A short note about Morozov's formula
数学中推论的英文简写
数学中推论的英文简写In mathematics, a theorem is a statement that has been proven to be true based on logical deduction from previously established propositions or axioms. A theorem is a fundamental building block of mathematical knowledge that helps to establish the validity of mathematical arguments and provides a foundation for further study and exploration.One common way to denote a theorem in mathematical writing is by using a numbering system. Theorems are typically numbered consecutively, and the numbering format may vary depending on the style guide or convention being followed. For example, a theorem may be denoted as Theorem 1.1 if it is the first theorem in Section 1 of a document. Subsequent theorems in the same section may be labeled Theorem 1.2, Theorem 1.3, and so on.In addition to numbering, theorems are often given names or titles to provide a concise description of the statement or to honor the mathematician who first proved it. For example, we have the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. Other famous theorems include the Bolzano-Weierstrass theorem, the Fundamental Theorem of Calculus, and Fermat's Last Theorem.When referencing a theorem in mathematical writing, it is common to use their short names or numbers along with appropriate abbreviations. The abbreviation "Thm." is commonly used to refer to a theorem, followed by either the theorem number or name. For example, one might write "According to Thm. 1.1" or "By thePythagorean theorem" to refer to a specific theorem in a proof. This helps to clearly identify the source of the statement and allows readers to easily locate the theorem if they wish to verify it.In addition to theorems, mathematics also relies heavily on lemmas, propositions, corollaries, and conjectures. These are all types of statements that play different roles in mathematical reasoning. Lemmas are auxiliary results that help in proving larger theorems. They are often used as intermediate steps or specialized cases of more general theorems. Propositions are statements that are not central to the main argument but provide additional information or support to theorems or lemmas. Corollaries are direct consequences or immediate extensions of theorems. Conjectures, on the other hand, are statements that are believed to be true but have not yet been proven. They serve as research problems and motivations for further investigation.When addressing these different types of statements, similar abbreviations can be used. For instance, "Lem." can stand for lemma, "Prop." for proposition, "Cor." for corollary, and "Conj." for conjecture. These abbreviations can help to efficiently convey information and maintain clarity in mathematical writing. Overall, the use of abbreviations is a common practice in mathematical writing and provides a concise and standardized way to refer to theorems and other related statements. It allows for efficient communication and understanding of mathematical arguments, while also respecting the historical and foundational contributions of mathematicians.。
A Short Proof of Jacobi's Formula for the Number of Representations of an Integer as a Sum
a r X i v :m a t h /9206203v 1 [m a t h .C O ] 3 J u n 1992A SHORT PROOF OF JACOBI’S FORMULA FOR THE NUMBER OF REPRE-SENTATIONS OF AN INTEGER AS A SUM OF FOUR SQUARES George E.Andrews 1,Shalosh B.Ekhad 2and Doron Zeilberger 2(Appeard in the Amer.Math.Monthly 100(1993),273-276)Diophantus probably knew,and Lagrange[L]proved,that every positive integer can be written as a sum of four perfect squares.Jacobi[J]proved the stronger result that the number of ways in which a positive integer can be so written 3equals 8times the sum of its divisors that are not multiples of 4.Here we give a short new proof that only uses high school algebra,and is completely from scratch.All infinite series and products that appear are to be taken in the entirely elementary sense of formal power series.The problem of representing integers as sums of squares has drawn the attention of many great mathematicians,and we encourage the reader to look up Grosswald’s[G]erudite masterpiece on this subject.The crucial part of our proof is played by two simple identities,that we state as one Lemma.Lemma :LetH n =H n (q )=1+q1−q 2...1+q n(1+q k )2H 2n H n +k H n −k =1,(a )n k =02(−q n +1)k H n=nk =−n(−q )k 2.(b )Proof:Let L 1(n )and L 2(n )be the left sides of (a)and (b)respectively,and let F 1(n,k ),andF 2(n,k )be the respective summands.Since both (a)and (b)obviously hold for n =0,it sufficesto prove that for every n ≥0,L 1(n +1)−L 1(n )=0,and L 2(n +1)−L 2(n )=2(−q )(n +1)2.To this end,we constructG 1(n,k ):=q n −k +1(1+q 2n +2)(1+q k )2(1+q n +k +1)1+q n +1F 2(n,k ),with the motive thatF 1(n +1,k )−F 1(n,k )=G 1(n,k )−G 1(n,k −1),F 2(n +1,k )−F 2(n,k )=G 2(n,k )−G 2(n,k −1),(1)which immediately imply them,by telescoping,upon summing from k=−n−1to k=n+1, and from k=0to k=n+1respectively.The two identities of(1)are purely routine,since dividing through by F1(n,k)and F2(n,k)respectively,lead to routinely-verifiable high-school-algebra identities.Dividing both sides of(a)by H4n and letting n→∞in(a)and(b)gives1+8∞k=1(−q)k(1+(−q)k)2.(2)The coefficient of a typical term q n on the left of(2)is the number of ways of writing n as a sum of four squares.It remains to show that the coefficient of q n in the sum on the right of(2)equals the sum of the divisors of n that are not multiples of4.Using the power-series expansion z/(1+z)2= ∞r=1(−1)(r+1)rz r,with z=(−q)k,and col-lecting like powers,the sum on the right side may be rewritten∞k=1∞ r=1(−1)(k+1)(r+1)r q kr=∞ n=1q n[ r|n(−1)(r+1)(n/r+1)r].The coefficient of q n above is a weighed sum of divisors r of n,where the coefficient of r is−1= +1−2if both r and n/r are even and+1otherwise,so the coefficient of q n isr|n r− r|n r,n/r even2r= d|n d− d|n4|d d= d|n4|d d.Thefinitary identities(a)and(b)combine to yield a singlefinitary identitynk=02(−q n+1)k(1+q k)2H n+kH n=(nk=−n(−q)k2)4,(3)which also immediately implies Jacobi’s theorem,by taking it“mod q n”for any desired n.Identity (3)makes it transparent that our proof only uses the potential infinity,not the ultimate one.The identities of the Lemma are examples of q-binomial coefficient identities,a.k.a terminat-ing basic hypergeometric series identities.The proof of such identities is now completely rou-tine[WZ][Z].The proof of the Lemma given here used the algorithm of[Z].Further applications of basic hypergeometric series to number theory can be found in[A1].An excellent modern reference to basic hypergeometric series is[GR].We conclude with some comments addressed mainly to the cognoscenti.Identities(a)and(b)are special cases of classical identities:(a)is a special case of Jackson’s theorem[GR,p.35,eq.(2.6.2)],and(b)is a special case of Watson’s q-analog of Whipple’s theorem([GR,p.35,eq.(2.5.1)],see also[A2,p.118,eq.(4.3)].)The discovery of (b)was motivated by[S1]and[S2].We see fairly clearly how to do the2-square theorem(a different instance of Jackson’s theorem replaces(a)); however the theorems for6and8squares apparently require(using this approach)some instance of the6Ψ6 summation theorem[GR,p.128,(5.3.1)](see[A1,pp.461-465]for details).Since we do not know afinitary analog of the6Ψ6summation,the question of a similar result for6and8squares is of interest. Acknowledgement:The referee made several helpful comments.References[A1]G.E.Andrews,Applications of basic hypergeometric functions,SIAM Rev.16(1974),441-484. [A2]G.E.Andrews,Thefifth and seventh order mock theta functions,Trans.Amer.Math.Soc., 293(1986),113-134.[GR]G.Gasper and M.Rahman,”Basic hypergeometric series”,Cambridge University Press,1990.[G]E.Grosswald,”Representations of integers as sums of squares”,Springer,New York,1985. [J]C.G.J.Jacobi,Note sur la d´e composition d’un nombre donn´e en quatre carre´e s,J.Reine Angew. Math.3(1828),191.Werke,vol.I,247.[L]grange,Nouveau M´e m.Acad.Roy.Sci.Berlin(1772),123-133;Oevres,vol.3,189-201. [S1]D.Shanks,A short proof of an identity of Euler,Proc.Amer.Math.Soc.,2(1951),747-749. [S2]D.Shanks,Two theorems of Gauss,Pacific.J.Math.,8(1958),609-612.[WZ]H.S.Wilf and D.Zeilberger,An algorithmic proof theory for multisum/integral(ordinary and ”q”)hypergeometric identities,Invent.Math.108(1992),575-633.[Z]D.Zeilberger,The method of creative telescoping for q-series,in preparation.Feb.1992;Revised:Aug.1992.。
《美丽心灵》完整中英文对照剧本
美丽心灵"1947年9月普林斯顿大学"Mathematicians won the war.是数学家赢得了世界大战Mathematicians broke the Japanese codes是数学家破解了日本密码and built the A bomb.也是数学家发明了原子弹Mathematicians... like you.就像你我一样的数学家The stated goal of the Soviets is global communism.苏联的终极目标是赤化全世界In medicine or economics,包括医学经济in technology or space,科技太空battle lines are being drawn.战争的界线已经划出To triumph, we need results.为了争取胜利我们需要结果Publishable, applicable results.需要可以发表和应用的结果Now who among you will be the next Morse?谁是未来的摩尔斯?The next Einstein?谁又是下一个爱因斯坦?Who among you will be the vanguard你们当中谁将是of democracy, freedom, and discovery?民主自由和发明的先锋?Today, we bequeath America's future今天我们将美国的未来into your able hands.交到你们的手中Welcome to Princeton, gentlemen.欢迎加入普林斯顿的行列It's not enough Hansen won the Carnegie Scholarship.汉森赢得卡内基奖学金居然还不满足No, he has to have it all for himself.他想要通吃It's the first time the Carnegie prize has been split. Hansen's all bent.这是第一次由两人共享这项奖学金汉森当然不爽Rumor is he's got his sights set on Wheeler Lab,听说他一心想进入惠勒研究室the new military think tank at M.I.T.就是麻省理工学院新的军事智囊团They're only taking one this year.今年他们只收一个人Hansen's used to being picked first.而汉森认为他是第一人选Oh, yeah, he's wasted on math.研究数学实在浪费他的才能He should be running for president.他应该去选总统There could be a mathematical explanation你的领带这样糟for how bad your tie is.一定有数学上的合理解释Thank you.谢谢你Neilson, symbol cryptography.我是尼尔森专攻符号密码学Neils here broke a Jap code.他破解过日本密码Helped rid the world of fascism.也曾帮助世界摆脱法西斯主义At least that's what he tells the girls, Neils?他都是这样对女孩子说的The name's Bender. Atomic physics.我叫宾达专攻原子物理学And you are? Am I late?你是哪位? 我迟到了吗?Yes. Yes, Mr. Sol.是的苏先生Oh, good. Hi.好! 你好Sol. Richard Sol.我是理查·苏The burden of genius.这就是天才的负担There he is. So many supplicants, and so little time. 他来了宴会这么多时间真不够用Mr. Sol. How are you, sir?苏先生您好您好Bender. Nice to see you.宾达很高兴看到你Congratulations, Mr. Hansen.汉森先生恭喜你Thank you.谢谢I'll take another.再给我一杯Excuse me?什么?A thousand pardons. I simply assumed you were the waiter. 真抱歉我以为你是服务生Play nice, Hansen.汉森给他留点面子Nice is not Hansen's strong suit.他才不会替人留面子Honest mistake.我真的是无心之过Well, Martin Hansen.想必你就是马丁·汉森It is Martin, isn't it?你是马丁吧?Why, yes, John, it is.没错我就是I imagine you're getting quite used to miscalculation.我猜你对误估已习以为常I've read your pre prints...我读过你的初稿...Both of them.两份都读过The one on Nazi ciphers,一份是纳粹党的密码文件and the other one on non linear equations,另一份是非线性方程式and I am supremely confident...我确信...that there is not a single seminal不管是哪一份or innovative idea in either one of them.都不具有发展和创新性Enjoy your punch.好好享受你的水果酒Gentlemen, meet John Nash,各位这位就是约翰·纳什the mysterious West Virginia genius.西弗吉尼亚州的神秘天才The other winner of the distinguished Carnegie Scholarship. 也是另一位著名的卡内基奖学金得主Bender. Of course.宾达当然了Oh, Christ.真要命!The prodigal roommate arrives.我就是你的浪子室友Roommate?室友?Oh, God, no.天呀饶了我吧!Did you know that having a hangover is...你知道所谓的宿醉...is not having enough water in your body就是体内没有足够的水份to run your Krebs cycles?去执行克雷布斯循环Which is exactly what happens to you这和因渴至死的情况when you're dying of thirst.完全一样So, dying of thirst...如此说来因渴至死的感觉...would probably feel pretty much like the hangover...可能和宿醉的感觉一样...that finally bloody kills you.反正最后仍逃不过一死John Nash?你是约翰·纳什吗?Hello.哈罗Charles Herman.我叫查尔斯·赫曼Pleased to meet you.很高兴认识你Well, it's official.正式宣布I'm almost human again.我已经恢复正常了Officer, I saw the driver who hit me.警官我看见撞到我的驾驶人His name was Johnny Walker.他的名字是"约翰走路"Well, I got in last night昨晚我去参加in time for...英文系办的...English department cocktails.鸡尾酒会Cock was mine,我当然是那只雄纠纠的鸡the tail belonged to a particularly lovely young thing至于那个美丽可爱的女孩with a passion for D.H...则是情欲大师劳伦斯的崇拜者D.H. Lawrence. You're not easily distracted, are you?你居然不怕被打扰I'm here to work.我是来工作的Hmmm, are you? Right.哎呀那当然I see. Crikey!我明白了Is my roommate a dick?你真是个无趣的家伙Listen. If we can't break the ice...既然打不破我们之间的冰山How about we drown it?干脆大醉一场如何?So what's your story?告诉我你的故事You the poor kid that never got to go to Exeter or Andover? 一个从没出过家门的穷小孩Despite my privileged upbringing,虽然我从小接受高等教育I'm actually quite well balanced.但身心却很平衡I have a chip on both shoulders.我的弱点是人际关系Maybe you're just better人与事比较起来with the old integers than you are with people.你恐怕比较会应付事My first grade teacher, she told me我的小学老师说过that I was born with two helpings of brain,我有两个脑袋but only half a helping of heart.却只有半颗心Really? Yeah.真的? 是呀Wow! She sounds lovely!哇她似乎挺可爱的The truth is that I...其实...I don't like people much.我并不喜欢人们And they don't much like me.他们也不喜欢我But why,怎么可能?with all your obvious wit and charm?你既风趣又有魅力Seriously, John.说真的Mathematics...讲到数学嘛...Mathematics is never going to lead you to a higher truth. 