Chapter 1 Bases for Lie algebras and a continuous CBH formula
爱伦坡小说中英文对照
爱伦坡⼩说中英⽂对照诗歌诗 Poetry哦，时代！哦，风尚！ O,Tempora! O,Mores!致玛格丽特To Margaret“致奥克塔维娅” To Octavia帖⽊⼉ Tamerlane歌 Song梦 Dreams亡灵 Spirits of the Dead模仿 Imitation“诗节”Stanzas"⼀个梦 A Dream“最快乐的⽇⼦” The Happiest Day 湖——致—— The Lake — To ——⼗四⾏诗——致科学 To Science阿尔阿拉夫 Al Aaraaf传奇 Romance埃德加·爱伦·坡致河—— To The River ——仙境 Fairy-Land“孤独” "Alone"“致艾萨克·利” To Isaac Lea伊丽莎⽩ Elizabeth⼀⾸离合诗 An Acrostic“咏乔·洛克” Lines on Joe Locke致海伦 To Helen以⾊拉费 Israfel睡美⼈The Sleeper不安的⼭⾕ The Valley of Unrest海中之城 The City in the Sea丽诺尔Lenore致乐园中的⼀位 To One in Paradise赞歌 Latin Hymn//Hymn谜 Enigma⼩夜曲 Serenade罗马⼤圆形竞技场 The Coliseum新婚⼩调 Bridal Ballad⼗四⾏诗——致桑特岛闹⿁的宫殿 The Haunted Palace⼗四⾏诗——静 Silence,a Sonnet征服者爬⾍ The Conqueror Worm梦境 Dream-Land尤拉丽——歌 Eulalie乌鸦 The Raven赠——的情⼈节礼物 A Valentine“深眠黄⼟” Deep in Earth致路易丝·奥利维亚·亨特⼩姐 To Miss Louise Olivia Hunter 致M.L.S—— To M. L. S尤娜路姆——⼀⾸歌谣 Ulalume — A Ballad⼀个谜 An Enigma钟 The Bells致海伦 To Helen梦中之梦 A Dream Within A Dream献给安妮 For Annie黄⾦国Eldorado致我的母亲 To My Mother安娜贝尔·李 Annabel Lee戏剧《波利希安》选场（⼀⾄五场未完） Scenes From 'Politian'哥特⼩说梅岑格施泰因Metzengerstein：孤僻暴戾的贵族宠爱壁毯中⾛下的红⾊魔马，纵容其⽣吃⼈⾁的故事。
The Little Prince
名著选读I believe that for his escape he took advantage of the migration of a lock of wild birds.On the morning of his departure he put his planet in perfect order.He carefully cleaned out his active volcanoes.He possessed two active volcanoes;and they were very convenient for heating his breakfast in the morning.He also had one volcano that was extinct.But,as he said,“One never knows!”So he cleaned out the extinct volcano,too.If they are well cleaned out,volcanoes burn slowly and steadily,without any eruptions.V olcanic eruptions are like fires in a chimney.On our earth we are obviously much too small to clean out our volcanoes.That is why they bring no end of trouble upon us.The little prince also pulled up,with a certain sense of dejection,the last little shoots of the baobabs.He believed that he would never want to return.But on this last morning all these familiar tasks seemed very precious to him.And when he watered the flower for the last time,and prepared to place her under the shelter of her glass globe,he realised that he was very close to tears.“Goodbye,”he said to the flower.But she made no answer.“Goodbye,”he said again.The flower coughed.But it was not because she had a cold.“I have been silly,”she said to him,at last.“I ask your forgiveness.Try to be happy...”He was surprised by this absence of reproaches.《小王子》是法国作家安托万·德·圣-埃克苏佩里于1942年写成的著名法国儿童文学短篇小说。
Lie algebras, in Infinite-dimensional Lie algebras and groups, Adv. Ser. in Math. Phys. 7,
C1;1;k g ' L0;k (0)
L1;k (1)
M1;k (1): Combining Theorem 4.6 and the statement (ii), we obtain the statement (i). Q.E.D.
References
H0 (g(0; 1; 1); L0 ;k (0)
Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan. Department of Mathematics, Faculty of Science, Osaka University,
[KK] Kac V.G. and Kazhdan D.A., Structure of representations with highest weight of in nite dimensional Lie algebras, Adv. in Math. 34, (1979), 97{108. [KnPZ] Knizhnik V. G., Polyakov A. M. and Zamolodchikov A.B., Fractal structure in 2D quantum gravity, Mod. Phys. Lett. A 3, (1988), 819{826. [KW1] Kac V.G. and Wakimoto M., Modular invariant representations of in nite dimensional Lie algebras and superalgebras, Proc. Natl. Acad. Soc., 35, (1988), 4956{4960. [KW2] Kac V.G. and Wakimoto M., Classi cation of modular invariant representations of ane Lie algebras, in In nite-dimensional Lie algebras and groups, Adv. Ser. in Math. Phys. 7, (1989), 138{177. [KWn] Kac V. G. and Wang W., Vertex operator superalgebras and their representations, Mathematical aspects of conformal and topological eld theories and quantum groups, 161{191, Contemp. Math. 175, Amer. Math. Soc. Providence, RI, (1994). [Mal] Malikov F.G., Verma modules over Kac-Moody algebras of rank 2, Leningrad Math. J., 2, No. 2, (1991), 269{286. [MFF] Malikov F.G., Feigin B.L. and Fuks D.B., Singular vectors in Verma modules over KacMoody algebras, Funkts. Anal. Prilozhen., 20, No. 2, (1988), 25{37. [MW] Mathieu P. and Walton M.A., Fractional-level Kac-Moody Algebras and Nonunitary Coset Conformal Theories, Prog. Theor. Phys. 102, (1990), 229-254. [MP] Moody R.V. and Pianzola A., Lie algebras with triangular decompositions, Wiley-Interscience Publ., New York, 1995. [P] Polyakov A. M., Quantum Gravity in two dimensions, Mod. Phys. Lett. A 2, (1987), 893-898. [Ray] Ray U., A Characterization of Lie superalgebras for a certain class of graded Lie superalgebras, preprint,to appear in Jour. Alg. [RW] Rocha-Caridi A. and Wallach N.R., Highest weight modules over graded Lie algebras: resolutions, ltrations and character formulas, Trans. Amer. Math. Soc. 20, (1983), 133{162. [TUY] Tsuchiya A., Ueno K. and Yamada Y., Conformal eld theory on moduli family of stable curves with gauge symmetry, in Integrable systems in quantum eld theory and statistical mechanics, Adv. Stud. Pure Math. 19, (1989), 459-566. [V] Verlinde E., Fusion rules and modular transformations in 2D conformal eld theory, Nucl. Phys. B 300, (1998), 360-376. (K. Iohara) (Y. Koga)
大学英语精读Book 1, unit 8 A Magician at Stretching a Dollar
都是天使们制作而成
由天上的上帝那里送来。
Background Information
1.The Chinese Most Important Festival: 1) In china what is the most important traditional festival? The Spring Festival 2) How do people celebrate it?
Christmas Day
Can you say something about Christmas Day ?
Text Analysis
1.The Author : Russell Baker(1925-) He is one of America‟s famous humorists and political satirists [ˈsætərɪst] (a person who writes satire讽刺作家). Baker won Pulitzer Prize 普利策奖 in 1979, and another in 1983, for the first volume of his autobiography(自传), Growing Up《成长》.
Background Information
It is said that on Christmas Eve he rides in his flying sleigh from house to house pulled by flying reindeer.
Background Information
有些时候只有母亲的爱 能分享我们未感到的喜悦
当我们梦寐以求的某样东西
美国文学试卷+答题纸+答案
2012-2013学年 第二学期 《美国文学》期末考试试卷（A 卷）专业：英语 年级：2010级 考试方式：闭卷 学分：2 考试时间：110分钟I ．Multiple Choices (每小题 1分，共20分)Directions: Select from the four choices of each item the one thatbest answers the question.1. Naturalism is evolved from realism when the author’s tone in writing becomes less serious and less sympathetic but more ironic and more_____________. A . rational B . humorous C. optimisticD . pessimistic2. Which of the following is not written by Ernest Hemingway, one of the best-known American authors of the 20th century? A. The Sun Also Rises B. The Old Man and the Sea C. Mosses from the Old ManseD. Hills Like White Elephant3. The Romantic writers would focus on all the following issues Except the __________ in the American history. A. individual feeling B. survival of the fittest C. strong imaginationD. return to nature4. Almost all Faulkner ’s heroes turned out to be tragic because__________. A. all enjoyed living in the declining American South.B. none of them was conditioned by the civilization and Social institutions.C. most of them were prisoners of the past.D. none were successful in their attempt to explain the inexplicable.5. As an autobiograp hical play, O’Neill’s ________ (1955) has gained its status as a world classic and simultaneously marks the climax of his literary career and the coming of age of American drama._.A. The Iceman ComethB. Long Day’s Journey into NightC. Beyond the HorizonD. Bound East for Cardiff6. Which of the following statements is right about Robert Frost’s poetry?A. He combined traditional verse forms with the difficult and highly ornamental language.B. He combined traditional verse forms with the pastoral language of the Southern area.C. He combined traditional verse forms with a simple spoken language, the speech of New England farmers.D. He combined traditional verse forms with the experimental.7. Edgar Allen Poe was characterized by his __________.A. psycho-analysisB. novels set in the WestC. free verseD. political pamphlets8. Which of the following is depicted as the mythical county in William Faulkner’s novels?A. CambridgeB. OxfordC. MississippiD. Yoknapatawpha9. ____________ was the first great American writer to write for pleasure rather than utility. He is considered to be founder of American literature by some critics.A. James Fenimore CooperB. Washington IrvingC. Ezra PoundD. Mark Twain10. We can perhaps summarize that Walt Whitman’s poems are characterized by all the following features except that they are _______________.A. lyrical and well-structuredB. conversational and crudeC. simple and rather crudeD. free-flowing11. The Grapes of Wrath by Steinbeck reveals the miserable lives of __________ .A. factory workersB. sailorsC. landless farm laborersD. veterans12. Among the American realistic writers, _________ focused his attention on the rising middle class and the way they lived.A. Herman MelvilleB. Henry JamesC. Mark TwainD. William Dean Howells13. Which of the following is a representative novel of naturalism by an American writer? 2A. Innocents AbroadB. McTeagueC. Daisy MillerD. The Grapes of Wrath14. The first symbol of self-made American man is _________.A. Benjamin FranklinB. Washington IrvingC. George WashingtonD. Mark Twain15. The Imagist writers followed three principles. They respectively are direct treatment, economy of expression and ________.A. local colorB. ironyC. clear rhythmD. blank verse16. Robert Frost is famous for his lyric poems. Which of the following lyric poems wasnot written by Robert Frost?A. “The Raven”B. “Stopping by Woods on a Snowy Evening”C. “After Apple-picking”D. “The Road Not Taken”17. “The lost generation”refers to the writers who relocated to Paris in the post WWⅠyears to reject to values of American materialism. All the following but ________are involved in this group.A. F. S. FitzgeraldB. Ernest HemingwayC. Theodore DreiserD. John Dos Passos18. The first settlers who became the founding fathers of the American nation were quite a few of them _________.A. AnglicansB. CatholicsC. NormansD. Puritans19. Which one of the following statements is applicable to the understanding of Transcendentalism?A. It is strongly influenced by social Darwinism.B. Belief in individualism, independence of mind, and self-reliance.C. Man has no free-will.D. It holds that determinism governs everything.20. In __________, Captain Ahab is obsessed with the revenge on a whale which shearedoff his leg on a previous voyage, and his crazy chasing of it eventually brings death to allon board the whaler except Ishmael, who survives to tell the tale.《美国文学》A卷第3页共18页4A. TypeeB. White JacketC. Moby DickD. Billy BuddII ．Explain the Following Literary Terms Briefly (每小题7分，共14分)Directions : Please write down the answers on the Answer Sheet.21. Local Colorism 22. Stream of ConsciousnessIII ．Identification of Fragments (每小题7分，共21分)Directions : Please give the name of the author and the title of the literary work from which it is taken and then briefly comment on itin English. Please write down the answers on the Answer Sheet.23. “‘That ’s right.’ He said; ‘I ’m no good now. I was all right. I had money. I ’m going to quit this,’ and, with death in his heart, he started down toward the Bowery. People had turned on the gas before and died; why shouldn ’t he? He remembered a lodging house where there were little, close rooms, with gas-jet in them, almost pre-arranged, he thought, for what he wanted to do, which rented for fifteen cents. Then he remembered that he had no fifteen cents.”24. “All day Buck brooded by the pool or roamed restlessly above the camp. Death, as a cessation of movement, as a passing out and away from the lives of the living, he knew, and he knew John Thornton was dead. It left a great void in him, somewhat akin to hunger, but a void which ached and ached, and which food could not fill.25. “Her skeleton was small and spare; perhaps that was why that would have been merely plumpness in another was obesity in her. She looked bloated, like a body long submerged in motionless water, and of that pallid hue.IV . Short Essay Questions (每小题10分，共 30 分)Directions : Please write down the answers on the Answer Sheet.《美国文学》A 卷 第5页 共18页26. The relationship between man and nature is a recurrent theme, perhaps one of the most important themes, in American literature. Write a short essay on it by contrasting tow or three American literary works, or two or three American literary movements, to tell what you know about their different views of nature. 27. Please make a comment on Eugene O ’Neil.28. Please briefly comment on Theodore Dreiser ’s novel Sister Carrie.V ．Appreciating a Literary Work (计 15 分)Directions:In this part, you are required to write a commentary paper in no less than 100 words. Please write it on the AnswerSheet .A Clean, Well-Lighted PlaceErnest HemingwayIt was very late and everyone had left the cafe except an old man who sat in the shadow the leaves of the tree made against the electric light. In the day time the street was dusty, but at night the dew settled the dust and the old man liked to sit late because he was deaf and now at night it was quiet and he felt the difference. The two waiters inside the cafe knew that the old man was a little drunk, and while he was a good client they knew that if he became too drunk he would leave without paying, so they kept watch on him."Last week he tried to commit suicide," one waiter said. "Why?""He was in despair." "What about?" "Nothing.""How do you know it was nothing?" "He has plenty of money."They sat together at a table that was close against the wall near the door of the cafe and looked at the terrace where the tables were all empty except where the old man sat in the shadow of the leaves of the tree that moved slightly in the wind. A girl and a soldier went by in the street. The street light shone on the brass number on his collar. The girl wore no head covering and hurried beside him."The guard will pick him up," one waiter said. "What does it matter if he gets what he's after?""He had better get off the street now. The guard will get him. They went by five minutes ago."The old man sitting in the shadow rapped on his saucer with his glass. The youngerwaiter went over to him."What do you want?"The old man looked at him. "Another brandy," he said."You'll be drunk," the waiter said. The old man looked at him. The waiter went away."He'll stay all night," he said to his colleague. "I'm sleepy now. I never get into bed before three o'clock. He should have killed himself last week."The waiter took the brandy bottle and another saucer from the counter inside the cafe and marched out to the old man's table. He put down the saucer and poured the glass full of brandy."You should have killed yourself last week," he said to the deaf man. The old man motioned with his finger. "A little more," he said. The waiter poured on into the glass so that the brandy slopped over and ran down the stem into the top saucer of the pile. "Thank you," the old man said. The waiter took the bottle back inside the cafe. He sat down at the table with his colleague again."He's drunk now," he said."He's drunk every night.""What did he want to kill himself for?""How should I know.""How did he do it?""He hung himself with a rope.""Who cut him down?""His niece.""Why did they do it?""Fear for his soul.""How much money has he got?" "He's got plenty.""He must be eighty years old.""Anyway I should say he was eighty.""I wish he would go home. I never get to bed before three o'clock. What kind of hour is that to go to bed?""He stays up because he likes it.""He's lonely. I'm not lonely. I have a wife waiting in bed for me.""He had a wife once too.""A wife would be no good to him now.""You can't tell. He might be better with a wife.""His niece looks after him. You said she cut him down.""I know." "I wouldn't want to be that old. An old man is a nasty thing.""Not always. This old man is clean. He drinks without spilling. Even now, drunk. Look at him.""I don't want to look at him. I wish