Zettl A new proof of the inequalities among SturmLiouville eigenvalues
《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。
怀疑真理的英语作文
怀疑真理的英语作文Title: Questioning Truth。
In the pursuit of understanding, we often findourselves standing at the crossroads of certainty and doubt, grappling with the elusive nature of truth. The human intellect, insatiable in its quest for knowledge,perpetually questions the validity of what is presented as truth. This perpetual questioning forms the cornerstone of intellectual growth and societal progress.At the heart of skepticism lies the recognition that truth is not a fixed entity but rather a dynamic concept, subject to interpretation and revision. From the philosophical inquiries of ancient thinkers to thescientific revolutions of modern times, humanity has continuously challenged established truths, unearthinglayers of complexity and nuance beneath the surface.One of the fundamental aspects of doubting truth is theacknowledgment of inherent biases and limitations in human perception and cognition. Our understanding of reality is filtered through the lens of our experiences, beliefs, and cultural upbringing, shaping the way we interpretinformation and discern truth from falsehood. This realization compels us to approach knowledge with humility, recognizing that our grasp of truth is always partial and subject to refinement.Furthermore, the very nature of truth itself is multifaceted, comprising subjective, objective, and intersubjective elements. While empirical evidence and logical reasoning provide a foundation for objective truths, subjective truths are deeply personal, influenced by individual perspectives and emotions. Intersubjective truths, on the other hand, emerge from shared experiences and collective agreements within societies.In contemporary society, the proliferation ofinformation through various media platforms has amplified the challenge of discerning truth from misinformation. The rise of fake news, echo chambers, and algorithmic biaseshas fueled skepticism towards the veracity of information presented to us. In such a landscape, critical thinking becomes indispensable, empowering individuals to question, analyze, and verify the information they encounter.Moreover, the quest for truth often entails grappling with paradoxes and ambiguities that defy easy resolution. The realm of metaphysics, ethics, and existential inquiries delves into questions that transcend empirical observation, prompting us to confront the limits of human understanding. It is in these existential quandaries that we find the most profound insights into the nature of truth and reality.Despite the inherent challenges and uncertainties surrounding the concept of truth, the act of questioning serves as a catalyst for intellectual growth and societal progress. By challenging entrenched beliefs and ideologies, we pave the way for innovation, tolerance, and greater social cohesion. Embracing doubt as a precursor to enlightenment, we embark on a perpetual journey towards deeper understanding and wisdom.In conclusion, skepticism towards truth is not a sign of ignorance but rather a testament to the richness of human intellect and curiosity. By questioning the veracity of what is presented as truth, we engage in a dialogue that transcends boundaries and fosters a more nuanced appreciation of reality. In the pursuit of truth, let us remain vigilant, humble, and open-minded, for it is through doubt that we inch closer towards the elusive essence of truth itself.。
Topics in Inequalities-Hojoo Lee
3 Homogenizations and Normalizations 3.1 Homogenizations . . . . . . . . . . . . . . . . . . . 3.2 Schur’s Inequality and Muirhead’s Theorem . . . . 3.3 Normalizations . . . . . . . . . . . . . . . . . . . . 3.4 Cauchy-Schwarz Inequality and H¨ older’s Inequality 4 Convexity 4.1 Jensen’s Inequality . . . . . 4.2 Power Means . . . . . . . . 4.3 Majorization Inequality . . 4.4 Supporting Line Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
希拉里和特朗普第一次辩论全文(英文)
希拉⾥和特朗普第⼀次辩论全⽂(英⽂)Round One: Achieving ProsperityThe question is about putting more money into the Americans’ pockets.First we have to build an economy that works for everyone, not just those at the top.That means we need more jobs, good jobs with rising incomes.I want us to invest in you. I want us to invest in your future.That means jobs in infrastructure and in mass manufacturing, in innovation and technology, clean renewable energy and in small businesses because most of the jobs will come from small business.We also have to make the economy fairer. That starts with raising the nationalminimum wage and also guarantee finely equal pay for women‘’s work.I also wane see more companies do profit sharing. If you helped make the profit you should be able to share them, not just the executives at the top.And I want us to do more to support people who are struggling to balance family and work. I’ve heard from so many of you about the difficulties you face and the stresses you' re under.And let’s have paid family leave, earned sick days. Let’s make sure we have affordable childcare and debt-free college. How are we gone do it ? We' re gonna do it by making the wealthy pay their fair share, close thecorporate loopholes.Our jobs are fleeing the country. They are going to Mexico. They are going to many other countries.You look at what China' s doing to our country in terms of making our product.They ' re devaluing their currency and there ody in’ours government to fight them. And we have a very good fight and we have a winning fight because they are using our country as a piggy bank to rebuild China and many other countries are doing the same thing. So we are losing our good jobs, so many of them.When you look at what' s happening in Mexico, one of my friend built a plantsaid it' s the eighth wonder of the world. They're building the big estsome of plants, anywhere in the world, some of the most sophisticated, some of thebest plants. With the United States, as you said, not so much.So Ford is leaving. You see, that, their small cars division, leaving. Thousands of jobs, leaving Michigan, leaving Ohio. They ' re all leaving. And we cannot allow it to happen anymore as far as childcare is concerned and so many other things.I think Hillary and I agree on that and we probably disagree on numbers and amounts and what we're going to do but perhaps we'll talk aboutthat later.But we have to stop our jobs from being stolen from us. We have to stop our companies from leaving the United States, and with it, firing all of their people.All you have to do is take a look at the Carrier air-conditioning in Indianapolis. They left fired 1,400 people. They are going to Mexico. So many, hundreds and hundreds companies are doing this. We cannot let it happen.Under my plan, I' ll be reducing taxes tremendously from 35% to 15% for companies, small and big businesses.That 's gonna be a job creator, like, we haven't seen since Ronald Regan. It going to be a beautiful thing to watch. Companies will come. They will build. They will expand. New companies will start. And I look very very much forward to doing it.We have to renegotiate our trade deals. We have to stop these countries from stealing our companies and our jobs.Round Two: Let's Talk About RaceRace remains a significant challenge in our country. Unfortunately race still determines too much. Often determines where people live. Determines what kind of education in their public schools they can get. And, yes, it determines how they’re treated in the criminal justice system.We’ve just seen those tragic examples in both Tulsa and Charlotte. And we’got to do several things at the same time.We have to restore trust between communities and the police.We have to work to assure that our police are using the best training, the best technique, that they' re well prepared to use force only when necessary.Everyone should be respected by the law. And everyone should respect the law. Right now that’s not the case in a lot of our neighborhoods. So I have, ever since the first day of my campaign, called for criminal justice reform.I 've laid out a platform that I think will begin to remedy the problems we have in the criminal justice system.But we also have to recognize that in addition to the challenges we face with policing, there are so many brave, good police officers who equally want reform.So we have to bring communities together in order to work on that as a mutual goal.And we’ve got to get guns out of the hands of the people who should not have them.The gun epidemic is the leading cause of death of young African-American men, more than the next 9 causes put together. So we have to do 2 things as I said. We have to restore trust. We have towork with the police. We have to make sure they respect the communities and the communities respect them.And we have to tackle the plague of gun violence which is a contributor to a lot of problems we are seeing today.First of all, Secretary Clinton dosen law and order. We need law and order. We don't wanna use a couple of words. And that ' t have it. We are not gonna'twanna use a couple of words. And that' t have it. We are not gonna'twanna use a couple of words. And that’t have it. We are not gonna'have a countryAnd when I look at what 's going on in Charlotte, the city I lovthe,city I have investments, when I look at what's going on through various parts of our country, whether it' s, I mean, I can just keep naming them all day long. We need law and order in our country.And I just got, today, the, eh you know, the endorsement, the fraternal order of police who just came in. We have endorsement from, I think, almost every police group, every, I mean, a lot percentage of them in the United States.We have a situation in which we have, eh, in the city, African-American, Hispanics are living in hell.Because it' s so dangerous. uYowalk down the street, you got shot.In Chicago, they' ve had thousands of shootings, thousands, since January 1st. Thousands of shootings.And I say: where is this? Is this a war-torn country. What are we doing? And we have to stop the violence. We have to bring back law and order in a place like Chicago where thousands of people have been killed, thousands, over the last number of years.In fact, almost 4,000 of people have been killed since Barack Obama became President. Over 4, almost 4,000 people have been killed in Chicago. We have to bring back law and order.Now, whether or not, in a place like Chicago, you do stop and frisk witch worked very well, Mary Julian is here, worked very well in New York. It brought the crime rate way down.But you take the gun away from criminals who shouldn't be having it. We have gangs roaming the street. And in many cases they' re illegally here, illegal immigrants. And they have guns. They shoot people.And we have to be very strong. We have to be very vigilant. We have to be, we have to know what we are doing.Right now our police, in many cases, are afraid to do anything. We have to protect our inner cities because African-American communities are being decimated by crime. Decimated!Round Three: Securing AmericaHillary:I think cyber security, cyber warfare will be one of the biggest challengesfacing the next president because clearly we are facing at this point 2 different kinds of ad versaries.There are the independent hacking groups that do, mostly, for commercial reasons to try to steal information that they can use to make money.But increasingly we are seeing cyber attacks coming from states, all kinds of states. The most recent and troubling of these are from Russia. There's no doubt now that Russia has used cyber attacks against all kinds of organizations in our country and I am deeply concerned about this.I know Donald' s, very, paise-worthy of Vladimir Putin. But Putin is playing a really tough long game here.And one of the things he's done is to let loose cyber attackers to hack into government files, to hack into personal files, hack into the Democratic National Committee.And we recently have learnt that, you know, this is one of their preferred methods of trying to wreak havoc and collect information.We need to make it very clear, whether it's Russia, China, Iran or anybody else.The United States has a much greater capacity and we are not gonna sit idly by and permit state actors to go after our information, our private sector information and our public sector information.And we are going to have to make it clear that we don' t want to use thekinds of tools that we have to engage in a different kind of warfare.But we will defend the citizens of this country. And the Russians need to understand that.I think they' ve been treating it as it is almost a probing. Howfar will we go?How much will we do? And that's why I was so, I was so shocked when Donald publicly invited Putin to hack into Americans. That is just unacceptable. That’s onereasonsofthewhy 15 national security officials who serve in Republican Information Administration have said that Donald is unfit to be the commander in chief.There’s comments like that really worry people who understand the threats that we face.Trump:Yeah, I do wanna say that I was just endorsed and more are coming the next week. It will be over 200 admirals, many of them are here, admirals and generals endorsed me to lead this country.That just happened and many more are coming. And I' m very proud of it. In addition I was just endorsed by ICE. They've never endorsed anybody before. On immigration I was just endorsed by ICE. I was just endorsed. 7,500 patrol agents.So when Secretary Clinton talks about this I' ll take the admirals and I' ll t generals, any day over the political hacks that I see that has led ourcountry so brilliantly over the last 10 years with their knowledge. Because look at the mess we' re in. Look at the mess we' re in. As long as there's the cyber, I agree to parts of what Secretary Clinton said. We should be better than anybody else and perhaps we’re not.I don't know if anybody knows that it's Russia that broke into the DNC. She saying Russia RussiaRussia. But I don't. Maybe it was. It could be Russia. But t could also be China. It could also be a lot of other people. Could also be somebody sitting on their bed that weighs 400 pounds, OK?You don' t know who broke into DNC. But what do we learn with DNC? We learnt that Bernie Sanders was taken advantage of by your people, by Debbie Wasserman Schultz. Look at what happened to her.But Bernie Sanders was taken advantage of. That's what we learned. Now, whether that was Russia, whether that was China, whether it was another country, we don't know, because the truth is, under President Obama we've lost control of things that we used to have control over. We came in with the Internet, we came up with the Internet, and I think Secretary Clinton and myself would agree very much, when you look at what ISIS is doing with the Internet, they're beating us at our own game. ISIS.So we have to get very, very tough on cyber and cyber warfare.It is —it is a huge problem. I have a son. He's 10 years old. He hascomputers.He is so good with these computers, it's unbelievable.The security aspect of cyber is very, very tough. And maybe it's hardly doable.But I will say, we are not doing the job we should be doing. But that's true throughout our whole governmental society.。