它永远不能领导你找到更高的真理And you know why?想知道原因吗?Because it's boring. It's really boring.因为它太枯燥无味了You know half these schoolboys are already published? 你知道半数以上的学生已经发表了他们的论文I cannot waste time with these classes...我不能在课堂上...and these books.和书本上再浪费时间Memorizing the weaker assumptions of lesser mortals! 去记住那些毫无说服力的假设吧!I need to look through...我必须要为博弈论to the governing dynamics.有所突破Find a truly original idea.找出它的原创理论来That's the only way I'll ever distinguish myself.那是唯一能让我出头的方法It's the only way that I'll ever...是唯一使我能成为...Matter.举足轻重的人物?Yes.是的All right, who's next?下一个是谁?No, I've played enough "go" for one day, thank you.我下够了今天不玩了Come on. I... I hate this game.来嘛我不喜欢围棋Cowards, all of you!你们都是胆小鬼None of you rise to meet my challenge?没人敢向我挑战Come on, Bender.宾达别这样嘛Whoever wins,so does his laundry all semester.谁赢阿苏就替他洗一学期的衣服Does that seem unfair to anyone else?你们不觉得这很不公平吗?Not at all.不觉得Look at him. Nash!你们看他纳什Taking a reverse constitutional?在研究反组织理论吗?I'm hoping to extract an algorithm我想找出一个演算法to define their movement.好替它们的活动下定义Psycho.神经病Hey, Nash, I thought you dropped out.你一直没去上课You ever going to go to class or...我以为你退学了...Classes will dull your mind.上课会使你的脑筋迟钝Destroy the potential for authentic creativity. 也会破坏创造的潜能Oh, oh, I didn't know that.这我倒不知道Nash is going to stun us all with his genius. 有一天纳什的才华会惊倒大家Which is another way of saying但这也可能是he doesn't have the nerve to compete.他不敢和我比赛的藉口You scared?怕了吗?Terrified.我被你吓呆了Mortified. Petrified.惊呆了窘呆了Stupefied... by you.而且是目瞪口呆No starch. Pressed and folded.咱们就一战定英雄Let me ask you something, John.我能不能问你一件事Be my guest, Martin.尽管问Bender and Sol here correctly completed宾达和阿苏已经完全无误的Allen's proof of Peyrot's Conjecture.证明出艾伦的推论Adequate work...尚差强人意啦...without innovation.可惜没有创新Oh. I'm flattered. You flattered?谢谢夸奖你呢? 我很感动你呢？Flattered.感动And I've got two weapons briefs而我有两样武器纲要under security review by the D.O.D.正由国防部审查中Derivative drivel.真叫我垂涎不已But Nash achievements: zero.而纳什的成就却是零I'm a patient man, Martin.我有耐心Is there an actual question coming?你到底想问什么?What if you never come up with your original idea? 假若你永远找不出原创理论How will it feel when I'm chosen for Wheeler...最后惠勒研究室选择了我...and you're not?你会有什么感觉?What if you lose?输是什么感觉?You should not have won.你不应该赢I had the first move, my play was perfect.是我先走的每一步都很完美The hubris of the defeated.这就是所谓的骄兵必败The game is flawed.这个游戏有缺点Gentlemen, the great John Nash.这就是伟大的约翰·纳什You've been in here for two days.你呆在这里已经两天了You know Hansen's just published another paper?汉森又发表了一篇报告I can't even find a topic for my doctorate.而我连博士论文的主题都没有Well, on the bright side, you've invented window art. 可是你却发明了橱窗艺术This is a group playing touch football.这是美式橄榄球队的比赛This is a cluster of pigeons fighting over bread crumbs. 这是一群鸽子在争夺面包屑And this here is a woman而这是一个女人who is chasing a man who stole her purse.在追逐偷她皮包的男人John, you watched a mugging.你居然目击抢劫案That's weird.真不可思议In competitive behavior someone always loses.在竞争的状态下总有人会输Well, my niece knows that, John,这点连我的小侄女都懂and she's about this high.而她只有这么点大See, if I could derive an equilibrium如果我能找出一种均衡where prevalence is a non singular event,在优势可逆的情况下where nobody loses,就会出现双赢的局面can you imagine the effect that would have你想这在有冲突的情况下会有多大的影响on conflict scenarios, and arms negotiations...像武器协商...When did you last eat?你什么时候吃过东西?When did you last eat?你什么时候吃的饭?Currency exchange?像货币交换You know, food.吃东西You have no respect for cognitive reverie, 你一点也不尊重幻想you know that?你知道吗?Yes.没错But pizza...但是说到比萨饼...Now, pizza I have enormous respect for. 我对它可是满腔的尊重And of course beer.当然还有啤酒I have respect for beer.我敬重啤酒I have respect for beer!我非常敬重啤酒Good evening, Neils. Hey, Nash.晚安尼尔森嗨纳什Who's winning? You or you?谁赢这个你还是另一个你Evening, Nash. Hey, guys.晚安纳什你们好Hey, Nash.纳什你好He's looking at you.他在看你Are you sure?你确定?Hey, Nash.嗨纳什Neils is trying to get your attention.尼尔森要你过去You're joking. Oh, no.你一定在开玩笑呢不是吧Go with God.勇敢的去Come back a man.一定要失身哟Fortune favors the brave.幸运与勇者同在Bombs away.炸他个屁滚尿流Gentlemen, might I remind you that my odds of success 各位让我提醒你们dramatically improve with each attempt?每试一次就会增加我一分胜算This is going to be classic.保证绝对精彩Maybe you want to buy me a drink.你是不是要请我喝杯酒I don't exactly know what I'm required to say为了想让你和我上床in order for you to have intercourse with me,我实在不知道该说些什么but could we assume that I said all that?你干脆就当我把该说的都说过了Essentially we're talking about fluid exchange, right?反正我们所谈的是有关液体交流的事So, could we just go straight to the sex?所以何不立刻切入性交的主题Oh, that was sweet.你可真鲜Have a nice night, asshole!再见啦你这个混蛋Ladies, wait!小姐们等一下I... I especially liked the bit about fluid exchange.我最喜欢液体交流那一段It was really charming.真的很棒Walk with me, John.约翰陪我走一走I've been meaning to talk with you.我一直想和你谈谈The faculty is completing mid year reviews.教授们已经完成了期中的审核We're deciding which placement applications to support. 我们正在决定如何分配学生...Wheeler, sir. That would be my first choice.教授到惠勒研究室是我的第一选择And actually, I don't really have a second choice, sir.也是唯一的选择我知道John, your fellows have attended classes.其他的人都按时上课They've written papers. They've published.不但写论文也已经发表了I'm still searching, sir, for my...教授我仍在研究我的...Your original idea, I know.你的原创理论?Governing dynamics, sir.是博弈论It's very clever, John, but I'm afraid很好可惜it's just not nearly good enough.还是不够好May I? Thank you.您的外套? 谢谢I've been working on manifold embedding.我一直在研究 "符合嵌入"My bargaining stratagems are starting to show some promise. 这个理论已经开始见到成效If you could just arrange another meeting,希望你能再次安排if you'd be kind enough, with Professor Einstein...我和爱因斯坦教授会面I've repeatedly asked you for that. Now, John.我曾再三请你替我安排约翰听我说I'd be able to show him my revisions on his...我会让他看修订后的...John.约翰Do you see what they're doing in there?你知道他们在做什么吗?Congratulations. Thank you so much.恭喜非常感谢Congratulations, Professor Max.教授恭喜你It's the pens.那支笔代表着Reserved for a member of the department会员们that makes the achievement of a lifetime.肯定了他的终生成就Now, what do you see, John?告诉我你看到了什么?Recognition.表扬Well done, Professor, well done.做得好教授谢谢Well, try seeing accomplishment.你看到贯彻了吗?Is there a difference?难道还有分别吗?John,约翰you haven't focused.你仍没有集中你的注意力I'm sorry, but up to this point,对不起到目前为止your record doesn't warrant any placement at all. 根据你的记录我不能保证你一定会有工作Good day.再见And my compliments to you, sir.我向你致意多谢Thank you so much.非常感谢I can't see it.怎么就是看不出来呢Jesus Christ, John.约翰你别这样I can't fail.我不能失败This is all I am.这是我的全部呀Come on, let's go out.咱们出去走走I got to get something done.我必须做些成绩出来I can't keep staring into space.光瞪着天空不是办法John, enough!够了我还是向他们屈服吧Got to face the wall,面对墙壁Fine, you want to do some damage? Fine...你想来硬的没问题But don't mess around. Do their classes.别浪费时间了上他们的课...Come on! Go on, bust your head! Kill yourself. 干脆敲破你的脑袋自杀算了John, do it. Don't mess around.好! 去呀别浪费时间了Bust your head!去撞你的脑袋呀Go on,bust that worthless head wide open.去啊把你那个没用的脑袋撞开Goddamn it, Charles!查尔斯你这个混蛋What the hell is your problem?!你到底有什么毛病It's not my problem.我没有毛病And it's not your problem.你也没有问题It's their problem.全是他们的问题Your answer isn't face the wall.面对墙壁不可能找到答案It's out there... where you've been working. 答案在你工作的地方That was heavy.那张桌子可真重That Isaac Newton fellow was right.看来牛顿的理论还挺正确的He was onto something. Clever boy.他可真厉害聪明的家伙Don't worry, that's mine.别担心是我的东西I'll come and get it in a minute.等会儿就去收拾Oh,God.我的天呀Incoming, gentlemen.美女来了Deep breaths.深呼吸Nash, you might want to stop shuffling纳什你最好your papers for five seconds.先休息几秒钟I will not buy you gentlemen beer.我不请你们喝啤酒Oh, we're not here for beer, my friend.我们不是来喝啤酒的Does anyone else feel she should be moving in slow motion? 你们同不同意她最好以慢镜头移动Will she want a large wedding, you think?她会要求一个盛大的婚礼吗?Shall we say swords, gentlemen? Pistols at dawn?要用剑决斗还是在黄昏时用手枪决斗?Have you remembered nothing?你们怎么都忘了Recall the lessons of Adam Smith,还记得现代经济学之父the father of modern economics.亚当·史密斯的理论"In competition...在竞争中individual ambition serves the common good."个人的野心往往会促进公共利益Exactly. Every man for himself, gentlemen.没错每个人都为自己着想And those who strike out are stuck with her friends.被三阵出局的人只能去约她的朋友I'm not gonna strike out.我绝不会被三阵的You can lead a blonde to water,你可以把美女带到水边but you can't make her drink.但你不能逼她喝水I don't think he said that...他好像没说过这句话Nobody move... She's looking over here.别动她在看我们了She's looking at Nash.她在看纳什Oh, God. He may have the upper hand now, 好吧可能他现在占有优势but wait until he opens his mouth.但等他一开口保证完蛋Remember the last time?还记得上次吗?Oh, yes, that was one for the history books. 对那一次可真鲜Adam Smith needs revision.亚当·史密斯需要修订他的理论了What are you talking about?你在说些什么?If we all go for the blonde...如果我们全去追那个美人...we block each other.结果一定全军覆没Not a single one of us is gonna get her.谁也得不到她So then we go for her friends,然后我们去找她的女朋友but they will all give us the cold shoulder 她们肯定会浇我们冷水because nobody likes to be second choice. 因为没人愿意屈居第二Well, what if no one goes for the blonde? 但是若没人去追那个金发美女We don't get in each other's way,那我们之间既互不侵犯and we don't insult the other girls.也没有羞辱到其他女孩That's the only way we win.只有这样大家才能赢That's the only way we all get laid.也只有这样才都有上床的机会Adam Smith said亚当·史密斯曾说过the best result comes...最好的结果...from everyone in the group doing是要能做到what's best for himself, right?分工和专业对不对?That's what he said, right?那是他说的对不对?Incomplete. Incomplete, okay?他的理论不完整Because the best result will come...因为最好的结果是...from everyone in the group团体中的每一个人doing what's best for himself and the group.都做对本身和团体最有利的事Nash, if this is some way for you to get the blonde on your own, 你想拿这套歪理去独占美人you can go to hell.门儿都没有Governing dynamics,gentlemen.各位这就是所谓的博弈论Governing dynamics. Adam Smith...博弈论亚当·史密斯...was wrong.他错了Oh, here we go. Careful, careful.又来啦小心点Thank you.谢谢你"C" of "S" equals "C" of "T".C(S)等于C(T)You do realize this flies in the face你知道这会推翻一百五十年来of a 150 years of economic theory?牢不可破的经济理论Yes, I do, sir.我知道That's rather presumptuous, don't you think?你不觉得太放肆了吗?It is, sir.是有一点Well, Mr. Nash,纳什先生with a breakthrough of this magnitude,由于你做出如此重大的突破I'm confident you will get any placement you like. 我相信你可以去任何你想去的地方Wheeler Labs,惠勒研究室they'll ask you to recommend two team members. 会请你介绍两位组员Yes!太棒了Stills and Frank are excellent choices.史提和法兰克应该很合适Sol and Bender, sir.我要阿苏和宾达Sol and Bender are extraordinary mathematicians. 阿苏和宾达都是非常优秀的数学家Has it occurred to you that Sol and Bender但你可曾想过might have plans of their own?他们俩可能另有打算Baby! Wheeler, we made it!我们终于能进惠勒研究室了Cheers, cheers, cheers!干杯干To...祝...Okay, awkward moment, gentlemen.各位尴尬的时刻来了Governing dynamics.博弈论Congratulations, John. Thanks.约翰恭喜你了谢谢Toast! To Wheeler Labs!干杯为惠勒研究室干杯To Wheeler!为惠勒研究室干杯"1953年五角大楼""五年后"General, the analyst from Wheeler Lab is here.将军惠勒研究室的分析家到了Dr. Nash, your coat?纳什博士你的外套Thank you, sir.谢谢你Doctor.博士General, this is Wheeler team leader Dr. John Nash.将军这位就是惠勒研究室的领导人约翰·纳什博士Glad you could come, Doctor.很高兴你能来Hello.你好Right this way.这边走We've been intercepting radio transmissions from Moscow. 我们截获由莫斯科发出的无线电报The computer can't detect a pattern,电脑无法查出它的模式but I'm sure it's code.但我认为绝对是密码Why is that, General?何以见得Ever just know something, Dr. Nash?就是有那种感觉你会吗?Constantly.经常会We've developed several ciphers.我们研发了几个密码索引If you'd like to review our preliminary data...要不要看看我们的初步资料Doctor?纳什博士?6 7 3 76 7 3 70 3 6...0 3 6...8 4 9 4.8 4 9 49 1 4 0 3 4.9 1 4 0 3 4I need a map.我需要一份地图40 6 13 0 8 67 46 9 0,40 6 13 0 8 67 46 9 0Starkey Corners, Maine.缅因州的斯达基角48 03 01,48 03 0191 26 35.91 26 35Prairie Portage, Minnesota.明尼苏达州的波吉特草原These are latitudes and longitudes.这些都是经纬线There are a least 10 others.至少还有十个They appear to be routing orders across the border into the U.S. 这似乎是进入美国边境的路线顺序Extraordinary.太惊人了Gentlemen,各位we need to move on this.我们需要做更进一步的研究Who's big brother?那位老大是谁?You've done your country a great service, son.你为国家贡献良多Captain! Yes, sir.上尉是的长官Accompany Dr. Nash.替纳什博士带路What are the Russians moving, general?将军俄国人有什么动向?Captain Rogers will escort you罗杰上尉会陪你to the unrestricted area, Doctor.到不受限区Thank you.谢谢Dr. Nash, follow me, please.纳什博士请跟我来It's Dr. Nash.是纳什博士All right.好的"麻省理工学院""惠勒国防研究室"Thank you, sir. Home run at the Pentagon?谢谢你先生在五角大楼打出全垒打了吗? Have they actually taken the word "classified"他们有没有把"机密"两个字out of the dictionary?从字典里删掉Oh, hi. The air conditioning broke again.嗨冷气机又坏了How am I supposed to be in here saving the world 我都快融化了if I'm melting?怎么去整救世界?Our hearts go out to you.约翰我们真替你叫屈You know, two trips to the Pentagon in four years. 四年中五角大楼两度叫我去That's two more than we've had.那你比我们多去了两次It gets better, John.好的还在后面Just got our latest scintillating assignment.我们才接到最新的伟大任务You know, the Russians have the H bomb,你们知道苏俄已经制出氢弹the Nazis are repatriating South America,纳粹的魔爪已伸向南美the Chinese have a standing army of 2.8 million, 中国有两百八十万的常备军and I am doing stress tests on a dam.而我却在替水坝做压力试验You made the cover of Fortune... again.但是你却二度上 "财富杂志"的封面Please note the use of the word "you", not "we". 请注意你用了"你" 而不是"我们"That was supposed to be just me.应该只有我一个人才对So not only do they rob me of the Fields medal,如今他们不但剥夺我得奖的机会now they put me on the cover of Fortune magazine 居然把我和这些小鼻子小眼的文人学者们with these hacks, these scholars of trivia.一起放在 "财富杂志" 的封面上John, exactly what's the difference约翰天才和才气横溢between genius and most genius?到底有什么不同?Quite a lot.差别大了He's your son.反正你总是有道理Anyway, you've got 10 minutes.你只剩下十分钟了I've always got 10 minutes.我的时间多得是Before your new class?