchy adopted the latter infinitesimals, which can be written in terms of e d arguments, when he intro-duced a concept of degree of infinitesimals (1823, 250; 1829, 325). Every infinitesimal of Cauchy’s is a vari-able in the parts that the present paperdiscusses.2A forerunner of the Technische Universit?at Berlin.54 BSHM BulletinHerman Amandus Schwarz, and some of them have been published in the original German by Dugac (1973). Noting the new aspects related to foundational concepts in analysis, full e d definitions of limit and continuous function, a new definition of derivative, and a modern definition of infinitesimals, Dugac considered that the nov-elty of Weierstrass’s lectures was incontestable (1978, 372, 1976, 6–7).3 After beginning his lectures by defining a variable magnitude, Weierstrass gave the definition of a function using the notion of correspondence. This brought him to the following important definition, which did not directly appear in Cauchy’s theory:(D1) If it is now possible to determine for h a bound d such thatfor all values of h which in their absolute value are smaller than d, f ðx þ hÞ f ðxÞ becomes smaller than any magnitude e, however small, then one says that infinitely small changes of the argument correspond to infinitely small changes of the function. (Dugac 1973, 119; English translation from Calinger 1995, 607)That is, Weierstrass defined not infinitely small changes of variables but ‘infinitelysmall changes of the arguments correspond(ing) to infinitely small changes of function’ that were presented in terms of e d inequalities. He founded his theory on this correspondence.Using this concept, he defined a continuous function as follows: (D2) If now a function is such that to infinitely small changes of the argument there correspond infinitely small changes of the function, one then says that it is a continuous function of the argument, or that it changes continuously with this argument. (Dugac 1973, 119–120; English translation from Calinger 1995, 607)So we see that in accordance with his definition of correspondence, Weierstrass actually defined a continuous function on an interval in terms of epsiloni cs. Since (D2) is derived by merely changing Cauchy’s term ‘produce’ to, it seems that Weierstrass took the idea of this definition from‘correspond’Cauchy. However, Weierstrass’s definition was given in terms of epsilonics, whileCauchy’s definition c an only be interpreted in these terms. Furthermore, Weierstrass achieved it without Cauchy’s limit.Luzten? (2003, 186) indicates that Weierstrass still used the concept of ‘infinitelysmall’ in his lectures. Until giving his definition of derivative, Weierstrass actuallya function continued to use the term ‘infinitesimally small’ and often wrote of ‘which becomes infinitely small with h’. But several instances of‘infinitesimallysmall’ appeared in forms of the relationships involving them. Definition (D1) gives the rela-tionship in terms of e d inequalities. We may therefore assume that Weierstrass’s lectures consistently used e d inequalities, even though his definitions were not directly written in terms of these inequalities.Weierstrass inserted sentences confirming that the relationships involving the term ‘infinitely small’ were defined in terms of e d inequalities as follows:ðhÞ is an (D3) If h denotes a magnitude which can assume infinitely small values, ’arbitrary function of h with the property that for an infinitely small value of h it3The present paper also quotes Kurt Bing’s translation included in Calinger’sClassics of mathematics.Volume 29 (2014) 55also becomes infinitely small (that is, that always, as soon as a definite arbitrary small magnitude e is chosen, a magnitude d can be determined such that for all values of h whose absolute value is smaller than d, ’ðhÞ becomes smaller than e).(Dugac 1973, 120; English translation from Calinger, 1995, 607)As Dugac (1973, 65) in dicates, some modern textbooks describe ’ðhÞ as infinitelysmall or infinitesimal.Weierstrass argued that the whole change of function can in general be decom-posed asDf ðxÞ ? f ðx þ hÞ f ðxÞ ? p:h þ hðhÞ; ð 1Þwhere the factor p is independent of h and ðh Þ is a magnitude that becomes infinitely small with h.4 However, he overlooked that such decomposition is not possible for all functions and inserted the term‘in general’. He rewrote h as dx.One can make the difference between Df ðxÞ and p:dx s maller than any magnitude with decreasing dx. Hence Weierstrass defined ‘differential’ as the changewhich a function undergoes when its argument changes by an infinitesimally small magnitude and denoted it as df ðxÞ. Then, df ðxÞ ? p:dx. Weierstrass pointed outthat the differential coefficient p is a function of x derived from f ðxÞ and called it a derivative (Dugac 1973, 120–121; English translation from Calinger 1995, 607–608). In accordance with Weierstrass’s definitions (D1) and (D3),he largelydefined a derivative in terms of epsilonics.Weierstrass did not adopt the term ‘infinitely small’ but directly used e dinequalities when he discussed properties of infinite seriesinvolving uniform conver-gence (Dugac 1973, 122–124). It may beinferred from the publishedportion of his notes that Cauchy’s limit has no place in Weierstrass’s lectures.Grattan-Guinness’s (1986, 228) description of the limit-avoiding character of his analysis represents this situation well.However, Weierstrass thought that his theory included most of the content of Cauchy’s theory. Cauchy first gave the definition of limits of variables andinfinitesi-mals. Then, he demonstrated notions and theorems that were written in terms of the relationships involving infinitesimals. From Weierstrass’s viewpoint,they were writ-ten in terms of e d inequalities. Analytical theory mainly examines properties of functions and series, which were described in the relationships involving Cauchy’s limits and infinitesimals. Weierstrass recognized this fact and had the idea of consis-tently developing his theory in terms of inequalities. Hence Weierstrass atfirst defined the relationships among infinitesimals in terms of e d inequalities. In accor-dance with this definition, Weierstrass rewrote Cauchy’sresults and naturally imported them into his own theory. This is a process that may be described as fol-lows: ‘Weierstrass completed the transformation away fromthe use of terms such as “infinitely small”’ (Katz 1998, 728).Weierstrass’s definition of limitDugac (1978, 370–372; 1976, 6–7) read (D1) as the first definition of limit withthe help of e d. But (D1) does not involve an endpoint thatvariables or functions4Dugac (1973, 65) indicated that ðhÞ corresponds to the modernnotion of oð1Þ. In addition, hðhÞ corre-sponds to the function that was introduced as ’ðhÞ in theformer quotation from Weierstrass’s sentences.。

