New Solvable and Quasi Exactly Solvable Periodic Potentials

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高三英语社会问题解决方案评估指标确定单选题50道

高三英语社会问题解决方案评估指标确定单选题50道

高三英语社会问题解决方案评估指标确定单选题50道1.Which aspect is the most important when evaluating a solution to social problems?A.EfficiencyB.CostC.SustainabilityD.Popularity答案:C。

本题考查社会问题解决方案评估指标中的重要性判断。

选项A“Efficiency( 效率)”很重要,但不是最重要的唯一标准。

选项B“Cost( 成本)”也只是一个方面。

选项D“Popularity( 受欢迎程度)”并非核心重要指标。

而选项C“Sustainability 可持续性)”对于长期解决社会问题至关重要,因为只有可持续的解决方案才能真正有效地解决问题而不会带来新的问题。

2.In assessing social problem solutions, what should be given top priority?A.Quick implementationB.Long-term effectivenessC.Low initial investmentD.High media attention答案:B。

选项A“Quick implementation(快速实施)”有一定好处,但如果不能长期有效也不行。

选项C“Low initial investment 低初始投资)”不是最重要的考量。

选项D“High media attention 高媒体关注度)”不是关键因素。

选项B“Long-term effectiveness(长期有效性)”在评估社会问题解决方案时应给予最高优先级,因为只有长期有效的方案才能真正解决社会问题。

3.Which factor is crucial in determining the importance of a social problem solution?A.Number of supportersB.Speed of resultsC.Impact on future generationsD.Immediate popularity答案:C。

你有什么天赋的英语作文

你有什么天赋的英语作文

In the vast expanse of human abilities, there exists a unique attribute that sets individuals apart from one anothertalent. When it comes to the English language, some individuals possess an innate aptitude that enables them to excel in various aspects of the language, from speaking and writing to understanding complex linguistic structures. This essay will delve into the concept of innate talent in English and explore how it manifests in different individuals.Firstly, it is essential to recognize that talent in English is not a monolithic concept. It can manifest in various forms. For some, it is the ability to articulate thoughts with eloquence and precision, while for others, it might be the capacity to grasp the nuances of the language quickly. Some individuals may have a natural flair for writing, crafting sentences that are both engaging and informative. Others might excel in listening and understanding spoken English, even when it is delivered at a rapid pace or with a heavy accent.One of the most apparent forms of English talent is linguistic intuition. This refers to the ability to understand and use the language correctly without having to consciously apply grammatical rules. Such individuals often find it easier to pick up new vocabulary and idiomatic expressions, making them adept at adapting to different dialects and styles of English.Another aspect of English talent is the capacity for creativity in language use. This is particularly evident in those who excel in writing. They can weave words together to create vivid imagery, compelling narratives, and persuasive arguments. Their writing not only conveys information but also engages the reader on an emotional level, making their work memorable and impactful.Moreover, talented English speakers often exhibit exceptional communication skills. They can express their thoughts clearly and concisely, making them effective in both formal and informal settings. This ability to communicate effectively is crucial in various professional and social contexts, where the ability to articulate ideas and persuade others can be a significant advantage.It is also worth noting that talent in English is not solely about native speakers. Many nonnative speakers demonstrate remarkable proficiency and mastery of the language. Their dedication to learning and understanding English, coupled with their natural aptitude, allows them to achieve high levels of fluency and accuracy.However, it is important to acknowledge that talent alone is not enough to guarantee success in mastering English. It must be complemented by hard work, practice, and acontinuous desire to improve. Talent can provide a head start, but it is the commitment to learning and refining ones skills that ultimately leads to mastery.In conclusion, the talent for English is a multifaceted gift that can manifest in various ways, from linguistic intuition to creative writing and effective communication. While it can provide an advantage, it is the combination of talent and dedication that truly sets apart those who excel in the English language. Recognizing and nurturing this talent, whether in oneself or in others, can lead to a deeper appreciation and mastery of the rich and diverse world of English.。

Quasi--exact solvability

Quasi--exact solvability
1991 Mathematics Subject Classi cation. Primary 17B81, 34L05, 35P05, 81Q05; Secondary 13A50, 17B56, 33E30, 81R05. Key words and phrases. Schrodinger operator, Lie algebra, di erential operator, quasi-exactly solvable. The rst author was supported in part by DGICYT Grant PS 89{0011. The second author was supported in part by an NSERC Grant. The third author was supported in part by NSF Grant DMS 89-01600. This paper is in nal form and no version of it will be submitted for publication elsewhere Typeset by AMS-TEX 1
Lie algebraic and Lie group theoretic methods have played a signi cant role in the development of quantum mechanics since its inception. In the classical applications, the Lie group appears as a symmetry group of the Hamiltonian operator, and the associated representation theory provides an algebraic means for computing the spectrum. Of particular importance are the exactly solvable problems, such as the harmonic oscillator or the hydrogen atom, whose point spectrum can be completely determined using purely algebraic methods. The fundamental concept of a \spectrum generating algebra" was introduced by Arima and Iachello, 4], 5], to study nuclear physics, and subsequently, by Iachello, Alhassid, Gursey, Levine, Wu and their collaborators, was also successfully applied to molecular dynamics and spectroscopy, 19], 22], and scattering theory, 1], 2], 3]. The Schrodinger operators amenable to the algebraic approach assume a \Lie algebraic form", meaning that they belong to the universal enveloping algebra of the spectrum generating algebra. Lie algebraic operators reappeared in the discovery of Turbiner, Shifman, Ushveridze, and their collaborators, 26], 28], 29], 33], of a new class of physically signi cant spectral problems, which they named \quasi-exactly solvable", having the property that a ( nite) part of the point spectrum can be determined using purely algebraic methods. This is an immediate consequence of the additional requirement that the hidden symmetry algebra preserve a nite-dimensional representation space consisting of smooth wave functions. In this case, the Hamiltonian restricts to a linear transformation on the representation space, and hence the associated eigenvalues can be computed by purely algebraic methods, meaning matrix eigenvalue calculations. Finally, one must decide the \normalizability" problem of whether the resulting \algebraic" eigenfunctions are square integrable and therefore represent true bound states of the system. Connections with conformal eld

社科英语Unit1翻译

社科英语Unit1翻译

Unit11.经济学家通常假设人是理性的。

理性的人们系统地,有目的地做最好的,他们可以实现他们的目标,考虑到可用的机会。

当你学习经济学,你会遇到公司决定雇佣多少工人,有多少他们的产品生产和销售利润最大化。

你也会遇到那些决定花多少时间工作和买什么商品和服务产生的收入来实现最高水平的满意度。

2.3.理性的人知道,生活中的决定很少是黑白分明的,但,从事实上涉及灰色阴影。

在晚餐时间,决定你,在禁食或吃得像猪,但是否采取额外的'斯波夫,马希尔'博塔托人。

当考试开始,你的决定不是吹他们或学习24小时一天,但是否花额外的一个小时来审查你的笔记,而不是看电视。

经济学家用"边际变化"一词来描述小的内特内塔帕调整到现有的行动计划。

请记住,边距意味着"边缘,所以边缘更改是围绕您正在执行的操作的边缘进行调整。

理性的人经常通过比较边际收益和边际成本来做出决策。

3、例如,考虑航空公司决定向飞行的乘客收取多少费用待机. 假设在美国驾驶一架 200 座的飞机花费航空公司在这种情况下,每个座位的平均费用为100,000美元/200美元,即500美元。

有人可能会得出这样的结论:航空公司绝不应该以较低的的价格出售机票。

500.In美元的事实,一个理性的航空公司往往可以找到办法,通过思考提高利润在边缘。

想象一下,一架飞机即将起飞,有10个空座位,和在登机口等候的候补乘客将支付300美元作为座位。

如果航空公司出售票?当然,它应该。

如果飞机有空座位,则增加一个座位的成本乘客很小。

虽然乘客乘坐飞机的平均费用为500美元,但边际成本只是花生袋和汽水罐的成本,额外的乘客将消耗。

只要候补乘客支付的比边际成本多,卖票是有利可图的。

4、边际决策有助于解释一些否则令人费解的经济现象。

这里有一个经典的问题:为什么水这么便宜,而钻石这么贵?人类需要水才能生存,而钻石是不必要的;出于某种原因,人们愿意为钻石支付比为钻石多得多的费用。

英美文学选读-英国-现代时期-练习题汇总(选择大题)

英美文学选读-英国-现代时期-练习题汇总(选择大题)