he would go home. He has no regard for those 6《美国文学》A 卷 第7页 共18页who must work."The old man looked from his glass across the square, then over at the waiters."Another brandy," he said, pointing to his glass. The waiter who was in a hurry came over."Finished," he said, speaking with that omission of syntax stupid people employ when talking to drunken people or foreigners. "No more tonight. Close now.""Another," said the old man."No. Finished." The waiter wiped the edge of the table with a towel and shook his head.The old man stood up, slowly counted the saucers, took a leather coin purse from his pocket and paid for the drinks, leaving half a peseta(西班牙货币单位) tip. The waiter watched him go down the street, a very old man walking unsteadily but with dignity."Why didn't you let him stay and drink?" the unhurried waiter asked. They were putting up the shutters. "It is not half-past two.""I want to go home to bed." "What is an hour?""More to me than to him." "An hour is the same.""You talk like an old man yourself. He can buy a bottle and drink at home." "It's not the same.""No, it is not," agreed the waiter with a wife. He did not wish to be unjust. He was only in a hurry."And you? You have no fear of going home before your usual hour?" "Are you trying to insult me?""No, hombre （老兄）, only to make a joke.""No," the waiter who was in a hurry said, rising from pulling down the metal shutters. "I have confidence. I am all confidence.""You have youth, confidence, and a job," the older waiter said. "You have everything.""And what do you lack?" "Everything but work.""You have everything I have.""No. I have never had confidence and I am not young." "Come on. Stop talking nonsense and lock up.""I am of those who like to stay late at the cafe," the older waiter said."With all those who do not want to go to bed. With all those who need a light for the night.""I want to go home and into bed.""We are of two different kinds," the older waiter said. He was now dressed to go home. "It is not only a question of youth and confidence although those things are very beautiful. Each night I am reluctant to close up because there may be some one who needs the cafe.""Hombre, there are bodegas open all night long.""You do not understand. This is a clean and pleasant cafe. It is well lighted. The light is very good and also, now, there are shadows of the leaves.""Good night," said the younger waiter."Good night," the other said. Turning off the electric light he continued the conversation with himself, It was the light of course but it is necessary that the place be clean and pleasant. You do not want music. Certainly you do not want music. Nor can you stand before a bar with dignity although that is all that is provided for these hours. What did he fear? It was not a fear or dread, It was a nothing that he knew too well. It was all a nothing and a man was a nothing too. It was only that and light was all it needed and a certain cleanness and order. Some lived in it and never felt it but he knew it all was nada （没有，虚无）y（所以）pues（既然，那么）nada y nada y pues nada. Our nada who art in nada, nada be thy name thy kingdom nada thy will be nada in nada as it is in nada. Give us this nada our daily nada and nada us our nada as we nada our nadas and nada us not into nada but deliver us from nada; pues nada. Hail nothing full of nothing, nothing is with thee. （这是一段模仿祷告词，其中的名词和动词都被虚无所取代，表明一切事物和行为都是虚无。
GTM目录
vol
书
名
1
2 Measure and Category
3
4 A Course in Homological Algebra
5 Categories for the Working Mathematician
6 Projective Planes
7 A Course in Arithmetic
8
9 Introduction to Lie Algebras and Representation Theory
10
11 Functions of One Complex Variable
12
13 Rings and Categories of Modules
14 Stable Mappings and Their Singularities
43 44 45 Probability Theory I 46 Probability Theory II 47 48 General Relativity for Mathematicians 49 50 Fermat Last Theorem 51 52 Algebraic Geometry 53 A Course in Mathematical Logic 54 55 56 57 58 p-adic Numbers, p-adic Analysis, and Zeta-Functions 59 60 Mathematical Methods of Classical Mechanics 61 Elements of Homotopy Theory 62 63 64 65 Differential Analysis on Complex Manifolds 66 Introduction to Affine Group Schemes 67 Local Fields 68 69 70 71 Riemann Surfaces 72 Classical Topology and Combinatiorial Group Theory 73 Algebras 74 Multiplicative Number Theory 75 76 Algebraic Geometry: Birational geometry of Algebraic Varieties 77 78 A Course in Universal Algebra 79 An Introduction to Ergodic Theory 80 81 82 Differential Forms in Algebraic Topology 83 Introduction to Cyclotomic Fields 84 A Classical Introduction to Modern Number Theory 85
美国algebra1 Chapter03
CHAPTER 388C HAPTERT ABLE OF C ONTENTS 3-1Using Letters to Represent Numbers3-2T ranslating Verbal Phrases Into Symbols3-3Algebraic T erms andVocabulary3-4Writing AlgebraicExpressions in Words3-5Evaluating AlgebraicExpressions3-6Open Sentences andSolution Sets3-7Writing FormulasChapter SummaryVocabularyReview ExercisesCumulative Review ALGEBRAIC EXPRESSIONS AND OPEN SENTENCES An express delivery company will deliver a letter or package locally,within two hours.The company has the following schedule of rates.In addition to the basic charge of $25,the cost is $3 per mile or part of a mile for the first 10 miles or less and $4.50 per mile or part of a mile for each additional mile over 10.Costs such as those described,that vary according to a schedule,are often shown by a formula or set of formulas.Formulas can be used to solve many different problems.In this c hapter,you will learn to write algebraic expressions and formulas,to use algebraic expressions and formulas to solve problems,and to determine the solution set of an open sentence.Using Letters to Represent Numbers89 Eggs are usually sold by the dozen,that is,12 in a carton.Therefore,we know that:In 1 carton,there are 12 ϫ1 or 12 eggs.In 2 cartons,there are 12 ϫ2 or 24 eggs.In 3 cartons,there are 12 ϫ3 or 36 eggs.In n cartons,there are 12 ϫn or 12n eggs.Here,n is called a variable or a placeholder that can represent different numbers from the set of whole numbers,{1,2,3,...}.The set of numbers that can replace a variable is called the domain or the replacement set of that variable.Recall that a numerical expression contains only numbers.An algebraic expression,such as 12n,however,is an expression or phrase that contains one or more variables.In this section,you will see how verbal phrases are translated into algebraic expressions,using letters as variables and using symbols to represent operations. Verbal Phrases Involving AdditionThe algebraic expression aϩb may be used to represent several different ver-bal phrases,such as:a plusb a added to b a is increased by bthe sum of a and b b is added to a b more than aThe word exceeds means “is more than.”Thus,the number that exceeds 5 by 2 can be written as “2 more than 5”or 5 ϩ2.Compare the numerical and algebraic expressions shown below.A numerical expression:The number that exceeds 5 by 2 is 5 ϩ2,or 7.An algebraic expression:The number that exceeds a by b is aϩb. Verbal Phrases Involving SubtractionThe algebraic expression a Ϫb may be used to represent several different ver-bal phrases,such as:a minusb a decreased by b b less than ab subtracted from a a diminished by b a reduced by bthe difference between a and bVerbal Phrases Involving MultiplicationThe algebraic expressions aϫb,a ؒb,(a)(b) and ab may be used to represent several different verbal phrases,such as:a timesb the product of a and b b multiplied by aThe preferred form to indicate multiplication in algebra is ab .Here,a product is indicated by using no symbol between the variables being multiplied.The multiplication symbol ϫis avoided in algebra because it can be con-fused with the letter or variable x .The raised dot,which is sometimes mistaken for a decimal point,is also avoided.Parentheses are used to write numerical expressions:(3)(5)(2) or 3(5)(2).Note that all but the first number must be in parentheses.In algebraic expressions,parentheses may be used but they are not needed:3(b )(h ) ϭ3bh .Verbal Phrases Involving Division The algebraic expressions a Ϭb and may be used to represent several differ-ent verbal phrases,such as:a divided by b the quotient of a and b The symbols a Ϭ4 and mean one-fourth of a as well as a divided by 4.Phrases and Commas In some verbal phrases,using a comma can prevent misreading.For example,in “the product of x and y ,decreased by 2,”the comma after y makes it clear that the x and y are to be multiplied before subtracting 2 and can be written as (xy )Ϫ2 or xy Ϫ2.Without the comma,the phrase,“the product of x and y decreased by 2,”would be written x (y Ϫ2).EXAMPLE 1Use mathematical symbols to translate the following verbal phrases into alge-braic language:Answersa.w more than 3 3 ϩwb.w less than 3 3 Ϫwc.r decreased by 2r Ϫ2d.the product of 5r and s 5rse.twice x ,decreased by 102x Ϫ10f.25,diminished by 4 times n 25 Ϫ4ng.the sum of t and u ,divided by 6h.100 decreased by twice (x ϩ5)100 Ϫ2(x ϩ5)t 1u 6a 4a b 90Algebraic Expressions and Open SentencesT ranslating Verbal Phrases into Symbols91EXERCISESWriting About Mathematics1.Explain why the sum of a and 4 can be written as aϩ4 or as 4 ϩa.2.Explain why 3 less than a can be written as aϪ3 but not as 3 Ϫa.Developing SkillsIn 3–20,use mathematical symbols to translate the verbal phrases into algebraic language.3.y plus 84.4 minus r5.7 times x6.x times 77.x divided by 108.10 divided by x9.c decreased by 610.one-tenth of w11.the product of x and y12.5 less than d13.8 divided by y14.y multiplied by 1015.t more than w16.one-third of z17.twice the difference of p and q18.a number that exceeds m by 419.5 times x,increased by 220.10 decreased by twice aIn 21–30,using the letter n to represent “number,”write each verbal phrase in algebraic language.21.a number increased by 222.20 more than a number23.8 increased by a number24.a number decreased by 625.2 less than a number26.3 times a number27.three-fourths of a number28.4 times a number,increased by 329.3 less than twice a number30.10 times a number,decreased by 2In 31–34,use the given variable(s) to write an algebraic expression for each verbal phrase.31.the number of baseball cards,if b cards are added to a collection of 100 cards32.Hector’s height,if he was h inches tall before he grew 2 inches33.the total cost of n envelopes that cost $0.39 each34.the cost of one pen,if 12 pens cost d dollarsA knowledge of arithmetic is important in algebra.Since the variables representnumbers that are familiar to you,it will be helpful to solve each problem by firstusing a simpler related problem;that is,relate similar arithmetic problems to thegiven algebraic one.EXAMPLE 1Represent each phrase by an algebraic expression.a .a distance that is 20 meters shorter than x metersb.a bill for n baseball caps,each costing d dollarsc.a weight that is 40 pounds heavier than p poundsd.an amount of money that is twice d dollarsSolution a.How to Proceed(1)Think of a similar problemThink of a distance that is in arithmetic:20 meters shorter than 50 meters.(2)Write an expression for this50 Ϫ20arithmetic problem:(3)Write a similar expression usingx Ϫ20Answerthe letter x in place of 50.b.How to Proceed(1)Think of a similar problemThink of a bill for 6 caps,each in arithmetic:costing 5 dollars.(2)Write an expression for this6(5)arithmetic problem.Multiplythe number of caps by the costof one cap:(3)Write a similar expressionnd Answerusing n and d :Note:After some practice,you will be able to do steps (1) and (2) mentally.c.(p ϩ40) pounds or (40 ϩp ) poundsd.2d dollarsAnswers a.(x Ϫ20) metersb.nd dollarsc.(p ϩ40) or (40 ϩp)poundsd.2d dollars92Algebraic Expressions and Open SentencesT ranslating Verbal Phrases into Symbols93 EXAMPLE 2Brianna paid 17 dollars for batteries and film for her camera.If the batteriescost x dollars,express the cost of the film in terms of x.Solution If Brianna had spent 5 dollars for the batteries,the amount that was left is found by subtracting the 5 dollars from the 17 dollars,(17 Ϫ5) dollars.Thiswould have been the cost of the film.If Brianna spent x dollars for the batter-ies,then the difference,(17 Ϫx) dollars would have been the cost of the film.Answer(17 Ϫx) dollarsNote:In general,if we know the sum of two quantities,then we can let x rep-resent one of these quantities and (the sum Ϫx) represent the other.EXERCISESWriting About Mathematics1.a.Represent the number of pounds of grapes you can buy with d dollars if each pound costsb dollars.b.Does the algebraic expression in part a always represent a whole number? Explainyour answer by showing examples using numbers.2.a.If x apples cost c cents,represent the cost of one apple.b.If x apples cost c cents,represent the cost of n apples.c.Do the algebraic expressions in parts a and b always represent whole numbers?Explain your answer.Developing SkillsIn 3–18,represent each answer in algebraic language,using the variable mentioned in the problem.3.The number of kilometers traveled by a bus is represented by x.If a train traveled 200 kilo-meters farther than the bus,represent the number of kilometers traveled by the train.4.Mr.Gold invested $1,000 in stocks.If he lost d dollars when he sold the stocks,represent theamount he received for them.5.The cost of a mountain bike is 5 times the cost of a skateboard.If the skateboard costs xdollars,represent the cost of the mountain bike.6.The length of a rectangle is represented by l.If the width of the rectangle is one-half of itslength,represent its width.7.After 12 centimeters had been cut from a piece of lumber,c centimeters were left.Represent the length of the original piece of lumber.94Algebraic Expressions and Open Sentences8.Paul and Martha saved 100 dollars.If the amount saved by Paul is represented by x,repre-sent the amount saved by Martha.9.A ballpoint pen sells for 39 cents.Represent the cost of x pens.10.Represent the cost of t feet of lumber that sells for g cents a foot.11.If Hilda weighed 45 kilograms,represent her weight after she had lost x kilograms.12.Ronald,who weighs c pounds,is d pounds overweight.Represent the number of poundsRonald should weigh.13.A woman spent $250 for jeans and a ski jacket.If she spent y dollars for the ski jacket,rep-resent the amount she spent for the jeans.14.A man bought an article for c dollars and sold it at a profit of $25.Represent the amountfor which he sold it.15.The width of a rectangle is represented by w meters.Represent the length of the rectangle ifit exceeds the width by 8 meters.16.The width of a rectangle is x centimeters.Represent the length of the rectangle if it exceedstwice the width by 3 centimeters.17.If a plane travels 550 kilometers per hour,represent the distance it will travel in h hours.18.If a car traveled for 5 hours at an average rate of r kilometers per hour,represent the dis-tance it traveled.19.a.Represent the total number of days in w weeks and 5 days.b.Represent the total number of days in w weeks and d days.Applying Skills20.An auditorium with m rows can seat a total of c people.If each row in the auditorium hasthe same number of seats,represent the number of seats in one row.21.Represent the total number of calories in x peanuts and y potato chips if each peanut con-tains 6 calories and each potato chip contains 14 calories.22.The charges for a telephone call are $0.45 for the first 3 minutes and $0.09 for each addi-tional minute or part of a minute.Represent the cost of a telephone call that lasts m min-utes when m is greater than 3.23.A printing shop charges a 75-cent minimum for the first 8 photocopies of a flyer.Additionalcopies cost 6 cents each.Represent the cost of c copies if c is greater than 8.24.A utility company measures gas consumption by the hundred cubic feet,CCF.The companyhas a three-step rate schedule for gas customers.First,there is a minimum charge of $5.00 per month for up to 3 CCF of gas used.Then,for the next 6 CCF,the charge is $0.75 perCCF.Finally,after 9 CCF,the charge is $0.55 per CCF.Represent the cost of g CCF of gas ifg is greater than 9.TermsA term is a number,a variable,or any product or quotient of numbers and vari-ables.For example:5,x ,4y ,8ab ,,and are terms.An algebraic expression that is written as a sum or a difference has morethan one term.For example,4a ϩ2b Ϫ5c has three terms.These terms,4a ,2b ,and 5c ,are separated by ϩand Ϫsigns.Factors of a TermIf a term contains two or more numbers or variables,then each number,eachvariable,and each product of numbers and variables is called a factor of theterm,or factor of the product.For example,the factors of 3xy are 1,3,x ,y ,3x ,3y ,xy ,and 3xy .When we factor whole numbers,we write only factors that areintegers.Any factor of an algebraic term is called the coefficient of the remainingfactor,or product of factors,of that term.For example,consider the algebraicterm 3xy :3 is the coefficient of xy3x is the coefficient of y 3y is the coefficient of x xy is the coefficient of 3When an algebraic term consists of a number and one or more variables,thenumber is called the numerical coefficient of the term.For example:In 8y ,the numerical coefficient is 8.In 4abc ,the numerical coefficient is 4.When the word coefficient is used alone,it usually means a numerical coef-ficient.Also,since x names the same term as 1x ,the coefficient of x is under-stood to be 1.This is true of all terms that contain only variables.For example:7 is the coefficient of b in the1 is the coefficient of b in the term b .term 7b .2.25 is the coefficient of gt in1 is the coefficient of gt in the term gt .the term 2.25gt .Bases,Exponents,and PowersYou learned in Chapter 2 that a power is the product of equal factors.A powerhas a base and an exponent.The base is one of the equal factors of the power.5c k 25Algebraic T erms and Vocabulary 95The exponent is the number of times the base is used as a factor.