哥德尔不完备定理英文原文英文回答：Gödel's incompleteness theorems are two mathematical theorems that demonstrate inherent limitations of axiomatic systems based on first-order logic. The theorems were published by Kurt Gödel in 1931 and are widely acknowledged as foundational results in mathematical logic.Theorem 1 (Incompleteness theorem): Any effectively axiomatizable theory capable of expressing basic arithmetic is either incomplete or inconsistent. That is, there are true statements about the natural numbers that cannot be proven within the theory.Theorem 2 (Undecidability theorem): No consistent, effectively axiomatizable theory capable of expressing basic arithmetic can decide all true statements about the natural numbers. That is, there are statements about the natural numbers that can neither be proven nor disprovenwithin the theory.Implications of Gödel's theorems:Limits of formal systems: Gödel's theorems demonstrate that no formal system can be both complete and consistent if it is capable of expressing basic arithmetic. This has profound implications for the foundations of mathematics and the limits of what can be proven within a given axiomatic system.Creativity and human intelligence: The incompleteness theorems suggest that there are mathematical truths that cannot be discovered through purely mechanical or algorithmic processes. This has led to speculation that human intelligence may involve non-computational elements that allow for creativity and insight.The nature of mathematics: Gödel's theorems have led to a deeper understanding of the nature of mathematics. They have helped to establish the distinction between provability and truth, and have raised questions about therole of intuition and human understanding in mathematical reasoning.中文回答：哥德尔不完备定理。

Anti-control of continuous-time dynamical systemsSimin Yu a ,Guanrong Chen b ,⇑a College of Automation,Guangdong University of Technology,Guangzhou 510006,China bDepartment of Electronic Engineering,City University of Hong Kong,Hong Kong SAR,Chinaa r t i c l e i n f o Article history:Received 3July 2011Received in revised form 2October 2011Accepted 2October 2011Available online 19October 2011Keywords:ChaosContinuous-time system Anti-controlGlobal boundednessLyapunov exponent placementa b s t r a c tBased on two basic characteristics of continuous-time autonomous chaotic systems,namely being globally bounded while having a positive Lyapunov exponent,this paper develops a universal and practical anti-control approach to design a general continuous-time autonomous chaotic system via Lyapunov exponent placement.This self-unified approach is verified by mathematical analysis and validated by several typical systems designs with pared to the common trial-and-error methods,this approach is semi-analytical with feasible guidelines for design and implementation.Finally,using the Shilnikov criteria,it is proved that the new approach yields a heteroclinic orbit in a three-dimensional autonomous system,therefore the resulting system is indeed chaotic in the sense of Shilnikov.Ó2011Elsevier B.V.All rights reserved.1.IntroductionAnti-control of chaos,or chaotification,refers to the desire of generating chaos from a non-chaotic system by a simple control input.For continuous-time dynamical systems,although several successful techniques have been developed for the task,such as time-delay feedback,topological conjugate mapping,and impulsive control [1–10],there are no very effective and universal methodologies available in the literature today.Most reports in the existing literature took a trial-and-error approach to anti-controlling continuous-time autonomous systems,through parameter tuning,numerical simula-tion and Lyapunov exponent calculation,which by no means provide unified theoretical guidelines for designers to follow [11–13].For discrete-time systems,namely for mappings,the situation is much more promising.Anti-control of discrete-time systems has developed several relatively complete theories and relatively mature techniques with analytic guidelines for the users supported by rigorous mathematical chaos theory [14–25].As a result,the traditional numerical approach of trial-and-error with computer simulation has literally become only a means of verification in general.The first milestone of a mathematically-rigorous anti-control theory and method was attributed to the Chen–Lai anti-control algorithm initiated in 1996[14–21],followed by the Wang–Chen chaotification scheme and the Shi–Chen theory of coupled-expanding maps developed in the 2000s [22–25].By comparison,it is very natural to ask whether or not the discrete-time anti-control methods can be directly modified and applied to the continuous-time setting.The answer is generally no ,because they are described by difference and differ-ential equations respectively,which have many essential distinctions.One prominent difference in point is the Lyapunov exponent placement:a discrete chaotic system can have all positive Lyapunov exponents but a continuous counterpart typ-ically needs to have positive,zero and negative Lyapunov exponents so as to stretch and fold the orbit flows in the phase space.Nevertheless,they still share many similarities and analogies in both system structure and dynamics.1007-5704/$-see front matter Ó2011Elsevier B.V.All rights reserved.doi:10.1016/sns.2011.10.001⇑Corresponding author.E-mail address:gchen@.hk (G.Chen).In this paper,based on the aforementioned two basic characteristics of continuous-time autonomous chaotic systems, namely being global bounded while having a positive Lyapunov exponent,a universal feedback controller design criterion is derived for anti-controlling continuous-time autonomous dynamical systems to become chaotic.First,general forms of uncontrolled and controlled systems and the feedback controller to be used are established such that the controlled system outputs are globally bounded while the controlled system has positive,zero and negative Lyapunov exponents.These generic forms allow a systematic design and parameter determination of the anti-controlled system,overcoming the time-consuming and uncertain trial-and-error parameter tuning disadvantages.To this end,the Shilnikov criteria are applied to a three-dimensional autonomous system as an example to show that the resulting anti-control system possesses a heteroclinic orbit therefore is chaotic in the sense of Shilnikov,which means the existence of Smale horseshoes.The rest of the paper is organized as follows.Section2describes the two questions to be investigated.Section3proposes the general criterion for anti-controlling continuous-time autonomous dynamical systems.Section4shows several design examples in general forms.Section5demonstrates by the Shilnikov criteria the existence of a heteroclinic orbit in the designed anti-controlled three-dimensional autonomous system,thereby proving the chaoticity of the resulting system. Section6concludes the investigation.2.Problem statementsConsider an n-dimensional continuous-time linear autonomous system_x¼Axð1Þwhere x=[x1,x2,...,x n]T with a real system matrixA¼a11a12ÁÁÁa1na21a22ÁÁÁa2n............a n1a n2ÁÁÁa nnB BB B@1C CC CAð2ÞIn modern control theory,a basic technical problem is:assuming that the origin of the uncontrolled system(1)is an unstable equilibrium,design a linear feedback controller for the system such that the origin of the controlled system be-comes asymptotically stable.On the contrary,a basic problem of anti-control theory is:assuming that the origin of the uncontrolled system(1)is an asymptotically stable equilibrium,design a simple nonlinear feedback controller f(r x,e)such that the controlled system _x¼AxþBfðr x;eÞð3Þbecomes chaotic,where B is a control matrix to be designed:B¼b11b12ÁÁÁb1nb21b22ÁÁÁb2n............b n1b n2ÁÁÁb nnB BB B@1C CC CAð4Þand the nonlinear feedback controllerfðr x;eÞ¼f1ðr1x1;e1Þf2ðr2x2;e2Þ...f nðr n x n;e nÞB BB B@1C CC CAð5Þwherer¼r10ÁÁÁ00r2ÁÁÁ0............00ÁÁÁr nB BB B@1C CC CAð6Þis the gain matrix,and e is an upper bound for the controller(5),which is also to be designed:e¼½e1;e2;...;e n Tð7Þ2618S.Yu,G.Chen/Commun Nonlinear Sci Numer Simulat17(2012)2617–2627In this paper,a universal approach to anti-control of continuous-time autonomous systems is proposed in a general form. Two tasks will be accomplished:(i)Design A,B,f(r x,e),r,and e such that the controlled system is dissipative and globally bounded,with positive,zeroand negative Lyapunov exponents.