Conformal restriction the chordal case
a rX iv:mat h /29343v2[mat h.PR]25Apr23Conformal restriction:the chordal case Gregory Lawler ∗Oded Schramm †Wendelin Werner ‡Abstract We characterize and describe all random subsets K of a given simply connected planar domain (the upper half-plane H ,say)which satisfy the “conformal restriction”property,i.e.,K connects two fixed boundary points (0and ∞,say)and the law of K conditioned to remain in a simply connected open subset H of H is identical to that of Φ(K ),where Φis a conformal map from H onto H with Φ(0)=0and Φ(∞)=∞.The construction of this family relies on the stochastic Loewner evolution processes with parameter κ≤8/3and on their distortion under conformal maps.We show in particular that SLE 8/3is the only random simple curve satisfying conformal restriction and relate it to the outer boundaries of planar Brownian motion and SLE 6.Keywords:Conformal invariance,restriction property,random fractals,SLE.MSC Classification:60K35,82B27,60J69,30C99Contents1Introduction3 2Preliminaries7 3Two-sided restriction10 4Brownian excursions15 5Conformal image of chordal SLE19 6Restriction property for SLE8/323 7Bubbles267.1Brownian bubbles (26)7.2Adding a Poisson cloud of bubbles to SLE (29)8One-sided restriction318.1Framework (31)8.2Excursions of reflected Brownian motions (31)8.3The SLE(κ,ρ)process (34)8.4Proof of Theorem8.4 (38)8.5Formal calculations (41)9Equivalence of the frontiers of SLE6and Brownian motion449.1Full plane SLE6and planar Brownian motion (44)9.2Chordal SLE6and reflected Brownian motion (46)9.3Chordal SLE6as Brownian motion reflected on its past hull..499.4Non-equivalence of pioneer points and SLE6 (50)9.5Conditioned SLE6 (52)10Remarks5221IntroductionConformalfield theory has been extremely successful in predicting the ex-act values of critical exponents describing the behavior of two-dimensional systems from statistical physics.In particular,in the fundamental papers [5,6],which were used and extended to the case of the“surface geometry”in[9],it is argued that there is a close relationship between critical planar systems and some families of conformally invariantfields.This gave rise to intense activity both in the theoretical physics community(predictions on the exact value of various exponents or quantities)and in the mathematical community(the study of highest-weight representations of certain Lie alge-bras).However,on the mathematical level,the explicit relation between the two-dimensional systems and thesefields remained rather mysterious.More recently,a one-parameter family of random processes called stochas-tic Loewner evolution,or SLE,was introduced[44].The SLEκprocess is ob-tained by solving Loewner’s differential equation with driving term B(κt), where B is one-dimensional Brownian motion,κ>0.The SLE processes are continuous,conformally invariant scaling limits of various discrete curves arising in the context of two-dimensional systems.In particular,for the models studied by physicists for which conformalfield theory(CFT)has been applied and for which exponents have been predicted,it is believed that SLE arises in some way in the scaling limit.This has been proved for site-percolation on the triangular lattice[46],loop-erased random walks[29] and the uniform spanning tree Peano path[29](a.k.a.the Hamiltonian path on the Manhattan lattice).Other models for which this is believed include the Ising model,the random cluster(or Potts)models with q≤4,and the self-avoiding walk.In a series of papers[23,24,25,26],the authors derived various prop-erties of the stochastic Loewner evolution SLE6,and used them to compute the“intersection exponents”for planar Brownian paths.This program was based on the earlier realization[32]that any conformally invariant process satisfying a certain restriction property has crossing or intersection expo-nents that are intimately related to these Brownian intersection exponents. In particular,[32]predicted a strong relation between planar Brownian mo-tion,self-avoiding walks,and critical percolation.As the boundary of SLE6is conformally invariant,satisfies restriction,and can be well understood,com-putations of its exponents yielded the Brownian intersection exponents(in particular,exponents that had been predicted by Duplantier-Kwon[15,14],3disconnection exponents,and Mandelbrot’s conjecture[34]that the Haus-dorffdimension of the boundary of planar Brownian motion is4/3).Sim-ilarly,the determination of the critical exponents for SLE6in[23,24,25] combined with Smirnov’s[46]proof of conformal invariance for critical per-colation on the triangular lattice(along with Kesten’s hyperscaling relations) facilitated proofs of several fundamental properties of critical percolation [47,28,45],some of which had been predicted in the theoretical physics literature,e.g.,[37,35,36,38,43].The main goal of the present paper is to investigate more deeply the restriction property that was instrumental in relating SLE6to Brownian mo-tion.One of our initial motivations was also to understand the scaling limit and exponents of the two-dimensional self-avoiding walk.Another motiva-tion was to reach a clean understanding of the relation between SLE and conformalfield theory.Consequences of the present paper in this direction are the subject of[18,19].See also[2,3]for aspects of SLE from a CFT perspective.Let us now briefly describe the conformal restriction property which we study in the present paper:Consider a simply connected domain in the complex plane C,say the upper half-plane H:={x+iy:y>0}.Suppose that two boundary points are given,say0and∞.We are going to study closed random subsets K of H such that:•1.The restriction measure Pαexists if and only ifα≥5/8.2.The only measure Pαthat is supported on simple curves is P5/8.It isthe law of chordal SLE8/3.3.The measures Pαforα>5/8can be constructed by adding to thechordal SLEκcurve certain Brownian bubbles with intensityλ,where α,λandκare related byα(κ)=6−κ2κ.4.For allα≥5/8,the dimension of the boundary of K defined under Pαisalmost surely4/3and locally“looks like”an SLE8/3curve.In particular, the Brownian frontier(i.e.,the outer boundary of the Brownian path) looks like a symmetric curve.As pointed out in[30],this gives strong support to the conjecture that chordal SLE8/3is the scaling limit of the infinite self-avoiding walk in the upper half-plane and allows one to recover(modulo this conjecture)the critical exponents that had been predicted in the theoretical physics literature(e.g., [36,16]).This conjecture has recently been tested[21,22]by Monte Carlo methods.Let us also mention(but this will not be the subject of the present paper,see[18,19])that in conformalfield theory language,−λ(κ)is the central charge of the Virasoro algebra associated to the discrete models(that correspond to SLEκ)and thatαis the corresponding highest-weight(for a degenerate representation at level2).To avoid confusion,let us point out that SLE6is not a chordal restriction measure as defined above.However,it satisfies locality,which implies a different form of restriction.We give below a proof of locality for SLE6, which is significantly simpler than the original proof appearing in[23].We will also study a slightly different restriction property,which we call right-sided restriction.The measures satisfying right-sided restriction sim-ilarly form a1-parameter collection P+α,α>0.We present several con-structions of the measures P+α.First,whenα≥5/8,these can be obtained from the measures Pα(basically,by keeping only the right-side boundary). Whenα∈(0,1),the measure P+αcan also be obtained from an appropriately reflected Brownian excursion.It follows that one can reflect a Brownian ex-cursion offa ray in such a way that its boundary will have precisely the law5of chordal SLE8/3.A third construction of P+α(valid for allα>0)is given by a process we call SLE(8/3,ρ).The process SLE(κ,ρ)is a variant of SLE where a drift is added to the driving function.In fact,it is just Loewner’s evolution driven by a Bessel-type process.The word chordal refers to connected sets joining two boundary points of a domain.There is an analogous radial theory,which investigates sets joining an interior point to the boundary of the domain.This will be the subject of a forthcoming paper[31].We now briefly describe how this paper is organized.In the preliminary section,we give some definitions,notations and derive some simple facts that will be used throughout the paper.In Section3,we study the family of chordal restriction measures,and show(1.1).Section4is devoted to the Brownian excursions.We define these measures and use a result of B.Vir´a g (see[49])to show that thefilling of such a Brownian excursion has the law P1.The key to several of the results of the present paper is the study of the distortion of SLE under conformal maps,for instance,the evolution of the image of the SLE path under the mappingΦ(as long as the SLE path remains in H),which is the subject of Section5.This study can be considered as a cleaner and more advanced treatment of similar questions addressed in[23]. In particular,we obtain a new short proof of the locality property for SLE6, which was essential in the papers[23,24,25].The SLE distortion behaviour is then also used in Section6to prove that the law of chordal SLE8/3is P5/8and is also instrumental in Section7,where we show that all measures Pαforα>5/8can be constructed by adding a Poisson cloud of bubbles to SLE curves.The longer Section8is devoted to the one-sided restriction measures P+α. As described above,we exhibit various constructions of these measures and show as a by-product of this description that the two-sided measures Pαdo not exist forα<5/8.A recurring theme in the paper is the principle that the law P of a random set K can often be characterized and understood through the function A→P[K∩A=∅]on an appropriate collection of sets A.In Section9we use this to show that the outer boundary of chordal SLE6is the same as the outer boundary(frontier)of appropriately reflected Brownian motion and the outer boundary of full-plane SLE6stopped on hitting the unit circle is the same as the outer boundary of Brownian motion stopped on hitting the unit circle.6We conclude the paper with some remarks and pointers to papers in preparation.2PreliminariesIn this section some definitions and notations will be given and some basic facts will be recalled.Important domains.The upper half plane{x+iy:x∈R,y>0} is denoted by H,the complex plane by C,the extended complex plane by ˆC=C∪{∞}and the unit disk by U.Bounded hulls.Let Q be the set of all bounded A⊂A∩H and H\A is(connected and)simply connected.We call such an A a bounded hull.The normalized conformal maps g A.For each A∈Q,there is a unique conformal transformation g A:H\A→H with g A(z)−z→0as z→∞. We can then define(as in[23])z(g A(z)−z).(2.1)a(A):=limz→∞First note that a(A)is real,because g A(z)−z has a power series expansion in1/z near∞and is real on the real line in a neighborhood of∞.Also note thata(A)=limy H(i y),(2.2)y→∞where H(z)=Im z−g(z) is the bounded harmonic function on H\A with boundary values Im z.Hence,a(A)≥0,and a(A)can be thought of as a measure of the size of A as seen from infinity.We will call a(A)the half-plane capacity of A(from infinity).The useful scaling rule for a(A),a(λA)=λ2a(A)(2.3) is easily verified directly.Since Im g A(z)−Im z is harmonic,bounded,and has non-positive boundary values,Im g A(z)≤Im z.Consequently,0<g′A(x)≤1,x∈R\A.(2.4) (In fact,g′A(x)can be viewed as the probability of an event,see Proposition 4.1.)7∗-hulls.Let Q∗be the set of A∈Q with0∈A.We call such an A a∗-hull. If A∈Q∗,then H=H\A is as the H in the introduction.The normalized conformal mapsΦA.For A∈Q∗,we defineΦA(z)= g A(z)−g A(0),which is the unique conformal transformationΦof H\A onto Hfixing0and∞withΦ(z)/z→1as z→∞.Semigroups.Let A be the set of all conformal transformationsΦ:H\A→H withΦ(0)=0andΦ(∞)=∞,where A∈Q∗.That is,A={λΦA:λ> 0,A∈Q∗}.Also let A1={ΦA:A∈Q∗}.Note that A and A1are both semigroups under composition.(Of course,the domain ofΦ1◦Φ2isΦ−12(H1) if H1is the domain ofΦ1.)We can consider Q∗as a semigroup with the product·,where A·A′is defined byΦA·A′=ΦA◦ΦA′.Note thata(A·A′)=a(A)+a(A′).(2.5) As a(A)≥0,this implies that a(A)is monotone in A.±-hulls.Let Q+be the set of A∈Q∗with A∩R⊂(0,∞).Letσdenote the orthogonal reflection about the imaginary axis,and let Q−={σ(A):A∈Q+}be the set of A∈Q∗with A∩R⊂(−∞,0).If A∈Q∗,then we can find unique A1,A3∈Q+and A2,A4∈Q−such that A=A1·A2=A4·A3. Note that Q+,Q−are semigroups.Smooth hulls.We will call A∈Q a smooth hull if there is a smooth curveγ:[0,1]→C withγ(0),γ(1)∈R,γ(0,1)⊂H,γ(0,1)has no self-intersections,and H∩∂A=γ(0,1).Any smooth hull in Q∗is in Q+∪Q−. Fillings.If A⊂H such that any path from z to∞inH\A.Similarly, F R H(A)denotes the union of A with the connected components ofH\A which do not intersect[0,∞).Also,for closed A⊂C, F C(A)denotes the union of A with the bounded connected components of C\A.Approximation.We will sometimes want to approximate A∈Q by smooth hulls.The idea of approximating general domains by smooth hulls is standard (see,e.g.,[17,Theorem3.2]).8Lemma2.1.Suppose A∈Q+.Then there exists a decreasing sequence of smooth hulls(A n)n≥1such that A= ∞n=1A n and the increasing sequence Φ′An(0)converges toΦ′A(0).Proof.The existence of the sequence A n can be obtained by various means,for example,by considering the image underΦ−1A of appropriately chosenpaths.The monotonicity ofΦ′An (0)follows immediately from the monotonic-ity of A n and(2.4).The convergence is immediate by elementary properties of conformal maps,sinceΦAnconverges locally uniformly toΦA on H\A. Covariant measures.Our aim in the present paper is to study measures on subsets of H.In order to simplify further definitions,we give a general definition that can be applied in various settings.Suppose thatµis a measure on a measurable spaceΩwhose elements are subsets of a domain D.Suppose thatΓis a set of conformal transformations from subdomains D′⊂D onto D that is closed under composition.We say thatµis covariant underΓ(orΓ-covariant)if for allϕ∈Γ,the measureµrestricted to the setϕ−1(Ω):={ϕ−1(K):K⊂D}is equal to a constant Fϕtimes the image measureµ◦ϕ−1.Ifµis afiniteΓ-covariant measure,then Fϕ=µ[ϕ−1(Ω)]/µ[Ω].Note that a probability measure P onΩisΓ-covariant if and only if for allϕ∈Γwith Fϕ=P[ϕ−1(Ω)]>0,the conditional law of P onϕ−1(Ω)is equal to P◦ϕ−1.Also note that ifµis covariant underΓ,then Fϕ◦ψ=FϕFψfor allϕ,ψ∈Γ, because the image measure ofµunderϕ−1is F−1ϕµrestricted toϕ−1(Ω),sothat the image underψ−1of this measure is F−1ϕF−1ψµrestricted toψ−1◦ϕ−1(Ω).Hence,the mapping F:ϕ→Fϕis a semigroup homomorphism fromΓinto the commutative multiplicative semigroup[0,∞).Whenµis a probability measure,this mapping is into[0,1].We say that a measureµisΓ-invariant if it isΓ-covariant with Fϕ≡1. Chordal Loewner chains.Throughout this paper,we will make use of chordal Loewner chains.Let us very briefly recall their definition(see[23]for details).Suppose that W=(W t,t≥0)is a real-valued continuous function. Define for each z∈g t(z)−W t,g0(z)=z.