(If a termis written without an exponent,the exponent is understood to be 1.)42ϭ4(4):base ϭ4exponent ϭ2power ϭ42ϭ16x 3= x (x )(x ):base ϭx exponent ϭ3power ϭx 335m :base ϭm exponent ϭ1power ϭm 1or m 5d 2= 5(d )(d ):base ϭd exponent ϭ2power ϭd 2An exponent refers only to the number or variable that is directly to its left,asseen in the last example,where 2 refers only to the base d .To show the product5d as a base (or to show any sum,difference,product,or quotient as a base),wemust enclose the base in parentheses.(5d )2ϭ(5d )(5d ):base ϭ5d exponent ϭ2power ϭ(5d )2(a + 4)2ϭ(a + 4)( a + 4):base ϭ(a + 4)exponent ϭ2power ϭ(a ϩ4)2Note that –d 4is not the same as (Ϫd )4.–d 4ϭ–1(d )(d )(d )(d ) is always a negative number.(Ϫd )4ϭ1(Ϫd )(Ϫd )(Ϫd )(Ϫd ) is always a positive number since the exponent is even.EXAMPLE 1For each term,name the coefficient,base,and exponent.Answersa.4x 5coefficient ϭ4base ϭx exponent ϭ5b.Ϫw 8coefficient ϭ–1base ϭw exponent ϭ8c.2p r coefficient ϭ2p base ϭr exponent ϭ1Note:Remember that coefficient means numerical coefficient ,and that 2pis a real number.Writing About Mathematics1.Does squaring distribute over multiplication,that is,does (ab )2= (a 2)(b 2)? Write (ab )2as (ab )(ab ) and use the associative and commutative properties of multiplication to justify your answer.2.Does squaring distribute over addition,that is,does (a ϩb )2ϭa 2ϩb 2? Substitute values for a and b to justify your answer.EXERCISES96Algebraic Expressions and Open SentencesDeveloping SkillsIn 3–6,name the factors (other than 1) of each product.3.xy4.3a5.7mn6.1st In 7–14,name,in each case,the numerical coefficient of x .7.8x8.(5 ϩ2)x 9.10.x 11.Ϫ1.4x 12.2 ϩ7x 13.3.4x 14.Ϫx In 15–22,name,in each case,the base and exponent of the power.15.m 216.Ϫs 317.t 18.(Ϫa )419.10620.(5y )421.(x ϩy )522.12c 3In 23–29,write each expression,using exponents.23.b ؒb ؒb ؒb ؒb24.p ؒr ؒr 25.a ؒa ؒa ؒa ؒb ؒb 26.7 ؒr ؒr ؒr ؒs ؒs 27.(6a )(6a )(6a )28.(a Ϫb )(a Ϫb )(a Ϫb )29.the fourth power of (m +2n )In 30–33,write each term as a product without using exponents.30.r 631.5x 432.4a 4b 233.(3y )5In 34–41,name,for each given term,the coefficient,base,and exponent.34.Ϫ3k35.Ϫk 336.p r 237.(ax )538.39.0.0004t 1240.41.(Ϫb )3Applying Skills42.If x represents the cost of a can of soda,what could 5x represent?43.If r represents the speed of a car in miles per hour,what could 3r represent?44.If n represents the number of CDs that Alice has,what could n Ϫ5 represent?45.If d represents the number of days until the end of the year,what could represent?46.If s represents the length of a side of a square,what could 4s represent?47.If r represents the measure of the radius of a circle,what could 2r represent?48.If w represents the number of weeks in a school year,what could represent?49.If d represents the cost of one dozen bottles of water,what could represent?50.If q represents the point value of one field goal,what could 7q represent?d 12w 4d 732a 4"2y 12x Algebraic T erms and Vocabulary 97In Section 1 of this chapter,we listed the words that can be represented by each of the four basic operations.We can use these same lists to write algebraic expressions in words and to write problems that can be represented by a given algebraic expression.For an algebraic expression such as 2n Ϫ3,n could be any real number.That is,associated with any real number n ,there is exactly one real number that is the value of 2n Ϫ3.However,if n and 2n Ϫ3 represent the number of cans of tuna that two customers buy,then n must be a whole number greater than or equal to 2 in order for both n and 2n Ϫ3 to be whole numbers.For this situation,the domain or replacement set would be the set of whole numbers.EXAMPLE 1If n represents the number of points that Hradish scored in a basketball game and 2n Ϫ3 represents the number of points that his friend Brad scored,describe in words the number of points that Brad scored.What is a possible domain for the variable n ?Solution The number of points scored is always a whole number.In order that 2n Ϫ3be a whole number,n must be at least 2.Answer The number of points that Brad scored is 3 less than twice the number thatHradish scored.A possible domain for n is the set of whole numbers greater than or equal to 2.EXAMPLE 2Molly earned d dollars in July and dollars in August.Describe in words the number of dollars that Molly earned in August.Answer In August,Molly earned 10 more than half the number of dollars that sheearned in July.EXAMPLE 3Describe a situation in which x and 12 Ϫx can be used to represent variable quantities.List the domain or replacement set for the answer.Solution If x eggs are used from a full dozen of eggs,there will be 12 Ϫx eggs left.Answer The domain or replacement set is the set of whole numbers less than or equalto 12.12d 1 1098Algebraic Expressions and Open SentencesAnother Solution The distance from my home to school is 12 miles.On my way to school,after Ihave traveled x miles,I have 12 Ϫx miles left to travel.Answer The domain or replacement set is the set of non-negative real numbers thatare less than or equal to 12.Many other answers are possible.Writing About Mathematics1.a.If 4 ϩn represents the number of books Ken read in September and 4 Ϫn representsthe number of books he read in October,how many books did he read in these two months?b.What is the domain of the variable n ?2.Pedro said that the replacement set for the amount that we pay for any item is the set of rational numbers of the form 0.01x where x is a whole number.Do you agree with Pedro?Explain why or why not.Developing SkillsIn 3–14:a.Write in words each of the given algebraic expressions.b.Describe a possible domain for each variable.3.By one route,the distance that Ian walks to school is d miles.By a different route,the dis-tance is d Ϫ0.2 miles.4.Juan pays n cents for a can of soda at the grocery store.When he buys soda from a machine,he pays n ϩ15 cents.5.Yesterday Alexander spent a minutes on leisurely reading and 3a ϩ10 minutes doing homework.6.The width of a rectangle is w meters and the length is 2w ϩ8 meters.7.During a school day,Abby spends h hours in class,hours at lunch and hours on sports.8.Jen spends d hours at work and hours driving to and from work.9.Alicia’s score for 18 holes of golf was g and her son’s score was 10 ϩg .10.Tom paid d cents for a notebook and 5d ϩ30 cents for a pen.11.Seema’s essay for English class had w words and Dominic’s had words.12.Virginia read r books last month and Anna read 3r Ϫ5 books.13.Mario and Pete are playing a card game where it is possible to have a negative score.Pete’s score is s and Mario’s score is s Ϫ220.34w 1 80d 12h3h 6EXERCISESWriting Algebraic Expressions in Words 99100Algebraic Expressions and Open Sentences14.In the past month,Agatha has increased the time that she walks each day from m minutesto 3mϪ10 minutes.Benjamin has 1 more tape than 3 times the number of tapes that Julia has.IfJulia has n tapes,then Benjamin has 3nϩ1 tapes.The algebraic expression 3nϩ1 represents an unspecified number.Only when the variable n is replaced by a specific number does 3nϩ1 become a spe-cific number.For example:If n ϭ10,then 3nϩ1 ϭ3(10) ϩ1 ϭ30 ϩ1 ϭ31.If nϭ15,then 3nϩ1 ϭ3(15) ϩ1 ϭ45 ϩ1 ϭ46.Since in this example,n represents the number of tapes that Julia has,onlywhole numbers are reasonable replacements for n.Therefore,the replacementset is the set of whole numbers or some subset of the set of whole numbers.When we substitute specific values for the variables in an algebraic expres-sion and then determine the value of the resulting expression,we are evaluatingthe algebraic expression.When we determine the number that an algebraic expression represents for specific values of its variables,we are evaluating the algebraic expression.EXAMPLE 1Evaluate 50 Ϫ3x when xϭ7.Solution How to Proceed(1)Write the expression:50 Ϫ3x(2)Replace the variable by its given value:50 Ϫ3(7)(3)Multiply:50 Ϫ21(4)Subtract:29Answer 29EXAMPLE 2Evaluate 2x 2Ϫ5x ϩ4 when:a.x = Ϫ7 b.x = 1.2Solution How to Proceeda.(1) Write the expression:2x 2Ϫ5x ϩ4(2)Replace the variable by the value Ϫ7:2(Ϫ7)2Ϫ5(Ϫ7) ϩ4(3)Evaluate the power:2(49) Ϫ5(Ϫ7) ϩ4(4)Multiply:98 ϩ35ϩ4(5)Add:137b.(1) Write the expression:2x 2Ϫ5x ϩ4(2) Replace the variable by the value 1.2:2(1.2)2Ϫ5(1.2) ϩ4(3) Evaluate the power:2(1.44) Ϫ5(1.2) ϩ4(4) Multiply: 2.88Ϫ6ϩ4(5) Add and subtract:0.88Answers a.137b.0.88EXAMPLE 3Evaluate when a ϭϪ4,n ϭ10,and d ϭ3.Solution How to Proceed(1)Write the expression:(2)Replace the variables with theirgiven values:(3)Simplify the expressions grouped by parentheses or fraction bar:(4)Multiply and divide:(5)Add:Answer25252135 1 26552135 1 27285 1(9)(3)2(24)5 1 (10 – 1)(3)2a5 1 (n 2 1)d 2a5 1 (n 2 1)dEvaluating Algebraic Expressions 101The values given for the variables can be stored in the calculator. ENTER:4103DISPLAY:ENTER:2 5 1DISPLAY:AnswerEXAMPLE 4Evaluate (2x)3Ϫ2x3when xϭ Ϫ0.40.Solution How to Proceed(1)Write the expression:(2x)3Ϫ2x3(2)Replace the variable by its given value:[2(Ϫ0.40)]3Ϫ2(Ϫ0.40)3(3)Simplify the expression within brackets:[Ϫ0.80]3Ϫ2(Ϫ0.40)3(4)Evaluate the powers:Ϫ0.512Ϫ2(Ϫ0.064)(5)Multiply:Ϫ0.512 ϩ0.128(6)Subtract:Ϫ0.384AnswerϪ0.38425255ENTERALPHA)؊ALPHA(؉،ALPHAENTERALPHASTOŁENTERALPHASTOŁENTERALPHASTOŁ(-)CalculatorSolution102Algebraic Expressions and Open SentencesWriting About Mathematics1.Explain why,in an algebraic expression such as 12ab ,12 is called a constant and a and b are called variables?2.Explain why,in step 2 of Example 1,parentheses were needed when x was replaced by its value.Developing SkillsTo understand this topic,you should first evaluate the expressions in Exercises 3 to 27 without a cal-culator.Then,store the values of the variables in the calculator and enter the given algebraic expres-sions to check your work.In 3–27,find the numerical value of each e a ϭ8,b ϭϪ6,d ϭ3,x ϭϪ4,and y ϭ0.5.3.5a4.5.0.3y6.a ϩ37.b Ϫ28.ax 29.10.5x Ϫ2y 11.7xy 312.ab Ϫdx 13.14.0.2d ϩ0.3b 15.16.(3y )217.18.a 2ϩ3d 219.(ay )320.x (y Ϫ2)21.4(2x ϩ3y )22.23.3y Ϫ(x Ϫd )24.2(x ϩy ) Ϫ525.(x Ϫd )526.(2a Ϫ5d )227.(2a )2Ϫ(5d )2Applying Skills28.At one car rental agency,the cost of a car for one day can be determined by using the alge-braic expression $32.00 ϩ$0.10m where m represents the number of miles driven.Determine the cost of rental for each of the following:a.Mike Baier drove the car he rented for 35 miles.b.Dana Morse drove the car he rented for 435 miles.c.Jim Szalach drove the car he rented for 102 miles.29.The local pottery co-op charges $40.00 a year for membership and $0.75 per pound for fir-ing pottery pieces made by the members.The algebraic expression 40 ϩ0.75p represents the yearly cost to a member who brings p pounds of pottery to be fired.Determine the yearly cost for each of the following:a.Tiffany is an amateur potter who fired 35 pounds of work this year.b.Nia sells her pottery in a local craft shop and fired 485 pounds of work this year.30.If a stone is thrown down into a deep gully with an initial velocity of 30 feet per second,thedistance it has fallen,in feet,after t seconds can be found by using the algebraic expression 16t 2ϩ30t .Find the distance the stone has fallen:a.after 1 second.b.after 2 seconds.c.after 3 seconds.12x (y 1 0.1)214x 2y34x325a 1 15b3bd 912x EXERCISESEvaluating Algebraic Expressions 103104Algebraic Expressions and Open Sentences31.The Parkside Bread Company sells cookies and scones as well as bread.Bread (b) costs$4.50 a loaf,cookies (c) cost $1.10 each,and scones (s) cost $1.50 each.The cost of a bakery order can be represented by 4.50bϩ1.10cϩ1.50s.Determine the cost of each of the fol-lowing orders:a.six cookies and two sconesb.three loaves of bread and one cookiec.one loaf of bread,a dozen cookies,and a half-dozen scones32.A Green Thumb volunteer can plant shrubbery at a rate of 6 shrubs per hour and aFriendly Garden volunteer can plant shrubbery at a rate of 8 shrubs per hour.The totalnumber of shrubs that g Green Thumb volunteers and f Friendly Garden volunteers canplant in h hours is given by the algebraic expression 6ghϩ8fh.Determine the number of shrubs planted:a.in 3 hours by 2 Green Thumb and 1 Friendly Garden volunteers.b.in 2 hours by 4 Green Thumb and 4 Friendly Garden volunteers.In this chapter,you learned how to translate words into algebraic expressions.The value of an algebraic expression depends on the value of the variables.When the values of the variables change,the value of the algebraic expressionchanges.For example,xϩ6 is an algebraic expression.The value of xϩ6depends on the value of x.If one value is assigned to an algebraic expression,an algebraic sentence is formed.These sentences may be formulas,equations,or inequalities.For exam-ple,when the value 9 is assigned to the algebraic expression xϩ6,we can writethe sentence “Six more than x is 9.”This sentence can be written in symbols asxϩ6 ϭ9.Every sentence that contains a variable is called an open sentence.xϩ6 ϭ93yϭ122nϾ0xϩ5 Յ8An open sentence is neither true nor false.The sentence will be true or false only when the variables are replaced by numbers from a domain or a replacement set,such as {0,1,2,3}.The numbers from the domain that make the sentence true are the elements of the solut ion set of the open sentence.A solution set,as seen below,cancontain one or more numbers or,at times,no numbers at all,from the replace-ment set.EXAMPLE 1Using the domain {0,1,2,3},find the solution set of each open sentence:a.xϩ6 ϭ9b.2nϾ0。
Hockey-冰球
Ice hockey was played for the first time in the Olympics in the Olympic Games of 1920.
• IJscolf, a game resembling colf on an ice-covered surface, was popular in the Low Countries between the Middle Ages and the Dutch Golden Age.
• In England, evidence of games of 'hockey on ice' (the name replaced "bandie ball"), played with a "bung" (a plug of cork or oak used as a stopper on a barrel) date back to the 1700s.
The belief that hockey was mentioned in a 1363 proclamation by King Edward III of England is based on modern translations of the proclamation, which was originally in Latin and explicitly forbade the games .
According to the Austin Hockey Association, the word "puck" derives from the Scots Gaelic puc or the Irish poc (to poke, punch or deliver a blow). "...The blow given by a hurler to the ball with his caman or hurley is always called a puck."
Algebra1中英版对照目录
分步和混合运算解一元一次方程 P92—P99 2—4：Solving Equations with Variables on Both Sides
变量在两边的一元一次方程的解法 P100—P106 2—5：Solving for a Variable 求变量的值（一元一次）P107—P113
指数幂的乘法运算（同底数幂相乘和积的乘方运算）P460—P466 7—4：Division Properties of Exponents 指数幂的除法 P467—P474 Quiz for Lessons 7-1 through 7-4 第七章 1—4 课小测试 P474—P475 7—5：Polynomials 多项式 P476—P483 7—6：Adding and Subtracting Polynomials 多项式的加减运算 P484—P491 7—7：Multiplying Polynomials 多项式的乘法（单项式与多项式相乘、多项
Algebra 1 与中文数学课本内容在顺序上的对照
Algebra 1
中教数学
CHAPTER 0---- To The Student 0—1：Geometry Formulas
几何公式（s quare，rec tangle,triangle,c irc le）Z3— Z4 0—2：Tree Diagrams 树状图 Z4—Z6 0—3：The Coordinate Plane 直角坐标系 Z7—Z8 0—4：Rounding and Estimating 四舍五入 Z9—Z11 0—5：Adding and Subtracting Decimals 小数的加减运算 Z12—Z13 0—6：Multiplying and Dividing Decimals 小数的乘除运算 Z14—Z16 0—7：Prime and Composite Numbers 素数与合数 Z17—Z18 0—8：Factoring 因数 Z19—Z20 0—9：GCF and LCM 最大公约数和最小公倍数 Z21—Z22
演示文稿数学专业英语第八讲附数学课程英文表达
4、L. Hormander “Linear Partial Differential Operators, ” I&II：偏微分方程的经典参考书； 5、A Course in Abstract Harmonic Analysis by Folland：高级的研究生调和分析教材； 6、Abstract Harmonic Analysis by Ross Hewitt：抽象调和分析的经典参考书； 7、Harmonic Analysis by Elias M. Stein：标准的研究生调和分析教材； 8、Elliptic Partial Differential Equations of Second Order by David Gilbarg：偏微分 方程的经典参考书； 9、Partial Differential Equations ，by Jeffrey Rauch：标准的研究生偏微分方程教材。
覆盖范围较广；
9、Elements of Homotopy Theory by G.W. Whitehead：高级、经典的代数拓扑参考 书。
第六页，共34页。
• 实分析、泛函分析：
1、Royden, Real analysis：标准研究生分析教材； 2、Walter Rudin, Real and complex analysis：标准研究生分析教材；
；
4、Principles of Algebraic Geometry by giffiths/harris：全面、经典的代数几何参考 书，偏复代数几何； 5、Commutative Algebra with a view toward Algebraic Geometry by Eisenbud：高
调代数参考书；
Combinatorial Hopf algebras and Towers of Algebras
a rXiv:071.3744v1[mat h.CO ]19Oct27COMBINATORIAL HOPF ALGEBRAS AND TOWERS OF ALGEBRAS NANTEL BERGERON,THOMAS LAM,AND HUILAN LI Abstract.Bergeron and Li have introduced a set of axioms which guarantee that the Grothendieck groups of a tower of algebras L n ≥0A n can be endowed with the structure of graded dual Hopf algebras.Hivert and Nzeutzhap,and independently Lam and Shimozono constructed dual graded graphs from prim-itive elements in Hopf algebras.In this paper we apply the composition of these constructions to towers of algebras.We show that if a tower L n ≥0A n gives rise to graded dual Hopf algebras then we must have dim(A n )=r n n !where r =dim(A 1). 1.Introduction This paper is concerned with the interplay between towers of associative alge-bras,pairs of dual combinatorial Hopf algebras,and dual graded graphs.Our point of departure is the study of the composition of two constructions:(1)the construc-tion of dual Hopf algebras from towers of algebras satisfying some axioms,due to Bergeron and Li [3];and (2)the construction of dual graded graphs from primitive elements in dual Hopf algebras,discovered independently by Hivert and Nzeutchap [8],and Lam and Shimozono:tower of algebras −→combinatorial Hopf algebra −→dual graded graph (1.1)The notion of a pair (Γ,Γ′)dual graded graphs is likely to be the least familiar.They were introduced by Fomin [5](see also [13])to encode the enumerative prop-erties of the Robinson-Schensted correspondence and its generalizations.The first arrow in (1.1)is obtained by using induction and restriction on the Grothendieck groups.The second arrow is obtained by using (some of the)structure constants of a combinatorial Hopf algebra as edge multiplicities for a graph.We review these constructions in Sections 2and 3.The notion of combinatorial Hopf algebra usedhere is related,but slightly different from the one in [1].The key example of all three classes of objects arises from the theory of symmet-ric functions.In [6],L.Geissinger showed that the ring Sym of symmetric functions is a graded self-dual Hopf ing the work of Frobenius and Schur,Zelevin-sky [14]interpreted the Hopf structurein terms of the Grothendieck groups of thetower of symmetric group algebras n ≥0C S n .Finally it follows from the classical work of Young that the branching rule for the symmetric group,or equivalently2NANTEL BERGERON,THOMAS LAM,AND HUILAN LIthe Pieri rule for symmetric functions,gives rise to the Young graph on the set of partitions.The Young graph is the motivating example of dual graded graphs.In recent years it has been shown that other graded dual Hopf algebras can be obtained from towers of algebras.In[11]Malvenuto and Reutenauer establish the duality between the Hopf algebra NSym of noncommutative symmetric functionsand the Hopf algebra QSym of quasi-symmetric functions.Krob and Thibon[9] then showed that this duality can be interpreted as the duality of the Grothendieckgroups associated with n≥0H n(0)the tower of Hecke algebras at q=0.For more examples,see[2,7,12].It is very tempting,as suggested by J.Y.Thibon,to classify all combinato-rial Hopf algebras which arise as Grothendieck groups associated with a tower of algebras n≥0A n.The list of axioms given by thefirst and last author in[3]guar-antees that the Grothendieck groups of a tower of algebras form a pair of graded dual Hopf algebras.This list of axioms is not totally satisfactory as some of the axioms are difficult to verify and the description is far from a classification.In this paper we present a very surprising fact which shows that towers of algebras giving rise to combinatorial Hopf algebras are much more rigid than they appear. Theorem1.1.If A= n≥0A n is a tower of algebras such that its associated Grothendieck groups form a pair of graded dual Hopf algebras,then dim(A n)=r n n! where r=dim(A1).The notion of“forming a pair of graded dual Hopf algebras”is made precise in Section2.The numbers r n n!will be familiar to experts in the theory of dual graded graphs–they count certain paths in a pair of dual graded graphs.The rigidity proved in Theorem1.1suggests that there may be a structure the-orem for towers of algebras which give rise to combinatorial Hopf algebras.In particular,to perform the inverse constructions of the arrows in(1.1),it suggests that one should study algebras related to symmetric groups(or wreath products of symmetric groups).There are many combinatorial Hopf algebras for which one may attempt to perform the inverse construction,but there are even more dual graded graphs.The general construction of[10]produces dual graded graphs from Bruhat orders of Weyl groups of Kac-Moody algebras and it is unclear whether there are Hopf algebras,or towers of algebras giving rise to these graphs.Acknowledgements.The second author would like to thank Mark Shimozono for the collaboration which led to the line of thinking in this paper.2.From towers of algebras to combinatorial Hopf algebrasWe recall here the work of Bergeron and Li[3]on towers of algebras.For B an arbitrary algebra we denote by B mod,the category of allfinitely generated left B-modules,and by P(B),the category of allfinitely generated projective left B-modules.For some category C of left B-modules(B mod or P(B))let F be the free abelian group generated by the symbols(M),one for each isomorphism class of modules M in C.Let F0be the subgroup of F generated by all expressions (M)−(L)−(N)one for each exact sequence0→L→M→N→0in C.The Grothendieck group K0(C)of the category C is defined by the quotient F/F0,an abelian additive group.For M∈C,we denote by[M]its image in K0(C).COMBINATORIAL HOPF ALGEBRAS AND TOWERS OF ALGEBRAS3 We then setG0(B)=K0(B mod)and K0(B)=K0(P(B)).For B afinite-dimensional algebra over afield K,let{V1,···,V s}be a complete list of nonisomorphic simple B-modules.The projective covers{P1,···,P s}of the simple modules V i’s is a complete list of nonisomorphic indecomposable projective B-modules.We have that G0(B)= s i=1Z[V i]and K0(B)= s i=1Z[P i].Letϕ:B→A be an injection of algebras preserving unities,and let M be a(left)A-module and N a(left)B-module.The induction of N from B to A is Ind A B N=A⊗ϕN,the(left)A-module A⊗N modulo the relations a⊗bn≡aϕ(b)⊗n, and the restriction of M from A to B is Res A B M=Hom A(A,M),the(left)B-module with the B-action defined by bf(a)=f(aϕ(b)).Let A= n≥0A n be a graded algebra over C with multiplicationρ:A⊗A→A. Bergeron and Li studiedfive axioms for A(we refer to[3]for full details):(1)For each n≥0,A n is afinite-dimensional algebra by itself with(internal) multiplicationµn:A n⊗A n→A n and unit1n.A0∼=C.(2)The(external)multiplicationρm,n:A m⊗A n→A m+n is an injective homo-morphism of algebras,for all m and n(sending1m⊗1n to1m+n).(3)A m+n is a two-sided projective A m⊗A n-module with the action defined by a·(b⊗c)=aρm,n(b⊗c)and(b⊗c)·a=ρm,n(b⊗c)a,for all m,n≥0,a∈A m+n,b∈A m,c∈A n and m,n≥0.(4)A relation between the decomposition of A n+m as a left A m⊗A n-module and as a right A m⊗A n-module holds.(5)An analogue of Mackey’s formula relating induction and restriction of modules holds.We say here that A= n≥0A n is a tower of algebras if it satisfies Conditions (1),(2)and(3).Condition(1)guarantees that the Grothendieck groups G(A)= n≥0G0(A n) and K(A)= n≥0K0(A n)are graded connected.Conditions(2)and(3)ensure that induction and restriction are well defined on G(A)and K(A),defining a mul-tiplication and comultiplication,as follows.For[M]∈G0(A m)(or K0(A m))and [N]∈G0(A n)(or K0(A n))we let[M][N]= Ind A m+n A m⊗A n M⊗N and∆([N])= k+l=n Res A k+l A k⊗A l N . The pairing between K(A)and G(A)is given by , :K(A)×G(A)→Z where [P],[M] = dim K Hom A n(P,M) if[P]∈K0(A n)and[M]∈G0(A n),0otherwise.Thus with(only)Conditions(1),(2),and(3),G(A)and K(A)are dual free Z-modules both endowed with a multiplication and comultiplication.Bergeron and Li[3]proveTheorem2.1.If a graded algebra A= n≥0A n over C satisfies Conditions(1)-(5)then G(A)and K(A)are graded dual Hopf algebras.In particular Theorem1.1applies to graded algebras which satisfy Conditions (1)-(5).Note that the dual Hopf algebras G(A)and K(A)come with distinguished4NANTEL BERGERON,THOMAS LAM,AND HUILAN LIbases consisting of the isomorphism classes of simple and indecomposable projective modules.3.From combinatorial Hopf algebras to dual graded graphsThis section recounts work of Fomin[5],Hivert and Nzeutchap[8],and Lam and Shimozono.A graded graphΓ=(V,E,h,m)consists of a set of vertices V,a set of(directed)edges E⊂V×V,a height function h:V→{0,1,...}and an edge multiplicity function m:V×V→{0,1,...}.If(v,u)∈E is an edge then we must have h(u)=h(v)+1.The multiplicity function determines the edge set:(v,u)∈E if and only if m(v,u)=0.We assume always that there is a single vertex v0of height0.Let Z V= v∈V Z·v be the free Z-module generated by the vertex set.Given a graded graphsΓ=(V,E,h,m)we define up and down operators U,D:Z V→Z V byUΓ(v)= u∈V m(v,u)u DΓ(v)= u∈V m(u,v)uand extending by linearity over Z.We will assume thatΓis locally-finite,so that these operators are well defined.A pair(Γ,Γ′)of graded graphs with the same vertex set V and height function h is called dual with differential coefficient r if we haveDΓ′UΓ−UΓDΓ′=r Id.We shall need the following result of Fomin.For a graded graphΓ,let f vΓdenote the number of paths from v0to v,where for two vertices w,u∈V,we think that there are m(w,u)edges connecting w to u.Theorem3.1(Fomin[5]).Let(Γ,Γ′)be a pair of dual graded graphs with differ-ential coefficient r.Thenr n n!= v:h(v)=n f vΓf vΓ′.Let H•= n≥0H n and H•= n≥0H n be graded dual Hopf algebras over Z with respect to the pairing .,. :H•×H•→Z.We assume that we are given dual sets of homogeneous free Z-module generators{pλ∈H•}λ∈Λand{sλ∈H•}λ∈Λ, such that all structure constants are non-negative integers.We also assume that dim(H i)=dim(H i)<∞for each i≥0and dim(H0)=dim(H0)=1,so that H0 and H0are spanned by distinguished elements the unit1.Let us suppose we are given non-zero homogeneous elementsα∈H1andβ∈H1of degree1.We now define a graded graphΓ(β)=(V,E,h,m)where V={sλ}λ∈Λand h:V→Z is defined by h(sλ)=deg(sλ).The map m:V×V→Z is defined bym(sλ,sµ)= pµ,βsλ = ∆(pµ),β⊗sλand E is determined by m.The graphΓ(β)is graded because of the assumption thatβhas degree1.Similarly,we define a graded graphΓ′(α)=(V′,E′,h′,m′) where V′=V,h′=h,andm′(sλ,sµ)= αpλ,sµ = α⊗pλ,∆(sµ) .The following theorem is due independently to Hivert and Nzeutchap[8]and Lam and Shimozono(unpublished).COMBINATORIAL HOPF ALGEBRAS AND TOWERS OF ALGEBRAS 5Theorem 3.2.The graded graphs Γ=Γ(β)and Γ′=Γ′(α)form a pair of dual graded graphs with differential coefficient α,β .Proof.We identify Z V with H •and note that U Γ(x )=βx where x ∈H •and we use the multiplication in H •.Also,D Γ′(x )= µ∈Λ α⊗p µ,∆x s µ= α,x (1) x (2).where ∆x = x (1)⊗x (2).Now observe that by our hypotheses on the degree of αand βthey are primitive elements:∆α=1⊗α+α⊗1and ∆β=1⊗β+β⊗1.We first calculateα,βx = ∆α,β⊗x = 1,β α,x + α,β 1,x = α,β 1,xand then computeD Γ′U Γ(x )=D Γ′(βx )=α,βx (1) x (2)+ α,x (1) βx (2) = α,β x +U ΓD Γ′(x )where to obtain α,β x in the last line we use ∆x =1⊗x +terms of other degrees .4.Proof of Theorem 1.1We are given a graded algebra A = n ≥0A n over C with multiplication ρsatis-fying Conditions (1),(2)and (3).Moreover we assume that the two Grothendieck groups G (A )and K (A )form a pair of graded dual Hopf algebras as in Section 2.Under these assumptions we show thatdim(A n )=r n n !where r =dim(A 1).Let H •=G (A )and H •=K (A ).Let {s (1)1=[S (1)1],...,s (1)t =[S (1)t ]}and{p (1)1=[P (1)1],...,p (1)t =[P (1)t ]}denote the isomorphism classes of simple and inde-composable projective A 1-modules,so that H 1= t i =1Z s (1)i and H 1= t i =1Z p (1)i .Define a i =dim(S (1)i )and b i =dim(P (1)i )for 1≤i ≤t .We set for the remainder of this paperα=t i =1a i p (1)i∈H 1and β=t i =1b i s (1)i ∈H 1.Since A 0∼=C ,we let s (0)1(respectively,p (0)1)be the unique simple (respectively,indecomposable projective)module representative in H 0(respectively,H 0).Simi-larly,let {s (n )i =[S (n )i ]}be all isomorphic classes of simple A n -modules and {p (n )i =[P (n )i ]}be all isomorphism classes of indecomposable projective A n -modules.The sets n ≥0{s (n )i }and n ≥0{p (n )i }form dual free Z -module bases of H •and H •.Now define Γ=Γ(β)and Γ′=Γ′(α)as in Section 3.Lemma 4.1.We havef s (n )j Γ=dim P (n )j and f s (n )jΓ′=dim S (n )j .6NANTEL BERGERON,THOMAS LAM,AND HUILAN LI Proof.We havem(s(n−1)i ,s(n)j)=tl=1b l c l,where c l is the number of copies of the indecomposable projective module P(1)l⊗P(n−1) i as a summand in Res A n A1⊗A n−1P(n)j.Note that s(0)1is the unit of H•andm(s(0)1,s(1)i)=b i=dim P(1)ifor all1≤i≤t.The dimension of an indecomposableprojective module P(n)jis given bydim P(n)j= i,l c l dim P(1)l⊗P(n−1)i = i m(s(n−1)i,s(n)j)dim P(n−1)i.By induction on n,we deduce that dim P(n)jis the number of paths from s(0)1tos(n)jinΓ.The claim forΓ′is similar.For anyfinite dimensional algebra B let{Sλ}λbe a complete set of simple B-modules.For eachλlet Pλbe the projective cover of Sλ.It is well known(see[4]) that we canfind minimal idempotents{e i}such that B= Be i where each Be i is isomorhpic to a Pλ.Moreover,the quotient of B by its radical shows that the multiplicity of Pλin B is equal to dim Sλ.This implies the following lemma. Lemma4.2.Let B be afinite dimensional algebra and{Sλ}λbe a complete set of simple B-modules.dim B= λ(dim Pλ)(dim Sλ),where Pλis the projective cover of Sλ.By Lemma4.2,we have r= t i=1a i b i= α,β .By Theorem3.2we may apply Theorem3.1to(Γ,Γ′).Using Lemma4.2and Lemma4.1,Theorem3.1says dim(A n)= i(dim P(n)i)(dim S(n)i)= i f s(n)iΓf s(n)iΓ′=r n n!.Remark4.3.If the tower consists of semisimple algebras A i thenΓ=Γ′so we obtain a self-dual graph.In this case the graph would be a weighted version of a differential poset in the sense of Stanley[13].If furthermore the branching of irreducible modules from A n to A1⊗A n−1is multiplicity free then we get a true differential poset.Remark4.4.The Hopf algebras H•and H•are not in general commutative and co-commutative.Thus in the definitions of Section3we could have obtained a different pair of dual graded graphs by setting m(sλ,sµ)= pµ,sλβ or m′(sλ,sµ)= pλα,sµ .References[1]M.Aguiar,N.Bergeron,and F.Sottile,Combinatorial Hopf algebras and generalized Dehn-Sommerville relations,Compos.Math.142(2006),no.1,1–30.[2]N.Bergeron,F.Hivert and J.Y.Thibon,The peak algebra and the Hecke-Clifford algebrasat q=bin.Theory Ser.A107-1(2004)1–19.[3]N.Bergeron and H.Li,Algebraic Structures on Grothendieck Groups of a Tower of Algebras,To appear.[arXiv:math/0612170].COMBINATORIAL HOPF ALGEBRAS AND TOWERS OF ALGEBRAS7 [4]C.Curtis and I.Reiner,Methods of representation theory.Vol.I.With applications tofinitegroups and orders,John Wiley&Sons,Inc.,New York,1990.[5]S.Fomin,Duality of graded graphs,J.Algebraic Combin.3(1994),no.4,357–404.[6]L.Geissinger,Hopf algebras of symmetric functions and class functions,Combinatoire etCombinatoire et repr´e sentation du groupe sym´e trique(Actes Table Ronde C.N.R.S.,Univ.Louis-Pasteur Strasbourg,Strasbourg,1976),pp.168–181.Lecture Notes in Math.,Vol.579, Springer,Berlin,1977.[7]F.Hivert,J.-C.Novelli and J.-Y.Thibon,Representation theory of the0-Ariki-Koike-Shojialgebras,to appear.[math.CO/0407218].[8]F.Hivert and eutchap,Dual Graded Graphs in Combinatorial Hopf Algebras,in prepa-ration.[9]D.Krob and J.Y.Thibon,Noncommutative symmetric functions.IV.Quantum linear groupsand Hecke algebras at q=0,J.Algebraic Combin.6-4(1997)339–376.[10]m and M.Shimozono,Dual graded graphs for Kac-Moody algebras,Algebra and NumberTheory,to appear.[math.CO/0702090].[11]C.Malvenuto and C.Reutenauer,Duality between quasi-symmetric functions and thesolomon descent algebra,J.Algebra177-3(1995)967–982.[12]A.N.Sergeev,Tensor algebra of the identity representation as a module over the Lie super-algebras GL(n;m)and Q(n),SR Sbornik51(1985)419–427.[13]R.Stanley,Differential posets,J.Amer.Math.Soc.1(1988),919–961.[14]A.V.Zelevinsky,Representations offinite classical groups.A Hopf algebra approach,LectureNotes in Mathematics869.Springer-Verlag,Berlin-New York,1981.(Nantel Bergeron)Department of Mathematics and Statistics,York University,To-ronto,Ontario M3J1P3,CANADAE-mail address:bergeron@mathstat.yorku.caURL:http://www.math.yorku.ca/bergeron(Thomas Lam)Department of Mathematics,Harvard University,Cambridge,,MA 02138.E-mail address:tfylam@URL:/~tfylam(Huilan Li)Department of Mathematics and Statistics,York University,Toronto, Ontario M3J1P3,CANADAE-mail address:lihuilan@mathstat.yorku.ca。