(ii)Apply the Shilnikov criteria to a three-dimensional autonomous system to show that the resulting controlled system has a heteroclinic orbit which is chaotic in the sense of Shilnikov.3.Anti-control principles and design criteriaConsider a given system as shown in(1)and its controlled setting as(3).The anti-control principles and design criteria are developed in this section.3.1.General principle of anti-controlIn this subsection,some general anti-control principles for system(3)are discussed.First,observe that,unlike discrete-time systems,for continuous-time systems one cannot simply set all of their Lyapunov exponents to be positive.Instead,all n-dimensional(n P3)autonomous chaotic systems have positive,zero and negative Lyapunov exponents.The following gives a necessary condition for the controlled system(3)to be chaotic.Theorem1.If the real parts of all eigenvalues of matrix A are negative,and ifsup06t<1k fðr x;eÞk6k e k<1ð8Þthen the orbits of the controlled system(3)are globally bounded,where kÁk is the Euclidean norm.Proof.It can be easily verified that the solution of(3)is given byxðtÞ¼expðA tÞÁxð0ÞþZ texp½AðtÀsÞ ÁBfðr xðsÞ;eÞd sð9Þwhere x(t)=(x1(t),x2(t),...,x n(t))T.Since the real parts of all eigenvalues of A are negative,there exist constants a,b>0such thatsup06t<1k expðA tÞk6a eÀb tð10ÞFurthermore,since f(r x,e)is uniformly bounded,by combining(3)and(9)one hassup 06t<1k xðtÞk6sup06t<1a eÀb tÁk xð0Þkþsup06t<1aÁk B kÁk e kZ teÀbðtÀsÞd s¼sup06t<1a eÀb tÁk xð0Þkþsup06t<1aÁk B kÁk e kbð1ÀeÀb tÞ6aÁk xð0ÞkþaÁk B kÁk e kb<1ð11ÞNamely,the orbits of system(3)are globally bounded,completing the proof of the theorem.h3.2.Design criteriaIn this subsection,based on the global boundedness discussed above,it is to design A,B,f(r x,e),r and e,such that the con-trolled system(3)is dissipative and possesses positive,zero and negative Lyapunov exponents,thereby becoming chaotic.The following gives a sufficient condition for the controlled system(3)to be chaotic.Theorem2.Consider the controlled system(3).If the followingfive conditions are all satisfied,then the system(3)is dissipative and chaotic.(i)Controller f(r x,e)is uniformly bounded.(ii)Real parts of all eigenvalues of matrix A are negative.(iii)For the diagonal entries of the Jacobian matrix of the controlled system evaluated at the origin x=0,namely,forJ 0¼J11J12ÁÁÁJ1nJ21J22ÁÁÁJ2n............Jn1Jn2ÁÁÁJ nnB BB B@1C CC CAx¼0¼a11þb11@f1ðr1x1;e1Þ1ÁÁÁa1nþb1n@f nðr n x n;e nÞn.........a n1þb n1@f1ðr1x1;e1Þ1ÁÁÁa nnþb nn@f nðr n x n;e nÞnB B@1C CAx¼0S.Yu,G.Chen/Commun Nonlinear Sci Numer Simulat17(2012)2617–26272619the following inequality holds:X n i¼1Jii¼X ni¼1a iiþb ii@f iðr i x i;e iÞ@x ix i¼0<0ð12Þ(iv)Choose B such that all its diagonal entries are zero and there is at most one nonzero entry in each of its rows,namely,B¼0ÁÁÁÁÁÁb1iÁÁÁÁÁÁÁÁÁ0ÁÁÁÁÁÁb2jÁÁÁÁÁÁb2j0ÁÁÁÁÁÁÁÁÁ.......................................ÁÁÁb nl 0B BB BB BB BB B@1C CC CC CC CC CAð13Þwhere main diagonal elements b pp=0(p=1,2,...,n)are all zeros.On each line,except b1i,b2j,b3k,...,b nl,other elements are zeros.b1i=b2j=b3k=ÁÁÁ=b nl=0,±1(but not all0),i,j,k,...,l are not equal to each other,i=2,3,...,n,j=1,3,...,n, k=1,2,4,...,n,...,l=1,2,...,nÀ1.(v)Choose f(r x,e),r and e such that the n roots of the algebraic polynomial j J0Àk I j=0satisfy the following four conditions:(1)There are nÀ2real roots c i(i=1,2,...,nÀ2)and one pair of complex conjugate roots k±=g±j x,satisfying the Shil-nikov inequalities j c i j>j g j.(2)The origin is a saddle-node of index2or3.(3)They enable the placement of positive,zero and negative Lyapunov exponents into the controlled system(3)by modi-fying both real and imaginary parts of the Jacobian eigenvalues.Proof.By Theorem1,it follows from conditions(i)and(ii)that the orbits of the controlled system(3)are globally bounded.Condition(iii)implies that the controlled system is dissipative.Thus,condition(iv)furthermore guarantees that the Jacobian matrix of the controlled system will never be diagonally dominant,therefore it is possible to enable the placement of positive,zero and negative Lyapunov exponents simultaneously.Finally,condition(v)guarantees that the origin is a saddle-node of index2or3.So,on the one hand,the controlled system has an expanding manifold therefore one can place positive Lyapunov exponents;on the other hand,there exist eigenvalues with negative real parts and the summation of all the negative real parts is large than that of all the positive real parts,therefore one can also place zero and negative Lyapunov exponents.Moreover,all these can be done by suitably choosing the parameters in A,B,f(r x,e),r and e,through modifying both real and imaginary parts of the Jacobian eigenvalues of the controlled system(3).This completes the proof of the theorem.hRemark1.Theorem2actually only provides some sufficient conditions for the design of a working anti-controller,but it does not tell how to choose those parameters.In this regard,the following three aspects need to be taken into consideration in the anti-controller design:(1)If the eigenvalues of the system matrix A do not satisfy the condition(ii)in the theorem,then one may consider adopt-ing an anti-controller of the form e AþBfðr x;eÞ,and choose e A such that Aþe A together satisfies the condition(ii).(2)The nonlinear feedback controller f(r x,e)can have the simple form of(5),namely,its every component f i(r i x i,e i)is onlya function of one state variable x i,i=1,2,...,n.