Translation of Taylor series into LFT expansions

"Series expansion" redirects here. For other notions of the term, see series.As the degree of the Taylor polynomial rises, it approaches the correct function. This image shows sin x and Taylor approximations, polynomials of degree 1, 3, 5, 7, 9, 11 and 13.The exponential function (in blue), and the sum of the first n+1 terms of its Taylor series at 0 (in red).In mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivativesat a single point. It may be regarded as the limit of the Taylor polynomials. Taylor series are named after English mathematician Brook Taylor. If the series is centered at zero, the series is also called a Maclaurin series, named after Scottish mathematician Colin Maclaurin.DefinitionThe Taylor series of a real or complex function f(x) that is infinitely differentiable in a neighborhood of a real or complex number a, is the power serieswhich in a more compact form can be written aswhere n! is the factorial of n and f (n)(a) denotes the n th derivative of f evaluated at the point a; the zeroth derivative of f is defined to be f itself and (x−a)0 and 0! are both defined to be 1.Often f(x) is equal to its Taylor series evaluated at x for all x sufficiently close to a. This is the main reason why Taylor series are important.In the particular case where a= 0, the series is also called a Maclaurin series.[edit] ExamplesThe Maclaurin series for any polynomial is the polynomial itself. The Maclaurin series for (1 −x)−1 is the geometric seriesso the Taylor series for x−1 at a = 1 isBy integrating the above Maclaurin series we find the Maclaurin series for −log(1 −x), where log denotes the natural logarithm:and the corresponding Taylor series for log(x) at a = 1 isThe Taylor series for the exponential function e x at a = 0 isThe above expansion holds because the derivative of e x is also e x and e0 equals 1. This leaves the terms (x− 0)n in the numerator and n! in the denominator for each term in the infinite sum.[edit] ConvergenceThe sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full period centered at the origin.The Taylor polynomials for log(1+x) only provide accurate approximations in the range −1 < x≤ 1. Note that, for x > 1, the Taylor polynomials of higher degree are worse approximations.Taylor series need not in general be convergent, but often they are. The limit of a convergent Taylor series of a function f need not in generalbe equal to the function value f(x), but often it is. If f(x) is equal to its Taylor series in a neighborhood of a, it is said to be analytic in this neighborhood. If f(x) is equal to its Taylor series everywhere it is called entire. The exponential function e x and the trigonometric functions sine and cosine are examples of entire functions. Examples of functions that are not entire include the logarithm, the trigonometric function tangent, and its inverse arctan. For these functions the Taylor series do not converge if x is far from a.Taylor series can be used to calculate the value of an entire function in every point, if the value of the function, and of all of its derivatives, are known at a single point. Uses of the Taylor series for entire functions include:1.The partial sums (the Taylor polynomials) of the series can be usedas approximations of the entire function. These approximations are good if sufficiently many terms are included.2.The series representation simplifies many mathematical proofs.Pictured on the right is an accurate approximation of sin(x) around the point a = 0. The pink curve is a polynomial of degree seven:The error in this approximation is no more than |x|9/9!. In particular, for |x| < 1, the error is less than 0.000003.In contrast, also shown is a picture of the natural logarithm function log(1 + x) and some of its Taylor polynomials around a = 0. These approximations converge to the function only in the region −1 < x≤ 1; outside of this region the higher-degree Taylor polynomials are worse approximations for the function. This is similar to Runge's phenomenon.Taylor's theorem gives a variety of general bounds on the size of the error in R n(x) incurred in approximating a function by its n th-degree Taylor polynomial.[edit] HistoryThe Greek philosopher Zeno considered the problem of summing an infinite series to achieve a finite result, but rejected it as an impossibility: the result was Zeno's paradox. Later, Aristotle proposed a philosophicalresolution of the paradox, but the mathematical content was apparently unresolved until taken up by Democritus and then Archimedes. It was through Archimedes's method of exhaustion that an infinite number of progressive subdivisions could be performed to achieve a finite result.[1] Liu Hui independently employed a similar method a few centuries later.[2]In the 14th century, the earliest examples of the use of Taylor series and closely-related methods were given by Madhava of Sangamagrama.[3] Though no record of his work survives, writings of later Indian mathematicians suggest that he found a number of special cases of the Taylor series, including those for the trigonometric functions of sine, cosine, tangent, and arctangent. The Kerala school of astronomy and mathematics further expanded his works with various series expansions and rational approximations until the 16th century.In the 17th century, James Gregory also worked in this area and published several Maclaurin series. It was not until 1715 however that a general method for constructing these series for all functions for which they exist was finally provided by Brook Taylor[4], after whom the series are now named.The Maclaurin series was named after Colin Maclaurin, a professor in Edinburgh, who published the special case of the Taylor result in the 18th century.[edit] PropertiesThe function e−1/x²is not analytic at x= 0: the Taylor series is identically 0, although the function is not.If this series converges for every x in the interval (a−r, a + r) and the sum is equal to f(x), then the function f(x) is said to be analytic in the interval(a−r, a+ r). If this is true for any r then the function is said to be an entire function. To check whether the series converges towards f(x), one normally uses estimates for the remainder term of Taylor's theorem. A function is analytic if and only if it can be represented as a power series; the coefficients in that power series are then necessarily the ones given in the above Taylor series formula.The importance of such a power series representation is at least fourfold. First, differentiation and integration of power series can be performed term by term and is hence particularly easy. Second, an analytic function can be uniquely extended to a holomorphic function defined on an open disk in the complex plane, which makes the whole machinery of complex analysis available. Third, the (truncated) series can be used to compute function values approximately (often by recasting the polynomial into the Chebyshev form and evaluating it with the Clenshaw algorithm).Fourth, algebraic operations can often be done much more readily on the power series representation; for instance the simplest proof of Euler's formula uses the Taylor series expansions for sine, cosine, and exponential functions. This result is of fundamental importance in such fields as harmonic analysis.Another reason why the Taylor series is the natural power series for studying a function f is given by the probabilistic interpretation of Taylor series. Given the value of f and its derivatives at a point a, the Taylor series is in some sense the most likely function that fits the given data.Note that there are examples of infinitely differentiable functions f(x) whose Taylor series converge, but are not equal to f(x). For instance, the function defined pointwise by f(x) = e−1/x² if x≠ 0 and f(0) = 0 is an example of a non-analytic smooth function. All its derivatives at x = 0 are zero, so the Taylor series of f(x) at 0 is zero everywhere, even though the function is nonzero for every x≠ 0. This particular pathology does not afflict Taylor series in complex analysis. There, the area of convergence of a Taylor series is always a disk in the complex plane (possibly with radius 0), and where the Taylor series converges, it converges to the function value. Notice that e−1/z²does not approach 0 as z approaches 0 along the imaginary axis, hence this function is not continuous as a function on the complex plane.Since every sequence of real or complex numbers can appear as coefficients in the Taylor series of an infinitely differentiable function defined onthe real line, the radius of convergence of a Taylor series can be zero.[5] There are even infinitely differentiable functions defined on the real line whose Taylor series have a radius of convergence 0 everywhere.[6]Some functions cannot be written as Taylor series because they have a singularity; in these cases, one can often still achieve a series expansion if one allows also negative powers of the variable x; see Laurent series. For example, f(x) = e−1/x² can be written as a Laurent series.[edit] List of Taylor series of some common functionsSee also List of mathematical seriesThe cosine function in the complex plane.An 8th degree approximation of the cosine function in the complex plane.The two above curves put together.Several important Maclaurin series expansions follow.[7] All these expansions are valid for complex arguments .Exponential function:Natural logarithm:Finite geometric series:Infinite geometric series:Variants of the infinite geometric series:Square root:Binomial series (includes the square root for α = 1/2 and the infinite geometric series for α = −1):with generalized binomial coefficientsTrigonometric functions:where the B s are Bernoulli numbers.Hyperbolic functions:Lambert's W function:The numbers B k appearing in the summation expansions of tan(x) and tanh(x) are the Bernoulli numbers. The E k in the expansion of sec(x) are Euler numbers.[edit] Calculation of Taylor seriesSeveral methods exist for the calculation of Taylor series of a large number of functions. One can attempt to use the Taylor series as-is and generalize the form of the coefficients, or one can use manipulations such as substitution, multiplication or division, addition or subtraction of standard Taylor series to construct the Taylor series of a function, by virtue of Taylor series being power series. In some cases, one can also derive the Taylor series by repeatedly applying integration by parts. Particularly convenient is the use of computer algebra systems to calculate Taylor series.[edit] First exampleCompute the 7th degree Maclaurin polynomial for the function.First, rewrite the function as.We have for the natural logarithm (by using the big O notation)and for the cosine functionThe latter series expansion has a zero constant term, which enables us to substitute the second series into the first one and to easily omit terms of higher order than the 7th degree by using the big O notation:Since the cosine is an even function, the coefficients for all the odd powers x, x3, x5, x7, . . . have to be zero.[edit] Second exampleSuppose we want the Taylor series at 0 of the function.We have for the exponential functionand, as in the first example,Assume the power series isThen multiplication with the denominator and substitution of the series of the cosine yieldsCollecting the terms up to fourth order yieldsComparing coefficients with the above series of the exponential function yields the desired Taylor series[edit] Taylor series as definitionsClassically, algebraic functions are defined by an algebraic equation, and transcendental functions (including those discussed above) are defined by some property that holds for them, such as a differential equation. For example the exponential function is the function which is everywhere equal to its own derivative, and assumes the value 1 at the origin. However, one may equally well define an analytic function by its Taylor series.Taylor series are used to define functions in diverse areas of mathematics. In particular, this is true in areas where the classical definitions of functions break down. For example, using Taylor series, one may defineanalytical functions of matrices and operators, such as the matrix exponential or matrix logarithm.In other areas, such as formal analysis, it is more convenient to work directly with the power series themselves. Thus one may define a solution of a differential equation as a power series which, one hopes to prove, is the Taylor series of the desired solution.