你现在得去上课了Can I not get a note from a doctor or something? 能不能请医生开张病假单You are a doctor, John, and no.抱歉这回可不行Now, come on, you know the drill,你知道这里的规定we get these beautiful facilities,学校既然给了我们这些设施M.I.T. gets America's great minds of today就要用当今美国最伟大的脑袋teaching America's great minds of tomorrow.去教未来伟大的脑袋做交换Now, have a nice day at school.好了祝你在学校里过的愉快.The bell's ringing.上课铃要响了The eager young minds of tomorrow.未来饥渴稚嫩的脑袋Can we leave one open, Professor?教授可以留一个窗户不要关吗?It's really hot, sir.实在太热了Your comfort comes second课堂的安静to my ability to hear my own voice.比你舒不舒服重要得多Personally,我个人认为I think this class will be a waste...这堂课不但浪费of your...你们的时间And what is infinitely worse...糟糕的是更浪费了...my time.我宝贵的时间However,不过...here we are. So...既然来了那就说清楚you may attend or not.来不来上课随你们的便You may complete your assignments at your whim.你可以喜欢了就完成你的作业We have begun.课这就开始了Miss.小姐Excuse me!打扰一下Excuse me!打扰一下We have a little problem.我们有点小小的问题It's extremely hot in here with the windows closed关上窗户这里会很热and extremely noisy with them open.开着却又太吵So, I was wondering if there was any way you could,我在想能不能请你们I don't know, maybe work someplace else for about 45 minutes? 先修别的地方大约45分钟就行了Not a problem. Thank you so much!没问题谢谢你们At a break! Got it!休息一下好的As you will find in multivariable calculus,你们会发现there is often...在多变性的微积分中...a number of solutions for any given problem. 往往一个难题会有多种解答As I was saying, this problem here黑板上的这个问题will take some of you many months to solve. 有些人可能会解上好几个月For others among you,更有些人it will take you the term of your natural lives. 可能要花上一辈子的时间Professor Nash.纳什教授William Parcher.我是威廉·帕彻Big brother,你所谓的老大at your service.在此听候你的吩咐What can I do for the Department of Defense? 我能为国防部做些什么?Are you going to to give me a raise?你是来替我加薪的吗?Let's take a walk.咱们散散步Impressive work at the Pentagon. Yes, it was. 你在五角大楼的工作让人钦佩的确如此Oppenheimer used to say,原子弹之父常说"Genius sees the answer before the question." "天才在问题发生前就已找到解答"You knew Oppenheimer?你认识奥本海默?His project was under my supervision.他的计划就在我的监督之下进行的Which project?哪个计划?That project.喔那个计划It's not that simple, you know?其实并不简单Well, you ended the war.可是你们仍然结束了战争We incinerated 150,000 people in a heartbeat. 我们杀死十五万人Great deeds come at great cost, Mr. Parcher. 要有牺牲才能完成伟业Well, conviction, it turns out,可是对旁观者而言is a luxury of those on the sidelines, Mr. Nash. 却不应该随便去断罪I'll try and keep that in mind.我会试着去记住它So, John, no family,据我所知你没家人no close friends...也没好友Why is that?怎么会这样子?I like to think it's because I'm a lone wolf.我宁愿想成自己是个独行侠But mainly it's because people don't like me. 但主要原因是人们不喜欢我Well, there are certain endeavors妙的是由于你缺少人际关系where your lack of personal connection在这里would be considered an advantage.却成为你的一大长处This is a secure area.这里是管制区They know me.他们认识我Have you ever been here?你曾来过这里吗?We were told during our initial briefing在刚来的时候that these warehouses were abandoned.他们说这是一栋废弃的仓库That's not precisely accurate.其实并不正确By telling you what I'm about to tell you,由于要告诉你以下的机密I am increasing your security clearance我特地把你参与机密的资格to top secret.升高至"最高机密"Disclosure of secure information can result in imprisonment. 泄露机密资料会去坐牢的Get it?懂吗?What operation?什么行动?"由战略室制作"Those are a good idea.这东西倒挺好用的This factory is in Berlin.这家工厂位于柏林.We seized it at the end of the war.战后我们把它关闭了Nazi engineers were attempting纳粹工程师to build a portable atomic bomb.原打算制造轻型原子弹The Soviets reached this facility before we did,苏联在我们之前到这里and we lost the damn thing.拿去所有的资料The routing orders at the Pentagon,五角大楼的路线顺序they were about this, weren't they?是不是和这有关?The Soviets aren't as unified as people believe.苏联的内部并没有真正的统一A faction of the Red Army calling itself Novaya Svobga,赤军集团称他们自己为"新自由""the New Freedom" has control of the bomb他们已经控制住原子弹and intends to detonate it on U.S. soil.而且打算在美国本土上引爆Their plan is to incur maximum civilian casualties.他们计划造成百姓们最大的伤害Man is capable of as much atrocity人类是非常as he has imagination.残酷的New Freedom has sleeper agents here in the U.S.新自由在美国派有间谍McCarthy is an idiot,麦卡锡议员是个白痴but unfortunately that doesn't make him wrong.但并不等于他的见解是错的New Freedom communicates to its agents新自由组织是通过报纸和杂志上的密码through codes imbedded in newspapers and magazines, 和他们的间谍传递消息and that's where you come in.所以我们需要借助你的才能You see, John,约翰你明白的what distinguishes you你与别人最大的不同is that you are,在于你是一个...quite simply,简单的说the best natural code breaker I have ever seen.你是我所见过最好的天生破码专家What exactly is it that you would like me to do?你到底要我干什么?Commit this list of periodicals to memory.记住上面所列的所有期刊Scan each new issue,仔细浏览每一期find any hidden codes,只要发现密码decipher them.就去破解它。
Genearlizations of Precupanu's Inequality for Orthornormal Families of Vectors in Inner Pro
In the right-hand side or in the left-hand side of (1.1) we have equality if and only if there are λ, µ ∈ R such that (1.2) λ x, a x
2
·x+µ
y, #43; µb) . 2
where some information about a third vector x is known: Theorem 2. Let (H ; ·, · ) be an inner product space over the real field R and x, y, z ∈ H such that: (1.8) | x, y | ≥ (1 − ε) x y , | x, z | ≥ (1 − ε) x z . z , where ε is a positive real number, reasonably small. Then √ (1.9) | y, z | ≥ max 1 − ε − 2ε, 1 − 4ε, 0 y x, a x, b x
In the complex case, apparently independent of Richard, M.L. Buzano obtained in [2] the following inequality (1.14) b + | a, b | · x 2, 2 provided x, a, b are vectors in the complex inner product space (H ; ·, · ) . In the same paper [6], Precupanu, without mentioning Buzano’s name in relation to the inequality (1.14), observed that, on utilising (1.14), one may obtain the following result of Moore type: | x, a x, b | ≤ a
Nonnegatively and Positively curved manifolds
a rXiv:077.391v1[mat h.D G]2J u l27NONNEGATIVELY AND POSITIVELY CUR VED MANIFOLDS BURKHARD WILKING The aim of this paper is to survey some results on nonnegatively and positively curved Riemannian manifolds.One of the important features of lower curvature bounds in general is the invariance under taking Gromov Hausdorfflimits.Cele-brated structure and finiteness results provide a partial understanding of the phe-nomena that occur while taking limits.These results however are not the subject of this survey since they are treated in other surveys of this volume.In this survey we take the more classical approach and focus on ”effective”re-sults.There are relatively few general ”effective”structure results in the subject.By Gromov’s Betti number theorem the total Betti number of a nonnegatively curved manifold is bounded above by an explicit constant which only depends on the dimension.The Gromoll Meyer theorem says that a positively curved open manifold is diffeomorphic to the Euclidean space.In the case of nonnegatively curved open manifolds,the soul theorem of Cheeger and Gromoll and Perelman’s solution of the soul conjecture clearly belong to the greatest structure results in the subject,as well.Also relatively good is the understanding of fundamental groups of nonnega-tively curved manifolds.A theorem of Synge asserts that an even dimensional orientable compact manifold of positive sectional curvature is simply connected.An odd dimensional positively curved manifold is known to be orientable (Synge),and its fundamental group is finite by the classical theorem of Bonnet and Myers.The fundamental groups of nonnegatively curved manifolds are virtually abelian,as a consequence of Toponogov’s splitting theorem.However,one of the ”effective”conjectures in this context,the so called Chern conjecture,was refuted:Shankar [1998]constructed a positively curved manifold with a non cyclic abelian funda-mental group.As we will discuss in the last section the known methods for constructing non-negatively curved manifolds are somewhat limited.The most important tools are the O’Neill formulas which imply that the base of a Riemannian submersion has nonnegative (positive)sectional curvature if the total space has.We recall that a smooth surjective map σ:M →B between two Riemannian manifolds is called a Riemannian submersion if the dual σad∗:T σ(p )B →T p M of the differential of σislength preserving for all p ∈M .Apart from taking products,the only other method is a special glueing technique,which was used by Cheeger,and more recently by Grove and Ziller to construct quite a few interesting examples of nonnegatively curved manifolds.By comparing with the class of known positively curved manifolds,the nonnega-tively curved manifolds form a huge class.In fact in dimensions above 24all known simply connected compact positively curved manifolds are diffeomorphic to rank 1symmetric spaces.Due to work of the author the situation is somewhat better in12BURKHARD WILKINGthe class of known examples of manifolds with positive curvature on open dense sets,see section4.Given the drastic difference in the number of known examples,it is somewhat painful that the only known obstructions on positively curved compact manifolds, which do not remain valid for the nonnegatively curved manifolds,are the above quoted results of Synge and Bonnet Myers on the fundamental groups.Since the list of general structure results is not far from being complete by now, the reader might ask why a survey on such a subject is necessary.The reason is that there are a lot of other beautiful theorems in the subject including structure results,but they usually need additional assumptions.We have subdivided the paper infive sections.Section1is on sphere theorems and related rigidity results,some notes on very recent significant developments were added in proof and can be found in section6.In section2,we survey results on compact nonnegatively curved manifolds,and in section3,results on open non-negatively curved manifolds.Then follows a section on compact positively curved manifolds with symmetry,since this was a particularly active area in recent years. Although we pose problems and conjectures throughout the paper we close the paper with a section on open problems.We do not have the ambition to be complete or to sketch all the significant historical developments that eventually led to the stated results.Instead we will usually only quote a few things according to personal taste.1.Sphere theorems and related rigidity results.A lot of techniques in the subject were developed or used in connection with proving sphere theorems.In this section we survey some of these results.We recall Toponogov’s triangle comparison theorem.Let M be a complete manifold with sectional curvature K≥κand consider a geodesic triangle∆in M consisting of minimal geodesics with length a,b,c∈R.Then there exists a triangle in the2-dimensional complete surface M2κof constant curvatureκwith side length a,b,c and the angles in the comparison triangle bound the corresponding angles in∆from below.1.1.Topological sphere theorems.We start with the classical sphere theorem of Berger and Klingenberg.Theorem1.1(Quarter pinched sphere theorem).Let M be a complete simply con-nected manifold with sectional curvature1/4<K≤1.Then M is homeomorphic to the sphere.The proof has two parts.Thefirst part is to show that the injectivity radius of M is at leastπ/2.This is elementary in even dimensions.In fact by Synge’s Theorem any even dimensional oriented manifold with curvature0<K≤1has injectivity radius≥π.In odd dimensions the result is due to Klingenberg and relies on a more delicate Morse theory argument on the loop space.The second part of the proof is due to Berger.He showed that any manifold with injectivity radius≥π/2and curvature>1is homeomorphic to a sphere.In fact by applying Toponogov’s theorem to two points of maximal distance,he showed that the manifold can be covered by two balls,which are via the exponential map diffeomorphic to balls in the Euclidean space.NONNEGATIVELY AND POSITIVELY CUR VED MANIFOLDS3 Grove and Shiohama[1977]gave a significant improvement of Berger’s theorem, by replacing the lower injectivity radius bound by a lower diameter bound.Theorem1.2(Diameter sphere theorem).Any manifold with sectional curvature ≥1and diameter>π/2is homeomorphic to a sphere.More important than the theorem was the fact the proof introduced a new con-cept:critical points of distance functions.A point q is critical with respect to the distance function d(p,·)if the set of initial vectors of minimal geodesics from q to p intersect each closed half space of T q M.If the point q is not critical it is not hard to see that there is a gradient like vectorfield X in a neighborhood of q.A vectorfield is said to be gradient like if for each integral curve c of X the map t→d(p,c(t)) is a monotonously increasing bilipschitz map onto its image.An elementary yet important observation is that local gradient like vectorfields can be glued together using a partition of unity.Proof of the diameter sphere theorem.We may scale the manifold such that its di-ameter isπ/2and the curvature is strictly>1.Choose two points p,q of maximal distanceπ/2,and let z be an arbitrary third point.Consider the spherical compar-ison triangle(˜p,˜q,˜z).We do know that the side length of(˜p,˜z)and(˜q,˜z)are less or equal toπ/2whereas d S2(˜p,˜q)=π/2.This implies that the angle of the triangle at˜z is≥π/2.By Toponogov’s theorem any minimal geodesic triangle with corners p,q,z in M has an angle strictly larger thanπ/2based at z.This in turn implies that the distance function d(p,·)has no critical points in M\{p,q}.Thus there is a gradient like vectorfield X on M\{p,q}.Furthermore without loss of generality Xis given on B r(p)\{p}by the actual gradient of the distance function d(p,·),where r is smaller than the injectivity radius.We may also assume X(z) ≤d(q,z)2for all z∈M\{p,q}.Then theflowΦof X exists for all future times and we can define a diffeomorphismψ:T p M→M\{q}as follows:for a unit vector v∈T p M and a nonnegative number t putψ(t·v)= exp(tv)if t∈[0,r]andψ(t·v)=Φt−r(exp(rv))if t≥r.Clearly this implies that M is homeomorphic to a sphere.There is another generalization of the sphere theorem of Berger and Klingenberg.A manifold is said to have positive isotropic curvature if for all orthonormal vectors e1,e2,e3,e4∈T p M the curvature operator satisfiesR(e1∧e2+e3∧e4,e1∧e2+e3∧e4)+R(e1∧e3+e4∧e2,e1∧e3+e4∧e2)>0 By estimating the indices of minimal2spheres in a manifold of positive isotropic curvature,Micallef and Moore[1988]were able to show thatTheorem1.3.A simply connected compact Riemannian manifold of positive isotropic curvature is a homotopy sphere.A simple computation shows that pointwise strictly quarter pinched manifolds have positive isotropic curvature.Thus the theorem of Micallef and Moore is a generalization of the quarter pinched sphere theorem.A more direct improvement of the quarter pinched sphere theorem is due to Abresch and Meyer[1996]. Theorem1.4.Let M be a compact simply connected manifold with sectional cur-vature14BURKHARD WILKING•M is homeomorphic to a sphere.