(3)The state equations of the anti-controlled system can be expressed as follows:_x1¼a11x1þa12x2þÁÁÁþa1n x nþb1i f iðr i x i;e iÞ_x2¼a21x1þa22x2þÁÁÁþa2n x nþb2j f jðr j x j;e jÞ_x3¼a31x1þa32x2þÁÁÁþa3n x nþb3k f kðr k x k;e kÞÁÁÁ_x n ¼a n1x1þa n2x2þÁÁÁþa nn x nþb nl f lðr l x l;e lÞ8>>>>>><>>>>>>:ð14ÞIt is noted that in order to ensure be able to place positive,zero and negative Lyapunov exponents,it is required to avoid possible diagonal dominant of the controlled system Jacobian matrix,i.e.,each component f i(r i x i,e i)should not appear in the corresponding state equation.For this purpose,system(14)should satisfy condition i–j–k–ÁÁÁ–l,i=2,3,...,n, j=1,3,...,n,k=1,2,4,...,n,...,l=1,2,...,nÀ1.On the contrary,for the discrete-time setting,such as for the Chen–Lai and Wang–Chen algorithms,the diagonal dominance property is acceptable,system(14)can only satisfy condition i=1, j=2,k=3,...,l=n.2620S.Yu,G.Chen/Commun Nonlinear Sci Numer Simulat17(2012)2617–26274.Design examplesBased on the general guidelines developed above,this section presents detailed design examples of typical three-,four-and five-dimensional continuous-time autonomous systems for anti-control of chaos,which all serve for different purposes of illustration.In these examples,the nonlinear feedback controller uses a simple piecewise linear saw-tooth function,yet it should be noted that the method developed here is universal in the sense that it works for all uniformly bounded nonlinear functions such as the modulo and sine functions,and it works for any finite-dimensional system although the design will become more complex as the dimension increases.4.1.Three-dimensional systemsStart from the general form of a three-dimensional autonomous system,_x1_x 2_x30B @1C A ¼a 11a 12a 13a 21a 22a 23a 31a 32a 330B @1C A x 1x 2x 3B @1C A þb 11b 12b 13b 21b 22b 23b 31b 32b 33B@1C A f 1ðr 1x 1;e 1Þf 2ðr 2x 2;e 2Þf 3ðr 3x 3;e 3ÞB@1C Að15ÞTake x 1as the anti-control state variable for feedback,and use a saw-tooth controller of the formf 1ðr 1x 1;e 1Þ¼e 1Ásawtooth ðpr 1ðx 1Àe 1=r 1Þ=e 1;p Þf 2ðr 2x 2;e 2Þ¼f 3ðr 3x 3;e 3Þ¼0&ð16Þwhere p =1,r 1=6.943,e 1=3.4715,and sup 06t <1k f 1(r 1x 1,e 1)k 6e 1=3.4715<1,with f 1(r 1x 1,e 1)as shown in Fig.1.GivingA ¼À0:4À1:952:051:95À2:30:05À2:050:051:7B@1C A ;B ¼000À100000B @1C Aand substituting them into (15)yields_x1¼À0:4x 1À1:95x 2þ2:05x 3_x 2¼1:95x 1À2:3x 2þ0:05x 3Àf 1ðr 1x 1;e 1Þ_x 3¼À2:05x 1þ0:05x 2þ1:7x 38><>:ð17ÞIt can be easily verified that the uncontrolled system matrix A has eigenvalues k 1=À0.3and k 2,3=À0.35±j 2.0,therefore the origin of the uncontrolled system is asymptotic stable.Furthermore,according to (17),the eigenvalues of the controlled system Jacobian at the origin are k 1=À4.3912and k 2,3=1.6956±j 1.6763,where the origin is a saddle-node of index 2.Thus,all conditions stated in Theorem 2are satisfied.Indeed,chaos can be generated as shown in Fig.2.As another example,in (14),use x 1,x 2,x 3as feedback control states,along with the saw-tooth feedback anti-controllerf i ðr i x i ;e i Þ¼e i Ásawtooth ðpr i ðx i Àe i =r i Þ=e i ;p Þð18Þwhere p =1,r 1=2.1,r 2=0.35,r 3=1.0,e 1=0.75,e 2=0.175,e 3=0.5,and it can be verified that sup 06t <1k f i (r i x i ,e i )k 60.75<1,i =1,2,3.Also,chooseA ¼À0:7À2:93:12:9À3:50:1À3:10:12:5B @1CA ;B ¼001À100010B@1C AS.Yu,G.Chen /Commun Nonlinear Sci Numer Simulat 17(2012)2617–26272621Substituting them into (15)gives_x1¼À0:7x 1À2:9x 2þ3:1x 3þf 3ðr 3x 3;e 3Þ_x 2¼2:9x 1À3:5x 2þ0:1x 3Àf 1ðr 1x 1;e 1Þ_x 3¼À3:1x 1þ0:1x 2þ2:5x 3þf 2ðr 2x 2;e 2Þ8><>:ð19ÞIt can be easily verified that the uncontrolled system matrix A has eigenvalues k 1=À0.5and k 2,3=À0.6±j 3.0,therefore the origin of the uncontrolled system is asymptotic stable.Furthermore,according to (19),the eigenvalues of the controlled system Jacobian at the origin are k 1=2.8978and k 2,3=0.5989±j 3.1697,where the origin is a saddle-node of index 2.Thus,all conditions stated in Theorem 2are satisfied.Indeed,chaos can be generated as shown by Fig.3.4.2.Four-dimensional systemsConsider a general form of a four-dimensional autonomous system,_x 1_x 2_x 3_x40B B B @1C C C A ¼a 11a 12a 13a 14a 21a 22a 23a 24a 31a 32a 33a 34a 41a 42a 43a 440B B B @1C C C A x 1x 2x 3x 40B B B @1C C C A þb 11b 12b 13b 14b 21b 22b 23b 24b 31b 32b 33b 34b 41b 42b 43b 44B B B @1C C C A f 1ðr 1x 1;e 1Þf 2ðr 2x 2;e 2Þf 3ðr 3x 3;e 3Þf 4ðr 4x 4;e 4ÞB B B@1C C C A ð20ÞGive x 1,x 2,x 3,x 4to be feedback anti-control state variables,with sine function controllerf i ðr i x i ;e i Þ¼e i Ásin ðr i x i Þð21Þwhere r 1=1,r 2=2,r 3=2,r 4=2,e 1=0.5,e 2=1,e 3=1,e 4=1,and it satisfies sup 06t <1k f i (r i x i ,e i )k 61<1,i =1,...,4.More-over,chooseA ¼À0:5À4:95:11:04:9À5:30:11:0À5:10:14:71:01:02:0À3:0À1:00B B B@1C CC A ;B ¼0100100000100À10B B B@1C C C AFig.2.The chaotic attractor generated by anti-controller using state variable x 1.Fig.3.The chaotic attractor generated by anti-controller using state variables x 1,x 2,x 3.Substituting them into (20)leads to_x 1¼À0:5x 1À4:9x 2þ5:1x 3þ1:0x 4þf 2ðr 2x 2;e 2Þ_x 2¼4:9x 1À5:3x 2þ0:1x 3þ1:0x 4þf 1ðr 1x 1;e 1Þ_x 3¼À5:1x 1þ0:1x 2þ4:7x 3þ1:0x 4þf 4ðr 4x 4;e 4Þ_x4¼1:0x 1þ2:0x 2À3:0x 3À1:0x 4Àf 3ðr 3x 3;e 3Þ8>>><>>>:ð22ÞThe uncontrolled system matrix A has four eigenvalues k 1=À1,k 2=À0.3,and k 3,4=À0.4±j 5.0,so the origin of the uncon-trolled system is asymptotically stable.It follows from (22)that the eigenvalues of the controller system Jacibian at the ori-gin are k 1=À15.2615,k 2=9.8142,k 3,4=1.6737±j 22.3207,so the origin is a saddle-node of index 3.Thus,all conditions inTheorem 2are satisfied,with chaotic attractor generated as shown in Fig.4.4.3.Five-dimensional systemsAgain,start from a general five-dimensional autonomous system_x1_x 2_x3_x 4_x 50B B B B B B @1C C C C C C A ¼a 11a 12a 13a 14a 15a 21a 22a 23a 24a 25a 31a 32a 33a 34a 35a 41a 42a 43a 44a 45a 51a 52a 53a 54a 550B B BB B B @1C C C C C C A x 1x 2x 3x 4x 50B B B B B B @1C C C C C C A þb 11b 12b 13b 14b 15b 21b 22b 23b 24b 25b 31b 32b 33b 34b 35b 41b 42b 43b 44b 45b 51b 52b 53b 54b 55B B B B B B @1C C C C C C A f 1ðr 1x 1;e 1Þf 2ðr 2x 2;e 2Þf 3ðr 3x 3;e 3Þf 4ðr 4x 4;e 4Þf 5ðr 5x 5;e 5Þ0BB BB BB@1C CC C C C Að23ÞGive x 1,x 2,x 3,x 4,x 5to be anti-control state variables,and use hyperbolic tangent feedback controllerf i ðr i x i ;e i Þ¼e i Átanh ðr i x i Þð24Þwhere r 1=1,r 2=3,r 3=4,r 4=2,r 5=2,e 1=5,e 2=15,e 3=10,e 4=10,e 5=10,with sup 06t <1k f i (r i x i ,e i )k 615<1,i =1, (5)TakeA ¼À0:5À4:95:11:01:04:9À5:30:11:01:0À5:10:14:71:0À1:01:02:0À3:0À1:0À1:0À1:01:01:01:0À1:0BB BB BB@1C C C C C C A ;B ¼01000100000001000001010BB BB BB@1C CC C C C AIt then follows that (23)becomes_x 1¼À0:5x 1À4:9x 2þ5:1x 3þ1:0x 4þ1:0x 5þf 2ðr 2x 2;e 2Þ8Fig.4.The chaotic attractor generated by anti-controller using state variables x 1,x 2,x 3,x 4.S.Yu,G.Chen /Commun Nonlinear Sci Numer Simulat 17(2012)2617–26272623index 2.It can be easily seen that all conditions of Theorem 2are satisfied,so a chaotic attractor can be generated as shown in Fig.5.5.Existence of a heteroclinic orbitIn this section,it is furthermore proved that in the three-dimensional setting,any anti-controlled system designed above,satisfying the Shilnikov criteria,indeed has a heteroclinic orbit,therefore is chaotic in the sense of Shilnikov [26,27].For a higher-dimensional system,as long as it has a three-dimensional subsystem that satisfies the Shilnikov criteria,it is consid-ered to be chaotic in the sense of Shilnikov.It follows from (15)–(17)that_x1¼a 11x 1þa 12x 2þa 13x 3_x 2¼a 21x 1þa 22x 2þa 23x 3Àf 1ðr 1x 1;e 1Þ_x 3¼a 31x 1þa 32x 2þa 33x 38><>:ð26Þwhere f 1(r 1x 1,e 1)=e 1Ásawtooth (p r 1x 1/e 1,p )is a saw-tooth function.By setting _x¼_y ¼_z ¼0in system (26),one obtains the equilibrium equationa 11x Ã1þa 12x Ã2þa 13x Ã3¼0a 21x Ã1þa 22x Ã2þa 23x Ã3¼f 1ðr 1x Ã1;e 1Þa 31x Ã1þa 32x Ã2þa 33x Ã3¼08><>:ð27ÞFig.5.The chaotic attractor generated by anti-controller using state variables x 1,x 2,x 3,x 4,x 5.LetD ¼a 11a 12a 13a 21a 22a 23a 31a 32a 33 ;D 1¼0a 12a 13f 1ðr 1x Ã1;e 1Þa 22a 230a 32a 33 ;D 2¼a 110a 13a 21f 1ðr 1x Ã1;e 1Þa 23a 310a 33 ;and D 3¼a 11a 120a 