[edit] Taylor series in several variablesThe Taylor series may also be generalized to functions of more than one variable withFor example, for a function that depends on two variables, x and y, the Taylor series to second order about the point (a, b) is:where the subscripts denote the respective partial derivatives.A second-order Taylor series expansion of a scalar-valued function of more than one variable can be compactly written aswhere is the gradient and is the Hessian matrix. Applyingthe multi-index notation the Taylor series for several variables becomesin full analogy to the single variable case.[edit] See also∙Taylor's theorem∙Linear approximation∙Power series∙Laurent series∙Holomorphic functions are analytic— a proof that a holomorphic function can be expressed as a Taylor power series ∙Newton's divided difference interpolation∙Difference engine∙Mean value theorem[edit] Notes1.^ Kline, M. (1990) Mathematical Thought from Ancient to Modern Times.Oxford University Press. pp. 35-37.2.^Boyer, C. and Merzbach, U. (1991) A History of Mathematics. John Wileyand Sons. pp. 202-203.3.^"Neither Newton nor Leibniz - The Pre-History of Calculus and CelestialMechanics in Medieval Kerala". MAT 314. Canisius College. Retrieved on 2006-07-09.4.^ Taylor, Brook, Methodus Incrementorum Directa et Inversa [Direct andReverse Methods of Incrementation] (London, 1715), pages 21-23(Proposition VII, Theorem 3, Corollary 2). Translated into English inD. J. Struik, A Source Book in Mathematics 1200-1800 (Cambridge,Massachusetts: Harvard University Press, 1969), pages 329-332.5.^ Exercise 12 on page 418 in Walter Rudin, Real and Complex Analysis.McGraw-Hill, New Dehli 1980, ISBN 0-07-099557-56.^ Exercise 13, same book7.^ Most of these can be found in (Abramowitz & Stegun 1970).。

On Taylor ' s formula for the resolvent of a complex matrixMatthew X. He a, Paolo E. Ricci b,_Article history:Received 25 June 2007Received in revised form 14 March 2008Accepted 25 March 2008Keywords: Powers of a matrixMatrix invariantsResolvent1.IntroductionAs a consequence of the Hilbert identity in [1], the resolvent R (A) =( A) 1of a nonsingular square matrix A( denoting the identity matrix) is shown to be an analytic function of the parameter in any domain D with empty intersection with the spectrum A of A. Therefore, by using Taylor expansion in a neighborhood of any fixed 0 D, we can find in [ 1] a representation formula for R (A) using all powers of R 0(A).In this article, by using some preceding results recalled, e.g., in 2[], we write down a representation formula using only a finite number of powers of R 0(A). This seems to be natural since only the first powers of R 0(A) are linearly independent.The main tool in this framework is given by the multivariable polynomials F k,n(v1,v2,...,v r ) ( n 1,0,1,... ; k 1,2,..., m r ) (see [ 2–6]), depending on the invariants (v1,v2,...,v r) of R (A)); here m denotes the degree of the minimal polynomial.2.Powers of matrices n a d F k,n functionsWe recall in this section some results on representation formulas for powers of matrices (see e.g. [2–6] and the references therein). For simplicity we refer to the case when the matrix is nonderogatory so that m r .Proposition 2.1. Let A be an r r(r 2) complex matrix, and denote by u1,u2,...,u r the invariants of A, and byrP( ) det( A) ( 1) j u j r j.j0its characteristic polynomial (by convention u0 1); then for the powers of A withnonnegative integral exponents the following representation formula holds true: A n F1,n i(u1,...,u7)A r 1F2,n 1(u1,u2,...,u r)A r 2F r,n 1(u1,u2, u r) . (2.1)The functions F k,n(u1, ,u r) that appear as coefficients in (2.1) are defined by the recurrence relationr1F k,n(u1, ,u r) u1F k,n 1(u1, ,u r) u2F k,n 2(u1, ,u r ) ( 1)r 1u r F k,n r (u1, ,u r)(k 1, ,r;n 1) (2.2) and initial conditions:F r k 1,h 2 (u1, ,u7) k,h, (k,h 1, ,r) . (2.3) Furthermore, if A is nonsingular (u r 0) , then formula (2.1) still holds for negative values of n, provided that we define the F k,n function for negative values of n as follows:F k,n(u1, ,u7) F r k 1, n r 2(u r 1 , ,u1 , 1)，(k 1, ,r;n 1).u r u r u73.Taylor expansion of the resolventWe consider the resolvent matrix R (A) defined as follows:R R (A) ( A) 1. (3.1) Note that sometimes there is a change of sign in Eq. (3.1), but this of course is not essential.It is well known that the resolvent is an analytic (rational) function of in every domain D of the complex plane excluding the spectrum of A, and furthermore it is vanishing at infinity so the only singular points (poles) of R (A) are the eigenvalues of A.In [6] it is proved that the invariants v1,v2, ,v r of R (A) are linked with those of A by the equationsl j r j l jv l ( ) ( 1)j( u j l j,(l 1,2, ,r). (3.2)j 0 l jAs a consequence of Proposition 2.1, and Eq. (3.2), the integral powers of R (A)can be represented as follows.Theorem 3.1 For every A and n N , Ar1R n (A) F r k,n 1(v 1( ),v 2( ), ,v r ( ))R k (A), (3.3)k0 where the v l ( ) (l 1,2, ,r) are given by Eq.(3.2). Denoting by (A) the spectral radius of A, for every , such that (A) min( , ), the Hilbert identity holds true(see [1]):R (A) R (A) ( )R (A)R (A). (3.4)Therefore for every A , we have Aand in generalk d R k (A) ( 1)k kR k 1(A),(k 1,2, ; A ) d so, for every 0 D,R ( A) can be expanded in the Taylor seriesR (A) ( 1)k kR k 01(A)(0)k ,k0 which is absolutely and uniformly convergent in D. Defining00 v 1 v 1( 0), ,v r v r ( 0),0 0 0 F k,n F k,n (v 1, ,v r ), where the v l ( ) are defined by Eq. (3.2), we can prove the following theorem.Theorem 3.2 The Taylor expansion (3.7) of the resolvent R (A) in a neighborhood of any regular point 0 can be written in the formTherefore we can derive as a consequence:Corollary 3.1 For every 0 A and L 1,2, r the series expansionsdR (A) d(3.5) (3.6)(3.7) (3.8)(3.9) r1 R (A) h0 0 ( 1)k F k0 r n,k ( 0)k R n 0 (A) . (3.10)Therefore, taking into account the initial conditions (2.3) we can write00R (A) ( 1)k F r,k (0)k ( 1)k F r 1,k (0)k R 0k 0 k 0 0 ( 1)k F 1,k ( 0)k R R 0 1,k0 so (3.10) holds true. The convergence of series expansions (3.11) is a trivialconsequence of the convergence of the initial expansion (3.7).4. Concluding remarksIt is worth noting that the resolvent R (A) is a keynote element for representing analytic functions of a matrix A. In fact,denoting by f (z) a function of the complex variable z , analytic in a domain containing the spectrum of A, and denoting by k (k 1,2 ,s) the distinct eigenvalues of A with multiplicitiesk ,the Lagrange –Sylvester formula (see [4]) is given bys k 1f (j)( ) f (A) f ( k ) k j ,k 1 j 0 jwhere k (0)k (k 1,2, ,s) is the projector associated with the eigenvalue k ,andk (j) ( k l A)j k (0),(k 1,2, ,s; j 0,1, , k 1). Denoting by k a Jordan curve, the boundary of the domain D k , separating a fixed 0 ( 1)k F l,k ( 0)k k0 (3.11)are convergent. Proof. Recalling (3.3), we can write 0 0 0R k 01 F 1,k R r 01 F 2,k R r 02F r,k ，k N)，R (A) R 00 0 0 F 1,1R R 01 F 2,1R R 0 2 F r,1 (0) 0 0 0 F 1,2 R r 01 F 2,2 R r 02F r,2 ( 0)2 ( 1)k 0 F 1,k R r 0 10 F 2,k R r 02 0 F r,k ( 0)k from all other eigenvalues, recalling the Riesz formula, it follows that1k 2 i k R (A)d . When k is only known approximately, this projector cannot be derived by using the residue theorem.In this case it is necessary to integrate R (A) along k (being possibly aGershgorin circle), by using the known representation of the resolvent (see [3]) or by substituting R (A) with its Taylor expansion, and assuming as initial point any 0 K inside D k .Which is the best formula depends on the relevant stability and computational cost. From the theoretical point of view,formulas (3.7), (3.10) and (4.1) seem to be equivalent from the stability point of view, since all require knowledge of invariants of the given matrix A. However, in our opinion, in the situation considered, Eq. (3.10) seems to be less expensive with respect to (3.7), since it requires one to approximate r series of elementary functions instead of an infinite series of matrices.AcknowledgementsWe are grateful to the anonymous referees for comments that led us to improve this paper.References[1] I. Glazman, Y. Liubitch, Analyse lin aire dans leés espaces de dimension finies:Manuel et probl meès, in: H. Damadian (Ed.), Traduit du russe par, Mir, Moscow, 1972.[2] M. Bruschi, P.E. Ricci, Sulle potenze di una matrice quadrata della quale sia noto ilpolinomio minimo, Pubbl. Ist. Mat. Appl. Fac. Ing. Univ. Stud. Roma, Quad. 13(1979) 9–18.[3] V.N. Faddeeva, Computational Methods of Linear Algebra, Dover Pub. Inc., NewYork, 1959.[4] F.R. Gantmacher, The Theory of Matrices, Vols. 1, 2 (K.A. Hirsch, Trans.), ChelseaPublishing Co., New York, 1959.[5] M. Bruschi, P.E. Ricci, Sulle funzioni Fk,n e i polinomi di Lucas di seconda speciegeneralizzati, Pubbl. Ist. Mat. Appl. Fac. Ing. Univ. Stud. Roma, Quad. 14(1979) 49–58.[6] M. Bruschi, P.E. Ricci, An explicit formula for f (A) and the generating function ofthe generalized Lucas polynomials, SIAM J. Math. Anal. 13 (1982)R (A)r 1 r k 1P( ) k 0 j 0( 1)j j A k , (4.1)。

taylor swift英语阅读理解Taylor Swift is an American singer-songwriter known for her narrative songwriting and emotional honesty. She was born on December 13, 1989, in Reading, Pennsylvania. Swift began singing at a young age and started writing her own songs at the age of 12.Swift's career gained momentum when she released her self-titled debut album in 2006. The album was a commercial success and produced several hit singles, including "Teardrops on My Guitar" and "Our Song." Swift's music is largely influenced by country-pop and she quickly became a prominent figure in the country music scene.In 2008, Swift released her second studio album, "Fearless," which spawned several chart-topping singles, such as "Love Story" and "You Belong with Me." The album earned Swift widespread recognition and critical acclaim. She became the youngest person to win the Grammy Award for Album of the Year at the age of 20.Over the years, Swift's sound has evolved and she has delved into different genres, incorporating elements of pop, synth-pop, and electronic music into her later albums. Some of her most popular and successful songs include "Shake It Off," "Blank Space," "Bad Blood," and "Lover."In addition to her successful music career, Swift has also acted in films such as "Valentine's Day" and "The Giver." She has been involved in various philanthropic endeavors and has been an advocate for causes such as education and disaster relief.Swift has won numerous awards throughout her career, including 11 Grammy Awards. She is known for her catchy melodies, relatable lyrics, and personal storytelling. Her albums often reflect her own experiences and relationships, which has contributed to her loyal fanbase.Overall, Taylor Swift has had a significant impact on the music industry and is recognized as one of the most influential and successful artists of her generation. Her talents as a singer, songwriter, and performer have garnered her widespread acclaim and she continues to evolve and redefine her sound.。

泰勒公式的证明及其应用数学与应用数学专业胡心愿［摘要］泰勒公式的相关理论是函数逼近论的基础.本文主要探索的是泰勒公式的一些证明方法，并对不同的证明方法进行相应的比较分析,在此基础上讨论泰勒公式在证明不等式、求函数极限、求近似值、求行列式的值、讨论了函数的凹凸性，判别拐点,判断级数敛散性等方面的应用.本文还针对多元函数的泰勒公式的推导和应用做了简单的论述。

［关键词］泰勒公式;不等式；应用；ProofofTaylor'sFormulaandItsApplicationMathematicsandApplicedMathematicsMajorHUXin-yuanAbstract：ThetheoryaboutTaylor'sFormulaisthebasiccontentofApproximationTheory。

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Keywords：Taylor'sFormula；inequality;application目录1泰勒公式。

A Comparative Study of Six Translated Versions ofReply to Li Shuyi from an Aesthetic PerspectiveI. IntroductionA. A brief introduction to the background of Reply to Li ShuyiReply to Li Shuyi is a poem written by Mao Zedong in memory of his beloved wife and his comrade. Li Shuyi was a teacher of Chinese in No. 10 Middle School of Changsha, who was a classmate and a good friend of Yang Kaihui--Mao Zedong’s first wife. In 1924, through Yang’s introduction, Li got to know, and later on married, Liu Zhixun; Liu was then one of Mao’s comrades-in-arms. In the summer of 1933, word came that Liu was killed in a battle. Weighed down with sadness, Li could not eat or sleep well; she later on wrote a ci-poem To the tune of Buddhist Dancers(《菩萨蛮》) in memory of her husband. It was not until the foundation of the People’s Republic of China that Li acquired the exact information about the death of her husband. On January 17, 1950, Li wrote to Mao, telling him how did Yang Kaihui die, and Mao replied to her on April 18.In January 1957, Mao’s poems (18 in all) were published in the first issue of the magazine Poetry in Peking; this reminded Li of a ci-poem On the Pillow to the tune of The Fair Lady Yu(《虞美人·枕上》) that Mao wrote to Yang when they first fell in love many years ago. However, Li could only remember the first two lines, she thus asked Mao for the whole piece, and in her letter, enclosing her own Buddhist Dancers on February 7. On May 11, Mao replied, “the poem Kaihui told you is not fine enough to be rewritten; here is enclosed a ci-poem Immortals to you. This sort of poetry is different from ancient poems about immortal, i.e. the poet himself does not appear in the poem. But in ci-poems about the seventh evening of theseventh moon”. The poem Mao mentioned in the letter is the poem above. On November 25, Mao agreed to publish the poem on condition that the original title Immortals is changed into To Li Shuyi. Thus, the poem was published in Journal of Hunan Normal College on New Year’s Day of 1958. In the 1963 edition of Poems of Chairman Mao, the title To Li Shuyi was changed into Reply to Li Shuyi.B. A brief introduction to “Triple Beauty” theoryChinese language is beautiful in three aspects: in sense so as to appeal to the heart, in sound so as to appeal to the ear and in form so as to appeal to the eye. Therefore, Mr. Xu Yuanchong, a well-known translator both home and abroad, proposed “Triple Beauty”theory in translation, that is, the translated verse should be as beautiful as the original in sense, sound and form.Beauty in sense means that the translated version should be faithful to the original in sense, and there should be no mistranslation, nor should there be over-translation or under-translation. Beauty in sense may come from historical associations, which will get lost in translation. However, the loss can be made up for by poetic diction. In order to bring out beauty in sense, we may choose the best possible words in the best possible order, borrow poetic diction from English and American poets and enhance beauty in sense by adding beauty in sound or in form so as to make up for the loss in either or both.Beauty in sound lies chiefly in rhythm and rhyme. In Chinese poetry, rhythm consists chiefly in even and oblique tones, which may be represented by strong and weak beats in translation, that is, a verse many be translated in iambics, trochaics, anapaestics or dactylics. Classical Chinese poetry consists chiefly of five-character and seven-characterlines. The former may be replaced by heroic couplets and the latter by Alexandrines. As to rhymes, it would be a happy translation if the English rhymes could more or less resemble the Chinese. Therefore, English metres can be used for representing Chinese tones, English rhymes can be used for replacing Chinese rhymes, while alliterations, assonances and repetitions can be used for bringing out the original beauty in sound.Beauty in form lies chiefly in line length and parallelism. Attention should be paid to line length and parallelism to bring out beauty in form. However, beauty in form is not as important as beauty in sound, still less than beauty in sense. Since beauty in sense is of first-rate importance, beauty in sound is of second-rate importance and beauty in form is of third-rate importance, we should do our best to make our version as beautiful as the original in the three aspects. If impossible, we may first leave out resemblance in form as in sound, but we should try our best to preserve the original beauty in sense.C. THESISThe thesis will give an in-depth analysis of Reply to Li Shuyi through the comparative study of six different translated versions with “Triple Beauty”theory as its basis. The ultimate aim lies in exploiting a relatively satisfactory way for the better representation of Mao’s poems, especially Reply to Li Shuyi and Chinese classical poetry so that it can maximally reproduce the original beauty which is so peculiarly unique to Chinese rhymed verse，and furthermore eliciting some reflections on poetry translation and translation study.The thesis will delve into the perspectives such as the translation of titles, tune names, puns and allusions, handling with images and figures of speech, as well asreproduction of meter, rhyme and equivalence in form. The thesis mainly emphasizes on how the poetic style, poetic flavor, imagery and culture conception of the original poem are preserved and refreshed in the translated versions, especially when the translator balances the manipulation on the dynamic equivalence and resemblance between the original and the translated piece from three aspects—sense, sound and form.II.The display of six different translated versionsA. The authorized versionREPLY TO LI SHU-YI-- to the tune of Tieh Lien HuaI lost my proud Poplar and you your Willow,Poplar and Willow soar to the Ninth Heaven.Wu Kang, asked what he can give,Serves them a laurel brew.The lonely moon goddess spreads her ample sleevesTo dance for these loyal souls in infinite space.Earth suddenly reports the tiger subdued,Tears of joy pour forth falling as mighty rain.(Poems of Mao Zedong 66)B. The version translated by Xu YuanchongTHE IMMORTALS--REPL Y TO LI SHUYITune:“BUTTERFLIES LINGERING OVER FLOWERS”You’ve lost your Willow and I’ve lost my Poplar proud, Their souls ascend the highest heaven, light as cloud.The Woodman, asked what he has for wine,Brings out a nectar of laurels divine.The lonely Goddess of the Moon, large sleeves outspread, Dances up endless skies for these immortal dead.From the earth comes the news of the Tiger o’erthrown,In a sudden shower their tears fly down.(Xu 88)C. The version translated by Gu ZhengkunREPLY TO LI SHUYITo the tune of Butterflies Love FlowersYou lost your darling Willow and I my Poplar proud,Both Poplar and Willow soar gracefully far above the cloud. They ask Wu Gang about what he has there,It’s the laurel wine that Wu offers them to share.The lonely goddess of them moon spreads her sleeves long,To console the loyal souls she dances in sky with a song. Suddenly the news about the tiger subdued comes from the earth, At once the rain pours down from our darlings’ tears of mirth. (Gu 143)D. The version translated by Zhao ZhentaoReply to Li ShuyiI lost my proud Yang, you your Liu.Like catkins of poplar and willow,They soared up lightly to the blue.Wu Gang was asked what he could offer;The god served an osmanthus brew.The lonely Goddess of the MoonDanced in her ample sleeves in heaven.Granting the loyal souls a boon.On earth the tiger got defeated.The news caused pours of tears so soon!(Zhao, 1980, 147)E. The version translated by Huang LongI was bereaved of my proud Poplar, and thou a Willow of thine;Poplar and Willow gently wing right up to the very heaven of heavens of nine. Wu Gang, upon an inquiry as to what’s available for his offer,Serves out with open hands osmanthus wine.Waving long flowing sleeves, Change E in solitude.Dance for these staunch souls in the myriad-li endless sky.At the sudden tiding of the Tiger on earth having been subdued,Burst into downpour of rain, their tears fly.(Huang 94)F. The version translated by NancyReply to Li Shu-yiI lost my proud Yang,You your dear Liu.—Light as yangliu in the wind, they soarStraight to the Realm beyond the blue.What, ah, has Wu Kang of the moon brought? In outstretched hands he bearsA cup of kwei blossom brew.Lone Chang-O steps out too,Our martyr-souls to entertain.Ample sleeves spreading, she whirls in dance Ten thousand leagues over the skyey main. There, comes word of Tiger Subdued on earth. Tears of joy fly—Lo, a downpour of celestial rain!(Lin 54)III. Comparative study of translated versionsA. Translation of the tune name and the titleAs an image, butterfly has two typical forms in the Chinese culture. One form is the meaning in the ci-poem “Die Lian Hua”(蝶恋花). This form gives readers a personalized meaning, which can remind the readers of the intimate relation between butterflies and flowers. Another form lies in a classic story named Liang Zhu(梁祝), in which the plot “Hua Die”(化蝶) occurs, implying eternal love. Both of these two forms represent the love between lovers, and this concept is well accepted by Chinese people.The tune name generally does not associate with the poetic content very closely, and most western readers could not understand the deep meaning and the culture background of the tune name. Since tune names are meaningless symbols for the westerners, some translators chose to omit the translation of the tune name, such as Nancy, Zhao Zhentao. On the contrary, the authorized version translated the tune name into Chinese phonetic alphabet, which will puzzle the western readers very much, because they do not know the meaning of these assembled alphabets, nor do they understand the culture background of the tune name without any notes. Gu Zhengkun translated the tune name as “butterflies love flowers”, which is better than the authorized version. His version provides the readers of the concept that the love and the relation between butterflies and flowers. However, love is a general term, and what it can offer to the readers is only inkling. Hence, there is no denying that Xu Yuanchong’s version is the best. In his version, he chose the word “linger”rather than the word “love”, which is more concrete, more dynamic, more appealing and more impressive. His version offers the readers with the vision that butterflies are flyingand lingering around the blossom of all kinds of flowers and are not willing to leave, which is much easier for the readers (especially the westerners) to be more aware of the essential meaning of the tune name.Concerned with the title, Xu Yuanchong’s version is characterized in the adding of “The Immortals”. In Mao Zedong’s reply to Li Shuyi of 1957, “The Immortals”were chosen by the poet as the title of the poem, thus Xu Yuanchong’s version seems more faithful to the original one. It will also help us to imagine the scene that the hero and the heroine are roaming in the heaven, being catered by the woodman—Wu Gang. “The Immortals” also speak out Mao Zedong’s hope that long live Yang Kaihui, long live Liu Zhixun, and long live all the heroes. Although they suffered a lot when they were alive, they will be very happy after their souls flying to the heaven, and their spirits will always inspire us to fight for freedom, for happiness and for justice. Xu Yuanchong also makes a concise note for the title, whereas Gu Zhengkun provides a note for the title with more than five hundred words. The note aims to explain the relationship between the hero, the heroine and the poet, but it may be too thorough for the readers to finish the note patiently. None of the other versions offer any notes.B. Translation of puns and comparative study of the first two linesPuns can improve expressive effects by using a word that has two meanings ,which is a common rhetoric speech both in English and Chinese with a long history. Since puns are deep-rooted in culture and language, translation of puns is always a demanding job.The words “yang” (poplar) and “liu” (willow)in this poem literally refer to the plants, originally two kinds of trees: poplar and willow. The subtlety here is that “poplar”(Yang) can suggest poplar flowers just as “willow” (Liu) can suggest willow catkin. Poplar in Chinese is pronounced as“yang”，the same as the pronunciation of the family name “yang(杨)”, while willow (liu) has the same pronunciation of the family name “liu(柳)”. In fact they refer to “Yang Kaihui”—Mao Zedong’s beloved wife and “Liu