(2.6) For each z∈Loewner evolution is defined as K t:={z∈κB t,where B is a standard one-dimensional Brownian motion,then the corresponding random Loewner chain is chordal SLEκ(SLE stands for stochastic Loewner evolution).3Two-sided restrictionIn this section we will be studying certain probability measures on a collection Ωof subsets of H.We start by definingΩ.Definition3.1.LetΩbe the collection of relatively closed subsets K of H such that1.K is connected,K is connected.A simple example of a set K∈Ωis a simple curveγfrom0to infinity in the upper half-plane.Ifγis just a curve from zero to infinity in the upper half-plane with double-points,then one can take K=F R H(γ)∈Ω,which is the set obtained byfilling in the loops created byγ.We endowΩwith theσ-field generated by the events{K∈Ω:K∩A=∅},where A∈Q∗.It is easy to check that this family of events is closed underfinite intersection,so that a probability measure onΩis characterized by the values of P[K∩A=∅]for A∈Q∗.Thus:Lemma3.2.Let P and P′be two probability measures onΩ.If P[K∩A=∅]=P′[K∩A=∅]holds for every A∈Q∗,then P=P′.It is worthwhile to note that theσ-field onΩis the same as the Borel σ-field induced by the Hausdorffmetric on closed subsets of2.P is A-covariant.3.There exists anα>0such that for all A∈Q∗,P[K∩A=∅]=Φ′A(0)α.4.There exists anα>0such that for all smooth hulls A∈Q∗,P[K∩A=∅]=Φ′A(0)α.Moreover,for eachfixedα>0,there exists at most one probability measure Pαsatisfying these conditions.Definition3.4.If the measure Pαexists,we call it the two-sided restriction measure with exponentα.Proof.Lemma3.2shows that a measure satisfying3is unique.A probability measure isΓ-covariant if and only if it isΓ-invariant.Therefore,1and2are equivalent.As noted above,any A∈Q∗can be written as A+·A−with A±∈Q±.Using this and Lemma2.1,we may deduce that conditions3and4are also equivalent.SinceΦ′λA(0)=Φ′A(0)for A∈Q∗,λ>0,3together withLemma3.2imply that P isΓ-invariant.BecauseΦ′A1·A2(0)=Φ′A1(0)Φ′A2(0),3also implies that for all A1,A2∈Q∗,P[K∩(A1·A2)=∅]=P[K∩A1=∅]P[K∩A2=∅],which implies1.Hence,it suffices to show that1implies4.Suppose1holds.Define the homomorphism F of Q∗onto the multiplica-tive semigroup(0,1]by F(A)=P[K∩A=∅].We also write F(ΦA)for F(A).Let G t(z)be the solution of the initial value problem∂t G t(z)=2G t(z)H.Note that this function can equivalently be defined as G t(z)= g t(z)−g t(0)=g t(z)+2t,where(g t)is the chordal Loewner chain driven by the function W t=1−2t.Hence,G t is the unique conformal map from H\K t onto H such that G t(0)=0and G t(z)/z→1when z→∞.(Here,K t is the evolving hull of g t.)Also,and this is why we focus on these functions G t,one has G t◦G s=G t+s in H\K t+s,for all s,t≥0.Since F is a homomorphism,11this implies that F (G t )=exp(−2αt )for some constant α≥0and all t ≥0,or that F (G t )=0for all t >0.However,the latter possibility would imply that K ∩K t =∅a.s.,for all t >0.Since t>0K t ={1}and 1/∈H \A and n A n is bounded away from 0and ∞.(Thisis very closely related to what is known as the Carath´e odory topology.)Now assume that A n →A ,where A n ,A ∈Q +.It is immediate that Φ′A n (0)→Φ′A (0),by Cauchy’s derivative formula (the maps may be extended to a neighborhood of 0by Schwarz reflection in the real line).Set A +n =ΦA n (A \A n )and A −n =ΦA (A n \A ).We claim that there is a constant δ>0and a sequence δn →0such thatA +n ∪A −n ⊂{x +iy :x ∈[δ,1/δ],y ≤δn }.(3.3)Indeed,since the map ΦA n ◦Φ−1A converges to the identity,locally uniformlyin H ,it follows (e.g.,from the argument principle)that for every compact set S ⊂H for all sufficiently large n ,S is contained in the image of ΦA n ◦Φ−1A ,which means that A +n ∩S =∅.Similarly,Φ−1A ◦ΦA n converges locally uniformly inthere is someǫ>0such that for infinitely many n P[K∩A−n]>ǫ.There-fore,with positive probability,K intersects infinitely many A−n.Since K is closed and(3.3)holds,this would then imply that P K∩[δ,1/δ] >0,a contradiction.Thus lim sup n→∞F(A n)≤F(A).A similar argument alsoshows that lim inf n→∞F(A n)≥F(A),and so lim n→∞F(A n)=F(A),and the continuity of F is verified.To complete the proof of the Lemma,we now show that A0is dense in Q+.Let A∈Q+.Set A′:=A∪[x0,x1],where x0:=inf(A∩R)and x1:=sup(A∩R).Forδ>0,δ<ΦA(x0)/2,let Dδbe the set of points in H with distance at mostδfrom[ΦA(x0),ΦA(x1)].Let Eδdenote theclosure of A∪Φ−1A (Dδ).It is clear that Eδ→A asδ→0+in the topologyconsidered above.It thus suffices to approximate Eδ.Note thatH withβ(0),β(s)∈R.We may assume thatβis parametrized by half-plane capacity from∞,so that a β[0,t] = 2t,t∈[0,s].Set g t:=gβ[0,t],Φt:=Φβ[0,t]=g t−g t(0),U t:=g t β(t) ,˜Ut:=U t−g t(0)=Φt β(t) ,t∈[0,s].By the chordal version of Loewner’s theorem,we have∂t g t(z)=2Φt(z)−˜U t +2(Φt(z)−˜U t)˜U t,Φ0(z)=z.(3.4)Since˜U t is continuous and positive,there is a sequence of piecewise constant functions˜U(n):[0,s]→(0,∞)such that sup{|˜U(n)t−˜U t|:t∈[0,s]}→0 as n→∞.LetΦ(n)t be the solution of(3.4)with˜U(n)t replacing˜U t.Then, clearly,Φ(n)s(z)→Φs(z)=ΦEδlocally uniformly inΦ′A+(0)are bounded away from zero,but limǫց0Φ′A(0)=0.AsΦA∗−◦ΦA+=ΦA=ΦA∗+◦ΦA−,(3.5) we haveΦ′A∗−(0)→0whenǫ→0.By applying F to(3.5)we getΦ′A∗−(0)α−Φ′A+(0)α=Φ′A∗+(0)αΦ′A−(0)α−.AsΦ′A∗−(0)=Φ′A∗+(0)(by symmetry),this means thatΦ′A∗−(0)α−α−staysbounded and bounded away from zero asǫց0,which givesα−=α.Since every A∈Q∗can be written as A+·A−this establishes4withα≥0.The caseα=0clearly implies K=∅a.s.,which is not permitted.This completes the proof.Let us now conclude this section with some simple remarks:Remark3.6.If K1,...,K n are independent sets with respective laws Pα1,...,Pαn,then the law of thefilling K:=F R H(K1∪...∪K n)of the union of the K j’s is Pαwithα=α1+···+αn becauseP[K⊂Φ−1(H)]=nj=1P[K j⊂Φ−1(H)]=Φ′(0)α1+···+αn.Remark3.7.Whenα<1/2,the measure Pαdoes not exist.To see this, suppose it did.Since it is unique,it is invariant under the symmetryσ: x+iy→−x+iy.Let A={e iθ:θ∈[0,π/2]}.Since K is almost surely connected and joins0to infinity,it meets either A orσ(A).Hence,symmetry implies thatΦ′A(0)α=P[K∩A=∅]≤1/2.On the other hand,one can calculate directlyΦ′A(0)=1/4,and henceα≥1/2.We will show later in the paper(Corollary8.6)that Pαonly exists for α≥5/8.Remark3.8.We have chosen to study subsets of the upper half-plane with the two special boundary points0and∞,but our analysis clearly applies to any simply connected domain O=C with two distinguished boundary points a and b,a=b.(We need to assume that the boundary of O is sufficiently nice near a and b.Otherwise,one needs to discuss prime ends in place of the distinguished points.)For instance,if∂O is smooth in the neighborhood of a and b,then for a conformal mapΦfrom a subset O′of O onto O,we getP[K∩(O\O′)=∅]=(Φ′(a)Φ′(b))α,14where P denotes the image of Pαunder a conformal map from H to O that takes a to0and b to∞.Remark3.9.The proof actually shows that weaker assumptions onΩare sufficient for the proposition.DefineΩb just asΩwas defined,except that Condition1is replaced by the requirements that K=∅andH\{0} such thatγ[0,1]∩R={γ(0),γ(1)}and and with positive P-probability K∩γ[0,1]=∅andγ[0,1]separates K inK and K is unbounded.Thus,P[Ω]=1. Using this fact,Lemma3.2may be applied,giving the remaining implication 3⇒2.4Brownian excursionsAn important example of a restriction measure is given by the law of the Brownian excursion from0to infinity in H.Loosely speaking,this is simply planar Brownian motion started from the origin and conditioned to stay in H at all positive times.It is closely related to the“complete conformal invariance”of(slightly different)measures on Brownian excursions in[32,27].Let X be a standard one-dimensional Brownian motion and Y an inde-pendent three-dimensional Bessel process(see e.g.,[41]for background on three-dimensional Bessel processes,its relation to Brownian motion condi-tioned to stay positive and stochastic differential equations).Let us briefly re-call that a three-dimensional Bessel process is the modulus(Euclidean norm) of a three-dimensional Brownian motion,and that it can be defined as the solution to the stochastic differential equation dY t=dw t+dt/Y t,where w is standard Brownian motion in R.It is very easy to see that(1/Y t,t≥t0)is15a local martingale for all t0>0,and that if T r denotes the hitting time of rby Y,then the law of(Y Tr+t ,t<T R−T r)is identical to that of a Brownianmotion started from r and conditioned to hit R before0(if0<r<R).Note that almost surely lim t→∞Y t=∞.The Brownian excursion can be defined as B t=X t+iY t.In other words, B has the same law as the solution to the following stochastic differential equation:dB t=dW t+i1Proof.LetΦ=ΦA.Suppose that W is a planar Brownian motion and Z is a Brownian excursion in H,both starting at z∈H\A.When Im(z)→∞, Im(Φ−1(z))=Im(z)+o(1).Hence,with a large probability(when R is large),a Brownian motion started from z∈I R(respectively,z∈Φ−1(I R)) will hitΦ−1(I R)(resp.,I R)before R.The strong Markov property of planar Brownian motion therefore shows that when R→∞,P[W hits I R before A∪R]∼P[W hitsΦ−1(I R)before A∪R].But sinceΦ◦W is a time-changed Brownian motion,andΦ:H\A→H,the right-hand is equal to the probability that a Brownian motion started from Φ(z)hits I R before R,namely,Im(Φ(z))/R.Hence,P[Z hits I R before A]=P[W hits I R before A∪R]Im(z)+o(1)when R→∞.In the limit R→∞,we getP[Z⊂H\A]=Im[Φ(z)]Im(z).(4.2)When z→0,Φ(z)=zΦ′(0)+O(|z|2)so thatP[B[0,∞)∩A=∅]=lims→0P[B[s,∞)⊂H\A]=lims→0E Im(Φ(B s))Figure R×[0,1].Using almost the same proof as in Proposition4.1(but keeping track ofthe law of the path),one can prove the following:Lemma4.2.Suppose A∈Q∗and B is a Brownian excursion in H starting at0.Then the conditional law of(ΦA(B(t)),t≥0)given B∩A=∅is the same as a time change of B.Finally,let us mention the following result that will be useful later on. Lemma4.3.Let P x+iy denote the law of a Brownian excursion B starting at x+iy∈y=a(A)yx2+y2 ,and the lemma readily follows.18Figure5.1:The various maps.Using Cauchy’s Theorem,for example,it is easy to see that the second statement of the lemma may be strengthened toy ∞−∞P x+iy B[0,∞)∩A=∅ dx=πa(A),y>sup{Im z:z∈A}.(4.3) 5Conformal image of chordal SLELet W:[0,∞)→R be continuous with W0=0,and let(g t)be the(chordal) Loewner chain driven by W satisfying(2.6).It is easy to verify by differen-tiation and(2.6)that the inverse map f t(z)=g−1t(z)satisfies2f′t(z)∂t f t(z)=−+o(z−1),z→∞,zwhere the coefficient a(t)depends on G and W t.Note that˜g t satisfies the Loewner equation∂t a(t)∂t˜g t(z)=where ˜Wt :=h t (W t ),h t :=˜g t ◦G ◦g −1t =g A t .(This follows from the proof of Loewner’s theorem,because ˜g t (˜K t +δ\˜K t )lies in a small neighborhood of ˜W t when δ>0is small.Also see [23,(2.6)].)The identity (2.5)gives a (g t (K t +∆t \K t ))=2∆t .The image of ˜K t +∆t \˜K t under˜g t is h t g t (K t +∆t \K t ) .The scaling rule (2.3)of a tells us that as ∆t →0+,the half-plane capacity of h t g t (K t +∆t \K t ) is asymptotic to h ′t (W t )2·2∆t .(The higher order derivatives of h t can be ignored,as follows from (2.2).Also see [23,(2.7)].)Hence,∂t a (t )=2h ′t (W t )2.(5.1)Using the chain rule we get[∂t h t ](z )=2h ′t (W t )2z −W t .(5.2)This formula is valid for z ∈H \g t (A )as well as for z in a punctured neighborhood of W t in R .In fact,it is also valid at W t with[∂t h t ](W t )=lim z →W t 2h ′t (W t )2z −W t =−3h ′′t (W t ).Computations of a similar nature appear (in a deterministic setting)in [11].Differentiating (5.2)with respect to z gives the equation[∂t h ′t ](z )=−2h ′t (W t )2h ′t (z )(z −W t )2−2h ′′t (z )2h ′t (W t )−4h ′′′t (W t )κdB tfor some measurable process b t adapted to the filtration of B t which satisfies t 0|b s|ds <∞a.s.for every t >0.20。
发明物英语作文
Invention is the cornerstone of human progress and has shaped the world we live in today.When writing an essay on an invention,its important to consider several key aspects:the purpose of the invention,its impact on society,and the process of its creation. Heres a detailed outline for an English essay on an invention:Title:The Marvel of Invention:Transforming LivesIntroduction:Hook the reader with a thoughtprovoking statement or question about the role of inventions in human history.Briefly introduce the chosen invention and its significance.Paragraph1:The Inventions GenesisDescribe the historical context in which the invention was created.Discuss the inventors background and motivations.Explain the initial challenges faced during the invention process.Paragraph2:The Functionality of the InventionDetail the inventions primary function and how it works.Use technical descriptions to explain the mechanics or principles behind the invention.Paragraph3:The Impact on SocietyDiscuss the immediate and longterm effects of the invention on society.Include examples of how the invention has improved or changed daily life,industry,or culture.Paragraph4:The Inventions EvolutionTrace the development of the invention over time,noting any significant improvements or adaptations.Discuss how the invention has been integrated into modern technology or practices.Paragraph5:Controversies and Ethical ConsiderationsAddress any controversies or ethical dilemmas associated with the invention. Consider the environmental,social,or health implications of the invention. Paragraph6:Personal ReflectionShare personal thoughts or experiences related to the invention.Reflect on how the invention has influenced ones perspective on technology and innovation.Conclusion:Summarize the main points discussed in the essay.Reiterate the inventions importance and its potential for future development.End with a call to action or a thoughtprovoking statement about the role of invention in shaping the future.Word Bank:Innovation:The introduction of new ideas,methods,or products.Prototype:An early model of a product used for testing and development. Paradigm shift:A fundamental change in approach or underlying assumptions. Technological advancement:Progress in the development and application of technology. Ethical dilemma:A situation that requires a choice between two or more conflicting values or principles.Sample Sentences:The invention of the steam engine by James Watt in the18th century was not just a mechanical breakthrough it was the catalyst for an industrial revolution.The advent of the internet has revolutionized communication,making the world a global village where information is shared instantaneously.While the benefits of genetic engineering are immense,it also raises ethical questions about the manipulation of life at its most fundamental level.Remember to use a variety of sentence structures and vocabulary to maintain the readers interest.Additionally,ensure that your essay is wellorganized,with a clear flow of ideas that logically progresses from one paragraph to the next.。