21a 22f 1r 1x Ã1;e 1ÀÁa 31a 320:Then,the above equilibrium equation has a solutionx Ã1¼D 1D ¼a 13a 32Àa 12a 33D f 1r 1x Ã1;e 1ÀÁx Ã2¼D 2D ¼a 11a 33Àa 13a 31D f 1r 1x Ã1;e 1ÀÁx Ã3¼D 3D ¼a 12a 31Àa 11a 32D f 1r 1x Ã1;e 1ÀÁ8>><>>:ð28ÞDenote k 1=D /(a 13a 32Àa 12a 33),k 2=D /(a 11a 33Àa 13a 31)and k 3=D /(a 12a 31Àa 11a 32),and then rewrite (28)ask 1x Ã1¼f 1r 1x Ã1;e 1ÀÁk 2x Ã2¼f 1r 1x Ã1;e 1ÀÁk 3x Ã3¼f 1r 1x Ã1;e 1ÀÁ8><>:ð29ÞBy the odd-symmetry of (29),one obtains the two equilibria,P 1x Ã1;x Ã2;x Ã3ÀÁand P 2Àx Ã1;Àx Ã2;Àx Ã3ÀÁ,which are closest to the ori-gin,as well as the distributions of the others,as shown in Fig.6.Now,computing P 1and P 2one can getx Ã1¼2e 1=ðr 1Àk 1Þx Ã2¼2k 1e 1=k 2ðr 1Àk 1Þx Ã3¼2k 1e 1=k 3ðr 1Àk 1Þ8><>:ð30ÞSince system (26)is piecewise linear,a straight line in the stable manifold E S (P 1)in the subspace corresponding to theequilibrium P 1x Ã1;x Ã2;x Ã3ÀÁis given byE S ðP 1Þ:x 1Àx Ã1l ¼x 2Àx Ã2m ¼x 3Àx Ã3nð31Þwhere l ,m ,n are constants;while a plane in the unstable manifold E U (P 1)is determined byE U ðP 1Þ:a x 1Àx Ã1ÀÁþb x 2Àx Ã2ÀÁþc x 3Àx Ã3ÀÁ¼0ð32Þwhere a ,b ,c are constants.Similarly,for another equilibrium P 2Àx Ã1;Àx Ã2;Àx Ã3ÀÁ,a straight line in the corresponding stable manifold E S (P 2)is given byE SðP 2Þ:x 1þx Ã1¼x 2þx Ã2¼x 3þx Ã3ð33Þand a plane in the unstable manifold E U (P 2)is described byE U ðP 2Þ:a x 1þx Ã1ÀÁþb x 2þx Ã2ÀÁþc x 3þx Ã3ÀÁ¼0ð34ÞThus,according to (31)–(34),the eigen-subspace corresponding to P 1and P 2can be specified,as shown in Fig.7As can be seen from Fig.7,the cross point of E S (P 1)and S :x 1=0is given byQ 1¼E S ðP 1Þ\S !Q 10;x Ã2Àm x Ã1;x Ã3Àn x Ã1ð35Þwhile the cross line between E U (P 1)and S :x 1=0isL 1¼E U ðP 1Þ\S !a x 1Àx Ã1ÀÁþb x 2Àx Ã2ÀÁþc x 3Àx Ã3ÀÁ¼0x 1¼0(ð36ÞS.Yu,G.Chen /Commun Nonlinear Sci Numer Simulat 17(2012)2617–26272625Similarly,E S (P 2)and S :x 1=0intersect at a point,Q 2¼E S ðP 2Þ\S !Q 20;Àx Ã2þm l x Ã1;Àx Ã3þn l x Ã1ð37Þand E U (P 2)and S :x 1=0intersect along a line,L 2¼E U ðP 2Þ\S !a x 1þx Ã1ÀÁþb x 2þx Ã2ÀÁþc x 3þx Ã3ÀÁ¼0x 1¼0(ð38ÞIt can be seen from Fig.7that if Q 1is located above L 2,then by (35)and (38),one hasax Ã1þb 2x Ã2Àm l x Ã1 þc 2x Ã3Àn l x Ã1¼0ð39ÞSimilarly,if Q 2is located above L 1,then by (36)and (37),the same equality (39)will hold automatically,which is clear fromthe symmetry about the origin as shown in Fig.7.Hence,if Q 2is located above L 1,then Q 1must be above L 2.If Q 1is located above L 2and also Q 2is located above L 1,then it means the controlled system (26)has a heteroclinic orbitconnecting the two equilibria P 1x Ã1;x Ã2;x Ã3ÀÁand P 2Àx Ã1;Àx Ã2;Àx Ã3ÀÁ.Fig.8.The dependence of d on parameter r 1.2626S.Yu,G.Chen /Commun Nonlinear Sci Numer Simulat 17(2012)2617–2627。

(0,2) 插值||(0,2) interpolation0#||zero-sharp; 读作零井或零开。

0+||zero-dagger; 读作零正。

1-因子||1-factor3-流形||3-manifold; 又称“三维流形”。

AIC准则||AIC criterion, Akaike information criterionAp 权||Ap-weightA稳定性||A-stability, absolute stabilityA最优设计||A-optimal designBCH 码||BCH code, Bose-Chaudhuri-Hocquenghem codeBIC准则||BIC criterion, Bayesian modification of the AICBMOA函数||analytic function of bounded mean oscillation; 全称“有界平均振动解析函数”。

BMO鞅||BMO martingaleBSD猜想||Birch and Swinnerton-Dyer conjecture; 全称“伯奇与斯温纳顿－戴尔猜想”。

B样条||B-splineC*代数||C*-algebra; 读作“C星代数”。

C0 类函数||function of class C0; 又称“连续函数类”。

CA T准则||CAT criterion, criterion for autoregressiveCM域||CM fieldCN 群||CN-groupCW 复形的同调||homology of CW complexCW复形||CW complexCW复形的同伦群||homotopy group of CW complexesCW剖分||CW decompositionCn 类函数||function of class Cn; 又称“n次连续可微函数类”。

Cp统计量||Cp-statisticC。

Vocabulary 集合与简易逻辑集合（集）set非负整数集the set of all non-negative integers自然数集the set of all natural numbers 正整数集the set of all positive integers 整数集the set of all integers有理数集the set of all rational numbers 实数集the set of all real numbers元素element属于belong to不属于not belong to有限集finite set无限集infinite set空集empty set包含inclusion, include包含于lie in子集subset真子集proper subset补集（余集）complementary set全集universe交集intersection并集union偶数集the set of all even numbers 奇数集the set of all odd numbers含绝对值的不等式inequality with absolute value 一元二次不等式one-variable quadratic inequality 逻辑logic逻辑联结词logic connective或or且and非not真true假false真值表truth table原命题original proposition逆命题converse proposition否命题negative proposition逆否命题converse-negative proposition 充分条件sufficient condition必要条件necessary condition充要条件sufficient and necessary condition……的充要条件是…… … if and only if …函数函数function自变量argument定义域domain值域range区间interval闭区间closed interval开区间open interval函数的图象graph of function映射mapping象image原象inverse image单调monotone增函数increasing function减函数decreasing function单调区间monotone interval反函数inverse function指数exponentn次方根n th root根式radical根指数radical exponent被开方数radicand指数函数exponential function对数logarithm常用对数common logarithm自然对数natural logarithm对数函数logarithmic function数列数列sequence of number项term通项公式the formula of general term 有穷数列finite sequence of number 无穷数列infinite sequence of number 递推公式recurrence formula等差数列arithmetic progression，arithmetic series公差common difference等差中项arithmetic mean等比数列geometric progression，geometric series公比common ratio等比中项geometric mean三角函数三角函数trigonometric function始边initial side终边terminal side正角positive angle负角negative angle零角zero angle象限角quadrant angle弧度radian弧度制radian measure角度制degree measure正弦sine余弦cosine正切tangent余切cotangent正割secant余割cosecant诱导公式induction formula正弦曲线sine curve余弦曲线cosine curve最大值maximum最小值minimum周期period最小正周期minimal positive period周期函数periodic function振幅amplitude of vibration频率frequency相位phase初相initial phase反正弦arc sine反余弦arc cosine反正切arc tangent平面向量有向线段directed line segment数量scalar quantity向量vector零向量zero vector相等向量equal vector共线向量collinear vectors平行向量parallel vectors向量的数乘multiplication of vector by scalar 单位向量unit vector基底base 基向量base vectors平移translation数量积inner product正弦定理sine theorem余弦定理cosine theorem不等式算术平均数arithmetic mean几何平均数geometric mean比较法method of compare 综合法method of synthesis 分析法method of analysis直线倾斜角angle of inclination斜率gradient点斜式point slope form截距intercept斜截式gradient intercept form两点式two-point form一般式general form夹角included angle线性规划linear programming 约束条件constraint condition 目标函数objective function可行域feasible region最优解optimal solution圆锥曲线曲线curve坐标法method of coordinate 解析几何analytic geometry笛卡儿Descartes标准方程standard equation一般方程general equation参数方程parameter equation 参数parameter圆锥曲线point conic椭圆ellipse焦点focus, focal points焦距focal length长轴major axis短轴minor axis离心率eccentricity双曲线hyperbola实轴real axis虚轴imaginary axis渐近线asymptote抛物线parabola准线directrixEnglish Chineseabbreviation 简写符号；简写abscissa 横坐标absolute complement 绝对补集absolute error 绝对误差absolute inequality 绝不等式absolute maximum 绝对极大值absolute minimum 绝对极小值absolute monotonic 绝对单调absolute value 绝对值accelerate 加速acceleration 加速度acceleration due to gravity 重力加速度; 地心加速度accumulation 累积accumulative 累积的accuracy 准确度act on 施于action 作用; 作用力acute angle 锐角acute-angled triangle 锐角三角形add 加addition 加法addition formula 加法公式addition law 加法定律addition law(of probability) （概率）加法定律additive inverse 加法逆元; 加法反元additive property 可加性adjacent angle 邻角adjacent side 邻边adjoint matrix 伴随矩阵algebra 代数algebraic 代数的algebraic equation 代数方程algebraic expression 代数式algebraic fraction 代数分式；代数分数式algebraic inequality 代数不等式algebraic number 代数数algebraic operation 代数运算algebraically closed 代数封闭algorithm 