Zhixun”—Li Shuyi’s husband respectively. The pun in this poem is a homophonic pun, which depends on sound identity. The puns here has two meanings, one is the literal meaning while the other is implied meaning. Although the poet actually refers to the revolutionary martyrs, he used the sounds of their surnames, which are also the names of two plants.Both of poplars and willows have been used by numerous poets and writers in ancient times as well as present times in China, so they both have their own symbolism in Chinese culture. They can be categorized to cultural images in some way, for if they were mentioned, the Chinese people can feel their essential meaning naturally. Poplar is the incarnation of honesty and rightness, while willow is the symbol of tenacity and diligency, and these characters are what Yang Kaihui and Liu Zhixun have.Willow’s pronunciation in Chinese is very close to a Chinese character “liu”(留), which means that somebody asks his intimates to stay with him rather than leave. When friends have to depart from each other, people often use willow to express their reluctance to leave. After Sui and Tang dynasty, Chinese people began to take willow as a symbol of farewell, which become a custom of China. Meanwhile, willow also has the function to arise people’s yearning towards their relatives and friends. In western culture, the drooping tresses of willow are also endowed with a sorrowful meaning. Willow, also called weeping willow in English, stands for grieves, especially grieves caused by the loss of the spouse orthe lover. Therefore, willow in both Chinese culture and English culture has the similar cultural connotation.The first image that comes into the reader s’ view is the “proud Poplar”, which will easily remind the readers of the heroic bearing of Yang Kaihui, who was always busy with her revolution industry. The image of willow also shows the poet’s regrets for the leave of the two heroes. These two images were translated in two kinds of way. One way is to translate according to the meaning, and the other way is to translate by the sound.In the authorized version, Xu Yuanchong’s version, Gu Zhengku’s version, and Huang Long’s version, the images are translated in the first way. Therefore, “proud Poplar” and “Willow” is employed. The readers can easily figure out that the images were personified, dignified and sublimated with the capitalization of the initial letters. The readers can also comprehend Mao Tse-tune’s proud feeling toward Yang Kaihui from the very word “proud”. Except for Xu Yuanchong’s version, the rest versions are all translated like “Poplar and Willow soar to the heaven”. It is common sense that all plants are firmly rooted in soil, never to say that they can soar or fly, which is illogical. On the contrary, Xu’s version is distinguished of “their souls”, which not only combines the surnames of the heroes and the plant names, echoes with “loyal souls” in the following lines, but also illuminates the kernel of the poem very well. Meanwhile, the cultural connotation is emphasized through the free translation.Zhao Zhentao’s version and Nancy’s version were finished in the second way. The readers can only know the surnames of the heroes who have passed away. Nancy did not enclose a note to introduce the background, and thus it is impossible for the westerners toperceive the melancholy hidden in the poem. Zhao Zhentao exploited a simile to make up his version and added the plants names into his version, which is better than Nancy’s, but still less wonderful than the other translators.In the first line, Huang Long’s version is the most impressive for his choice of words. “B ereaved” means that somebody lost a close friend or relative because they have passed away, hence the word can better illustrate the poet’s sadness for losing his beloved wife with more affection than the word “lost”. In addition, the poet is a bereaved husband, while Li Shuyi is a bereaved wife. Thus, the very word “bereaved”forms another pun, which illustrates the original poem sublimely. “Thou”and “thine”all came from the Middle English, which is more vivid and more brilliant than the other s’choice, for it preserves more ancientry in the original poem compared with other versions.In the first line, only Xu Yuanchong chose to use present perfect. This tense revealed that the bereaving of Yang Kaihui laid huge impacts on Mao Zedong, and the impacts last for such long time that even after more than thirty years Mao still can not relieve.The second lines depicted how the souls of the heroes fly to heaven. The phrase “q ing yang” (轻飏) of the Chinese is original, which was translated into different versions, such as “soar” in authorized version, Gu Zhengkun’s version and Zhao Zhentao’s version; “ascend light as cloud”in Xu Yuanchong’s version; “gently wing”in Huang Long’s version. T he character “yang” carries a definite wind-image in Chinese which is absent in its English equivalent word “soar”. To bring out the full image effect of the line, “Light as willows in the wind they soar” is actually a more adequate and faithful rendering than “Like willows they soar” as found in current translations.Thus Nancy’s version andZhao’s version is better than the authorized version and Gu’s version. Xu’s version is very vivid, “light as clouds”manifests the springiness of the souls when they were flying. Huang Long’s version can help us to imagine that the heroes are flying with wings like the angles, which is also very beautiful. “The transferred meaning of the word “wing” can be paraphrased in an ancient Chinese legend, which says people can ascend to the heaven and become immortal. This legend in Chinese called “yu hua”(羽化)” (Huang 95). Thus, the very word “wing” can echo with the original title THE IMMORTALS.“Chong xiao jiu(重霄九)” is “jiu chong xiao(九重霄)”in Chinese，which means the highest place of the sky. Mao Zedong’s reversing of the phrase sequence is for the sake that he prefers to make an end rhyme through the reversing. The authorized version translated this image as “the Ninth Heaven”. Obviously, this is a literal translation. Every person knows heaven is a place of great happiness, delight, and pleasure, which is the abode of God, angels as well as the souls of those who are granted salvation, and people there will get everlasting bliss. However, “the Ninth Heaven” may lead to ambiguity and mislead the westerners, because whether there is a “ninth heaven”in western culture is argued. “An ancient Greek named Hipparchus once pointed out that the heaven has nine layers with the ninth heaven being crystalline sphere”(Zhao, 8, 1979); Dante’s Divine Comedy also mentioned the empyrean, a word used as a name for the firmament, the dwelling place of God and the blessed, as well as the source of light. Whereas, “there are also some people holding a different idea, and they believe that the highest heaven in the western culture is the seventh heaven, in which God and angels live” (Hua 207) and (Chen 576). Thus we can conclude, the phrase “Ninth Heaven” is a choice, but it will not be thebest choice. Xu Yuanchong’s “the highest heaven” and Huang Long’s “the very heaven of heavens”both are faithful to the original poem and are better than “the Ninth Heaven”; Zhao Zhentao’s “the blue”and Nancy’s “the Realm beyond the blue”are translated according the meaning, which can be easily understood by the westerners; Gu Zhengkun’s “far above the cloud” also preserved the original meaning very well.C.Translation of allusionsAllusions are one of the most brilliant essences in both Chinese and English，which are succinct in form but profound in meaning. Allusions are reference to famous persons, things or event that writers suppose are familiar to their readers. This suppose is on the basis that their readers share the knowledge or belief with them having a common historical, cultural and literary heritage, which can enable the readers to identify the allusions and to understand their essence.In this poem, there are three allusions in total, that is, the allusion of Wu Gang(吴刚) and Chang-E(嫦娥), which are reference to Chinese myths, and the allusion of “tiger subdued”(伏虎), which is reference to a religious legend.Wu Gang is a mythical figure in Chinese culture. He was a native of the West River in the Han dynasty, who sought after the way of Immortality. Owing to a casual offence of divine law, he was condemned to the Sisyphean labor of cutting a huge sweet osmanthus tree in the moon, and the tree is 5,000 feet tall, which healed itself after every stroke of Wu Gang’s axe. Thus the term of servitude inflicted upon him is permanent. The legend of Wu Gang was expressed perfectly by the combination of the word “the woodman”and the word “divine”in Xu Yuanchong’s version, meanwhile, “woodman”also introduces theidentity of Wu Gang. Hazes, guesses and obscurities caused by different understanding with different cultural backgrounds are reduced in a great way due to the adoption of domestication strategy. Other translators chose to translate the allusion with its sound in Chinese character. Literariness of the original is not spoiled with the allusion being implicit in the target language. Although target language readers’acceptability is challenged, foreignization will raise westerners’ curiosities and transmit the brilliant Chinese culture to the world.Chang-E is another mythical figure, who fled to the moon, living an eternal solitary life after secretly taking the elixir of her husband Hou Yi(后羿), the Archer ,who had received the elixir from the Queen of the West(西王母). Chang-E in the authorized version, Xu’s version, Gu’s version and Zhao’s version is all translated as “the lonely moon goddess”. In western culture, there are also moon goddess such as “Diana”, “Helen”, “Phoebe”and “Artemis”, which are well accepted by the westerners. If Chang-E is translated like those names, the cultural image of the original poem will disappear, because none of those images have the meaning of solitude which is a special feature of Chang-E, meanwhile, in western culture moon is a goddess itself. Therefore, the version “moon goddess”will make the westerners feel the familiarity, but this version will mislead the westerners in another way, for the moon goddess in their mind is different from that in our mind. However, after adding the word “lonely”, meaning of solitude is conveyed. Huang Long’s version and Nancy’s version chose to keep the image as it was, which is foreignization. Readers can understand the background of the image after referring to the notes.