•n is even and the cohomology ring H∗(M,Z2)is generated by one element.It is a well known result in topology that the Z2cohomology rings of spaces which are generated by one element are precisely given by the Z2-cohomology rings of rank1symmetric spaces RP n,CP n,HP n,Ca P2and S n,cf.[Zhizhou,2002].The proof of Theorem1.4has again two parts.Abresch and Meyerfirst establish that the injectivity radius of M is bounded below by the conjugate radius which in turn is bounded below byπ.From the diameter sphere theorem it is clear that without loss of generality diam(M,g)≤π(1+10−6).They then establish the horse shoe inequality,which was conjectured by Berger:for p∈M and any unit vector v∈T p M one hasd(exp(πv),exp(−πv))<π.In particular exp(πv)and exp(−πv)can be joined by a unique minimal geodesic. Once the horse shoe inequality is established it is easy to see that there is a smooth map f:RP n→M n such that in odd dimensions the integral degree is1and in even dimensions the Z2-degree is1.The theorem then follows by a straightforward cohomology computation.The horse shoe inequality relies on a mixed Jacobifield estimate.We only state the problem here in a very rough form.Let c be a normal geodesic in M and J a Jacobifield with J(0)=0.Suppose that at time t0=2πNONNEGATIVELY AND POSITIVELY CUR VED MANIFOLDS5 The work of Weiss[1993]goes in a different direction.He uses the fact that a quarter pinched sphere M n has Morse perfection n.A topological sphere M n is said that to have Morse perfection≥k if there is a smooth mapΨ:S k→C∞(M,R) satisfyingΨ(−p)=−Ψ(p),and for each p∈S k the functionΨ(p)is a Morse function with precisely two critical points.It is not hard to see that a quarter pinched sphere has Morse perfection n.Weiss used this to rule out quite a few of the exotic spheres by showing that their Morse perfection is<n.He showed that in dimensions n=4m−1any exotic sphere bounding a parallelizable manifold has odd order in the group of exotic spheres unless the Morse perfection≤n−2.By Hitchin there are also exotic spheres with a non-vanishingα-invariant,and thus these spheres do not even admit metrics with positive scalar curvature,see the survey of Jonathan Rosenberg.Similar to the quarter pinched sphere theorem,one can also strengthen the assumptions in the diameter sphere theorem in order to get a differentiable sphere theorem.This was carried out by Grove and Wilhelm[1997].Theorem1.6.Let M be an n-manifold with sectional curvature≥1containing (n−2)-points with pairwise distance>π/2.Then M is diffeomorphic to a sphere.If one has only k points with pairwise distance>π/2,then Grove and Wilhelm obtain restrictions on the differentiable structure of M.With a slight variation of the proof of Grove and Wilhelm one can actually get a slightly better result.Let M be an inner metric space.We say that M has a weak 2-nd packing radius≥r if diam(M)≥r.We say it has a weak k-th packing radius ≥r if there is a point p∈M such that∂B r(p)is connected and endowed with its inner metric has weak(k−1)-th packing radius≥r.Theorem1.7.Let(M,g)be an n-manifold with sectional curvature≥1and weak (k+1)-th packing radius>π/2.Then there is a family of metrics g t(t∈[0,1) with sectional curvature≥1and g0=g such that(M,g t)converges for t→1to an n-dimensional Alexandrov space A satisfying:If k≥n,then A is isometric to the standard sphere.If k<n,then A is given by the k-th iterated suspensionΣk A′of an n−k-dimensional Alexandrov space A′.Corollary1.8.Letε>0.A manifold with sectional curvature≥1and diameter >π/2also admits a metric with sectional curvature≥1and diameter>π−ε.As in the paper of Grove and Wilhelm,one can show in the situation of Theo-rem1.7that there is a sequence of positively metrics˜g i on the standard sphere with curvature≥1such that(S n,˜g i)i∈N converges to A as well.In particular,Grove and Wilhelm showed that an affirmative answer to the following question would imply the differentiable diameter sphere theorem.Question1.9(Smooth stability conjecture).Suppose a sequence of compact n-manifolds(M k,g k)with curvature≥−1converges in the Gromov Hausdorfftopol-ogy to an n-dimensional compact Alexandrov space A.Does this imply that for alllarge k1and k2the manifolds M k1and M k2are diffeomorphic?By Perelman’s stability theorem it is known that M k1and M k2are homeomor-phic for all large k1and k2,see the article of Vitali Kapovitch in this volume. Sketch of the proof of Theorem1.7.Let p,q∈M be points such that d(p,q)>π/2+εfor someε>0.We claim that we canfind a continuous family of metrics6BURKHARD WILKINGwith g0=g and K t≥1such that(M,g t)converges for t→1to the suspension of ∂Bπ/2(p).We consider the suspension X of M,i.e.,X=[−π/2,π/2]×M/∼where the equivalence classes of∼are given by p+:={π/2}×M,p−:={−π/2}×M and the one point sets{(t,p)}for|t|=π/2.Recall that X endowed with the usual warped product metric is an Alexandrov space with curvature≥1.We consider the curve c(t)=((1−t)π/2,p)as a curve in X,r(t):=π/2+ε(1−t) and the ball B r(t)(c(t))⊂X.Put N t:=∂B r(t)(c(t)).Since X\B r(t)(c(t))is strictly convex and N t is contained in the Riemannian manifold X\{p±}for all t=1,it follows that N t is an Alexandrov space with curvature≥1for all t∈[0,1].Clearly N0is up to a small scaling factor isometric to M.Moreover N1is isometric to the suspension of∂Bπ/2(p)⊂M.Using that N t is strictly convex in the Riemannian manifold X\{p±}for t∈[0,1),it follows that the family N t can be approximated by a family of strictly convex smooth submanifolds˜N t⊂X\{p±},t∈[0,1).Furthermore,one may assume that lim t→1N t=N1=lim t→1˜N t.We found a family of metrics g t of curvature>1such that(M,g t)converges to the suspension of∂Bπ/2(p).We may assume that∂Bπ/2(p)has weak k-th packing radius>π/2and k≥2.We now choose a curve of points q t∈M converging for t→1to a point on the equator q1∈∂Bπ/2(p)of the limit space such that there is a point q2in∂Bπ/2(p) whose intrinsic distance to q1is>π/2.We now repeat the above construction for all t∈(0,1)with(M,g,p)replaced by (M,g t,q t).This way we get for each t an one parameter family of smooth metrics g(t,s)with K≥1which converges for s→1to the suspension of the boundary of Bπ/2(q t)⊂(M,g t).It is then easy to see that one can choose the metrics such that they depend smoothly on s and t.Moreover,after a possible reparameterization of g(s,t)the one parameter family t→g(t,t)converges to the double suspension of the boundary of Bπ/2(q1)⊂∂Bπ/2(p).Clearly the theorem follows by iterating this process.We recall that to each Riemannian manifold(M,g)and each point p∈M one can assign a curvature operator R:Λ2T p M→Λ2T p M.We call the operator2-positive if the sum of the smallest two eigenvalues is positive.It is known that manifolds with2-positive curvature operator have positive isotropic curvature. Theorem1.10.Let(M,g)be a compact manifold with2-positive curvature opera-tor.Then the normalized Ricciflow evolves g to a limit metric of constant sectional curvature.In dimension3the theorem is due to Hamilton[1982].Hamilton[1986]also showed that the theorem holds for4-manifolds with positive curvature operator. This was extended by Chen to4-manifolds with2-positive curvature operator.In dimension2it was shown by Hamilton and Chow that for any surface the normalized Ricciflow converges to limit metric of constant curvature.In dimensions above4 the theorem is due to B¨o hm and Wilking[2006].For n≥3the proof solely relies on the maximum principle and works more generally in the category of orbifolds.We recall that a family of metrics g t on M is said to be a solution of the Ricci flow if∂NONNEGATIVELY AND POSITIVELY CUR VED MANIFOLDS7 Hamilton showed that if one represents the curvature operator R with respect to suitable moving orthonormal frames,then∂R=R2+R#.dtSketch of the proof of Theorem1.10.We let S2B so(n) denote the vectorspace of algebraic curvature operators satisfying the Bianchi identity.We call a continuous family C(s)s∈[0,1)⊂S2B(so(n))of closed convex O(n)-invariant cones of full dimension a pinching family,if(1)each R∈C(s)\{0}has positive scalar curvature,(2)R2+R#is contained in the interior of the tangent cone of C(s)at R forall R∈C(s)\{0}and all s∈(0,1),(3)C(s)converges in the pointed Hausdorfftopology to the one-dimensionalcone R+I as s→1.The argument in[B¨o hm and Wilking,2006]has two parts.One part is a general argument showing for any pinching family C(s)(s∈[0,1))that on any compact manifold(M,g)for which the curvature operator is contained in the interior of C(0)at every point the normalized Ricciflow evolves g to a constant curvature limit metric.In the proof of this result onefirst constructs to such a pinching family a pinching set in the sense Hamilton which in turn gives the convergence result.The harder problem is actually to construct a pinching family with C(0)being the cone of2-nonnegative curvature operators.Here a new tool is established.It is a formula that describes how this ordinary differential equation R′=R2+R# changes under O(n)-equivariant linear transformations.By chance the transforma-tion law is a lot simpler than for a generic O(n)-invariant quadratic expression.The transformation law often allows to construct new ODE-invariant curvature cones as the image of a given invariant curvature cone under suitable equivariant linear transformation l:S2B so(n) →S2B so(n) .This in turn is used to establish the existence of a pinching family.1.3.Related rigidity results.Wefirst mention the diameter rigidity theorem of Gromoll and Grove[1987]Theorem1.11(Diameter rigidity).Let(M,g)be a compact manifold with sec-tional curvature K≥1and diameter≥π/2.Then one of the following holds:a)M is homeomorphic to a sphere.b)M is locally isometric to a rank one symmetric space.The original theorem allowed a potential exceptional case•M has the cohomology ring of the Cayley plane,but is not isometric to the Cayley plane.8BURKHARD WILKINGThis case was ruled out much later by the author,see[Wilking,2001].The proof of the diameter rigidity theorem is closely linked to the rigidity of Hopffibrations which was established by Gromoll and Grove[1988]as well Theorem1.12(Rigidity of Hopffibrations).Letσ:S n→B be a Riemannian sub-mersion with connectedfibers.Thenσis metrically congruent to a Hopffibration. In particular thefibers are totally geodesic and B is rank one symmetric space.Similarly to the previous theorem,the original theorem allowed for a possible exception,Grove and Gromoll assumed in addition(n,dim B)=(15,8).Using very different methods,the rigidity of this special case was proved by the author in [Wilking,2001].This in turn ruled out the exceptional case in the diameter rigidity theorem as well.Sketch of the proof of the diameter rigidity theorem.The proof of the diameter rigid-ity theorem is the most beautiful rigidity argument in positive curvature.One as-sumes that the manifold is not homeomorphic to a sphere.Let p be a point with N2:=∂Bπ/2(p)=∅.One defines N1=∂Bπ/2(N2)as the boundary of the distance tube Bπ/2(N2)around N2.It then requires some work to see that N1and N2are totally geodesic submanifolds without boundary satisfying N2=∂Bπ/2(N1).Not both manifolds can be points,since otherwise one can show that M is homeomorphic to a sphere.If one endows the unit normal bundleν1(N i)with its natural connection metric,then Grove and Gromoll show in a next step that the mapσi:ν1(N i)→N j,v→exp(π/2v)is a Riemannian submersion,{i,j}={1,2}. Furthermoreσi restricts to a Riemannian submersionν1q(N i)→N j for all q∈N i.In the simply connected case one shows that N i is simply connected as well,i= 1,2.By the rigidity of submersions defined on Euclidean spheres(Theorem1.12) we deduce that N i is either a point or a rank one symmetric space with diameter π/2.Going back to the definition of N1,it is then easy to see that N1={p}.Using thatσ1:S n−1→N2is submersion with totally geodesicfibers,one can show that the pull back metric exp∗p g on Bπ/2(0)⊂T p M is determined byσ1.Thus M is isometric to a rank one symmetric space.In the non simply connected case one can show that either the universal cover is not a sphere and thereby symmetric or dim(N1)+dim(N2)=n−1.In the latter case it is not hard to verify that M has constant curvature one.Since the proof of the differentiable sphere theorem for manifolds with2-positive curvature follows from a Ricciflow argument it is of course not surprising that it has a rigidity version as well.Theorem1.13.A simply connected compact manifold with2-nonnegative curva-ture operator satisfies one of the following statements.