黎曼猜想英语
黎曼猜想英语The Riemann Hypothesis, named after the 19th-century mathematician Bernhard Riemann, is one of the most profound and consequential conjectures in mathematics. It is concerned with the distribution of the zeros of the Riemann zeta function, a complex function denoted as $$\zeta(s)$$, where $$s$$ is a complex number. The hypothesis posits that all non-trivial zeros of this analytical function have their real parts equal to $$\frac{1}{2}$$.To understand the significance of this conjecture, one must delve into the realm of number theory and the distribution of prime numbers. Prime numbers are the building blocks of arithmetic, as every natural number greater than 1 is either a prime or can be factored into primes. The distribution of these primes, however, has puzzled mathematicians for centuries. The Riemann zeta function encodes information about the distribution of primes through its zeros, and thus, the Riemann Hypothesis is directly linked to understanding this distribution.The zeta function is defined for all complex numbers except for $$s = 1$$, where it has a simple pole. For values of $$s$$ with a real part greater than 1, it converges to a sum over the positive integers, as shown in the following equation:$$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$$。
涌现优于权威英文原文
涌现优于权威英文原文"Emergence Trumps Authority"In today's rapidly changing and interconnected world, the concept of emergence is gaining increasing attention as a more effective way to tackle complex problems and drive innovation. Emergence refers to the phenomenon where new and unexpected patterns, properties, or behaviors emerge from the interactions of simpler elements within a system. This stands in stark contrast to the traditional top-down approach of authority, where decisions and solutions are handed down from a single source of power.The main advantage of emergence over authority is its ability to harness the collective intelligence and creativity of a group. Instead of relying on the expertise of a few individuals at the top, emergence draws on the diverse perspectives and experiences of many. This leads to more robust and innovative solutions, as well as greater buy-in and support from those involved in the process.Furthermore, emergence is better suited to navigate the complexities and uncertainties of modern challenges. With the pace of change accelerating and the interdependencies of various systems becoming more evident, no single authority figure can possibly possess all the knowledge and insight needed to address the diverse and evolving issues we face. In contrast, emergence allows for a more organic and adaptive approach, where solutions can emerge and evolve over time as new information and perspectives come to light.Additionally, emergence encourages participation and empowerment, as individuals feel a sense of ownership and responsibility for the outcomes of the collective efforts. This can lead to increased motivation, collaboration, and resilience within the group, as well as a greater sense of satisfaction and fulfillment for all involved.While authority certainly has its time and place, especially in situations requiring clear direction and decisive action, the benefits of emergence cannot be overlooked. By recognizing and harnessing the power of emergence, organizations, communities, and individuals can better adapt to the complexities and uncertainties of our modern world and drive more effective and sustainable solutions. Ultimately, "Emergence Trumps Authority."。
科学发现需要怀疑和坚持 作文
科学发现需要怀疑和坚持作文英文回答:Scientific discovery is a process of ongoing questioning, exploration, and refinement. It requires adeep-seated skepticism and a relentless pursuit of evidence. The scientific method is built upon the foundation of doubt, as it is through challenging existing knowledge and assumptions that new insights can be gained.The ability to question and challenge is essential for scientific progress. It forces scientists to examine their own beliefs and biases, and to consider alternative explanations for their observations. This critical thinking is vital for avoiding confirmation bias, the tendency to seek out evidence that supports one's existing beliefswhile ignoring evidence that contradicts them.Skepticism, however, must be balanced with a strong commitment to empirical evidence. Scientists must bewilling to abandon their cherished theories if the evidence no longer supports them. This requires a degree of intellectual humility and a willingness to admit when one is wrong.The history of science is replete with examples of discoveries that were made as a result of questioning and persistence. The heliocentric model of the solar system,for example, was initially met with great resistance, but eventually prevailed due to the overwhelming evidence inits favor. Similarly, the germ theory of disease wasinitially ridiculed, but eventually became accepted as the basis for modern medicine.These examples demonstrate the transformative power of scientific inquiry. By embracing doubt and relentlessly pursuing evidence, scientists have made countless discoveries that have revolutionized our understanding of the world and improved our lives.中文回答:科学发现是一个持续不断的质疑、探索和完善的过程。
Decay of singular values of the Gramians of infinite-dimensional systems
Decay of singular values of the Gramians of infinite-dimensional systemsMark R.Opmeer1Abstract—We show that the Gramians of a control systemwith an analytic semigroup,control and observation operatorsthat are not too unbounded and which havefinite-dimensionalinput and output spaces have singular values which decayexponentially in the square root.As a corollary it is shown thatthe Hankel singular values of such control systems also decayexponentially in the square root.Another corollary shows thatsolutions of algebraic Riccati equations for such systems alsohave singular values which decay exponentially in the squareroot.I.I NTRODUCTIONThe Hankel singular values of a control system˙x(t)=Ax(t)+Bu(t),y(t)=Cx(t),provide fundamental information about how well the systemcan be approximated by lower dimensional ones.If G is atransfer function with Hankel singular values(σk)∞k=1,thenfor any transfer function G n of McMillan degree n there holds:σn+1≤ G−G n ∞,where · ∞is the H∞norm.Moreover,there exists a transfer function G n of McMillan degree n such thatG−G n ∞≤∞k=n+1σk.See for example[10,page346and Lemma10.5.3]for the finite-dimensional case and[12]for the general case.A popular model reduction method is balanced truncation[15],[6],[8],[10,Chapter9],[28,Chapter7].To implement this method,two Lyapunov equations have to be solved to obtain the controllability and observability Gramians of the system.For large scale systems,for computational feasibil-ity,usually low rank approximations are computed instead [13],[3].A fundamental question is when such low rank approximations exist.To answer this question,it is necessary and sufficient to analyze the decay of the singular values of the Gramians.The same question regarding low rank approximations is relevant for algebraic Riccati equations instead of Lyapunov equations.The above two questions are closely related since the Han-kel singular values of a system are equal to the square roots of the singular values of the product of the controllability and observability Gramians.Also the singular values of the solution of a Riccati equation can be related to the singular values of a Gramian.1Mark Opmeer is with the Department of Mathematical Sciences,University of Bath,BA27AY Bath,United Kingdom m.opmeer@There are various articles which deal with the above ques-tions in the context offinite-dimensional systems[21],[1], [9].However,the obtained bounds depend on the condition number of the matrix A.Since for discretizations of partial differential equations this condition number converges to infinity as the discretization is refined,these bounds are not very satisfactory.In[18],the infinite-dimensional case was considered and it was shown that for a large class of infinite-dimensional systems the Hankel singular values decay superpolynomially.This implies a result on super-polynomial decay of the singular values of the Gramians. Recently,in[11],it was shown that the singular values of the Gramians of certain infinite-dimensional systems decay at a rate exponential in the square root.However,in this result from[11],a nonstandard definition of singular value is used. This coincides with the standard definition of singular value only if B and C are bounded.In particular,the results of [11]can only be used to deduce decay at a rate exponential in the square root for the Hankel singular values in case B and C are bounded.In this article we prove decay of the singular values(in the standard sense)of the Gramians at a rate exponential in the square root for a large class of infinite-dimensional systems (which includes many systems of interest with unbounded B and C),improving the results of[11].From this we conclude decay at a rate exponential in the square root for the Hankel singular values for this class of systems,thereby improving the results of[18].We also show that for a wide range of algebraic Riccati equations the singular values of the positive semidefinite solution decay at a rate exponential in the square root.We now briefly outline the method of proof.The observ-ability Gramian L is explicitly given byL=∞e A∗t C∗C e At dt.(1) As in[11],we use sinc quadrature on this integral to obtain an explicit approximation of the Gramian of the required rank.In the case of unbounded C,the integrand is unbounded at t=0.What is essentially done in[11]is that instead A−αLA−α=∞e A∗t A−αC∗CA−αe At dt,is considered.Under the assumption that CA−αis bounded for someα∈[0,1](which is a reasonable assumption for applications to PDEs),the integrand is no longer unbounded at t=0.However,the additional factors A−αlead to the nonstandard singular values of L considered in[11]. Contrary to what is done in[11],we deal directly with the2015 European Control Conference (ECC) July 15-17, 2015. Linz, Austriasingularity of the integrand at t=0in(1).Using the theory of analytic semigroups it can be shown that,for the class of systems considered,the integrand has what is called an integrable algebraic singularity at t=0,i.e.it behaves like t−γforγ∈[0,1)near t=0.Since sinc quadrature still gives a convergence rate exponential in the square root in the presence of such a singularity,the decay rate follows.II.A NALYTIC CONTROL SYSTEMSIn this section we define the class of systems to which our results apply and prove some properties for this class of systems.Wefirst recall the definition of an exponentially sta-ble analytic semigroup.Standard references for this include [24],[2],[5],[20].Definition1:Let X be a Hilbert space,letδ∈(0,π/2], and let∆δbe the open sector∆δ:={z∈C:z=0,|arg(z)|<δ}.The family of operators{T(t)∈L(X):t∈∆δ}is called an exponentially stable analytic semigroup if1)z→T(z)is analytic on∆δ,2)T(0)=I and T(z+w)=T(z)T(w)for all z,w∈∆δ,3)there exist M≥1andω>0such that for all z∈∆δT(z) ≤M e−ωRe(z),4)for all x∈X there holds limz→0z∈∆δT(z)x=x.Remark2:We note that most books consider the caseω= 0of bounded analytic semigroups.It is easily seen that if S is a bounded analytic semigroup,then T(z):=e−ωz S(z) is an exponentially stable analytic semigroup in the sense of Definition1.Conversely,if T is an exponentially stable analytic semigroup in the sense of Definition1,then S(z):= eωz T(z)is a bounded analytic semigroup.An analytic semigroup is strongly continuous and there-fore has a generator A(in some sense the semigroup equals z→e Az).A crucial and well-known property of analytic semigroups is the following.Since most books only consider the case of real z in the desired estimate,we give the proof. Lemma3:Let T be an exponentially stable analytic semi-group as in Definition1.Denote its generator by A.Then for all a∈[0,1)there exists a constant M a≥1such that for all z∈∆aδAT(z) ≤M a|z|e−ωRe(z).Proof:We adapt the proof from[2,Theorem3.7.19] (which only considers the case of real z).We have AT(z)= T (z)and by Cauchy’s integral formula for the derivative we therefore haveAT(z)=1i2π|w−z|=rT(w)(w−z)2dw,provided that r>0is such that{w∈C:|w−z|≤r}⊂∆δ. Estimating this integral givesAT(z) ≤M e−ωRe(z)r,so thatzAT(z) ≤M e−ωRe(z)|z|r.We can choose r:=|z|sin(εδ),whereε:=1−a∈(0,1].We then havezAT(z) ≤Msin(εδ)e−ωRe(z).Therefore,definingM a:=Msin(δ−aδ),we obtain the desired result.As in[24,Lemma 3.10.9]we obtain a corollary onfractional powers.Corollary4:Let T be an exponentially stable analyticsemigroup as in Definition1.Denote its generator by A.Letα∈[0,1].Then for all a∈[0,1)there exists a constantM a≥1such that for all z∈∆aδAαT(z) ≤M a|z|αe−ωRe(z).Proof:By[24,Lemma3.9.8]there exists a constantC>0such that for all y∈D(A)Aαy ≤C y 1−α Ay α.Let x∈X and note that then T(z)x∈D(A)and thereforeby the aboveAαT(z)x ≤C T(z)x 1−α AT(z)x α.Now using the bounds from Definition1and Lemma3wehaveAαT(z)x ≤CM1−αe−(1−α)ωRe(z)Mαa|z|αe−αωRe(z),so that withM a:=CM1−αMαa,we obtain the desired estimate.Theorem5:Let T be an exponentially stable analyticsemigroup as in Definition1.Denote its generator by A.LetY be a Hilbert space,letα∈[0,1]and let C:D(A)→Ybe such that CA−α∈L(X,Y).Then for all a∈[0,1)there exists a constant M a≥1such that for all z∈∆aδCT(z) L(X,Y)≤M a|z|αe−ωRe(z).Proof:We have,using Corollary4for the secondinequality,CT(z) L(X,Y)≤ CA−α L(X,Y) AαT(z) L(X)≤ CA−α L(X,Y)M a|z|αe−ωRe(z),so that withM a:= CA−α L(X,Y) M a,we obtained the desired inequality.Remark 6:From Theorem 5it follows that,for A and C as in that theorem and with α∈[0,1/2),the observability Gramian L :=∞T (t )∗C ∗CT (t )dt,is a well-defined bounded operator on X .See e.g.[24,Chapter 10]and [26,Chapter 5]for more information on Gramians for infinite-dimensional systems.Remark 7:Let A and C be as in Theorem 5.For x,y ∈D (A )the function f :∆δ→Cf (z ):= CT (z )x,CT (¯z )y ,is holomorphic on ∆δand by Theorem 5for all a ∈[0,1)there exists a constant M a ≥1such that for all z ∈∆aδ|f (z )|≤M 2a|z |2αe −2ωRe(z ) x y .Note that with L the observability Gramian from Remark 6Lx,y = ∞f (t )dt.III.S INC QUADRATUREWe quote the following result from [14]regarding sincquadrature.Theorem 8:Assume that for ϑ∈(0,π/2),the complex-valued function f is analytic on the sector∆ϑ:={z ∈C :z =0,|arg(z )|≤ϑ},and is such thatsup r ≥0ϑ−ϑ|f (r e iθ)|r dθ<∞,(2) ∞|f (r e iϑ)|dr <∞,(3)and such that there exist M,β,γ>0,such that|f (x )|≤Mx γ−1x ∈(0,1),e −βx x ∈(1,∞).