I.Multiple Choice(40 points in all, 1 for each)Select from the four choices of each item the one that best answers the question or completes the statement. Write the corresponding letter A, B, C or D on the answer sheet.chapter22.Which of the following is NOT a typical feature of Modernism?A.To elevate the individual and inner being over the social being.B.To put the stress on traditional values.C.To portray the distorted and alienated relationships between man and hisenvironment.D.To advocate a conscious break with the past.(024)19.Modernism takes the irrational philosophy and()as its theoretical base.A. the theory of psycho-analysisB. Darwin’s evolutionary theoryC. the French symbolismD. Utilitarianism(057)17.______________ is the most outstanding stream of consciousness novelist, with ___________ as his encyclopedia – like masterpiece .A.James Joyce, Ulysses B.E.M. Foster, A Passage to IndiaC.D.H. Lawrence, Sons and Lovers D.Virginia Woolf, Mrs Dalloway(074)15.All of the following are stream –of- consciousness novels EXCEPT________. A.Pilgrimage B.UlyssesC.Mrs. Dalloway D.Tess of the D’ Urbervilles(084)?21. In the play The Importance of Being Earnest by Oscar Wilde,the upper — class people are described all of the following EXCEPT ______.A. corruptB. snobbishC. hypocriticalD. ambitious(087)17. After the First World War, there appeared the following literary trends ofmodernism EXCEPT ______.A. expressionismB. surrealismC. stream of consciousnessD. black humour(094)18. The masterpieces of critical realism in the early 20th century are the three trilogiesof ______.A. Galsworthy's Forsyte novelsB. Hardy' s Wessex novels(094)C. Greene's Catholic novelsD. Woolf's stream-of-consciousness novels19. In the mid - 1950s and early 1960s, there appeared ―______‖ who demonstrated aparticular disillusion over the depressing situation in Britain and launched a bitter protest against the outmoded social and political values in their society.5 A. The Beat Generation B. The Lost GenerationC. The Angry Young MenD. Black Mountain Poets(094)16. The rise of _____ and new science greatly incited modernist writers to make new explorations on human natures and human relationships.4A. the existentialistic ideaB. the irrational philosophyC. scientific socialismD. social Darwinism(097)22. The 20th century has witnessed a great achievement in English poetry, which aremainly represented by the following EXCEPT _____.3a. Thomas Hardyb. Ezra Poundc. T. S. Eliotd. Lord Byron (浙0210)25. Which of the following is James Joyce's masterpiece?a. Dublinersb. A Portrait of the Artist as a Young Manc. Ulyssesd. Finnegans Wake(浙0210)20.The following are English stream-of-consciousness novels EXCEPT ______.2A. PilgrimageB. UlyssesC. Mrs.DallowayD. A Passage to Inida9. In the late nineteenth century, modernism flushed in English literature. Unlike modernist poets and novelists, modern dramatists ______.1A. showed not only satirical attitude to bourgeois class, but also optimistic emotion toward lifeB. did not make so many innovations in techniques and formsC. inherited the romantic fuzziness and self-indulged emotionismD. took the irrational philosophy and the theory of psycho-analysis as its theoretical base1 George Bernard Shaw21.___is considered to be the best-known English dramatist since Shakespeare, and his representative works are plays inspired by social criticism.A.Richard SheridanB.Oliver GoldsmithC.Oscar WildeD.Bernard Shaw(024)1.Mrs. Warren’s Profession is one of George Bernard Shaw’s plays. What is Mrs.Warren’s profession then ?[A]Real estate. [B]Prostitution.[C]House-keeping. [D]Farming. (034)21.George Bernard Shaw’s play, Mrs. Warren’s Profession is a grotesquely realistic exposure of the().A. slum landlordismB. political corruption in EnglandC. economic oppression of womenD. religious corruption in England(054)4.George Bernard Shaw’s play _______ established his position as the leading playwright of his time. 5A.Widowers’ Houses B.Too True to Be GoodC.Mrs. Warren’s Profession D.Candida(084)5.George Bernard Shaw’s play Mrs. Warren’s Profession is about______.A. slum landlordismB. the economic oppression of womenC. the political corruption in EnglandD. the religious corruption in England(087)12. Among the following writers ______ is considered to be the best—known English dramatist since Shakespeare.A. Oscar WildeB. John GalsworthyC. W. B. YeatsD. George Bernard Shaw(087)18. George Ber nard Shaw’s _____is a better play of the later period, with the author’s almost nihilistic bitterness on the subjects of the cruelty and madness of WWI and the aimlessness and disillusion of the young.4A. Too True to Be GoodB. Mrs. Warren’s ProfessionC. Widowers’HousesD. Fanny’s First Play(097)15. George Bernard Shaw’ s ______ is a grotesquely realistic exposure of slum landlordism.3A. Widower’ s HouseB. Mrs. Warren’ s ProfessionC. The Apple CartD. Getting Married(104)7. George Be rnard Shaw’s ______ explored his idea of ―Life Force‖, the power that would create superior beings to be equal to God and to solve all the social, moral, and metaphysical problems of human society.2A. Man and SupermanB. The Apple CartC. PygmalionD. Too True to Be Good(107)4. As a realistic dramatist, George Bernard Shaw is concerned with political, economic, moral, or religious problems in his works. The general mood he expressed in his plays is ______.1A. indignationB. satisfactionC. optimismD. pessimism2 T. S. Eliot19. ―When the evening is spread out against the sky (034) 5Like a patient etherized upon a table.‖(T. S. Eliot, ―The Love Song of J. Alfred Prufrock‖)What does the image in the quoted lines suggest? _______.[A]Violence [B]Horror [C]Inactivity[D]Indifference (034)17.―For I have known them all already, known them all—/Have known the evenings, mornings, afternoons,/I have measured out my life with coffee spoons.‖The above lines are taken from().A. Wordsworth’s ―The Solitary Reaper‖B. Eliot’s―The Love Song of J. Alfred Prufrock‖ (054)C. Coleridge’s―Kubla Khan‖D. Yeats’s―The Lake Isle of Innisfree‖20.The beginning of ―The Love Song of J.Alfred Prufrock‖ moves from a series of fairly concrete physical settings—a cityscape( the famous―patient etherized upon a table‖)and several interiors (women’s arms in the lamplight, coffee spoons,fireplaces)—to a series of vague ocean images. It aims to convey().A. Prufrock’s emotional distance from the world as he comes to recognize his second-rate statusB. Prufrock’s eagerness to meet his dating loverC. Prufrock’s reluctance to meet his dating loverD. Prufrock’s excitement about the modern world(057)20.Which of the following poems by T.S. Eliot is hailed as a landmark and a model of the 20th century English poetry?A.Poems 1909-1925 B.The Hollow ManC.Prufrock and Other Observations D.The Waste Land(074)3.T.S.Eliot’s most important single poem _______has been hailed as a landmark and a model of the 20th-century English poetry. 4A.The Hollow Man B.The Waste LandC.Murder in the Cathedral D.Ash Wednesday(084)22. T. S. Eliot's most popular verse play is ______.3A. Murder in the CathedralB. The Cocktail PartyC. The Family ReunionD. The Waste Land(094)22. The Waste Land by T. S. Eliot is a poem concerned with the _____ breakup of a modern civilization in which human life has lost its meaning, significance and purpose.A. spiritualB. religiousC. politicalD. physical(097)1. T. S. Eliot’ s ______ bearing a strong thematic resemblance to The Waste Land, is generally regarded as the darkest of Eliot’ s poems.2A. ―Gerontion‖B. ―Prufrock‖C. Murder in the CathedralD. The Hollow Men (104)1. T. S. Eliot’s ______ is a poem of dramatic monologue and a prelude to The WasteLand, helping to point up the continuity of Eliot’s thinking.1A. ―Prufrock‖B. ―Gerontion‖C. The Hollow MenD. Four Quartets (107)3 D. H. Lawrence4. The statement ―A demanding mother turns away from her husband and gives all her affection to her sons‖ sums up the main plot of D. H. Lawrence′s.5[A]Lady Chatterley’s Lover[B]Women in love[C]Sons and Lovers [D]The Plumed Serpent(034)20.The major concern of ______ fiction lies in the tracing of the psychologicaldevelopment of his characters and in his energetic criticism of the dehumanizing effect of the capitalist industrialization on human nature.wrence'sB.J.Galsworthy'sC.W.Thackeray’sD.T.Hardy’s (024)21. ―He was silent with conceit of his son. Mrs. Morel sniffed, as if it were nothing.‖(Sons and Lovers by wrence)From the above quotation, we can see that Mrs. More l’s attitude to her husband is ______ .4A. sincerely warmB. genuinely kindC. seemingly angryD. merely contemptuous(044)22.The story starting with the marriage of Paul’s parents Walter Morel and Mrs. Morel must be().A. Thomas Hard y’s Tess of the D’Urbervilles(054)B. D. H. Lawrence’s Sons and LoversC. George Eliot’s MiddlemarchD. Charlotte Bronte’s Jane Eyre22.The major concern of ______ fiction lies in the tracing of the psychological development of his characters and in his energetic criticism of the dehumanizing effect of the capitalist industrialization on human nature.A.John Galsworthy’s B.Thomas Hardy’s C.D.H.Lawrence’s D.Charles Dickens’(084)17. In Modern English literature, the literary interest of _____ lay in the tracing of the psychological development of his characters and in his energetic criticism of the dehu-manizing effect of the capitalist industrialization on human nature.3A. George Bernard ShawB.T.S. EliotC. Oscar WildeD.D.H. Lawrence(097)18. D. H. Lawrence’ s ______ is a remarkable novel in which the individual consciousness is subtly revealed and strands of themes are intricately wound up.2A. Sons and LoversB. The RainbowC. Women in LoveD. Lady Chatterley’ s Love(104)4. D. H. Lawren ce’s autobiographical novel is ______.A. The RainbowB. Women in LoveC. Sons and LoversD. Lady Chatterley’s Lover(107)10. D. H. Lawrence’s artistic tendency is mainly ______ , which combines dramaticscenes with an authoritative commentary.1A. romanticismB. realismC. naturalismD. modernism(107)23. In his novels, Laurence made a bold psychological exploration of various humanrelationships, especially those between _____, with a great frankness.a. man and natureb. man and societyc. man and womand. all of the above(浙0210)II. Reading Comprehension (16 points in all, 4 for each)Read the quoted parts carefully and answer the questions in English. Write your answers in the corresponding space on the answer sheet.George Bernard Shaw42. The following quotation is from Mrs. Warren’s Profession:VIVIE: [ intensely interested by this time] No; but why did you choose that business?Saving money and good management will succeed in any business.MRS. WARREN: Yes, saving money. But where can a woman get the money to save in any other business? Could you save out of four shillings a weekand keep yourself dressed as well? Not you. Of course, if you’ re a plainwoman and cant earn anything more ; or if you have a turn for music, orthe stage, or newspaper - writing ; that’s different...Questions :41A. Identify the playwright of the above quotation.B. What business do you think Mrs. Warren is involved in?C. What's the theme of the play?(097)42. A. George Bernard ShawB. ProstitutionC. The economic oppression of women. By describing Mrs. Warren and her sisters’sufferings, Shaw wants to show that in the dark capitalist society, honest women were forced to be prostitutes and were therefore looked down upon as shameful dirty women. The play reveals the social causes of prostitution and thus sharply attacks the capitalism.T. S Eliot42.―And when I am formulated, sprawling on a pin,When I am pinned and wriggling on the wall.Then how should beginTo spit out all the butt-ends of my days and ways.‖Questions:42A.Identify the poem and the poet.B.What does the phrase ―butt-ends‖ mean?C.What idea does the quoted passage express? (024)(42.A.T.S.Eliot:―The Love Song of J.Alfred Pruforck.‖B.The ends of cigarettes, meaning trivial things here.C.Here, Prufrock's inability to do anything against the society he is in is made strikingly clear by using a sharp comparison. Prufrock imagines himself as a kind of insect pinned on the wall and struggling in vain to get free. This image vividly shows Prufrock's current predicament.) .(024)42. The following quotation is from one of the poems by T. S. Eliot:No! I am not Prince Hamlet, nor was meant to be;Am an attendant lord, one that will doTo swell a progress, start a scene or twoAdvise the prince; no doubt, an easy tool,Deferential, glad to be of use,Politic, cautious, and meticulous,Full of high sentence, but a bit obtuse;Questions:A. Identify the title of the poem from which the quoted part is taken.B. Who's the speaker of the quoted lines?C. What does the first line show about the speaker? (094)42. A. ―The Love Song of J. Alfred Prufrock‖B. J. Alfred PrufrockC. Prufrock is conscious of the fact that he is like Hamlet in some respects. But heis sensible enough that he cannot be compared with Hamlet.42. Let us go then, you and I,When the evening is spread out against the skyLike a patient etherized upon a table;Let us go, through certain half- deserted streets,The muttering retreatsOf restless nights in one -night cheap hotelsAnd sawdust restaurants with oyster- shells:(The lines above are taken from ―The Love Song of J. Alfred Prufrock‖ by T. S Eliot. )Questions:A. What does the poem present?B. What form is the poem composed in?C. What does the poem suggest? (104)42. A. presenting the meditation of an aging young man over the business ofproposing marriage;B. in a form of dramatic monologueC. suggesting an ironic c ontrast between a pretended ―Love song‖ and aconfession of the speaker’s incapability of facing up to love and to life in a sterile upper-class world.III. Questions and Answers (24 points in all, 6 for each)Give a brief answer to each of the following questions in English. Write your answers in the corresponding space on the answer sheet.B. Shaw45.It is said that B. Shaw’s play, Mrs. Warren’s Profession, has a strong realistic theme, which fully reflects the dramatist’s Fabianist idea. Try to su mmarize this theme briefly. (044)45. A. The play reveals that guilt for prostitution lies more upon the social systemthan the immoral woman.B. In the play, Shaw shows clearly that all human sufferings are consequences ofthe cruel economic exploitation, which is pursued shamelessly by the so-called respectable members of the society through the lowest and the dirtiest means.46.It is said that B. Shaw’s play, Mrs. Warren’s Profession, has a strong realistic theme, which fully reflects the dramatist’s Fab ianist idea. Try to summarize this theme briefly.(084)46. A. As one of the influential members of the Fabian Society, Shaw regarded theestablishment of socialism by the emancipation of land and industrial capital from individual and class ownership as the final goal.B. As a realistic dramatist, he took the modern social issues as his subjects. Most ofhis plays are concerned with political, economic, moral, or religious problems.C. Mrs. Warren’s Profession is a play about the economic oppression of women.46. What are the main features of Bernard Shaw’s plays with regard to the theme,characterization and plot?(097)4546. A. Structurally and thematically, Shaw followed the great tradition of realism. As arealistic dramatist, he took the modern social issues as his subjects with the aim of directing social reforms.B. One feature of Shaw’s characterization is that he makes the trick of showing upone character vividly at the expense of another. Another feature is that Shaw’s characters are the representatives of ideas, points of view that shift and alter during the play.C. Shaw’s plays have plots, but they do not work by plots.45. What are the features of George Bernard Shaw’s characterization in his plays?(107)45. A. One feature of Shaw’s character ization is that he makes the trick of showing upone character vividly at the expense of another.B. Another feature is that Shaw’s characters are the representatives of ideas, pointsof view that shift and alter during the play.T.S.Eliot46. The Waste Land is T.S.Eliot’s most important single poem.Try to state the theme and the significance of the poem briefly. (087) 4646. A. Theme: the poem is concerned with the spiritual breakup of a moderncivilization in which human life has lost its meaning, significance andpurpose.B. Significance: The poem has been hailed as a landmark and a model of the20th-century English poetry, comparable to Wordsworth’s Lyrical Ballads. IV. Topic Discussion(20 points in all, 10 for each)Write no less than 150 words on each of the following topics in English in thecorresponding space on the answer sheet.chapter49. Define modernism in English literature. Name two major modernistic Britishwriters and list one major work by each.(097)4949. A. Modernism rose out of skepticism and disillusion of capitalism.B. The French symbolism appearing in the late 19th century, heralded modernism.After WWI, all kinds of literary trends of modernism appeared: expressionism, surrealism, futurism, etc.C. Modernism takes the irrational philosophy and the theory of psycho-analysis asits theoretical base.D. Modernism is, in many aspects, a reaction against realism.E. James Joyce: Ulysses; T. S. Eliot : The Waste Land.。

曼昆经济学原理第六版英文答案

曼昆经济学原理第六版英文答案

曼昆经济学原理第六版英文答案【篇一:曼昆《经济学原理(微观经济学分册)》第6版课后习题详解(1~2章)】class=txt>课后习题详解第一篇导言第1章经济学十大原理一、概念题1. 稀缺性稀缺性是指在给定的时间内,相对于人的需求而言,经济资源的供给总是不足的,也就是资源的有限性与人类的欲望无限性之间的矛盾。

2. 经济学经济学是研究如何将稀缺的资源有效地配置给相互竞争的用途,以使人类的欲望得到最大限度满足的科学。

其中微观经济学是以单个经济主体为研究对象,研究单个经济主体面对既定资源约束时如何进行选择的科学;宏观经济学则以整个国民经济为研究对象,主要着眼于经济总量的研究。

3. 效率效率是指人们在实践活动中的产出与投入比值或者是效益与成本比值,比值大效率高,比值小效率低。

它与产出或收益大小成正比,与投入或成本成反比。

4. 平等平等是指人与人的利益关系及利益关系的原则、制度、做法、行为等都合乎社会发展的需要,即经济成果在社会成员中公平分配的特性。

它是一个历史范畴,按其所产生的社会历史条件和社会性质的不同而不同,不存在永恒的公平;它也是一个客观范畴,尽管在不同的社会形态中内涵不同对其的理解不同,但都是社会存在的反映,具有客观性。