算法系统; 规则系统alternate angle （交）错角alternate segment 内错弓形alternating series 交错级数alternative hypothesis 择一假设; 备择假设; 另一假设altitude 高；高度；顶垂线；高线ambiguous case 两义情况；二义情况amount 本利和；总数analysis 分析；解析analytic geometry 解析几何angle 角angle at the centre 圆心角angle at the circumference 圆周角angle between a line and a plane 直 与平面的交角angle between two planes 两平面的交角angle bisection 角平分angle bisector 角平分线 ；分角线angle in the alternate segment 交错弓形的圆周角angle in the same segment 同弓形内的圆周角angle of depression 俯角angle of elevation 仰角angle of friction 静摩擦角; 极限角angle of greatest slope 最大斜率的角angle of inclination 倾斜角angle of intersection 相交角；交角angle of projection 投射角angle of rotation 旋转角angle of the sector 扇形角angle sum of a triangle 三角形内角和angles at a point 同顶角angular displacement 角移位angular momentum 角动量angular motion 角运动angular velocity 角速度annum(X% per annum) 年（年利率X%）anti-clockwise direction 逆时针方向；返时针方向anti-clockwise moment 逆时针力矩anti-derivative 反导数; 反微商anti-logarithm 逆对数；反对数anti-symmetric 反对称apex 顶点approach 接近；趋近approximate value 近似值approximation 近似；略计；逼近Arabic system 阿刺伯数字系统arbitrary 任意arbitrary constant 任意常数arc 弧arc length 弧长arc-cosine function 反余弦函数arc-sin function 反正弦函数arc-tangent function 反正切函数area 面积Argand diagram 阿根图, 阿氏图argument (1)论证; (2)辐角argument of a complex number 复数的辐角argument of a function 函数的自变量arithmetic 算术arithmetic mean 算术平均；等差中顶；算术中顶arithmetic progression 算术级数；等差级数arithmetic sequence 等差序列arithmetic series 等差级数arm 边array 数组; 数组arrow 前号ascending order 递升序ascending powers of X X 的升幂assertion 断语; 断定associative law 结合律assumed mean 假定平均数assumption 假定；假设asymmetrical 非对称asymptote 渐近asymptotic error constant 渐近误差常数at rest 静止augmented matrix 增广矩阵auxiliary angle 辅助角auxiliary circle 辅助圆auxiliary equation 辅助方程average 平均；平均数；平均值average speed 平均速率axiom 公理axiom of existence 存在公理axiom of extension 延伸公理axiom of inclusion 包含公理axiom of pairing 配对公理axiom of power 幂集公理axiom of specification 分类公理axiomatic theory of probability 概率公理论axis 轴axis of parabola 拋物线的轴axis of revolution 旋转轴axis of rotation 旋转轴axis of symmetry 对称轴back substitution 回代bar chart 棒形图；条线图；条形图；线条图base （1）底；（2）基；基数base angle 底角base area 底面base line 底线base number 底数；基数base of logarithm 对数的底basis 基Bayes' theorem 贝叶斯定理bearing 方位（角）；角方向（角）bell-shaped curve 钟形图belong to 属于Bernoulli distribution 伯努利分布Bernoulli trials 伯努利试验bias 偏差；偏倚biconditional 双修件式; 双修件句bijection 对射; 双射; 单满射bijective function 对射函数; 只射函数billion 十亿bimodal distribution 双峰分布binary number 二进数binary operation 二元运算binary scale 二进法binary system 二进制binomial 二项式binomial distribution 二项分布binomial expression 二项式binomial series 二项级数binomial theorem 二项式定理bisect 平分；等分bisection method 分半法；分半方法bisector 等分线 ；平分线Boolean algebra 布尔代数boundary condition 边界条件boundary line 界（线）；边界bounded 有界的bounded above 有上界的；上有界的bounded below 有下界的；下有界的bounded function 有界函数bounded sequence 有界序列brace 大括号bracket 括号breadth 阔度broken line graph 折线图calculation 计算calculator 计算器；计算器calculus (1) 微积分学; (2) 演算cancel 消法；相消canellation law 消去律canonical 典型; 标准capacity 容量cardioid 心脏Cartesian coordinates 笛卡儿坐标Cartesian equation 笛卡儿方程Cartesian plane 笛卡儿平面Cartesian product 笛卡儿积category 类型；范畴catenary 悬链Cauchy sequence 柯西序列Cauchy's principal value 柯西主值Cauchy-Schwarz inequality 柯西 - 许瓦尔兹不等式central limit theorem 中心极限定理central line 中线central tendency 集中趋centre 中心；心centre of a circle 圆心centre of gravity 重心centre of mass 质量中心centrifugal force 离心力centripedal acceleration 向心加速度centripedal force force 向心力centroid 形心；距心certain event 必然事件chain rule 链式法则chance 机会change of axes 坐标轴的变换change of base 基的变换change of coordinates 坐标轴的变换change of subject 主项变换change of variable 换元；变量的换characteristic equation 特征(征)方程characteristic function 特征(征)函数characteristic of logarithm 对数的首数; 对数的定位部characteristic root 特征(征)根chart 图；图表check digit 检验数位checking 验算chord 弦chord of contact 切点弦circle 圆circular 圆形；圆的circular function 圆函数；三角函数circular measure 弧度法circular motion 圆周运动circular permutation 环形排列; 圆形排列; 循环排列circumcentre 外心；外接圆心circumcircle 外接圆circumference 圆周circumradius 外接圆半径circumscribed circle 外接圆cissoid 蔓叶class 区；组；类class boundary 组界class interval 组区间；组距class limit 组限；区限class mark 组中点；区中点classical theory of probability 古典概率论classification 分类clnometer 测斜仪clockwise direction 顺时针方向clockwise moment 顺时针力矩closed convex region 闭凸区域closed interval 闭区间coaxial 共轴coaxial circles 共轴圆coaxial system 共轴系coded data 编码数据coding method 编码法co-domain 上域coefficient 系数coefficient of friction 摩擦系数coefficient of restitution 碰撞系数; 恢复系数coefficient of variation 变差系数cofactor 余因子; 余因式cofactor matrix 列矩阵coincide 迭合；重合collection of terms 并项collinear 共线collinear planes 共线面collision 碰撞column (1)列；纵行；(2) 柱column matrix 列矩阵column vector 列向量combination 组合common chord 公弦common denominator 同分母；公分母common difference 公差common divisor 公约数；公约common factor 公因子；公因子common logarithm 常用对数common multiple 公位数；公倍common ratio 公比common tangent 公切commutative law 交换律comparable 可比较的compass 罗盘compass bearing 罗盘方位角compasses 圆规compasses construction 圆规作图compatible 可相容的complement 余；补余complement law 补余律complementary angle 余角complementary equation 补充方程complementary event 互补事件complementary function 余函数complementary probability 互补概率complete oscillation 全振动completing the square 配方complex conjugate 复共轭complex number 复数complex unmber plane 复数平面complex root 复数根component 分量component of force 分力composite function 复合函数; 合成函数composite number 复合数；合成数composition of mappings 映射构合composition of relations 复合关系compound angle 复角compound angle formula 复角公式compound bar chart 综合棒形图compound discount 复折扣compound interest 复利；复利息compound probability 合成概率compound statement 复合命题; 复合叙述computation 计算computer 计算机；电子计算器concave 凹concave downward 凹向下的concave polygon 凹多边形concave upward 凹向上的concentric circles 同心圆concept 概念conclusion 结论concurrent 共点concyclic 共圆concyclic points 共圆点condition 条件conditional 条件句；条件式conditional identity 条件恒等式conditional inequality 条件不等式conditional probability 条件概率cone 锥；圆锥（体）confidence coefficient 置信系数confidence interval 置信区间confidence level 置信水平confidence limit 置信极限confocal section 共焦圆锥曲congruence (1)全等；(2)同余congruence class 同余类congruent 全等congruent figures 全等图形congruent triangles 全等三角形conic 二次曲 ; 圆锥曲conic section 二次曲 ; 圆锥曲conical pendulum 圆锥摆conjecture 猜想conjugate 共轭conjugate axis 共轭conjugate diameters 共轭轴conjugate hyperbola 共轭(直)径conjugate imaginary / complex number 共轭双曲conjugate radical 共轭虚/复数conjugate surd 共轭根式; 共轭不尽根conjunction 合取connective 连词connector box 捙接框consecutive integers 连续整数consecutive numbers 连续数；相邻数consequence 结论；推论consequent 条件；后项conservation of energy 能量守恒conservation of momentum 动量守恒conserved 守恒consistency condition 相容条件consistent 一贯的；相容的consistent estimator 相容估计量constant 常数constant acceleration 恒加速度constant force 恒力constant of integration 积分常数constant speed 恒速率constant term 常项constant velocity 怛速度constraint 约束；约束条件construct 作construction 作图construction of equation 方程的设立continued proportion 连比例continued ratio 连比continuity 连续性continuity correction 连续校正continuous 连续的continuous data 连续数据continuous function 连续函数continuous proportion 连续比例continuous random variable 连续随机变量contradiction 矛盾converge 收敛convergence 收敛性convergent 收敛的convergent iteration 收敛的迭代convergent sequence 收敛序列convergent series 收敛级数converse 逆(定理)converse of a relation 逆关系converse theorem 逆定理conversion 转换convex 凸convex polygon 凸多边形convexity 凸性coordinate 坐标coordinate geometry 