(4)Define for m ∈N 0h := 2πϑγm,N :=1hlogγβmh +1 .Then there exists a M>0such that for all m ∈N 0 ∞0f (x )dx −h N k =−me khf e kh≤ M e −√2πϑγm .Proof:This is [14,Theorem 3.9]with the modification of N for exponential decay from [14,page 77].The following lemma gives a sufficient condition for the conditions from Theorem 8to hold.Lemma 9:If,for ϑ∈(0,π/2),the complex-valued func-tion f is analytic on the sector∆ϑ:={z ∈C :z =0,|arg(z )|≤ϑ},and is such that there exist M,β>0and γ∈(0,1]suchthat for all z ∈∆ϑ|f (z )|≤M |z |γ−1e −βRe(z ),then f satisfies (2),(3)and (4).Proof:That the condition implies (4)is clear.Note that for θ∈[−ϑ,ϑ]there holds|f (r e iθ)|≤Mr γ−1e −βr cos θ≤Mr γ−1e −βr cos ϑ.It follows that (2)holds sinceϑ−ϑ|f (r e iθ)|r dθ≤2ϑMr γe −βr cos ϑ,which is uniformly bounded in r ≥0since γ>0and sinceϑ∈(0,π/2)and β>0we have βcos ϑ>0.It also follows that (3)holds since ∞0|f (r e iϑ)|dr ≤ ∞Mr γ−1e −βr cos ϑdr ≤M 10r γ−1dr +M ∞1e −βr cos ϑdr,and the first integral on the right-hand side is finite since γ>0and the second integral on the right-hand side is finite since βcos ϑ>0.Remark 10:Alternatively,sinc quadrature with regard to a bullet-shaped region rather than with regard to a sector can be used to obtain the same convergence rate under slightly weaker assumptions (which are not relevant for our application);see [25,Example 4.2.11].IV.T HE MAIN RESULTWe now combine the results from Sections II and III to prove our main result.Recall that the singular values of an operator T :X →X are defined for k ∈N byσk (T ):=inf { T −T k :rank (T k )<k }.Theorem 11:Let T be an exponentially stable analytic semigroup as in Definition 1.Denote its generator by A .Let Y be a finite-dimensional Hilbert space,let α∈[0,1/2)and let C :D (A )→Y be such that CA −α∈L (X ,Y ).Denote the observability Gramian of the pair A,C by L .Then there exist M,c >0such that for all k ∈Nσk (L )≤M e −c √k .Proof:Let x,y ∈D (A ).By Remark 7,the function f :∆δ→C defined byf (z ):= CT (z )x,CT (¯z )y ,satisfies the conditions of Lemma 9(with γ:=1−2α,β:=2ωand ϑ:=aδwith a ∈(0,1))and therefore the conditions of Theorem 8.With N and h as in Theorem 8,define for m ∈N 0the operator L m :X →X byL m :=hN k =−me kh T e kh ∗C ∗CT e kh .Note that since e kh>0and the range of T(t)is in D(A) for t>0,the operator L m is well-defined and bounded. As L m is the sum of m+N+1operators of rank at most dim Y,we haverank(L m)≤(m+N+1)dim Y.Since N is bounded by a constant times m,it follows that there exists a ∈N such that for all m∈Nrank(L m)< m.We note thatL m x,y =hNk=−me kh fe kh,andLx,y =∞f(x)dx,so that by Theorem8,there exists a M such that| Lx,y − L m x,y |≤ M e−√2πϑγm.Using that the constant in Remark7is of the form a constant independent of x and y times x times y and keeping track of the constants in the proof of Theorem8in[14],we deduce that M= M1 x y ,where M1is independent of x and y.We infer that| Lx,y − L m x,y |≤ M1e−√2πϑγm x y . Recall that for a self-adjoint operator T there holdsT =supx∈D: x =1| T x,x |,where D is a dense subspace.Applying this with T:=L−L m and D=D(A)givesL−L m ≤ M1e−√2πϑγm.Since,as mentioned above,rank(L m)< m,it follows thatσ m(L)≤ M1e−√2πϑγm.Standard arguments using monotonicity of the singular val-ues then give that there exists M,c>0such that for allk∈Nσk(L)≤M e−c √.The following corollary proves decay of the Hankel sin-gular values.Corollary12:Let T be an exponentially stable analytic semigroup as in Definition1.Denote its generator by A. Let U and Y be Hilbert spaces,at least one of which is finite-dimensional.Letα∈[0,1/2)and let C:D(A)→Y be such that CA−α∈L(X,Y).Letβ∈[0,1/2)and let B:U→D(A∗) be such that B∗A−β∗∈L(X,U).Let the Hankel operator of the triple(A,B,C)be given byH:L2(0,∞;U)→L2(0,∞;Y),(Hu)(t)=∞0CT−1(t+s)Bu(s)ds,where T−1is the extension of the semigroup T to the spaceD(A∗) .Then there exist M,c>0such that for all k∈Nσk(H)≤M e−c√k.Proof:Denote the observability Gramian of the systemby L C and the controllability Gramian of the system byL B.Assume that Y isfinite-dimensional(if U isfinite-dimensional,then the roles of L B and L C below have to bereversed).The assumptions imply that L B is bounded and that L C iscompact.It follows that the product L B L C is compact.It alsofollows from the assumptions that H is compact.Thereforethe nonzero singular values of H equal the square roots ofthe nonzero eigenvalues of the product L B L C(this can beproven as in[4,Lemma8.2.9]).The nonzero eigenvaluesof L B L C equal the nonzero eigenvalues of L1/2BL C L1/2Band since this operator is self-adjoint,these are equal to thenonzero singular values of L1/2BL C L1/2B.So for all k∈Nσk(H)=σk(L1/2BL C L1/2B).It therefore suffices to show that there exists M, c>0suchthat for all k∈Nσk(L1/2BL C L1/2B)≤ M e− c√.If X∈L(X)has rank at most k,then L1/2BXL1/2Balso hasrank at most k.We further haveL1/2BL C L1/2B−L1/2BXL1/2B≤ L1/2BL C−X L1/2B= L B L C−X .We conclude that for all k∈Nσk(L1/2BL C L1/2B)≤ L B σk(L C).The singular value bound for L C proven in Theorem11thengives the desired result.We now give a corollary regarding Riccati equations.Notethat condition(5)below is in particular satisfied in the“standard case”Q NN∗R=[I00I].It is also satisfied for“strictly bounded real”and“strictly positive real”systems.Corollary13:Let T be an exponentially stable analyticsemigroup as in Definition1.Denote its generator by A.Let U be a Hilbert space and let Y be afinite-dimensionalHilbert space.Letα∈[0,1/2)and let C:D(A)→Y besuch that CA−α∈L(X,Y).Letβ∈[0,1/2)and let B:U→D(A∗) be such that B∗A−β∗∈L(X,U).Let D∈L(U,Y).Let the weighting operatorQ NN∗R∈L(Y×U)be self-adjoint and such that there exists aε>0such thatfor all u∈L2(0,∞;U)∞Q NN∗Ry(t)u(t),y(t)u(t)dt≥ε∞u(t) 2+ y(t) 2dt,(5)where y is the output for initial condition zero and input ufor the system˙x(t)=Ax(t)+Bu(t),y(t)=Cx(t)+Du(t).Then the Riccati equationXA+A∗X+C∗QC=(B∗X+D∗QC+N∗C)∗·(D∗QD+D∗N+N∗D+R)−1·(B∗X+D∗QC+N∗C), has a unique positive semidefinite solution X∈L(X)and this solution is such that there exist M,c>0such that forall k∈Nσk(X)≤M e−c √k.Proof:It follows from[24,Theorem 5.7.3]that (A,B,C,D)generate a regular linear system.The existence of a unique positive semidefinite solution of the Riccati equation then follows from[27]or[23].We note that for the special class of systems considered here,the Riccati equation given above coincides with the Riccati equations from[27] and[23](see[22]).From[23,Proposition7.2]we obtain thatX=Ψ∗Q−(Q F+N)R−1(F∗Q+N∗)Ψ,(6)whereΨ∈L(X,L2(0,∞;Y))is the output map,F∈L(L2(0,∞;U),L2(0,∞;Y))is the input-output map and R∈L(L2(0,∞;U),L2(0,∞;U))is the Popov-Toeplitz operator defined byR:=R+N∗F+F∗N+F∗Q F.We note that the observability Gramian of the system equals L:=Ψ∗Ψ.By Theorem11we have that the singular values of L decay exponentially in the square root.Since the nonzero singular values ofΨequal the square roots of the nonzero eigenvalues ofΨ∗Ψ,and sinceΨ∗Ψis self-adjoint implies that its eigenvalues equal its singular values,we see that the singular values ofΨdecay exponentially in the square root.By(6)we have that X equals a bounded operator times an operator whose singular values decay exponentially in the square root.It follows that the singular values of X decay exponentially in the square root.To see this,consider a bounded operator S and an operator T whose singular values decay exponentially in the square root.Then there exist M,c>0such that for all k∈N there exists an operator T k of rank less than k such thatT−T k ≤M e−c √k.It follows that ST k has rank less than k and thatST−ST k ≤ S T−T k ≤ S M e−c √,so that the singular values of ST decay exponentially in the square root.Remark14:We note that a result similar to Corollary13 also holds when the system is exponentially stabilizable(in a suitable sense)rather than exponentially stable.However,a proof of that result requires some infinite-dimensional control theory which is beyond the scope of this article;therefore this result will be presented elsewhere.V.E XAMPLESFor some examples where the conditions of Theorem11 and Corollary12are satisfied we refer to[18,Section3]. In this section we give some examples where the singular values do not decay at a rate exponential in the square root, highlighting the importance of the assumptions in Theorem 11and Corollary12.Wefirst consider what may happen if U and Y are allowed to be infinite-dimensional.Example15:If A is a negative self-adjoint operator witha bounded inverse on a Hilbert space X and C=I(with therefore Y=X),then the observability Gramian equals −12A−ing this,by a suitable choice of X and A, the singular values of the observability Gramian can made equal to any desired sequence of nonnegative numbers.By choosing B=I,the same is true for the Hankel singular values.Note that the assumptions of Theorem11and Corollary 12are not satisfied in this case because both Y and U are allowed to be infinite-dimensional in this example.The other assumptions of Theorem11and Corollary12are satisfied. As the following example shows,even for one-dimensional Y and U arbitrary singular values can be obtained.Example16:This example is taken from[17].Let(σk)∞k=1be a strictly decreasing sequence of positive num-bers.Definec k:=σkk,A ij:=−c i c jσi+σj.Then c defines a bounded operator C: 2(N)→C by Cx:= x,c and A defines a bounded negative self-adjoint operator on 2(N).The observability Gramian of the pair A,C is represented by the diagonal matrix with diagonal entriesσk. Therefore the singular values of the observability Gramianequal the given sequence(σk)∞k=1.If we choose B=C∗, then the Hankel operator of the triple A,B,C has singularvalues equal to the given sequence(σk)∞k=1.In this example all the conditions of Theorem11and Corollary12are satisfied except for the exponential stability assumption.The semigroup generated by A is strongly stable,but not exponentially stable.With a slightly more complicated construction the sameconclusions can be drawn if(σk)∞k=1is a non-increasing sequence of positive numbers,see[16].We note that this example can be easily adapted to show that anyfinite non-increasing sequence of positive numbers(σk)nk=1can occur as the singular values of the Gramian of a finite-dimensional system with dim X=n and dim U=1 and as the sequence of nonzero Hankel singular values of a system with dim X=n and dim U=1=dim Y.The following example illustrates that B and C should not be“too unbounded”for the singular values to decay at a rate exponential in the square root.Example17:We consider the following one-dimensionalheat equation on the unit interval∂w ∂t =∂2w∂ξ2,w (t,0)=−u (t ),w (t,1)=0,y (t )=∂w∂ξ(t,0).In this case CA −αis not in L (X ,Y )for any α<3/4.Therefore,one of the conditions of Theorem 11is not satisfied;the other conditions of Theorem 11are satisfied.It can be shown that the observability Gramian is unbounded.The Hankel operator is also unbounded.In particular,all the singular values of the observability Gramian and all the Hankel singular values are infinite.The next example is from [19].It illustrates the importance of the analyticity of the semigroup.Example 18:Consider the following first order damped hyperbolic PDE with dynamic boundary control and obser-vation.∂w∂t =−∂w ∂ξ−εw,t >0,ξ∈(0,1),w t (t,0)+w (t,0)=u (t ),z t (t )+z (t )=w (t,1),y (t )=z (t ).The transfer function of this system isG (s )=1(s +1)2e −(s +ε).This system satisfies all the assumptions of Corollary 12(with α=β=0)except that the exponentially stable semigroup generated by A isn’t ing the explicit description of the transfer function,it follows from [7]that for (σk )∞k =1the Hankel singular valuesk 2σk →C =0.In particular,the Hankel singular values of this system do not decay at a rate exponential in √k .R EFERENCES[1]Athanasios C.Antoulas,Danny C.Sorensen,and Yunkai Zhou.Onthe decay rate of Hankel singular values and related issues.Systems Control Lett.,46(5):323–342,2002.[2]Wolfgang Arendt,Charles J.K.Batty,Matthias Hieber,and FrankNeubrander.Vector-valued Laplace transforms and Cauchy problems ,volume 96of Monographs in Mathematics .Birkh¨a user Verlag,Basel,2001.[3]Peter Benner and Jens Saak.Numerical solution of large and sparsecontinuous time algebraic matrix Riccati and Lyapunov equations:a state of the art survey.GAMM-Mitt.,36(1):32–52,2013.[4]Ruth F.Curtain and Hans Zwart.An introduction to infinite-dimensional linear systems theory .Springer-Verlag,New York,1995.[5]Klaus-Jochen Engel and Rainer Nagel.One-parameter semigroups forlinear evolution equations .Springer-Verlag,New York,2000.[6]Dale F.Enns.Model reduction with balanced realizations:An errorbound and a frequency weighted generalization.In Proc.CDC ,pages 127–132,1984.[7]Keith Glover,James Lam,and Jonathan R.Partington.Rationalapproximation of a class of infinite-dimensional systems.I.Singular values of Hankel operators.Math.Control Signals Systems ,3(4):325–344,1990.[8]Keith Glover.All optimal Hankel-norm approximations of linearmultivariable systems and their L ∞-error bounds.Internat.J.Control ,39(6):1115–1193,1984.[9]Lars Grasedyck.Existence of a low rank or H -matrix approximantto the solution of a Sylvester equation.Numer.Linear Algebra Appl.,11(4):371–389,2004.[10]Michael Green and David J.N.Limebeer.Linear robust control .Prentice Hall,Englewood Cliffs,New Jersey,1995.[11]Luka Grubiˇs i´c and Daniel Kressner.On the eigenvalue decay ofsolutions to operator Lyapunov equations.Systems Control 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了不起的盖茨比英文论文正稿