“Tiger subdued”is a religious allusion, which says there was a feral hungry tiger always staying and roaring at the outside of a temple. An arhat of the temple perceived the hunger of the tiger, so he fed the tiger with some food divided from his own food, and years later the feral tiger was tamed by the arhat. In this poem, “tiger subdued” alludes to the ultimate overthrow of the Kuomintang(KMT国民党) rule in 1949. All the translators chose tiger to express the cruel tyranny of KMT and the ferocity of reactionaries. Most of them chose the verb “subdue” to illustrate the success of conquering of the enemies, Zhao Zhentao’s “defeat” also conveys the same idea, and Xu Yuanchong’s “overthrown” is for the concerns of forming an end rhyme.D.Translation of the action verb “peng (捧)”A subtler case exists with the action verb “peng” in line four. “Peng” means ho lding something with both hands, which carries an implied cup-image or flask-image in addition to a hand-image. To translate it simply as “present”, “serve” “offer” is to strip off all the image richness of the original for a barren prosaic presentation. An express indication of the implied pictures of hands and cup is almost imperative. Huang Long’s “with open hands” emphasizes the unlimited respect for the loyal souls, which also has the means of “generosity”, so this version is very appropriate.E.The choice between domestication and foreignizationIn Chinese legend, the trees on the moon are sweet osmanthuses. When wine on the moon is mentioned, people will associate the wine with “gui hua jiu”(桂花酒) naturally，and the vintage must be prepared for the gods in the moon. In fact, “gui hua jiu” is made of common white wine with sweet osmanthus flowers soaked in it, and a plurality of dayslater the wine will have the flavor of sweet osmanthus.The authorized version and Gu’ version translated “gui hua jiu” into “laurel wine”, which is domestication. The first stanza of the whole ci-poem is about how did the spirit of the heroes go to the moon and how are they generously catered by Wu Gang and Change-E, which sings high praise for the revolutionary spirit of the heroes. Therefore, we are certain that “gui hua jiu” can not be ordinary wine. On the contrary, it is a symbol of heroes and honors, and “l aurel” annotates this meaning accurately, because in western culture laurel represents honors and glories. Those most brilliant poets are called “laurel poet”, too. However, laurel is a Mediterranean evergreen tree, also called “bay”, “bay laurel”and “sweet bay”, which has aromatic, simple leaves and small blackish berries, so we are clear that laurel does not grow in China. In addition, according to Zhao Zhentao’s research, “the flowers of laurel can not be used as flavors”(Zhao, 25, 1978). Therefore, “laurel wine”contradicts with the original poem in sense of “flower”, and “gui hua jiu” can not be laurel wine. Translation like this will mislead the westerners.Xu Yuanchong also chose domestication when translating “gui hua jiu” and the word “nectar”chosen by him is very appropriate. Nectar is the drink of gods in the stories of ancient Greece, while “gui hua jiu”is the drink of the celestial beings in the stories of Chinese myths. The derivation meaning of nectar is death-overcoming, and the drink was used for confering immortality, while “gui hua jiu” also has the same function: Wu Gang served the heroes with “gui hua jiu”, which also implies the spirit of the brave departed will last forever.Zhao Zhentao translated “gui hua jiu”as “osmanthus brew”, while Huang Longtranslated it as “osmanthus wine”. Their difference lies in the choice of “brew” and “wine”. “Wine”is a beverage made of ferment, while “brew”is a beverage made by boiling, steeping, or mixing of various ingredients. Therefore, Zhao’s version seems more accurate. However, in Zhao’s version, “the god” and “osmanthus brew” coexist in one sentence, that means, a mythical word and a science word are arranged together, which is very unharmonious. There are also some other translators who translate “gui hua jiu”as “cassia wine”like Engle, but versions like those are too accurate to retain the romantic charm.Nancy chose foreignlization as the method to translate the image, which retains the ethnic flavors in a relative way. The readers will understand the image better with reference to the note.parative study of the second stanza of the poemIn the fifth line, “shu(舒)” is an important verb. Most translators translated this verb as “spread”or “outspread”. “Spread”means to open something so that it covers a huge area, in this sense, versions like those seem to be very faithful to the original poem, however, compared with Huang Long’s “wave”, those versions seem to be less brilliant. “Wave” means to hold something and move it from side to side, and the track made of this movement is like the wave in the sea. If Chang-E would like to wave her sleeves, the first thing she has to do is to spread her sleeves, therefore, “wave” can convey the meaning of “spread”, while “spread” can not convey the meaning of “wave”. Only with the action of spreading the sleeves, the dance of Chang-E can not be beautiful, but if “waving of the sleeves”is added, the dance will be terrific. Therefore, “wave”is a more thorough and。

Transfer of Taylorist ideas to China,1910-1930sStephen L.MorganDepartment of Management,The University of Melbourne,Melbourne,AustraliaAbstractPurpose –Management is a “hot field”in China,yet little has been written in English about the history of management in China.Contrary to contemporary management literature,the paper aims to show that Chinese entrepreneurs and managers were exposed to modern management ideas from the early twentieth century.The paper is an initial exploration of the transfer of managerial knowledge to China,especially Scientific Management,during the interwar period.Design/methodology/approach –Draws on Chinese journal articles and books from 1910-1930s,supplemented with archive materials and secondary sources in Chinese and English.Findings –Chinese industrialists,officials and academics were attracted to Taylor’s ideas of scientific management during the 1920s and 1930s,which were experimented with on a wider scale than is commonly realized.The interest in “new”management extended beyond industrialists and industry officials to reportage in the popular press.Research limitations/implications –Future research should consider first how new ideas about management and organization were implemented on the shopfloor in individual Chinese enterprises,and second examine the role of social networks constituted by native place,industry ties and professional association membership in the diffusion of managerial ideas among the Chinese business elite of the period.Originality/value –The paper shows that the transfer to China of modern management as an ideas system was not a recent phenomenon,but part of a century-long process of transfer and adaptation of western management theory and practice.Keywords China,Management history,Business history,Scientific managementPaper type Research paper IntroductionEconomic reforms in China have transformed economy and society,including the management of business.Modern management is a “hot field”for young Chinese,and the shelves of bookshops in China are filled with translated and local management titles.Texts on general management,along with accounting,human resource,strategic and marketing management abound,not to mention dozens of titles on how to make your fortune in the stock market and the biographies of rich self-made business people.Ten years ago there were few titles –25years ago none at all.The current issue and full text archive of this journal is available at/1751-1348.htmAn earlier version of the paper was presented at the 2003meeting of The Business History Conference and appeared in the selected proceedings,Business and Economic History Online.Research for this paper was supported in part by a Faculty Research Grant,Faculty of Economics and Commerce,the University of Melbourne.The author would like to thank the staff of the East Asian Collection,Baillieu Library,the Shanghai Municipal Library,the Shanghai Municipal Archive and the ILO Archive,Geneva,in locating rare Chinese and other materials that have contributed to the writing of paper.Advice from anonymous referees and the Editor was most welcome.JMH12,4408Journal of Management History Vol.12No.4,2006pp.408-424q Emerald Group Publishing Limited 1751-1348DOI 10.1108/17511340610692761Little wonder,then,that most writers on contemporary Chinese management have a kind of historical amnesia.Western and Chinese authors appear unaware–or at best forgetful–of the rich experience of earlier management practice(Dai,2003;Peng et al., 2001;Li and Tsui,2002;Tsui and Lau,2002;Tang and Ward,2003).This early experience includes the adaptation of western management in the interwar years(1918-37)in a business environment that was not so dissimilar from China today,marked by active markets and vigorous competition among domestic and foreign-controlledfirms.The management theory of the day was scientific management,or Taylorism,based on the ideas of the American engineer Taylor(1856-1915).Chinese industrialists,government officials and business academics were attracted to the ideas of scientific management (kexue guanli fa)for the advancement of China,most notably in the sphere of labor or personnel management.These early Chinese interpreters of scientific management were not alone;they were part of an international movement for industrial efficiency and workplace reform that spanned America,Europe and Japan(Merkle,1980).My paper is a modest contribution to exploring the transfer of management know-how–what we might call soft technologies–to China in the early twentieth century,focused on scientific management.Taylorism had a wider currency in China before1949than is recognized.One of the very few mentions of Taylorism in China is a recent study of the origin of the Chinese post-1950danwei(work unit)institution of labor management(Frazier,2002).This paper is focused primarily on the translation and transfer of Taylorist ideas to China,the interpretation of several proponents of Taylorism,and includes brief sketches of the introduction of Taylorism by some prominent business men.Although the paper is focused on the interwar years,the study is relevant to the transfer of management theory and practice to contemporary China.The expression “scientific management”is frequently invoked to raise the competitiveness of Chinese enterprises today much as it was by the management pioneers of the1920s and1930s. Scientific management has been promoted since the early1980s by the Chinese Enterprise Management Association(Warner,1987,p.76)and by no lesser a person than former president,Jiang Zeming,who said in2000:Scientific management(kexue guanli)not only needs to embrace the management of state affairs,the economy,society and culture,but also embrace the management of every branch of industry and government;...To strengthen and improve management of society,we must promote an agenda for the formation of all-encompassing scientific management systems and mechanisms(cited in Xu and Lao,2001,p.3).Both then and now“scientific”is a term loaded with value,infused with the sense of advanced modernity,that is juxtaposed to a native“backward tradition”of past practices.For those in the1920s,the specific content of the inherited“tradition”was managerial practices of Chinese-style unlimited partnerships;for the reform period,the content is the legacy of the Soviet-planning model.There is more to a historical study of scientific management in China than an arcane analogy or historical resonance with contemporary enterprise reform.A focus on scientific management lets us escape the1949-50divide as representing two quite different narratives of management experience in China,the earlier period marked by nativist traditions and the later period stamped with Soviet models.Such a view is at best dubious,as it suggests the earlier period was unaffected by non-Chinese ideas of management and that China was untouched by the Taylorist industrial efficiency Taylorist ideasto China,1910-1930s409movement that enveloped the world in the 1920s and 1930s.The American-and Japanese-educated industrial instructors at the Rong Family enterprises,one of the leading industrial groups of interwar China,were probably acquainted with Taylor,yet a recent study does not mention the ideas that informed these programs (Cochran,2000,pp.128-31).Many multinational enterprises entered China and brought their managerial practices with them,not least the Americans (Wilkins,1970;Wilkin,1974).Similarly,we need to reassess the idea that post-1949management was a fundamental break with the past.While Soviet management ideas were important for industrial organization in the 1950s (Kaple,1994),the rupture with the past wrought by the new communist regime in industrial management is exaggerated (Walder,1986,pp.30-35).The memory and learned experiences of past management practices were unlikely to have disappeared over night.Recent studies of the origin of the danewi identify its precursors in private-,Nationalist-and Communist-run enterprises of the 1930s and1940s (Lu¨and Perry,1997;Bian,2002;Frazier,2002).After all,even Lenin was a fan of Taylor –Soviet planning took lessons from Taylorism (Merkle,1980,pp.103-35passim;Scoville,2001).The paper is an initial exploration of China’s encounter with western management thought and practice.The remainder of the paper discusses the introduction of scientific management to China,focused on key promoters,organizations and publications,and largely deals with the “rhetoric”or ideology of interwar business leaders’encounter with Taylorism as an ideas system (Sturdy,2004).The paper concludes with