5. 机会成本机会成本是指将一种资源用于某种用途,而未用于其他用途所放弃的最大预期收益。

其存在的前提条件是:①资源是稀缺的;②资源具有多种用途;③资源的投向不受限制。

6. 理性人理性人是指系统而有目的地尽最大努力去实现其目标的人,是经济研究中所假设的、在一定条件下具有典型理性行为的经济活动主体。

7. 边际变动边际变动是指对行动计划的微小增量调整。

8. 激励激励是指引起一个人做出某种行为的某种东西。

9. 市场经济市场经济是指由家庭和企业在市场上的相互交易决定资源配置的经济,而资源配置实际上就是决定社会生产什么、生产多少、如何生产以及为谁生产的过程。

10. 产权产权是指个人拥有并控制稀缺资源的能力,也可以理解为人们对其所交易东西的所有权,即人们在交易活动中使自己或他人在经济利益上受益或受损的权力。

2024 年高考英语(新课标II卷)阅读理解中派生词总结

2024 年高考英语(新课标II卷)阅读理解中派生词总结

permit v. 允许;准许 → permission n. 允许;准许;许可
Байду номын сангаас
survive v. 生存;存活;继续存在 → survival n. 生存;存活;幸存
B 篇 communicate v. 交流;沟通 → communication n. 表达;交流;交际
long adj. 长的;长时间的,长久的 → length n. 长度
3/4
七选五
accept v. 接受;领受;相信 → acceptable adj. 可接受的 → unacceptable adj. 不能接受地 minimal adj. 最低的;最小限度的 → minimize/minimise v. 使减到最少;小看;最小化 mindful adj. 留心的;考虑到 → mindfully adv. 注意地;谨慎地 create v. 创造,创作,创建 → creative adj. 有创造力的,创造性的 concentrate v. 集中(注意力);(使)聚集于 → concentrated adj. 集中的;浓缩的 differ v. 相异,有区别 → different adj. 不同的,有区别的,有差异的 tour n. 旅行,旅游,观光 → tourism n. 旅游业,观光业 amaze v. 使惊奇,使惊诧 → amazed adj. 大为惊奇的,惊讶的 relative adj. 相比较而言的,相对的 → relatively adv. 相当地,相对地
origin n. 起源,源头 → original adj. 原来的,起初的,最早的;首创的
submit v. 提交,呈递(文件、建议等);屈服 → submission n. 提交(物),呈递(物);投降,屈服

选择性必修第三册 Unit 4 A glimpse of the future(教师版)

 选择性必修第三册 Unit 4   A glimpse of the future(教师版)

Ⅰ.阅读单词——会意1.artificial adj.人造的,人工的2.artificial intelligence人工智能3.susceptible adj.易受影响的4.automation n.自动化5.leap n.剧变6.horizon n.地平线7.reckon v.认为,以为8.cell n.细胞9.crater n.(物体坠落、炸弹爆炸等在地上造成的)坑10.agile adj.敏捷的,灵活的11.slope v.倾斜,成斜坡12.hop n.跳跃13.squeak v.发出吱吱声14.tendon n.腱15.wavelength n.波长16.mutt n.杂种狗17.frown v.皱眉18.positronic adj.正电子的Ⅱ.重点单词——记形1.victim n.受害者2.client n.客户,客人,顾客3.cite v.引用,引述4.bound adj.可能的5.mechanical adj.机械的6.desperate adj.拼命的,绝望的7.clue n.线索,提示Ⅲ.拓展单词——悉变1.humanity n.人类→human n.人adj.人的;有人情味的2.assistant n.助手,助理→assist v.帮助→assistance n.帮助3.potentially ad v.潜在地→potential adj.潜在的;可能的n.潜力4.capacity n.能力,才能;容量→capable adj.有能力的;能干的5.analyse v.分析→analysis n.分析6.regulate v.控制,管理→regulation n.规章制度,规则7.illegal adj.非法的,违法的→legal adj.合法的8.immoral adj.不道德的,道德败坏的→moral adj.道德的9.assess v.评价,评定→assessment n.评价;评估10.cycle n.自行车v.骑自行车→cyclist n.骑自行车的人11.demand n.& v.需要,需求→demanding adj.要求高的;苛求的12.tightly ad v.紧紧地→tight adj.紧的→tighten v.(使)变紧;加强13.faulty adj.有故障的,有缺陷的→fault n.缺陷;过错14.consistent adj.一贯的,一致的→consist v.由……组成;在于1.cautious /'kɔːʃəs/adj.小心的;谨慎的→caution /'kɔːʃn/n.谨慎;小心2.comply /kəm'pla I/v i.遵守;服从comply with遵从;服从3.abundance /ə'bʌndəns/n.大量;丰盛;充裕4.viable /'va Iəbl/adj.可实施的;切实可行的5.bypass /'ba I pɑːs/v t.避开;绕过n.旁道;小路6.pledge /pledʒ/v t.保证给予;正式承诺;发誓n.誓约;捐款承诺7.controversy /'kɒntrəvɜːsi;kən'trɒvəsi/n.争论;争议8.flawless /'flɔːləs/adj.完美的;无瑕的9.prestigious /pre'st I dʒəs/adj.有威望的;声誉高的10.buzz /bʌz/n.喧闹;嘈杂声;嗡嗡声v i.发出嗡嗡声;充满兴奋;闹哄哄Ⅳ.背核心短语1.bank on依靠,指望2.on the horizon将要发生3.be bound to很有可能,肯定会4.happen to发生在……身上5.so far 迄今为止6.in contrast相反7.hang out闲逛8.be susceptible to易受……的影响9.wipe out摧毁;毁灭10.lie in 在于11.agree on就……达成一致12.go through 履行(某程序);经历(苦难等);完成;通过;检查;用完13.be replaced with被……取代14.change one’s mind 改变主意15.in the long run从长远来看16.around the corner很近;即将发生Ⅴ.悟经典句式1.Every recommendation she gives me is great since she can predict if I will like a particular restaurant.(since 引导原因状语从句)她给我的每一个建议都很棒,因为她能预测我是否会喜欢某家餐馆。

新发展研究生英语1学生用语UNIT8SOLITUDE翻译

新发展研究生英语1学生用语UNIT8SOLITUDE翻译

Emerson says, "Now and then a man exquisitely made can live alone, and must; but coop up most men and you undo them." Solitude tries a man in a way society does not; it throws him upon his own resources, and if these resources be m eager, if the ground he occupies in and of himself be poor and narrow, he willhave a sorry time of it.爱默生说“有时候,一个至精之人能够也必须独孤地生活,但是孤独禁锢了多数人而却得不到解脱”,有别于社会,孤独用另一种方式试探一个人。

它让人诉诸于自身的资源,如果人的这些资源寥寥,如果他所拥有或掌控的范围狭小匮乏,孤独将会带给他悲伤的时光。

Hence we readily attribute some extra virtue to those persons who voluntarily embrace solitude, who live alone in the country or in the woods, or in the mountains, and find life sweet. We know they cannotlive without converse, without society of some sort, and we credit them with the power of invoking it from themselves, or else of finding more companionship with dumb things than ordinary mortals.因此,那些自愿拥抱孤独的人,他们或独居于乡村,或隐迹于树林山涧而甘之如饴,我们乐意把他们归之于某些高风亮节。

欧洲文化入门考试内容归纳中文版

欧洲文化入门考试内容归纳中文版

《欧洲文化入门》本作者绪论《欧洲文化入门》由于其内容庞杂,琐碎,因而是一门学习起来比较困难的课程。

其实大家大可不必担心,只要我们潜下心去,找出里面的规律和线索,这门课并不难攻克。

我们要牢记文化的五分法:一、社会历史(包括政治、经济、宗教、历史) 二、哲学三、文学四、科学五、艺术(包括绘画、雕塑、建筑和音乐),以记忆每个时代的各要点为主,理解纵向的变迁为辅,后者主要的作用时帮助我们更好的记住前者。

《欧洲文化入门》的考试大致包括以下几种题型:四选一,填空,判断,简答题,名词解释,论述题。

选择题:这种题型可考查考生的记忆、理解、判断、推理分析,综合比较,鉴别评价等多种能力,评分客观,故常被应用。

在答题时,如果能瞬时准确地把正确答案找出来最好,假如没有把握,就应采用排除法,即应从排除最明显的错误开始,把接近正确答案的备选项留下,再分析比较强以逐一否定最终选定正确答案。

填空题:这种题型常用于考核考生准确记忆的“再现”能力,在答题时,无论有几个空,回答都应明确、肯定,不能含糊其辞,填空题看似容易实则难,最好的应对办法是对英语语言知识中最基本的知识、概念、原理等要牢记。

名词解释:这种题型一般针对英语专业自考本科段课程中的基本概念、专业名词进行命题,主要考核考生的识记、理解能力。

在答题时,答案要简明、概括、准确,如分值较大,可简要扩展。

简答题:这种题型一般围绕基本概论、原理及其联系进行命题,着重考核考生对概念、史实、原理的掌握、辨别和理解能力。

在答题时,既不能像名词解释那样简单,也不能像论述题那样长篇大论,答案要有层次性,列出要点,并加以简要扩展就可以。

论述题:这种题型一般从试卷编制的全局出发,能从体现考试大纲中的重点内容和基本问题的角度来命题,着重考核考生分析、解决实际问题的能力,考核考生综合应用能力和创见性。

在答题时,要仔细审题,列出答案要点,然后对要点逐一展开叙述,此时考生应发挥自己的真知灼见,要在深度,广度上下功。

希拉里竞选英文演讲稿:为何我们需要更具体和实际的改革

希拉里竞选英文演讲稿:为何我们需要更具体和实际的改革

希拉里竞选英文演讲稿:为何我们需要更具体和实际的改革Ladies and gentlemen,It is with great pleasure that I speak to you today about the need for substantive and practical reform in our great nation. As we approach the upcoming election, it is essential that we evaluate our count ry’s current state and the challenges we face as a society.The past few years have been some of the most contentious in our nation’s history, marked by political and social polarization. We have witnessed an increase in hate crimes, a rise in economic inequality, and a decrease in social mobility. We are facing a healthcare crisis, a climate crisis, and an immigration crisis. These issues are not new, but they have been exacerbated by a lack of leadership and a failureto implement meaningful reform.As a candidate for president, I am committed to finding practical solutions to the challenges we face. This means addressing the root causes of these problems and working to create a more equitable, just, and prosperous society. My vision for America is one where every person has theopportunity to succeed, where no one is left behind, andwhere we stand united as Americans.One of the most pressing issues of our time is healthcare. Too many Americans lack access to affordable healthcare, and those who do have coverage are often burdened by skyrocketing costs. We need to reform our healthcare system to provide universal coverage, reduce costs, and ensure that everycitizen has access to the care they need. We must alsoaddress the opioid epidemic, mental health, and other healthcare challenges that are affecting families across our nation.Another area of focus must be climate change. Thescientific consensus is clear; we are facing an unprecedented threat to our planet, and urgent action is required. As president, I will bring the full force of the federal government to bear on this issue, investing in renewable energy, reducing greenhouse gas emissions, and collaborating with our global partners to combat this global crisis.We must also address economic inequality. Too many Americans are struggling to make ends meet, while our wealthiest citizens are seeing their fortunes growexponentially. We need to ensure that the economy works for everyone, not just a select few. This means investing in education and job training, raising the minimum wage, and expanding access to affordable housing.In addition to these issues, we must also reform our immigration system. The current administration’s policies have resulted in families being separated, children being detained, and the exploitation of immigrant workers. We needto create a pathway to citizenship for undocumented immigrants, protect the rights of refugees and asylum-seekers, and reform our visa program to attract the best and brightest from around the world.These are just a few examples of the challenges we face, and the need for real, substantive solutions. But we cannot achieve this without first coming together as a nation. We must reject the politics of division and work towards a more inclusive, compassionate, and united America. We must respect one another’s differences and work towards a common goal of creating a better future for us all.In conclusion, I am committed to creating pragmatic and achievable policies that will address the issues facing ournation today. I believe in the power of the American people to come together and build a brighter tomorrow. The challenges we face are significant, but with hard work, dedication, and a commitment to progress, we can overcome them. Thank you.。

The Future of Shopping

The Future of Shopping

标题:The Future of Shopping.作者: Rigby, Darrell1,2全文字数: 5088主题语:*ELECTRONIC commerce *RETAIL industry*SELF-service (Economics) *SHOPPING*INTERNET sales *CONSUMER behavior*TELESHOPPING *ONE-stop shopping*TARGET marketing *CONSUMERS' preferences*INTERNET marketing *SOFTWARE摘要:When the dot-com bubble burst 10 years ago, the ensuing collapse wiped out half of all online retailers. Today, ecommerce is well established and much digital retailing is now highly profitable. As it evolves, digital retailing is quickly morphing into something so different that it requires a new name: omnichannel retailing. The name reflects the fact that retailers will be able to interact with customers through countless channels-websites, physical stores, kiosks, direct mail and catalogs, call centers, social media, mobile devices, gaming consoles, televisions, networked appliances, home services, and more. If traditional retailers hope to survive, they must embrace omnichannel retailing and also transform the one big feature internet retailers lack-stores- from a liability into an asset. They must turn shopping into an entertaining, exciting, and emotionally engaging experience by skillfully blending the physical with the digital. They must also hire new kinds of talent, move away from outdated measures of success, and become adept at rapid test-and-learn methodologies. A successful omnichannel strategy should not only guarantee a retailer's survival-no small matter in today's environment-but also deliver a revolution in customers' expectations and experiences [ABSTRACT FROM AUTHOR]在十年前互联网泡沫破灭后紧接着发生的是将近一半的网络零售商被筛选掉。

英语翻译二级口译实务模拟试题及答案解析(2)