Part One Introduction1.1 Research Background and SignificanceAs “the spokesman of a crucial and revealing period in the culturalfield of his country”,(Wang Weiping, 2004:57) Fitzgerald knows the societyclearly due to his rich personal experience and creates many characterslike Gatsby out of the American life. These novels describe the disillusionment of the American dream in the Jazz Age. And the reason isalways that the transformations of the values of traditional morality makepeople not believe in any hero but pursue individual consumption andenjoyment. Fitzgerald deeply feels the spiritual emptiness and moraldecadence under the superficial splendor and prosperity in that age, sohe is called the spokesman of the Jazz Age. Most of his works describe theexpectation, dissatisfaction and disappointment of the American younggeneration in the 1920s. One distinguishing feature of his works is thatthe close relationship between his personal experiences and works. Nearlyall the characters in his novel can find the original shapes in realitysometimes even himself, especially in The Great Gatsby.1.2 Motivation and ObjectiveConsidering disillusion of the American dream, different people havedifferent understandings. “In its board sense, it refers to the ideal ofa nation such as democracy, equality and freedom. While in its narrow sense,it refers to the pursuit of obtaining success of life”. (Wang Weiping,2004:57) Essentially, the American dream is a confident desire forperfection by means of progress. But the history and reality of Americanhave proved that “the American dream” is, to some extent, a kind ofillusion.The great Gatsby is one of the representative works that reflects theillusory nature of the American dream.Part Two Gatsby’s American Dream Gatsby is the representative figure of a self-made man in the twenties. He is born in a poor family in the west of America, and his parents are shiftless and unsuccessful farm people. He comes to the big city—New York where the young men are inspired to make the most of their opportunities to get the key to be successful. They believe that they can be successful men in history, just like Benjamin Franklin, who has been ma n “of humble origin, narrow fortune, small advantages, and self-taught.”(Chen Qing, 2006:18) The poor boy is inspired to do the great deeds by the example of Franklin, like Gatsby. So in the end, Gatsby gets wealth and fame by industry. Some people may think that Gatsby has fulfilled his dream; it is obvious that Gatsby’s dream is a symbol of the American dream for wealth and youth. However, Gatsby’s real dream is to win back his first love Daisy but not to get the money only. Gatsby genuinely has a belief that money can buy anything, innocently thinking that his wealth can erase the last five years of his and daisy’s l ove and reunite them at the original point of 1917. He falls in love with the beautiful and wealthy girl of the upper class and he wants to enter into the upper class through his efforts. Although he devotes his whole life to win the wealth and position, he fails totally at last. Why does he fail? In the thesis the reasons for his failure would be discussed.Part Three Reasons for the Disillusion3.1Gatsby’s Dream is out of Connection with the RealityGay Gatsby is a great young man who has many superficial characters. He tries his best to realize his dreams, but he fails at last. The reason is that he is completely an American dreamer, a man of great imagination and extraordinary hope. He is willing to do anything to gain the social status he thinks necessary to win back Daisy who i s the “golden girl”in his dream. His relentless quest for Daisy demonstrates that Gatsby has an outstanding ability to transform his dreams into reality. But actually, his great dream is swallowed up by the meanness of the upper class and his “golden girl”—Daisy, which are the superficial reasons for his failure. And the deep reason is that his dream is out of connection with the reality. In fact, the people in the twenties do n’t believe in the values of traditional morality any longer and they have their philosophy of life—to “seize every day” and “enjoy every moment”.(Chen Qing, 2006:1) The differences in the understanding between Daisy and Gatsby narrates the distance between the dream and reality; while competition between Gatsby and Tom expresses the strange points of the material wealth and how fragile of the pursuit of the spirit. And the difference between Gatsby and upper class people reflects the downfall of the American society. All these contradictions and conflicts make people realize the fragile of the idealism when the dream confronts the reality. The pursuit of the spiritual life can not realize in the society which only pays a great attention to the material life.3.2 Gatsby’s Love for Daisy is IdealisticAt the last of the novel, he expresses himself “I love only one girl forever”, (F.Scott Fitzgerald, 2004:143) which represents a kind of idealism. But this kind of idealistic love can not be realized, because the girl whom he loves is just a superficial and vulgar woman in reality. For Daisy, love should be built on the wealth and high position and she has accustomed to this sense of security, which can make her live happily. But Gatsby who is born into a poor family gains his wealth and position criminally; he can not give Daisy the security which she has accustomed to. And Daisy will not love anyone who can not give her this kind of security. So in front of Daisy, Gatsby’s real love is an idealistic dream, and this dream can not be realized forever.Just as Americans have endued America with meaning through their dreamsfor their own lives, Gatsby instills Daisy with a sort of idealized perfection that she neither deserves nor possesses. Just as the American dream in the roaring twenties is ruined by the unworthiness of its goal, money and pleasure, Gatsby’s dream is ruined by the unworthiness of its goal, Daisy. “He knew that Daisy was extraordinary, but he didn’t realize just how extraordinary a ‘nice’ girl could be.”(Chen Qing, 2006:25) In Gatsby’s eyes, Daisy is a noble fairy, but she is a superficial and vulgar woman in reality.Gatsby’s love and the American dream connect closely because his first love Daisy is the embodiment of his dream. Daisy is the symbol of wealth and position, and marrying Daisy means entering into the upper class, so he devotes his whole life to win her back. But he fails at last, that is because he can not recognize Daisy clearly and his blindness leads to his death in the end.3.2.1Daisy in RealityDaisy is born into the upper class and grows up in wealthy conditions, so she depends on wealth from her birth and only responds to surface. When Gatsby shows her around his splendid house, she admires what she sees. Later, Gatsby takes out a pile of shirts and throws them one by one before her. She likes these beautiful shirts so much that she even cries out, “They’re such beautiful shirts”, “It makes me sad because I’ve never seen such—such beautiful shirts before”. (F.Scott Fitzgerald, 2004:124) “Daisy has n’t any practical value. The people whom Daisy likes are also beautiful without any true value.”(Chen Qing, 2006:15) When she takes part in Gatsby’s party, she s ees many super stars that can only be seen on TV, which makes her very interested and excited. “Perhaps you k now that lady”, (F.Scott Fitzgerald, 2004:140) Gatsby indicates a gorgeous woman who sits in state under a white-plum tree. Daisy stares and says“She’s lovely”,“I’ve never met so many celebri ties, I like that man—what was his name?”.( F.Scott Fitzgerald, 2004:141)“Daisy likes to arrest men’s attention and to get men’s compliments in order that she can depend on them. Maybe it is because s he couldn’t make a living independently, so she has to attach herself to a man, which is her way to live in this world.”(Chen Qing, 2006:15) Daisy marries Tom because of his large wealth and high social position. When Gatsby leaves her, she is very sad and even wants to kill herself. But with the arrival of Tom, her life changes immediately, and then she decides to forget Gatsby, and marries with Tom who has great wealth and high social position.From this aspect we can see that, for Daisy, love is just an interesting game, whereas money and position are the most important things in it. For money and position, she forgets Gatsby quickly, and devotes herself to another wealthy man no matter she loves him or not. But in fact, she will “love” Gatsby if he can make her believe that she can live a wealthy and safe life together with him. Because they have such two different kinds of disillusions towards each other, Daisy can never understand Gatsby’s love for her and thus is not worthy of being loved by Gatsby.After the marriage, the husband Tom treats Daisy badly. Daisy knows that Tom B doesn’t care about her and ha s a mistress outside, so she needs a chance to complain her unfortunate life. Therefore, when Nike visits them, she complains about her misfortune to him. Nick feels very uncomfortable when he hears the complaint from Daisy because he can not understand Daisy. He says “It seemed to me that the thing for Daisy to do was to rush out of the house, child in arms—but apparently there were no such intentions in her head.”(F.Scott Fitzgerald, 2004:27) Actually, Daisy can do nothing, she will not leave Tom. Without Tom she can not enjoy the wealthy life any longer. But she cannot control her fate and she feels sad for herself. So she wants to change her fate with someone’s help. Just at this very time, his first love Gatsby appears as a wealthy man and calls her back. It seems to be a good chance for her to make a decision to leave her bad husband. But instead she refused him, just because Gatsby’s background makes herlack of safety. She destroys Gatsby’s dream totally and goes away with Tom in a hurry. Daisy has to go on living with Tom, which is her fate, so in reality she is also an unfortunate woman. To this point, we can say that at this time, she still doesn’t deserve the love of Gatsby.3.2.2 Daisy in Gatsby’s DreamNo matter how other people may think about Daisy, in the mind of great Gatsby, she should be cherished forever. Although Gatsby and Daisy have departed for five years, Daisy is a lways living in Gatsby’s memory and she has turned into an perfect fairy as time passes by. For Gatsby, if he can marry with Daisy, he can enter into the upper class naturally and gain the identity in the upper class. Maybe it is because in Gatsby’s mind, Daisy has been endowed with a kind of special value. For this special vale, Gatsby would rather devote his whole life to winning her back. At this time, Daisy in reality is not important for him. Because Daisy has become the noble embodiment of the upper class in Gatsby’s dream.Gatsby always thinks the reason why Daisy leaves him is that he is poor and has a low social position when he is young, and he totally believes that he can win Daisy back if he can gain the wealth and high social position. With this faith in his mind, Gatsby determines to win Daisy back, making her the single goal of all his dreams and the symbol of everything he desired. So he earns large wealth by efforts and then buys a splendid house across the bay of Daisy’s and g ives parties day and night on every weekend just want to arrest her coming.Actually, Five years later when Gatsby sees Daisy again, he feels that she is different from the Daisy he dreams of day and night. In reality, Daisy has already lost her magical power and becomes a common woman. Later, Gatsby invites Daisy to take part in his party. And after she leaves, Gatsby feels very disappointed because he thinks that Daisy can’t really understand him any longer. It seems that Gatsby is so disappointed that he will not love her any longer. But Gatsby is really a great dreamer. Hejust pursues every thing in his dream and because Daisy in his dream has turned into a perfect fairy, so Daisy in reality is not so important for him; he would rather believe that Daisy still loves him.In fact, Daisy doesn’t really love anyone else. But Gatsby does not think so. He loves Daisy and he also believes that Daisy also loves him. So people think that Gatsby is a son of God. “He was a son of God—a phrase which, if it means anything, means just that—and he must be about his F ather’s business, the of a vast, vulgar, and meretricious beauty.”(F.Scott Fitzgerald, 2004:131) From this sentence, we can know that he makes Daisy as an embodiment of his dream, without thinking what Daisy is in reality. Actually, Both Nick and Gatsby notice her special voice, and are d escribed as “full of money” and even a “deathless song”.(F.Scott Fitzgerald, 2004:59) Her voice is seemed to jingle with a sound of wealth to Gatsby. For Daisy, it seems that wealth is more important than love itself. So Gatsby can’t get true love from her no matter he is poor or wealthy. This indicates that Gatsby’s dream for Daisy will be inevitable destructed.3.3 Gatsby’s Dream for the Upper Class is IdealisticFitzgerald deeply feels the transformations of the values o f traditional morality and people will not admire any self-made hero longer and they only care about individual enjoyment. So in a society with spiritual emptiness and moral decadence, Gatsby with much imagination is doomed to fail.Compared with the wealthy people, Gatsby is different in nature. He has dreams and ambitions, and tries his best to realize them. But he can not see the upper class clearly. The upper class in Gatsby’s dream is different from the one in reality.And his dream can not be realized in this kind of upper class nowadays.3.3.1 The Upper Class People in Gatsby’s MindIn Gatsby’s mind, if he has money, he can get into the upper classnaturally, and being an upper class member, he should be a moral person who has many virtues. So he stresses himself with the self-improvement and hard work; it can be proved easily in Gatsby’s schedule: No wasting timeNo more smoking or chewingBath every other dayRead one improving book or magazine per weekSave$5.00[crossed out] $3.00 per weekBe better to parents (F.Scott Fitzgerald, 2004:231)Gatsby’s li st of self-improving resolves is similar with Benjamin Franklin’s rules for self-improvement on eighteenth-century. Gatsby learns to him, and decides to be a decent person like him. In Nick’s eyes, “he was a handsome young man about thirty years old and dressed very well. He spoke very politely and it is a little funny to me. Most men in his age were not so polite. He spoke as if he is carefully choosing his words.”(F.Scott Fitzgerald, 2004:42) He does his best to be the same hero as Benjamin Franklin, believing that he can also be successful if he sets strict demands on himself.3.3.2 The Upper Class People in RealityBut in fact, people in that age do not believe in any hero of the American dream. They only care about themselves and enjoy themselves without considering the feelings of other people. They just like to pursue the enjoyment. And they have their philosophy of life—to “seize every day”and “enjoy every moment”. (Chen Qing, 2006:1) That can be proved easily in the novel of The Great Gatsby.In the party of the upper class, everyone needn’t know the name of anyone else and talks friendly as if they have known each other well for a long time. P eople taking part in Gatsby’s parties don’t know h im at all and even don’t see him before. They will not appreciate Gatsby who holds party for them, but making the rumors for him. In their eyes, Gatsbyis just a subordinate. Maybe, some of them do not care about who he is. They come here just because they want to enjoy themselves, and they need these big parties to prove their positions, while Gatsby also need them to raise his social position. They have mutual needs, so the parties can be held on every weekend. But nobody will remember him after they go away from his big parties. “A ll over the party, people were laughing, talking and drinking. But all the happiness seemed so empty.”