•The normalized Ricciflow evolves the metric to a limit metric which is upto scaling is isometric to S n or CP n/2.•M is isometric to an irreducible symmetric space.•M is isometric to nontrivial Riemannian product.Of course in the last case the factors of M have nonnegative curvature operators. By Theorem2.2(M,g)admits a possibly different metric g1such that(M,g1)is locally isometric to(M,g)and(M,g1)isfinitely covered by a Riemannian product T d×M′where M′is simply connected and compact.This effectively gives a reduction to the simply connected case.NONNEGATIVELY AND POSITIVELY CUR VED MANIFOLDS9 The theorem has many names attached to it.Of course Theorem1.10(Hamilton [1982,1986],B¨o hm and Wilking[2006])enters as the’generic’case.This in turn was used by Ni and Wu[2006]to reduce the problem from2-nonnegative curvature operators to nonnegative curvature operators.One has to mention Gallot and Meyer’s[1975]investigation of manifolds with nonnegative curvature operator using the Bochner technique.Berger’s classification of holonomy groups,as well as Mori’s [1979],Siu and Yau’s[1980]solution of the Frankel conjecture are key tools.Based on this Chen and Tian[2006]proved convergence of the Ricciflow for compact K¨a hler manifolds with positive bisectional curvature.Sketch of a proof of Theorem1.13.Considerfirst the case that the curvature op-erator of M is not nonnegative.We claim that then the Ricciflow immediately evolves g to a metric with2-positive curvature operator.We consider a short time solution g(t)of the Ricciflow and let f:[0,ε)×M→R, denote the function which assigns to(t,p)the sum of the lowest two eigenvalues of the curvature operator of(M,g(t))at p.Wefirst want to show that f(t,·)is positive somewhere for small t>0.We may assume that f(0,p)=0for all p.It is straightforward to check that f satisfies∂f(λ1+λ2) R+t(R2+R#) .∂t|t=0+From the invariance of2-nonnegative curvature operators it is known that q(R)≥0. In fact a detailed analysis of the proof shows that q(R)≥2(λ1(R))2.In the present situation we deduce by afirst order argument that f(t,p)becomes positive somewhere for small t>0.Now it is not hard to establish a strong maximum principle that shows that f(t,·)is everywhere positive for small t>0,see Ni and Wu[2006].In other words(M,g t)has2-positive curvature operator for t>0and the result follows from Theorem1.10.We are left with the case that the curvature operator of(M,g)is nonnegative. Essentially this case was already treated by Gallot and Meyer using the Bochner technique,see[Petersen,2006].We present a slightly different argument following Chow and Yang(1989).Using Hamilton’s[1986]strong maximum principle one deduces that for t>0the curvature operator of(M,g t)has constant rank and that the kernel is parallel.Thus either R t is positive or the holonomy is non generic.We may assume that M does not split as a product.Hence without loss of generality M is irreducible with non generic holonomy.Since(M,g t)clearly has positive scalar curvature Berger’s classification of holonomy groups implies that Hol(M)∼=U(n/2),Sp(1)Sp(n/4)unless(M,g)is a symmetric space.In the case of Hol(M)∼=Sp(1)Sp(n)we can employ another theorem of Berger[1966]to see that M is up to scaling isometric to HP n/4,since in our case the sectional curvature of (M,g t)is positive.In the remaining case Hol(M)=U(n/2)it follows that M is K¨a hler and(M,g t)has positive(bi-)sectional curvature.By Mori[1979]and Siu and Yau’s[1980]solution of the Frankel conjecture M is biholomorphic to CP n/2. In particular,M admits a K¨a hler Einstein metric.Due to work of Chen and Tian [2006]it follows,that the normalized Ricciflow on M converges to the Fubini study metric which completes the proof.pact nonnegatively curved manifoldsThe most fundamental obstruction to this date is Gromov’s Betti number theo-rem.10BURKHARD WILKINGTheorem2.1(Gromov,1981).Let M n be an n-dimensional complete manifold with nonnegative sectional curvature,and let F be afield.Then the total Betti number satisfiesb(M,F):=ni=0b i(M,F)≤1010n4.Gromov’s original bound on the total Betti number was depending double expo-nentially on the dimension.The improvement is due to Abresch[1987].However, this bound is not optimal either.In fact Gromov posed the problem whether the best possible bound is2n,the total Betti number of the n-dimensional torus.The statement is particularly striking since the nonnegatively curved manifolds in a fixed dimension≥7have infinitely many homology types with respect to integer coefficients.More generally Gromov gave explicit estimates for the total Betti num-bers of compact n-manifolds with curvature≥−1and diameter≤D.The proof is an ingenious combination of Toponogov’s theorem and critical point theory. Sketch of the proof of Theorem2.1.The most surprising part in the proof is a def-inition:Gromov assigns to every ball B r(p)⊂M afinite number called the corank of the ball.It is defined as the maximum over all k such that for all q∈B2r(p) there are points q1,...,q k withd(q,q1)≥2n+3r,d(q,q i+1)≥2n d(p,q i)and q i is a critical point of the distance function of q in the sense of Grove and Shiohama.One can show as follows that the corank of a ball is at most2n:Choose a minimal geodesic c ij from q i to q j,i<j and minimal geodesic c i from q to q i, i=1,...,k.Since q i is a critical point we canfind a possibly different minimal geodesic˜c i from q to q i such that the angle of the triangle(˜c i,c j,c ij)based at q i is ≤π/2.Therefore L(c j)2≤L(c ij)2+L(c i)2.Applying Toponogov’s theorem to the triangle(c i,c j,c ij)gives that the angleϕij between c i and c j satisfies tan(ϕij)≥2n. Thusϕij≥π/2−2−n.The upper bound on k now follows from an Euclidean sphere packing argument in T q M.By reverse induction on the corank,one establishes an estimate for the content of a ball cont(B r(p))which is defined as the dimension of the image of H∗(B r(p)) in H∗(B5r(p)).A ball B r(p)with maximal corank is necessarily contractible in B5r(p)since for some q∈B2r(p)the distance function of q has no critical points in B8r(q)\{q}.This establishes the induction base.It is immediate from the definition that corank(Bρ(q))≥corank(B r(p))for all q∈B3r/2(p)and allρ≤r/4. In the induction step one distinguishes between two cases.In thefirst case,one assumes that corank(Bρ(q))>corank(B r(p))for all q∈B r(p)andρ:=r4n .Thus for some point x∈B2ρ(q)there is no criticalpoint of the distance function of x in B8r(x)\B2−n+3r(x).This implies that one can homotop B r(p)to a subset of B r/4(x)in B5r(p).From this it is not hard to deduce that cont(B r/4(x))≥cont(B r(p)).We have seen above corank(B r/4(x))≥corank(B r(p)).One can now apply the same argument again with B r(p)replaced。
Contents
On the Common Structure of Bohmian Mechanics and the Ghirardi–Rimini–Weber TheoryValia Allori∗,Sheldon Goldstein†,Roderich Tumulka‡,and Nino Zangh`ı§June2,2007AbstractBohmian mechanics and the Ghirardi–Rimini–Weber theory provide opposite resolutions of the quantum measurement problem:the former postulates addi-tional variables(the particle positions)besides the wave function,whereas thelatter implements spontaneous collapses of the wave function by a nonlinear andstochastic modification of Schr¨o dinger’s equation.Still,both theories,when un-derstood appropriately,share the following structure:They are ultimately notabout wave functions but about‘matter’moving in space,represented by eitherparticle trajectories,fields on space-time,or a discrete set of space-time points.The role of the wave function then is to govern the motion of the matter.PACS:03.65.Ta.Key words:quantum theory without observers;Bohmian me-chanics;Ghirardi–Rimini–Weber theory of spontaneous wave function collapse;primitive ontology;local beables.Dedicated to GianCarlo Ghirardi on the occasion of his70th birthday Contents1Introduction2∗Department of Philosophy,Davison Hall,Rutgers,The State University of New Jersey,26Nichol Avenue,New Brunswick,NJ08901-1411,USA.E-mail:vallori@ †Departments of Mathematics,Physics and Philosophy,Hill Center,Rutgers,The State Uni-versity of New Jersey,110Frelinghuysen Road,Piscataway,NJ08854-8019,USA.E-mail:old-stein@‡Mathematisches Institut,Eberhard-Karls-Universit¨a t,Auf der Morgenstelle10,72076T¨u bingen, Germany.E-mail:tumulka@everest.mathematik.uni-tuebingen.de§Dipartimento di Fisica dell’Universit`a di Genova and INFN sezione di Genova,Via Dodecaneso33, 16146Genova,Italy.E-mail:zanghi@ge.infn.it2Bohmian Mechanics4 3Ghirardi,Rimini,and Weber53.1GRWm (7)3.2GRWf (8)3.3Empirical Equivalence Between GRWm and GRWf (10)4Primitive Ontology114.1Primitive Ontology and Physical Equivalence (12)4.2Primitive Ontology and Symmetry (14)4.3Without Primitive Ontology (18)4.4Primitive Ontology and Quantum State (20)5Differences between BM and GR W215.1Primitive Ontology and Quadratic Functionals (22)5.2Primitive Ontology and Equivariance (23)6A Plethora of Theories246.1Particles,Fields,and Flashes (24)6.2Schr¨o dinger Wave Functions and Many-Worlds (26)7The Flexible Wave Function287.1GRWf Without Collapse (28)7.2Bohmian Mechanics With Collapse (30)7.3Empirical Equivalence and Equivariance (31)8What is a Quantum Theory without Observers?33 1IntroductionBohmian mechanics(BM)and the Ghirardi–Rimini–Weber(GRW)theory are two quan-tum theories without observers,and thus provide two possible solutions of the mea-surement problem of quantum mechanics.However,they would seem to have little in common beyond achieving the goal of describing a possible reality in which observers wouldfind,for the outcomes of their experiments,the probabilities prescribed by the quantum formalism.They are two precise,unambiguous fundamental physical theories that describe and explain the world around us,but they appear to do this by employing opposite strategies.In Bohmian mechanics(Bohm,1952;Bell,1966;D¨u rr et al.,1992; Berndl et al.,1995)the wave function evolves according to the Schr¨o dinger equation but is not the complete description of the state at a given time;this description involves further variables,traditionally called‘hidden variables,’namely the particle positions.In the GRW theory(Pearle,1976;Ghirardi et al.,1986;Bell,1987a;Bassi and Ghirardi, 2003),in contrast,the wave functionψdescribes the state of any physical system com-pletely,butψcollapses spontaneously,thus departing from the Schr¨o dinger evolution. That is,the two theories choose different horns of the alternative that Bell formulated as his conclusion from the measurement problem(Bell,1987a):‘Either the wave function, as given by the Schr¨o dinger equation,is not everything,or it is not right.’The two theories are always presented almost as dichotomical,as in the recent paper by Putnam(2005).Our suggestion here is instead that BM and GRW theory have much more in common than one would expect atfirst sight.So much,indeed,that they should be regarded as being close to each other,rather than opposite.The differences are less profound than the similarities,provided that the GRW theory is understood appropriately,as involving variables describing matter in space-time.These variables we call the primitive ontology(PO)of the theory,and they form the common structure of BM and GRW.The gain from the comparison with BM is the insight that the GRW theory can,and should,be understood in terms of the PO.We think this view in terms of the PO provides a deeper understanding of the GRW theory in particular,and of quantum theories without observer in general.To formulate more clearly and advertise this view is our goal.After recalling what Bohmian mechanics is in Section2,we introduce two concrete examples of GRW theories in Section3.These examples