consideration of the theoretical issues of the transfer of management ideas to China for future research into the practice of adaptation,the interaction of “nativist”Chinese practices and foreign managerial ideologies.The translation of Taylor into ChineseScientific management was more than a cult of efficiency,though it is commonly associated with time and motion studies,incentive wage systems,and an emphasis on efficiency at the expense of the humanity of the worker.Taylorism was instead a complex set of ideas and values.It intertwined a focus on labor efficiency,product quality,technical training and education that emphasized cooperative harmony between labor and capital (Guillen,1994;Nelson,1980,1992;Wren,1987).The ideas were progressive for worker welfare and public administration (Nyland,1996;Schachter,1989).Taylor saw his ideas as nothing less than “a complete mental revolution”in work and social relations (Taylor,1972,p.27).For Chinese managers of the interwar years scientific management was more than yet another western import.It was for them the most advanced statement of management philosophy and practice.Mu Xiangyu (1876-1943;also known as Mu Ouchu or H.Y.Mo)is credited with the introduction of scientific management to China.Already established in business,at age 33in 1909Mu went to the USA to study,obtaining a bachelor of science in agriculture from the University of Illinois,1913,and a master of science in agriculture at Texas A&M College,1914(Boorman,1967-71,Vol.3.pp.38-40).During his studies he came across scientific management.In April 1914,Mu wrote to Taylor asking permission to translate into Chinese The Principles of Scientific Management ,which had been published in 1911.Taylor responded enthusiastically,according to the letter reproduced after the Preface of the Mu’s 1916translation (Daierluo [Taylor],1916,np):JMH 12,4410Answering your letter of April23rd,it will give me the very greatest pleasure to have you translate my book–The Principles of Scientific Management–into Chinese.I am sending you,under separate cover a copy of each of my books and also a copy of thetranslation of The Principles of Scientific Management into Japanese,which may interest you.Will be very greatly interested to hear of the success of your translation into Chinese.If you happen to be near Philadelphia it will give me great Pleasure to see you at my house and also to show you the application of The Principles of Scientific Management in some of the shops in Philadelphia.I might add that this book has been translated into the following languages:Italian,French,German,Russian,Lettish,Dutch,Spanish and Japanese.Yours sincerely,Fred W.Taylor On his return to China,Mu set up the Deda Cotton Mill in Shanghai and translated Taylor’s book,in collaboration with Dong Dongsu(Mu,1989,p.51).As with many western economic and technical concepts,the earlier Japanese translation guided adaptation into Chinese.Access to the Japanese edition published in1912under the title(in Chinese pinyin transliteration)Xueli de shiye guanli fa(Scientific industry management methods)explains some of the peculiarities in the choice of words in many texts on scientific management published during the1920-30s.The Japanese at the time used the characters xueli de(J:gakuriteki),which conveyed the sense of “theoretical principles”,to represent the meaning of“scientific”now rendered kexue (J:kagakuteki),and which Mu used in his translation–“Gongchang shiyong”xueli de shiye guanli fa[“Applied factory”scientific industrial management methods].Mu added the words“applied factory”to the title to indicate to Chinese industrialists the practical scope of Taylor’s ideas(Xu and Lao,2001,p.95)[1].Although Mu produced thefirst full-length translation of a Taylor work,he was not thefirst to have had contact with Taylorism.Other Chinese who had studied abroad were also involved in disseminating Taylorist ideas,taking positions in business, government and universities(Xu and Lao,2001,pp.95-96).The earliest published account of scientific management was by Yang Quan(better known as Yang Xingfo), whose essay“personnel efficiency”(renshi zhi xiaolu)appeared in the journal Kexue (Science)in1915(Xu and Lao,2001,pp.96,114-15;Xu,1991,p.1309)[2].Yang studied in the USA1912-18,taking an engineering degree at Cornell University and a MBA at Harvard University.At Cornell,Yang took Kimball’s elective course on works administration,thefirst course on scientific management at an American university based(Witzel,2001,Vol.2,pp.536-537).Mu Xiangyue was not content only to translate Taylor.He also introduced new management concepts to his businesses,focused mostly on personnel.He was an extraordinary activist in business and public affairs.Besides the Deda Cotton Mill,Mu set up the Housheng Cotton Mill(1916)and the Zhengzhou Yufeng Cotton Mill(1919). He organised the China Cotton Improvement Society and wrote a pamphlet on cotton growing(1916);chaired the committee on cotton growing of the Federation of Chinese Cotton Mills(1919);established the Chinese Yarn and Cloth Exchange in Shanghai (1920),the central commodity exchange for cotton;set up a bank to assist poor Taylorist ideasto China,1910-1930s411students;and served on government committees and was a deputy minister of industry in the 1930s (Xu,1991,pp.1521-1522;Boorman,1967-71,Vol.3,pp.38-40).The three managerial problems that concerned Mu most were the lack of technical competency among managers,the wasteful use of labor,and that the “brutal treatment”of ordinary workers disincentivized them (Liu,2001a,b,pp.86-87).These problems mostly stemmed from the indirect management of the labor process,notablythe use of the contract labor (baogong ),an internal contract system similar to those found in Britain and the USA in the nineteen century (Littler,1982).An enterprise recruited directly only managerial and senior technical staff,and subcontracted to foremen or intermediary labor contractors the hiring of workers along with their payment,training,management and housing.The indirect management of labor at the time reflected the imperfections of the labor market for the recruitment and control of mostly unskilled workers of rural origin (Wright,1981;Honig,1983).With the increased capital intensity of enterprises during the interwar years and more focus on efficiency,the contract system became an impediment to improving technical efficiency,product quality and worker morale.Mu first required his managers to have technical training.He secondly required the contractors to possess a minimum technical competency,to recruit workers that met his criteria,and to report to him daily on the state of the workshop floor.Coupled with detailed operating and disciplinary rules for all employees,Mu reduced the shop-floor power of contractors and raised productivity and quality (Liu,2001a,pp.87-88).Neither Mu’s management initiatives nor his translation of Taylor attracted interest from other industrialists in the late 1910s.China was then in the midst of the World War One boom when foreign competition for Chinese firms was much reduced.Profits were high;efficiency was not a priority –in the 1920s that changed.The economy slumped and an increasingly militant working class voiced demands for higher wages and better conditions,which with the return of foreign firms and their products,increased market competitive pressure.Scientific management offered improved competitiveness through increased productivity,better labor-capital relations,and more motivated workers (Liu,2001a,pp.88-89).Several enterprises experimented with Taylorist-inspired methods in the 1920s.These include the Rong family Shenxin No.3Cotton Mill,the Kangyuan Can Factory,the Yongtai Silk Filature,the Shanghai Huasheng Electrical Company,and Commercial Press (Liu,2001a,b,pp 89-90;p.10).Commercial press and the Kangyuan factory were held up as model companies (Anon,1931).At Commercial press,with nearly 5,000staff,Taylorist methods were claimed to have raised productivity (chan neng )2.5fold,while wages rose 20-30percent and employee discipline improved (Anon,1931).Despite these early initiatives,experiments with Taylorism and other management practices were few until the Nanjing Decade (1928-37),though further research in company archives may prove otherwise.China’s press in the 1920s carried many articles on new trends in politics,economics and other social science disciplines,including those related to management.The widely read popular journal Dongfang zazhi (Eastern Miscellany)followed closely the political,social and intellectual currents of America,Europe and Japan,and their impact on China.Over the decade to 1930,articles on management-related topics in the journal included,to cite but a handful,a review of scientific management,a translated essay on industrial psychology,a discussion ofJMH 12,4412Henry Ford’s personnel management,and a survey of the industrial rationalization movement(You,1922;Sangdaike,1924;Zhi,1928;Huang,1930;Li,1930;Liu,1930). The China Institute of Scientific ManagementThe world-wide interest in managerial innovation and technical efficiency spawn many organizations for the promotion of scientific management,such as the Taylor Society in the USA and Ueno Yochi’s Industrial Efficiency Institute(Sangyo noritsu kenkyuju)in Japan(Merkle,1980;Warner,1994;Tsutsui,2001).In Geneva,an International Management Institute(IMI)was set up in1927to promote scientific management[3].Following a letter from the IMI to the Nationalist Government of China,the Minister for Industry and Commerce Kong Xiangxi(1881-1967;also known as H.H.Kung)in May1930convened a meeting of Shanghai industrialists,who agreed to form the Chinese Industry and Commerce Management Association(Zhongguo gongshang guanli xiehui,hereafter abbreviated CICMA).At the meeting Kong called for the promotion of scientific management to cultivate more skilled personnel who could overcome the problems of China’s“young and backward industry”and its inadequacies in management,technical skill and organization(Anon,1930a).A week later the preparatory committee changed the name to the Chinese Scientific Management Association(Zhongguo kexue guanli xiehui)to emphasize the promotion of scientific management.The founding conference of the association in late June, however,adopted the original Chinese name,CICMA(Liu,2001a,b,pp.90-92,pp.2-4), though the official name in English was“The China Institute of Scientific Management”(Xu and Lao,2001,pp.117,119).The CICMA founding conference elected a12-member board of directors,chaired by Kong Xiangxi,which included Mu Xiangyue,Yang Xingfo and two of the most prominent businessmen of inter-war China,Liu Hongsheng and Rong Zongjing(Anon, 1930b).The aims of the association were,firstly,to collect research materials on scientific management and the problems of industrial rationalization,and secondly,to discuss,publish and put into practice methods to improve management in China.Kong told the conference their mission was three fold:to improve“personnel administration”, emphasising a“service morality”and spirit of cooperation among“managers and the managed”;to improve production skills and reduce waste;and to foster the growth of national industry for“the benefit of the masses”(Liu,2001b,p.3;Anon,1930b).The association embarked on an elaborate program of research and activism,which included eight research committees.They comprised the administration(jingying), personnel(renshi),accounting(kuiji),finance(licai),general affairs(shiwu),factory operations(changwu),marketing(tuixiao)and planning(sheji)committees(Xu and Lao,2001,pp.119-120;Anon,1930c).Kong was an enthusiastic promoter of the National Government’s corporatist mission to bind the industrial-business classes to the State’s development goals. From late1930through1931,Kong oversaw initiatives to extend the reach of the CICMA.In November1930,the government convened a National Industry and Commerce Congress(Quanguo gongshang huiyi)(Liu,2001a;Xu and Lao,2001, pp.149-153;Anon,1930d).CICMA members who attended the congress were successful in carrying a resolution for the promotion of scientific management and industrial rationalization.It called on the Nanjing Government