(F.Scott Fitzgerald, 2004:40) In the society which people don’t care about others and just knowing to enjoy them. Gatsby, who does n’t really understand the true value of wealth in real life, can not live longer and his dreams can not be realized.3.4 Destroyer of Gatsby’s Dream—Tom3.4.1 Tom in RealityTom is the representative person in the upper class. In the novel, he is the direct destroyer of Gatsby’s dream. Both Gatsby and Tom possess wealth, but they are very different. Gatsby earns his wealth by diligence and discipline and uses his wealth to realize his great dreams, while Tom do not have any large goals and lives happily on the wealth that his father has created. In the end, Tom wins and goes away with Daisy in a hurry, while Gatsby i s killed by the husband of Tom’s mistress, George Wilson. From this aspect, we can see that Tom is a vicious man in reality. Maybe it is because he is born into a wealthy family and has been spoiled. Tom is a former football player at Yale and enjoys the high status, but he is an arrogant, cruel person who does nothing important everyday, just playing with cars and racing horses. Only in this way can he win his own ego. It is obvious that he does not have any moral standards which he demands for the people around him. He is the representative person of the upper class in reality which has replaced American idealism.Tom is also an immoral person who just knows to ask others to be honest. In order to meet his desire, Tom has an affair with the fleshly woman Myrtleand keeps an apartment in the city for their dating. But he is not upset at all and he even opens their relationship in public. He treats Daisy badly, however, when he knows that Daisy and Gatsby are having an affair. He even becomes outraged and forced them to confront it in the room at the hotel where he irritates Gatsby to win his wife back and thus destroys his rival’s dreams. From this aspect, we can also see that Tom actually is a man of violence, because he tells the group of people that Gatsby’s wealth is criminally obtained and thus he is cheating Daisy all the time. Then he forces Daisy to change the declaration that she has never loved him. To be honest, Daisy does not love Tom, she loves Tom for his wealth and will be reluctant to leave him, for he can provide her with security and the lifestyle to which she is accustomed. The victory of Tom reflects that Gatsby can not really enter into the upper class. For the upper class people like Tom, they will never take Gatsby as their friend but a subordinate.Tom is also a careless person who will not be worried about the sufferings he causes. When problems arise, he will run to his money and safe situation, leaving it to be dealt with by others and will never take the responsibility. After the death of Tom’s mistress Myrtle, Tom tells her husband that Gatsby is the killer and then runs away with Daisy quickly until the affair finishes.3.4.2 The Differences between Gatsby and TomComparing with Tom, Gatsby is really a strange hero of the Jazz Age. Because he creates wealth by diligence and discipline but he doesn’t really understand the true value of wealth in real life, that is, “wealth makes people enjoy life”. (Chen Qing, 2006:16) He does n’t know this point, so he still lives a simple life. “His bedroom was the simplest room of all—except where the dresser was garnished with a toilet set of pure dull gold.”(F.Scott Fitzgerald, 2004:122)He holds big parties not to enjoy himself, but to arrest his first love Daisy’s coming.“When the party is over and the guests go away, Gatsby soaks in large loneliness” (F.ScottFitzgerald, 2004:78) and at this moment he is himself.Tom is a representative person of the realism, while Gatsby is a representative person of the idealism. Tom is a realist who lives in reality while Gatsby is an idealist who lives in his dreams. Gatsby’s dream is out of connection with the reality, so his dream can not be realized.Part Four ConclusionAlthough Gatsby’s dream fails, his greatness still rests on his dream, which distinguishes him from the other disillusioned people, who have lost the capacity to wonder and to dream. But as time goes by, Gatsby reveals himself to be a man who stakes everything on his dreams, unaware that his dreams are out of connection with the reality and unworthy of him.To sum up, Gatsby’s dream is incompatible with reality. Gatsby is a great dreamer who just lives in his dreams, without considering the changes of the society. In his dream, every thing is beautiful: Daisy is a perfect fairy who can give him hopes; the upper class people are all decent ones who can cooperate with him very well. But in reality, every thing is different: his dreaming girl can not really understand Gatsby’s love for her; while the upper class people are the disillusioned ones who just make Gatsby as a subordinate. Facing the reality, Gatsby does not adjust himself to suit the society, but to pursuit his dreams which are out of connection with the reality. At last, it leads his dream to be a disillusion.Considering ourselves, we are the students who are going to devote ourselves into the society. Some of us may have great dreams and are ambitiously hoping to realize them immediately, just like Gatsby. But we must remind ourselves every moment that we are living in reality but not in our dreams. Dream is always very beautiful while the reality is always impersonal, it can not be changed as what we want to. So we should suit ourselves with the changes of the society and make achievement step by step. If we can persist in the above principle and work hard everyday, I am surewe will succeed and then fulfill our dreams eventually.Bibliography[1] Cheng Qing, The Disillusionment of The American dream in The GreatGatsby and Tender Is The Night [J]. Shanghai International Studies University, June 2006.[2] F.Scott Fitzgerald, The great Gatsby [M], 2004[3] Wang Weiping, Wang ping, The Great Gatsby—The reflection of theillusory nature of the American dream [J]. Journal of Grangxi University Wuzhou Brabch, 2004[4] 籍晓红.菲茨杰拉德的“美国梦”主题—了不起的盖茨比和夜色温柔之比较[J],晋东南师专科学校学报,2002[5] 菲茨杰拉德(著),巫宁坤(译).了不起的盖茨比[M],漓江,2003[6] 伯香,龙江.美国经典小说赏析[J],大学,2005.8[7] 成功.美国梦的开始与破灭—解读菲茨杰拉德的了不起的盖茨比[J], 师学院学报, 2006[8] 婷, 红霞, 贾丽丽.“美国梦”的一曲挽歌—评了不起的盖茨比[J], 职业技术学院学报, 2007。
杰斐逊思想英语作文
杰斐逊思想英语作文Thomas Jefferson's Enduring Legacy: The Principles of American DemocracyThomas Jefferson, one of the Founding Fathers of the United States, was a man of remarkable intellect and vision. His ideas and principles have had a lasting impact on the American democratic system and continue to shape the nation's political and social landscape even centuries after his time. As a statesman, philosopher, and political theorist, Jefferson's contributions to the establishment and preservation of American democracy are undeniable.At the heart of Jefferson's political philosophy was a deep-rooted belief in individual liberty and the inherent rights of all people. He famously proclaimed in the Declaration of Independence that "all men are created equal" and that they are endowed with "unalienable Rights" to "Life, Liberty and the Pursuit of Happiness." This fundamental principle of equality and natural rights formed the cornerstone of his vision for the new nation.Jefferson's commitment to individual liberty extended beyond mere rhetoric. He was a staunch advocate for the separation of church and state, arguing that religious freedom was essential for the protection of civil liberties. He believed that the government should not have the power to dictate or favor any particular religious denomination, as this would infringe upon the rights of the people to freely practice their faith or lack thereof.Furthermore, Jefferson's distrust of centralized power and his preference for a decentralized, federalist system of government have had a lasting impact on the American political landscape. He believed that power should be distributed among the states and the federal government, with the states retaining a significant degree of autonomy and decision-making authority. This system of checks and balances, as it came to be known, was designed to prevent the concentration of power in the hands of a few and to ensure that the rights of the people were protected.Jefferson's vision for America also included a strong emphasis on education and the cultivation of an informed and engaged citizenry. He believed that a well-educated populace was essential for the maintenance of a healthy democracy, as it would enable citizens to make informed decisions and participate actively in the political process. To this end, he advocated for the establishment of public education systems and the promotion of scientific and intellectualpursuits.In addition to his political philosophy, Jefferson's legacy as a statesman is equally remarkable. As the third President of the United States, he oversaw the Louisiana Purchase, which doubled the size of the country and paved the way for westward expansion. He also established the United States Military Academy at West Point, which has played a crucial role in the development of the nation's armed forces.However, it is important to acknowledge that Jefferson's legacy is not without its complexities and contradictions. As a slave-owning plantation owner, he grappled with the inherent tensions between his professed ideals of liberty and equality and the institution of slavery that he personally upheld. This contradiction has been a source of ongoing debate and reflection among historians and scholars, underscoring the need to critically examine the nuances and complexities of historical figures and their legacies.Despite these complexities, Jefferson's ideas and principles have endured and continue to shape the American democratic experience. His emphasis on individual liberty, the separation of church and state, and the distribution of power have become fundamental tenets of the American political system. Moreover, his commitment to education and the cultivation of an informed citizenry has had alasting impact on the nation's educational landscape and the civic engagement of its people.In conclusion, Thomas Jefferson's legacy as a Founding Father and political theorist is one that continues to resonate with Americans and people around the world. His ideas and principles have stood the test of time, serving as a guiding light for the ongoing pursuit of a more just, equitable, and democratic society. As we navigate the challenges and complexities of the modern era, it is crucial that we continue to draw inspiration from Jefferson's vision and work towards the realization of the ideals he so passionately championed.。
科学发现需要怀疑与坚持的作文
科学发现需要怀疑与坚持的作文英文回答:Science is all about questioning and persisting. As a scientist, I have to constantly doubt the existing theories and hypotheses, and be open to new possibilities. For example, when I was conducting an experiment to test a hypothesis, I encountered unexpected results that contradicted my initial assumptions. Instead of giving up,I persisted in exploring the reasons behind the unexpected outcomes. This led me to discover a new phenomenon that had not been previously documented. This experience taught me the importance of skepticism and perseverance in scientific research.中文回答:科学就是要怀疑与坚持。
作为一名科学家,我必须不断地质疑现有的理论和假设,同时也要对新的可能性持开放态度。
举个例子,当我进行实验来验证一个假设时,我遇到了出乎意料的结果,与我最初的假设相矛盾。
但我没有放弃,而是坚持探索这些意外结果背后的原因。
这让我发现了一个以前未曾记录的新现象。
这个经历让我明白了在科学研究中怀疑和坚持的重要性。
英语作文致胜法典
英语作文致胜法典Captivating the reader from the very first sentence is the golden rule of writing a winning English essay. It's about painting a vivid picture with words, weaving anarrative that not only informs but also captivates the imagination. The art of crafting an essay that stands outlies in its structure, clarity, and the seamless flow of ideas.Start with a hook that grabs attention – a compelling question, a startling fact, or a bold statement. This sets the tone and draws the reader into your world of words. Following this, the introduction should succinctly lay out the thesis statement, a clear and concise declaration of your position or argument.The body paragraphs are the heart of your essay, where each one should focus on a single main idea that supports your thesis. Use strong topic sentences to guide the reader, and follow with evidence, examples, or analysis that strengthens your argument. Remember, coherence is key; transition words and phrases should be used to link your ideas and maintain the flow.Varying your sentence structure is crucial for maintaining reader interest. Short, punchy sentences can be as effective as longer, more complex ones. The use of rhetorical devices such as metaphors, similes, and analogiescan add depth and richness to your writing.In the conclusion, restate your thesis and summarize the main points without introducing new information. End with a strong closing statement that leaves a lasting impression, reinforcing your argument and providing a sense of closure.Finally, the secret to a winning essay is revision. Proofread for grammar, punctuation, and spelling errors. Ensure that your essay is free of clichés and that your voice is clear and consistent throughout. With these elements in place, your English essay will not only meet the criteria but will also engage and persuade your audience, securing its place as a piece of writing that truly stands out.。
The Zenith of Zeal Reaching New Heights
The Zenith of Zeal Reaching New Heights The Zenith of Zeal Reaching New Heights The concept of reaching new heights in life is often associated with the idea of achieving the zenith of zeal. This phrase encompasses the idea of reaching the peak of one's passion, enthusiasm, and energy in pursuit of a goal or aspiration. It signifies a sense of unwavering determination and relentless drive to push oneself beyond limitations and boundaries. The journey towards reaching new heights is often filled with challenges, obstacles, and setbacks, but it is the unwavering zeal that propels individuals to overcome these hurdles and continue striving towards their aspirations. From a personal perspective, reaching new heights in life can be a deeply fulfilling and transformative experience. It requires a deep sense of self-awareness, clarity of purpose, and a strong belief in one's abilities. The journey towards reaching new heights often involves stepping out of one's comfort zone, taking risks, and embracing uncertainty. It demands resilience, perseverance, and a willingness to learn from failures and setbacks. However, the sense of accomplishment and growth that comes from pushing oneself to new heights is unparalleled. It instills a profound sense of confidence, self-belief, and empowerment, which can positively impact various aspects of one's life. On a professional level, reaching new heights often translates to achieving significant milestones, making groundbreaking advancements, and leaving a lasting impact in one's field of work. It involves setting ambitious goals, fostering innovation, and continuously pushing the boundaries of what is deemed possible. The pursuit of reaching new heights in a professional capacity often requires collaboration, teamwork, and a collective effort towards a shared vision. It involves inspiring and motivating others to join in the journey towards achieving extraordinary feats and making meaningful contributions to the world. From a societal perspective, individuals who reach new heights often serve as sources of inspiration and role models for others. Their stories of perseverance, determination, and success have the power to ignite a spark of motivation and drive within others. They demonstrate that with unwavering zeal and relentless effort, one can overcome adversities and achieve remarkable feats. Their accomplishments contribute to the collective progress and advancement of society, paving the way for newpossibilities and opportunities for future generations. The journey towards reaching new heights is not without its share of challenges and adversities. It often involves facing moments of self-doubt, fear of failure, and the pressure to meet expectations. However, it is during these moments of struggle and uncertainty that one's zeal is truly put to the test. It is the ability to harness one's inner strength, resilience, and determination that enables individuals to navigate through these challenges and emerge stronger and more resilient. Ultimately, the pursuit of reaching new heights in life is a deeply personal and transformative journey. It requires a combination of passion, perseverance, and an unwavering belief in one's abilities. It is a journey that not only leads to personal growth and fulfillment but also has the potential to inspire and uplift others. As individuals continue to strive towards their aspirations with unwavering zeal, they pave the way for new possibilities, advancements, and contributions that have the power to shape the world in meaningful ways. The zenith of zeal represents the pinnacle of human potential and the extraordinary heights that individuals can reach when they are driven by passion, purpose, and unwavering determination.。