involve rather different choices of crucial variables,describing matter in space-time,and give us a sense of the range of possibilities for such variables.We discuss in Section4the notion of the primitive ontology(PO)of a theory(a notion introduced in(D¨u rr et al.,1992))and connect it to Bell’s notion of‘local beables’(Bell,1976).In Section4.1we relate the primitive ontology of a theory to the notion of physical equivalence between theories.We stress in Section4.2the connection,first discussed in(Goldstein,1998),between the primitive ontology and symmetry properties,with particular concern for the generalization of such theories to a relativistically invariant quantum theory without observers.In Section4.3 we argue that a theory without a primitive ontology is at best profoundly problematical. We proceed in Sections5to an analysis of the differences between GRW(with primitive ontology)and BM,and in Section6we discuss a variety of possible theories.We consider in Section7.1a‘no-collapse’reformulation of one of the GRW theories and in Section7.2 a‘collapse’interpretation of BM.These formulations enable us to better appreciate the common structure of BM and the GRW theories,as well as the differences,as we discuss in Section7.3.We conclude in Section8with a summary of this common structure.2Bohmian MechanicsBohmian mechanics is a theory of(nonrelativistic)particles in motion.The motion of a system of N particles is provided by their world lines t→Q i(t),i=1,...,N,where Q i(t)denotes the position in R3of the i-th particle at time t.These world lines are determined by Bohm’s law of motion(Bohm,1952;Bell,1966;D¨u rr et al.,1992;Berndl et al.,1995),dQ i dt =vψi(Q1,...,Q N)=m iImψ∗∇iψψ∗ψ(Q1...,Q N),(1)where m i,i=1,...,N,are the masses of the particles;the wave functionψevolves according to Schr¨o dinger’s equationi ∂ψ∂t=Hψ,(2)where H is the usual nonrelativistic Schr¨o dinger Hamiltonian;for spinless particles it isof the formH=−Nk=1 22m k∇2k+V,(3)containing as parameters the masses of the particles as well as the potential energy function V of the system.In the usual yet unfortunate terminology,the actual positions Q1,...,Q N of the par-ticles are the hidden variables of the theory:the variables which,together with the wave function,provide a complete description of the system,the wave function alone provid-ing only a partial,incomplete,description.From the point of view of BM,however,this is a strange terminology since it suggests that the main object of the theory is the wave function,with the additional information provided by the particles’positions playing a secondary role.The situation is rather much the opposite:BM is a theory of particles; their positions are the primary variables,and the description in terms of them must be completed by specifying the wave function to define the dynamics(1).As a consequence of Schr¨o dinger’s equation and of Bohm’s law of motion,the quan-tum equilibrium distribution|ψ(q)|2is equivariant.This means that if the configuration Q(t)=(Q1(t),...,Q N(t))of a system is random with distribution|ψt|2at some time t, then this will be true also for any other time t.Thus,the quantum equilibrium hypoth-esis,which asserts that whenever a system has wave functionψt,its configuration Q(t) is random with distribution|ψt|2,can consistently be assumed.This hypothesis is not as hypothetical as its name may suggest:the quantum equilibrium hypothesis follows, in fact,by the law of large numbers from the assumption that the(initial)configuration of the universe is typical(i.e.,not-too-special)for the|Ψ|2distribution,withΨthe(ini-tial)wave function of the universe(D¨u rr et al.,1992).The situation resembles the way Maxwell’s distribution for velocities in a classical gas follows from the assumption that the phase point of the gas is typical for the uniform distribution on the energy surface.As a consequence of the quantum equilibrium hypothesis,a Bohmian universe,even if deterministic,appears random to its inhabitants.In fact,the probability distributions observed by the inhabitants agree exactly with those of the quantum formalism.To begin to understand why,note that any measurement apparatus must also consist of Bohmian particles.Calling Q S the configuration of the particles of the system to be measured and Q A the configuration of the particles of the apparatus,we can write for the configuration of the big Bohmian system relevant to the analysis of the measurement Q=(Q S,Q A).Let us suppose that the initial wave functionψof the big system is a product stateΨ(q)=Ψ(q S,q A)=ψ(q S)φ(q A).During the measurement,thisΨevolves according to the Schr¨o dinger equation,and in the case of an ideal measurement it evolves toΨt= αψαφα,whereαruns through the eigenvalues of the observable measured,φαis a state of the apparatus in which the pointer points to the valueα,andψαis the projection ofψto the appropriate eigenspace of the observable.By the quantum equilibrium hypothesis,the probability for the random apparatus configuration Q A(t)to be such as to correspond to the pointer pointing to the valueαis ψα 2.For a more detailed discussion see(D¨u rr et al.,1992, 2004b).3Ghirardi,Rimini,and WeberThe theory proposed by Ghirardi,Rimini and Weber(1986)is in agreement with the predictions of nonrelativistic quantum mechanics as far as all present experiments are concerned(Bassi and Ghirardi,2003);for a discussion of future experiments that may distinguish this theory from quantum mechanics,see Section V of(Bassi and Ghirardi, 2003).According to the way in which this theory is usually presented,the evolution of the wave function follows,instead of Schr¨o dinger’s equation,a stochastic jump process in Hilbert space.We shall succinctly summarize this process as follows.Consider a quantum system described(in the standard language)by an N-‘particle’1 wave functionψ=ψ(q1,...,q N),q i∈R3,i=1,...,N;for any point x in R3(the‘center’of the collapse that will be defined next),define on the Hilbert space of the system thecollapse operatorΛi(x)=1(2πσ2)3/2e−(b Q i−x)22σ2,(4)where Q i is the position operator of‘particle’i.Hereσis a new constant of nature of order of10−7m.Letψtbe the initial wave function,i.e.,the normalized wave function at some time t0arbitrarily chosen as initial time.Thenψevolves in the following way: 1We wish to emphasize here that there are no particles in this theory:the word‘particle’is used only for convenience in order to be able to use the standard notation and terminology.1.It evolves unitarily,according to Schr¨o dinger’s equation,until a random timeT1=t0+∆T1,so thatψT1=U∆T1ψt,(5)where U t is the unitary operator U t=e−i Ht corresponding to the standard Hamil-tonian H governing the system,e.g.,given by(3)for N spinless particles,and∆T1 is a random time distributed according to the exponential distribution with rate Nλ(where the quantityλis another constant of nature of the theory,2of order of 10−15s−1).2.At time T1it undergoes an instantaneous collapse with random center X1andrandom label I1according toψT1→ψT1+=ΛI1(X1)1/2ψT1ΛI1(X1)1/2ψT1.(6)I1is chosen at random in the set{1,...,N}with uniform distribution.The center of the collapse X1is chosen randomly with probability distribution3P(X1∈dx1|ψT1,I1=i1)= ψT1|Λi1(x1)ψT1dx1= Λi1(x1)1/2ψT12dx1.(7)3.Then the algorithm is iterated:ψT1+evolves unitarily until a random time T2=T1+∆T2,where∆T2is a random time(independent of∆T1)distributed according to the exponential distribution with rate Nλ,and so on.In other words,the evolution of the wave function is the Schr¨o dinger evolution in-terrupted by collapses.When the wave function isψa collapse with center x and label i occurs at rater(x,i|ψ)=λ ψ|Λi(x)ψ (8) and when this happens,the wave function changes toΛi(x)1/2ψ/ Λi(x)1/2ψ .Thus,if between time t0and any time t>t0,n collapses have occurred at the times t0<T1<T2<...<T n<t,with centers X1,...,X n and labels I1,...,I n,the wavefunction at time t will beψt=L F n t,tψtL F n t,tψt(9)where F n={(X1,T1,I1),...,(X n,T n,I n)}andL F n t,t0=U t−TnΛIn(X n)1/2U Tn−T n−1ΛIn−1(X n−1)1/2U Tn−1−T n−2···ΛI1(X1)1/2U T1−t0.(10)2Pearle and Squires(1994)have argued thatλshould be chosen differently for every‘particle,’with λi proportional to the mass m i.3Hereafter,when no ambiguity could arise,we use the standard notations of probability theory, according to which a capital letter,such as X,is used to denote a random variable,while the the values taken by it are denoted by small letters;X∈dx is a shorthand for X∈[x,x+dx],etc.Since T i ,X i ,I i and n are random,ψt is also random.It should be observed that—unless t 0is the initial time of the universe—also ψt 0should be regarded as random,being determined by the collapses that occurred at times earlier that t 0.However,given ψt 0,the statistics of the future evolution of thewave function is completely determined;for example,the joint distribution of the first n collapses after t 0,with particle labels I 1,...,I n ∈{1,...,N },isP X 1∈dx 1,T 1∈dt 1,I 1=i 1,...,X n ∈dx n ,T n ∈dt n ,I n =i n |ψt 0=λn e −Nλ(t n −t 0) L f n t n ,t 0ψt 0 2dx 1dt 1···dx n dt n ,(11)with f n ={(x 1,t 1,i 1),...,(x n ,t n ,i n )}and L f n t n ,t 0given,mutatis mutandis ,by (10).This is,more or less,all there is to say about the formulation of the GRW theory according to most theorists.In contrast,GianCarlo Ghirardi believes that the descrip-tion provided above is not the whole story,and we agree with him.We believe that,depending on the choice of what we call the primitive ontology (PO)of the theory,there are correspondingly different versions of the theory.We will discuss the notion of prim-itive ontology in detail in Section 4.In the subsections below we present two versions of the GRW theory,based on two different choices of the PO,namely the matter density ontology (in Section 3.1)and the flash ontology (in Section 3.2).3.1GR WmIn the first version of the GRW theory,denoted by GRWm ,the PO is given by a field:We have a variable m (x,t )for every point x ∈R 3in space and every time t ,defined bym (x,t )=N i =1m i R 3N dq 1···dq N δ(q i −x ) ψ(q 1,...,q N ,t ) 2.(12)In words,one starts with the |ψ|2–distribution in configuration space R 3N ,then obtains the marginal distribution of the i -th degree of freedom q i ∈R 3by integrating out all other variables q j ,j =i ,multiplies by the mass associated with q i ,and sums over i .GRWm was essentially proposed by Ghirardi and co–workers in (Benatti et al.,1995);4see also (Goldstein,1998).The field m (·,t )is supposed to be understood as the density of matter in space at time t .Since these variables are functionals of the wave function ψ,they are not ‘hidden variables’since,unlike the positions in BM ,they need not be specified in addition to the 4They first proposed (for a model slightly more complicated than the one considered here)that the matter density be given by an expression similar to (12)but this difference is not relevant for our purposes.wave function,but rather are determined by it.Nonetheless,they are additional ele-ments of the GRW theory that need to be posited in order to have a complete description of the world in the framework of that theory.GRWm is a theory about the behavior of afield m(·,t)on three-dimensional space. The microscopic description of reality provided by the matter densityfield m(·,t)is not particle-like but instead continuous,in contrast to the particle ontology of BM.This is reminiscent of Schr¨o dinger’s early view of the wave function as representing a continuous matterfield.But while Schr¨o dinger was obliged to abandon his early view because of the tendency of the wave function to spread,the spontaneous wave function collapses built into the GRW theory tend to localize the wave function,thus counteracting this tendency and overcoming the problem.A parallel with BM begins to emerge:they both essentially involve more than the wave function.In one the matter is spread out continuously,while in the other it is concentrated infinitely many particles;however,both theories are concerned with matter in three-dimensional space,and in some regions of space there is more than in others.You mayfind GRWm a surprising proposal.You may ask,was it not the point of GRW—perhaps even its main advantage over BM—that it can do without objects beyond the wave function,such as particle trajectories or matter density?Is not the dualism present in GRWm unnecessary?That is,what is wrong with the version of the GRW theory,which we call GRW0,which involves just the wave function and nothing else?We will return to these questions in Section4.3.To be sure,it seems that if there was nothing wrong with GRW0,then,by simplicity,it should be preferable to GRWm. We stress,however,that Ghirardi must regard GRW0as seriously deficient;otherwise he would not have proposed anything like GRWm.We will indicate in Section4.3why we think Ghirardi is correct.To establish the inadequacy of GRW0is not,however,the main point of this paper.3.2GR WfAccording to another version of the GRW theory,which wasfirst suggested by Bell (1987a,1989),then adopted in(Kent,1989;Goldstein,1998;Tumulka,2006a,b;Allori et al.,2005;Maudlin,forthcoming),and here denoted GRWf,the PO is given by‘events’in space-time calledflashes,mathematically described by points in space-time.This is,admittedly,an unusual PO,but it is a possible one nonetheless.In GRWf matter is neither made of particles following world lines,such as in classical or Bohmian mechanics, nor of a continuous distribution of matter such as in GRWm,but rather of discrete points in space-time,in factfinitely many points in every bounded space-time region, see Figure1.In the GRWf theory,the space-time locations of theflashes can be read offfrom theFigure1:A typical pattern offlashes in space-time,and thus a possible world according to the GRWf theoryhistory of the wave function given by(9)and(10):everyflash corresponds to one of the spontaneous collapses of the wave function,and its space-time location is just the space-time location of that collapse.Accordingly,equation(11)gives the joint distribution of thefirst nflashes,after some initial time t0.Theflashes form the setF={(X1,T1),...,(X k,T k),...