love and logic课文翻译
love and logic课文及翻译Love and logic: The story of a fallacy爱情与逻辑：谬误的故事1 I had my first date with Polly after I made the trade with my roommate Rob. That year every guy on campus had a leather jacket, and Rob couldn't stand the idea of being the only football player who didn't, so he made a pact that he'd give me his girl in exchange for my jacket. He wasn't the brightest guy. Polly wasn't too shrewd, either.在我和室友罗伯的交易成功之后，我和波莉有了第一次约会。
那一年校园里每个人都有件皮夹克，而罗伯是校足球队员中唯一一个没有皮夹克的，他一想到这个就受不了，于是他和我达成了一项协议，用他的女友换取我的夹克。
他可不那么聪明，而他的女友波莉也不太精明。
2 But she was pretty, well-off, didn't dye her hair strange colors or wear too much makeup. She had the right background to be the girlfriend of a dogged, brilliant lawyer. If I could show the elite law firms I applied to that I had a radiant, well-spoken counterpart by my side, I just might edge past the competition.但她漂亮而且富有，也没有把头发染成奇怪的颜色或是化很浓的妆。
Ordinarydifferentialequation
Ordinary differential equationIn mathematics, an ordinary differential equation (or ODE ) is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable.A simple example is Newton's second law of motion, which leads to the differential equationfor the motion of a particle of constant mass m . In general, the force F depends upon the position x(t) of the particle at time t , and thus the unknown function x(t) appears on both sides of the differential equation, as is indicated in the notation F (x (t )).Ordinary differential equations are distinguished from partial differential equations, which involve partial derivatives of functions of several variables.Ordinary differential equations arise in many different contexts including geometry, mechanics, astronomy and population modelling. Many famous mathematicians have studied differential equations and contributed to the field,including Newton, Leibniz, the Bernoulli family, Riccati, Clairaut, d'Alembert and Euler.Much study has been devoted to the solution of ordinary differential equations. In the case where the equation is linear, it can be solved by analytical methods. Unfortunately, most of the interesting differential equations are non-linear and, with a few exceptions, cannot be solved exactly. Approximate solutions are arrived at using computer approximations (see numerical ordinary differential equations).The trajectory of a projectile launched from a cannon follows a curve determined by an ordinary differential equation that is derived fromNewton's second law.Existence and uniqueness of solutionsThere are several theorems that establish existence anduniqueness of solutions to initial value problemsinvolving ODEs both locally and globally. SeePicard –Lindelöf theorem for a brief discussion of thisissue.DefinitionsOrdinary differential equationLet ybe an unknown function in x with the n th derivative of y , and let Fbe a given functionthen an equation of the formis called an ordinary differential equation (ODE) of order n . If y is an unknown vector valued function,it is called a system of ordinary differential equations of dimension m (in this case, F : ℝmn +1→ ℝm ).More generally, an implicit ordinary differential equation of order nhas the formwhere F : ℝn+2→ ℝ depends on y(n). To distinguish the above case from this one, an equation of the formis called an explicit differential equation.A differential equation not depending on x is called autonomous.A differential equation is said to be linear if F can be written as a linear combination of the derivatives of y together with a constant term, all possibly depending on x:(x) and r(x) continuous functions in x. The function r(x) is called the source term; if r(x)=0 then the linear with aidifferential equation is called homogeneous, otherwise it is called non-homogeneous or inhomogeneous. SolutionsGiven a differential equationa function u: I⊂ R→ R is called the solution or integral curve for F, if u is n-times differentiable on I, andGiven two solutions u: J⊂ R→ R and v: I⊂ R→ R, u is called an extension of v if I⊂ J andA solution which has no extension is called a global solution.A general solution of an n-th order equation is a solution containing n arbitrary variables, corresponding to n constants of integration. A particular solution is derived from the general solution by setting the constants to particular values, often chosen to fulfill set 'initial conditions or boundary conditions'. A singular solution is a solution that can't be derived from the general solution.Reduction to a first order systemAny differential equation of order n can be written as a system of n first-order differential equations. Given an explicit ordinary differential equation of order n (and dimension 1),define a new family of unknown functionsfor i from 1 to n.The original differential equation can be rewritten as the system of differential equations with order 1 and dimension n given bywhich can be written concisely in vector notation aswithandLinear ordinary differential equationsA well understood particular class of differential equations is linear differential equations. We can always reduce an explicit linear differential equation of any order to a system of differential equation of order 1which we can write concisely using matrix and vector notation aswithHomogeneous equationsThe set of solutions for a system of homogeneous linear differential equations of order 1 and dimension nforms an n-dimensional vector space. Given a basis for this vector space , which is called a fundamental system, every solution can be written asThe n × n matrixis called fundamental matrix. In general there is no method to explicitly construct a fundamental system, but if one solution is known d'Alembert reduction can be used to reduce the dimension of the differential equation by one.Nonhomogeneous equationsThe set of solutions for a system of inhomogeneous linear differential equations of order 1 and dimension ncan be constructed by finding the fundamental system to the corresponding homogeneous equation and one particular solution to the inhomogeneous equation. Every solution to nonhomogeneous equation can then be written asA particular solution to the nonhomogeneous equation can be found by the method of undetermined coefficients or the method of variation of parameters.Concerning second order linear ordinary differential equations, it is well known thatSo, if is a solution of: , then such that:So, if is a solution of: ; then a particular solution of , isgiven by:. [1]Fundamental systems for homogeneous equations with constant coefficientsIf a system of homogeneous linear differential equations has constant coefficientsthen we can explicitly construct a fundamental system. The fundamental system can be written as a matrix differential equationwith solution as a matrix exponentialwhich is a fundamental matrix for the original differential equation. To explicitly calculate this expression we first transform A into Jordan normal formand then evaluate the Jordan blocksof J separately asTheories of ODEsSingular solutionsThe theory of singular solutions of ordinary and partial differential equations was a subject of research from the time of Leibniz, but only since the middle of the nineteenth century did it receive special attention. A valuable but little-known work on the subject is that of Houtain (1854). Darboux (starting in 1873) was a leader in the theory, and in the geometric interpretation of these solutions he opened a field which was worked by various writers, notably Casorati and Cayley. To the latter is due (1872) the theory of singular solutions of differential equations of the first order as accepted circa 1900.Reduction to quadraturesThe primitive attempt in dealing with differential equations had in view a reduction to quadratures. As it had been the hope of eighteenth-century algebraists to find a method for solving the general equation of the th degree, so it was the hope of analysts to find a general method for integrating any differential equation. Gauss (1799) showed, however, that the differential equation meets its limitations very soon unless complex numbers are introduced. Hence analysts began to substitute the study of functions, thus opening a new and fertile field. Cauchy was the first to appreciate the importance of this view. Thereafter the real question was to be, not whether a solution is possible by means of known functions or their integrals, but whether a given differential equation suffices for the definition of a function of the independent variable or variables, and if so, what are the characteristic properties of this function.Fuchsian theoryTwo memoirs by Fuchs (Crelle, 1866, 1868), inspired a novel approach, subsequently elaborated by Thomé and Frobenius. Collet was a prominent contributor beginning in 1869, although his method for integrating a non-linear system was communicated to Bertrand in 1868. Clebsch (1873) attacked the theory along lines parallel to those followed in his theory of Abelian integrals. As the latter can be classified according to the properties of the fundamental curve which remains unchanged under a rational transformation, so Clebsch proposed to classify the transcendent functions defined by the differential equations according to the invariant properties of the corresponding surfaces f = 0 under rational one-to-one transformations.Lie's theoryFrom 1870 Sophus Lie's work put the theory of differential equations on a more satisfactory foundation. He showed that the integration theories of the older mathematicians can, by the introduction of what are now called Lie groups, be referred to a common source; and that ordinary differential equations which admit the same infinitesimal transformations present comparable difficulties of integration. He also emphasized the subject of transformations of contact.A general approach to solve DE's uses the symmetry property of differential equations, the continuous infinitesimal transformations of solutions to solutions (Lie theory). Continuous group theory, Lie algebras and differential geometry are used to understand the structure of linear and nonlinear (partial) differential equations for generating integrable equations, to find its Lax pairs, recursion operators, Bäcklund transform and finally finding exact analytic solutions to the DE.Symmetry methods have been recognized to study differential equations arising in mathematics, physics, engineering, and many other disciplines.Sturm–Liouville theorySturm–Liouville theory is a theory of eigenvalues and eigenfunctions of linear operators defined in terms of second-order homogeneous linear equations, and is useful in the analysis of certain partial differential equations.Software for ODE solving•FuncDesigner (free license: BSD, uses Automatic differentiation, also can be used online via Sage-server [2])•VisSim [3] - a visual language for differential equation solving•Mathematical Assistant on Web [4] online solving first order (linear and with separated variables) and second order linear differential equations (with constant coefficients), including intermediate steps in the solution.•DotNumerics: Ordinary Differential Equations for C# and [5] Initial-value problem for nonstiff and stiff ordinary differential equations (explicit Runge-Kutta, implicit Runge-Kutta, Gear’s BDF and Adams-Moulton).•Online experiments with JSXGraph [6]References[1]Polyanin, Andrei D.; Valentin F. Zaitsev (2003). Handbook of Exact Solutions for Ordinary Differential Equations, 2nd. Ed.. Chapman &Hall/CRC. ISBN 1-5848-8297-2.[2]/welcome[3][4]http://user.mendelu.cz/marik/maw/index.php?lang=en&form=ode[5]/NumericalLibraries/DifferentialEquations/[6]http://jsxgraph.uni-bayreuth.de/wiki/index.php/Differential_equationsBibliography• A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations (2nd edition)", Chapman & Hall/CRC Press, Boca Raton, 2003. ISBN 1-58488-297-2• A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, Handbook of First Order Partial Differential Equations, Taylor & Francis, London, 2002. ISBN 0-415-27267-X• D. Zwillinger, Handbook of Differential Equations (3rd edition), Academic Press, Boston, 1997.•Hartman, Philip, Ordinary Differential Equations, 2nd Ed., Society for Industrial & Applied Math, 2002. ISBN 0-89871-510-5.•W. Johnson, A Treatise on Ordinary and Partial Differential Equations (/cgi/b/bib/ bibperm?q1=abv5010.0001.001), John Wiley and Sons, 1913, in University of Michigan Historical Math Collection (/u/umhistmath/)• E.L. Ince, Ordinary Differential Equations, Dover Publications, 1958, ISBN 0486603490•Witold Hurewicz, Lectures on Ordinary Differential Equations, Dover Publications, ISBN 0-486-49510-8•Ibragimov, Nail H (1993), CRC Handbook of Lie Group Analysis of Differential Equations Vol. 1-3, Providence: CRC-Press, ISBN 0849344883.External links•Differential Equations (/Science/Math/Differential_Equations//) at the Open Directory Project (includes a list of software for solving differential equations).•EqWorld: The World of Mathematical Equations (http://eqworld.ipmnet.ru/index.htm), containing a list of ordinary differential equations with their solutions.•Online Notes / Differential Equations (/classes/de/de.aspx) by Paul Dawkins, Lamar University.•Differential Equations (/diffeq/diffeq.html), S.O.S. Mathematics.• A primer on analytical solution of differential equations (/mws/gen/ 08ode/mws_gen_ode_bck_primer.pdf) from the Holistic Numerical Methods Institute, University of South Florida.•Ordinary Differential Equations and Dynamical Systems (http://www.mat.univie.ac.at/~gerald/ftp/book-ode/ ) lecture notes by Gerald Teschl.•Notes on Diffy Qs: Differential Equations for Engineers (/diffyqs/) An introductory textbook on differential equations by Jiri Lebl of UIUC.Article Sources and Contributors7Article Sources and ContributorsOrdinary differential equation Source: /w/index.php?oldid=433160713 Contributors: 48v, A. di M., Absurdburger, AdamSmithee, After Midnight, Ahadley,Ahoerstemeier,AlfyAlf,Alll,AndreiPolyanin,Anetode,Ap,Arthena,ArthurRubin,BL,BMF81,********************,Bemoeial,BenFrantzDale,Benjamin.friedrich,BereanHunter,Bernhard Bauer, Beve, Bloodshedder, Bo Jacoby, Bogdangiusca, Bryan Derksen, Charles Matthews, Chilti, Chris in denmark, ChrisUK, Christian List, Cloudmichael, Cmdrjameson, Cmprince, Conversion script, Cpuwhiz11, Cutler, Delaszk, Dicklyon, DiegoPG, Dmitrey, Dmr2, DominiqueNC, Dominus, Donludwig, Doradus, Dysprosia, Ed Poor, Ekotkie, Emperorbma, Enochlau, Fintor, Fruge, Fzix info, Gauge, Gene s, Gerbrant, Giftlite, Gombang, HappyCamper, Heuwitt, Hongsichuan, Ht686rg90, Icairns, Isilanes, Iulianu, Jack in the box, Jak86, Jao, JeLuF, Jitse Niesen, Jni, JoanneB, John C PI, Jokes Free4Me, JonMcLoone, Josevellezcaldas, Juansempere, Kawautar, Kdmckale, Krakhan, Kwantus, L-H, LachlanA, Lethe, Linas, Lingwitt, Liquider, Lupo, MarkGallagher,MathMartin, Matusz, Melikamp, Michael Hardy, Mikez, Moskvax, MrOllie, Msh210, Mtness, Niteowlneils, Oleg Alexandrov, Patrick, Paul August, Paul Matthews, PaulTanenbaum, Pdenapo, PenguiN42, Phil Bastian, PizzaMargherita, Pm215, Poor Yorick, Pt, Rasterfahrer, Raven in Orbit, Recentchanges, RedWolf, Rich Farmbrough, Rl, RobHar, Rogper, Romanm, Rpm, Ruakh, Salix alba, Sbyrnes321, Sekky, Shandris, Shirt58, SilverSurfer314, Ssd, Starlight37, Stevertigo, Stw, Susvolans, Sverdrup, Tarquin, Tbsmith, Technopilgrim, Telso, Template namespace initialisation script, The Anome, Tobias Hoevekamp, TomyDuby, TotientDragooned, Tristanreid, Twin Bird, Tyagi, Ulner, Vadimvadim, Waltpohl, Wclxlus, Whommighter, Wideofthemark, WriterHound, Xrchz, Yhkhoo, 今古庸龍, 176 anonymous editsImage Sources, Licenses and ContributorsImage:Parabolic trajectory.svg Source: /w/index.php?title=File:Parabolic_trajectory.svg License: Public Domain Contributors: Oleg AlexandrovLicenseCreative Commons Attribution-Share Alike 3.0 Unported/licenses/by-sa/3.0/。
Lesson-3---Groundless-Beliefs
TexPtaArntalIysIi:s Types/Causes of groundless beliefs (paras. 2–24)
Section 1. Result of environment in early life. (paras. 2–3)
e.g. bare feet the bare branches of winter trees. The fridge was completely bare.