英语翻译二级口译实务模拟试题及答案解析(2)

英语翻译二级口译实务模拟试题及答案解析(2)(1/2)Part ⅠInterpret the following passages from English into Chinese. You will hear this signal to tell you when you start interpreting. Now let's begin.第1题My wife Nane and I are both extremely happy to be with you today. I feel truly proud to belong to this extraordinary class of 2004, and I am pleased to see that so many parents and family members were here today. The day belongs to them, too. Without their constant support, understanding and sacrifice, none of us could have achieved what we have. For me, to receive a degree from Harvard is a very great honor indeed. There are few countries in the world whose leaders in public life, business, science and the humanities have not had some association with Harvard—and no country that has not benefited from Harvard´s outstanding contributions to human knowledge. //You have invited me, I know, not as an individual, but as Secretary-General of the United Nations. You are saying that the United Nations matters, and that you want to hear what we have to say. Are you fight in believing that the UN matters? I think you are, because the UN offers the best hope of a stable world and a broadly equitable world order, based on generally accepted rules. That statement has been much questioned in the past year. But recent events have reaffirmed, and even strengthened, its validity. A rule-based system is in the interest of all countries—especially today. Globalization has shrunk the world. The very openness, which is such an important feature of today´s most successful societies, makes deadly weapons relatively easy to obtain, and terrorists relatively difficult to restrain. //Today, the strong feel almost as vulnerable to the weak as the weak feel vulnerable to the strong. So it is in the interest of every country to have international rules and to abide by them. And such a system can only work if, in devising and applying the rules, the legitimate interests of all countries are accommodated, and decisions are reached collectively. That is the essence of multilateralism, and the founding principle of the United Nations. All great American leaders have understood this. That is one of the things that make this country such a unique world power. America feels the need to frame its policies, and exercise its leadership, not just in the light of its own particular interests, but also with an eye to international interests, and universal principles. //Among the finest examples of this was the plan for reconstructing Europe after World War Ⅱ, which General Marshall announced here at Harvard in 1947. That was one part of a larger-scale and truly statesmanlike effort, in which Americans joined with others to build a new international system—a system which worked, by and large, and which survives, in its essentials, nearly 60 years later. During those 60 years, the United States and its partners developed the United Nations, built an open world economy, promoted human rights and decolonization, and supported the transformation of Europe into a democratic, cooperative community of states, such that war between them has become unthinkable. //下一题(2/2)Part ⅠInterpret the following passages from English into Chinese. You will hear this signal to tell you when you start interpreting. Now let's begin.第2题If you thought multimedia was something to be enjoyed in the privacy of your home, think again. Banks are on the frontier of the "information superhighway" because they spend more on the technology than any other type of civilian business.Take the case of J. P. Morgan, America´s fourth largest bank by assets. It has developed a system whereby deals and documents can be finalized quickly on the computer screen with the help of an electronic pen. Its securities analysts in London and traders in Tokyo can talk to each other via the same screen. And clients´trust can be built up, and deals completed, faster than via a telephone line which carries no pictures.The new electronic gizmos are currently being introduced into Morgan´s trading departments in New York, but eventually they will be used around the world-Aisa included. They make it economically possible to establish small dealing rooms in capitals such as Kuala Lumpur and Bangkok, while concentrating Morgan´s expensive back-office functions in Singapore.Morgan´s pioneering effort illustrates how United States banks are using high technology and large amounts of capital to lever their way into Asian markets.Rivals in Europe and Japan are doing so too, but they do not have the same access to the vast pool of saving available to American banks. US Pension Fund assets, for example, total US $4. 4 trillion, more than three times the size of Japan´s.US institutions are in the best position to act as a bridge between the growing capital demands of Asia and the supply of investment from the rest of the world. The bridge, of course, could wobble badly, as it did in the Latin American debt crisis of the 1980s, but this is hard to imagine in the 1990s when the economic fundamentals in Asia look so favorable.It took American banks almost a decade to emerge from the Latin American rubble, but they are now formidable competitor. They have written off nonperforming loans and cut payrolls far more boldly than their Japanese counterparts, which are still dogged by soured loans to spendthrift property speculators in Japan.The US commercial banks´ toughest rivals in Asian cross-border business are more likely to be their investment-bank compatriots rather than the Japanese, and the capital markets of Asia, as elsewhere, will be their battleground.The big five US banks —Citibank, Bank of America, Chemical, Morgan and Chase Manhattan, enjoy the advantage of being big. Compared with firms such as Salomon Brothers and Goldman Sachs, the big five are bigger in most senses of the word. They have more capital, more staff and more branches worldwide through which to distribute corporate issues.What remains to be seen is whether they have trading and deal-making ability to compete with investment hanks.上一题下一题(1/2)Part ⅡInterpret the following passages from Chinese into English. You will hear this signal to tell you when you start interpreting. Now let's begin.第3题下面你将听到一段回忆邓小平同志的发言。

九年级音乐发展英语阅读理解25题

九年级音乐发展英语阅读理解25题

九年级音乐发展英语阅读理解25题1<背景文章>Classical music has a long and rich history. It originated in Europe during the Middle Ages and Renaissance. Classical music is known for its complex harmonies, beautiful melodies, and strict musical forms.Some of the most famous classical composers include Wolfgang Amadeus Mozart, Ludwig van Beethoven, and Johann Sebastian Bach. Mozart was a child prodigy who composed some of the most beautiful and memorable music in history. Beethoven, on the other hand, is known for his powerful and emotional music. Bach is considered one of the greatest composers of all time, known for his complex fugues and chorales.Classical music has had a profound influence on later music genres. It has inspired many composers and musicians over the centuries. The beauty and complexity of classical music continue to attract audiences around the world.The main styles of classical music include symphonies, concertos, sonatas, and operas. Symphonies are large-scale works for orchestra, while concertos feature a solo instrument accompanied by an orchestra. Sonatas are usually for solo piano or a small ensemble, and operas combine music, drama, and singing.Classical music has also had a significant impact on culture and society. It has been used in movies, television shows, and advertisements. It is often performed in concert halls and festivals, attracting audiences of all ages.In conclusion, classical music is a treasure of human civilization. Its beauty and complexity continue to inspire and delight people around the world.1. Mozart is known as a ___.A. great singerB. child prodigyC. famous painterD. talented actor答案:B。

Quantum Coherence in an Exactly Solvable One-dimensional Model with Defects

Quantum Coherence in an Exactly Solvable One-dimensional Model with Defects

a r X i v :c o n d -m a t /9505082v 1 18 M a y 1995Quantum Coherence in an Exactly Solvable One-dimensionalModel with DefectsP.Schmitteckert (∗),P.Schwab and U.EckernInstitut f¨u r Physik,Universit¨a t AugsburgD-86135Augsburg,Germany(18.5.1995)Abstract .–Using the Quantum Inverse Scattering Method we construct an integrable Heisenberg-XXZ-model,or equivalently a model for spinless fermions with nearest-neighbour interaction,with defects.Each defect in-volves three sites with a fine tuning between nearest-neighbour and next-nearest-neighbour terms.We investigate the finite size corrections to the ground state energy and its dependence on an external flux as a function of a parameter ν,characterizing the strength of the defects.For intermediate values of ν,both quantities become very small,although the ground state wavefunction remains extended.(∗)Electronic mail:Peter.Schmitteckert@physik.uni-augsburg.de PACS:71.27.+a —Strongly correlated electron systems.71.30.+h —Metal-insulator transitions.75.10.Jm —Quantized spin models.Introduction .–Three recent experiments have demonstrated that persistent currents,peri-odic in the magnetic flux,exist in mesoscopic metal [1]and semiconductor [2]rings at very low temperatures.Surprisingly,though the current is found to be small,of the order of ∼ev F /L for single rings (v F is the Fermi velocity,and L the circumference),it is still two orders of magnitude larger than expected theoretically,at least for the metal rings studied in [1].In the latter,the electron motion is diffusive,i.e.the elastic mean free path is much smaller than the circumference.While it is well established that the Coulomb interaction gives an important contribution to the current for a measurement on an ensemble of rings[3],the interaction effect in single rings,is far from being understood theoretically.In this article,we consider a one-dimensional,interacting model in the presence of amagneticflux,or equivalently,with twisted boundary conditions.We introduce very special“defects”into the model describing spinless fermions with nearest-neighbour interaction.Despite this inhomogeneity,the model remains integrable and we present exact results forthefinite size corrections to the ground state energy,and its dependence on the magneticflux,as a function of a parameterνcharacterizing the strength of the defects.Clearly,ourinvestigation does not provide an answer to the questions raised by the experiments(there,the number of transverse channels is much larger than one).Instead,our work is closelyrelated to,and an extension of,various recent theoretical studies[4–9]of quantum coherencein strongly interacting electron systems.Construction of the model.–Using the Quantum Inverse Scattering Method(QISM),weconstruct our model from the R and L matrices of the Heisenberg-XXZ-model on an inho-mogeneous lattice as,for example,described in[10].The central equation of the QISM is theYang-Baxter equation,which guarantees that a scattering process factorizes in two-particlescattering processes and does not depend on the order of these.In order to construct amodel with defects,we allow that the local L n matrix depends,in addition to the spec-tral parameterλ,on a parameterνn,L n(λ)=L(λ+νn).The transfer matrix is givenby T(λ)=Tr M n=1L(λ+νn),where M denotes the number of lattice sites.To include twisted boundary conditions,we multiply the L M matrix of the Heisenberg-XXZ-ring withexp(iφˆσz/2).The Hamiltonian is then given as the logarithmic derivative of the transfer matrix withrespect toλ,at a specific value[10].In particular,in the special case in which allνn=0,we obtain the usual XXZ-model,which can be transformed to a spinless fermion modelby a Jordan-Wigner transformation.For a general set of parameters,{νn},it is difficultto determine the Hamiltonian explicitly,with one exception,namely where there are nodefects on neighbouring sites,i.e.νnνn+1=0for all n.This is the situation we study inthe following.As an illustration,consider a vanishing nearest-neighbour interaction,and asingle defect at the site n1characterized by the parameterν.The resulting Hamiltonian isgiven byH=H0+H I n1(ν)=−Mn=1 c+n c n+1+c+n+1c n +H I n1(ν)(1)H I n1(ν)=(1−1where nℓdenotes the location of a defect with strengthνnℓ.An illustration is given in Fig.1. The expression for the Hamiltonian in the presence of afinite nearest-neighbour interaction is more lengthy but similar in structure,i.e.a defect located at nℓaffects the lattice sites nℓ−1,nℓ,and nℓ+1only[11].Single defect,no interaction.–As is apparent from Eq.(2),forν=0,the Hamiltonian re-duces to H0,i.e.the standard single-band tight-binding model(the hopping matrix element is chosen to be unity).In the opposite limit,ν=∞,the lattice site n1is cut out of the ring.As a result,the model represents free fermions on a ring of M−1sites,however,with an additional phase factor e iδ1,δ1=π/2,for the hopping between n1−1and n1+1,plus one uncoupled site.We emphasize that the parameters,cosh−1(ν)and tanh(ν),as well as the phase factorδ1=π/2,arefine-tuned in the following sense:a generic impurity breaks translational invariance and lifts the degeneracies of the single-particle spectrum,which are found atφ=0,±π.While our defects also break translational invariance,this symmetry is replaced by another,of not as clear physical origin.As a result wefind that even when changingν,no degeneracies are lifted—they only occur at different,ν-dependent values of φ.The correspondingν-dependent symmetry operators can be constructed[10].The localization of electronic states is another,well established phenomenon,in one-dimensional disordered systems.In Fig.2,we plot the squared modulus of the wave-function for the single-particle level lowest in energy.Clearly,for the integrable case,the wavefunction is extended though reduced at the defect.Allowing,however,the phaseδ1to be different fromπ/2,which corresponds to the non-integrable case,wefind a drastically different behaviour with a clear localization of the wave-function near the defect.Several defects,finite interaction.–The results for afinite nearest-neighbour interaction, and in the presence of several defects,are obtained from the Bethe equations,which we derive from the algebraic Bethe ansatz,with the resultcosh(λj−iη)cosh(λj+νnℓ+iη)=e iφNk=1k=j sinh(λj−λk−2iη)in the energy spectrum,which is quite different from the observation that in mesoscopic physics,thermodynamic and transport properties can vary considerably by moving only one impurity by a few lattice constants.In the following,we simplify our defect model further,by assuming that|νnℓ|=ν, and choosing an equal number of defects of strength+νand−ν.With this choice,the ground state energy is an even function of the phaseφ.We define x=lim r,M→∞(r/M) to be the density of defects.Assuming in addition N=M/2,i.e.a halffilled band,we calculate in afirst step the ground state energy per lattice site in the thermodynamic limit, e∞=lim M→∞e M,e M≡E M/ing the standard method[10,12],wefinde∞=−sin2(2η) ∞−∞ρ∞(λ)2(5)ρ0∞(λ)= 2(π−2η)cosh[πλ/(π−2η)] −1(6)ρ∞(λ)=(1−x)ρ0∞(λ)+x ρ0∞(λ+ν)+ρ0∞(λ−ν) /2.(7) The last equation relates the density of roots for the infinite system,ρ∞,to the corresponding density of roots of the homogeneous system,ρ0∞.Forν=0,clearly,ρ∞=ρ0∞,and we recover the results of[12].On the other hand,forν→∞,it follows from(7)thatρ∞=(1−x)ρ0∞; as a direct consequence,the integral in Eq.(5)is simply multiplied by the factor(1−x) compared to the homogeneous case,and e∞is easily calculated.Apparently,for all values ofν,the ground state energy e∞is a linear function of the defect concentration(and it is independent ofφ,i.e.the boundary conditions).The leading orderfinite size corrections can be determined with the help of the Wiener-Hopf technique,as described in[4],with the resulte M(φ)−e∞=−(π2/6)sin(2η)/(π−2η)4πη 1Finally,we consider the phase sensitivity of the ground state energy,similiar to Refs.[4-6],where the homogeneous system was studied.Here we choose M=103and x=0.2,and we plot[e M(π)−e M(0)]M2as a function ofνin Fig.4for differentfillings N/M.Close to halffilling(N=440,490),the dependence onνis very similar to the result described above for thefinite size corrections,i.e.the phase sensitivity is very small betweenν≈4andν≈8. In this context,note that Eq.(8)implies that[e M(π)−e M(0)]/[e M(0)−e∞]=−3π/(4η); but note also that different∆’s have been chosen in Figs.3and4.The asymptotic(largeν)results shown in Fig.4can be explained as follows:in this limit (and for the given parameters),200sites are cut out from the system of1000sites.But,as discussed above,the occupancy of these sites still enters into the Hamiltonian.The defect phases,δℓ,are found to be±2η,and as a result,wefind additional cusps in the energy-phase relation,for example,forηslightly less thanπ/4,atφ=±4η.Consequently,the number of discontinuities in the persistent current,I=−∂E M/∂Φ,increases,implying a reduction of the phase average of the current over half a period,i.e.e M(π)−e M(0),compared to its ν=0value.For lowfilling,N≪M,on the other hand,the interaction is unimportant and we may apply the free electron result,I=−(ev F/L)φ/π,which implies that E M(π)−E M(0)≈π2N/M2for small N.(In our units,L=M and¯h v F=2sin(πN/M).)Comparing the limitsν=0andν→∞,the phase sensitivity is thus approximately enhanced in the latter case by the factor(1−x)−2,which is apparent in Fig.4.Conclusions.–We have presented exact results for thefinite size corrections to the ground state energy and itsflux sensitivity for an one-dimensional,interacting model in the pres-ence of defects.Through its construction,the model remains integrable.The defects are “magnetic”in the sense that a defect triangle,compare Fig.1,encloses afinite magnetic flux.This means that time reversal symmetry is broken and,for general{νn},the energy is not an even function ofφ.Surprisingly,the energy spectrum is independent of the spatial distribution of defects,and wefind neither level repulsion nor localized states,which are considered to be generic properties of a“real”impurity.We believe that the absence of these effects is strongly related to the integrability of the model.Nevertheless,thefinite size corrections and the phase sensitivity,i.e.the persistent current,can become very small.***We thank K.-H.H¨o ck for interesting discussions and useful comments on the manuscript.P.Schmitteckert acknowledgesfinancial support through a graduate student fellowship by the state of Bavaria.REFERENCES[1]L´e vy L.P.,Dolan G.,Dunsmuir J.and Bouchiat H.,Phys.Rev.Lett.,64(1990)2074;L´e vy L.P.,Physica B,169(1991)245;Chandrasekhar V.,Webb R.A.,Brady M.J.,Ketchen M.B.,Gallagher W.J.and Kleinsasser A., Phys.Rev.Lett.,67(1991)3578.[2]Mailly D.,Chapelier C.and Benoit A.,Phys.Rev.Lett.,70(1993)2020.[3]Ambegaokar V.and Eckern U.,Phys.Rev.Lett.,65(1990)381;67(1991)3192;Schmid A.,Phys.Rev.Lett.,66(1991)80.[4]Hamer C.J,Quispel G.R.W.and Batchelor M.T.,J.Phys.A,20(1987)5677.[5]Sriam Shastry B.and Sutherland B.,Phys.Rev.Lett.,65(1990)1833;65(1990)243.[6]Alcaraz F.C.,Barber M.N.and Batchelor M.T.,Phys.Rev.Lett.,58(1987)771;Ann.Phys.(New York),182(1988)280.[7]Zotos X.,Preloˇv sek P.and Sega S.,Phys.Rev.B,42(1990)8445.[8]Bariev R.Z.,Kl¨u mper A.,Schadschneider A.and Zittartz J.,J.Phys A:Math.Gen.,26(1993)1249.[9]Castelle H.,Zotos X.and Preloˇv sek P.,Phys.Rev.Lett.,74(1995)972.[10]Korepin V.E.,Bogoliubov N.M.and Izergin A.G.,Quantum Inverse ScatteringMethod and Correlation Functions,Cambridge University Press(1993);Takhtadzhan L.A.and F addeev L.D.,Russ.Math.Surveys,34:5(1979)11.[11]Schmitteckert P.,Schwab P.and Eckern U.,to be published.[12]Yang C.N.and Yang C.P.,Phys.Rev.,150(1966)321;Phys.Rev.,150(1966)327.List of FiguresFigure1:Graphical representation of a defect at the site nℓ,compare Eq.(2).Note that forν→0(ν→∞)the dashed(dotted)lines representing the corresponding hopping contributions are effectively cut.Figure2:Squared modulus of the wavefunction of the lowest energy eigenstate for the non-interacting limit in the presence of a single defect(ν=1)at the site n1=50 (M=100).For the integrable case(δ1=π/2),the eigenstate is extended,though reduced at the defect,while a detuning of the defect phase(δ1=π/2)leads to a localization of the wavefunction.Figure3:Finite size corrections to the ground state energy,(e M−e∞)M2,as a function of the defect parameterνfor halffilling,N=M/2.The system size is varied from M=102to M=104;the results for large systemsfit perfectly with the analytical expression,Eq.(8)forφ=0.The nearest-neighbour interaction is chosen to be ∆=0.5,and the defect density is x=0.2.Figure4:Phase sensitivity of the ground state energy,[e M(φ=π)−e M(0)]M2,as a function ofνfor differentfillings,i.e.numbers of fermions,N(M=103,∆=0.2).。