}(with T1<T2<...).In Bell’s words:[...]the GRW jumps(which are part of the wave function,not somethingelse)are well localized in ordinary space.Indeed each is centered on a par-ticular spacetime point(x,t).So we can propose these events as the basis ofthe‘local beables’of the theory.These are the mathematical counterpartsin the theory to real events at definite places and times in the real world(asdistinct from the many purely mathematical constructions that occur in theworking out of physical theories,as distinct from things which may be realbut not localized,and distinct from the‘observables’of other formulationsof quantum mechanics,for which we have no use here).A piece of matterthen is a galaxy of such events.(Bell,1987a)That is,Bell’s idea is that GRW can account for objective reality in three-dimensional space in terms of space-time points(X k,T k)that correspond to the localization events (collapses)of the wave function.Note that if the number N of the degrees of freedom in the wave function is large,as in the case of a macroscopic object,the number offlashes isalso large(ifλ=10−15s−1and N=1023,we obtain108flashes per second).Therefore, for a reasonable choice of the parameters of the GRWf theory,a cubic centimeter of solid matter contains more than108flashes per second.That is to say that large numbers of flashes can form macroscopic shapes,such as tables and chairs.That is how wefind an image of our world in GRWf.Note however that at almost every time space is in fact empty,containing noflashes and thus no matter.Thus,while the atomic theory of matter entails that space is not everywhere continuouslyfilled with matter but rather is largely void,GRWf entails that at most times space is entirely void.According to this theory,the world is made offlashes and the wave function serves as the tool to generate the‘law of evolution’for theflashes:equation(8)gives the rate of theflash process—the probability per unit time of theflash of label i occurring at the point x.For this reason,we prefer the word‘flash’to‘hitting’or‘collapse center’: the latter words suggest that the role of these events is to affect the wave function, or that they are not more than certain facts about the wave function,whereas‘flash’suggests rather something like an elementary event.Since the wave functionψevolves in a random way,F={(X k,T k):k∈N}is a random subset of space-time,a point process in space-time,as probabilists would say.GRWf is thus a theory whose output is a point process in space-time.53.3Empirical Equivalence Between GR Wm and GR WfWe remark that GRWm and GRWf are empirically equivalent,i.e.,they make always and exactly the same predictions for the outcomes of experiments.In other words,there is 5An anonymous referee has remarked that according to GRWf with the original parameters,in a single living cell there might occur as few as oneflash per hour,so that the cell is empty of matter for surprisingly long periods,quite against our intuition of a cell as a rather classical object.We make a few remarks to this objection.First,one should of course be careful with the language:there is presumably no cell in GRWf,though the structure of the wave function(on configuration space—even though there are no configurations)might suggest otherwise.Second,it all depends on the choice of the parametersλandσ,and,as long as experiments have notfixed their values,this cell argument may indeed be an argument for a choice different from GRW’s original one(say,with largerλand larger σ).We do not wish to argue here for any particular choice.Third,while most people might expect a cell to be real in much the same way as(say)a cat,one would not necessarily expect this of a single atom.Thus,it seems quite conceivable that,at some critical scale between that of atoms and that of cats,the ontological character of objects changes—as indeed it does in GRWf because of the limited resolution of matter given by the space-time density offlashes(e.g.,in water approximately oneflash every20micrometers every second).The cell example shows that the critical scale in GRWf is larger than one might have expected,and thus that GRWf is a mildly quirky picture of the world.But this mild quirkiness should be seen in perspective.In comparison,many other views about quantum reality are heavily eccentric,as they propose that reality is radically different from what we normally think it is like:e.g.,that there exist parallel worlds,or that there exists no matter at all,or that reality is contradictory in itself.no experiment we could possibly perform that would tell us whether we are in a GRWm world or in a GRWf world,assuming we are in one of the two.This should be contrasted with the fact that there are possible experiments(though we cannot perform any with the present technology)that decide whether we are in a Bohmian world or in a GRW world.The reason is simple.Consider any experiment,which isfinished at time t.Consider the same realization of the wave function on the time interval[0,t],but associated with different primitive ontologies in the two worlds.At time t,the result gets written down,encoded in the shape of the ink;more abstractly,the result gets encoded in the position of some macroscopic amount of matter.If in the GRWf ontology,this matter is in position1,then theflashes must be located in position1;thus,the collapses are centered at position1;thus,the wave function is near zero at position2;thus,by(12) the density of matter is low at position2and high at position1;thus,in GRWm the matter is also in position1,displaying the same result as in the GRWf world.We will discuss empirical equivalence again in Section7.3.4Primitive OntologyThe matter densityfield in GRWm,theflashes in GRWf,and the particle trajectories in BM have something in common:they form(what we have called)the primitive ontology of these theories.The PO of a theory—and its behavior—is what the theory is fundamentally about.It is closely connected with what Bell called the‘local beables’:[I]n the words of Bohr,‘it is decisive to recognize that,however far thephenomena transcend the scope of classical physical explanation,the accountof all evidence must be expressed in classical terms’.It is the ambition ofthe theory of local beables to bring these‘classical terms’into the equations,and not relegate them entirely to the surrounding talk.(Bell,1976)The elements of the primitive ontology are the stuffthat things are made of.The wave function also belongs to the ontology of GRWm,GRWf and BM,but not to the PO: according to these theories physical objects are not made of wave functions6.Instead, the role of the wave function in these theories is quite different,as we will see in the following.In each of these theories,the only reason the wave function is of any interest at all is that it is relevant to the behavior of the PO.Roughly speaking,the wave function tells the matter how to move.In BM the wave function determines the motion of the 6We would not go so far as Dowker and Herbauts(2005)and Nelson(1985),who have suggested that,physically,the wave function does not exist at all,and only the PO exists.But we have to admit that this view is a possibility,in fact a more serious one than the widespread view that no PO exists.particles via equation(1),in GRWm the wave function determines the distribution of matter in the most immediate way via equation(12),and in GRWf the wave function determines the probability distribution of the futureflashes via equation(11).We now see a clear parallel between BM and the GRW theory,at least in its versions GRWm and GRWf.Each of these theories is about matter in space-time,what might be called a decoration of space-time.Each involves a dual structure(X,ψ):the PO X providing the decoration,and the wave functionψgoverning the PO.The wave function in each of these theories,which has the role of generating the dynamics for the PO,has a nomological character utterly absent in the PO.This difference is crucial for understanding the symmetry properties of these theories and therefore is vital for the construction of a Lorentz invariant quantum theory without observers,as we will discuss in Section4.2.Even the Copenhagen interpretation(orthodox quantum theory,OQT)involves a dual structure:what might be regarded as its PO is the classical description of macro-scopic objects which Bohr insisted was indispensable—including in particular pointer orientations conveying the outcomes of experiments—with the wave function serving to determine the probability relations between the successive states of these objects.In this way,ψgoverns a PO,even for OQT.An important difference,however,between OQT on the one hand and BM,GRWm,and GRWf on the other is that the latter are fully precise about what belongs to the PO(particle trajectories,respectively continuous matter density orflashes)whereas the Copenhagen interpretation is rather vague,even noncommittal,on this point,since the notion of‘macroscopic’is an intrinsically vague one:of how many atoms need an object consist in order to be macroscopic?And,what exactly constitutes a‘classical description’of a macroscopic object?Therefore,as the example of the Copenhagen interpretation of quantum mechan-ics makes vivid,an adequate fundamental physical theory,one with any pretension to precision,must involve a PO defined on the microscopic scale.4.1Primitive Ontology and Physical EquivalenceTo appreciate the concept of PO,it might be useful to regard the positions of particles, the mass density and theflashes,respectively,as the output of BM,GRWm and GRWf, with the wave function,in contrast,serving as part of an algorithm that generates this output.Suppose we want to write a computer program for simulating a system (or a universe)according to a certain theory.For writing the program,we have to face the question:Which among the many variables to compute should be the output of the program?All other variables are internal variables of the program:they may be necessary for doing the computation,but they are not what the user is interested in.In the way we propose to understand BM,GRWm,and GRWf,the output of the program,the result of the simulation,should be the particle world lines,the m(·,t)。
Single Variable Calculus_中国大学mooc课后章节答案期末考试题库2023年