to direct all provincial and municipal government agencies responsible for industry and commerce to organise Taylorist ideasto China,1910-1930s413branches of the CICMA and implement scientific management methods within their jurisdictions.Secondly,the resolution required the ministry to direct business to set work performance standards and to reward workers who exceeded the benchmarks.Thirdly,the concluding manifesto of the congress agreed to support research into and implementation of scientific management in China.In January 1931,the Ministry of Industry (Shiyebu ,successor to the Ministry of Industry and Commerce (Gongshangbu ),still headed by Kong)invited provincial and municipal administrations to implement the resolution (Liu,2001a).While the response of these government authorities need further research,the next six years to the eve of the Sino-Japanese War in 1937saw wide dissemination of scientific management and other management ideas,and adoption in degrees of new management methods among some larger enterprises.Dissemination of TaylorismThe CICMA used the Ministry of Industry’s journal Gongshang ban yuekan (official English title:Semi-Monthly Economics Journal )as its publishing vehicle for the promotion of scientific management between 1930and 1934.Nearly every issue had news about or an article on scientific management.From May 1934to July 1937,the CICMA published its own journal,the Gongshang guanli yuekan [Industry and Commerce Management Monthly ;the official English title was The Scientific Management Monthly ].Among early publishing efforts of the association were a special issue of the ministry’s journal in July 1931and three volumes of an Anthology of Chinese Business Management (Zhongguo gongshang guanli congkan )(Liu,2001b).The special issue comprised five articles in the main section,summarised in Table I.Under the Investigation Column (Diaocha lan ),a regular feature of the journal,was a report on the implementation of scientific management at Commercial press and the Kangyuan Can Factory,which had become exemplars for the scientific management movement in China (Anon,1931).Wang Yunwu,general manager of Commercial press,wrote an article on personnel management (Wang,1931),and Zhao Xiyu’s discussed the investigation of employees for recruitment and remuneration purposes based on the system at Commercial Press (Zhao,1931).Other articles focused on opportunities for small-and medium-sized enterprises (Kong,1931),cost accounting (Zhou,1931),and time and motion studies (Yin,1931).Personnel matters were a major focus of the CICMA,directed towards improving the quality of managerial staff and technical efficiency of the workforce.Scientific management was perceived as a means to address specific institutional constraints such as the persistence of internal labor contracts and the low educational level of most industrial workers.Articles on personnel management (renshi guanli )accounted for 39percent of articles in the CICMA’s monthly journal.This focus reflected the interest of many promoters of scientific management,in particular the CICMA editorial board.The board included He Qingru,for example,an American-trained personnel specialist who ran a related journal,Renshi guanli yuekan (Personnel Management Monthly ),the voice of the Chinese Personnel Management Association (Zhongguo renshi guanli xuehui )(Liu,2001b,p.6,note 16;Xu and Lao,2001,p.121)[4].The personnel articles discussed recruitment,education and training,various wage systems,employee discipline systems,employee welfare and insurance schemes.Essays on general management and production management accounted for 24articles each in the CICMA’s monthly journal,and financial management was coveredJMH 12,4414A u t h o r A r t i c l e t i t l e C o n t e n t s u m m a r y W a n g Y u n w u P e r s o n n e l m a n a g e m e n t (p p .1-18)7D i s c u s s e d t h e i m p a c t o f p e r s o n n e l s y s t e m s o n t h e e c o n o m i c s a n d a d m i n i s t r a t i o n o f f a c t o r i e s ;t h e p s y c h o l o g y o f w o r k e r s ;s e t t i n g w o r k s t a n da r d s (b e nc h m a r k s );e m p l o y e e t ra i n i n g ,h e a l t h a n d i n s u r a n c e ,a n d w e lf a r e (p e n s i o n s ,e t c );l a b o r -c a p i t a l r e l a t i o n s ;a n d e m p lo y e e m o t i v a t i o n K o n gS h i ’e S ci e n ti fic m a na g e m e n t a n d s m a l l -s c a l e e n t e r p r i s e s (p p .19-30)A n o v e r v i e w o fi nd u s t r i a l i s m ,m a n a ge me n t i n n o v a t i on a n d T a y l o r ’s i d ea s ;s t r e s s e d t h a t s c i e n t i fic m a n a g e m e n t w a s m o r e t h a n t i m e a n d m o t i o n s t u d i e s ,w o r k s t a n d a r d s a nd c o s t a c c o u n ti n g f o r i m p r o v e d e f fic i e n c y –i t w a s “a fun d a m e n t a l r e v o l u ti o n i n t h e p s yc h o l o g y o f l a b o r a nd c a p it a l f o r c o o p e r a tive pr o d u cti o n”;d i scusse din ter n a ti o n a l m a nagem en t tr ends ;s c i en t i fic m a n a g e m e n t w a s n o t j u s t f o r l a r g e fir m s –b u s i n e s s g r o u p s a n d g o v e r n m e n t s c a n m a k e i t a c c e s s i b l e t o s m a ll fir m s Z h a o X iy u T h e a i m s a n d m e t h o d s f o r em p l o y e ei nv e s ti g at i o n (pp .31-56)B a s e d o n Z h a o ’s e x p e r i e n c e o f c o n d u c t i n g a n i n v e s t ig a t i o n o f t h e e m p l o y e e s a t C o m m e r c i a l p r e s s .T h e a i m o f w o r k e r i n v e s t i g a t io n sw a st oo b t a i ni n f o r m a ti o n a bo u t t he at t it u d e s ,e xp e r i e n c e a n d s k i l l s o f w o r k e r s t o e n h a n c e r e c r u i t m e n t a n d r e m u n e r a ti o n d e c is i o n m a k i ng ;d i s c u s s e d t h ede s i g n ,a d m i n i s t r a t io n a n d a n a l y si s o f th e su r v e y i n s t r u m e n t Z h o u Z i a n C o s t a c c o u n t i n g u n d e r s c i e n t i fic m a n a g e m e n t (p p .57-70)I n t ro d u c e d t h e i d e a s o f T ay l o r ,G a n t t ,E m e r s o n a n d H a r r i s o n o nth e u se ofa c co u ntin g c on tr o l s t o m a n ag e b e tt e r m a t e r ia l s a n d r a i s e e f fic i e n c y o f o p e r a t i o n s ,i n c l u d i n g s t a t i s t i c a lr e p o r t i n g p r o c e d u r e s Y i n M i n g l u M o ti o n s t u d i e s a n d t i m e m e a s u r em e n t s (p p.71-91)I n t r o d u c e d F .B .a n d L.M .G il b r e t h a d a p t a t i o n s o f T a y l o r ’s w or k -s t u d i e s ;p r a c t i c a l m e t h o d s f o r t h e c o n d u c t o f t im e a n d m o t i o n s t u d i e s ;w h y t i m e a n d m o t i on s tu d i es ar e n e c es s a r y ;an de x a m p l es o ft im ea n d m ot i o n s t ud ie sA n o n y m o u s (In v e s ti g ati on c ol u m n )I m p lem e n t a t i o n o f s c i e n t i fic m a n a g e m e n t i n f a c t or ie s–I nv e s t i g a t io n.(p p.169-90)(1)C o m m er c i alP re ss (2)K ang y ua nC a n Fa c to ry D e t a i l s o f t h e s c i e n t i fic m a n a g e m e n t s y s te m s a tC om m e r ci al p r e s s inS h a n g h a i,i n t r o d u c e d b y t h e g e n e r al m a n ag e r ,Y u n w u ,a nd attheK a n gyu a nC a n F ac t o r y ,i n t r od u ce d b yown e r,K an gyua n S o u rc e :Go ngsh a ngbanyueka n ,“Kex ueguanlizh u a n h ao ”(S peciali s sueon Sc ient ificMan a g ement ),3(15July1931)Table I.Analysis of the special issue on scientific managementTaylorist ideas to China,1910-1930s 415。

Taylor's Formula and the Study of Extrema1. Taylor's Formula for MappingsTheorem 1. If a mapping Y U f →: from a neighborhood ()x U U = of a point x in a normed space X into a normed space Y has derivatives up to order n -1 inclusive in U and has an n-th order derivative()()x f nat the point x, then()()()()()⎪⎭⎫ ⎝⎛++++=+n n n h o h x f n h x f x f h x f !1,Λ (1)as 0→h .Equality (1) is one of the varieties of Taylor's formula, written here for rather general classes of mappings.Proof. We prove Taylor's formula by induction. For1=nit is true by definition of ()x f ,.Assume formula (1) is true for some N n ∈-1.Then by the mean-value theorem, formula (12) of Sect. 10.5, and the induction hypothesis, we obtain.()()()()()()()()()()()()()⎪⎭⎫ ⎝⎛=⎪⎭⎫ ⎝⎛=⎪⎪⎭⎫ ⎝⎛-+++-+≤⎪⎭⎫ ⎝⎛+++-+--<<nn n n n n h o h h o h h x f n h x f x f h x f h x f n h f x f h x f 11,,,,10!11sup !1x θθθθθΛΛ，as 0→h .We shall not take the time here to discuss other versions of Taylor's formula, which are sometimes quite useful. They were discussed earlier in detail for numerical functions. At this point we leave it to the reader to derive them (see, for example, Problem 1 below). 2. Methods of Studying Interior ExtremaUsing Taylor's formula, we shall exhibit necessary conditions and also sufficient conditions for an interior local extremum of real-valued functions defined on an open subset of a normed space. As we shall see, these conditions are analogous to the differential conditions already known to us for an extremum of a real-valued function of a real variable.Theorem 2. Let R U f →: be a real-valued function defined on an open set U in a normed space X and having continuous derivatives up to order 11≥-k inclusive in a neighborhood of a point U x ∈and a derivative()()x f kof order k at the point x itself.If()()()0,,01,==-x f x f k Λand()()0≠x f k , then for x to be an extremum of the function f it is:necessary that k be even and that the form ()()k k h x fbe semidefinite,andsufficient that the values of the form()()k k h x fon the unit sphere 1=h be bounded awayfrom zero; moreover, x is a local minimum if the inequalities()()0>≥δk k h x f ,hold on that sphere, and a local maximum if()()0<≤δk k h x f ,Proof. For the proof we consider the Taylor expansion (1) of f in a neighborhood of x. The assumptions enable us to write()()()()()k k k h h h x f k x f h x f α+=-+!1where ()h α is a real-valued function, and ()0→h α as 0→h . We first prove the necessary conditions. Since ()()0≠x f k , there exists a vector00≠h on which ()()00≠kk h x f . Then for values of thereal parameter t sufficiently close to zero,()()()()()()kk k th th th x f k x f th x f 0000!1α+=-+()()()k k k k t h th h x f k ⎪⎭⎫ ⎝⎛+=000!1αand the expression in the outer parentheses has the same sign as()()kk h x f 0.For x to be an extremum it is necessary for the left-hand side (and hence also the right-handside) of this last equality to be of constant sign when t changes sign. But this is possible only if k is even.This reasoning shows that if x is an extremum, then the sign of the difference ()()x f th x f -+0 is the same as that of ()()kk h x f 0for sufficiently small t; hence in that case there cannot be twovectors0h , 1hat which the form ()()x f kassumes values with opposite signs.We now turn to the proof of the sufficiency conditions. For definiteness we consider the case when()()0>≥δk k h x ffor 1=h . Then()()()()()kk k h h h x f k x f h x f α+=-+!1()()()k k k h h h h x f k ⎪⎪⎪⎭⎫ ⎝⎛+⎪⎪⎭⎫ ⎝⎛=α!1 ()kh h k ⎪⎭⎫ ⎝⎛+≥αδ!1and, since ()0→h α as 0→h , the last term in this inequality is positive for all vectors0≠h sufficiently close to zero. Thus, for all such vectors h,()()0>-+x f h x f ,that is, x is a strict local minimum.The sufficient condition for a strict local maximum is verified similiarly.Remark 1. If the space X is finite-dimensional, the unit sphere ()1;x S with center at X x ∈, being a closed bounded subset of X, is compact. Then the continuous function()()()()kk i i i i k k h h x f h x f ⋅⋅∂=ΛΛ11 (a k-form) has both a maximal and a minimal value on ()1;x S . Ifthese values are of opposite sign, then f does not have an extremum at x. If they are both of the same sign, then, as was shown in Theorem 2, there is an extremum. In the latter case, a sufficient condition for an extremum can obviously be stated as the equivalent requirement that the form()()k k h x fbe either positive- or negative-definite.It was this form of the condition that we encountered in studying realvalued functions on n R .Remark 2. As we have seen in the example of functions R R f n →:, the semi-definitenessof the form()()k k h x fexhibited in the necessary conditions for an extremum is not a sufficientcriterion for an extremum.Remark 3. In practice, when studying extrema of differentiable functions one normally uses