解析几何；坐标几何coordinate system 坐标系系定理；系；推论coplanar 共面coplanar forces 共面力coplanar lines 共面co-prime 互质; 互素corollary 系定理; 系; 推论correct to 准确至；取值至correlation 相关correlation coefficient 相关系数correspondence 对应corresponding angles (1)同位角；(2)对应角corresponding element 对应边corresponding sides 对应边cosecant 余割cosine 余弦cosine formula 余弦公式cost price 成本cotangent 余切countable 可数countable set 可数集countably infinite 可数无限counter clockwise direction 逆时针方向；返时针方向counter example 反例counting 数数；计数couple 力偶Carmer's rule 克莱玛法则criterion 准则critical point 临界点critical region 临界域cirtical value 临界值cross-multiplication 交叉相乘cross-section 横切面；横截面；截痕cube 正方体；立方；立方体cube root 立方根cubic 三次方；立方；三次(的)cubic equation 三次方程cubic roots of unity 单位的立方根cuboid 长方体；矩体cumulative 累积的cumulative distribution function 累积分布函数cumulative frequecy 累积频数；累积频率cumulative frequency curve 累积频数曲 cumulative frequcncy distribution 累积频数分布cumulative frequency polygon 累积频数多边形；累积频率直方图curvature of a curve 曲线的曲率curve 曲线curve sketching 曲线描绘(法)curve tracing 曲线描迹(法)curved line 曲线curved surface 曲面curved surface area 曲面面积cyclic expression 输换式cyclic permutation 圆形排列cyclic quadrilateral 圆内接四边形cycloid 旋输线; 摆线cylinder 柱；圆柱体cylindrical 圆柱形的damped oscillation 阻尼振动data 数据De Moivre's theorem 棣美弗定理De Morgan's law 德摩根律decagon 十边形decay 衰变decay factor 衰变因子decelerate 减速decelaration 减速度decile 十分位数decimal 小数decimal place 小数位decimal point 小数点decimal system 十进制decision box 判定框declarative sentence 说明语句declarative statement 说明命题decoding 译码decrease 递减decreasing function 递减函数；下降函数decreasing sequence 递减序列；下降序列decreasing series 递减级数；下降级数decrement 减量deduce 演绎deduction 推论deductive reasoning 演绎推理definite 确定的；定的definite integral 定积分definition 定义degenerated conic section 降级锥曲线degree (1) 度; (2) 次degree of a polynomial 多项式的次数degree of accuracy 准确度degree of confidence 置信度degree of freedom 自由度degree of ODE 常微分方程次数degree of precision 精确度delete 删除; 删去denary number 十进数denominator 分母dependence (1)相关; (2)应变dependent event(s) 相关事件; 相依事件; 从属事件dependent variable 应变量; 应变数depreciation 折旧derivable 可导derivative 导数derived curve 导函数曲线derived function 导函数derived statistics 推算统计资料; 派生统计资料descending order 递降序descending powers of x x的降序descriptive statistics 描述统计学detached coefficients 分离系数(法) determinant 行列式deviation 偏差; 变差deviation from the mean 离均差diagonal 对角线diagonal matrix 对角矩阵diagram 图; 图表diameter 直径diameter of a conic 二次曲线的直径difference 差difference equation 差分方程difference of sets 差集differentiable 可微differential 微分differential coefficient 微商; 微分系数differential equation 微分方程differential mean value theorem 微分中值定理differentiate 求...的导数differentiate from first principle 从基本原理求导数differentiation 微分法digit 数字dimension 量; 量网; 维(数)direct impact 直接碰撞direct image 直接像direct proportion 正比例direct tax, direct taxation 直接税direct variation 正变(分)directed angle 有向角directed line 有向直线directed line segment 有向线段directed number 有向数direction 方向; 方位direction angle 方向角direction cosine 方向余弦direction number 方向数direction ratio 方向比directrix 准线Dirichlet function 狄利克来函数discontinuity 不连续性discontinuous 间断(的);连续(的); 不连续(的) discontinuous point 不连续点discount 折扣discrete 分立; 离散discrete data 离散数据; 间断数据discrete random variable 间断随机变数discrete uniform distribution 离散均匀分布discriminant 判别式disjoint 不相交的disjoint sets 不相交的集disjunction 析取dispersion 离差displacement 位移disprove 反证distance 距离distance formula 距离公式distinct roots 相异根distincr solution 相异解distribution 公布distributive law 分配律diverge 发散divergence 发散(性)divergent 发散的divergent iteration 发散性迭代divergent sequence 发散序列divergent series 发散级数divide 除dividend (1)被除数；(2)股息divisible 可整除division 除法division algorithm 除法算式divisor 除数；除式；因子divisor of zero 零因子dodecagon 十二边形domain 定义域dot 点dot product 点积double angle 二倍角double angle formula 二倍角公式double root 二重根dual 对偶duality (1)对偶性; (2) 双重性due east/ south/ west /north 向东/ 南/ 西/ 北dynamics 动力学eccentric angle 离心角eccentric circles 离心圆eccentricity 离心率echelon form 梯阵式echelon matrix 梯矩阵edge 棱；边efficient estimator 有效估计量effort 施力eigenvalue 本征值eigenvector 本征向量elastic body 弹性体elastic collision 弹性碰撞elastic constant 弹性常数elastic force 弹力elasticity 弹性element 元素elementary event 基本事件elementary function 初等函数elementary row operation 基本行运算elimination 消法elimination method 消去法；消元法ellipse 椭圆ellipsiod 椭球体elliptic function 椭圆函数elongation 伸张；展empirical data 实验数据empirical formula 实验公式empirical probability 实验概率；经验概率empty set 空集encoding 编码enclosure 界限end point 端点energy 能; 能量entire surd 整方根epicycloid 外摆线equal 相等equal ratios theorem 等比定理equal roots 等根equal sets 等集equality 等(式)equality sign 等号equation 方程equation in one unknown 一元方程equation in two unknowns(variables) 二元方程equation of a straight line 直线方程equation of locus 轨迹方程equiangular 等角(的)equidistant 等距(的)equilateral 等边(的)equilateral polygon 等边多边形equilateral triangle 等边三角形equilibrium 平衡equiprobable 等概率的equiprobable space 等概率空间equivalence 等价equivalence class 等价类equivalence relation 等价关系equivalent 等价(的)error 误差error allowance 误差宽容度error estimate 误差估计error term 误差项error tolerance 误差宽容度escribed circle 旁切圆estimate 估计；估计量estimator 估计量Euclidean algorithm 欧几里德算法Euclidean geometry 欧几里德几何Euler's formula 尤拉公式；欧拉公式evaluate 计值even function 偶函数even number 偶数evenly distributed 均匀分布的event 事件exact 真确exact differential form 恰当微分形式exact solution 准确解；精确解；真确解exact value 法确解；精确解；真确解example 例excentre 外心exception 例外excess 起exclusive 不包含exclusive disjunction 不包含性析取exclusive events 互斥事件exercise 练习exhaustive event(s) 彻底事件existential quantifier 存在量词expand 展开expand form 展开式expansion 展式expectation 期望expectation value, expected value 期望值；预期值experiment 实验；试验experimental 试验的experimental probability 实验概率explicit function 显函数exponent 指数exponential function 指数函数exponential order 指数阶; 指数级express…in terms of… 以………表达expression 式；数式extension 外延；延长；扩张；扩充extension of a function 函数的扩张exterior angle 外角external angle bisector 外分角external point of division 外分点extreme point 极值点extreme value 极值extremum 极值face 面factor 因子；因式；商factor method 因式分解法factor theorem 因子定理；因式定理factorial 阶乘factorization 因子分解；因式分解factorization of polynomial 多项式因式分解fallacy 谬误FALSE 假(的)falsehood 假值family 族family of circles 圆族family of concentric circles 同心圆族family of straight lines 直线族feasible solution 可行解；容许解Fermat's last theorem 费尔马最后定理Fibonacci number 斐波那契数；黄金分割数Fibonacci sequence 斐波那契序列fictitious mean 假定平均数figure (1)图(形)；(2)数字final velocity 末速度finite 有限finite dimensional vector space 有限维向量空间finite population 有限总体finite probability space 有限概率空间finite sequence 有限序列finite series 有限级数finite set 有限集first approximation 首近似值first derivative 一阶导数first order differential equation 一阶微分方程first projection 第一投影; 第一射影first quartile 第一四分位数first term 首项fixed deposit 定期存款fixed point 定点fixed point iteration method 定点迭代法fixed pulley 定滑轮flow chart 流程图focal axis 焦轴focal chord 焦弦focal length 焦距focus(foci) 焦点folium of Descartes 笛卡儿叶形线foot of perpendicular 垂足for all X 对所有Xfor each /every X 对每一Xforce 力forced oscillation 受迫振动form 形式；型formal proof 形式化的证明format 格式；规格formula(formulae) 公式four leaved rose curve 四瓣玫瑰线four rules 四则four-figure table 四位数表fourth root 四次方根fraction 分数；分式fraction in lowest term 最简分数fractional equation 分式方程fractional index 分数指数fractional inequality 分式不等式free fall 自由下坠free vector 自由向量; 自由矢量frequency 频数；频率frequency distribution 频数分布；频率分布frequency distribution table 频数分布表frequency polygon 频数多边形；频率多边形friction 摩擦; 摩擦力frictionless motion 无摩擦运动frustum 平截头体fulcrum 支点function 