Phrases: with your bare hands bare(v.)your soul (to sb.): to tell your deepest
Be established: to be formulated
But if the staunchest Roman Catholic and the staunchest Presbyterian had been exchanged when infants,...(3)
Presbyterian When infants: (elliptical) when they were infants
The number of propositions met with: the number of propositions encountered or experienced
Class as groundless: to classify as groundless
They rest upon mere tradition, or on somebody’s bare assertion unsupported by ever a shadow of proof …(1)
《西方正典》的附录书单5
D. The Chaotic Age: A Canonical Prophecy混乱时代：预言经典（中）Germany德国" Hugo von Hofmannsthal 霍夫曼斯塔尔Poems and Verse Plays 诗歌和诗剧Selected Prose 散文选Selected Plays and Libretti 戏剧和歌词选" Rainer Maria Rilke 里尔克Selected Poetry (including the Duino Elegies) 诗选（包括《杜伊诺哀歌》）The Sonnets to Orpheus致俄耳甫斯十四行The Notebooks of Malte Laurids Brigge马尔特"劳里茨"布里格笔记New Poems: First Part and Other Part" Hermann Broch 布洛赫/布洛克The Sleepwalkers 梦游者The Death of Virgil 维吉尔之死Hugo von Hofmannsthal and His Time 霍夫曼斯塔尔和他的时代" Georg Trakl 特拉克尔Selected Poems 诗选" Gottfried Benn戈特弗里德"本Selected Poems 诗选" Franz Kafka 卡夫卡Amerika 美国The Complete Stories 短篇小说全集The Blue Octavo Notebook 蓝色八开本笔记The Trial 审判The Diaries 日记The Castle 城堡Parables, Fragments, Aphorisms 寓言，片段，格言" Bertolt Brecht 布莱希特Poems, 1913-1956 诗集，1913-1956The Threepenny Opera 三毛钱歌剧The Good Woman of Setzuan四川好人Mother Courage and Her Children大胆妈妈和她的孩子们Galileo 伽利略The Caucasian Chalk Circle高加索灰阑记" Arthur Schnitzler阿尔图尔"施尼茨勒Plays and Stories 戏剧和故事" Frank Wedekind 魏特金Lulu Plays 露露Spring Awakening 春之醒" Karl Krauss卡尔"克劳斯The Last Days of Mankind 人类的末日" Günter Eich君特"艾希Moles 鼹鼠" Thomas Mann 托马斯"曼The Magic Mountain 魔山Stories of Three Decades 三十年间小说集Joseph and His Brothers约瑟和他的兄弟们Doctor Faustus 浮士德Confessions of Felix Krull, Confidence Man大骗子菲利克斯"克鲁尔的自白" Alfred D鯾lin 德布林Berlin Alexanderplatz 柏林亚历山大广场" Hermann Hesse 黑塞The Glass Bead Game (Magister Ludi) 玻璃球游戏Narcissus and Goldmund纳尔齐斯和歌尔德蒙" Robert Musil 罗伯特"穆齐尔Young T鰎less青年特尔勒斯The Man Without Qualities 没有个性的人" Joseph Roth约瑟夫"洛特The Radetzky March 拉德茨基进行曲" Paul Celan 保罗"策兰Poems 诗集" Thomas Bernhard 托马斯"伯恩哈特Woodcutters 木刻家" Heinrich B鰈l海因里希"伯尔Billiards at Half-Past Nine九点半钟的台球" Ingeborg Bachmann英格褒"巴赫曼In the Storm of Roses 玫瑰风暴" Hans Magnus Enzensberger恩岑斯贝格Poems for People Who Don't Read Poems 给不读诗者的诗" Walter Benjamin 本雅明Illuminations 启迪" Robert Walser 罗伯特"瓦尔泽Selected Stories 故事选" Christa Wolf克里斯蒂"沃尔夫Cassandra 卡桑德拉" Peter Handke彼得"汉特克Slow Homecoming 迟归" Max Frisch马克斯"费里施I'm Not Stiller 我不是斯蒂勒Man in the Holocene 全新世的人类" Günter Grass 君特"格拉斯The Tin Drum 铁皮鼓The Flounder 比目鱼" Friedrich Dürrenmatt 迪伦马特The Visit 老妇还乡" Johannes Bobrowski约翰内斯"波勃罗夫斯基Shadow Lands 阴影大地Russia俄罗斯" Anna Akhmatova 阿赫玛托娃Poems 诗集" Leonid Andreyev 安德烈耶夫Selected Tales 故事选" Andrey Bely 别雷Petersburg 彼得堡" Osip Mandelshtam 曼德尔斯塔姆Selected Poems 诗选" Velimir Khlebnikov赫列勃尼科夫The King of Time 时间之王" Vladimir Mayakovsky 马雅可夫斯基The Bedbug and Selected Poetry 臭虫及诗选" Mikhail Bulgakov 布尔加科夫The Master and the Margarita 大师与玛格丽特" Mikhail Kuzmin 库兹明Alexandrian Songs 亚历山大之歌" Maksim Gorky 高尔基Reminiscences of Tolstoy, Chekhov, and Andreev 回忆托尔斯泰，契诃夫和安德烈耶夫Autobiography 自传" Ivan Bunin 蒲宁Selected Stories 故事选" Isaac Babel 巴别尔Collected Stories 故事集" Boris Pasternak 帕斯捷尔纳克Doctor Zhivago 日瓦戈医生Selected Poems 诗选" Yury Olesha 阿廖莎Envy 嫉妒" Marina Tsvetayeva 茨维塔耶娃Selected Poems 诗选" Mikhail Zoshchenko 左琴科Nervous People and Other Satires 紧张的人和其他讽刺故事" Andrei Platonov 普拉东诺夫The Foundation Pit 地坑" Aleksandr Solzhenitsyn 索尔仁尼琴One Day in the Life of Ivan Denisovich 伊凡"杰尼索维奇的一天The Cancer Ward 癌症楼The Gulag Archipelago 古拉格群岛August 1914 1914年8月" Joesph Brodsky 布洛茨基A Part of Speech: Poems 言语的一部分：诗歌Scandinavia斯堪的纳维亚" Isak Dinesen 迪内森Winter's Tales 冬天的故事Seven Gothic Tales 7个哥特故事" Martin Andersen Nex 尼克索Pelle the Conqueror 征服者贝莱" Knut Hamsun 哈姆生Hunger 饥饿Pan 潘" Sigrid Undset西格丽德"温塞特Kristin Lavransdatter克丽丝汀"拉芙兰斯达忒" Gunnar Ekel鰂埃盖洛夫Guide to the Underworld阴间指南" Tomas Transtr鰉er 特朗斯特罗姆Selected Poems 诗选" P鋜Lagerkvist拉格奎斯特/拉格尔克维斯特Barabbas大盗巴拉巴" Lars Gustafsson 古斯塔夫森Selected Poems 诗选Serbo-Croat塞尔维亚-克罗地亚" Ivo Andric伊沃"安德里奇The Bridge on the Drina德里纳河上的桥" Vasko Popa瓦斯科"波帕Selected Poems 诗选" Danilo Kis达尼洛"基什A Tomb for Boris Davidovich 鲍里斯"达维多维奇的坟墓Czech捷克" Karel Capek 恰佩克War with the Newts 鱿鱼之乱R.U.R. R.U.R" Vaclav Havel 哈维尔Largo Desolato悲情声声慢" Milan Kundera 昆德拉The Unbearable Lightness of Being 生命中不能承受之轻" Jaroslav Seifert雅罗斯拉夫"塞弗尔特Selected Poetry 诗选" Miroslav Holub米洛斯拉夫"赫鲁伯The Fly 苍蝇Polish波兰" Bruno Schulz布鲁诺"舒尔茨The Street of Crocodiles 鳄鱼街Sanatorium Under the Sign of the Hourglass 砂制时镜下的疗养院" Czeslaw Milosz 米沃什Selected Poems 诗选" Witold Gombrowicz 贡布罗维奇Three Novels 小说三种" Stanislaw Lem斯坦尼斯拉夫"莱姆The Investigation 调查Solaris飞向太空/索拉里斯星/太阳系" Zbigniew Herbert齐别根纽"赫伯特Selected Poems 诗选" Adam Zagajewski亚当"扎加耶夫斯基Tremor 战栗Hungarian匈牙利" Attila József 约瑟夫Perched on Nothing's Branch 栖息在乌有之枝" Ferenc Juhasz菲兰克"茱哈兹Selected Poems 诗选" Laszlo Németh 内迈特Guilt 有罪Modern Greek当代希腊" C. P. Cavafy 卡瓦菲Collected Poems 诗集" George Seferis 塞弗里斯Collected Poems 诗集" Nikos Kazantzakis 卡赞扎基斯The Greek Passion 希腊激情The Odyssey: A Modern Sequel 奥德赛：现代续篇" Yannis Ritsos扬尼斯"里索斯Exile and Return 放逐与回归" Odysseas Elytis埃利蒂斯What I Love: Selected Poems 我所爱：诗选" Angelos Sikelianos西凯里阿诺斯Selected Poems 诗选Yiddish意第绪语" Sholem Aleichem肖洛姆"阿莱赫姆Tevye the Dairyman and The Railroad Stories 记日记者特维与铁路故事The Nightingale 夜莺" Mendele Mokher Seforim 塞弗里姆The Travels and Adventures of Benjamin the Third 便雅悯三世的旅行与历险" I. L. Peretz 佩雷兹Selected Stories 诗选" Jacob Glatstein 雅各"格莱斯坦Selected Poems 诗选" Moshe-Leib Halpern 哈尔佩尔恩Selected Poems 诗选" H. Leivick (Leivick Halpern) 莱维克Selected Poems 诗选" Israel Joshua Singer 以色列"约书亚"辛格The Brothers Ashkenazi 亚实基拿兄弟们Yoshe Kalb 约瑟"卡尔布" Chaim Grade格拉达The Yeshiva经学院" S. Ansky S.安斯基The Dybbuk 附鬼" Mani Leib 莱布Selected Poems 诗选" Sholem Asch 肖洛姆"阿施East River 东河" Isaac Bashevis Singer艾萨克"巴什维斯"辛格Collected Stories 故事集In My Father's Court 在我父亲的庭院里The Manor, the Estate, the Family Moskat 庄园，产业，莫斯卡特一家Satan in Goray撒旦在戈雷/撒旦在哥瑞Hebrew希伯来语" Hayyim Nahman Bialik比利亚克Shirot Bialik: The Epic Poems 施罗特"比利亚克：史诗诗歌" S. Y. Agnon 艾格农In the Heart of the Seas 在海的心脏Twenty-One Stories 故事二十一篇" Aharon Appelfeld阿亥龙"阿佩菲尔德The Immortal Bartfuss 不朽的巴特法斯Badenheim 1939 1939年的巴登海姆" Yaakov Shabtai雅可夫"萨巴泰Past Continuous 昔日绵绵" Yehuda Amichai 阿米亥Selected Poetry 诗选Travels 旅行" A. B. Yehoshua A. B.耶和舒华A Late Divorce 迟来的离婚" Amos Oz 阿摩斯"奥兹A Perfect Peace 完美的和平" T. Carmi T.卡尔米At the Stone of Losses 迷路石" Nathan Zach 扎克Selected Poems 诗选" Dalia Ravikovitch 拉维科维奇A Dress of Fire 火装" Dan Pagis 丹"帕吉斯Selected Poems 诗选" David Shahar 大卫"沙哈尔The Palace of Shattered Vessels 破碎器皿之宫" David Grossman大卫"格罗斯曼See Under: Love徵之于：爱" Yoram Kaniuk 坎纽克His Daughter 他的女儿Arabic阿拉伯语" Najib Mahfuz 纳吉布"马哈福兹Midaq Alley麦达格巷Fountain and Tomb 喷泉与公寓Miramar米拉玛尔公寓" Adunis 阿杜尼斯Selected Poems 诗选" Mahmud Darwish玛哈穆德"达维什The Music of Human Flesh 众生音乐" Taha Husayn 侯赛因An Egyptian Childhood 埃及童年Latin America拉丁美洲" Rubén Dário 鲁文"达里奥Selected Poetry 诗选" Jorge Luis Borges 博尔赫斯The Aleph and Other Stories 阿莱夫和其他故事Dreamtigers (The Maker) 梦虎Ficciones 杜撰集Labyrinths 迷宫A Personal Anthology 自选集" Alejo Carpentier 卡彭铁尔Explosion in a Cathedral 教堂大爆炸The Lost Steps 消失的足迹Reasons of State 国家的理由The Kingdom of This World人间王国/这个世界的王国" Guillermo Cabrera Infante吉列尔莫"卡夫雷拉"因方特Three Trapped Tigers三只忧伤的老虎View of Dawn in the Tropics 热带黎明景色" Severo Sarduy塞维洛"萨尔度Maitreya弥勒" Reinaldo Arenas雷纳尔多"阿里纳斯The Ill-Fated Peregrinations of Fray Servando 弗雷"塞尔万多的背运旅程" Pablo Neruda 聂鲁达Canto General 漫歌Residence on Earth 土地的居民Twenty Love Poems and a Song of Despair 二十首情诗和一支伤心的歌Fully Empowered 全权Selected Poems 诗选" Nicolás Guillén 纪廉Selected Poems 诗选" Octavio Paz 帕斯The Collected Poems 诗集The Labyrinth of Solitude 孤独的迷宫" César Vallejo 巴列霍Selected Poems 诗选Spain, Take This Cup from Me 西班牙，为我拿开这杯苦酒" Miguel Angel Asturias 阿斯图里亚斯Men of Maize 玉米人" José Lezama Lima何西"雷萨马"利马Paradiso 天堂" José Donoso何西"多诺索The Obscene Bird of Night 淫秽的夜鸟" Julio Cortázar 科塔萨尔Hopscotch 跳房子All Fires the Fire万火归一火Blow-up and Other Stories 放大及其他小说" Gabriel García Márquez 加西亚"马尔克斯One Hundred Years of Solitude 百年孤独Love in the Time of Cholera 霍乱时期的爱情" Mario Vargas Llosa 巴尔加斯"略萨The War of the End of the World世界末日之战" Carlos Fuentes 卡洛斯"富恩特斯A Change of Skin 换皮Terra Nostra 我们的土地" Carlos Drummond de Andrade卡洛斯"德拉蒙德"德"安德拉德Travelling in the Family 居家旅行The West Indies西印度群岛" C. L. R. James C. L. R.詹姆斯The Black Jacobins 黑色雅各宾派The Future in the Present 现在的未来" V. S. Naipaul 奈保尔A Bend in the River 河弯A House for Mr. Biswas 毕斯沃斯先生的房子" Derek Walcott德雷克"沃尔科特Collected Poems 诗集" Wilson Harris 威尔逊"哈里斯The Guyana Quartet 圭亚那四重奏" Michael Thelwell 迈克尔"瑟尔维尔The Harder They Come 越爱越认真" Aimé Césaire埃梅"塞萨尔Collected Poetry 诗集Africa非洲" Chinua Achebe阿契贝Things Fall Apart 瓦解Arrow of God 神箭No Longer at Ease 动荡" Wole Soyinka 索因卡A Dance of the Forest 森林之舞" Amos Tutuola阿摩斯"图图欧拉The Palm-Wine Drinkard and His Dead Palm-Wine Tapster in the Dead's Town棕榈酒鬼，以及他在死人镇的死酒保" Christopher Okigbo 克里斯托弗"奥基博Labyrinths, with Path of Thunder 迷宫，以及雷路" John Pepper Clark (-Bekederemo) 约翰"佩柏"克拉克Casualties: Poems 伤亡：诗篇" Ayi K. Armah 阿马The Beautyful Ones Are Not Yet Born 美丽者还没有诞生" Wa Thiong'o Ngugi 恩古吉A Grain of Wheat 一粒麦种" Gabriel Okara 加布里埃尔"奥卡拉The Fisherman's Invocation 渔夫的符咒" Nadine Gordimer 纳丁"戈迪默Collected Stories 小说集" J. M. Coetzee J. M.库切Foe 仇敌" Athol Fugard阿索尔"富加德A Lesson from Aloes 阿罗斯的教训" Léopold S. Senghor 莱奥波德"桑戈尔Selected Poems 诗选India (in English)印度英语文学" R. K. Narayan R. K.纳拉扬The Guide 向导" Salman Rushdie 拉什迪Midnight's Children 午夜的孩子" Ruth Prawer Jhabvala 杰布瓦拉Heat and Dust 热与灰Canada加拿大" Malcolm Lowry 马尔克姆"劳利Under the Volcano 在火山下" Robertson Davies 罗伯逊"戴维斯The Deptford Trilogy德普特福德三部曲The Rebel Angels 反叛的天使" Alice Munro 爱丽丝"芒罗Something I've Been Meaning to Tell You有件事我一直想告诉你" Northrop Frye诺斯罗普"弗莱Fables of Identity同一的寓言" Anne Hébert 安娜"埃贝尔Selected Poems 诗选" Jay Macpherson 杰伊"麦克弗森Poems Twice Told 两次讲述的诗篇" Margaret Atwood 玛格丽特"阿特伍德Surfacing 浮现" Daryl Hine戴瑞"海因Selected Poems 诗选。
the libido for the ugly 全文讲义
Pennsylvania State Profile
State Unique Name The Keystone State
Capital City
Harrisburg
Location
40.27605 N, 076.88450 W
Bird
Ruffed Grouse
Border States
Delaware - Maryland - New Jersey - New York -
Unit Seven
Libido for the
Ugly
-- Henry Louis Mencken (1880-1956)
Henry Louis Mencken (1880–1956)
• an American journalist, essayist, magazine editor, satirist, acerbic critic of American life and culture, and a student of American English. Known as the "Sage of Baltimore", he is regarded as one of the most influential American writers and prose stylists of the first half of the 20th century
dragon master 英文 24册
dragon master 英文 24册Title: A Comprehensive Overview of the Dragon Master English 24-Book SeriesIntroduction:The Dragon Master English 24-Book Series is a highly acclaimed collection of books designed to enhance language learning for individuals of all ages. This article aims to provide a detailed exploration of the series, highlighting its key features and benefits. The discussion will be divided into five main points, each consisting of several sub-points, followed by a comprehensive conclusion.I. Point 1: Engaging Storylines1.1 Captivating Plots: The Dragon Master series offers a range of captivating storylines that appeal to readers of various interests and age groups.1.2 Diverse Characters: The books feature a diverse cast of characters, allowing readers to engage with different perspectives and experiences.1.3 Thematic Depth: Each book explores a unique theme, providing readers with valuable insights and fostering critical thinking skills.II. Point 2: Language Acquisition2.1 Vocabulary Expansion: The series introduces a wide range of vocabulary, aiding readers in expanding their word bank and improving language proficiency.2.2 Grammar Reinforcement: Through carefully crafted sentences and dialogues, the books reinforce grammar rules and help learners develop a strong foundation in English.2.3 Language Skills Development: The series focuses on developing all language skills, including reading, writing, listening, and speaking, through various interactive activities and exercises.III. Point 3: Cultural Awareness3.1 Cultural Context: The Dragon Master series incorporates cultural elements from different countries, providing readers with a deeper understanding of diverse cultures and traditions.3.2 Global Perspectives: The books promote a global mindset by featuring stories and characters from around the world, fostering empathy and cultural sensitivity.3.3 Appreciation of Diversity: By showcasing characters from different backgrounds, the series encourages readers to embrace diversity and promotes inclusivity.IV. Point 4: Interactive Learning Tools4.1 Illustrations and Visual Aids: The series includes vibrant illustrations and visual aids that enhance comprehension and engage visual learners.4.2 Interactive Exercises: Each book offers a variety of interactive exercises, such as quizzes and puzzles, to reinforce learning and make the experience enjoyable.4.3 Audio Accompaniment: The series provides audio recordings of the texts, enabling learners to improve their pronunciation and listening skills.V. Point 5: Progression and Adaptability5.1 Gradual Difficulty Increase: The books are designed with a progressive difficulty level, allowing learners to gradually advance their language skills.5.2 Adaptability to Different Levels: The Dragon Master series offers books for learners of various proficiency levels, ensuring that each reader can find suitable material.5.3 Supplementary Materials: The series provides additional resources, such as workbooks and teacher's guides, to support educators and facilitate classroom integration.Conclusion:The Dragon Master English 24-Book Series is a comprehensive language learning resource that combines engaging storylines, language acquisition tools, cultural awareness, interactive learning, and adaptability. With its diverse content and well-structured approach, this series offers an effective and enjoyable way to enhance English language skills. Whether used for self-study or classroom instruction, the Dragon Master series is a valuable asset for language learners of all ages and proficiency levels.。