only the first or second differentials. If the uniqueness and type of extremum are obvious from the meaning of the problem being studied, one can restrict attention to the first differential when seeking an extremum, simply finding the point x where ()0,=x f 3. Some Examples Example 1. Let()()RR C L ;31∈ and()[]()R b a C f ;,1∈.In other words,()()321321,,,,u u u L u u u αis a continuously differentiable real-valued function defined in 3Rand ()x f x α a smoothreal-valued function defined on the closed interval []R b a ⊂,. Consider the function()[]()R R b a C F →;,:1(2)defined by the relation()[]()()f F R b a C f α;,1∈ ()()()R dx x f x f x L ba∈=⎰,,, (3)Thus, (2) is a real-valued functional defined on the set of functions()[]()R b a C ;,1.The basic variational principles connected with motion are known in physics and mechanics. According to these principles, the actual motions are distinguished among all the conceivable motions in that they proceed along trajectories along which certain functionals have an extremum. Questions connected with the extrema of functionals are central in optimalcontrol theory. Thus, finding and studying the extrema of functionals is a problemof intrinsic importance, and the theory associated with it is the subject of a large area of analysis - the calculus of variations. We have already done a few things to make the transition from the analysis of the extrema of numerical functions to the problem of finding and studying extrema of functionals seem natural to the reader. However, we shall not go deeply into the special problems of variational calculus, but rather use the example of the functional (3) to illustrate only the general ideas of differentiation and study of local extrema considered above.We shall show that the functional (3) is a differentiate mapping and find its differential. We remark that the function (3) can be regarded as the composition of the mappings()()()()()x f x f x L x f F ,1,,= (4)defined by the formula()[]()[]()R b a C R b a C F ;,;,:11→(5)followed by the mapping[]()()()R dx x g g F R b a C g ba∈=∈⎰2;,α (6)By properties of the integral, the mapping 2Fis obviously linear and continuous, so thatits differentiability is clear. We shall show that the mapping1Fis also differentiable, and that()()()()()()()()()()x h x f x f x L x h x f x f x L x h f F ,,3,2,1.,,,∂+∂=(7)for()[]()R b a C h ;,1∈.Indeed, by the corollary to the mean-value theorem, we can write in the present case()()()ii iu u u L u u u L u u u L ∆∂--∆+∆+∆+∑=32131321332211,,,,,,()()()()()()∆⋅∂-∆+∂∂-∆+∂∂-∆+∂≤<<u L u L u L u L u L u L 3312211110sup θθθθ()()ii i i u L u u L i ∆⋅∂-+∂≤=≤≤=3,2,110max max 33,2,1θθ (8)where ()321,,u u u u = and ()321,,∆∆∆=∆.If we now recall that the norm ()1c fof the function f in()[]()R b a C ;,1is⎭⎬⎫⎩⎨⎧c c f f ,,max (wherecfis the maximum absolute value of the function on the closed interval []b a ,), then,setting x u =1,()x f u =2, ()x f u ,3=, 01=∆, ()x h =∆2, and ()x h ,3=∆, we obtain from inequality (8),taking account of the uniform continuity of the functions ()3,2,1,,,321=∂i u u u L i , on boundedsubsets of3R , that()()()()()()()()()()()()()()()()x h x f x f x L x h x f x f x L x f x f x L x h x f x h x f x L bx ,,3,2,,,0,,,,,,,,max ∂-∂--++≤≤()()1c h o = as()01→c hBut this means that Eq. (7) holds.By the chain rule for differentiating a composite function, we now conclude that the functional (3) is indeed differentiable, and()()()()()()()()()()⎰∂+∂=b adx x h x f x f x L x h x f x f x L h f F ,,3,2,,,,, (9)We often consider the restriction of the functional (3) to the affine space consisting of the functions()[]()R b a C f ;,1∈that assume fixed values ()A a f =, ()B b f = at the endpoints of theclosed interval []b a ,. In this case, the functions h in the tangent space ()1f TC , must have the value zero at the endpoints of the closed interval []b a ,. Taking this fact into account, we may integrate by parts in (9) and bring it into the form()()()()()()()()⎰⎪⎭⎫⎝⎛∂-∂=b a dx x h x f x f x L dx d x f x f x L h f F ,3,2,,,,,(10)of course under the assumption that L and f belong to the corresponding class ()2C .In particular, if f is an extremum (extremal) of such a functional, then by Theorem 2 we have()0,=h f Ffor every function()[]()R b a C h ;,1∈such that ()()0==b h a h . From this and relation (10)one can easily conclude (see Problem 3 below) that the function f must satisfy the equation()()()()()()0,,,,,3,2=∂-∂x f x f x L dxdx f x f x L (11)This is a frequently-encountered form of the equation known in the calculus of variations as the Euler-Lagrange equation.Let us now consider some specific examples. Example 2. The shortest-path problemAmong all the curves in a plane joining two fixed points, find the curve that has minimal length.The answer in this case is obvious, and it rather serves as a check on the formal computations we will be doing later.We shall assume that a fixed Cartesian coordinate system has been chosen in the plane, in which the two points are, for example, ()0,0 and ()0,1 . We confine ourselves to just the curves that are the graphs of functions()[]()R C f ;1,01∈assuming the value zero at both ends ofthe closed interval []1,0 . The length of such a curve()()()⎰+=12,1dx x f f F (12)depends on the function f and is a functional of the type considered in Example 1. In this case the function L has the form()()233211,,u u u u L +=and therefore the necessary condition (11) for an extremal here reduces to the equation()()()012,,=⎪⎪⎪⎭⎫⎝⎛+x f x f dx dfrom which it follows that()()()常数≡+x fx f 2,,1 (13)on the closed interval []1,0 Since the function21uu + is not constant on any interval, Eq. (13) is possible only if()≡x f ,const on []b a ,. Thus a smooth extremal of this problem must be a linear function whosegraph passes through the points ()0,0 and ()0,1. It follows that ()0≡x f , and we arrive at the closed interval of the line joining the two given points. Example 3. The brachistochrone problemThe classical brachistochrone problem, posed by Johann Bernoulli I in 1696, was to find the shape of a track along which a point mass would pass from a prescribed point 0Ptoanother fixed point1Pat a lower level under the action of gravity in the shortest time.We neglect friction, of course. In addition, we shall assume that the trivial case in whichboth points lie on the same vertical line is excluded. In the vertical plane passing through the points 0Pand1Pwe introduce a rectangularcoordinate system such that 0Pis at the origin, the x-axis is directed vertically downward,and the point1Phas positive coordinates ()11,y x .We shall find the shape of the track amongthe graphs of smooth functions defined on the closed interval []1,0x and satisfying the condition ()00=f ,()11y x f =. At the moment we shall not take time to discuss this by no means uncontroversial assumption (see Problem 4 below). If the particle began its descent from the point0Pwith zero velocity, the law of variationof its velocity in these coordinates can be written asgxv 2= (14)Recalling that the differential of the arc length is computed by the formula()()()()dx x f dy dx ds 2,221+=+=(15)we find the time of descent()()()⎰+=12,121x dx xx f gf F (16)along the trajectory defined by the graph of the function ()x f y =on the closed interval []1,0x .For the functional (16)()()1233211,,u u uu u L +=,and therefore the condition (11) for an extremum reduces in this case to the equation()()()012,,=⎪⎪⎪⎪⎪⎭⎫⎝⎛⎪⎭⎫ ⎝⎛+x f x x f dx d , from which it follows that()()()xc x fx f =+2,,1 (17)where c is a nonzero constant, since the points are not both on the same vertical line. Taking account of (15), we can rewrite (17) in the formx c dsdy= (18)However, from the geometric point of viewϕcos =ds dx ,ϕsin =dsdy (19)where ϕ is the angle between the tangent to the trajectory and the positive x-axis.By comparing Eq. (18) with the second equation in (19), we findϕ22sin 1cx =(20)But it follows from (19) and (20) thatdx dy d dy =ϕ,2222sin 2sin c c d d tg d dx tg d dx ϕϕϕϕϕϕϕ=⎪⎪⎭⎫ ⎝⎛==,from which we find()b c y +-=ϕϕ2sin 2212(21)Settinga c =221 and t =ϕ2, we write relations (20) and (21) as()()bt t a y t a x +-=-=sin cos 1 (22)Since 0≠a , it follows that 0=x only for πk t 2=,Z k ∈. It follows from the form of thefunction (22) that we may assume without loss of generality that the parameter value 0=t corresponds to the point()0,00=P . In this case Eq. (21) implies 0=b , and we arrive at thesimpler form()()t t a y t a x sin cos 1-=-= (23)for the parametric definition of this curve.Thus the brachistochrone is a cycloid having a cusp at the initial point0Pwhere thetangent is vertical. The constant a, which is a scaling coefficient, must be chosen so that the curve (23) also passes through the point1P .Such a choice, as one can see by sketching thecurve (23), is by no means always unique, and this shows that the necessary condition (11) foran extremum is in general not sufficient. However, from physical considerations it is clear which of the possible values of the parameter a should be preferred (and this, of course, can be confirmed by direct computation).泰勒公式和极值的研究1.映射的泰勒公式定理1 如果从赋范空间X 的点x 的邻域()x U U =到赋范空间Y 的映射Y U f→:在U中有直到n-1阶(包括n-1在内)的导数，而在点x 处有n 阶导数。

taylor swift高中英语阅读Taylor Swift's High School Reading ExperienceTaylor Swift, the beloved country-turned-pop superstar, has always been known for her captivating songwriting and her ability to connect with her fans on a deeply personal level. However, what many may not know is the crucial role that her high school reading experiences played in shaping her as an artist and a storyteller.Growing up in Wyomissing, Pennsylvania, Taylor was an avid reader from a young age. She was particularly drawn to classic literature, with authors like F. Scott Fitzgerald, Ernest Hemingway, and Jane Austen becoming her literary companions throughout her adolescence. These early literary influences can be seen woven into the fabric of her music, from the vivid imagery and emotive language in her lyrics to the intricate narrative structures that have become her signature.One of the most significant books that left a lasting impression on Taylor was "The Great Gatsby" by F. Scott Fitzgerald. She has spoken extensively about her love for this classic novel, describing it as a "work of art" that has deeply influenced her approach to storytelling.The themes of unrequited love, the pursuit of the American Dream, and the disillusionment of the upper class resonated profoundly with the young Taylor, and she has often cited Jay Gatsby as a character that has informed her understanding of the complexities of human relationships.Another seminal work that shaped Taylor's literary sensibilities was Ernest Hemingway's "The Sun Also Rises." The rawness and stripped-down beauty of Hemingway's prose struck a chord with the aspiring songwriter, who has often spoken about the importance of concision and clarity in her own writing. The novel's exploration of disillusionment, loss, and the search for meaning in a post-war world mirrored the themes that would later become central to Taylor's own artistic journey.In addition to these classic works of literature, Taylor also developed a deep appreciation for the writings of Jane Austen. The intricate social dynamics, the nuanced character studies, and the themes of love and heartbreak that permeate Austen's novels have all left an indelible mark on Taylor's songwriting. She has even gone so far as to incorporate direct references to Austen's work into her music, such as the line "I'm a nightmare dressed like a daydream" from her hit song "Blank Space," which echoes the famous opening line of Austen's "Pride and Prejudice."It is clear that Taylor Swift's high school reading experiences played a pivotal role in shaping her as an artist and a storyteller. The rich tapestry of literary influences that she has woven into her music has not only resonated with her legions of fans but has also solidified her reputation as one of the most compelling and insightful songwriters of her generation. Whether she is channeling the tragic grandeur of Gatsby or the subtle nuances of Austen's heroines, Taylor's literary prowess shines through in every note and every word, a testament to the enduring power of the written word.。