希拉里和特朗普第一次辩论全文(英文)

希拉里和特朗普第一次辩论全文(英文)

希拉⾥和特朗普第⼀次辩论全⽂(英⽂)Round One: Achieving ProsperityThe question is about putting more money into the Americans’ pockets.First we have to build an economy that works for everyone, not just those at the top.That means we need more jobs, good jobs with rising incomes.I want us to invest in you. I want us to invest in your future.That means jobs in infrastructure and in mass manufacturing, in innovation and technology, clean renewable energy and in small businesses because most of the jobs will come from small business.We also have to make the economy fairer. That starts with raising the nationalminimum wage and also guarantee finely equal pay for women‘’s work.I also wane see more companies do profit sharing. If you helped make the profit you should be able to share them, not just the executives at the top.And I want us to do more to support people who are struggling to balance family and work. I’ve heard from so many of you about the difficulties you face and the stresses you' re under.And let’s have paid family leave, earned sick days. Let’s make sure we have affordable childcare and debt-free college. How are we gone do it ? We' re gonna do it by making the wealthy pay their fair share, close thecorporate loopholes.Our jobs are fleeing the country. They are going to Mexico. They are going to many other countries.You look at what China' s doing to our country in terms of making our product.They ' re devaluing their currency and there ody in’ours government to fight them. And we have a very good fight and we have a winning fight because they are using our country as a piggy bank to rebuild China and many other countries are doing the same thing. So we are losing our good jobs, so many of them.When you look at what' s happening in Mexico, one of my friend built a plantsaid it' s the eighth wonder of the world. They're building the big estsome of plants, anywhere in the world, some of the most sophisticated, some of thebest plants. With the United States, as you said, not so much.So Ford is leaving. You see, that, their small cars division, leaving. Thousands of jobs, leaving Michigan, leaving Ohio. They ' re all leaving. And we cannot allow it to happen anymore as far as childcare is concerned and so many other things.I think Hillary and I agree on that and we probably disagree on numbers and amounts and what we're going to do but perhaps we'll talk aboutthat later.But we have to stop our jobs from being stolen from us. We have to stop our companies from leaving the United States, and with it, firing all of their people.All you have to do is take a look at the Carrier air-conditioning in Indianapolis. They left fired 1,400 people. They are going to Mexico. So many, hundreds and hundreds companies are doing this. We cannot let it happen.Under my plan, I' ll be reducing taxes tremendously from 35% to 15% for companies, small and big businesses.That 's gonna be a job creator, like, we haven't seen since Ronald Regan. It going to be a beautiful thing to watch. Companies will come. They will build. They will expand. New companies will start. And I look very very much forward to doing it.We have to renegotiate our trade deals. We have to stop these countries from stealing our companies and our jobs.Round Two: Let's Talk About RaceRace remains a significant challenge in our country. Unfortunately race still determines too much. Often determines where people live. Determines what kind of education in their public schools they can get. And, yes, it determines how they’re treated in the criminal justice system.We’ve just seen those tragic examples in both Tulsa and Charlotte. And we’got to do several things at the same time.We have to restore trust between communities and the police.We have to work to assure that our police are using the best training, the best technique, that they' re well prepared to use force only when necessary.Everyone should be respected by the law. And everyone should respect the law. Right now that’s not the case in a lot of our neighborhoods. So I have, ever since the first day of my campaign, called for criminal justice reform.I 've laid out a platform that I think will begin to remedy the problems we have in the criminal justice system.But we also have to recognize that in addition to the challenges we face with policing, there are so many brave, good police officers who equally want reform.So we have to bring communities together in order to work on that as a mutual goal.And we’ve got to get guns out of the hands of the people who should not have them.The gun epidemic is the leading cause of death of young African-American men, more than the next 9 causes put together. So we have to do 2 things as I said. We have to restore trust. We have towork with the police. We have to make sure they respect the communities and the communities respect them.And we have to tackle the plague of gun violence which is a contributor to a lot of problems we are seeing today.First of all, Secretary Clinton dosen law and order. We need law and order. We don't wanna use a couple of words. And that ' t have it. We are not gonna'twanna use a couple of words. And that' t have it. We are not gonna'twanna use a couple of words. And that’t have it. We are not gonna'have a countryAnd when I look at what 's going on in Charlotte, the city I lovthe,city I have investments, when I look at what's going on through various parts of our country, whether it' s, I mean, I can just keep naming them all day long. We need law and order in our country.And I just got, today, the, eh you know, the endorsement, the fraternal order of police who just came in. We have endorsement from, I think, almost every police group, every, I mean, a lot percentage of them in the United States.We have a situation in which we have, eh, in the city, African-American, Hispanics are living in hell.Because it' s so dangerous. uYowalk down the street, you got shot.In Chicago, they' ve had thousands of shootings, thousands, since January 1st. Thousands of shootings.And I say: where is this? Is this a war-torn country. What are we doing? And we have to stop the violence. We have to bring back law and order in a place like Chicago where thousands of people have been killed, thousands, over the last number of years.In fact, almost 4,000 of people have been killed since Barack Obama became President. Over 4, almost 4,000 people have been killed in Chicago. We have to bring back law and order.Now, whether or not, in a place like Chicago, you do stop and frisk witch worked very well, Mary Julian is here, worked very well in New York. It brought the crime rate way down.But you take the gun away from criminals who shouldn't be having it. We have gangs roaming the street. And in many cases they' re illegally here, illegal immigrants. And they have guns. They shoot people.And we have to be very strong. We have to be very vigilant. We have to be, we have to know what we are doing.Right now our police, in many cases, are afraid to do anything. We have to protect our inner cities because African-American communities are being decimated by crime. Decimated!Round Three: Securing AmericaHillary:I think cyber security, cyber warfare will be one of the biggest challengesfacing the next president because clearly we are facing at this point 2 different kinds of ad versaries.There are the independent hacking groups that do, mostly, for commercial reasons to try to steal information that they can use to make money.But increasingly we are seeing cyber attacks coming from states, all kinds of states. The most recent and troubling of these are from Russia. There's no doubt now that Russia has used cyber attacks against all kinds of organizations in our country and I am deeply concerned about this.I know Donald' s, very, paise-worthy of Vladimir Putin. But Putin is playing a really tough long game here.And one of the things he's done is to let loose cyber attackers to hack into government files, to hack into personal files, hack into the Democratic National Committee.And we recently have learnt that, you know, this is one of their preferred methods of trying to wreak havoc and collect information.We need to make it very clear, whether it's Russia, China, Iran or anybody else.The United States has a much greater capacity and we are not gonna sit idly by and permit state actors to go after our information, our private sector information and our public sector information.And we are going to have to make it clear that we don' t want to use thekinds of tools that we have to engage in a different kind of warfare.But we will defend the citizens of this country. And the Russians need to understand that.I think they' ve been treating it as it is almost a probing. Howfar will we go?How much will we do? And that's why I was so, I was so shocked when Donald publicly invited Putin to hack into Americans. That is just unacceptable. That’s onereasonsofthewhy 15 national security officials who serve in Republican Information Administration have said that Donald is unfit to be the commander in chief.There’s comments like that really worry people who understand the threats that we face.Trump:Yeah, I do wanna say that I was just endorsed and more are coming the next week. It will be over 200 admirals, many of them are here, admirals and generals endorsed me to lead this country.That just happened and many more are coming. And I' m very proud of it. In addition I was just endorsed by ICE. They've never endorsed anybody before. On immigration I was just endorsed by ICE. I was just endorsed. 7,500 patrol agents.So when Secretary Clinton talks about this I' ll take the admirals and I' ll t generals, any day over the political hacks that I see that has led ourcountry so brilliantly over the last 10 years with their knowledge. Because look at the mess we' re in. Look at the mess we' re in. As long as there's the cyber, I agree to parts of what Secretary Clinton said. We should be better than anybody else and perhaps we’re not.I don't know if anybody knows that it's Russia that broke into the DNC. She saying Russia RussiaRussia. But I don't. Maybe it was. It could be Russia. But t could also be China. It could also be a lot of other people. Could also be somebody sitting on their bed that weighs 400 pounds, OK?You don' t know who broke into DNC. But what do we learn with DNC? We learnt that Bernie Sanders was taken advantage of by your people, by Debbie Wasserman Schultz. Look at what happened to her.But Bernie Sanders was taken advantage of. That's what we learned. Now, whether that was Russia, whether that was China, whether it was another country, we don't know, because the truth is, under President Obama we've lost control of things that we used to have control over. We came in with the Internet, we came up with the Internet, and I think Secretary Clinton and myself would agree very much, when you look at what ISIS is doing with the Internet, they're beating us at our own game. ISIS.So we have to get very, very tough on cyber and cyber warfare.It is —it is a huge problem. I have a son. He's 10 years old. He hascomputers.He is so good with these computers, it's unbelievable.The security aspect of cyber is very, very tough. And maybe it's hardly doable.But I will say, we are not doing the job we should be doing. But that's true throughout our whole governmental society.。