函数function of function 复合函数；迭函数functional notation 函数记号fundamental theorem of algebra 代数基本定理fundamental theorem of calculus 微积分基本定理gain 增益；赚；盈利gain perent 赚率；增益率；盈利百分率game （1）对策；（2）博奕Gaussian distribution 高斯分布Gaussian elimination 高斯消去法general form 一般式；通式general solution 通解；一般解general term 通项generating function 母函数; 生成函数generator (1)母线; (2)生成元geoborad 几何板geometric distribution 几何分布geometric mean 几何平均数；等比中项geometric progression 几何级数；等比级数geometric sequence 等比序列geometric series 等比级数geometry 几何；几何学given 给定；已知global 全局; 整体global maximum 全局极大值; 整体极大值global minimum 全局极小值; 整体极小值golden section 黄金分割grade 等级gradient (1)斜率；倾斜率；(2)梯度grand total 总计graph 图像；图形；图表graph paper 图表纸graphical method 图解法graphical representation 图示；以图样表达graphical solution 图解gravitational acceleration 重力加速度gravity 重力greatest term 最大项greatest value 最大值grid lines 网网格线group 组；grouped data 分组数据；分类数据grouping terms 并项；集项growth 增长growth factor 增长因子half angle 半角half angle formula 半角公式half closed interval 半闭区间half open interval 半开区间harmonic mean (1) 调和平均数; (2) 调和中项harmonic progression 调和级数head 正面（钱币）height 高（度）helix 螺旋线hemisphere 半球体；半球heptagon 七边形Heron's formula 希罗公式heterogeneous (1)参差的; (2)不纯一的hexagon 六边形higher order derivative 高阶导数highest common factor(H.C.F) 最大公因子；最高公因式；最高公因子Hindu-Arabic numeral 阿刺伯数字histogram 组织图；直方图；矩形图Holder's Inequality 赫耳德不等式homogeneous 齐次的homogeneous equation 齐次方程Hooke's law 虎克定律horizontal 水平的；水平horizontal asymptote 水平渐近线horizontal component 水平分量horizontal line 横线 ；水平线horizontal range 水平射程hyperbola 双曲线hyperbolic function 双曲函数hypergeometric distribution 超几何分布hypocycloid 内摆线hypotenuse 斜边hypothesis 假设hypothesis testing 假设检验hypothetical syllogism 假设三段论hypotrochoid 次内摆线idempotent 全幂等的identical 全等；恒等identity 等（式）identity element 单位元identity law 同一律identity mapping 恒等映射identity matrix 恒等矩阵identity relation 恒等关系式if and only if/iff 当且仅当；若且仅若if…, then 若….则；如果…..则illustration 例证；说明image 像点；像image axis 虚轴imaginary circle 虚圆imaginary number 虚数imaginary part 虚部imaginary root 虚根imaginary unit 虚数单位impact 碰撞implication 蕴涵式；蕴含式implicit definition 隐定义implicit function 隐函数imply 蕴涵；蕴含impossible event 不可能事件improper fraction 假分数improper integral 广义积分; 非正常积分impulse 冲量impulsive force 冲力incentre 内力incircle 内切圆inclination 倾角；斜角inclined plane 斜面included angle 夹角included side 夹边inclusion mapping 包含映射inclusive 包含的；可兼的inclusive disjunction 包含性析取；可兼析取inconsistent 不相的(的)；不一致(的) increase 递增；增加increasing function 递增函数increasing sequence 递增序列increasing series 递增级数increment 增量indefinite integral 不定积分idenfinite integration 不定积分法independence 独立；自变independent equations 独立方程independent event 独立事件independent variable 自变量；独立变量indeterminate (1)不定的；(2)不定元；未定元indeterminate coefficient 不定系数；未定系数indeterminate form 待定型；不定型index,indices 指数；指index notation 指数记数法induced operation 诱导运算induction hypothesis 归纳法假设inelastic collision 非弹性碰撞inequality 不等式；不等inequality sign 不等号inertia 惯性；惯量infer 推断inference 推论infinite 无限；无穷infinite dimensional 无限维infinite population 无限总体infinite sequence 无限序列；无穷序列infinite series 无限级数；无穷级数infinitely many 无穷多infinitesimal 无限小；无穷小infinity 无限(大)；无穷(大)inflection (inflexion) point 拐点；转折点inherent error 固有误差initial approximation 初始近似值initial condition 原始条件；初值条件initial point 始点；起点initial side 始边initial value 初值；始值initial velocity 初速度initial-value problem 初值问题injection 内射injective function 内射函数inner product 内积input 输入input box 输入inscribed circle 内切圆insertion 插入insertion of brackets 加括号instantaneous 瞬时的instantaneous acceleration 瞬时加速度instantaneous speed 瞬时速率instantaneous velocity 瞬时速度integer 整数integrable 可积integrable function 可积函数integral 积分integral index 整数指数integral mean value theorem 积数指数integral part 整数部份integral solution 整数解integral value 整数值integrand 被积函数integrate 积；积分；......的积分integrating factor 积分因子integration 积分法integration by parts 分部积分法integration by substitution 代换积分法；换元积分法integration constant 积分常数interaction 相互作用intercept 截距；截段intercept form 截距式intercept theorem 截线定理interchange 互换interest 利息interest rate 利率interest tax 利息税interior angle 内角interior angles on the same side of the transversal 同旁内角interior opposite angle 内对角intermediate value theorem 介值定理internal bisector 内分角internal division 内分割internal energy 内能internal force 内力internal point of division 内分点interpolating polynomial 插值多项式interpolation 插值inter-quartile range 四分位数间距intersect 相交intersection (1)交集；(2)相交；(3)交点interval 区间interval estimation 区间估计；区域估计intuition 直观invalid 失效；无效invariance 不变性invariant (1)不变的；(2)不变量；不变式inverse 反的；逆的inverse circular function 反三角函数inverse cosine function 反余弦函数inverse function 反函数；逆函数inverse cosine function 反三角函数inverse function 反函数；逆映射inverse mapping 反向映射；逆映射inverse matrix 逆矩阵inverse problem 逆算问题inverse proportion 反比例；逆比例inverse relation 逆关系inverse sine function 反正弦函数inverse tangent function 反正切函数inverse variation 反变(分)；逆变(分) invertible 可逆的invertible matrix 可逆矩阵irrational equation 无理方程irrational number 无理数irreducibility 不可约性irregular 不规则isomorphism 同构isosceles triangle 等腰三角形iterate (1)迭代值; (2)迭代iteration 迭代iteration form 迭代形iterative function 迭代函数iterative method 迭代法jet propulsion 喷气推进joint variation 联变(分)；连变(分)kinetic energy 动能kinetic friction 动摩擦known 己知L.H.S. 末项L'Hospital's rule 洛必达法则Lagrange interpolating polynomial 拉格朗日插值多项代Lagrange theorem 拉格朗日定理Lami's law 拉密定律Laplace expansion 拉普拉斯展式last term 末项latent root 本征根; 首通径lattice point 格点latus rectum 正焦弦; 首通径law 律；定律law of conservation of momentum 动量守恒定律law of indices 指数律；指数定律law of inference 推论律law of trichotomy 三分律leading coefficient 首项系数leading diagonal 主对角线least common multiple, lowest common multiple (L.C.M) 最小公倍数；最低公倍式least value 最小值left hand limit 左方极限lemma 引理lemniscate 双纽线length 长(度)letter 文字；字母like surd 同类根式like terms 同类项limacon 蜗牛线limit 极限limit of sequence 序列的极限limiting case 极限情况limiting friction 最大静摩擦limiting position 极限位置line 线；行line of action 作用力线line of best-fit 最佳拟合line of greatest slope 最大斜率的直 ；最大斜率line of intersection 交线line segment 线段linear 线性；一次linear convergence 线性收敛性linear differeantial equation 线性微分方程linear equation 线性方程；一次方程linear equation in two unknowns 二元一次方程；二元线性方程linear inequality 一次不等式；线性不等式linear momentum 线动量linear programming 线性规划linearly dependent 线性相关的linearly independent 线性无关的literal coefficient 文字系数literal equation 文字方程load 负荷loaded coin 不公正钱币loaded die 不公正骰子local maximum 局部极大(值)local minimum 局部极小(值)locus, loci 轨迹logarithm 对数logarithmic equation 对数方程logarithmic function 对数函数logic 逻辑logical deduction 逻辑推论；逻辑推理logical step 逻辑步骤long division method 长除法loop 回路loss 赔本；亏蚀loss per cent 赔率；亏蚀百分率lower bound 下界lower limit 下限lower quartile 下四分位数lower sum 下和lower triangular matrix 下三角形矩阵lowest common multiple(L.C.M) 最小公倍数machine 机械Maclaurin expansion 麦克劳林展开式Maclaurin series 麦克劳林级数magnitude 量；数量；长度；大小major arc 优弧；大弧major axis 长轴major sector 优扇形；大扇形major segment 优弓形；大弓形mantissa 尾数mantissa of logarithm 对数的尾数；对数的定值部many to one 多个对一个many-sided figure 多边形many-valued 多值的map into 映入map onto 映上mapping 映射marked price 标价Markov chain 马可夫链mass 质量mathematical analysis 数学分析mathematical induction 数学归纳法mathematical sentence 数句mathematics 数学matrix 阵; 矩阵matrix addition 矩阵加法matrix equation 矩阵方程matrix multiplication 矩阵乘法matrix operation 矩阵运算maximize 极大maximum absolute error 最大绝对误差maximum point 极大点maximum value 极大值mean 平均(值)；平均数；中数mean deviation 中均差；平均偏差mean value theorem 中值定理measure of dispersion 离差的量度。