纪伯伦《先知》中英文对照
纪伯伦《先知》中英文对照纪伯伦《先知》中英文对照版内容提要:书中的哲理可以说是纪伯伦经积年累月爱与睿智的结晶。
其深邃广博几乎已达到人类精神修养所能及的最高境界，《先知》之所以可历久弥新，不受时间的限制，因为，它直接表达了“真理”，纪伯伦视《先知》为其一生最在的成就，他曾经说过:“我想，自我从黎巴嫩山构思《先知》一书开始，我就已和它寸步不离了，它仿佛是我身体的一部分……在我完成四年之后才将其付印，因为我想要确定，非常地确定，书中的每一个文字都必须是我的最佳贡献。
这一本奇妙的著作，它满足了个别心灵的不同需求，哲学家认为它是哲学，诗人称是诗，青年则说:“这里有一切蕴含在我心中的东西。
”老年人说:“我曾不断追寻，但却不知追寻为何,但在我垂暮之年，我在这本书中找到我的宝藏。
”作者简介:卡里?纪伯伦于1883年1月6日生于黎巴嫩一处名叫布雪里的地方。
布雪里位于称巴嫩的“圣谷”瓦第?卡地沙悬崖旁的平原之上，卡地沙以其丰沛的水源和青绿的柏树林闻名，当地居民称这引起柏树林为”上帝的柏树林”，而今，人们称纪伯伦家旁边的柏树为“神圣的柏树”。
纪伯伦的童年便是在“神圣的柏树”下度过的。
纪伯伦生长在一个宗教气息浓厚的家庭中。
母亲卡蜜拉是一位民龙教派牧师的女儿，美丽聪慧，多才多艺。
在和纪伯伦的父亲结婚之前，是一寡妇，育有一子——彼得。
嫁给纪伯伦的父亲之后，所生的头一胎便是纪伯伦，后来又陆续生下两个女儿——苏妲娜和玛丽安娜。
童年时期，纪伯伦的母亲亲自教他阿拉伯文和法文，以后，又请家庭教师教他英文。
1888年纪伯伦随母亲和哥哥彼得与两个妹移居美国波士顿，父亲为了守护家中的产业仍然留在故乡。
到达美国之后，得彼以经营杂货店维持生计，而纪伯伦继续求学。
在学校中，纪伯伦的表现相当优异。
1897年纪伯伦返回黎巴嫩以便接受祖国的教育，于是他讲入贝鲁特的一所教会大学就读，继续研读阿拉伯文和法文，并且选修了医学、国际法及宗教史和音乐等课程。
1898年，暑假期间，纪伯伦随父亲旅游中东各地。
3.GenderTheory-IASSCS
3. Gender TheoryOverviewIn this lecture we will focus on the difference between sex and gender, and review the emergence of the study of gender as a discipline.ObjectivesBy the end of this topic you will have:•Reflected on your own understandings about gender•Gained an understanding about the difference between different theoretical constructions of gender•Focused on the development of gender identity, and considered gender diversity.•Considered the implications of this for your own practice.Key ConceptsSocial construction, binaries, sex, gender, male/female, masculine/feminine, gender theory, queer theory, systemic discriminationRequired readingCalasanti, TM & Slevin, KF (2001). Chapters 1 & 2 in Gender, Social Inequalities and Aging. AltaMira press, Walnut Creek CA. pp 1 – 27.Butler, J. (1990) Chapter 1, Subjects of sex/gender/desire in Gender Trouble: Feminism and the subversion of identity pp 1-25. Routledge, NY.Further readingLeonard W. (2005). Queer occupations: Development of Victoria’s gay, lesbian, bisexual, transgender and intersex health and wellbeing action plan. Gay &Lesbian Issues in Psychology Journal 1 (3) p. 92Peters J. (2005). Gender prac: Gender as performance, not gender theory. Gay & Lesbian Issues in Psychology Journal 1 (3) p. 98Activity: What is gender?Objective: To explore discourses about gender and sex.•Write down all the words and phrases that come to mind when you hear the word ‘gender’.•Now write down all the words that come to mind when you hear the word ‘sex’.•When you have exhausted your ideas, sort and categorise them as follows: •Group all the words that apply mostly to women together•Group all the words that apply mostly to men.•Look at these groups of words. Are there similarities? Differences?•Are any words or concepts left over? If there are, why don’t they belong to one of the above categories above?Journal work: Reflect on the results of this activity and make notes in your journal. Complete the following sentence stems:•I learned that I …•I was surprised that I …•In future I will …Lecture Notes‘Gender’ refers to the socially constructed roles, responsibilities, identities and expectations assigned to men and women. It contrasts with the fundamental biological and physiological differences between males and females, which are known as secondary sex characteristics. Gender roles differ between cultures and communities and over time.For many, gender is always thought about in binary terms: man/woman;masculine/feminine. Expectations of women and men are limited by these binaries, and are communicated through sex role stereotyping. These stereotypes limit gender appropriate behaviour to a range of rigid roles which are assigned to women and men on the basis of their gender, for example, ‘women are nurturers’, ‘men are aggressors’. These role expectations are subtle and deeply ingrained, however there is great diversity in how individuals express their gender which frequently does not conform to these stereotypes. Not all women fit the stereotypical expectations of femininity, not all men fit those qualities associated with masculinity. Transgender people feel that they have been born into a body in which their gender identity and their physical sex are not coherent. The terms transgender and transsexual are both used to describe this phenomenon, generally, transsexual people have had surgical intervention to change their physical characteristics to match their gender identity, while transgender people have not. Many people who identify as transgender choose to live their gender identity rather than their physical or genetic sex. The transition from one gender to another can be painful and difficult. Transgender people are not transvestites. Transvestites are people who cross-dress, that is, they choose from time to time to wear the clothes of the opposite gender. Because one area in which women have greater freedom of expression than men is in how they dress,women are rarely called transvestites when they choose to wear what was traditionally known as men’s clothing.Activity:Think about the following statements and reflect on the implications of each one for your work.•Women and men are not all the same. Age, race, ability/ disability, culture, language, class, sexual orientation and access to resources (among otherdifferences) all factors that influence access to resources and services.•Workers, policy makers and planners all bring their own bias to their work.This will be influenced by life experience, gender, class, culture, education,economic status and other factors.•Older women, particularly those from marginalised groups such as indigenous people, culturally and linguistically diverse communities andlesbians, often do not have economic equality with men from similarbackgrounds and status. They are under represented in decision makingprocesses, therefore measures are necessary to ensure that their voices areheard.•To be responsive to the diversity of women and men in the community it is important to consult with the people who will be affected by the policies/programs.Journal work: In your journal, write about your reflections, and then identify three things you can do to change/improve your practice in relation to ensuring that stereotypes and assumptions are not used about gender roles and expressions with older people.Gender TheoryIn the lecture on sexuality theory, we discussed how heterosexuality was so naturalised that it was not named until after those practices which deviated from the heterosexual norm were categorised. In many way, gender is similar; the binary construction of male/female and the correspondingly appropriate masculine and feminine behaviours and role were, until recently, so naturalised that to think outside of this binary was almost impossible.In the 1960s, second wave feminism started to critique essentialist assumptions about gender, and in the academy, feminist scholars started to develop feminist theory based on a theoretical or philosophical analysis of women’s liberation politics, and a field of study called women’s studies emerged.Feminist politics and women’s studies put a spotlight on the inequities between women and men in almost all societies. In the West, women campaigned to gain equal rights and opportunities and in the developing world, a ‘gender analysis’ was applied to aid and development funding. While there are vast differences between the needs of women in developed and developing nations, and different approaches have been applied, by using a gender lens, many inequalities between men and women werehighlighted and work was undertaken to try to address or change the status quo. However, in this focus on inequalities between women and men, the term ‘gender’ somehow became synonymous with ‘women’.It was not until the 1980s that the field of masculinity studies emerged in the academy. Masculinity (or men’s) studies focused on the construction of masculinity, and used feminist theory to analyse the ways in which gender and power operate in the lives of men and develop masculinity theory. This new field of study has started to highlight the ways in which gender inequities affect not only women, but also men. More recently, a multidisciplinary field of study has emerged that examines the cultural representations and the lived experience of being male or female. This allows for an analysis of gender for both women and men, and is used to interrogate the phenomena in a wide range of disciplines.Queer theoryQueer theory comes out of gender, gay and lesbian and feminist studies. It is influenced by the work of post structuralists such as Foucault and Derrida (among others). Queer theory explores categories of gender and sexuality and challenges the notion of identity. As a field of study, it focuses on the construction of identity and examines how normalisation is implicated in its construction.Simone de Beauvoir observed that ‘one is not born a woman, but becomes one’. Similarly, for Butler (1990) gender is not so much something one is, as something one does. She uses the concept of performativity – the aspect of discourse that has the capacity to produce what it names – as setting the limits of ‘normality’ and‘otherness’. According to Butler (1993):Performative acts are forms of authoritative speech: most performatives for instance are statements that, in uttering also perform a certain action and exercise a binding power … If the power of discourse to produce that which it names is linked with the question of performativity, then the performative is one domain in which power acts as discourse (Butler, 1993, p. 225).By this she means that the repeated normalising of certain social positions – such as heterosexuality, masculinity or femininity – naturalises the acts, thereby acting to set limits on what is normal and excluding anything outside of these limits as deviant.In this theory, gender is an embodied discourse. It has no relation to biological truths about the sexed body, and exists only in discourse. Gender has a history that exists beyond the individual who enacts (or resists) the roles expected from the category to whom s/he has been assigned.IdentityPsychology (clinical and academic), sociology and other social sciences, as well as literary and cultural theory have all contributed to developing theories on identity formation. Esterberg (1997) posited that:… at least ten different meanings [of identity, have been] used by scholars,ranging from the sense of oneself as continuous, existing throughout time, to a sense of oneself as belonging to a group or having shared group membership.Some see identities as something essential, tangible, and real, inherent in theself; yet others see identities as shifting, constructed, a matter of creatingmeaning from social categories and coming to attach labels to oneself. Thebody of academic and popular literature on identity is so large that it nearlydefies categorization. Perhaps no term has been used so much in recent yearsor become so popular – both in academic and in lay worlds – as identity(Esterberg, 1997, p. 14).This quote draws attention to the way in which debates about the origins of identity are often constructed as being produced by either nature or nurture, which reduces what is essentially a complex process to a simple binary (Dyson, 2007). Psychologists such as Erikson defined identity as the development of individual personality involving a stable core sense of self, within the context of the social milieu. Here notions of stability and interiority are central. However, for sociologists: “… identities are not something deep down inside the individual but are located in the interaction between the individual and society. Identities, thus, are always in process” (Esterberg, 1997, p. 14 – 15). Ussher (1997) represented this debate asmaterial/discursive (see Lecture One), arguing that the two are always placed in opposition to each other, which is counterproductive (Dyson, 2007).Assumptions about older people’s gender and sexual identity beingmasculine/feminine and heterosexual can lead to marginalising those who do not conform to these assumed norms. These issues will be addressed in more detail in Lecture 10. In planning, policy development and service delivery it is important not to make assumptions about identity, but to acknowledge the possibility that applying normative assumptions about gender or identity to everyone can be harmful to some.A gender sensitive approachAs has been pointed out earlier in this lecture, the socialised roles of masculinity and femininity are not innate, but socially constructed:The forces that construct gender roles embed men and women in relations ofsubordination and dominance… In spite of the relative advantages afforded to men under the social and political system that socialised us, many men feeldisadvantaged compared to women’ (Pease 1999).A gender sensitive approach is not about creating competition between individual women and men to equalise access to resources and services. It is about recognising that we live in a system that creates competition between women and men and that men are socialised to maintain that system through domination and aggression. A gender sensitive approach works on a number of levels•Analysing and understanding the system that creates gender roles and stereotypes;•Understanding both personal values and systemic biases about