金钱的坏处 英语作文

金钱的坏处 英语作文

Money,often regarded as the lifeblood of modern society,has the potential to bring both great benefits and significant drawbacks.While it is true that financial resources can provide comfort,security,and opportunities,it is also important to consider the negative aspects that can arise from an overemphasis on monetary wealth.This essay will explore the various ways in which an excessive focus on money can have detrimental effects on individuals and society as a whole.1.Materialism and Loss of Values:One of the primary issues associated with the pursuit of wealth is the tendency towards materialism.When money becomes the ultimate goal,people may begin to prioritize possessions over personal relationships,experiences,and intangible values such as kindness and community.This shift in priorities can lead to a loss of connection with what truly matters in life,resulting in a sense of emptiness and dissatisfaction.2.Stress and Health Issues:The constant chase for more money can also lead to increased stress levels.The pressure to earn more,invest wisely,and maintain a certain lifestyle can take a toll on an individuals mental and physical health.Studies have shown that high levels of stress can contribute to a range of health problems,including heart disease,depression,and anxiety.3.Inequality and Social Division:Money can be a source of social division,as the gap between the rich and the poor continues to widen.This economic inequality can lead to resentment and social unrest,as those with less feel marginalized and left behind.It can also perpetuate a cycle of poverty, as those with fewer resources struggle to access quality education,healthcare,and other opportunities that could improve their circumstances.4.Corruption and Unethical Behavior:The influence of money can sometimes lead individuals and organizations to engage in corrupt practices.The desire for financial gain can override moral and ethical considerations,resulting in actions such as bribery,fraud,and exploitation.This not only harms the individuals involved but also undermines trust in institutions and the rule of law.5.Environmental Degradation:The pursuit of wealth can also have negative environmental consequences.The drive for economic growth often leads to overconsumption and the depletion of natural resources. Industries that prioritize profit over sustainability can contribute to pollution, deforestation,and climate change,with longterm impacts on the planet and its inhabitants.6.Loss of Personal Freedom:Individuals who are overly focused on accumulating wealth may find that their personal freedom is compromised.The need to work long hours or engage in activities solely for financial gain can limit the time available for leisure,family,and personal development. This can lead to a life that feels unfulfilling and constrained by the pursuit of money.7.Interpersonal Conflicts:Money can also be a source of conflict within relationships.Disputes over financial matters can strain personal connections,leading to resentment and the breakdown of trust. The desire for wealth can also lead to competitiveness and jealousy among friends and family members,further damaging these important relationships.Conclusion:While money is undoubtedly a useful tool in modern society,it is crucial to recognize its potential to cause harm when it becomes an end in itself.A balanced approach,where financial stability is pursued without sacrificing personal values,relationships,and wellbeing,is essential for a fulfilling and harmonious life.It is through this lens that we can appreciate the true value of money as a means to an end,rather than the end itself.。

Honeywell HC900 Hybrid Controller产品说明书

Honeywell HC900 Hybrid Controller产品说明书

HC900 Hybrid ControllerWhen you need more than just discrete controlProduct Note - Positional Proportional OutputBackground:Many processes under control today have requirements that are best satisfied using energy regulating devices that are directly connected to electromechanical motor actuators. Electromechanical actuators have been in use in industry for control system regulation for more than 50 years, and there is a large installed base of these devices. Many control system upgrade projects focus on the electronic hardware located in the control room, but fail to include motor actuators in the project, particularly if they are functioning acceptably. Although newer “Smart” actuators can offer numerous maintenance and asset management advantages, they are often overlooked when upgrade projects are planned.The motor actuator used for regulation uses a bi-directional motor that holds its last position when deactivated. Many actuators use a position sensing device to provide a signal to the control system of the actual actuator position. One of the most common sensors used for this function, because of its low cost, simple design and rugged performance, is a variable resistor or slidewire. Since the slidewire is a passive electronic device, one of the requirements of the control system is to provide an excitation voltage in order to obtain a position output signal. Although the slidewire is a cost effective and rugged device, it does have a mechanical attribute that is subject to wear over time. For this reason, some actuator suppliers offer electronic position sensors that can supply a voltage or current without the dependency on wear component hardware. For these devices, control systems typically only require a high level analog measurement capability and an algorithm to position the motor. Algorithms used to position actuators with feedback sensors do not have a standard terminology for their identification but a few of the most common names used are: Position Proportional Outputs (PPO), Position Adjusting Type (PAT), and Motor Positioner (MP). Honeywell uses the PPO terminology.When knowing the actual motor position in a control system is not important, many actuators are supplied without a position sensor. The control systems used to regulate these actuators depend on a special algorithm that simulates an internal feedback to substitute for the actual feedback signal. The most commonly recognized algorithm for this type of control is the “Three-Position-Step” algorithm.Both the Position Proportional Output (PPO) and the Three Position Step Control (TPSC) can provide excellent process control when properly engineered into the control system.Problem Statement:Many first time application engineers approach position proportional control as a very simple task. Just measure the analog signal representing position, compare it to the desired position, and if the value is lower than required, turn a digitaloutput ON to increase the position, and if the value is too high, turn the decreasing output ON. To keep the motor from oscillating, add some dead-band, or said another way, compromise position accuracy to improve motor life. This is typically not more than a few program statements in a PLC.Before putting the system on-line the following problems must be addressed:1. The passive slidewire position sensor of the actuator requires electrical excitation, (typically DC power source with avoltage low enough not to cause the resistive element to heat up or burn out). If the controller does not provide this voltage, an external power supply must be used.Once on-line, other problems surface:2. Once the slidewire is powered and the actuator is moved to its 0% and 100% travel limits based on the setting of themotor limit switches, a common observation is that the limit switches stop the motor before the slidewire contact reaches its 0% and 100% voltage limits. If the power supply used to excite the slidewire was 5 volts, the feedback voltage measured between 0% and 100% may typically be between ~0.3V and ~4.8V. A measurement of feedback based on a 5V span would have an error of 0.5V or a 10% error. To correct this error the actual span value of the feedback signal as measured on a 5V analog input range must be re-spanned to have 0.3V and 4.8V represent 0% and 100% travel of the motor.3. Once the feedback is addressed, positioning the motor becomes the next issue. If a simple compare algorithm isused to determine when to increase or decrease the motor, an oscillation of the motor is guaranteed. This is because it takes time to measure the analog feedback, process the compare function, and turn an output OFF. During this time the motor is still moving. When the motor is finally deactivated, it will have already passed the compare point that initiated the action to turn off. The amount of overshoot is based on the algorithm processing time and motor speed. Also, if the update rate of the algorithm is not consistent or varies with processor loading, the amount of overshoot will also vary.To avoid this overshoot condition, a timed output approach is typically used to position the motor. With this approach, if a 15 second motor is at its 0% position and a new request comes in for a 50% position, the output algorithm would turn ON its output for 7.5 seconds. This feature adds a new complexity, to measure the time for full scale motor travel and use this value in the output algorithm calculation.4. Another variable that must be considered is the amount of motor over-travel that may occur after power to the motor isremoved. To minimize this condition, many actuators employ a gearing or braking system, but it can seldom be reduced to zero. To optimize control of this variable, a combination of timed pulses and a dead-band are typically used.When fully implemented the final program often requires significantly more work than initially planned. The Solution:The HC900 “Position Proportional Output” (PPO)The HC900 Position Proportional Output is designed to provide optimum actuator position control with a minimum of setup complexity. From a standard analog input range for feedback slidewire measurements to a fully automatic feedbackcalibration feature, the HC900 PPO algorithm excels. Once on-line, the PPO algorithm provides additional benefits with a scaled output of the actual motor position that may be used on displays or integrated into the control strategy, and diagnostics routines that run in the background that detect actuator problems. The following are a few of the HC900 PPO algorithm highlights and benefits.1. Standard analog ranges embedded in the PPO algorithm to measure slidewire resistances between 100 and 1000ohms. (No external power supply required. Uses a simple 3-wire connection to the standard universal AI module.) Voltage ranges of 0 to 5V, 0 to 1V, and current ranges 0 to 20mA , 4 to 20mA are also supported for feedback inputs.2. Flexibility to use any type of digital output module with the PPO algorithm, AC, DC or Relay. (No special outputmodules or incompatible voltages.)3. Complete actuator control setup from a single PPO function block including analog input hardware and range, digitaloutput hardware, actuator feedback scaling and motor speed, actuator sensitivity (dead-band). ( Easy to setup, easy to troubleshoot.)4. Fully automatic feedback signal calibration. Scales the actual feedback signal values to represent 0 to 100% outputand calculates motor speed. (The easy choice for easy to use.)5. Semi-automatic feedback calibration (user initiated step-by-step process) and manual feedback calibration (usermanually moves the motor) also supported. (Offers flexibility when required.)6. User entered dead-band support for values from 0.5% to +/- 5.0%. Ideal for actuators without braking action.7. Input scaling support is included to allow reverse scaling for heating/cooling applications using Duplex control (0 to 50,50 to 0).8. User entered output limit values supported. For users who want to limit actual actuator travel to values less than 0and 100%.9. Actuator position output pin provided for displays or connection to other function blocks.10. Feedback failure detect output pin provided. Digital output turns ON if the algorithm detects a failed slidewire.11. Automatically defaults to Three-Position–Step control on feedback failure.12. Fault output pin turns ON upon detection of a failed or stalled motor.The HC900 controller can support up 64 PPO algorithms, or two for each PID loop. The algorithms also execute independently from the PID algorithms, allowing them to accept setpoint signals from other sources. For example, a single manually entered variable from an OI overview display could be used to set the position of multiple actuators simultaneously. Another application might connect two actuators to the same PID algorithm with the actuators operating in series, applying unique input scaling to each actuator. These are only a few examples of the control flexibility afforded users with the HC900 PPO algorithm.How to Order:The PPO algorithm is offered in the HC900 controller at no additional cost in CPU versions 900C51-0011, 900C52-0011, 900C31-0011, 900C32-0011 and later versions. If direct slidewire measurements are required, analog input module 900A01-0002 is required. This module is a direct replacement for module 900A01-0001 in all applications.Use the HC900 PPO algorithm in retrofit applications and in new applications when accurate positioning of drive actuators is required.Typical Position Proportional Configuration and Wiring:Analog Input Module AC Output Module。

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a rXiv:q uant-ph/99644v114J u n1999UICHEP-TH/99-3,IP/BBSR/99-10,February 9,2008New Solvable and Quasi Exactly Solvable Periodic Potentials Avinash Khare a,1and Uday Sukhatme b,2a)Institute of Physics,Sachivalaya Marg,Bhubaneswar 751005,Orissa,India b)Department of Physics,University of Illinois at Chicago,Chicago,IL 60607-7059,U.S.A.Abstract Using the formalism of supersymmetric quantum mechanics,we obtain a large number of new analytically solvable one-dimensional periodic potentials and study their properties.More specif-ically,the supersymmetric partners of the Lam´e potentials ma (a +1)sn 2(x,m )are computed for integer values a =1,2,3,....For all cases (except a =1),we show that the partner potential is distinctly different from the original Lam´e potential,even though they both have the same energy band structure.We also derive and discuss the energy band edges of the associated Lam´e po-tentials pm sn 2(x,m )+qm cn 2(x,m )/dn 2(x,m ),which constitute a much richer class of periodic putation of their supersymmetric partners yields many additional new solvable and quasi exactly solvable periodic potentials.1.Introduction:The energy spectrum of electrons on a lattice is of central importance in condensed matter physics. In particular,knowledge of the existence and locations of band edges and band gaps determines many physical properties.Unfortunately,even in one dimension,there are very few analytically solvable periodic potential problems in quantum mechanics.The aim of this paper is to extend the small currently known set of analytically solvable periodic potentials.For a potential with period L,one is seeking solutions of the Schr¨o dinger equation subject to the Bloch conditionψ(x)=e ikLψ(x+L),(1) where k denotes the crystal momentum.The spectrum shows energy bands whose edges correspond to kL=0,π,that is the wave functions at the band edges satisfyψ(x)=±ψ(x+L).For periodic potentials,the band edge energies and wave functions are often called eigenvalues and eigenfunctions, and we will also use this terminology.The classic text book example which is used to demonstrate band structure is the Kronig-Penney modelV(x)=∞n=−∞V0δ(x−nL).It should be noted that the band edges for the Kronig-Penney model can only be computed by solving a transcendental equation.Another well studied class of periodic potentials isV(x)=pm sn2(x,m),p≡a(a+1).(2)Here sn(x,m)is a Jacobi elliptic function of real elliptic modulus parameter m(0≤m≤1)with period4K(m).For simplicity,from now onward,we will not explicitly display the modulus parameter m as an argument of Jacobi elliptic functions[1].The elliptic function potentials of eq.(2)have a period L=2K(m),and will be referred to as Lam´e potentials,since the corresponding Schr¨o dinger equation is called Lam´e’s equation[2,3].It is well known that for any integer value a=1,2,3,...,the corresponding Lam´e potential(2)has a bound bands followed by a continuum band[2,3].All band edge energies and wave functions are analytically known.At this point it is worth recalling that supersymmetric quantum mechanics(SUSYQM)has proved useful in discovering many,new,analytically solvable potentials on both the full as well as the half line[4].It is then natural to enquire if one can also use similar techniques to discover new solvable periodic potentials.In this paper,we demonstrate that this is indeed possible.2Our work is inspired by several recent papers[5,6,7,8]which discuss various general aspects of SUSYQM for periodic potentials.In particular,Dunne and Feinberg[5]defined and developed the concept of“self-isospectral”periodic potentials in detail.A one dimensional potential V−(x)of period L is said to be self-isospectral if its supersymmetric partner potential V+(x)is just the original potential upto a discrete transformation-a translation by any constant amount,a reflection,or both.A common example is translation by half a period,in which case the condition for self-isospectrality isV+(x)=V−(x−L/2).(3) It is easily checked that if the superpotential W satisfiesW(x)=−W(x−L/2),(4)then condition(3)immediately follows.In this sense,any self-isospectral potential is rather uninter-esting,since application of the SUSYQM formalism[4]to it just yields a discrete transformation and basically nothing new.We have recently pointed out[9]that the Lam´e potentials given in eq.(2)are not self-isospectral for a≥2,and hence SUSYQM generates new exactly solvable periodic problems. This point is further developed in detail in this paper.We expand our discussion to the band edges and wave functions of a much richer class of periodic potentials given bycn2(x)V(x)=pm sn2(x)+qmThe associated Lam´e potentials given by eq.(5)can also be re-written in the alternative formV(x)=pm sn2(x)+qm sn2(x+K(m)),(6)since[1]√1−m/dn(x). sn(x+K)=cn(x)/dn(x),cn(x+K)=−It is clear from(6)that potentials(p,q)and(q,p)have the same energy spectra with wave functions shifted by K(m).Therefore,it is sufficient to restrict our attention to p≥q.Before actually solving the Schr¨o dinger equation for the associated Lam´e potential(5),let us make a few general comments.Throughout this paper,we have chosen units with¯h=1,and taken the particle mass in the Schr¨o dinger equation to be1/2.Note that in the limit when the elliptic modulus parameter m=0,the potential vanishes and one has a rigid rotator problem of period2K(0)=π,whose energy eigenvalues are at E=0,1,4,9,...with all the nonzero values being two-fold degenerate.On the other hand,the limit m→1is much trickier since K(m)tends to infinity and the periodic nature of the potential is obscured.The Schr¨o dinger equation forfinding the eigenstates for an arbitrary periodic potential is called Hill’s equation in the mathematics literature[3].A general property of Hill’s equation is the oscillation theorem which states that for a potential with period L,the band edge wave functions arranged in order of increasing energy E0≤E1≤E2≤E3≤E4≤E5≤E6≤...are of period L,2L,2L,L,L,2L,2L,....The corresponding number of wave function nodes in the interval L are0,1,1,2,2,3,3,...and the energy band gaps are given by∆1≡E2−E1,∆2≡E4−E3,∆3≡E6−E5,....We shall see that the expected m=0limit and the oscillation theorem are very useful in identifying if all band edge eigenstates have been properly determined or if some have been missed.The plan of the paper is as follows.In Sec.2,we briefly review the basic ideas of SUSYQM.A detailed discussion of Lam´e potentials and their supersymmetric partners is given in Sec.3.Solutions of the Schr¨o dinger equation for the associated Lam´e potentials are presented in Sec. 4.Many key new results are summarized in Table3.It is shown that the locus of quasi exactly solvable problems [10,11]in the(p,q)plane are parabolas about the line p=q.Our solutions are valid for any real choice of the parameters a,b(recall p=a(a+1),q=b(b+1)).Integer and half-integer values of a,b,including the very interesting special case a=b=integer,are treated in detail in Sec. 5.In most cases,the application of SUSYQM gives new solvable periodic potentials,many of which are illustrated in thefigures.Finally,Sec.6contains some concluding remarks.42.Supersymmetric Quantum Mechanics Formalism:The supersymmetric partner potentials V±(x)are defined in terms of the superpotential W(x)byV±(x)=W2(x)±W′(x).(7) The corresponding Hamiltonians H±can be factorized asH−=A+A,H+=AA+,(8) whereA=ddx+W(x),(9)so that the spectra of H±are nonnegative.It is also clear that on the full line,both H±cannot havezero energy modes since bothψ(±)given byψ(±)(x)=exp(± x W(y)dy),(10) cannot be simultaneously normalized.On the other hand,when the superpotential W(x)is periodic(W(x+L)=W(x))then the potentials V−(x)and V+(x)are isospectral-their spectra match completely,including the zero modes,and SUSY is unbroken providedLW(y)dy=0.(11)It is worth noting that in this case bothψ(±)belong to the Hilbert space.Thus in this case even though SUSY is unbroken,the Witten index is zero[5].The condition(11)is trivially satisfied in case W(x) is an odd function of x and throughout this paper we shall only consider superpotentials W whichare odd function of x.Further,using the known eigenfunctionsψ(−)n(x)of V−(x)one can immediatelywrite down the corresponding un-normalized eigenfunctionsψ(+)n(x)of V+(x).In particular,from eq.(10)it follows that the ground state of V+(x)is given byψ(+) 0(x)=1dx+W(x) ψ(−)n(x),(n≥1).(13) 5Thus by starting from an exactly solvable periodic potential V−(x),one gets another isospectral periodic potential V+(x).As emphasized previously,if V−(x)is not self-isospectral,then V+(x)is a new solvable periodic potential!m´e Potentials(p,0)and Their Supersymmetric Partners:The supersymmetric quantum mechanics formalism of the previous section will now be applied to the Lam´e potentials ma(a+1)sn2(x,m).Analytic solutions are known for integer values of a[2],and the supersymmetric partner potentials can be readily computed.Wefirst discuss the results for small integer values of a,and then present some eigenstate results for arbitrary integer values of a.m´e potentials with a=1,2,3:a=1:The a=1Lam´e potential V−=2m sn2(x)−m is known to be self-isospectral[5]since its SUSY partner satisfies V+(x)=V−(x−K(m)).Both V+(x)and V−(x)have one energy band ranging from energy0to energy1−m,with a continuum starting at energy1[2].Note that at m=0one has energy eigenvalues at0,1as expected for a rigid rotator and as m→1,one gets V−(x)→1−2sech2x, the band width1−m vanishes as expected,and one has an energy level at E=0.a=2:For the a=2case,the Lam´e potential(2)has2bound bands and a continuum band.The energies and wave functions of thefive band edges are well known[2,3].The lowest energy band ranges from2+2m−2δto1+m,the second energy band ranges from1+4m to4+m and the continuum starts at energy2+2m+2δ,whereδ=√dx logψ(−)(x)=6m sn(x)cn(x)dn(x)and hence the partner potential V+(x)for the potential V−(x)given in eq.(14)isV+(x)=−V−(x)+72m2sn2(x)cn2(x)dn2(x)dn(x)[2m+δ1+11−15m sn2(x)]1−m+4m2,(19) andV+(x)=−V−(x)+2m2sn2(x)cn2(x)[2m+δ1+1−5m sn2(x)]2.(20)Clearly,the potential V−(x)is not self-isospectral.In fact,V−(x)and V+(x)are distinctly different periodic potentials which have the same seven band edges corresponding to three bound bands and a continuum band[2].In Fig.2we have plotted the potentials V±(x)corresponding to a=3for several different values of the parameter m.The values of m are0.5,0.8,0.998.It is clear from thefigure that the potentials V+(x)and V−(x)have different shapes and are far from being ing eqs.(12)and(13)and the known eigenstates of V−(x),we can immediately compute all the7band7edges corresponding to the known3bound bands and a continuum band[2,3].For example,the ground stateψ(+)is given byψ(+) 0(x)=1dn(x)[1+2m+δ1−5m sn2(x)].(21)The wave functions for the remaining six states are similarly written down by using eq.(13).These are shown in Table2.The band edge energies for the a=3Lam´e potential(12,0)as a function of the elliptic modulus parameter m are plotted in Fig.3.Note that at m=0one has energy eigenvalues at 0,1,4,9as expected for a rigid rotator and as m→1,one gets V−(x)→9−12sech2x,the band widths vanish as expected,and one has three energy levels at E=0,5,8with a continuum above E=9.B.Results for general integer values of a:The extension to higher values of a is straightforward.It is possible to make several general comments about the form of the band edge wave functions for the partner potentials V+(x).This is most conveniently done by separately discussing the cases of even and odd values of a.a=even integer:For a even,say a=2N,it is known[2]that there are N+1solutions of the form F N(sn2x),and N solutions each of the three formssn x cn x F N−1(sn2x),sn x dn x F N−1(sn2x),cn x dn x F N−1(sn2x).Here F r denotes a polynomial of degree r in its argument.The ground stateψ−0(x)(which is the lower edge of the lowest band)is of the form F N(sn2x).It is easily checked using eq.(13)that the corresponding partner potential V+(x)has N solutions each of the four formsdn x G N(sn2x)ψ−0(x),cn x G N(sn2x)ψ−0(x),while the ground state is given byψ+0(x)=1/ψ−0(x).a=odd integer:For a odd,say a=2N+1,it is known[2]that the Lam´e potentials have N+1 solutions each of the three formssn x F N(sn2x),cn x F N(sn2x),dn x F N(sn2x)and N solutions of the formsn x cn x dn x F N−1(sn2x).8The ground stateψ−0(x)is of the form dn x F N(sn2x).We can then easily deduce that the corresponding partner potentials V+(x)will have N+1solutions each of the two formssn x G N+1(sn2x),ψ−0(x)and N solutions each of the two formsdn x G N+1(sn2x),ψ−0(x)while as usual,the ground state is given byψ+0(x)=1/ψ−0(x).In summary,for integral a,Lam´e potentials with a≥2are not self isospectral.They have distinct supersymmetric partner potentials even though both potentials have the same(2a+1)band edge eigenvalues.4.Associated Lam´e Potentials(p,q)and Their Supersymmetric Partners:In contrast to the Lam´e potentials discussed above,there seems to be no systematic treatment of associated Lam´e potentials in the literature.Therefore,we willfirst devote some time to discuss the properties of associated Lam´e potentials,show that they are quasi exactly solvable and then proceed to construct and study their isospectral supersymmetric partner potentials.A.Description of associated Lam´e potentials:As mentioned before,we will refer to the associated Lam´e potentials given by eq.(5)or the equivalent form eq.(6)as the(p,q)potential.The special cases p=0,as well as q=0,correspond to ordinary Lam´e potentials.In general,for any value of p and q,the associated Lam´e potentials have a period2K(m)since sn(x+2K)=−sn(x),cn(x+2K)=−cn(x),dn(x+2K)=dn(x).However,for the special case p=q,eq.(6)shows that the period is K(m).From a physical viewpoint, if one thinks of a Lam´e potential(p,0)as due to a one-dimensional regular array of atoms with spacing 2K(m),and“strength”p,then the associated Lam´e potential(p,q)results from two alternating types of atoms spaced by K(m)with“strengths”p and q respectively.If the two types of atoms are identical [which makes p=q],one expects a potential of period K(m).Extrema(defined for this discussion as either local or global maxima and minima)of associated Lam´e potentials are easily found by setting dV(x)/dx=0.This givessn(x)cn(x)[p dn4(x)−q(1−m)]=0.9Extrema occur when(i)sn(x)=0,that is x=0,±2K(m),±4K(m),...;(ii)cn(x)=0,that is x=±K(m),±3K(m),...;(iii)dn4(x)=(1−m)q/p.At the points specified by(i)and(ii),one always has extrema and V(x)has values pm and qm.In addition,since dn4(x)has a minimum value (1−m)2and a maximum value unity[1],condition(iii)yields additional extrema provided(1−m)2≤(1−m)q/p≤1.For givenfixed values of q and m,this condition has a solution provided p lies in the critical rangeq(1−m)≤p≤q/(1−m).Alternatively,for givenfixed values of p and q with p≥q,condition(iii)has a solution provided m is greater than the critical value1−q/p.The associated Lam´e potentials for q=2,m=0.5and several values of p are plotted in Fig.4(a). In the critical range of p values1≤p≤4,one expects additional extrema,and these are clearly seen in Fig.4(a).In general the period is2K(0.5)=3.708,but for p=q=2,the period K(0.5)is evident.Note that as p increases,any given extremum changes character.For example,at x=0,as p increases,one goes from a maximum to a local minimum to an absolute minimum.In Fig.4(b)we have plotted associated Lam´e potentials for p=4,q=2and several values of m.As expected from the above discussion,one always sees extrema at the points specified by conditions(i)and(ii),and additional extrema coming from condition(iii)are evident for m≥1/2.B.Solutions of the associated Lam´e equation-parabolas of solvability:The associated Lam´e equation is just the Schr¨o dinger equation for the potential in eq.(5).−d2ψdn2(x)−E]ψ=0.(22) On substitutingψ(x)=[dn(x)]−b y(x),(23) it is easily shown that y(x),satisfies the Hermite elliptic equation[3]sn(x)cn(x)y′′(x)+2bmOn further substitutingsn(x)=sin t,y(x)≡z(t),(26) one obtains Ince’s equation(1−m sin2t)z′′(t)+(2b−1)m sin t cos t z′(t)+[λ−(a+1−b)(a+b)m sin2t]z(t)=0,(27) which is a well known QES equation[3].In particular,on substitutingcos t=u,z(t)≡w(u)w(u)=∞ n=0u n R ncases,one can obtain the superpotential and hence the partner potential V+and enquire if V−is self-isospectral.For example,consider the case of p=a(a+1),q=a(a−1)in which case W is given byW≡−ψ′0(x)dn(x),(29)so that the corresponding partner potentials areV−=(a−1)amcn2(x)dn2(x)+m(a−1)a sn2(x)−ma2.(30) It is easily seen that these partner potentials satisfy eq.(3),are consequently self-isospectral and SUSY gives nothing new in this case.It is amusing to note that the superpotential W obtained here was in fact discussed in ref.[5](see their eq.(32)).Let us now consider the SUSY partner potential computed from the ground state for the p= a(a+1),q=(a−2)(a−3)case.It is given by(see Table3)ψ0(x)=[m(a−1)−1−δ1+m(2a−1)sn2(x)](dn(x))a−2,(31) whereδ1=dn(x)−2m(2a−1)sn(x)cn(x)dn(x)dn2(x)−2−m(a2−2a+2)+2δ1,(33)V+(x)=−V−(x)+2W2(x).(34) It is easily checked that these potentials are not self-isospectral since they do not satisfy the condition (3).Thus one has discovered a whole class of new elliptic periodic potentials V+(x)as given by eq.(34)for which three states are analytically known no matter what a is.In particular,the energy eigenfunctions for V+of these three states are easily obtained by using the corresponding energy eigenstates of V−as given in Table3and using eqs.(12)and(13).5.Associated Lam´e Potentials with Special Values of p and q:We shall now discuss associated Lam´e potentials(a(a+1),b(b+1)),where a and b are either both positive integers or half-integers.In most cases,we show that although several band edge energies are12exactly known from Table3,one usually does not know all the band edge energies,that is one has a quasi exactly solvable problem.However,in the special case of p=q(a=b=integer),we show that all the band edge eigenstates can be obtained and one has an exactly solvable periodic problem.A.a,b=integer,a=b:First,let us note that the Lam´e potentials(a(a+1),0)are in this category when a=integer and b=0.For example,when a=3,one has the(12,0)potential.We see from Fig.5that two parabolas of solvability pass through the point(12,0).From Table3it follows that3band edges of period2K(m)are obtained from q=(a−2)(a−3)and4band edges of period4K(m)are obtained from q=(a−3)(a−4).Altogether,arranging in order of increasing nodes,one has7band edges with periods2K,4K,4K,2K,2K,4K,4K with0,1,1,2,2,3,3nodes respectively.There are no missing states,and as discussed in Sec.3A,this gives three bound bands and a continuum band.As a second example with q=0,consider the(6,2)associated Lam´e potential,that is p=6,q=2. In this case,taking a=2,one can getfive band edges from Table3-one solution of period2K is obtained from q=a(a−1),while the remaining four solutions of period4K are obtained from q=(a−3)(a−4).The eigenvalues and eigenfunctions are given in Table4along with the number of nodes in one period2K.It is clear that there are two solutions of period2K with2nodes in the interval2K which have to be present but have not been obtained.This is also clear from the m=0 limit,since the energies from Table5are0,1,1,9,9and the states at4,4are missing.Thus,this is a QES problem.Fig.6illustrates the(6,2)associated Lam´e potential and its supersymmetric partner for three choices of m.The self-isospectral nature of the(6,2)potential is evident from Fig.6-it also follows from eqs.(30)with a=2.The band edge energies for the(6,2)associated Lam´e potential as a function of the elliptic modulus parameter m is shown in Fig.7.The two unobtained band edges of period2K will have energies E=4at m=0and E=3at m=1.Let us now discuss the general associated Lam´e potential(a(a+1),b(b+1)).Without any loss of generality let us assume that a>ing Table3,we obtain(a−b)states of period2K(4K)for q=[a−(a−b)][a−(a−b−1)]for(a−b)odd(even),and(a+b+1)states of period4K(2K)for q=[a−(a+b+1)][a−(a+b)]for(a−b)odd(even).It can be established that some states are missing by looking at the node structure as well as the m=0limit.Hence we again have a QES problem.13B.a =b =integer:Let us now discuss the special case of p =q =a (a +1),a =1,2,....In this case the associated Lam´e potential (5)has period K ,rather than 2K .It then follows from the oscillation theorem that with increasing energy,the band edges must have periods K,2K,2K,K,K,...and in the m =0limit the eigenvalues must go to E =0,4,16,36,...with all nonzero eigenvalues being doubly degenerate.It is easy to check from Table 5that one case for which we already have exact results is when p =q =2.In particular,consider the special case a =1,for which V −(x )of eq.(33)takes the formV −(x )=2m sn 2(x )+2m cn 2(x )1−m.(35)Using Table 5,we can calculate three energy eigenvalues and eigenfunctions of V −taking a =1in q =(a −2)(a −3).These are given in Table 5.Whereas the ground state is of period K ,the next two states in Table 5indeed have period 2K .Using a =1in eqs.(31)to (34),we find that the corresponding SUSY partner potential isV +(x )=2−m −2√1−mm 2sn 2(x )cn 2(x )1−m ]2.(36)Are the potentials V ±(x )self-isospectral?Using the relationssn(x +K (m )/2)=(1+√1−m sn(x )+cn(x )dn(x )1−m ,(37)cn(x +K (m )/2)=(1+√1−m )1/2cn(x )−sn(x )dn(x )1−m,(38)dn(x +K (m )/2)=(1−m )1/4 (1+√dn 2(x )+√the three eigenvalues in a closed simple form.In particular consider an ansatz of the formy=A+B sn2x+D sn4x.(40)On substituting this ansatz in eq.(24)it is easy to show that the energy eigenvalueλ(=E−m(a−4)2) must obey the cubic equationλ3+[28m−20−12am]λ2+[64−304m+160ma+32m2(a−2)(a−3)]λ−64m(2a−3)(2−2m+ma)=0.(41) The solution of this equation is in general quite lengthy but in the special case of a=2this cubic equation is easily solved yielding three eigenvalues in a compact form.On combining them with the two levels given in Table3,we obtain the eigenvalues and eigenfunctions of all thefive band edges for the case p=q=6.These are given in Table6.We have also verified that thesefive eigenstates in ascending order of energy indeed have periods K,2K,2K,K,K respectively and taht the energy eigenvalues have expected limits at m=0.In particular the associated Lam´e potential V−(x)isV−(x)=6m sn2(x)+6mcn2(x)dn2(x),δ8=dn(x)+2sn(x)cn(x)and eqs.(12)and(13).In Fig.9,we plot the band edge energies for the(6,6)potential as a function of the elliptic modulus parameter m.It is also clear from here that even the higher associated Lam´e potentials with p=q=12,20,... which have7,9,...band edges are also exactly solvable in principle and none of them will be self-isospectral,so that in each case one obtains a new exactly solvable periodic potential.In particular, for p=q=n(n+1)there will be(2n+1)band edges in both V±(x)whose energy eigenvalues can be obtained from Table3when q has the form[n−2n][n−(2n+1)].Out of the(2n+1)band edges in V−(x),(n+1)solutions(including the ground state)have the form F n(sn2x).On the other hand,as far as the(2n+1)solutions of the partner potential dn n xV+are concerned,there are n states each of the two formssn x cn xG n(sn2x),dn2n−1xψ−0(x)while the ground state(i.e.the lower edge of the lowest band)is given byψ+0(x)=1/ψ−0(x).C.a,b=half-integer:Let us now specialize to the case when both a,b are half integral with a>b.As an illustration, let usfirst consider the case of a=3/2,b=1/2so that p=15/4,q=3/4.In this case,the oscillation theorem requires band edges with periods2K,4K,4K,2K,2K,ing Table3and Fig.5,we see one gets three eigenstates when q=(a−2)(a−3)with a=3/2,all with period2K.The ground.To understand this state is at E0=9m4degeneracy better,let us go along the parabola of solvability P2given by q=(a−2)(a−3).The band gap is given by∆2≡|−2+m+21−m+m2−2+m for a=2and √∆2=2states have the same period2K(4K)as the n−k states when n−k is odd(even).It turns out that the n−k solutions are in fact common in both and so we only obtain the energy of the n+k+2band edges and all of them have the same period2K(4K)depending on if n−k is odd(even)so that it is only a QES problem and not an exactly solvable problem as one is unable to obtain a single eigenstate with period4K(2K)in case n−k is odd(even).We would like to point out some of the pecularities of the spectrum in these cases.For example, in case(p,q)=(35/4,3/4),(63/4,3/4),(99/4,3/4)...then onefinds that3,4,5,...QES energy levels of period4K,2K,4K,...respectively are analytically known of which the one at the highest energy is doubly degenerate.As an illustration,in Table7we have given the4QES energy eigenstates all of period2K for the(63/4,3/4)potential.The interesting point about this case is that the partner potentials V±(x)are not self-isospectral and hence one has discovered a new QES potential where4 band edges of period2K and the corresponding eigenfunctions are explicitly known.Of these,the one at E=16+m2sn(x)cn(x)[12m sn2(x)−2−5m−√References[1]For the properties of Jacobi elliptic functions,see,for example,I.S.Gradshteyn and I.M.Ryzhik,Table of Integrals,Series and Products(Academic Press,1980).The modulus parameter m is often called k2in the mathematics literature.The related complementary quantity(1−m) is often called k′2.[2]F.M.Arscott,Periodic Differential Equations(Pergamon,Oxford,1981);E.T.Whittaker andG.N.Watson,A Course of Modern Analysis(Cambridge Univ.Press,Cambridge,1980).[3]W.Magnus and S.Winkler,Hill’s Equation(Wiley,New York,1966).[4]See,for example,F.Cooper,A.Khare and U.P.Sukhatme,Phys.Rep.251,267(1995).[5]G.Dunne and J.Feinberg,Phys.Rev.D57,1271(1998).[6]G.Dunne and J.Mannix,Phys.Lett.B428,115(1998).[7]H.Braden and A.Macfarlane,J.Phys.A18,3151(1995).[8]Y.Brihaye and M.Godart,J.Math.Phys.34,5283(1993);Y.Brihaye and S.Braibant,J.Math.Phys.34,2107(1993).[9]U.Sukhatme and A.Khare,E-print quant-ph/9902072(1999).[10]A.Turbiner,Comm.Math.Phys.118,467(1988).[11]hveridze,Quasi Exactly Solvable Models in Quantum Mechanics(Institute of Physics,Bristol,1993).[12]Some isolated solutions of the Schr¨o dinger equation for Lam´e potentials with half-integer valuesof a are also known[2],but they have certain peculiar features which require further study.18Table CaptionsTable1:The eigenvalues and eigenfunctions for the5band edges corresponding to the a=2 Lam´e potential V−which gives(p,q)=(6,0)and its SUSY partner V+.Here B≡1+m+δand √δ≡1−m+4m2;δ2≡√4−7m+4m2.The potentials V±have period L=2K(m)and their analytic forms are given by eqs.(19)and(20)respectively.The periods of various eigenfunctions and the number of nodes in the interval L are tabulated.Table3:Eigenvalues and eigenfunctions for various associated Lam´e potentials(p,q)with p= a(a+1)and q=(a−n+1)(a−n)for n=1,2,3,....The periods of various eigenfunctions and the number of nodes in the interval2K(m)are tabulated.Hereδ4≡4−7m+2ma+m2(a−2)2;δ6≡ 9−9m+m2(a−2)2.Table4:Thefive eigenvalues and eigenfunctions for the self-isospectral associated Lam´e potential corresponding to a=2,b=1which gives(p,q)=(6,2).The potential is V−(x)=6m sn2(x)+ 2m cn2(x)16−16m+m2.The number of nodes in one period K(m)of the potential is tabulated.Table7:Energy eigenvalues and eigenfunctions for the associated Lam´e potential corresponding19。

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