怀疑与真理英语作文
怀疑与真理英语作文We all know the "experiments on the Leaning Tower of Pisa".How can a feather fall as fast as an iron ball?So Aristotle concluded that how fast the objects fall is determined by their weight, and the heavier the object, the faster it falls.Galileo doubted skeptical and conducted a "oblique experiment", found the "free fall law".Later, the famous "Leaning Tower of Pisa experiment" was a sensation.(I am not sure if the "Leaning Tower of Pisa experiment" was done by Galileo himself.)Modern people know that this is due to the difference between the feather and the iron ball falling speed is due to the role of air resistance, but in Aristotle's age, this conclusion is undoubtedly very advanced, is an indisputable truth.It was because no one doubted it at that time that Aristotle's conclusion became the truth, and because Galileo doubted it, it overthrew the "truth".Doubt is the best tool to uncover the truth.Doubt arises because the scope of human cognition is limited.After doubt and test and demonstration, the truth surfaced step by step, so that the scope of human cognition gradually expanded.However, we are not sure whether "truth" istruth.Just as Aristotle suggested that force was the cause of the motion of the object, it was the truth; Galileo reversed his conclusion that effort was not the cause of the motion; then Newton perfected Galileo's conclusion that force was the cause of the change of the motion.Doubt keeps man close to the truth, but are you sure that Newton's law must be true?The truth of yesterday is overturned today, and will the truth of today certainly survive to tomorrow?Indeed, the universe is so large that man cannot understand the whole mystery, which is bound to limit the truth of mankind at this stage.As modern people believe, the accepted natural and natural "truth" is exactly the same as the "truth" that the ancients thought that they had concluded wereindisputable.Perhaps posterity will see as inconceivable what we conclude as we see the "truth" of our predecessors.From this perspective, human beings can only be infinitely close but have access to truth.People constantly find the truth applicable at a, the truth applicable at b... and even at y, z, and even someone can find the common truth applicable from a to z, but beyond these 26 letters, break into the unknown field, who is sure that the truth is still the "truth", to know that the air resistance is unknown to Aristotle.Is the "false truth" useless?No.Before the emergence of new truths, these conclusions apply to the time and space of today, and play a vital role in human production and life.So why doubt it?Because doubt can get a more universal truth, so that people can understand the world more, improve life, we can not wait until the successful invention of home appliances to find electricity.I hope we can use doubt as a compass to find our light in the dark night of exploring the truth.。
Sturm-Liouville问题特征值间不等式的证明
Sturm-Liouville问题特征值间不等式的证明杨春梅;杨树生;张晓军【摘要】考虑带有正的首项系数和权函数的自伴正则的Sturm-Liouville问题.构造了一个二阶线性群SL(2,R)中的矩阵,利用分离边界条件下特征值增减性和集合的连通性,证明了文献中给出的分离边界条件和耦合边界条件特征值间不等式.%In view of the self-adjoint regular Sturm-Liouville problems with the positive leading coeffi-cient and weight function ,and the matrix structured in the linear group of two order SL(2 ,R) ,the paper gives the proof for inequality among eigenvalues for separated and coupled boundary conditions provided by published literature ,by using the increase-decrease characteristics of eigevalue and connectedness of set and the separated boundary conditions .【期刊名称】《内蒙古师范大学学报(自然科学汉文版)》【年(卷),期】2017(046)006【总页数】5页(P781-784,787)【关键词】自伴正则Sturm-Liouville问题;分离和耦合边界条件;特征值间不等式【作者】杨春梅;杨树生;张晓军【作者单位】内蒙古工业大学理学院 ,内蒙古呼和浩特 010051;河套学院理学系 ,内蒙古巴彦淖尔 015000;河套学院理学系 ,内蒙古巴彦淖尔 015000【正文语种】中文【中图分类】O175.3考虑定义在区间(a,b)上的带有正的首项系数和权函数的自伴正则的Sturm-Liouville问题,即-(py′)′+qy=λ wy,(1)其中 -∞≤a<b≤∞; 1/p,q,w∈L((a,b),R), p>0, w>0,(2)对方程(1)的任何解y,令Y(t)那么边界条件由线性方程组(3)给出,其中复数λ称为(1)的谱参数.L((a,b),R)表示定义在区间(a,b)上的Lebesgue 可积实函数空间,表示复数域C中秩为2的2×4矩阵集合.A*表示复矩阵A的共轭转置矩阵.众所周知,该问题的特征值可以排列形成趋于+∞的非单调递减数列λ0,λ1,λ2,λ3,…,使得一个特征值出现在数列中的次数等于它的重数.因此,当满足条件(2)的微分方程(1)固定后,对于每一个n∈N0={0,1,2,…},存在一个定义在自伴边界条件空间上的函数λn.本文构造了一个二阶线性群SL(2,R)中的矩阵,利用分离边界条件下特征值增减性和集合的连通性,以及文献[1-3]中给出的自伴边界条件空间中关于特征值连续性的结果和一些耦合边界条件与分离边界条件特征值间的关系,给出文献[3]中的分离边界条件和耦合边界条件特征值间不等式的一种新证法.首先证明了耦合边界条件下的特征值不等式非常接近于分离边界条件下特征值不等式,其次证明了对于一般的耦合自伴的边界条件下的不等式可以从耦合边界条件一些集合的连通性推出.1 预备知识对于每个λ∈C,令φ11(·,λ)和φ12(·,λ)分别是满足初始条件(a,λ)=1确定的微分方程(1)的解.用φ21表示用φ22表示令(a,b),λ∈C.这里Φ(·,λ)在a点和b点的值分别由Φ(·,λ)在a点的右极限和b点的左极限确定.对于每一个t∈(a,b),Φ(t,λ)是λ的一个整矩阵函数.此外,对于t∈(a,b)和λ∈R,Φ(t,λ)∈SL(2,R).由文献 [4]中的定理1.3知,包含方程(1)和(3)的自伴正则的Sturm-Liouville问题有无数个特征值,它们都是实的且有下界.此外,特征值是给定问题的特征函数Δ(λ)=det (A+B Φ(b,λ))的零点,且没有有限聚点.由文献 [5]中的定理4.16知,包含方程(1)和(3)的Sturm-Liouville问题的每一个特征值的代数重数和几何重数是相等的.所以本文没有区别这两个概念,重数可以是代数重数或几何重数.此外,在给定的区间上计算给定问题的特征值时,根据它们的重数,总是假设它们的特征值是可数的.根据文献 [5],取商空间GL(2,C)(C)作为自伴边界条件空间,即每一个边界条件(3)的线性方程组的系数矩阵的等价类,由线性方程组(3)表示的边界条件记作 [A|B].通常像A这样的大写字母也用来表示边界条件.自伴的实的边界条件空间包括实的分离边界条件和形如 [K|-I],K∈SL(2,R)的实的耦合边界条件.在引言中提到,对于每一个n∈N0={0,1,2,…},存在一个定义在自伴边界条件空间上的函数λn.对于中的元素A,函数λn记作λn(A),而且对θ∈(-π,π],K∈SL(2,R),函数λn记作λn(eiθK).对于K∈SL(2,R),用{σn(K);n∈N0}表示分离自伴边界条件(4)的特征值的集合.对于K∈SL(2,R),用{τn(K);n∈N0}表示分离自伴边界条件(5)的特征值的集合.引理1[1] 令K∈SL(2,R),记ΔK是包含微分方程(1)和边界条件 [K|-I]的Sturm-Liouville问题的特征函数,则: (a) 对任意θ∈(-π,π],特征值{λn(eiθK);n∈N0}是实方程ΔK(λ)=2-2 cos θ的根; (b) 对于满足0<ΔK(λ*)<4的任意λ*∈R,有对任意λ∈{σ0(K),σ1(K),σ2(K),…,τ0(K),τ1(K),τ2(K),…},有ΔK(λ)≤0或ΔK(λ)≥4.引理2[3] 对于任意两个正常数c和ε,存在一个λ*使得对于任意λ≤λ*,有φ11(b,λ)≥c,φ12(b,λ)>0,φ21(b,λ)≥c,φ22(b,λ)>0,及下面关于在边界条件空间上关于λn的连续性的结果出自文献 [2]中的命题3.10和3.18,用来描述λn的不连续性.引理3 令n∈N0,则作为自伴的实的边界条件空间中函数λn在除了中的点外,在空间中的每个点处都是连续的,而且上的限制也是连续的.分离边界条件的一般形式为其中α∈[0,π),β∈(0,π].引理4[3] 作为区间(α,β)上的函数λn(Sα,β)在区域[0,π)×(0,π]上是连续的,关于α是严格单调递减的,关于β是严格单调递增的.此外,对于每一个α∈[0,π),对于每一个2 特征值间的不等式新证法下面给出文献 [3]中的特征值间的一个不等式的一种新证法.这些结果和本文的其余部分,对任何k≥2的整数和k个数c1,c2,…,ck,记号{c1,c2,…,ck}的含义是c1,c2,…,ck中的每一个.定理1[3] 固定微分方程(1)满足(3),令K∈SL(2,R),若k12≤0且k22>0,则对于任意θ∈(-π,π),θ≠0,有τ0(K)≤λ0(K)<λ0(eiθK)<λ0(-K)≤{σ0(K),τ1(K)}≤λ1(-K)<λ1(eiθK)<λ1(K)≤{σ1(K),τ2(K)}≤λ2(K)<λ2(eiθK)<λ2(-K)≤{σ2(K),τ3(K)}≤λ3(-K)<λ3(eiθK)<λ3(K)≤{σ3(K),τ4(K)}≤….证明由引理1的(a)和(b),只需证明,对于集合κ={L∈SL(2,R);l12≤0,l22>0}中的任意K,有τ0(K)≤λ0(K)<λ0(-K)≤{σ0(K),τ1(K)}≤λ1(-K)<λ1(K)≤{σ1(K),τ2(K)}≤λ2(K)<λ2(-K)≤{σ2(K),τ3(K)}≤λ3(-K)<λ3(K)≤{σ3(K),τ4(K)}≤….(7)构造二阶线性群SL(2,R)中的矩阵由于线性方程组的初等行变换是线性方程组的同解变换,故对系数矩阵(A|B)作初等行变换不改变方程组的解,即边界条件 [A|B]也可以作初等行变换,于是有(8)(9)对于每一个j∈N0,由(8)式、(9)式和引理3,有(10)对于K∈SL(2,R),用{τn(K);n∈N}表示分离自伴边界条件的特征值的集合,且(11)对于K∈SL(2,R),用{σn(K);n∈N}表示分离自伴边界条件的特征值的集合,且(12)对于每一个j∈N0,由(11)式、(12)式和引理4,有(13)由引理4,有λ0(S0,π/2)<λ0(S0,3π/4)<λ0(S0,π)<λ1(S0,π/2)<λ1(S0,3π/4)<λ1(S0,π)< (14)固定一个奇数n≥5,当h→0+时,由引理4中λn(Sα,β)的连续性,以及(10)式和(12)-(14)式,有τ0(Kh)<{λ0(Kh),λ0(-Kh)}<σ0(Kh)<τ1(Kh)<{λ1(Kh),λ1(-Kh)}<σ1(Kh)<…<τn-1(Kh)<{λn-1(Kh),λn-1(-Kh)}<σn-1(Kh)<τn(Kh)<{λn(Kh),λn(-Kh)}<σn(Kh)<τn+1(Kh)<λn+1(Kh).(15)对任意K∈κ,由ΔK(λ)=2-k22φ11(b,λ)+k21φ12(b,λ)+k12φ21(b,λ)-k11φ22(b,λ)和引理2,有从而有因此,在区间(-∞,λ0(K))内有ΔK(λ)<0,故当h→0+时,由(15)式和引理1中的(a),有τ0(K)<λ0(Kh)<λ0(-Kh)<σ0(Kh)<τ1(Kh)<λ1(-Kh)<λ1(Kh)<σ1(Kh)<…<τn-1(Kh)<λn-1(Kh)<λn-1(-Kh)<σn-1(Kh)<τn(Kh)<λn(-Kh)<λn(Kh)<σn(Kh)<τn+1(Kh)<λn+1(Kh).(16)结合ΔK(λ)<0及引理1中的(a)和(c),对于任意λ∈{τ0(Kh),τ2(Kh),…,τn+1(Kh),σ1(Kh),σ3(Kh),…,σn(Kh)},有ΔKh(λ)≤0,(17)且对于任意λ∈{τ1(Kh),τ3(Kh),…,τn(Kh),σ0(Kh),σ2(Kh),…,σn-1(Kh)},有ΔK(λ)≥4.(18)由(4)式、(5)式和引理4,对于每个j∈N0,τj和σj是κ上的连续函数.因此,由(17)式、(18)式和引理1中的(c)及集合κ的连通性,对于每一个K∈κ及任意λ∈{τ0(K),τ2(K),…,τn+1(K),σ1(K),σ3(K),…,σn(K)},有ΔK(λ)≤0,(19)且对于任意λ∈{τ1(K),τ3(K),…,τn(K),σ0(K),σ2(K),…,σn-1(K)},有ΔK(λ)≥4.(20)由引理3,对于每一个j∈N0,λj在κ上是连续的,所以由(17)式、(19)式、(20)式和σj,λj,τj在κ上的连续性及κ的连通性,对于任意K∈κ,有{τ0(K),λ0(K)}<{λ0(-K),σ0(K),τ1(K),λ1(-K)}<{λ1(K),σ1(K),τ2(K),λ2(K)}<…<{λn-1(-K),σn-1(K),τn(K),λn(-K)}<{λn(K),σn(K),τn+1(K),λn+1(K)}.(21)因为n≥5的奇数n是任意的,对于任意的这样的K,有{τ0(K),λ0(K)}<{λ0(-K),σ0(K),τ1(K),λ1(-K)}<{λ1(K),σ1(K),τ2(K),λ2(K)}<{λ2(-K),σ2(K),τ3(K),λ3(-K)}<{λ3(K),σ3(K),τ4(K),λ4(K)}<….(22)结合ΔK(λ)<0及(21)式和(20)式,可推出(6)式.参考文献:[1] Qingkai Kong,Qunlin,Hongyou Wu,et al. New Proof of the Inequalities Among Sturm-Liouville Eigenvalues [J]. Pan American Math [J]. 2000(2):1-10.[2] Kong Q Wu,Zettl H A. Dependence of Then-th Strum-liouville Eigenvalues on the Problem [J]. Journal of DifferentialEquations,1999,156:328-354.[3] Zhong Wang,HongYou Wu. Geometric Aspects of Strurm-Liouville Problems,VI.Arcs of Boundary conditions for Equalities in Eigenvalue Inequalities [J]. Pacific Journal of Applied Mathematics,2007,1(1).[4] Eastham M S P,Kong Q,Wu H,et al. Inequalities Among Eigenvalues of Strum-Liouville Problems [J]. Inequalities & Appl,1999(3):25-43.[5] Kong Q,Wu H,Zettl A. Geometric Aspects of Strurm-Liouville Problems,I.Structures on Spaces of Boundary Cconditions [J]. Proceedings of The Royal Society of Edinburgh,2000,130A:561-589.。
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§1. Notation and Prerequisite Results For each λ ∈ C, let φ11 (·, λ) and φ12 (·, λ) be the solutions to (0.1) determined by the initial conditions (1.1) φ11 (a, λ) = 1, (pφ11 )(a, λ) = 0 and φ12 (a, λ) = 0, (pφ12 )(a, λ) = 1,
where (0.3) (0.4) −∞ ≤ a < b ≤ ∞, 1/p, q, w ∈ L((a, b), R), A 0 −1 1 0 A∗ = B p, w > 0 a.e. on (a, b), 0 −1 1 0 B∗,
(A | B ) ∈ M∗ 2×4 (C),
and λ ∈ C is the so called spectral parameter of (0.1). Here L((a, b), R) denotes the space of Lebesgue integrable real functions on (a, b), M∗ 2×4 (C) stands for the set of 2 by 4 matrices 1
Consider a self-adjoint regular Sturm-Liouville problem (SLP) with a positive leading coefficient and a positive weight function, i.e., (0.1) (0.2) −(py ) + qy = λwy on (a, b), y (a) (py )(a) (A | B ) = 0, y (b) (py )(b)
of the problem and hence do not have a finite accumulation point. Thus, as mentioned in the introduction, the eigenvalues of the problem can be ordered to form a non-decreasing sequence (1.5) λ 0 , λ1 , λ2 , λ 3 , · · ·
over C with rank 2, and A∗ is the complex conjugate transpose of the complex matrix A. It is well-known that the eigenvalues of the problem can be ordered to form a non-decreasing sequence (0.5) λ 0 , λ1 , λ 2 , λ3 , · · ·
A New Proof of the Inequalities among Sturm-Liouville Eigenvalues
QINGKAI KONG, QUN LIN, HONGYOU WU and ANTON ZETTL
Abstract. We consider self-adjoint regular Sturm-Liouville problems with positive leading coefficients and weight functions. A new proof of the inequalities among the eigenvalues for separated boundary conditions and those for coupled boundary conditions established recently by three of the authors with M. S. P. Eastham is given. This new proof does not assume that any special case of the inequalities has been proven. 1991 Mathematics Subject Classification: Primary 34B24; Secondary 34L05, 34L15. Key words: self-adjoint regular Sturm-Liouville problems, separated and coupled boundary conditions, inequalities among eigenvalues.
approaching +∞ so that the number of times an eigenvalue appears in the sequence is equal to its multiplicity. Note that by Theorem 4.16 in [ 6 ], the algebraic and geometric multiplicities of each eigenvalue of the SLP consisting of (0.1) and (0.2) are equal. So, in this paper we are not going to distinguish these two concepts and the word multiplicity will be used for either of them. Moreover, when counting the eigenvalues of the problem in a given interval, we will always assume that the eigenvalues are counted according to their multiplicities. 3
approaching +∞ so that the number of times an eigenvalue appears in the sequence is equal to its multiplicity. Hence, when the differential equation (0.1) satisfying (0.3) is fixed, there is a function λn defined on the space of self-adjoint boundary conditions (BC’s) for each n ∈ N0 =: {0, 1, 2, · · · }. In [ 4 ], for each coupled self-adjoint BC, two separated self-adjoint BC’s were identified and some inequalities among the eigenvalues for these three BC’s were established. These inequalities generalize the well known classical ones among the eigenvalues for the periodic BC, the Dirichlet BC and the Neumann BC (see, for example, [ 3 ] and [ 2 ] for the case of smooth coefficients and weight functions and [ 9 ] for the general case of integrable coefficients and weight functions). Moreover, they play an important role in the forthcoming update of the Fortran code SLEIGN2, in which they are used not only to bracket the eigenvalues for coupled BC’s but also to determine their indices. In this paper, we present a new proof of these inequalities. This new proof, unlike the original one given in [ 4 ], does not use the fact that the inequalities are already proved for the periodic case, and hence shorten the whole proof of the inequalities. To obtain the new proof, we combine some continuity results about the eigenvalues considered as functions on the the space of self-adjoint BC’s shown in [ 7 ] and some relations among the eigenvalues for coupled BC’s and those for separated BC’s. These relations are direct consequences of the manifold structure on the space of self-adjoint BC’s introduced in [ 6 ]. The main idea of this new proof is as follows: one first shows the inequalities for a coupled self-adjoint BC very close to the set of separated self-adjoint BC’s, then the inequalities for a general coupled self-adjoint BC can be deduced from the connectedness of some sets of coupled self-adjoint BC’s. This paper is organized as follows. In Section 1, we mention some basic results, describe the space of self-adjoint BC’s, and recall certain continuity results about eigenvalues. Section 2 is devoted to the new proof of the inequalities among eigenvalues. Throughout this paper, we always fix the differential equation (0.1) and assume that it satisfies (0.3). 2