gender, and how these effect us as workers and the people with whom we work;•Striving to achieve equity and justice for consumers and for workers.Gender and PolicyGender is a critical consideration in all areas of policy and program planning and development, and there is no area that impacts on women and men in exactly the same way. There has been a tendency to make ‘gender’ synonymous with ‘women’. Until the women’s health movement of the 1970s, biomedicine used a male model as the norm in research, and the investigation and recognition of gender differences were restricted to reproductive issues. This was essentially seen as the female domain, with the focus being on pregnancy and reproduction. Although there is increased recognition of gender differences in health status and service usage, there are many gaps in understanding of how to measure gender differences in health status. A gender sensitive approach to policy, research and practice would provide a gendered picture of health status and recognise the differences between, and diversity of, women and men by actively listening to the voices of consumers and working to achieve equity in outcomes.Gender Inclusive AnalysisSeeks to understand the differences and disparities in the roles that women and men play, the power imbalances in their relations, their needs, constraints and opportunities and the impact of these differences on their lives. In health, a gender analysis determines how these differences determine their differential exposure to risk, access to the benefits of technology, information, resources and health care, and the realisation of rights. It is important to understand how the lives of women and men differ in family life, labour force participation, economic status, educational background, subjective experience of violence and participation in decision making roles.Systemic DiscriminationSystemic discrimination, is often enshrined in the institutions that govern our lives. It stems from assumptions and stereotypes about women and men, as well as about class, age, race, ethnicity and ability. While roles and expectations may differ between cultures and over time, the process by which we learn them is so much part of our everyday experience, that to most people it is invisible. This then is the discrimination that is called systemic, because it is so pervasive as to constitute a system under which we live. There are a number of factors that impact differently on the lives of women and men. The more gender awareness is incorporated into your work, the more you will become aware of these factors.Journal WorkReflect on the ways in which you address gender issues in your work.•In what ways might you be able to apply a gender lens to your work?•How can you ensure that people who do not conform to normative gender identity are not ignored and have their needs met?Write about this in your journal.。
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Chapter1Bases for Lie algebras and a continuous CBH formulaMatthias Kawski1Department of Mathematics and StatisticsArizona State University,Tempe,Arizona85287-1804,U.S.A.,/˜kawski,kawski@1.1Description of the problemMany time-varying linear systems˙x=F(t,x)naturally split into time-invariant geometric components and time-dependent parameters.A special case are non-linear control systems that are affine in the control u,and specified by analytic vectorfields on a manifold M n˙x=f0(x)+mk=1u k f k(x).(1.1)It is natural to search for solution formulas for x(t)=x(t,u)that,separate the time-dependent contributions of the controls u from the invariant,geometric role of the vectorfields f k.Ideally,one may be able to a-priori obtain closed-form expressions for theflows of certain vectorfields.The quadratures of the control might be done in real-time,or the integrals of the controls may be considered new variables for theoretical purposes such as path-planning or tracking.To make this scheme work,one needs simple formulas for assembling these pieces to obtain the solution curve x(t,u).Such formulas are nontrivial since in general the vectorfields f k do not commute:already in the case of linear 1Supported in part by NSF-grant DMS00-7236912CHAPTER1.BASES FOR LIE ALGEBRAS systems,exp(sA)·exp(tB)=exp(sA+tB)(for matrices A and B).Thus the desired formulas not only involve theflows of the system vectorfields f i,but also theflows of their iterated commutators[f i,f j],[[f i,f j],f k],and so on.Using Hall-Viennot bases H for the free Lie algebra generated by m indeter-minates X1,...X m,Sussmann[22]gave a general formula as a directed infinite product of exponentialsx(T,u)=→H∈Hexp(ξH(T,u)·f H).(1.2)Here the vectorfield f H is the image of the formal bracket H under the canonical Lie algebra homomorphism that maps X i to f ing the chronological product (U∗V)(t)= T0U(s)V′(s)ds,the iterated integralsξH are defined recursively byξXk(T,u)= T0u k(t)dt andξHK=ξH∗ξK(1.3) if H,K,HK are Hall words and the left factor of K is not equal to H[9,22].(In the case of repeated left factors,the formula contains an additional factorial.) An alternative to such infinite exponential product(in Lie group language:“coordinates of the2nd kind”)is a single exponential of an infinite Lie series (“coordinates of the1st kind”).x(T,u)=exp( B∈BζB(T,u)·f B)(1.4)It is straightforward to obtain explicit formulas forζB for some spanning sets B of the free Lie algebra[22],but much preferable are series that use bases B, and which,in addition,yield as simple formulas forζB as(1.3)does forξH.Problem1.Construct bases B={B k:k≥0}for the free Lie algebra on afinite number of generators X1,...X m such that the corresponding iterated integral functionalsζB defined by(1.4)have simple formulas(similar to(1.3)), suitable for control applications(both analysis and design).The formulae(1.4)and(1.2)arise from the“free control system”on the free associative algebra on m generators.Its universality means that its solutions map to solutions of specific systems(1.1)on M n via the evaluation homomor-phism X i→f i.However,the resulting formulas contain many redundant terms since the vectorfields f B are not linearly independent.Problem2.Provide an algorithm that generates for anyfinite collection ofanalytic vectorfields F={f1,...,f m}on M n a basis for L(f1,...,f m)together with effective formulas for associated iterated integral functionals.Without loss of generality one may assume that the collection F satisfies the Lie algebra rank condition.i.e.L(f1,...,f m)(p)=T p M at a specified initial point p.It makes sense tofirst consider special classes of systems F,e.g.which1.2.MOTIVATION AND HISTORY OF THE PROBLEM3 are such that L(f1,...,f m)isfinite,nilpotent,solvable,etc.The words simple and effective are not used in a technical sense in problems1and2(as in for-mal studies of computational complexity),but instead refer to comparison with the elegant formula(1.3),which has proven convenient for theoretical studies, numerical computation,and practical implementations.1.2Motivation and history of the problem Series expansions of solution to differential equations have a long history.El-ementary Picard iteration of the universal control system˙S= m i=1X i u i on the free associative algebra over{X1,...,X m}yields the Chen Fliess series[5, 11,21].Other major tools are Volterra series,and the Magnus expansion[14], which groups the terms in a different way than the Fliess series.The main draw-back of the Fliess series is that(unlike its exponential product expansion(1.2)) nofinite truncation is the exact solution of any approximating system.A key innovation is the chronological calculus of1970s Agrachev and Gamkrelidze[1]. However,it is generally not formulated using explicit bases.The series and product expansions have manifold uses in control beyond simple computation of integral curves,and analysis of reachable sets(which includes controllability and optimality).These include state-space realizations of systems given in input-output operator form[8,20],output tracking and path-planning.For the latter,express the target or reference trajectory in terms of theξorζ,now considered as coordinates of a suitably lifted system(e.g.free nilpotent)and invert the restriction of the map u→{ξB:B∈B N}or u→{ζB:B∈B N}(for somefinite subbasis B N)to afinitely parameterized family of controls u,e.g.piecewise polynomial[7]or trigonometric polynomial[12,17].The Campbell-Baker-Hausdorffformula[18]is a classic tool to combine prod-ucts of exponentials into a single exponential e a e b=e H(a,b)where H(a,b)=a+b+12[a,b]+112[a,[a,b]]−112[b,[a,b]+....It has been extensively used fordesigning piecewise constant control variations that generate high order tangent vectors to reachable sets,e.g.for deriving conditions for optimality.However, repeated use of the formula quickly leads to unwieldily expressions.The ex-pansion(1.2)is the natural continuous analogue of the CBH formula–and the problem is tofind the most useful form.The uses of these expansions(1.2)and(1.4)extend far beyond control,as they apply to any dynamical systems that split into different interacting com-ponents.In particular,closely related techniques have recently found much attention in numerical analysis.This started with a search for Runge-Kutta-like integration schemes such that the approximate solutions inherently satisfy algebraic constraints(e.g.conservation laws)imposed on the dynamical sys-tem[3].Much effort has been devoted to optimize such schemes,in particular minimizing the number of costly function evaluations[16].For a recent survey see[6].Clearly the form(1.4)is most attractive as it requires the evaluation of only a single(computationally costly)exponential.The general area of noncommuting formal power series admits both dynam-4CHAPTER1.BASES FOR LIE ALGEBRASical systems/analytic and purely algebraic/combinatorial approaches.Alge-braically,underlying the expansions(1.2)and(1.4)is the Chen series[2],which is well known to be an exponential Lie series,compare[18],thus guaranteeing the existence of the alternative expansionsw⊗w!=exp B∈BζB⊗B !=→ B∈B exp(ξB⊗B)(1.5) w∈Z∗Thefirst bases for free Lie algebras build on Hall’s work in the1930s on commu-tator groups.While several other bases(Lyndon,Sirsov)have been proposed in the sequel,Viennot[23]showed that they are all special cases of generalized Hall bases.Underlying their construction is a unique factorization principle, which in turn is closely related to Poincare-Birckhoff-Witt bases(of the uni-versal enveloping algebra of a Lie algebra)and Lazard elimination.Formulas for the dual PBW basesξB have been given by Sch¨u tzenberger,Sussmann[22], Grossman,and Melancon and Reutenauer[15].For an introductory survey see [11],while[15]elucidates the underlying Hopf algebra structure,and[18]is the principal technical reference for combinatorics of free Lie algebras.1.3Available related resultsThe direct expansion of the logarithm into a formal power series may be sim-plified using symmetrization[18,22],but this still does not yield well-defined “coordinates”with respect to a basis.Explicit,but quite unattractive formulas forthefirst14coefficientsζH in the case of m=2and a Hall basis are calculated in[10].This calculation can be automated in a computer algebra system for terms of considerably higher order, but no apparent algebraic structure is discernible.These results suffice for some numerical purposes,but they don’t provide much structural insight.Several new algebraic structures introduced in[19]lead to systematic for-mulas forζB using spanning sets B that are smaller than those in[22],but are not bases.These formulas can be refined to apply to Hall-bases,but at the cost of loosing their simple structure.Further recent insights into the underlying algebraic structures may be found in[4,13].The introductory survey[11]lays out in elementary terms the close con-nections between Lazard elimination,Hall-sets,chronological products,and the particularly attractive formula(1.3).These intimate connections suggest that to obtain similarly attractive expressions forζB one may have to start from the very beginning by building bases for free Lie algebras that do not rely on recursive use of Lazard elimination.While it is desirable that any such new bases still restrict to bases of the homogeneous subspaces of the free Lie alge-bra,we suggest consider balancing the simplicity of the basis for the Lie algebra and structural simplicity of the formulas for the dual objectsζB.In particular, consider bases whose elements are not necessarily Lie monomials,but possibly nontrivial linear combinations of iterated Lie brackets of the generators.Bibliography[1]A.Agrachev and R.Gamkrelidze,“Chronological algebras and nonstation-ary vectorfields”,Journal Soviet Math.,17,pp.1650–1675(1979).[2]K.T.Chen,“Integration of paths,geometric invariants and a generalizedBaker-Hausdorffformula”,Annals of Mathematics,65,pp.163–178(1957).[3]P.Crouch and R.Grossman,“The explicit computation of integration algo-rithms andfirst integrals for ordinary differential equations with polynomial coefficients using trees”,Proc.Int.Symposium on Symbolic and Algebraic Computation,pp.89-94,ACM Press(1992).[4]A.Dzhumadil’daev,“Non-associative algebras without unit”,Comm.Alge-bra(2002)(to appear).[5]M.Fliess,“Fonctionelles causales nonlin´e aires et indetermin´e es noncommu-tatives”,Bull.Soc.Math.France,109,pp.3–40(1981).[6]A.Iserles,“Expansions that grow on trees”,Notices AMS,49,pp.430-440(2002).[7]G.Jacob,“Motion planning by piecewise constant or polynomial inputs”,Proc.NOLCOS,Bordeaux(1992).[8]B.Jakubczyk,“Convergence of power series along 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