Single Variable Calculus_西北工业大学中国大学mooc课后章节答案期末考试题库2023年1.If f (x) and g (x) are differentiable on (a, b), 【图片】and f (x) > 0, g (x) > 0,x∈(a, b), then when x∈(a, b), we have答案:2.For what values of a and b will 【图片】be differentiable for all values of x?答案:a=-1/2, b=13.The evaluation of integral【图片】(where x>1) is答案:4.Find the derivative of【图片】答案:5.Find the centroid of a thin, flat plate covering the “triangular” region i n thefirst quadrant bounded by they-axis, the parabola【图片】, and the line【图片】.答案:6.If【图片】, find the limit of g(x) as x approaches the indicated value.答案:7.Find the derivative of the function below at x=0,【图片】答案:8.【图片】is答案:-1/329.If f (x) is continuous and F′(x) = f(x), then答案:10.Find the volume of the solid generated by revolving the region bounded bythe curve【图片】and the lines【图片】about【图片】.答案:11.The mean value【图片】that satisfies the Rolle’s Theorem on the function【图片】is答案:12.The critical number of 【图片】is ( )答案:0 and 213.Which statement is true?【图片】答案:A14.If【图片】,then【图片】答案:15.Evaluate【图片】.答案:16.The integtral of【图片】is答案:17.When x approaches infinity, the limit of【图片】is答案:18.The evaluation of integral【图片】is答案:19.If【图片】has continuous second-order derivative, and【图片】, then答案:20.Find the length of the enclosed loop【图片】shown here. The loop starts at【图片】and ends at【图片】.【图片】答案:21.The height of a body moving vertically is given by 【图片】, with s in metersand t in se conds. The body’s maximum height is ( )答案:22.If f (x) is increasing and f(x) > 0, then答案:23. A rock climber is about to haul up 100 N of equipment that has been hangingbeneath her on 40 m of rope that weighs 0.8 newton per meter. How much work will it take? (Hint: Solve for the rope and equipment separately, thenadd.)答案:24.The integral of【图片】is答案:25.Expand【图片】by partial function答案:26.Assume that u is a function of x and v is the derivative of u, then thederivative of arcsin(u) is答案:27.Find the center of mass of a thin plate covering the region bounded below bythe parabola 【图片】and above by the line 【图片】, if the density at the point 【图片】is 【图片】.答案:28.Find the limit【图片】答案:-129.Find the length of the curve【图片】, from【图片】 to【图片】.答案:53/630.Find the volume of the solid generated by revolving the regions bounded bythe curve 【图片】and line 【图片】about the x-axis.答案:31.Find the total area of the shaded region in the following picture.【图片】答案:4/332.The total area between the region 【图片】and the x-axis is答案:33.Which statement is NOT true?答案:34.Calculate【图片】答案:-135.The second derivative of the function y=secx is ( )答案:36.If gas in a cylinder is maintained at a constant temperature T, the pressure Pis related to the volume V by a formula of the form 【图片】in which a, b, n, and R are constants. Then【图片】答案:37.If【图片】then【图片】.答案:38.Calculate 【图片】The limit is ( )答案:139.Find the tangent to the folium of descartes 【图片】at the point (3,3)答案:x+y=640.Let 【图片】The tangent line to the graph of g(x) at (0,0) is ( ).答案:x-axis41.Find the derivative of the function below at x=0, 【图片】答案:It does not exist42.Find【图片】答案:43.The average value of 【图片】over theinterval [【图片】] is答案:44.Find the average rate of change of the function【图片】over the giveninterval [2,3]答案:1945.For【图片】 find the number【图片】 by using the two steps learned in 2.3.答案:0.0546.The linearization of the function 【图片】at x=1 is ( ).答案:47.If and only if x=ln(y),y=e^x.答案:正确48.Find the derivative of the function【图片】答案:49.Find the derivative of the function 【图片】It is ( )答案:50.If f (x) is an antiderivative of【图片】then【图片】答案:51.If f ′(x ) < 0, f ′′(x ) < 0, x∈(a, b), then the graph of f (x) on (a, b) is答案:decreasing and concave down.52.If【图片】, find【图片】.答案:753.At what points are the function【图片】 continuous?答案:Discontinuous at odd integer multiples of , but continuous at all other x.54.On what interval is the function 【图片】continuous?答案:55.On what interval is the function【图片】continuous?答案:56.【图片】【图片】and【图片】答案:0, 357.Suppose that the functionf(x)is second order continuous differentiable, and【图片】,【图片】. Therefore,【图片】答案:58.When x approaches 0, the limit of【图片】is答案:59.Find the area of the surface generated by revolving the curve 【图片】aboutthe x-axis to generate a solid.答案:60.Find the average rate of change of the function【图片】 over the giveninterval [0,2]答案:161.Find the limit of the function【图片】 and is the function continuous at thepoint being approached?答案:The limit is 0 and the function is continuous at62.The integral of [x/(x^2+1)]dx is答案:1/2[ln(x^2+1)]+C63.When x approaches 0, the limit of (1+3x)^(1/x) is答案:e^364.When x approaches infinity, the limit of x^(1/x) is答案:165.When x approaches infinity, for two functions f(x) and g(x), the limit off(x)/g(x) is infinity, and the limit of g(x)/f(x) is 0, thus a relationship between their growth rates can be said that答案:Function f(x) grouws faster than g(x).66. A function f is called a One-to-One function if it never takes on the same valuetwice.答案:正确67.The integtral of [e^(2x+1)]dx is答案:1/2[e^(2x+1)]+C68. A force of 2 N will stretch a rubber band 2 cm (0.02 m). Assuming thatHooke's Law applies, how far will a 4-N force stretch the rubber band?答案:4 cm69.Find the area of the surface generated by revolving the curve【图片】aboutthey-axis.答案:70.Which statement is true?答案:71.Which statement is false?答案:72.Find the integration formula of the solid volume generated by the curve 【图片】, the x-axis, and the line 【图片】revolved about the x-axis by the shell method.答案:73.Find the integration formula of the area of the region bounded above by thecurve 【图片】, below by the curve 【图片】, on the left by 【图片】, and on the right by 【图片】.答案:74.If 【图片】is continuous on [-1,1] and the average value is 2, then 【图片】答案:475. A cubic function is a polynomial of degree 3; that is, it has the form 【图片】，where a≠0. Then ( ) is false.答案:x=1 is critical number when the cubic function has only one criticalnumber.76.The graph of【图片】has ( )asymptotes.答案:377.If 【图片】then答案:78.The average value of【图片】on【图片】is答案:79.If f (x) is continuous on (−1, 1), and【图片】then答案:80.The derivative of the function【图片】 is答案:81.The function 【图片】has ( )答案:A. neither a local maximum nor a local minimum82.Find the derivative of function【图片】答案:83.Find y' , if【图片】答案:84.The derivative of 【图片】is( )答案:85.Let【图片】,Then【图片】答案:18x(x+1)86.At what points, is the function 【图片】continuous?答案:A. Discontinuous only when x= 3 or x= 187.Find the derivative of x(e^x).答案:e^x(x+1)88.The integral of (1/x)dx is答案:ln|x|+C89.Find the area of the surface generated by revolving the curve 【图片】aboutthe y-axis to generate a solid.答案:90.Find the length of the curve【图片】.答案:7ing the trapezoidal rule to estimate the integralwith n=4 steps【图片】答案:0.70500。
Local systems of vertex operators, vertex superalgebras and modules
(1.2)
z − z1 z1 − z −1 a(z1 )b(z ) − (−1)|a(z )||b(z )| z0 δ b(z )a(z1 ) . (1.3) z0 −z0
This is our desired “product” formula. Noticing that the expression of the right hand side of (1.3) is not well defined for arbitrary a(z ), b(z ) ∈ (EndM )[[z, z −1 ]], we restrict our attention to the subspace F (M ) consisting of those formal series a(z ) satisfying the truncation condition, i.e., a(z )u ∈ M ((z )) for any u ∈ M ; note that we must have A ⊆ F (M ). Then F (M ) is a subspace on which the right-hand side of (1.3) is well d acts. Let I (z ) = idM be the identity operator of M . Then defined and on which D = dz we obtain a quadruple (F (M ), I (z ), D, Y ). But (F (M ), D, I (z ), Y ) is not a vertex algebra. To a certain extent, vertex algebras look like commutative associative algebras with identity because of the “commutativity” and the “associativity.” Just as there is no canonical largest commutative associative algebra associated to a vector space, there is no (canonical) universal or largest vertex algebra associated to M , but there are maximal vertex algebras inside the space F (M ), what we call “local systems of vertex operators.” 3
An Isoperimetric Inequality With Applications To Curve Shortening
2L2A
’71"
>2
fv p
ds.
Adding these two inequalities yields
LA
(L + L)A
> (. pds
which proves inequality A. Inequality B follows from lemma 1.
Remark. H. Jacobowitz has observed that the bone shaped nonconvex curve in Figure shows that inequality 13 does not hold for all simple closed curves. The curve is made of straight lines and circular arcs with curvature 1/R. The quantity xds is approximately 2r/R. If the shank of the bone is long enough so that the area and length of the ends can be neglected, then the ratio L/A 1/e. When e is small enough inequality B fails and an attempt to shorten such a curve by the process discussed in the next section leads to an initial increase in the isoperimetric ratio.
of
A note on the law of large numbers for fuzzyvariables∗Robert Full´e r rfuller@ra.abo.fiEberhard Triesch triesch@math2.rwth-aachen.de AbstractThis short note a counterexample showing that Williamson’s theorem on the law of large numbers for fuzzy variables under a general triangular norm extension principle isnot valid.The objective of this note is to provide a counterexample to Theorem1in Williamson’s paper[1].To save space,we essentially use the same notation as in[1]and we do not repeat the statement of the theorem.Let t(u,v):=uv(product norm)and define the sequence of fuzzy numbers(X i)∞i=1by their membership functions as follows:µXi (x):= 1−|x|i,if−1≤x≤1,0,otherwiseThenαXi =βXi=0for all i and Theorem1states that the membership functionsµZNofthe arithmetic means Z N=(1/N) N i=1X i converge pointwise(as N→∞)to the function µgiven byµ(x):= 1,for x=0,0,otherwiseat least on(−1,0)∩(0,1).However,we will show that in the open interval(-1,1),the functionsµZNare bounded from below by some strictly positive function(not depending on N).To see this,recall that the T-arithmetic meansµZNare defined byµZN (z)=supx1+...x N=N zNi=1µX i(x i).For z∈(−1,1),we can thus estimateµZN (z)≥Ni=1µX i(z)=N i=1(1−|z|i)≥∞ i=1(1−|z|i).By Euler’s pentagonal number theorem(see,e.g.,[2],p.312),the infinite product is well known to converge for|z|<1to the following series:f(|z|)=1+∞n=1(−1)n(|z|(3n2−n)/2+|z|(3n2+n)/2).The value of the infinite product is thus positive on the interval(−1,1).∗Thefinal version of this paper appeared in:Fuzzy Sets and Systems,55(1993)235-236.References[1]R.C.Williamson,The law of large numbers for fuzzy variables under a general tri-angular norm extension principle,Fuzzy Sets and Systems,41(1991)55-81.[2]T.M.Apostol,Introduction to Analytic Number Theory,(Springer Verlag,Berlin-Heilderberg-New York,1976).。
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r =0
xr i xr+1
yi y s+1
(2.8)
Introduce the Schur polynomials (corresponding to hook diagramms): 1 ··· 1 xi1 · · · xin+1 . . . . det . . xn−1 · · · xn−1 i1 in+1 xr · · · xr i1 in+1 Sr (xi1 , . . . , xin+1 ) := 1 ··· 1 xi1 · · · xin+1 det . . . . . . xn i1 · · · xn in+1
References
[1] A. Morozov, “Pair correlator in the Itzykson–Zuber Integral”, Modern Phys. Lett. A 7, no. 37 3503–3507 (1992). [2] M. Bertola, B. Eynard, ”Mixed Correlation Functions of the Two-Matrix Model”, SPHT T03/028, CRM-2961 (2003). J. Phys. A36 (2003) 7733-7750, xxx, hepth/0303161. [3] S. L. Shatashvili, “Correlation Functions in the Itzykson–Zuber Model”, Comm. Math. Phys. 154 421–432 (1993). [4] C. Itzykson and J.B. Zuber, “The planar approximation (II)”, J. Math. Phys. 21, 411 (1980). [5] M.L. Mehta, Random Matrices, 2nd edition, (Academic Press, New York, 1991). 4
1
Introduction
The main result of this paper is given in 2.13. In [1], A. Morozov proposed a formula for correlation functions of unitary matrices with Itzykson-Zuber’s type measure: |Uji|2
U (N )
:=
1 I (X, Y )
dU |Uji |2 e tr (XU
U (N )
†Y
U)
(1.1)
where dU is the Haar measure over the unitary group U (N ) (appropriately normalized, so that 1.4 below holds with prefactor 1), and X and Y are two given diagonal complex matrices: X = diag(x1 , . . . , xN ) , Y = diag(y1 , . . . , yN ) (1.2) and the normalization factor I (X, Y ) is the so-called Harish-Chandra-Itzykson-Zuber integral: I (X, Y ) :=
SPhT-T04/077
A short note about Morozov’s formula
arXiv:math-ph/0406063v1 25 Jun 2004
B. Eynard 1 Service de Physique Th´ eorique de Saclay, F-91191 Gif-sur-Yvette Cedex, France.
n+1
Sr (xi1 , . . . , xin+1 ) =
r =0
1 x − xik k =1
(2.10)
Inserting that into 1.7, one gets: 1 1 I (X, Y ) |Uji |2 x − x y − y i j i,j =
U (N )
U (N )
†Y
dU tr
Abstract The purpose of this short note, is to rewrite Morozov’s formula for correlation functions over the unitary group, in a much simpler form, involving the computation of a single determinant.
(2.12)
3
or: 1 1 U U† x−X y−Y det E − = 1−
1 1 E y− x −X Y
tr
det E 1 1 E E −1 = 1 − det 1 − x−X y−Y
(2.13)
3
Concluding remarks
From that expression of Morozov’s formula, it is rather easy to recover any individual correlator by taking residues:
U)
=
· · · xn in+1
n yρ (i1 )
n · · · yρ (in+1 )
where SN is the smmetric group of rank N .
2
Rewriting Morozov’s formula
The purpose of this short note is to rewrite this formula in a much simpler form, in a way very similar to what was done in [2]. For any two complex numbers x and y (such that |x| > max |xi | and |y | > max |yi |), consider the choice: 1 ai = = x − xi
N −1 x ℓ y ρ (ℓ ) n=0 x ℓ y ρ (ℓ ) n+1
1 ǫ(ρ)e = ∆(X )∆(Y ) ρ∈S
N
(−1) 1−
i=1 N
n
= =
1 ǫ(ρ)e ∆(X )∆(Y ) ρ∈S
N
i1 <i2 <...<in+1 N
1 x − xik k =1
n+1
l=1
1 y − yρ(il )
† Uij Uji = Res Res x→ xi y → xj
tr
1 1 U U† x−X y−Y
(3.14)
Notice also that 2.13 is very similar to what was found in [2] after integration over X and Y . Notice that by expanding the determinant in 2.13 along its last column, one can find a recursion relation relating U (N ) to U (N − 1) integrals, which is equivalent to Shatashvili’s approach [3]. Aknowledgements: the author wants to thank M. Bertola and J. Harnad for discussions about that topic. The author thinks that this short note should be seen as an addendum to the article [2].
r −n
=
a1 ≤a2 ≤...≤ar −n k =1
xiak =
j1 +···+jn+1 =r −n
n+1 1 xj i1 · · · xin+1 . (2.9)
j
The formal generating function of these Schur polynomials is:
∞
1 xr+1
1 1 exi yj x − xi y − yj
Thus, Morozov’s formula can be rewritten: dU tr
U (N )
1 1 U U† x−X y−Y
e
tr (XU † Y U )
det E − det E − =
1 1 E y− x −X Y
∆(X )∆(Y )
i<j
det E ∆(X )∆(Y )
(1.4)
(1.5)
(xi − xj )
,
∆(Y ) :=
i<j
(y i − y j ) .
(1.6)
Here we shall assume that det E = 0 and ∆(X ) = 0 and ∆(Y ) = 0. Morozov’s formula was proven in [2], from Shatashvili’s formula [3]. Morozov’s formula was originaly written as follows, for any arbitrary sequences of complex numbers ai , bj one has: