年轻时喜欢读数学家传记,尤其是自传或回忆录。当时可以说,凡是有汉译本的我都读过,一共也没有几本。相比之下,物理学家的传记要比数学家多多了。虽然我也喜欢读,但未必能把有汉译本的都读过。当然,后来没有汉译本的数学家传也看过些,仍然不多。 今天看到丘成桐回忆录的电子版,很开心。作者是丘成桐和 Steve Nadis 。书名为《一生之形:对宇宙隐秘几何的数学家追寻 ( The Shape of a Life: One Mathematician’s Search for the Universe’s Hidden Geometry ) 》。由耶鲁大学出版社 2019 年出版。 在书中,丘成桐谈了他的生活,包括数学。他自称有三个家,在哈佛旁、在北京城、在数学中。尤其是数学,是他的普世护照,超越了距离、语言和文化,让他瞬间走遍全球。这真是令人羡慕的境界。 除前言和尾声外,全书分 12 章。依次为:奔波青年 (Itinerant Youth) ,生活继续 (Life Goes On) ,来到美国 (Coming to America) ,近观卡拉比 (In the Foothills of Mount Galabi) ,进军巅峰 (The March to the Summit) ,回乡之路 (The Road to Jiaoling) ,特殊之年 (A Special Year) ,阳光圣地亚哥的弦与波 (Strings and Waves in Sunny San Diego) ,加盟哈佛 (Harvard Bound) ,亲履中土 (Getting Centered) ,超越庞卡莱 (Beyond Poincare) ,两种文化之间 (Between Two Cultures) 。书中还有不少图,既有与名家如陈省身杨振宁霍金合影,也有各种几何图形。 合作者 Steve Nadis 似乎是职业作家,与丘成桐还合作过 The Shape of Inner Space 和 From the Great Wall to the Great Collider 等。我都没有读过。 现在有这么多书看,我都想退休了。如稼轩词所谓 声名少日畏人知 老去行藏与愿违 山草旧曾呼远志 故人今又寄当归 何人可觅安心法 有客来观杜德机 却笑使君那得似 清江万顷白鸥飞 欲读书系列博文 欲读书之《希腊罗马名人传》 ( 部分已读 ) 欲读书之《英国史》 ( 未读 ) 欲读书之《大街》和《名士风流》 ( 已读 ) 欲读书之大卫·洛奇的教授小说 ( 部分已读 ) 欲读书之希腊神话和其他 ( 部分已读 ) 欲读书之《徐霞客游记》 ( 部分已读 ) 欲读书之《品园》 ( 已读 ) 欲读书之《东坡乐府笺》 ( 已读 ) 欲读书之《应物兄 ( 上下册 ) 》 ( 已读 ) 欲读书之《全宋词》
作者:蒋迅 奥巴马向凯瑟润·强森授予总统自由勋章 Source: 白宫 2015年11月16日,美国总统奥巴马宣布了总统自由勋章获奖人名单,其中之一是NASA数学家凯瑟润·强森。这位被称为“穿裙子的计算机”的妇女没有证明过什么猜想,也没有以她命名的定理,那么她是怎样一位数学家呢?让我们来读一读她的故事吧。 本文发表在《航天员》2015年第1期上。 凯瑟润·强森( Katherine Johnson )女士 Source: NASA 现在当人们谈到“Computer”时,理所当然地把它想成是一个电子设备,人们更倾向于相信计算机的结果而不是由人工计算的结果。但是在电子计算机出现的初期时,人们更相信人工计算,而不是电子设备。那时候甚至有一个职业就是“computer”。今天要讲的就是一个“computer”的故事。这个“computer”就是凯瑟润·强森( Katherine Johnson )。作为一位女性非洲裔空间科学家和数学家,凯瑟润·强森女士是值得一篇传记的,因为她的勇气和毅力以及她对NASA早期科学计算和电子计算机时代的特殊贡献应该成为在技术行业的女性和少数民族学习的榜样。 凯瑟润女士出生于1918年8月26日。她原来随父母姓“寇尔曼”(Coleman)。她的母亲原来是一个教师,后来辞职在家专心抚养子女。他的父亲是一个农民,只有小学六年级文化。但是他坚持要让自己的孩子都能上大学,他总是对他的孩子们说:“你们会进大学的”。但她根本不知道大学是什么。她的父亲还说过:“你像其他人一样好,但你不比其他人更好。”她记住了。从此没有自卑感,因为她“跟别人一样的好,虽然没有更好”。在她的老家,非洲裔孩子最多只能上到八年级。於是,父亲把全家搬到离乡村老家125英里之外的“西维吉尼亚有色人种学院”(West Virginia Colored Institute)所在地印斯逖突特市( Institute, West Virginia ),为的就是让他的子女们可以接受教育。在那里,父亲把妻子留下来,照顾孩子们,他自己则返回老家继续去赚钱。他必须外出找兼职工作,他去做过伐木工,终於找到了一份旅馆的清洁工工作,一个月的工资是100美元,可以维持一家的支出。而他们则在每个暑假回到老家去帮助父亲。这样的生活一直持续了八年。最后他的四个孩子都上了大学。 凯瑟润上过的西维吉尼亚大学 从小,凯瑟润就喜欢数学,可以轻易地解代数方程。她相信自己的数学天赋来自她的父亲。她记得父亲特别会心算。只要给他口头出一个算术题,他就能心算出答案来。他看着一棵树,就可以说出他能从中得到多少木材。凯瑟润6岁上小学时直接进入二年级,又跳过五年级,14岁就从西维吉尼亚州立高中毕业。在学校期间,老师们都知道她有极强的求知欲望。人们时常看到她回家的路上和校长一起。她从校长那里得知了很多恒星和星系。1932年,她进入“西维吉尼亚州立学院”(West Virginia State College)学院。“西维吉尼亚州立学院”就是原来的“西维吉尼亚有色人种学院”,现在已改名为“西维吉尼亚州立大学”( West Virginia State University )。她还得到了全额奖学金。一开始,她并不知到自己的数学天赋,她选择的是法语专业。在大一的时候,她遇到了一位识才的数学教师克雷敦( W.W. Schiefflin Claytor )博士。克雷敦是美国历史上第三位非裔数学博士(1933年)。他对她说,“如果我教的哪门课里没有你的话,我一定要找到你。”就这样,在大二大下半年时,她又加上了一个数学专业。克雷敦专门开了一门课“解析几何”(analytic geometry),教师里只有师生两人。现在的公立学校恐怕难以想象会为一个学生开一门课。克雷敦在这里工作不到四年,正好是凯瑟润在那里学习的四年。但就是这门课和这位老师为美国航天事业做出了意想不到的贡献。这是后话。她记得,老师走进教室开始讲课。他知道她学到哪里了,下面该讲什么了。克雷敦说她是做研究数学家的材料。“我将帮你做好准备,”老师对她说。她问老师:“那是一个什么职业?”老师没有正面回答,只是说,“那是需要你自己去发现的事情”。所以从那时起,她就开始梦想着当一名研究数学家。 凯瑟润的毕业照 1937年,18岁的凯瑟润从“西维吉尼亚州立学院”获得法语和数学两个学士学位,并获得 拉丁文学位荣誉 ( summa cum laude )。大学毕业后,她开始在维吉尼亚、西维吉尼亚和马里兰当中小学教师(1936年至1952年)。她还记得她的起薪是每月65美元。当时美国正处於大萧条期间,工作很不容易找。有一天,她突然受到了一个电报,说如果她能弹钢琴,就可以雇佣她。她不知道这个电报是从哪里来的,於是求救于她在大学时的老师。原来有一所小学需要一位法语教师,但还必须能弹钢琴。她的老师记得她会弹钢琴,就把她推荐给那所小学了。就这样她教了两年的法语和音乐。 在去那所维吉尼亚的学校的路上,凯瑟润第一次感受到了种族歧视。公交车在进入维吉尼亚时,司机要求所有的有色人种都到后排去。凯瑟润一动不动,直到司机有礼貌地请到她。出发前,她母亲就警告过:“记住,你是去维吉尼亚。” 1939年她与一位教师结婚。从她的访谈中可知,其实他们早就结婚,但一直没有公开,因为那时美国对已婚妇女当教师是有歧视的。他们一共生了三个孩子,但仍不想放弃自己的梦想。这也是他父亲的期待,因为他希望凯瑟润能成为“比小学教师高一些的老师”。新婚一年后,她上了以白人为主的“西维吉尼亚大学”数学系做研究生。她是校园里唯一的女性黑人学生,因此常常招来另类的目光。尽管她不在乎这些,但还是由於她的丈夫得了癌症,她不得不放弃学业,再次当老师以维持家庭开支。先是替先生完成课时,然后转到其它几所中小学。 一次偶然的机会让她证明了她其实比别人更好。1952年暑假里,她和全家到纽波特纽斯市去参加姐姐的婚礼,纽波特纽斯市离NASA兰利中心非常近。从姐夫那里,她得知NASA正在招收黑人妇女,他们希望招到数学家。为了实现凯瑟润的梦想,她和丈夫立即决定搬到纽波特纽斯市。不过,当年的名额已经满了。她不得不等待一年。在此期间,她在维吉尼亚州纽波特纽斯市的一所公立学校做代课教师。 因为不能确定NASA的工作,她也申请了一些其他的工作。就在NASA正式通知她去上班的前一天,她也受到了当地一所学院的职位,是一个项目的主任。她必须做出一个决断。她曾经梦想过当大学老师,这也是她父亲对她的期待,但她更想当一名数学家。那所学院的院长提出给她的工资与NASA持平,希望她能接受。但是当研究数学家的愿望让她放不下NASA的机会。她毅然决定接受NASA的职位,尽管那只是一个初级计算者(Junior Computer)的职位。 1953年6月,她被NACA(NASA的前身)兰利中心作为合同工雇员雇佣,工作就是与其他一群妇女进行数据处理。NASA早在1935年就有这样一群妇女,不过那时都是白人。1940年代开始才开始有了黑人。专门招募黑人是美国联邦政府平权运动一种努力。按照那个年代的规定,NASA把她们分为两个小组。两组做着完全一样的事情。她所在的小组有十三四个人,都是黑人,都是大学毕业。她把这群人称为是“穿裙子的虚拟计算机”。她们要做的就是读飞机的黑盒子里的数据,然后进行数学计算,比如算出速度和压力,然后画到图表上。这些数据可以帮助NASA工程师进行空气动力学研究。事实上,她的真正职称就是“计算者”(Computer),这个词在当时不是我们现在意义上的电子计算机。当时电子技术才开始使用。人们普遍更相信数学家的大脑。而NASA更希望妇女来完成这件事,因为他们认为妇女会比较细心,不易出错。有一次,一架飞机奇怪地失事了。通过解读飞机的黑盒子,她们发现飞机下方有大量的气流,到了飞机无法承受的地步。从此以后就规定了两架飞机的纵向距离至少要1000英尺。 凯瑟润回想起上大学的时候自己不知道研究数学家是干什么的。老师说她必须自己去发现。她开始有些明白了。也许这原来就是老师说的那个职业?这个职业很适合她,她开始喜爱她的工作。 卫星轨道计算公式 更大的机会还在等待凯瑟润。上班两个星期后的一天,凯瑟润被叫到一个全部是男性工程师的飞行力学研究室里去帮忙。凯瑟润在解析几何上的知识使得她的那些男性长官看到了她的价值,最后竟然“忘记”让她回到那群妇女之中去。“我们写出了我们自己的课本,因为那时根本没有关于空间的课本。”他们从自己所知道的一点点知识开始写,经常需要找出以前的教科书去查几何学定理。她说她有些幸运。但其实,这个幸运来自她坚实的几何知识的积累。这真要感谢那位“西维吉尼亚州立学院”的数学教授。 凯瑟润与同事合影(1970年) 尽管那时候种族歧视和对妇女的歧视还比较普遍,凯瑟润尽量忽略这样的事情。另一方面,在NASA,种族歧视不严重。大家都在忙禄于科研,完成任务是第一位的。“我知到有种族隔离,但我没有感觉。对我没有影响。如果真有,你就去适应它。”她坚持要参加从来没有女性参加过的一些会议。她会问,她能否去。对方说,还没有妇女参加过。她就接着问:有法律禁止妇女吗?没有。“OK,我要去”。於是她开始参加相关的会议。她说自己做 了这些工作,因此有权参加这些会议。 1961年5月5日,NASA宇航员艾伦·谢泼德(Alan Shepard)乘坐自由7号载人飞船进入太空,整个飞行历时15分28秒。这是他的飞行轨道。 1958年,NACA改名为NASA,而凯瑟润则与工程师们一起成为了“空间任务组”(Space Task Force)的一员。凯瑟润从“飞行力学研究室”调到“太空飞行器控制研究室”。她的职称已经是“航天技师”(Aerospace Technologist)。那时候的轨道模型都是二维抛物线型的,也就是我们通常说的弹道再入,所以比较简单。当说到一个载人舱要落在一个什么地方,人们其实需要知道的是再入点和再入倾角。她需要做的就是从指定的降落点一点一点退回去,直到找到那个轨道上的再入点。只要误差在 ε 允许范围之内,就不会出那个范围。“但是如果再入倾角差上几度或者速度差一点,那就回不来了,”她说。“这是我的长处”,她自信地说。她负责计算了1959年第一位进入太空的美国宇航员艾伦·谢泼德(Alan Shepard)的再入轨道和他在1961年“水星计划”中的发射窗口。她为宇航员们绘制了在万一电表失灵情况下使用根据恒星来实现的导航图。后来的飞行越来越复杂,变量也越来越多,不但要考虑飞船的位置,从哪里起飞,在哪里降落,地球和月亮的自转等,还要考虑飞船的姿态控制、降落伞、反向火箭等等因素。人工计算越来越不现实。1962年,NASA买了一台一间屋子大小的计算机,第一次用电子计算机计算了 约翰·格伦 ( John Glenn )的轨道。计算机可以算得又多又快,但人们对这项新技术还是不太放心,特别是格伦不放心。格伦要求NASA指派凯瑟润验证电子计算机的数据是否正确。“如果她得到计算机得到的结果,我就去。”凯瑟润强调是集体努力的结果。但是显然是凯瑟润的计算能力和结果准确度帮助NASA建立了对新电子技术的信任。 NASA早期人工计算机 凯瑟润也开始使用电子计算机进行计算了。1969年,她计算了“阿波罗11号飞船”飞往月球的轨道。这个计算要更复杂得多,其中包括月球落地舱和轨道舱的对接。当尼尔阿姆斯特朗踏上月球的时刻,她正在波科诺山(Pocono Mountains)上参加一个联谊会议。她在那里观看了电视。她跟全世界收看电视的人们一样激动,但她又显得格外平静。对她来说这似乎只是一个例行公事罢了。其实她这个“计算机”当时非常的紧张。“我做了全部的计算,我知道我的计算是正确的,”她说,“但是就像开车一样,任何事情都可能发生。我不希望有任何事情发生。”在“阿波罗计划”结束后,她继而参加航天飞机计划,地球资源卫星和火星任务等。 凯瑟润一共在26篇科学论文里署名。在1960年代,一般不会有“女性计算机”能够被作为共同作者写上论文的。所以她能够被当作共同作者之一具有重要意义。比如下面这篇NASA技术报告: “The Determination of Azimuth Angle at Burnout for Placing a Satellite over a Selected Earth Position” 1960. Authors: T.H. Skopinski, Katherine G. Johnson 凯瑟润获得过很多褒奖:1967年的“NASA月球轨道器和执行小组奖”(Lunar Orbiter Spacecraft and Operations team award - for pioneering work in the field of navigation problems supporting the five spacecraft that orbited and mapped the moon in preparation for the Apollo program)和“阿波罗集体成就奖”(Apollo Group Achievement Award),其中第二项奖使她得到了在“阿波罗11号”上飞行过的300面国旗中的一面;1971年、1980年和1984年至1986年的“NASA兰利研究中心特别成就奖”(NASA Langley Research Center Special Achievement award);1998年纽约明代尔州立学院名誉博士学位;1999年西维吉尼亚州立学院杰出校友;2006年首都学院名誉博士,2010年奥多明尼昂大学名誉博士。 1956年凯瑟润的丈夫因脑癌去世。1959年,她与美国空军中校詹姆斯·强森(James A. Johnson)结婚。从此随夫姓强森至今。1986年,她从NASA退休。退休后,她和她的丈夫居住在纽波特纽斯市一个老年公寓里。她有六个孙子和孙女,及四个重孙。她平时弹钢琴、打桥牌,或做拼图游戏。她在一家教会的唱诗班里歌唱了五十多年。有时,她会到学校去演讲,或在NASA频道里接受中小学生的提问。她与孩子们交流,让他们知道通过数学和科学,他们能有什么样的机会。有一次,一位小女孩的提问是:“你还活着?”原来她的照片早就印在了课本上。孩子们觉得上了课本的人一定是历史上过去的人了。 谈到男孩和女孩的区别,她说,“我认为区别在於求知的欲望。对我来说,知道为什么是重要的。有些女孩子不愿意问问题,但如果你好奇的话,你需要问出来。世界上没有愚蠢的问题。女孩子的能力和男孩子一样地强,有时候她们有更多的想象力。” 对於教育,她也有自己的看法。她说,有的老师只教答案。她自己不是这样。她会教:问题是什么,如何下手,如果下手合适,你就能得到答案。有的老师只教那些会出现在考试中的内容,这样教学生只会应付考试。应该教问题的背景和如何解题。如果一次做不出来,两次就可以做出来。她说,她不教答案。 “运气是准备和机会的结合。”她的故事印证了她的这句话。“我在兰利找到了我要寻找的东西,”她说。“这就是一个’研究数学家‘要做的事情。33年来,我每天高兴地去工作。我从来没有哪天起来说我今天不想去工作。” 凯瑟润与前宇航员、NASA前副局长梅尔文合影 Source: twitter 凯瑟润工作照(1980年) 凯瑟润正在工作(1980年) 凯瑟润在计算机终端(1980年) 这是笔者【NASA人的故事】系列中的一篇。请到 这里 继续阅读
图灵测试,如果仅仅作为一个非形式化描述的开放的哲学问题,那么,各种观点层出不穷也就在所难免;如果仅仅作为一个可形式化描述的封闭的科学问题,那么,除了在美国获得博士学位的英国数学家图灵和在英国获得博士学位的美国分析哲学家塞尔各自的观点所使用的书面语言应该重新审视之外,在下以科学的严格约束条件加以限制之后,再分别就小字符的直接形式化方法和大字符的间接形式化方法对它们加以规范约束之后,即可分别得到基于算数语言的图灵测试(1)、基于中文语言的图灵测试(3)、基于数字和文字(汉字)的广义双语的图灵测试(2),在各自相应的约束条件下(这是科学技术可以做到的),都可以在严格限定的约束条件下顺利通过图灵测试。这就是在下考虑过的三类孪生图灵机设计方案的独特价值、作用和意义。 附录1:The Turing test is a test of a machine's ability to exhibit intelligent behavior equivalent to, or indistinguishable from, that of a human. Alan Turing proposed that a human evaluator would judge natural language conversations between a human and a machine that is designed to generate human-like responses. The evaluator would be aware that one of the two partners in conversation is a machine, and all participants would be separated from one another. The conversation would be limited to a text-only channel such as a computer keyboard and screen so that the result would not be dependent on the machine's ability to render words as speech. If the evaluator cannot reliably tell the machine from the human (Turing originally suggested that the machine would convince a human 70% of the time after five minutes of conversation), the machine is said to have passed the test. The test does not check the ability to give correct answers to questions, only how closely answers resemble those a human would give. The test was introduced by Alan Turing in his 1950 paper Computing Machinery and Intelligence, while working at The University of Manchester (Turing, 1950; p. 460). It opens with the words: I propose to consider the question, 'Can machines think?' Because thinking is difficult to define, Turing chooses to replace the question by another, which is closely related to it and is expressed in relatively unambiguous words. Turing's new question is: Are there imaginable digital computers which would do well in the imitation game? This question, Turing believed, is one that can actually be answered. In the remainder of the paper, he argued against all the major objections to the proposition that machines can think. In the years since 1950, the test has proven to be both highly influential and widely criticised, and it is an essential concept in the philosophy of artificial intelligence. https://en.m.wikipedia.org/wiki/Turing_test 附录2:The Chinese room Main article: Chinese room John Searle's 1980 paper Minds, Brains, and Programs proposed the Chinese room thought experiment and argued that the Turing test could not be used to determine if a machine can think. Searle noted that software (such as ELIZA) could pass the Turing Test simply by manipulating symbols of which they had no understanding. Without understanding, they could not be described as thinking in the same sense people do. Therefore, Searle concludes, the Turing Test cannot prove that a machine can think. Much like the Turing test itself, Searle's argument has been both widely criticised and highly endorsed. Arguments such as Searle's and others working on the philosophy of mind sparked off a more intense debate about the nature of intelligence, the possibility of intelligent machines and the value of the Turing test that continued through the 1980s and 1990s. 附录3:Consciousness vs. the simulation of consciousness Main article: Chinese room See also: Synthetic intelligence The Turing test is concerned strictly with how the subject acts – the external behaviour of the machine. In this regard, it takes a behaviourist or functionalist approach to the study of the mind. The example of ELIZA suggests that a machine passing the test may be able to simulate human conversational behaviour by following a simple (but large) list of mechanical rules, without thinking or having a mind at all. John Searle has argued that external behaviour cannot be used to determine if a machine is actually thinking or merely simulating thinking. His Chinese room argument is intended to show that, even if the Turing test is a good operational definition of intelligence, it may not indicate that the machine has a mind, consciousness, or intentionality. (Intentionality is a philosophical term for the power of thoughts to be about something.)
徐晓 ,帅哥一枚,物理学家,科学网博主; 老曹, 曹广福 ,衰哥一位,数学家,科学网博主。 你应该同数学家约会吗? 汤雅·科凡诺娃; 翻译:白露为霜 格雷厄姆·马斯特(Graham Masterton)的经典《如何让你的男人在床上狂野》(How to Drive Your Man Wild in Bed)里有一章讲解如何选择一个恋人。它标出男人身上你需要谨慎对待的“红旗”(red flag)。他将这些潜在的不良苗头列成很长的一个表,其中包括前一星期没洗澡以及说话时只谈他自己。 不好的特征的列表还包括哪些职业要避免。你能猜出排在名单上的第一位的职业吗? OK,我想你应该能够猜出,因为是我在写这件事。在书的第64页赫然列出: “避免,作为群体,数学家......”(Avoid, on the whole, mathematicians…) 我算是不听这一劝告的专家了:事实上,我已经同三个数学家结过婚,并同X个数学家约会过。这并不一定是因为我就这么喜欢数学家,我只是没遇上其他人。 当我还是一个学生时候,我有一个理论:数学家同物理学家是不同的。理论是基于我连续参加的两个数学物理会议。第一个是针对数学的而第二个是物理的。第一个是非常安静,而第二个则是豪饮和派对。所以我做出判断,数学家内向而物理学家则外向。我敢肯定,我的第二任丈夫选择了一个错误的专业,因为他喜欢喝酒和聚会。 到如今,很多年之后,我遇到了更多的数学家,而且我要告诉你,他们是多种多样的。说数学家是一种类型是不准确和不公平的描述。我知道的一个数学家甚至成为色情电影的明星。我写这篇文章是为了那些对同数学家约会感兴趣的女孩。我不是在谈论数学专业的学生,我说的是做严肃的研究工作的数学家。我的忠告是什么呢? 我也有几句提醒的话。但请保持在脑子里,它们并不一定适用于所有的数学家。 第一,许多数学家,像我的第一任丈夫,都非常热爱数学。我很佩服这种奉献,但这意味着,他们计划在星期六晚上做数学,宁愿在自己的办公桌上渡过假期。如果他们日程表里每年只能安排一次音乐会,这对我来说是不够的。当然,这一条适用于任何痴迷于他的工作的人。 第二,有些数学家认为他们非常聪明,比其他许多人更聪明。他们将自信从数学扩大到其他领域。他们以专家的姿态开始进入生物、政治、个人关系,等等,其实他们真的不知道自己在说啥。 第三,有些数学家只关注数学世界,以至于他们看不到周围的一切。关于这类数学家有个笑话: “外向的数学家和内向的数学家的区别是什么?外向的看着你的鞋子,而不是他自己的鞋”。 的确,我遇到过很多这样的数学家。你认为他们的妻子是在抱怨丈夫没有注意到自己的新发型?这种琐碎的事情是不值得一提。他们的妻子抱怨的是自己的丈夫没有注意到,家里的家具被拿去抵债了,或者,宠物猫已经死了,取而代之的是一条狗。我的第三任丈夫就是如此。在我们的婚姻的某个时期,我发现他根本不知道我的眼睛是啥颜色。他也不知道他自己的眼睛是啥颜色。他不是色盲,只是无动于衷。于是我请求他帮我做件事:在心里记住我的眼睛的颜色,他做到了。我的朋友艾琳甚至提出为这样的数学家的妻子们建立一个支持小组(support group)。 当然你对这些特点需要注意,数学家还是有让我喜欢的地方。许多数学家的确非常聪明。这意味着,与他们交谈很有趣。另外,我喜欢能够被某件事情所驱动的人,因为它显示了激情的容量(capacity for passion)。 数学家往往是公开和直接的。许多数学家,像我一样,说假话说不来。就因为这个原因,我不玩Mafia了(白露为霜注:一种源于苏联时代的游戏)。我更喜欢那些有话直说,不躲躲藏藏的人。 有一些数学家有一种无辜纯洁的气质,那让我想起普希金的诗剧“莫扎特和萨列里”(Mozart and Salieri)中莫扎特的角色的话语:天才和邪恶是不相容的两件事,不是吗? - 我觉得这可以延伸到数学家。许多数学家忙于探索数学的奥秘,他们对策划和玩“游戏”不感兴趣。 英文原文: Should You Date a Mathematician? The book “How to Drive Your Man Wildin Bed by Graham Masterton has a chapter on how to choose a lover. It highlights red flags for men who need to be approached with caution. There is a whole list of potentially bad signs, including neglecting to shower in the previous week and talking only about himself. The list of bad features also includes professions to avoid. Can you guess the first profession on the list? OK, I think you should be able to meta-guess given the fact that I am writing about it. Indeed, the list on page 64 starts: “Avoid, on the whole, mathematicians…” I am an expert on NOT avoiding mathematicians: in fact, I’ve married three of them and dated x number of them. That isn’t necessarily because I like mathematicians so much; I just do not meet anyone else. When I was a student I had a theory that mathematicians are different from physicists. My theory was based on two conferences on mathematical physics I attended in a row. The first one was targeted for mathematicians and the second for physicists. The first one was very quiet, and the second one was all boozing and partying. So I decided that mathematicians are introverts and physicists are extroverts. I was sure then that my second husband chose a wrong field, because he liked booze and parties. By now, years later, I’ve met many more mathematicians, and I have to tell you that they are varied. It is impossible and unfair to describe mathematicians as a type. One mathematician even became the star of an erotic movie. I write this essay for girls who are interested in dating mathematicians. I am not talking about math majors here, I am talking about mathematicians who do serious research. Do I have a word of advice? I do have several words of caution. While they don’t apply to all mathematicians, it’s worth keeping them in mind. First, there are many mathematicians who, like my first husband, are very devoted to mathematics. I admire that devotion, but it means that they plan to do mathematics on Saturday nights and prefer to spend vacation at their desks. If they can only fit in one music concert per year, it is not enough for me. Of course, this applies to anyone who is obsessed by his work. Second, there are mathematicians who believe that they are very smart. Smarter than many other people. They expand their credibility in math to other fields. They start going into biology, politics and relationships with the charisma of an expert, when in fact they do not have a clue what they are talking about. Third, there are mathematicians who enjoy their math world so much that they do not see much else around them. The jokes are made about this type of mathematician: “What is the difference between an extroverted mathematician and an introverted one? The extroverted one looks at your shoes, rather than at his own shoes.” Yes, I have met a lot of mathematicians like that. Do you think that their wives complain that their husbands do not notice their new haircuts? No. Such triviality is not worth mentioning. Their wives complain that their husbands didn’t notice that the furniture was repossessed or that their old cat died and was replaced by a dog. My third husband was like that. At some point in my marriage I discovered that he didn’t know the color of my eyes. He didn’t know the color of his eyes either. He wasn’t color-blind: he was just indifferent. I asked him as a personal favor to learn the color of my eyes by heart and he did. My friend Irene even suggested creating a support group for the wives of such mathematicians. While you need to watch out for those traits, there are also things I like about mathematicians. Many mathematicians are indeed very smart. That means it is interesting to talk to them. Also, I like when people are driven by something, for it shows a capacity for passion. Mathematicians are often open and direct. Many mathematicians, like me, have trouble making false statements. I stopped playing —Mafia— because of that. I prefer people who say what they think and do not hold back. There is a certain innocence among some mathematicians, and that reminds me of the words of the Mozart character in Pushkin’s poetic drama, Mozart and Salieri: —And genius and villainy are two things incompatible, aren’t they?— I feel this relates to mathematicians as well. Many mathematicians are so busy understanding mathematics, they are not interested in plotting and playing games.
Peter Scholze生于1987年12月11日,2013年晋升为德国波恩大学数学系教授,他也是德国目前最年轻的教授。 因解决Weight-monodromy猜想的特殊情形而获2015年美国数学学会的Cole奖。 他发表论文的情况: Perfectoid spaces and their Applications , Proceedings of the ICM 2014. The pro-étale topology for schemes (with Bhargav Bhatt), to appear in Proceedings of the conference in honour of Gérard Laumon, 2013. On torsion in the cohomology of locally symmetric varieties , Preprint, Bonn, 2013. Perfectoid spaces: A survey , Current Developments in Mathematics, 2012. Moduli of p-divisible groups (with Jared Weinstein), Cambridge Journal of Mathematics 1 (2013), 145-237. p-adic Hodge theory for rigid-analytic varieties , Forum of Mathematics, Pi, 1, e1, 2013. Perfectoid spaces , Publ. math. de l'IHéS 116 (2012), no. 1, 245-313. On the cohomology of compact unitary group Shimura varieties at ramified split places (with Sug Woo Shin), J. Amer. Math. Soc. 26 (2013), no. 1, 261-294. The Langlands-Kottwitz method and deformation spaces of p-divisible groups , J. Amer. Math. Soc. 26 (2013), no. 1, 227-259. The Local Langlands Correspondence for GL_n over p-adic fields , Invent. Math. 192 (2013), no. 3, 663--715. The Langlands-Kottwitz method for some simple Shimura varieties , Invent. Math. 192 (2013), no. 3, 627--661. The Langlands-Kottwitz method for the modular curve , Int. Math. Res. Not. 2011, no. 15, 3368-3425.
(COMMENTfrom Richard Qian: If there is one humanbrain in the world whose one is most similar to mine, or say, mine is mostsimilar to his, he is The Dr. Zhang. Noone knew me, however, I will be one ofthe famous scientist in the world TOMORROW at that moment, there are only three American professors knew me. I willlearn to be in the peace from Dr. Zhang.) 作者:汤涛 来源:中国科学报 发布时间: 2013-7-19 7:59:38 http://news.sciencenet.cn/htmlnews/2013/7/280149.shtm 张益唐:孤独的数学家 “我的心很平静。我不大关心金钱和荣誉,我喜欢静下来做自己想做的事情。” (COMMENTfrom Richard Qian: If there is one humanbrain in the world whose one is most similar to mine, or say, mine is mostsimilar to his, he is The Dr. Zhang. Noone knew me, however, I will be one ofthe famous scientist in the world TOMORROW at that moment, there are only three American professors knew me. I willlearn to be in the peace from Dr. Zhang. Today, I cry with tear for him with bitter and joy. I hope couple of months later, he would prefer to cry for me, his unique brother.) 张益唐的故事之所以特别轰动的原因在于,作出巨大数学贡献的他已经接近 60 岁,之前只是个默默无闻的讲师。 2012 年 7 月 3 日,在一个阳光明媚的下午,张益唐在科罗拉多州好友齐雅格家后院抽烟, 20 多分钟里他有如神明启示般的想出了主要思路,找到了别人没有想到的特别突破口。 2013 年 4 月 17 日,一篇数论论文被投递到纯粹数学领域最著名的刊物《数学年刊》。不到 1 个月,论文所涉及领域的顶级专家罕有地暴露自己审稿人的身份,信心十足地向外界宣布:这是一个有历史性突破的重要工作,文章漂亮极了。这位评审人就是当 今最顶级的解析数论专家亨利•伊万尼克。 顶级专家的高度评价被科学界的泰斗级期刊《自然》敏锐地捕捉到了; 2013 年 5 月 13 日, 《自然》催生了一次历史性的哈佛演讲。这篇文章的作者、一个学术界的“隐形侠”,第一次站在世界最高学府的讲台上,并告诉世人:我走进了世纪数学猜想的大 门!哈佛的讲台下面座无虚席,连过道上都站满了人。演讲内容被即时传到网上,网上不少人在刷新网页等待最新消息。 2013 年 5 月 14 日,《自然》在“突破性新闻”栏目里,宣布一个数学界的重大猜想被敲开了大门。 5 月 18 日,《数学年刊》创刊 130 年来最快接受论文的纪录诞生了。 世界震动了! 5 月 20 日,《纽约时报》大篇幅报道了这个华人学者的工作。文中引用了刚刚卸 任《数学年刊》主编职务的彼得•萨纳克的讲话:“这一工作很深邃,结论非常深刻。” 5 月 22 日,老牌英国报纸《卫报》刊登文章,文章的标题是:鲜为人知的 教授在折磨了数世纪数学精英的大问题上迈进了一大步。印度主流报纸把作出这一非凡贡献的人,与印度历史上最伟大的天才数学家拉马努金相媲美。 这位作出重大数学突破的就是张益唐,由于对数学界最著名的猜想之一孪生素数猜想的破冰性工作,使他从默默无闻的大学讲师跻身于世界重量级数学家的行列。 这是一个永久的疑问:为什么要研究数学猜想?短视地回答这个问题很困难。纯粹数学的研究很 像体育比赛。刘翔跑得那么快有什么用?世界短跑纪录的刷新、跳高纪录的刷新到底有什么用?但这并不妨碍每四年一次的奥运会。很多数学大猜想的突破很像顶尖 高手的棋艺对决,是世界纪录的突破。 孪生素数猜想 变大海捞针为泳池捞针 远在中古时代,人类社会就产生了自然数的概念,人们也因此创立了一个古老而漂亮的数学分支:数论。数论里面一个重要的概念就是素数,指的是那些只能被 1 和其自身整除的数,比如 5 、 7 、 11 、 19 等。 张益唐所做的工作和素数有关,尤其和所谓的孪生素数有关。孪生素数是指差为 2 的素数对,即 p 和 p+2 同为素数。前几个孪生素数分别是( 3 , 5 )、( 5 , 7 )、( 11 , 13 )、( 17 , 19 )等。 100 以内有 8 个孪生素数对; 501 到 600 间只 有两对。随着数的变大,可以观察到的孪生素数越来越少。 2011 年,人们发现目前为止最大的孪生素数共有 20 多万位数。但这个数后面再多找一对孪生素数都 要花至少两年的时间。 那么会不会有一天再也找不到新的孪生素数对呢?数学家认为答案是否定的。几百年前就有个孪生素数猜想:有无穷多个素数 p ,使得 p 与 p+2 同为素数。但至今人们都不知如何证明这个猜想。 张益唐在《数学年刊》上发表的这篇题为《素数间的有界距离》的文章,证明了存在无数多个素数对( p, q ),其中每一对中的素数之差,即 p 和 q 的距离,不超过七千万。 如何理解张益唐的结果呢?诺丁汉大学物理教师安东尼奥•帕蒂拉举了个有趣的例子:假如在素 数王国里素数只能找邻近的同类结婚,那 3 、 5 、 7 、 11 这种小素数找对象都很容易。但是素数越大,对象就越难找。但是根据张益唐的发现,素数和下一个素数 的距离,应该小于或等于七千万。孤独的数字不会持续孤独下去,总有另一个素数与之匹配。换言之,对于“大龄光棍”素数来说,七千万步之内,必有芳草。 七千万听起来是个巨大的数字,但在数学上只是一个常数而已。虽然它和孪生素数猜想的距离为 2 的结果还有十万八千里,但用张益唐的方法把七千万缩短到几百以内也是指日可待的事情。实际上,在文章被公布于众后,短短的一个月以内,七千万就被菲尔茨 奖获得者陶哲轩发起的网上讨论班缩小到六万多。 张益唐起到的作用就是把大海捞针的力气活缩短到在水塘里捞针,而他给出的方法还可以把水塘捞针轻松变为游泳池里捞针。也许最后变成在碗里捞针还需要一些再创新的工作。但给出了这一伟大框架已经是让全世界数学家瞠目结舌的壮举了。 非凡探索路 演绎一个数学神话 张益唐的故事之所以特别轰动的原因在于,作出巨大数学贡献的他已经接近 60 岁,之前只是个 默默无闻的讲师。为了潜心研究数学,他几乎把自己与世隔绝,在美国的偏远省份“潜伏”下来。他的妹妹曾在网上发寻人启事寻找哥哥。当时在美国当教授的老同 学给他妹妹回了个电邮,表示他哥哥健康地活着,在钻研数学呢。 张益唐于 1955 年出生于北京。他 1978 年考进了北京大学数学系。北大 1977 年没有招生,所以他是北大数学系“文革”后恢复高考的第一批学生。 1978 年第 1 期《人民文学》发表了作家徐迟的报告文学《哥德巴赫猜想》,讲述了数学家陈 景润刻苦钻研在哥德巴赫猜想研究上取得重大突破的真实故事,一时间陈景润和哥德巴赫猜想变得家喻户晓。像那个时代很多有志青年一样,张益唐也是被徐迟的文 章、被陈景润的故事、被哥德巴赫猜想引导到数学系,以致终身投入到数学中去。 4 年的北大学习为张益唐打下了坚实的数学基础。那时的北大教书育人之风极强,最顶尖的教师 都在讲台上耕耘。北大也有很多眼界很高的老师,学富五车,但不轻易落手写小文章,可谈起大问题颇为津津乐道,这让年轻的张益唐“中毒”匪浅。这也奠定了他 一辈子只做大问题、不为小问题折腰的风格。张益唐也是 1978 级公认的数学学习尖子。 张益唐 1982 年毕业后跟随著名数论专家潘承彪读了 3 年的硕士。潘承彪的哥哥就是大名鼎鼎的山东大学前校长,因在哥德巴赫猜想方面的工作而闻名的潘承洞院士。潘氏兄弟也是北大数学系校友,毕业后在各自的岗位上做出了非凡的精彩。 张益唐总是说在潘承彪的指导下他在北大打下了非常扎实的数论基础。 1985 年,张益唐来到了位于美国的名校普渡大学读博士,成为抗日名将孙立人和物理学家邓稼先的校友。 但张益唐在普渡的六七年是不堪回首的时光。他在美国的导师是代数专家莫宗坚。张益唐的研究 课题是导师的专长——雅可比猜想,但苦干了 7 年,得到的结果乏善可陈。眼界极高的张益唐不屑把博士论文结果整理出来发表。更糟糕的是,他和导师的关系糟得 一塌糊涂。这里有学术上的冲突,也有性格上的不和。 因为博士论文的结果没有发表,加上导师连一封推荐信都不愿意写,张益唐毕业后连个博士后的工作都没有找到。 一面要继续做数学,一面还要糊口。毕业后的前六七年他干过很多杂活,包括临时会计、餐馆帮手、送外卖。你能想象一代北大数学才子、数学博士数年间在快餐店、在唐人街餐馆打工的情形吗?看到这里,你是否对“天将降大任于斯人也,必先苦其心志,劳其筋骨”有更深刻的理解呢? 1999 年后,张益唐又回到了学校,到美国的新罕布什尔大学做助教、讲师。新罕布什尔大学 是成立于 1866 年的一所综合性公立大学。虽然教学量比较大,比起研究系列的教授、副教授的工资性价比低很多,但能回到学校,做自己驾轻就熟的事情,还能 利用图书馆、办公室作研究,对一个胸有大志的数学人来说,应该是非常满足的了。 在新罕布什尔大学的 14 年是张益唐研究的黄金期。不需要研究经费,凭自己坚实的数学功底, 充满智慧的大脑,以及潜心钻研的精神,他终于演绎出数学史上的一个神话。 2012 年 7 月 3 日,在一个阳光明媚的下午,张益唐在科罗拉多州好友齐雅格家后院 抽烟, 20 多分钟里他有如神明启示般的想出了主要思路,找到了别人没有想到的特别突破口。 校友情深 助千里马奔腾 张益唐的成功路上有众多的朋友帮助,特别是北大校友的帮助。 一位北大化学系的校友在上世纪 90 年代开了几家赛百味连锁店。他听北大校友说张益唐在逆境 中还在作数学的大问题,很想资助张益唐,但又怕被拒绝。所以他就想了一个点子,每个季度请张益唐来帮助给这些连锁店报税,让张益唐用简单数学来得到较为轻 松的报酬,同时有较多时间去研究数学大问题。 张益唐一辈子的转折点是落脚新罕布什尔大学。促成这件事的有两个主要人物,他们是北大数学系 1980 级的校友唐朴祁和葛力明。 毕业于湘潭一中的唐朴祁是 1980 年湖南省高考状元,是张益唐在北大时的系友、普渡大学读 博士时的同学。 1999 年初,已经在美国大计算机公司工作的唐朴祁去纽约参加学术年会时,找到在纽约打工的张益唐,聊到自己在计算机网络研究中遇到的一个 数学难题。大约 3 周以后,张益唐居然想出了解决问题的基本思想,最后产生了两人的一个软件合作专利。据说这个专利已经在计算机网络基础设施领域有广泛应 用。三个星期啃下一个有广泛实际用途的计算机算法难题,让张益唐顿觉宝刀不老,信心大增。唐朴祁也对老友的数学实战功夫印象深刻。 同年晚些时候唐朴祁与在新罕布什尔大学工作的葛力明见面,他提到张益唐的强大分析实力和当 时的艰难处境。作为学长的张益唐不仅做过他们的习题课老师,也是上世纪 80 年代他们自己组织的大学生讨论班上的常客。此时已是大学教授的葛力明似乎更有条 件帮一下他们的朋友和老师。这次会面时,唐朴祁已经不知道张益唐的准确工作地点。经过一番周折,葛力明在美国南方的一个赛百味快餐店联系上了张益唐,两三 天后,张益唐就来到新罕布什尔大学了。每过几天,张益唐都会说,有进展,应该很快就出来了。他是指自己正在攻克的一两个世界难题。但时间过得很快,两个 月、三个月,两年、三年…… 14 年后,张益唐轰动性的工作终于横空出世了。 当然,在美国大学里要留一个没有多少学术资历的人 14 年肯定不是一件简单的事,中间也有酸甜苦辣的故事。这里的主要帮手还是系里的明星教授葛力明。 葛力明过去的 10 年一半时间在中国科学院数学院工作,教书育人,深得国内同行的好评;同时由于在研究领域的国际声誉,他也是新罕布什尔大学数学系的大教授。难能可贵的是,作为学弟,在执迷于数学的学长最困难的时候,他真正做到了出手相助。 思考张益唐 释放学术研究正能量 张益唐成功很重要的一点是淡定,宠辱不惊。在朋友开的赛百味快餐店帮忙,他可以一丝不苟。在大学任教,年近 60 还只是个讲师,在一般人看来无疑是失败,甚至是潦倒的,但他处之泰然,不改其志。 难能可贵的是逆境之中他还是一如既往地作大问题。作大问题的人不需要太多,但不能没有!张益唐的精神及成就,对中国科学界是极大的正能量,也是对目前浮躁的科研环境的一种鞭策。 2013 年 5 月 20 日,耶鲁大学法学教授斯蒂芬•卡特在《彭博》上撰文《可以是电影明星的数学家》,他认为张益唐的励志故事是一个很好的电影题材。网上也有人建议文学家、编剧、导演们可以把张益唐的故事搬上银幕,拍出比《美丽心灵》更美的电影。 张益唐做过学生会主席,具有演讲天才,喜欢文学、音乐,是 NBA 球赛的铁杆球迷,还可以喝一斤二锅头没感觉。他应该是新时代数学家的好代言人。 成名后的张益唐仍像过去一样低调淡定。他说:“我的心很平静。我不大关心金钱和荣誉,我喜欢静下来做自己想做的事情。” 张益唐自己想做的事情是什么呢?他还在瞄着迄今未解决的另一个大猜想。我们希望他能够在平静中再创神话。 (作者系香港浸会大学教授) 《中国科学报》 (2013-07-19 第 5 版 人物周刊 ) 相关文章 我的同学张益唐 张益唐是北大 1978 级学生,我是低他两届的北大数学系系友。加上研究生期间,我们在燕园里共同度过了 6 年时光。 在大学期间,张益唐数学成绩是很有口碑的。沉寂多年后,他在孪生素数问题上作出了巨大突破,大家都替他高兴。 这几年他在国外的艰苦和成功,有我的同学唐朴祁、葛力明的重要帮助,张益唐的同班同学、好友沈捷是我多年的学术合作者,从他们 3 位那里我得到了很多关于张益唐出国后的资料,在此表示感谢。 张益唐有很多故事,比如他能把班上所有同学的生日轻松记下来,每年都会给好朋友发电邮祝贺生日;只作大问题,对小文章毫无兴趣,以至毕业后找不到数学方面的工作,等等。 参加 2013 年 5 月武汉大学数学学院组织的科学计算国际会议期间,大家都在热议张益唐的数学突破和他这些年的坎坷。几位北大校友,包括沈捷、上海交通大学数学 系主任金石、北京大学数学院常务副院长张平文,都觉得张益唐对数学的孜孜追求有很强的正能量,值得宣传鼓舞。在他们的鼓励下,还有唐朴祁、葛力明的大力支 持下,我动笔写了这篇文章。 张益唐是我们这个时代的骄傲,他甘于寂寞、专注科研的精神值得我们学习,希望本文能够给读者传递一些正能量。(汤涛) 《中国科学报》 (2013-07-19 第 5 版 人物周刊 )
我是个数学的门外汉,所以我的问题可能很幼稚,各位请包涵。在图一中,有 A 、 B 、 C 三种情况。二维的曲线代表一个由不同观点的人构成的封闭社会,线上每个点代表一个人(我们不去考虑曲线上点的无限性,考虑它为有限点构成)。曲线中间的黑色圆点,代表一个社会决定,比如说政权的选择。图中的箭头线,代表每个人发出的意见或观点,箭头线不等长,如果等长,只是例外。箭头线的长度,与每个人对黑点的支持程度成反比:线越短,支持度越高;线越长,反对程度越大。 对于图一 A ,我把它认为是“独裁社会”,也就是说,黑点的位置 X 由极少数人 (红色部分包含的曲线)决定。图一 B ,我用来代表民主社会,红色部分代表大多数民意,姑且说是 51% 的民意。其中的黑点 Y 代表了多数民意的决定,但不能代表所有人的意见。图一 C ,我企图用来代表“中庸社会”。其中的黑点 Z ,比图一的 Y 点要靠右一些,理论上,它应该是一个能够代表曲线上所有点意见的最佳位置。 在我们认识的社会中,民主是一个群体社会的概念,它是一个社会体系,有很多的内容,但它的一个基本内容是多数人的意志。一个民主社会体系可以通过选票这个机制来表达多数人的意志。尽管民主社会的选择,通常具有趋中的现象,但还不是中庸。中庸在传统中国文化中,它通常是对一个人的行为而言,中不偏,庸不易。对一个人的选择,没法实行民主。但对一个社会来说,中庸之道可以是一个选择吗?或者说,图一 C 中的 Z 点不仅在数学上成立,在社会机制上也可以实行。它不可能是所有人同等赞同或反对的一个社会决定,但这个决定代表了所有人的最大利益。这就是传说中的共产主义吗? 问题: 1 )从数学上如何表现出点 Y 和 Z 的最佳位置? 2 )图 1B 的数学模型,如果有的话,它的社会实践可以通过选票来实现。但从 社会学的角度看, Y 点一定是最理想的社会状况吗?为什么? 3 ) Z 点的最佳位置在数学上如果能找到,从社会结构上可以实行吗?或者说,如 何能到达一个最佳的社会决定,它顾及到了一个社会中所有人的意见? 4 )从社会学的角度看,维持图一 A 的情况,是否需要更多的能量?为什么?
请教数学家( 7 ) 多元向量除法 草原所所长,法人代表,侯向阳博士 及其他领导,工作人员: 师弟所长, 在这封电子邮件里,我们继续讨论:系统演替趋势的定义和多元向量除法的定义。我在科学网博文“多元向量乘法的几何意义”: http://blog.sciencenet.cn/blog-333331-545120.html 中提出多元向量 ( m - 向量 )乘 积的定义: 若向量 A i 、 B i 是 m 元 向量, i=1,2,…m, 则它们(对应分量相乘)的‘向量积’ C i = A i * B i = ( a 1 b 1 , a 2 b 2 , a 3 b 3 , ... , a m b m )也是 m 元向量。 ‘向量积’的几何解释是位于对角线上以对应分量为边的 m 个矩形的面积。 并进一步把这样定义的‘向量积’与‘点积’、‘差积’做了比较,说‘向量积’简练地保留了分量的信息,没有冗余,是同一空间的 m 元向量。 乘积是同一空间的 m 元向量的专业说法是: 这样定义的乘法是‘封闭的’,它们的积仍然是同一空间的向量,所以这样定义的‘向量乘法’有逆运算,多元向量除法运算。 多元向量除法的定义是:分量的商做商的分量( Bai, 1997 ) 我们在‘草原退化趋势分析及动态监测方程组’,方程( 2 )中,使用了多元向量除法定义。 若多元向量没有除法,则草原演替趋势的定义便不能成立。 而趋势定义的成立与应用,又从反面证明了多元向量除法定义的重要意义:使多元系统动态分析可解,有唯一确定的解。 参考文献: Multi-Dimensional Sphere Model and Vegetation Instantaneous Trend Analysis T. JAY BAI1, TOM COTTRELL2, DUN-YUAN HAO3, TALA TE4, and ROBERT J. BROZKA5 (Ecological Modelling, 97/1-2. http://blog.sciencenet.cn/home.php?mod=spaceuid=333331do=blogid=449959 918 猜想 http://blog.sciencenet.cn/home.php?mod=spaceuid=333331do=blogid=531295
http://news.sciencenet.cn/htmlnews/2007111975942705194534.html 张寿武:一位天才加幸运的数学家 Recently, I read this story about Shou-Wu, Zhang. There is a interesting detail about the importance of papers. Before the year of 1989, Faltings seemed to be not interested in Zhang's project. When Zhang completed a manuscript and showed to Faltings, he got a friendly response from the later. The saying of President Xi, Empty talk can lead a country astray( "空谈误国"), seems to be correct in many fields.
说课:《魔方和数学建模》 VS 群星灿烂的匈牙利数学家 上一回说: 在 1974 年前后,英国、美国和日本都发明出了类似的转动魔方, 为什么只有匈牙利人鲁毕克的魔方能风靡全球? 2009 年英国《每日电讯》报道,匈牙利人的魔方在全世界已经销售了 3.5 亿 个。如果一个魔方赚 1 美元,匈牙利靠魔方就赚了 3.5 亿美元 。 YouTube 有 39,600 个关于魔方的视频,其总点击率可想而知。 http://www.telegraph.co.uk/lifestyle/4412176/Rubiks-Cube-inventor-is-back-with-Rubiks-360.html ( 英国每日电讯 ) 为什么只有匈牙利人鲁毕克的魔方能风靡全球,火爆世界? 这与匈牙利的数学及其文化有关。实际上,这就又回到了我们的《魔方和数学建模》第 1 讲,即魔方的文化内涵。 《魔方和数学建模》第一讲:魔方的文化内涵 http://v.163.com/special/cuvocw/mofangheshuxue.html (网易播放) http://www.icourses.edu.cn/details/10425V002 (爱课程播放) 匈牙利到底有什么样的数学及其文化呢? 还是先让我们看看匈牙利有多少世界级别的伟大数学家吧! 先看看出自匈牙利的两位世界数学掌门人: 冯 · 卡门( Theodore von Kármán , 1881–1963) ,伟大数学家; 冯 · 诺伊曼( John von Neumann , 1903–1957) ,伟大数学家。 冯 · 卡门 在 匈牙利大学本科毕业 ,在德国获得博士学位,后来去了美国,是我国著名科学家钱学森的导师, 开创了数学和基础科学在航空和航天以及其他技术领域的应用,被誉为 “ 航空航天时代的科学奇才 ” ; 冯 · 诺伊曼 是小学、中学、大学和博士都毕业于匈牙利的大学, 开创了现代计算机理论和博弈科学 。 需要强调的是, 冯 · 卡门 在 匈牙利大学本科毕业, 冯 · 诺伊曼在匈牙利学习一直到博士毕业。 匈牙利有 5 位数学家获得国际数学大奖,他们是: 沃尔夫奖: Paul Erdős ( 1983 ), Peter Lax ( 1987 ), László Lovász ( 1999 ), Raoul Bott ( 2000 ); 阿贝尔奖: Endre Szemerédi ( 2012 )。 以上的奖都是数学奖,此外,匈牙利还出过 13 位诺贝尔奖获得者。 较早有名的匈牙利数学家还有: Farkas Bolyai (1775–1856) ; János Bolyai (1802–1860) ; Gyula K ő nig (1849–1913) ; József Kürschák (1864–1933) ; Farkas Gyula (1847 - 1930) ; Lipót Fejér (1880–1959) ; Frigyes Riesz (1880 - 1956) ; George Pólya (1888–1985) ,等等。 关于匈牙利数学家的资料,引用了在美国出版的 《匈牙利的艺术和科学》( Hungarian Arts and Sciences ,1848-2000 ) 。 图 1 和图 2 是 Abel 奖官方网站介绍匈牙利的部分数学家的网页截图。 图 2 Abel 奖官方网站介绍和提及 11 位匈牙利数学家 由此可见,匈牙利拥有辉煌的数学,其数学文化又怎么样呢? 匈牙利的数学文化(含奥数文化)和数学教育 据网络文章 CREATING A CULTURE OF PROBLEM SOLVING 和著作 Hungarian Arts and Sciences ( 1848-2000 )介绍,匈牙利不但有很多著名的数学家,还有不少著名的数学教育家。下面是美国数学学会网站为本科生推荐的匈牙利布达佩斯 15 周学习数学项目。 Budapest Semesters in Mathematics(BSM) 是专门为美国和加拿大本科生培训数学的项目。 Budapest Semester in Math - A 15-week mathematics study abroad program in Budapest, Hungary. Students take mathematics classes taught in English. http://www.maa.org/students/undergrad/ http://www.budapestsemesters.com/ 下面列出了六位匈牙利著名的数学教育家,详细介绍其中的两位,即第一位和第六位。 从 1896 年到 1914 年, László Rátz 任 KMaL 杂志 ( High School Mathematics and Physics Journal )主编。这是一本面向 对数学和物理特别感兴趣 的中学生的数学和物理期刊,创刊于 1894 年 ,一直办到现在,是世界上非常有名的中学数学和物理期刊,图 9 是该杂志的英文版合订本。 Paul Erdős 是匈牙利家喻户晓的数学家,也是 20 世纪最伟大的数学家。如前所述, Paul Erdős 于 1983 年获得 沃尔夫奖。 Paul Erdős 曾经和全世界 485 个人合作,发表过 1475 篇学术论文。 有一部关于 Paul Erdős 的电影记录片,题目叫 N is a Number ,还有 两本描述 Paul Erdős 的书,一本书名为 My Brain Is Open , 另一本书名叫 The Man Who Loved Only Numbers ,如图 10 和图 12 所示。 以上两本关于 Paul Erdős 数学生涯的书,穿插了很多数学游戏。因为, Paul Erdős 的主要研究领域是数论、组合学和图论等,这也是匈牙利为北美大学生培训的数学内容。 2009 年,美国数学学会出版了一本书,题目叫《伟大数学家的著名游戏》,作者 是 塞尔维亚 (The Republic of Serbia) 的一所大学( UNIVERSITY OF NIŠ )的数学教授( Miodrag S. Petkovic ),图 13 是这本书的封面。 图 13 美国数学学会出版的《伟大数学家的著名游戏》 《伟大数学家的著名游戏》一书,从古到今重点列出了 62 位世界伟大的数学家,中国有一位,是宋代杨辉(约 1238 -约 1298 );匈牙利有三位,他们是: George Pólya (1888 – 1985) ,有名著 How to Solve It (1945) ; John von Neumann ( 1903 – 1957) ,奠基计算机理论和博弈科学; Paul Erdős (1913 – 1996) ,主要贡献领域:数论,组合学和图论等。 由此可见,匈牙利拥有众多的群星灿烂的伟大数学家,游戏数学更是他们的强项,魔方从匈牙利出发,征服了全世界绝不是偶然的。 博友相关主题: 1) 杰出的匈牙利科学家群(武夷山) http://blog.sciencenet.cn/home.php?mod=spaceuid=1557do=blogid=2106 2) 匈牙利人为何大师频出? 精选 (徐坚) http://blog.sciencenet.cn/home.php?mod=spaceuid=63234do=blogid=283564
INTERNATIONAL SEMINAR ON HISTORY OF MATHEMATICS (NEW DELHI: 2012) (A Brief Report) A two-day International Seminar on History of Mathematics was organized by Ramjas College, University of Delhi recently . It was organized jointly with the Indian Society for History of Mathematics (ISHM) and Co-sponsored by the International Commission on the History of Mathematics (ICHM). The event focused on celebrating the National Mathematical Year to commemorate the 125th Birth Year of Genius Srinivasa Ramanujan. It covered all aspects of the history of mathematics, and in particular, the ancient Indian history of the subject. A large number of mathematicians participated in the deliberations, including leading experts from 16 countries including USA, U.K., France, Italy, Israel, Iran, Brazil Japan, China, Egypt, Denmark, South Africa and Nepal, apart from many eminent scholars from India. The venue of the conference was Ramjas College, University of Delhi. At the outset Principal Rajendra Prasad, Ramjas College (and Chairman, Local Organizing Committee) Welcomed the Delegates, followed by the opening remarks by Dr. Man Mohan (Seminar Convener). He gave a brief account of the Seminar and related organizational efforts .over the past one year. He mentioned that the inputs from Professors G E Andrews and Bruce C Brendt provided the platform for this Seminar. A small video clipping on Ramanujan was also shown. At this stage Professor June Barrow-Green (Open Univerusity, U.K.) read out the message of the President of the International Commission on the History of Mathematics. This was followed by the remarks highlighting the importance of the Seminar by Professor S G Dani (President, ISHM) and the remarks of Ambassador Balkrishna Shetty, the Guest of Honor. The Seminar was inaugurated by Professor Dinesh Singh, Vice Chancellor, University of Delhi. He congratulated the Ramjas College for organizing this International seminar. He expressed that it is difficult to find examples that can parallel the story of Ramanujan in the annals of Mathematical lore. Professor Dinesh Singh, Vice Chancellor, University of Delhi inaugurates the Seminar The inaugural session ended with a vote of thanks by Dr. Ruchika Verma, the Co-Convener. There were in all eleven lively sessions including the concurrent ones. The academic sessions began with the Talk by Professor S G Dani, President of the Indian Society for History of Mathematics. He explained the Role of Mathematical Societies in the early development of mathematics in India. Ambassador Balkrishna Shetty, IFS (Retd.), Diplomatic Mentor, Indian Council for World affairs, discussed about Connecting Dots: Meaning, Mathematical Development and Teaching. The following talks and papers were presented: A. H Siddiqi, Gautam Buddha University, NOIDA , “Functional Analysis in Historical Prospective and its Development in India” A.K. Agarwal, Panjab University, Chandigarh , “Ramanujan's congruence properties of partition function” Ajay Kumar, Dept. of Mathematics, D.U., “The Uncertainty Principle: A Mathematical Survey” Anjing QU, North Western University , China , “Genius in Mathematics: revisiting the history of Galois Theory” Anuradha Rajkonwar Chetiya, Ramjas College , “Tracing the Origin of Probability” Asim Kumar Majumdar, Visva-Bharati, Sriniketan, Birbhum, W.B , “Contribution of Sri Radhanath Sikdar and Sir Ashutosh Mukherjee as mathematicians” 4 Athanase Papadopoulos , Strashurg Cedex , France ,”Spherical Geometry: Some Milestones” BhuDev Sharma, JIIT University , NOIDA , “Srinivasa Ramanujan - Some Remarkable Life Events” CHEN Yiwen, Northwest University, Xi’an, China , “Mathematical communication of Chinese Scientific and Technological Journals in the early 20th century” D. S. Hooda, Jaypee University of Engg. and Technology, GUNA , Raghogarh, “Vedic and Ancient Indian Mathematics” Dinesh Singh, Vice Chancellor, University of Delhi , “A History of Functional Analysis: The Role of Fourier, Lebesgue and Hilbert” Ekaratna Acharya, Tribhuvan University, Saraswati Campus, Kathmandu, Nepal , “Naya Raj Pant’s Explanation of a formula of Ganita Kaumudi of Narayana Pandita” Geetha S Rao, Ramanujan Inst. for Adv. Study in Maths, Chennai ,” Tracing the development of Approximation Theory” Govind Singh, Kumaun University , Nainital , “Varahamihira: A Versatile Intellectual” Gregg De Young, The American University in Cairo, Egypt , “The Unintended Textbook: Muhammad Barakat’s Commentary on Book I of the Elements” Jayant Shah, Northeastern University, Boston, MA, USA , “Absence of Indian astronomy in Dayan li of Yixing” Jens Hyrup, Philosophy and Sc. Studies, Roskilde University, Denmark , “Sanskrit-Prakrit interaction in elementary mathematics as reflected in Arabic, Catalan and Italian formulations of the rule of three” June Barrow-Green, Editor, Historia Mathematica, Executive Committee Member-ICHM , Open University, U.K.,“Poincaré’s Last Geometric Theorem and its Legacy” K Ramsubramanian, IIT Bombay, Mumbai , “The significance of asak ṛ t-karma in finding the manda-kar ṇ a” K. Srinivasa Rao, Director, Srinivasa Ramanujan Academy of Maths Talent, Chennai , “Gauss, Ramanujan and hypergeoemtric series “ Kim Plofker, Union College, Schenectady NY, USA, “Treatises and tables: evolution of Sanskrit mathematical astronomy in the early modern period” Madan Lal Ghai, Mathematics, Punjabi University, Patiala , “Ancient Indian Mathematics” Min Bahadur Shreshta, Tribhuvan University, Kirtipur, Kathmandu, Nepal , “Paradigm change in philosophy of mathematics” Mita Darbari, St. Aloysius College, Jabalpur, “Pythagoras: Who was he? Niccolò Guicciardini, Editor, Historia Mathematica EC Member -ICHM, Italy , “Open issues in the new historiography of European early modern mathematics” Oscar Joo Abdounur, Rio Claro , Brazil , “The late creation of the university system in Brazil” Pankaj Kumar, Ramjas College , “Mathematics before the Vedic Period” Praveen Kumar, Ramjas College , “Fire Altars in Ancient India and their Mathematical Relation” R S Kaushal, Dept. of Physics, D.U ., “Ermakov-lewis-ray-reid system of coupled nonlinear differential equations: historical perspective” Rudra Hari Gyawali, Tribhuvan University, Sanothimi Campus, Bhaktapur, Nepal, “ Symbolic and Digital Logic” S L Singh, 21, Govind Nagar, Rishikesh ,”Resonance of Mathematics of Vedic Tradition” Swaminath Mishra, Walter Sisulu Univ Mthatha, South Africa , “Mathematics: Pure and Applied - Some Historical Perspectives“ 5 TANG Quan, Maths and Inf Sc, Xianyang Normal Univ., Xianyang, China , “Time difference algorithm of solar eclipse theory in ancient China and India” V. M. Mallayya, Mohandas College of Eng. Technology, Trivandrum , “Geometrical Approach to Indeterminate Equations” Wang Chang, Northwest University, Xi’an, China , “Fréchet’s Doctoral Thesis and Some Ideas” XU Chuansheng, Linyi University, Shandong Province, China , “A Research on The St. Petersburg Probability School ’s Probability Theorem” Yukio-Ohashi, Shibuya-ku Tokyo , Japan ,” Eccentric Model of the Solar Orbit in China” ZHAO Jiwei, Northwest University, Xi’an, Shaanxi Province, China , “Finding the Area of Sphere’s Surface: Bhaskara II and Archimedes” Summaries of all talks and papers, along with the CV and photo of each author, were available in the 132 page 4-color Souvenir that was released on the occasion. It contained useful information about the host city of Delhi, articles on its history, culture and seats of learning besides some articles on the history of mathematics. It also contained the information about the organizing institutes and the names and addresses of the participants. A Session in Progress A special session on Srinivasa Ramanujan was also held with a Panel Discussion and Interaction with students. The theme was "Relevace of Ramanujan in Mathematical Education Today". The Panelists included Professor S. G. Dani, Ambassador Balkrishna Shetty, Professors Anjing Qu, K. Srinivasa Rao, A. K. Agarwal, Kim Plofker, June Barrow-Green, Min Bahadur Shreshta, and Athanase Papadopoulos. Other highlights of the Seminar included screening a documentary Enigma of Srinivasa Ramanujan shown by Prof. K Srinivasa Rao, Chennai The delegates enjoyed a pre-dinner cultural programme by students and invited artists.. The illuminated College Building Appreciated by all, it was held in the College Lawns which was well illuminated and decorated in the classic tradition. The Valedictory function was chaired by Principal Rajendra Prasad. Participants freely expressed their comments and observations and gave fruitful suggestions for the future. Foreign and Indian delegates lauded the untiring work done by Dr. Man Mohan and his colleagues. The Co-convener, Dr. Ruchika Verma utilized this opportunity to present a vote of thanks and expressed gratitude to all those who helped in organizing this Seminar. She Principal Raendra Prasad at the Valedictory Function appreciated the support provided by the Principal, Dr. Rajendra Prasad and the Vice Chancellor, Professor Dinesh Singh, right from the beginning. She in particular thanked her colleagues Praveen, Pankaj and Jyotish for compiling the beautiful Souvenir, Munesh Chakravortty and M Ojit Kumar for excellent venue arrangements; Neelima, Virender and Shikha for managing registration; and Anuradha, Vishal and Bhuwan for organising the cultural program. The expenses on organizing the Seminar were met by • Collaboration amount given by the International Commission on the rHistory of Mathematics (ICHM) • Grants received from the National Board of Higher Mathematics (NBHM), the Council of Historical Research (ICHR), the Department of Science and technology (DST), the Council of Scientific Industrial Research (CSIR), and the Indian National Science Academy (INSA). • Advertisements in the Souvenir. • Delegation Fees. The Conference was well attended by about 150 persons and was a great success. In fact Organizers deserve all praise for so successfully arranging such a large scale International Seminar on history of Mathematics in India at a time when things seemed difficult in view of the prevailing tense international scenario. ReportofRamjasSeminar.pdf
武夷山先生曾有一篇博文,介绍了一首诗,据说英国数学家李特尔伍德喜欢。当时看过武先生博文后,特意记了下来。 武夷山博文:英国数学家李特尔伍德最喜欢的几句诗 http://blog.sciencenet.cn/blog-1557-451091.html 最近给大一新生讲授一门新课,无固定教材。花了很多时间,精心准备了内容,认认真真去讲,却有学生不来上课,很失望。想到那首诗,有感而发,也试着翻译一下。(见笑了!俺知道诗的翻译很有规矩,行家不必深究,让业余的人也玩玩。) 原诗: With them the Seed of Wisdom did I sow, And with my own hand labour'd it to grow: And this was all the Harvest that I reap'd -- "I came like Water and like Wind I go." 我的译文: 我播撒智慧, 我辛勤耕耘, 而收获却似梦幻, 来如水兮去如风。
可能受生物钟影响,北京时间 6 点多我就睡醒了。我一般不睡懒觉,醒了就起床。洗澡、洗衣服、看书等,直到有人敲门,原来是学生让去喝早茶。随后他们几人也陆续起床了。 大约印度时间 8 点,数学系副主任 J.S.Sikka 来接我们去吃早饭。此时他用白布紧紧裹着脸,起初认为他是否脸上受伤了,因而也不便多问。有意思的是,在会场他展露出了浓密的满脸胡须,问我是否认识他。至今也不知他当时为何裹着脸。 早餐比德里大学招待所的丰盛多了,有位老服务员对我们中国人很是热情,忙着给我们送上新油炸出的大饼,一些糊糊状的东西好像也干净不少。 两辆专用巴士把代表们送到会场—— Tagore Auditorium 。入口处有几位姑娘热情欢迎参会嘉宾,和国内不同的是,她们微笑着用大米、谷物等给每个人额头上点上红色印记,戴上花环,品尝冰糖和一种植物种子(略有苦味)。由于是第一次享受这样的礼遇,我们都很兴奋和激动,忙于照相纪念。 由于注册费 150 美元早已汇来,因而仅需领取代表证和有关资料即可。所发资料主要有两本: souvenir 主要是拉马努金、组织单位和组织人员的介绍,还有一本是 Book of abstracts 。 在 Souvenir 中,扉页就是拉马努金的画像,并写着:“ An equation for me has no meaning unless it expresses a thought of God. ” 封三也是拉马努金的画像,旁边写着:“ A tribute to the mathematical genius ” , “ He may lived a short life but he lived tall ”。 Tagore Auditorium 是一座大型综合性现代化建筑,造型独特、典雅,具有会议、电影、文艺演出、多功能会展、培训、礼仪接待等功能。场馆大门口外是 Tagore 塑像, 小广场富于时代气息和文化品位。中心会议厅呈阶梯状,设有 1000 余个高档软椅座位,台口两侧配大型电子屏幕显示器,多个特种电脑灯。主席台台面宽 20 余米,高 10 余米,与 ISHM-2012 相比,显得高大、宏伟和现代化多了。巨幅会标树立在主席台后方,会标以拉马努金的画像为背景,写着大会的有关主要信息,居中大字为“ INAUGURAL FUNCTION ”,主席台两侧为拉马努金的画像宣传版。主席台上设有 12 个席位。 开幕式原定印度时间 10:00 ,结果是 11:00 多才开始。印度人不守时是世界闻名的,他们拖后约定时间一小时很正常。然而这次推迟是因为省长 Jagannath Pahadia 的缘故,昨晚看到的持枪警察是其卫兵。对于姗姗来迟的省长大人,主持人要求全体代表起立欢迎,首先映入眼帘的是其持枪卫兵,省长身后紧跟着秘书和若干会议组织者。当省长落座后,还有两个人站在其身后,一个身着军装另一个穿便服, 4 个卫兵持枪分别站在主席台两边的后侧(在中国政治局常委出席会议也未这样戒备)。 开幕式最初程序还是分别给主席台上就座的人员献花,佩戴花环,接着是歌手唱歌,其歌声很是动听,可惜不知其意。然后依次主席台上领导讲话,最后讲话者是省长。他说: The modern age is the age of basic sciences which have blessed us with many comforts in our life. Science has conquered time and space and brought different economics under the umbrella of globalization. The year 2012 is also being celebrated as International year of Mathematics and 125 th Birth anniversary of the renowned mathematician Srinivasa Ramanujian is also being celebrated world wide. 的确,拉马努金不仅是印度的骄傲,也是整个世界的骄傲。他用独特思维诠释了近现代数学思想,连哈代都自愧不如。在其设计的数学天赋评分表中,哈代给自己打了 25 分,给李尔伍德打了 30 分,给希尔伯特打了 80 分,而给拉马努金却打了 100 分。 茶歇之后依次是 S.G.Dani 和 H.P.Dikshit 的两个 45 分钟大会报告,他们风趣幽默、态度认真、治学严谨,报告很是精彩。 报告结束后,已是印度时间下午 2:00 。午餐设在 Faculty House ,由于人员集中,两辆巴士很快就挤满了人,大会组织人员看到我们几个中国和欧美代表不上车有些不解,大概他们认为巴士车还可以再挤上一些人。后来他们用私家车把我们带到餐厅。 我们就餐时,宾馆警卫进行了较为隆重的欢送省长大人仪式。省长出生于 1932 年,今年已经 80 岁了。他还是该大学名誉校长。 省长一行的撤离,腾出宾馆不少房间,因而下午我们就搬到这里了。曲老师和 C 师妹各住一单间,他们的房间设施不错,分内外间,里间是卫生间和衣柜,外间有超薄电视机、沙发,大床看上去也豪华不少。 余下我们四人住在两个房间,后来发现这两个房间是通着的,因而交流起来很是方便。可房间设施较差,两个房间只有一个热水器、一台旧电视,且由于在四层大多时间自来水上不去,因而也就无法洗澡。 下午的报告仍在 Tagore Auditorium 举行。由于时间原因,有 2 个大会报告推迟到 23 日。因每个人都被安排了报告,曲老师叮嘱我们要认真准备报告、好好修改课件。曲老师和 C 师妹、 Z 都带着笔记本电脑,因而修改起来也较为方便。 晚饭前是文艺演出,组织者专门搭建了舞台,观众席也不是露天的,好像是用薄薄的纱罩在几根立柱围成的场地上方。会议请了两套演出班子,分别是男、女歌手带领其乐队,据说那个男歌手是印度著名歌唱家。 演出期间,丰盛的晚餐正在准备中,实际上奶茶早在下午就开始熬了。为了照顾我们,女主任还专门请来了炒面大师,为我们准备中国饭食。原来我从未喝过奶茶,加之这几天缺水喝,品尝着用陶罐盛着的热乎乎奶茶,感觉真是香甜呀!
匈牙利无家无工作的浪漫型大数学家 最近为了回答魔方的一些(数学文化和游戏文化)问题,俺查阅了很多关于匈牙利的文献(数学和游戏)。 1974 年,匈牙利人鲁毕克发明出了转动魔方, 1980 年前后,魔方风靡世界。 转动魔方于上世纪 70 年代诞生,这不是一个偶然事件。这与世界游戏文化(含数学文化)的沉淀有关。(适当的时候展开论述,有事实有根据) 匈牙利人的魔方能风靡世界,也不是偶然的。这与匈牙利的数学文化(游戏文化)有关。(适当的时候展开论述,有事实有根据) 今天简单说说匈牙利的一位浪漫数学家,叫 爱多士( Paul Erd ő s )。 这位数学家浪漫到什么程度呢? 他没有家,没有工作。 爱多士就像是一位云游诗人,他 美在四海云游, 不带分文在手, 为了他的吃和住, 掏出数学诗歌一首。 他补他的旧衣, 他修他的鞋底, 用数论, 用图论, 还有数学界的小道消息。 匈牙利有一位很有名的诗人,叫裴多菲( 1823 - 1849 ),其著名的诗: 生命诚可贵,爱情价更高;若为自由故,二者皆可抛 ! 为了追求自由,可以不要爱情,也可以不要生命。 诗人是这么说的,也是这么做的。普通人可以说到,但是做不到。 无独有偶,匈牙利有一位很有名的数学家,叫 爱多士( Paul Erd ő s , 1913 - 1996 )。 爱多士( Paul Erd ő s )没有家,没有工作,只有数学。 爱多士( Paul Erd ő s) 是世界上最伟大数学家之一,其浪漫尺度不亚于诗人 裴多菲 。 有一部纪录短片和两本传记描述 爱多士( Paul Erd ő s )数学人生(参见下图)。 匈牙利是数学强国,匈牙利有很多的数学家,这可能与匈牙利的数学文化有关系。 什么是匈牙利的数学文化呢? 这个问题可不好回答。 但是,数学文化里必须要有足够浓的游戏数学文化,否则就算不上是什么先进的数学文化。 2009 年,美国数学学会出版了一本书,题目叫《伟大数学家的著名游戏》,作者 是 塞尔维亚 (The Republic of Serbia) 的一所大学( UNIVERSITY OF NIŠ )的数学教授( Miodrag S. Petkovic )。 塞尔维亚是匈牙利的邻邦,他们的文化似乎也应该一衣带水。 《伟大数学家的著名游戏》重点提到了一位中国的数学家,就是宋代的杨辉。对于匈牙利的数学家,却重点提到了三位,典型是 爱多士( Paul Erd ő s )。
天才是寂寞的,关于天才的、 武夷山老师的 博文和其博主也是寂寞的。该博文被科学网热门,评论者寥寥有几,似乎无数学方面的评论。 武夷山的博文“两种天才”见: http://blog.sciencenet.cn/blog-1557-586566.html 午后困,天气闷热,生活在人间的“天才”感觉很无聊,思绪如天才般跳跃 。 于是,翻来闲书Nasar的关于Nash的《美丽心灵》,品味一下数学家眼里 的天才。 英文版 Prologue 第二页(Touchstone版第12页): Geniuses, the mathematician Paul Halmos wrote, "are of two kinds: the ones who are just like all of us, but very much more so, and the ones who, apparently, have anextra human spark. We canall run, and some of us can run the mile in less than 4minutes; but there is nothing that most of us can do that compares with the creation of the Great G-minor Fugue." Nash's genius was of that mysterious variety more often associated with music and art than with the oldest of all sciences. It wasn't merely that his mind worked faster, that his memory was more retentive, or that his power of concentration was greater. The flashes of intuition were nonrational. Like other great mathematical intuitionists -Georg Friedrich Bernhard Riemann, Jules Henri Poincare, Srinivasa Ramanujan-Nash saw the vision first, constructing the laborious proofs long afterward. But even after he'd try to explain some astonishing result, the actual route he had taken remained a mystery to others who tried to follow his reasoning. Donald Newman, a mathematician who knew Nash at MIT in the 1950's, used to say about him that "everyone else would climb a peak by looking for a path somewhere on the mountain. Nash would climb another mountain altogether and from that distant peak would shine a searchlight back onto the first peak". 中文见王尔山的翻译本第二页。 我们总结一下数学家眼里的天才,参考王本。Halmos说,存在两类天才,第一种就是武夷山博文里提到的普通天才,象你我一样,但要更加卓越;第二种(还)具有非同寻常的智慧灵光。他进一步具体解释为:大家都能跑,其中一些人可以在5分钟里跑完2千米,但是我们中间大多数人根本不能取得和巴赫伟大的G小调赋格曲媲美的成就。 我们关心的是第二种 的天才 。 Nash的天才是 神秘的,让人联想到音乐和艺术。不仅仅他的心智运转快,记忆力更加出众,或者更能集中注意力,这些都是能看到的,可以合理解释的。存在神秘的、 常人或局外旁观者 不能解释的是: 产生伟大作品的直觉火花 。 它是神秘的、不能用常理解释。 它来自何处?如何被产生的,怎样运作的? 这种神秘性,和音乐家工作方式类似。导致天才产生伟大工作的是:思想火花,即 非同寻常的智慧灵光 。 Nasar进一步指出:Nash的天才特征是他在解决数学问题前先有(看)vision。这个vision,我的理 解就是大局观和框架,在解决问题前,要构思出 框架( framework)。如同对 一件艺术品和建筑, 全局观和框架对研究工作(作品)很重要。Nash数学家同事解释他另外的显著特征是:不循规蹈矩的工作或思考方式。Nash解决问题方式是至上而下的:“其他人通常会在山上寻找攀登顶峰的道路。Nash却干脆爬上另外一座山,再反过来从那遥远的山峰用探照灯照射这座山。“(王本) 伟大工作是衡量一个人是否是天才的标准。天才的伟大工作来自于:智慧灵光和特有的工作方式。所谓 智慧灵光,即灵感,往往来自于发现表面看来毫不相干的事情上,存在 的 联系。 所有这些都来自于经验、积累以及运气。运气包括在某个时期和谁工作、听了谁的讲座、看了谁的文章等,比如Nash博士期间的工作和Neumann有关联,两者都在普林斯顿,尽管不同(后者在高等研究院,前者在大学数学系,Nash曾拜访过N大牛)。 当然所有这些都是建立在第一类普通天才所具备的共性基础上的,比如持久的集中力和坚韧的意志品质等。以上的论述 似乎 否定了天才要具有不同寻常的物质基础,从而武夷山老师的问题也就没有了意义。 借用鲁迅的名言,天才就是除了喝咖啡以外,所有时间都用在工作上了。
英国大数学家凯莱( Arthur Cayley )多产、多能而且多趣——下面是数学史家 Eves 总结的几点(见《数学圈》): 1 )凯莱的数学作品数量与柯西不相上下,两位可能都是仅次于欧拉的人。但谁是第二呢,要统计了二人的出版物才好说。 2 )凯莱开始并不以数学为生,在接受剑桥 Sadlerian 教授之前做过 14 年的律师。但他总是小心翼翼做法律,免得它受数学兴趣的干扰。在 14 年律师期间,他发表了 200 多页的数学论文。 3 )凯莱极爱读小说,旅游时读,开会前读,在任何零星的时间里读。他的一生读过几千部小说,不光英文,还有希腊文、法文、德文和意大利文。 4 )凯莱是真正英国传统的登山爱好者,常去欧洲大陆登山和长途旅行。据说他讲过,他喜欢爬山的原因是,虽然登山艰辛而劳累,但征服一座山峰的愉悦就感觉像解决了一道数学难题或完成了一个复杂的数学理论。他还说,登山很容易获得那种快感。 5 ) 1842 年, 21 岁的凯莱从剑桥三一学院毕业,获得数学荣誉学位考试的优等成绩,同年,在更难的史密斯奖竞赛中名列第一。 6 )凯莱喜欢画画,特别是水彩画,是一位出色的水彩画家。 7 )在数学家中凯莱是最冷静和尖锐的,但他不光是数学家,他是数学家与自然爱好者的复合体。 8 ) 1883 年,在英国科学促进会主席的就职演讲中,凯莱表达了如下的观点:“很难确定为现代数学划定一个疆域。‘疆域’一词是不对的:我说的是一个充满了美妙风景的疆域——不是平淡无奇的大平原,而是远远浮现在眼前的美丽乡村,那儿的山坡和沟谷、溪流和岩石、森林和鲜花,都等着我们去探幽入微。但正如其他事物的美一样,数学的美也只能感觉而不能解释。”这些话不是学究式的空谈,而是真切反映了对自然的亲近。 【关于那个就职演说,还有一段关于欧几里得几何的话,好像与很多数学家和物理学家的观点不同,值得我们再反思一下: It is well known that Euclid 's twelfth axiom, even in Playfair 's form of it, has been considered as needing demonstration: and that Lobachevsky constructed a perfectly consistent theory, wherein this axiom was assumed not to hold good, or say a system of non-Euclidean plane geometry. My own view is that Euclid 's twelfth axiom in Playfair 's form of it does not need demonstration, but is part of our experience - the space, that is, which we become acquainted with by experience, but which is the representation lying at the foundation of all external experience. Riemann 's view ... is that, having ' in intellectu ' a more general notion of space ( in fact a notion of non-Euclidean space ) , we learn by experience that space ( the physical space of our experience ) is, if not exactly, at least to the highest degree of approximation, Euclidean space. But suppose the physical space of our experience to be thus only approximately Euclidean space, what is the consequence which follows? Not that the propositions of geometry are only approximately true, but that they remain absolutely true in regard to that Euclidean space which has been so long regarded as being the physical space of our experience. 我们知道那是欧几里得第五公设,老凯说 12 ,不知为什么。所谓 Playfair 形式( John Playfair , 17428 - 1819 ,是苏格兰数学家),就是我们在平面几何里学过的那种形式: 过直线外一点,有且仅有一条直线与它平行 。(欧几里得原来的表述形式很“模糊”: 如果一条线段 两条直线 相交,在某一侧的内角和小于两直角 和,那么这两条直线在不断延伸后,会在内角和小于两直角和的一侧相交。)】
Donald Rubin John L. Loeb Professor of Statistics 所在单位 Harvard University 发文量 Publications: 321 | 被引量 Citations: 40103 | G-指数 G-Index: 199 | H-指数 H-Index: 62 研究领域: Interests: Psychiatry Psychology , Statistics , Education 科研合作 Collaborated with 1553 co-authors from 1971 to 2011 ; Cited by 54925 authors 个人主页 http://www.stat.harvard.edu/faculty_page.php?page=rubin.html
可怜的mathematical physicists 里外不是人,物理学家认为这是数学,数学家认为是物理。。。。 而事实上,最美的部分,也恰恰在数学与物理最深层次的结合部。。。 The so-called "mathematical physicists" , are scientists that in words of Feynman are "neither good physicists nor good mathematicians" . Described by Feynman below: I am not getting anything out of the meeting. I am learning nothing. Because there are no experiments this field is not an active one, so few of the best men are doing work in it. The result is that there are hosts of dopes here (126) and it is not good for my blood pressure: such inane things are said and seriously discussed here that I get into arguments outside the formal sessions (say, at lunch) whenever anyone asks me a question or starts to tell me about his "work". The "work" is always: (1) completely un-understandable, (2) vague and indefinite, (3) something correct that is obvious and self evident, but a worked out by a long and difficult analysis, and presented as an important discovery, or, a (4) claim based on the stupidity of the author that some obvious and correct fact, accepted and checked for years, is, in fact, false (these are the worst: no argument will convince the idiot), (5) an attempt to do something probably impossible, but certainly of no utility, which it is finally revealed at the end, fails (dessert arrives and is eaten), or (6) just plain wrong. There is great deal of "activity in the field" these days, but this "activity" is mainly in showing that the previous "activity" of somebody else resulted in an error or in nothing useful or in nothing promising. It is like a lot of worms trying to get out of a bottle by crawling all over each other. Remind me not to come to any more gravity conferences! References ↑ Richard P. Feynman, "Feynman lectures on physics" ↑ Quoted from Feynman's letter to his wife while attending Gravity Conference in 1962 in Warsaw, Poland published in book "What Do You Care What Other People Think" , page 91
2004 年 5 月 24 日 , Michael Atiyah 和 Isadore Singer 在奥斯陆获 Abel 奖时,接受了丹麦 Aalborg 大学 Martin Raussen 和挪威 Trondheim 科技大学 Christian Skau 的访问( Notices of The AMS , 2005, Vol. 52, No.2: 223-231 ), Atiyal-Singer 指标定理 是 20 世纪的数学珍宝,连接了几何与分析,也沟通了数学和物理—— A 更喜欢说它是一个理论, S 说它是一个新的起点,就像我们爬上一座高山,发现了过去所在的高原。他们原来没想到,这个定理会给物理学带来那么大的影响—— Perhaps it should not have been a surprise because it used a lot of geometry and also quantum mechanics in a way, à la Dirac. 关于数学为什么契合物理学, Singer 讲了一个小故事:他曾在费曼( Feynman )的讨论班讲反常,老费的博士后们总是想拿坐标来计算。老费告诉他们, 物理学定律是独立于坐标系的。没听 Singer 讲吗,他就没用坐标来描写这种情形 。独立于坐标,就意味着几何。 Singer 是用坐标独立来说明数学(几何)与物理学的自然联系,其实那 也是物理学“自觉几何化”的原因。 相对性原 理假定物理定律与坐标无关,就把物理学几何化了(狭义相对论的 Minkowski 几何和广义相对论的 Riemann 几何)。今天,那种独立性更进一步:物理学应该与时空背景无关,我们 需要一个“背景独立的理论”—— 如 Smolin 说的, 背景独立形式并不仅仅是不同的语言,而有可能表达了确定理论的原理和定律,迄今所研究的一切都将作为近似从它们推导出来。 尽管超弦代表了大多数人的声音, Atiyah 也欣赏少数派的 Alain Connes 和 Roger Penrose 。 Singer 喜欢弦,说它是一个和谐的整体,而且 K 理论 进来了;他也欣赏 Connes 的 非对易几何 ,那是几何量子 化需要的,也许还能 解释黑洞和大爆炸呢。 关于物理学对数学的影响, A 猜想, 量子物理也许会影响数论 ,特别是 Riemann 猜想 。他与 Wiles (证明了 Fermat 大定理 )讨论 过,但 W 不以为然。 S 则认为也许从统计物理学会生出一个 统计拓 扑学 ,那样的话,我们 就不必去数几何体的空穴数或 Betti 数 了…… 未来究竟会发生什么激动人心的事情呢? A 认为这个 问题“从定义上说”是没有答案的—— 假如我们能预言,它就不会那么令人激动了。
微分几何经过种种的融合后将会是多姿多彩的,但是它能否有足够丰富的结构来迎合近代物理时空量子化的需要? 我看不行! 数学有的时候确实领先物理,但归根结底,数学是落后于物理的,而且是远远落后。。。 好的数学,都是从物理中来的,比如微积分之于牛顿! 新几何不能指望数学家,数学家靠不住了,就得自己动手! math is too hard formathematicians... physic is too hard forphysicist... “Physics is to mathematics like sex is to masturbation.” —Richard Feynman
到二十世纪末,人们对「信号」这个词的理解已经发生了微妙的变化。如果在二十世纪上半叶的时候提到一个信号,人们还倾向于将它理解为一个连续的函数。而到下半叶,信号已经越来越多地对应于一个离散的数组。毫无疑问,这是电子计算机革命的后果。 在这样的情形下,「不确定性原理」也有了新的形式。在连续情形下,我们可以讨论一个信号是否集中在某个区域内。而在离散情形下,重要的问题变成了信号是否集中在某些离散的位置上,而在其余位置上是零。数学家给出了这样有趣的定理: 一个长度为 N 的离散信号中有 a 个非零数值,而它的傅立叶变换中有 b 个非零数值,那么 a+b ≥ 2√N。 也就是说一个信号和它的傅立叶变换中的非零元素不能都太少。毫无疑问,这也是某种新形式的「不确定性原理」。 在上面的定理中,如果已知 N 是素数,那么我们甚至还有强得多的结论(它是 N. Chebotarev 在 1926 年证明的一个定理的自然推论): 一个长度为素数 N 的离散信号中有 a 个非零数值,而它的傅立叶变换中有 b 个非零数值,那么 a+b N。 不幸的是这里「素数」的条件是必须的。对于非素数来说,第二条命题很容易找到反例,这时第一条命题已经是能够达到的最好结果了。 这些定理有什么用呢?如果它仅仅是能用来说明某些事情做不到,就像它字面意思所反映出的那样,那它的用处当然相对有限。可是——这无疑是辩证法的一个好例证——这样一系列宣称「不确定」的定理,事实上是能够用来推出某些「确定」的事实的。 设想这样一种情况:假定我们知道一个信号总长度为 N,已知其中有很大一部分值是零,但是不知道是哪一部分(这是很常见的情形,大多数信号都是如此),于此同时,我们测量出了这个信号在频域空间中的 K 个频率值,但是 KN (也就是我们的测量由于某些原因并不完整,漏掉了一部分频域信息)。有没有可能把这个信号还原出来呢? 按照传统的信号处理理论,这是不可能的,因为正如前面所说的那样,频域空间和原本的时空域相比,信息量是一样多的,所以要还原出全部信号,必须知道 全部的频域信息,就象是要解出多少个未知数就需要多少个方程一样。如果只知道一部分频域信息,就像是只知道 K 个方程,却要解出 N 个未知数来,任何一个学过初等代数的人都知道,既然 KN,解一定是不唯一的。 但是借助不确定性原理,却正可以做到这一点!原因是我们关于原信号有一个「很多位置是零」的假设。那么,假如有两个不同的信号碰巧具有相同的 K 个频率值,那么这两个信号的差的傅立叶变换在这 K 个频率位置上就是零。另一方面,因为两个不同的信号在原本的时空域都有很多值是零,它们的差必然在时空域也包含很多零。不确定性原理(一个函数不能在频域 和时空域都包含很多零)告诉我们,这是不可能的。于是,原信号事实上是唯一确定的! 这当然是一个非常违反直觉的结论。它说明在特定的情况下,我们可以用较少的方程解出较多的未知数来。这件事情在应用上极为重要。一个简单的例子是医 学核磁共振技术(很多家里有重病患者的朋友应该都听说过这种技术)。核磁共振成像本质上就是采集身体图像的频域信息来还原空间信息。由于采集成本很高,所 以核磁共振成像很昂贵,也很消耗资源。但是上述推理说明,事实上核磁共振可以只采集一少部分频域信息(这样成本更低速度也更快),就能完好还原出全部身体 图像来,这在医学上的价值是不可估量的。 在今天,类似的思想已经被应用到极多不同领域,从医学上的核磁共振和 X 光断层扫描到石油勘测和卫星遥感。简而言之:不确定性可以让测量的成本更低效果更好,虽然这听起来很自相矛盾。 糟糕的是,本篇开头所描述的那个不确定性定理还不够强,所能带来的对频域测量的节省程度还不够大。但是数学上它又是不可改进的。这一僵局在本世纪初 被打破了。E. Candès 和陶哲轩等人证明了一系列新的不确定性原理,大大提高了不等式的强度,付出的代价是……随机性。他们的定理可以粗略叙述为: 一个长度为 N 的离散信号中有 a 个非零数值,而它的傅立叶变换中有 b 个非零数值,那么 a+b 以极大概率不小于 N/√(log N) 乘以一个常数。 这里的「极大概率」并不是一个生活用语,而是一个关于具体概率的精确的数学描述。换言之,虽然在最倒霉的情况下不确定性可以比较小,但是这种情况很罕见。一般来说,不确定性总是很大。于是可以带来的测量上的节约也很大。 这当然也是一种「不确定性原理」,而且因为引入了随机性,所以在某种意义上来说比原先的定理更「不确定」。在他们的工作的基础上,一种被称为「压缩 感知」的技术在最近的五六年内如火如荼地发展起来,已经成为涵盖信号处理、信息提取、医学成像等等多个工程领域的最重要的新兴工程技术之一。 不过,这些后续的发展估计是远远超出海森堡的本意了。
英国数学家李特尔伍德最喜欢的几句诗 武夷山 英国数学家 John Edensor Littlewood ( 1885 - 1977 ) 的文集《李特尔伍德杂录》 ( Littlewood’s Miscellany ) 是由Béla Bollobás (1943 -, 生于匈牙利的英国数学家 ) 编辑的。Béla Bollobás 在该书的“序言”中说,李特尔伍德最喜欢《鲁拜集》的第 28 首: With them the Seed of Wisdom did I sow, And with my own hand labour'd it to grow: And this was all the Harvest that I reap'd -- "I came like Water and like Wind I go." 我的译文如下: 我将智慧的种子播撒, 亲手劳作,促其长大。 我收获的一切转瞬成空: “我来如水,去如风”。 然后,我找到了郭沫若的译文 (http://5352919.blog.hexun.com/10118936_d.html) : 我也学播了智慧之种, 亲手培植它渐渐葱茏; 而今我所获得的收成 —— 只是 “ 来如流水,逝如风。 ”
数学圈里流行过一句俏皮话: Old math teachers never die, they just multiply . 表面的意思是,数学老师从来不死,而只是在“繁殖”。 “ die ”还有“ 中止 ”的意 思——所以,那句话的“数学”意义在于,当除 不尽的时候,就只好改做乘法。类似的拿数学家和数学名 词“开涮”的趣话还 有: Old math teachers never die, they just reduce to lowest terms; Old mathematicians never die, they just disintegrate; Old mathematicians never die, they just go off on a tangent; Old mathematicians never die, they just lose some functions. 仔细想来,这些调侃有着十分的道理——数学家们虽然“不死”,却落到了最低的份儿,崩溃了,走自己的路,残疾了。 关于 go off on a tangent ,除了“跑题儿”(甚至 “顾左右而言他”; tangent 即 “ diverging from an original purpose or course ”), The Oxford American College Dictionary 还解为 a completely different line of thought or action ,忽略 了“跑” 的过程,结果就有 点儿“独持偏见,一意孤行”了。
我所知道的计算流体力学(CFD) (1) Jameson的故事 Jameson是当今CFD届的超级大牛。偶的超级偶像哦。Jameson是个英国人,出生在军人世家。从小随老爹驻守印度。于是长大了也抗起枪到海外保卫日不落帝国,军衔是Second Lieutenant。无奈“日不落”已落,皇家陆军已经不需要他了。大概有什么立功表现把,退役后就直接进了剑桥大学。在那里拿到博士学位。辗转间从英国来到了美国,从工厂又到了学校。成了Princeton的教授。在那里提出了著名的中心差分格式和有限体积法。就是在这里,发表了他那篇著名的中心差分离散的有限体积法。中心差分格式,大家都知道,是二阶,但是稳定范围特别小,Pe不能超过2,于是就得加人工粘性(一听这名字,数学家就倔嘴巴,不科学嘛),这是大学生都知道的事,怎么加就是学问了。Jameson用二阶项做背景粘性,用四阶项抑制激波振荡(也亏他想得出来),配合他提出的有限体积法,获得了极大的成功,很快风靡世界,工程界几乎无一例外在使用他的方法,原因很简单,他的方法乐百氏,而且又有相当精度。从此大行于市,座上了P大的航空系系主任,也确立了CFD界第一大牛人的地位。 Jameson发文章有个特点,喜欢发在小会议上或者烂杂志上,反正是SCI检索不到地方。包括后来关于非结构网格,多重网格等等经典的开创性文章,都是这样。(如果按照清华的唯SCI论的评判标准,我估计在清华最多只能给他评一个副教授当当。)牛牛的人总是遭人忌妒,哪里都这样。看着Jameson的有限体积方法这么受欢迎,有些人就红眼了。于是说,有限体积方法不错,可惜只适合于定常问题计算,非定常计算就不怎么样嘛。Jameson那里能容忍别人对他的得意之做胡说。于是,灵机一动,想出了一个双时间尺度的方法,引进一个非物理时间,把非定常问题变成了一个定常问题计算,还真好使,又风靡世界,从此天下太平。97年,Jameson年龄到了,就从P大退休了,结果又被聘请到Standford大学当Thomas V. Jones Professor搞起了湍流来。前不久偶导师见他回来,对欧们边摇头边说,“几年不见,老得快不行了”,言下之意,我们如果想多活几年,不要去搞什么湍流。 (2) Steven A. Orszag的故事 Steven A. Orszag是一个天才级别的人物啦。在直接数值模拟,谱方法,湍流模型等等许多方面都有开创性的贡献。天才嘛,总是有缺陷的,不是生活不能自理,就是不懂得处理人际关系。前者还好办,只是lp不舒服,后者嘛,让同事和同行不舒服,可麻烦就大了。不幸的是,Orszag属于后者。对于他的恃才傲物,有人早就恨得牙根痒痒,报复的机会终于来了。 三十年前,湍流模型的先驱们,是通过数值试验,再连懵带猜的确定下了双方程湍流模型的参数。20年前,Orszag突发奇想,能否用RNG(重整化群理论)从理论上推导这些参数呢?RNG理论在相变上取得了很大的成功,发明者也在81年获得了 Nobel奖。牛人就是牛人很快居然真从理论上推出了这些参数。这下湍流模型界可炸开了锅,这岂不是要砸掉很多人的饭碗?这不等于说那些老家伙几十年前的工作一钱不值么?这帮大学霸可不是省油的灯。环顾地球之大,Orszag居然找不到一本杂志愿意接受他这篇文章。Orszag这个郁闷呀,这个生气呀,好歹庵也是绝代高手嘛,昨这么不给面子呢?他一气之下干脆自己扛杆旗,办份杂志,自己当主编,自己出版,看谁说闲话。1986年,《Journal of Scientific Computing》终于开张了。第一篇文章就是“Renormalization Group nalysis of Turbulence: I Basic Theory”。这篇文章很快获得了大家的广泛认同。但是对RNG的攻击并没有到此为止。偶看到最搞笑的是一个牛牛(不想提他的名字了)在AIAA J. 上的一篇文章。当然是吹自己的模型计算比标准双方程模型多么多么的好。都已经比较结束了,他还觉得不过瘾,话锋一转,把RNG模型胡算一把,然后一桶狂批,还煞有介事的分析为啥算不好。其实我倒觉得,既然RNG能够从理论上推导出他们当年胡乱搞出来的参数,不正是对他们工作的证明么?能够从完全黑暗的世界寻找到这些参数,这除了天才,还能说什么呢? (3) Godunov的故事 Godunov大家都晓得吧,迎风类型格式的开山鼻祖。二十世纪CFD的数值方法基本上是沿着他老人家开创的Godunov类型格式的方向发展。连如今大姥级的Roe,van Leer都要发文章pmp,毕竟他们都是靠着老大发家的嘛。他座上老大宝座的屠龙刀-Godunov格式,实际上是1954年他25岁时候的博士论文。老板上课时候曾经讲,当时不知道为啥他得罪了苏维埃政府要砍他的头,于是他一着急,弄出了这把屠龙宝刀,拣回了小命(不过这个传闻,我没有找到相关的文献得以证实,好在我相信偶老板读的书比我多,二来嘛本来就是八卦系列也无所谓了)。 我现在就来讲讲有根有据的东西,老大是怎么弄出这把屠龙刀的。1954年春天,苏联的第一台电子计算机“Strela”就将送到老大当时所在的单位Keldish Institute of athematics,上级要求他们弄几个格式来算一算。当时一个叫Zhukov的人就弄出了一个东西。这家伙也算是个牛人了,弄出来的这个东西,同1年后 P.D Lax的CFD奠基性名著中提出的东西是完全一样的。可惜呢,这家伙数学不好,他是连蒙带猜弄出来的,尤其是为了自圆其说的那几个假设,现在回过头来看根本就是错误的,是推不出这个结果的。当时为了弥合这个问题,就请来了Godunov看能不能解决这个问题。结果一发不可收拾,居然就借此搞出了 Godunov格式。后来老大回忆刀,幸好当时他没有看到Lax的文章,要是看了,压根就不会有Godunov格式了。(If I would have read Lax’s paper a year earlier, “Godunov’s Scheme” would never have been created.)这么重大的贡献得发文章让大家都晓得才行呀。老大于是一毕业就四处投杂志,他先投了一家叫Applied Mathematics and Mechanics的杂志,杂志居然把他据了,理由是,老大的工作是一个纯粹的数学工作,没有做任何关于力学的研究。老大一想也对,他本来就是数学家嘛,于是他改投一个纯数学的杂志,谁知道,没过多久,又被退稿了,这次的理由是,老大的工作是一个纯力学的研究,没有任何关于数学的内容。老大当场晕倒。后来老大又投了几家还是不中,这下没有办法了,老大只好找后门,托他的老板Petrovskii了,正好老板是Mathematicheskii Sbornik杂志的编辑,终于在1959年,毕业四年后这篇文章发表在了这个杂志。 (4) Van Leer的故事 Van Leer 原先同Roe关系非常的好。后来Roe发表了著名的后来用他名字命名的Roe格式,Van Leer就有点座不住了。因为他一直相信他比Roe高明那么一点点。于是他决心超过Roe。当时迎风格式在应用上有两个发展方向,一个是Roe格式为代表的通量差分分裂类型,令一个就是矢通量差分类型,典型代表就是Steger-Warming格式。很快van Leer找到了突破口,他注意到Steger-Warming格式有个不大不小的缺陷,通量分裂是不可微的,这在计算激波时候,有可能发生过冲现象。于是 van Leer对此做了一番改造,提出了一个满足可微条件的分裂。van Leer兴高采烈地投到杂志社,然而令他失望的是,杂志社把他给拒绝了。他可受不了了,于是自己掏钱,飞到西伯利亚,向Godunov求教。Godunov看过后大加赞赏。这下可乐坏van Leer。既然老大首肯了,谁还敢说不字,这篇文章顺利出版。后来这个格式就用van Leer本人的名字命名并流行起来,终于,他还是跟Roe平起平坐了。 (5) Batchelor的故事 Batchelor 是GI Taylor之后,剑桥学派的领袖。不过他其实并不是英国人,而是澳大利亚人。他从小在墨尔本长大。第二次世界大战其间,在从事了一个航空相关的课题研究中,他对湍流研究产生了浓厚的兴趣,尤其是GI Taylor三十年代关于湍流研究的工作。于是他就给Taylor写信,想做他的research student。Taylor很快同意了。Batchelor是一个很跋扈的人,说话颇有些像黑社会的老大的风范。他有一个死党和跟屁虫。他非常想让这个跟屁虫跟他一块到英国去研究湍流,省得他一个人寂寞。这个死党呢,大学学的是跟湍流八竿子打不着的核物理。这并不要紧,Batchelor充分发挥了他黑社会老大般的威严对他说,“跟我到英国找Taylor研究湍流去吧!”这个铁杆兄弟也不含糊,立刻说,好,跟老大走。不过走前,你回答我两个问题:谁是 G.I. Taylor? 湍流是什么玩艺?前一个问题好回答,后一个问题,Batchelor究竟是怎么回答的,是威逼利诱,还是晓之以理动之以情说服的,大家一直为这个问题争论了几十年。总之,最后两人都去了英国。见了Taylor呢,两人都失望了,原来Taylor已经不搞湍流了,全力搞什么水下爆炸之类的跟军事有关的课题(估计这个来钱)。好在大师就是大师,让这两个年轻人自编自导自己去折腾,在旁边指导指导。最后两人都成为大师。Batchelor的这个小兄弟究竟是谁呢?呵呵,就是大名鼎鼎的AA Townsend。这个故事再次说明跟好一个老大是多么重要亚。 Batchelor曾经一度以为可以在他手上终结湍流问题。所以那段时间,在湍流研究上特别努力,结果当然是大失所望。Batchelor被湍流折磨得心力憔悴,50年代后期以后逐渐把精力从科研转移到了写书,创办应用数学力学系和JFM杂志上来。前面文章说了,为了多活几年不要搞湍流,这个故事则告诉我们,为了不郁闷,生活充满阳光,也不要搞湍流。另一个被湍流折磨死掉的大牛就是量子力学里面的Heisenberg。年轻的时候,靠着他的天才禀赋,胡乱猜了一个湍流解获得了博士学位,后半生被湍流研究折磨而死,临终时候都念念不忘。用《大话西游》里面的话来说应该是怎么来着?我猜中了这个开头,可是却猜不到这个结局。 (6) Von Neumann的故事 Von Neumann是天才里面的天才。据说他6岁能心算8位数除法,8岁时已掌握了微积分,12岁时能读波莱尔的著作《函数论》……。有一次,冯·诺伊曼对他的朋友说:"我能背诵《双城记》"。人家就挑了几章作试验,果然他-一背诵如流。他对于圆周率π的小数位数,自然对数的底e的数值以及多位数的平方数和立方数……四十年代的时候,Von Neumann在曼哈顿计划里面主要负责数值计算工作,他的另外两个同事就是费米和费曼。牛人在一起当然就喜欢比一比。需要做一个复杂的数值计算时,他们三人立即一跃而起。费米呢,上了点年纪,就拉计算尺计算,费曼呢,年轻人喜欢接受新事物,就用台式计算机,而冯·诺伊曼啥都不用,总是用心算。可是冯·诺伊曼往往第一个先算出来,当然这三位杰出学者所得出的最后答数总是非常接近的。(好啦,好啦,俺实在不愿继续写他的非凡事迹了,越写越自卑,越写越郁闷。)也就是在这段时间,Von Neumann提出了CFD上面非常有名的Neumann稳定性分析。这个现在本科生都晓得的东西,在当时被美国军方列为高度军事机密,这一保密就是十年。俺每次读到这段的时候,常常想起哈里森.福特的《夺宝奇兵》的最后一个镜头。【说到这里,顺便扯远一点,很多人,包括数学系人都认为Neumann稳定性分析为无条件稳定的格式,就意味着计算时间步长选取是不受限制的,这个认识是不正确的。Neumann稳定只保证格式的对幅度是保真的,但是并不保证是保相位的,相位的误差的累积也足以把一个结果改得面目全非】 前面讲过了一个让同事不爽的天才,而Von Neumann则属于让lp不爽的天才。某天lp让他上班途中顺便仍包垃圾,结果中午回来的时候,他又把垃圾带回来了,而他的公文包被他当垃圾扔了。另外一次,lp回来后,Von Neumann问她,我的水杯在那里呢,我找了一下午都没有找到。Lp大叫,天啦,我们在这个房子里面生活了十五年!天才的才气往往同寿命成反比,Von Neumann也不例外,刚过50多点点就去世了。应了俺本科上铺曾经爱说得一句话,天才是两头燃烧的蜡烛,明亮,但不会长久。 (7) Kuchemann的故事 今天要讲的是关于Kuchemann的故事。一看这名字就知道是德国人,1930年19岁的他进入了当时世界上最NB的大学Goettingen大学。起初他不是学流体的,而是理论物理的,他的导师就大牛M. Born。如果希特勒不上台,也许他会沿着理论物理学的道路走下去。然而1933年希特勒上台,推行歧视犹太人政策改变了这一切,Goettingen大学里面同犹太人沾亲带故的人纷纷远走他乡,这也包括了Born。为此Kuchemann郁闷坏了,因为他找不到一个他看得上眼的大师级的导师。于是他翻开 G大的研究生招生手册,翻来翻去,终于找到了一个没有走的大牛——近代流体力学大师Prandtl。于是他就拜Prandtl为师,改学空气动力学起来。在Prandtl和Tollmien(发现T-S波的那个大牛)的指导下,25岁就获得了博士学位。 欧一直怀疑Kuchemann是个种族主义者,即使不是,也肯定是欧洲至上主义者。这家伙特别瞧不起美国这个暴发户。二战后随着美国的崛起和欧洲的衰落,欧洲科学家纷纷踏上移民美国的之路,美国屡次三番的邀请他去,他就是不去,他说他是欧洲人,他要呆在欧洲,于是他宁可去了英国,也不去美国。他在英国一直呆到1976年去世。 他老人家最大的贡献是两个,一个是实用的脱体涡流型,在他之前人们都认为机翼只能采用附着流型,涡分离是必须避免的。有了他的理论,现在高速飞行很常用的前缘三维分离涡产生涡升力的细长机翼才得以实现(可笑的是,中国的气动教科书直到现在还在以附着流型为例,用白努力方程给学生解释升力产生的原因)。他的第二个重大贡献就是压缩波产生升力的高超声速流型,也就是现在称为乘波体的飞行器。可惜在他有生之年没有能够看到这个流型的应用。直到今年3月27日,美国采用他的乘波体方案以超燃冲压发动机为动力的的X-43A飞行成功,实现了7马赫数的w稳定飞行,一举打破了SR71在40年前创下的3.3马赫的飞行记录。 他老人家还说过一句,让所有从事CFD工作的人们需要永远永远铭记的话:每一种具体的理论或数值方法都是暂时的,而对流动本质的理解却是永恒的。
读书笔记 2月间,断断续续在读Masha Gessen的新书《 Perfect Rigor + 》(出版社:HOUGHTON MIFFLIN HARCOURT. BOSTON/NEWYORK 2009)。20世纪俄罗斯的数学呈现在眼前,有一种颠覆感,自己有限的历史知识向逆推:20世纪中,俄罗斯数学对中国数学的影响,俄罗斯的数学。 书的介绍: The true story of a mathematical mystery , a million--dollar prize, and the fate of genius in today's world. 对我来说, 这本书特殊之一,作者并没有亲自采访到Grigory Perelman,但她呈现他成长的历史背景和他的成长过程,活生生的, it's great。 她本身是一位在俄罗斯成长起来的数学天才,高中时代随父母移民美国,现居住于俄罗斯。新邻居刚好是来自俄罗斯的犹太人科学家家庭,他们家也有好多书和电影,更增加了我读这本书的兴趣。 书中所写俄罗斯数学家的生活: “Classical music and male bonding, mathematics and sports, poetry and ideas added up to Kolgorov's vision of the ideal man and the ideal school.” P39 Before the Iron Curtain sealed off the Soviet Union from the rest of the world, Kolmogorov and Alexandrov had done some traveling.......They imported all they could:books, music, ideas. "Interestin that this idea of a truly beloved friend seems to be purely Aryan: The Greeks and the Bermans seem always to have had it ." Alexandrov wrote toKolgorov in 1931, a few years before the reference to Aryans would have had a different connotation. "The theory of a lone friend is a difficult to fulfill in the contemporary world. " Kolmogorov lamented in response. "The wife will always have pretensions to that role,but it would be too sad to consent to this . In Aristotle's times, these two sides of the issuenever came into contact: the wife was one thing, and the friend quite another. " Kolmogorov brought back from Germany collections of verse by Goethe, who would always be his favorite poet. In all their letters to each other, Kolmogorov and Alexandrovinclded detailed reports of concerts attended and music heard, and when vinyl record became available, they started collecting them. ALexandrov hosted weekly classic-musicevenings at the university; he would play records and lecture on the music and the composers; ater Alexandrov's death, Kolmogorov---already nearing eighty and crippled byParkinson's disease---took over as host. 这竟然与几天后所读数学家Freeman Dyson的文章 Birds and Frogse中所写相同,真是君子所见: “In Russia, Mathematicians and composers and film-producers talk to one another,walk together in the snow on winter nighs, sit together over a bottle of wine, and share each other's thoughts.” 我爱这句话,我爱这篇文章,发誓要把它翻译出来! Dyson写道: "To end this talk, I come back to Yuri Manin and his book Mathematics. The book is mainly about mathematics. It may come as a surprise to Western readers that he writeswith equal eloquence about other subjucts such as the collective unconscious, the origin of human language, the psychology of autism, and the role of the trickster in themythology of many cultures. To his compatriots in Russia, such many-sided interests and expertise would come as no surprise. Russian intellectuals maintain the proud tradition of the old Russina intelligentsia, with scientists and poets and artists and musicians belonging to a single community. They are still today, as we see them in the plays of Chekhov, a groupof indalists bound together by their alienation from a superstitious society and a capricious government. In Russia, Mathematicians and composers and film-producers talk to oneanother, walk together in the snow on winter nighs, sit together over a bottle of wine, and share each other's thoughts. " Masha Gessen书中的介绍: Drawing on interviews with Perelman's teachers, classmates, coaches, teammates, and colleagues in Russia and the United States--and informed by her own background as amath whiz raised in Russia---Gessen set out to uncover the nature of Perelman's genius. What she found was a mind of almost superhuman computational power. Perelman was notintuitive or deeply imaginative as some mathemeaticians are, but his clear and rigorous thinking enabled him to pursue mathematical concepts to their logical(sometimes distant)end. Yet this very strength turend out to be his undoing. As Gessen discovered, such a mind is unable to cope with the messy reality of human affairs. When the jealous, rivalries, andpassions of life intruded on his platonic ideal, Perelman began to withdraw--first from the world of mathematics and then, increasingly, from the world im general. In telling his story,Masha Gessen has constructed a gripping and tragic tale sheds rare light on the unique burden of genius. P27 How to make a mathematician: Similarly, he(Rukshin) told me several times that his teaching methods could be reproduced, and had been, to rather spectacular results: his students made money by trainingmath competitors all over the former Soviet bloc. But other times he told me he was a magician, and these were the times he seemed most sincere. "There are several stages ofteaching ," he said. "There are the student, apprenticeship stages, like in the medieval guild. Then there are the craftsman, the master--these are the stages of mastery. Thenthere is the art stage. But there is beyond the art stage. This is the witchcraft stage. A sort of magic. It's a question of charisma and all sorts of other things." It may also have been that Rukshin was more driven than any coach before or since. He did some research work in mathematics, but mathematics seemed to be almost a sidelineof his life's work: creating world-class mathematics competitiors. That kind of single-minded passion can look and feel very much like magic. Magicians need willing, impressionalbe subjects to work theri craft. P 37 A beautiful School Whatever the reason, his not being a part of the military effort left Kolmogorov free to devote his considerable energies to creating the world ofr mathematicians that he hadenvisioned since he was a young man. Kolmogorov and Alexandrov both hailed from Luzitania, Luzin's magic land of mathematics, and they sought to re-create it at their dacha(俄国的乡间邸宅) outside of Moscow, where they would invite their students for days of walking, cross--country skiing, Listening to music, and discussing their mathematical projects. (dacha This involved a retreat to a country dacha or secluded library, there to write or grow vegetables and be isolated from the world beyond.追求自由的人逃离到远离都市的乡村别墅或隐秘的图书馆,在那里写作、种菜与世隔绝。) "The way our graduate group interacted with Kolmogorov was almost classically Greek, " said one of the countless memoirs published by his students; Virtually everyone who hadcontact with Kolmogorov seemed to have been moved to write about him. 'Through the woods or along the shore of the Klyazma River the muscular mathematician would bemoving brisky, on foot or on skis, surrounded by young people. The shy students would be rushing behind him. He talked almost without stopping---although, unlike perhaps theancient Greeks, he talked less of mathematics and more of other things." Kolmogorov believed that a mathematician who aspired to greatness had to be well versed in music, the visual arts, and poetry,and --no less important---he had to be sound ofbody. The mix of influences that shaped Kolmogorov's idea of a good mathematical education would have been an odd combination anywhere, but in the Soviet Union in the middle ofthe twentieth century, It was extraordinary almost beyond belief. Kolmogorov haided from a wealthy Russian family that founded a school of its own in Yaroslavl....... In 1922, Kolmogorov---nineteen, a student at Moscow University, and already an emerging mathematician in his own right---started teaching mathematics at an experimentalschool in Moscow. Incredibly, the school was modeled after the Dalton School, the famous New York City institution immortalized by, among others, Woody Allen in the filmManhattan. The Dalton Plan, which lay at the foundatin of both the Dalton School and the Potylikha Exempary Experimental School where Kolmogorov taught, called for anindividual insturction plan for every student. Each child would map out his own path for the month and proceed to work independently. So every student spent of his school time athis desk, or going to the small school libraries to get a book, or writing something." Kolmogorov recalled in his final interview. "The instructor would be sitting in the corner, reading,and the students would approach him in turn to show what they ahd done." This might have been the first sighting of the figure of the insturctor reading quietly behnd his desk;decades later, the math-club coach would take up this position. P39 Before the Iron Curtain sealed off the Soviet Union from the rest of the world, Kolmogorov and Alexandrov had done some traveling.......They imported all they could: books,music, ideas. "Interestin that this idea of a truly beloved friend seems to be purely Aryan: The Greeks and the Bermans seem always to have had it ." Alexandrov wrote to Kolgorovin 1931, a few years before the reference to Aryans would have had a different connotation. "The theory of a lone friend is a difficult to fulfill in the contemporary world. "Kolmogorov lamented in response. "The wife will always have pretensions to that role,but it would be too sad to consent to this . In Aristotle's times, these two sides of the issuenever came into contact: the wife was one thing, and the friend quite another. " Kolmogorov brought back from Germany collections of verse by Goethe, who would always be his favorite poet. In all their letters to each other, Kolmogorov and Alexandrovincleded detailed reports of concerts attended and music heard, and when vinyl record became available, they started collectiing them. ALexandrov hosted weekily classic-musicevenings at the university; he would play records and lecture on the music and the composers; ater Alexandrov's death, Kolmogorov---already nearing eighty and crippled byParkinson's disease---took over as host. Classical music and male bonding, mathematics and sports, poetry and ideas added up to Kolgorov's vision of the ideal man and the ideal school.
1900年Hilbert提出23大未解决的数学问题,激励了无数青年数学家的研究热情,数学在20世纪取得了极大的进步。2010年4月10日, 哈佛大学召集十余位“Big thinker”,征集社会科学难题及重要课题,并在Facebook广泛调查征求意见,列出十大难题。上周,美国NSF公布征集他们将支持的未解决的社 会科学难题,让社会学家提出"grand challenge questions that are both foundational and transformative"。 相关链接: Social science lines up its biggest challenges'Top ten' crucial questions set research priorities for the field. 哈佛大学问题征集Hard Problems in Social Science 哈佛大学社会科学未解决难题top ten 1. How can we induce people to look after their health? 2. How do societies create effective and resilient institutions, such as governments? 3. How can humanity increase its collective wisdom? 4. How do we reduce the ‘skill gap’ between black and white people in America? 5. How can we aggregate information possessed by individuals to make the best decisions? 6. How can we understand the human capacity to create and articulate knowledge? 7. Why do so many female workers still earn less than male workers? 8.How and why does the ‘social’ become ‘biological’? 9.How can we be robust against ‘black swans’ — rare events that have extreme consequences? 10.Why do social processes, in particular civil violence, either persist over time or suddenly change? ---转自陈泉兄的分享
中国学术评价网学术不端行为评议团公告(第6号) --关于方舟子抄袭案(第2号) (24292 bytes) Posted by: 柯华 Date: January 23, 2011 07:00PM 中国学术评价网学术不端行为评议团已对洪荞网友举报方舟子“数学史上一个大恩怨的真相”一文涉嫌抄袭英国University of St Andrews数学系教授John J O'Connor 和 Edmund F Robertson发表在自己主持的网站上的系列文章“Nicolo Tartaglia”、“Lodovico Ferrari”、“Scipione del Ferro”和“Girolamo Cardano”一案进行了评议,认定方舟子的文章确系抄袭之作,现将评议书和抄袭剽窃认定证书予以公布,同时将其抄送相关机构。 中国学术评价网版主 柯华 2011年1月23日(北京时间) 文件编号:学评网201101241号 抄送对象: 被抄袭人John O'Connor、被抄袭人Edmund Robertson、《经济观察报》、《中国青年报》冰点周刊主编徐百科、《中国青年报》冰点周刊科学版编辑杨芳、《中国青年报》新闻热点、中国科技大学校友会、党政办、生命科学院、团委、研究生院、新闻中心、福建省云霄县县长信箱、云霄县委宣传部网站、云霄一中、云霄一中校友会、福建省图书馆读者活动中心、中国新闻网编辑部、中央电视台、新华社总编室、学术批评网版主、密歇根州立大学学术诚信办公室、密歇根州立大学主管研究生工作副校长、密歇根州立大学学生报纸主编、被抄袭人Stanton Braude、被抄袭期刊现任主编、美联社、《科学》杂志新闻在线、《自然》杂志新闻编辑、《自然》杂志Asia-Pacific correspondent、《纽约时报》新闻部、美国《剽窃》(Plagiary )电子杂志编辑、科学诚信网 China Academic Integrity Review 方舟子抄袭案(第2号) 评议书 2010年12月23日,中国学术评价网版主柯华博士就洪荞网友举报方舟子“数学史上一个大恩怨的真相”一文抄袭英国University of St Andrews数学系教授John J O'Connor 和 Edmund F Robertson一事(见【方舟子涉嫌抄袭剽窃】公示第二号,链接: )召集本评议团进行评议。本评议团由三人组成,分别为美国机械和航空学博士、美国专业工程师(professional engineer),计算机硕士、IT工作者和中国某研究院电子技术研究员。 本评议团采用民事诉讼的“优势证据”标准,认真审核了举报材料,确认举报人提供了可靠、直接和具体的证据,尽到了自己的举证责任。根据这些证据,本评议团认定,方舟子的“数学史上一个大恩怨的真相”一文(发表于2006年9月23日《经济观察报》)是根据John J O'Connor 和 Edmund F Robertson发表在自己主持的网站( )上的系列文章“Nicolo Tartaglia”、“Lodovico Ferrari”、“Scipione del Ferro”和“Girolamo Cardano”翻译、改写而成。但是,方舟子在文章中没有说明自己的资料来源。中国学术评价网在组成评议团之前,曾将举报材料送达方舟子,请他做出辩解或提出反驳。但是,方舟子对此没有作出任何回应。 本评议团采用国际社会普遍接受的“抄袭”定义,即将他人作品或者作品的片段窃为己有属于抄袭剽窃。据此,本评议团认定,方舟子的上述行为属跨语际抄袭,是不可接受的学术不端行为。 《中华人民共和国著作权法》和大不列颠及北爱尔兰联合王国的版权法规Copyright, Designs and Patents Act 1988对于合理、合法翻译外文作品有相似的界定,即未经著作权人许可,以改编、翻译等方式使用该作品的,属于侵犯版权行为。本评议团敦请柯华先生就方舟子侵权行为通知有关机构。 此致 中国学术评价网版主柯华博士 中国学术评价网 学术不端行为评议团全体成员 2011年1月22日 关于我们 中国学术评价网由分布在世界各地的中国学者自发组成,旨在保护中国学者免受来自跨国网络恐怖、暴力团伙的人格侮辱和人身攻击,保护其职业生涯和家庭生活免遭肆意破坏。我们为学者发表自己的意见和观点提供平台。目前,我们致力于对方舟子现象的研究,对方舟子的不端及非法行为进行记录、揭发、评议和举报。 China Academic Integrity Review The Verdict January 22, 2011 As a result of the allegation by a registered member of China Academic Integrity Review, Hong Qiao, that Dr. Shimin Fang (aka Fang Zhouzi) plagiarized the serial articles by Drs. John J O'Connor and Edmund F Robertson of University of St Andrews, this panel is convened by Dr. Ke Hua, the coordinator of China Academic Integrity Review, to assess whether the accusation is true. The panel consists of three members, they are a Ph.D. in mechanics and aeronautics, a Master in computer science, and a research professor in electronics respectively. Before this assessment, Dr. Fang was offered an opportunity to defend himself and he did not response to the offer. Nevertheless, the panel has made an independent and careful examination of the material evidence and makes the following finding: Dr. Fang’s Chinese article, The Truth behind a Great Feud in Mathematics History (《数学史上一个大恩怨的真相》), published on Sept. 23, 2006, in a Chinese newspaper, Economic Observer (《经济观察报》), is a translated version of articles written by Drs. John J O'Connor and Edmund F Robertson. Dr. Fang did not acknowledge this fact in his article. The articles being plagiarized are: Based upon the copyright laws of both China and the United Kingdom, as well as the consensus definition of plagiarism, hold by the government agencies, academic institutions, and professional organizations around the world, this panel has unanimously reached the following verdict: The allegation is true, and Dr. Fang did commit plagiarism. As for the copyright violation issue, this panel urges Dr. Ke Hua to notify the related parties and agencies. The Academic Misconduct Assessment Panel China Academic Integrity Review About Us China Academic Integrity Review (AIR-China) is formed by a group of Chinese scholars from all over the world after the world-astonishing event involving internationally-acclaimed urologist Xiao Chuanguo and self-assumed science cop Fang Zhouzi. Our mission is to safeguard Chinese scholars’ human dignity, academic reputation, and legal rights from harassment, intimidation, threats, and terror by a certain transnational internet group, as well as from unwarranted and baseless attacks by laypersons who are not in the academic circle but use anonymous posts on the internet and/or sensational journalism to belittle Chinese scholars' achievements. We provide a platform for scholars to express their views on related issues. 抄袭剽窃认定证书中文版: 抄袭剽窃认定证书英文版: 【方舟子涉嫌抄袭剽窃】公示第二号(举报人:洪荞) 【说明: 2010年12月15日下午7:02(北京时间),本人以《就〈数学史上一个大恩怨的真相〉一文涉嫌抄袭的通知》为题,给方舟子发出如下邮件: 方舟子先生台鉴: 我是“中国学术评价”网站“方舟子系列”专题“抄袭剽窃”专辑主持人。今收到网友洪荞的文章,《“真相”的真相》,其中认为您在2006年9月23日《经济观察报》上发表的《数学史上一个大恩怨的真相》一文,涉嫌抄袭英国University of St Andrews 的两位数学教授在数学史网站上发表的系列文章。 经认真核对,仔细比较,本人认为洪荞网友的指控成立。按照“中国学术评价”网站《抄袭剽窃案例认定程序》(见: ),本人现将洪荞网友的文章转发给您,请您务必在三天内为自己的行为作出解释或者辩护。本人将根据您的回复,决定是否将其提交本网站评议团裁决。逾期不予回复,此案将自动按照《抄袭剽窃案例认定程序》处理。 特此告知。 敬颂 著祺! 亦明 谨上 2010年12月15日 至今,三日期限已到,但方舟子仍未回信。根据本网站《抄袭剽窃案例认定程序》,现将洪荞网友的举报文章公布出来,提请版主召集评议团就此举报是否成立予以评议。同时,欢迎诸位网友对此案踊跃发表自己的意见。 亦明 《中国学术评价网•方舟子系列专题•抄袭剽窃专辑》主持人 2010年12月18日】 “真相”的真相 洪荞 2010/12/12 2006年9月23日方舟子在《经济观察报》发表了一篇题为《数学史上一个大恩怨的真相》的科普文章。后来文章改名《被冤枉的数学家》,收录在《爱因斯坦信上帝吗?——方舟子解读科学史著名谜团》一书中。通过下面的比较我们可以看到这篇文章其实是十足的抄袭之作。 这次被抄袭的是 University of St Andrews 的两位数学教授。文章来自两位教授的数学史网站。在网站上两位教授说:我们非常欢迎各位使用我们准备的材料,但使用时要提到我们是原作者。我们也非常欢迎各位把我们准备的材料翻译成其它文字,但翻译时要提到我们是原作者 。 “真相”一文讲的是三次方程求解的争论史。上述网站对四位相关人物每人都有一份精彩的介绍 。“真相”所讲的故事几乎全部来自 。此外, “真相”还用到了 中的几小段以及 中的各一句。在这次抄袭中,方舟子的做法是维持 的整体结构,但砍掉一些句子。在谈及到其它三位人物时,方舟子从相应的文章中抽取一到几句与现有部分混到一起。如果刨掉了那些可以被肯定是抄袭的句子外,“真相”剩下的部分就只是一些段落间的连接句或可有可无的评论句。这样的句子真假难辨,因此全部罗列如下,由读者自己判断。换句话说,“真相”一文的整体构思是从 照搬过来的,而它的原创句子不会超出下面几句。 第一段: 头尾是连接句,中间是网文《数学和数学家的故事》的复述。 第二段:“这个流行版本从总体到细节都是错误的”, “而且也留下了有关这一争执的著作。后人对此事的看法在很大程度上就是受塔塔利亚一面之词的影响” 。 第三段:“塔塔利亚与卡当之间并未进行过数学比赛,和塔塔利亚比赛的另有其人。在当时的意大利,两个数学家进行解题比赛成了风气,方式是两人各拿出赌金,给对方出若干道题,30天后提交答案,解出更多道题的人获胜,胜者赢得全部赌金。” 第四段:“当时经常出现的比赛题目是三次方程,因为三次方程的解法还未被发现”,“塔塔利亚欣喜若狂” 。 第五段: 无。 第六段:“卡当把武林秘笈拿到手,并没有就对塔塔利亚翻脸,但塔塔利亚却像许多泄密者一样” 。 第七段:“卡当与塔塔利亚不同,热衷于通过著书立说发布新发现来赢得名利”。 第八段: 无。 第九段:“决定要为主人讨回公道”, “万一输了脸可就丢大了”。 第十段:“费拉里可谓占尽了天时地利人和”。 第十一段:“看来那个时候并没有禁止拖欠教师工资的规定” 。 第十二段:“只有卡当得以长寿,活到了75岁,不过他本来可以活得更长” 。 第十三段: 这段应该是原创 。 需要指出的是“抄一小段也是抄”的原则是可以用到这个案例上的。“真相”的第五段完全来自 。反之, 中的下面一段完完全全被抄进了“真相”的第九段: So Tartaglia replied to Ferrari, trying to bring Cardan into the debate. Cardan, however, had no intention of debating with Tartaglia. Ferrari and Tartaglia wrote fruitlessly to each other for about a year, trading the most offensive personal insults but achieving little in the way of resolving the dispute. Suddenly in 1548, Tartaglia received an impressive offer of a lectureship in his home town, Brescia. To clearly establish his credentials for the post, Tartaglia was asked to journey to Milan and take part in the contest with Ferrari. 事实上,“真相”并不只是抄了一小段。读完下面对比的读者不难得出“真相”是大面积抄袭的结论。 最后让我们以“真相”的结尾来结尾: 不过事实的真相毕竟难以掩盖,尤其是在信息发达的今天,更是如此。 数学史上一个大恩怨的真相 •方舟子• 数学史上著名的一个大恩怨许多人在中学学解方程时都听老师讲过。故事说,文艺复兴时期意大利数学家塔塔利亚发现了三次方程的解法,秘而不宣。一位叫卡当的骗子把解法骗到了手,公布出来,并宣称是他自己发现的。塔塔利亚一气之下向卡当挑战比赛解方程,大获全胜,因为塔塔利亚教他时留了一招。不过至今这些公式还被称作卡当公式,而塔塔利亚连名字都没有留下来,塔塔利亚只是一个外号,意大利语意思是“结巴”。网上广为流传的一篇《数学和数学家的故事》长文就是这么介绍的。 这个流行版本从总体到细节都是错误的。塔塔利亚不仅留下了名字(真名尼科洛•方塔纳),而且也留下了有关这一争执的著作。后人对此事的看法在很大程度上就是受塔塔利亚一面之词的影响。 塔塔利亚与卡当之间并未进行过数学比赛,和塔塔利亚比赛的另有其人。在当时的意大利,两个数学家进行解题比赛成了风气,方式是两人各拿出赌金,给对方出若干道题,30天后提交答案,解出更多道题的人获胜,胜者赢得全部赌金。 塔塔利亚很热衷于参加这种比赛,并多次获胜。Tartaglia gradually acquired a reputation as a promising mathematician by participating successfully in a large number of debates. 当时经常出现的比赛题目是三次方程,因为三次方程的解法还未被发现。意大利博洛尼亚数学家费罗发现了三次方程的一种特殊形式“三次加一次”的解法,临死前传给了学生费奥。费奥的数学水平其实很差,得到费罗的秘传后便吹嘘自己能够解所有的三次方程。塔塔利亚也自称能够解三次方程,于是两人在1535年进行了比赛。塔塔利亚给费奥出了30道其他形式的三次方程,把费奥给难住了。费奥则给塔塔利亚出了30道清一色的“三次加一次”方程题,认定塔塔利亚也都解不出来。塔塔利亚在接受费奥挑战的时候,的确还不知道如何解这类方程题。据说是在最后一天的早晨,塔塔利亚在苦思冥想了一夜之后,突然来了灵感,发现了解法,用了不到两个小时就全部解答了。塔塔利亚欣喜若狂,宽宏大量地放弃了费奥交的赌金。 The first person known to have solved cubic equations algebraically was del Ferro but he told nobody of his achievement. On his deathbed, however, del Ferro passed on the secret to his (rather poor) student Fior. ... and Fior had only been shown by del Ferro how to solve one type, namely 'unknowns and cubes equal to numbers'... Fior began to boast that he was able to solve cubics and a challenge between him and Tartaglia was arranged in 1535. In fact Tartaglia had also discovered how to solve one type of cubic equation ... Tartaglia submitted a variety of different questions, exposing Fior as an, at best, mediocre mathematician. Fior, on the other hand, offered Tartaglia thirty opportunities to solve the 'unknowns and cubes' problem since he believed that he would be unable to solve this type, as in fact had been the case when the contest was set up. However, in the early hours of 13 February 1535, inspiration came to Tartaglia and he discovered the method to solve 'squares and cubes equal to numbers'. Tartaglia was then able to solve all thirty of Fior's problems in less than two hours. ... Tartaglia did not take his prize for winning from Fior, however, the honour of winning was enough. 当时担任米兰官方数学教师的卡当听说了此事,通过他人转告塔塔利亚,希望能够知道解法,遭到塔塔利亚的拒绝。于是卡当直接给塔塔利亚写信,暗示可以向米兰总督推荐塔塔利亚。 At this point Cardan enters the story. As public lecturer of mathematics at the Piatti Foundation in Milan, ... he contacted Tartaglia, through an intermediary, ... asked to be shown the method, promising to keep it secret. Tartaglia, however, refused. An incensed Cardan now wrote to Tartaglia directly, ... hinting that he had been discussing Tartaglia's brilliance with the governor of Milan, Alfonso d'Avalos, the Marchese del Vasto, who was one of Cardan's powerful patrons. 在威尼斯当穷教师的塔塔利亚一见有高升的机会,态度大变,于1539年3月动身前往米兰,受到卡当的热情招待。在卡当苦苦哀求,并向上帝发誓绝不泄密后,塔塔利亚终于向卡当传授了用诗歌暗语写成的解法。卡当把“武林秘笈”拿到手,并没有就对塔塔利亚翻脸,但塔塔利亚却像许多泄密者一样,马上就后悔了,无心再在米兰求发展,匆忙赶回威尼斯。在那一年卡当出版了两本数学著作,塔塔利亚都细细研读,一方面很高兴卡当没有在著作中公布三次方程解法,一方面又觉得自己受了卡当的欺骗,在给卡当的信中把这两本书嘲笑了一番,断绝了与卡当的交情。 On receipt of this letter, Tartaglia radically revised his attitude, ... So, in March 1539, Tartaglia left Venice and travelled to Milan. ... Cardan attended to his guest's every need and soon the conversation turned to the problem of cubic equations. Tartaglia, after much persuasion, agreed to tell Cardan his method, if Cardan would swear never to reveal it, ... and Tartaglia divulged his formula in the form of a poem ... Anxious now to leave Cardan's house, he obtained from his host, a letter of introduction to the Marchese and left to seek him out. Instead though, he turned back for Venice, wondering if his decision to part with his formula had been a mistake ... Cardan published two mathematical books later that year and, as soon as he could get copies, Tartaglia checked to make sure his formula was not included. Though he felt a little happier to find that the formula was not included in the texts, when Cardan wrote to him in a friendly manner Tartaglia rebuffed his offer of continued friendship and mercilessly ridiculed his books on the merest trivialities. 卡当在获得塔塔利亚的解法后,在其基础上很快就发现所有的三次方程的解法。次年,卡当18岁的秘书费拉里在三次方程解法的基础上又发现了四次方程的解法。卡当与塔塔利亚不同,热衷于通过著书立说发布新发现来赢得名利。但是他和费拉里发现的解法都是建立在塔塔利亚的解法基础上的,根据卡当立下的誓言,塔塔利亚不公布其解法,他们的解法就不得公布。 Based on Tartaglia's formula, Cardan and Ferrari, his assistant, made remarkable progress finding proofs of all cases of the cubic and, even more impressively, solving the quartic equation. It was soon clear to Cardan that his secretary was an exceptionally gifted young man ... Ferrari ... when he was eighteen years old, he began to teach. ... Cardan and Ferrari made remarkable progress on the foundations that Tartaglia had unwillingly given them. They ... eventually were able to extend solutions discovered in these special cases. Ferrari discovered the solution of the quartic equation in 1540 ... but it relied on the solution of cubic equations so could not be published before the solution of the cubic had been published. However, there was no way to make this public without the breaking the sacred oath made by Cardan. 而塔塔利亚显然是想把其解法当成赢得比赛的秘密武器,丝毫也没有想公布出来的迹象。 Tartaglia made no move to publish his formula ... Tartaglia probably wished to keep his formula in reserve for any upcoming debates. 这让卡当很苦恼。 1543年,卡当和费拉里前往博洛尼亚,见到在那里接替费罗当数学教授的费罗的女婿,后者向他们出示了费罗的手稿,证明费罗在塔塔利亚之前就已经发现了解法。这使卡当如释重负,觉得没有必要再遵守誓言,于是在1545年出版的著作《大术》中公布了三次方程和四次方程的解法。为了避免被指控剽窃,卡当在书中特别提到了费罗和塔塔利亚的贡献。 Cardan and Ferrari travelled to Bologna in 1543 and learnt from della Nave that it had been del Ferro, not Tartaglia, who had been the first to solve the cubic equation. Cardan felt that although he had sworn not to reveal Tartaglia's method surely nothing prevented him from publishing del Ferro's formula. In 1545 Cardan published Artis magnae sive de regulis algebraicis liber unus, or Ars magna as it is more commonly known, which contained solutions to both the cubic and quartic equations and all of the additional work he had completed on Tartaglia's formula. Del Ferro and Tartaglia are credited with their discoveries, as is Ferrari, and the story written down in the text. Cardan and Ferrari satisfied della Nave ... and della Nave showed them in return the papers of the late del Ferro, proving that Tartaglia was not the first to discover the solution of the cubic. del Ferro ... kept a notebook in which he recorded his most important discoveries. This notebook passed to del Ferro's son-in-law Hannibal Nave ... Hannibal Nave took over del Ferro's lecturing duties at the University of Bologna 但是这并没有减轻塔塔利亚对他的憎恨。塔塔利亚在第二年出版了一本书,在书中揭露卡当背信弃义,淋漓尽致地对卡当进行人身攻击。卡当此时由于《大术》一书已名满天下,不想和塔塔利亚计较,但费拉里决定要为主人讨回公道,在公开信中对塔塔利亚反唇相讥,向塔塔利亚提出比赛挑战。塔塔利亚对此很不情愿,因为和无名小辈比赛即使赢了也没有什么好处,万一输了脸可就丢大了。塔塔利亚在给费拉里的回信中,要求由卡当来应战。但是卡当仍不予理会。塔塔利亚和费拉里来来回回打了一年的笔墨官司,仍然没有解决争端。到1548年事情出现转机,塔塔利亚的家乡布雷西亚向塔塔利亚提供了一份报酬不薄的教职,条件是塔塔利亚必须去和费拉里比赛解决争端。 Tartaglia was furious when he discovered that Cardan had disregarded his oath and his intense dislike of Cardan turned into a pathological hatred. The following year Tartaglia published a book, New Problems and Inventions which clearly stated his side of the story and his belief that Cardan had acted in extreme bad faith. For good measure, he added a few malicious personal insults directed against Cardan. Ars Magna had clearly established Cardan as the world's leading mathematician and he was not much damaged by Tartaglia's venomous attacks. Ferrari, however, wrote to Tartaglia, berating him mercilessly and challenged him to a public debate. Tartaglia was extremely reluctant to dispute with Ferrari, still a relatively unknown mathematician, against whom even a victory would do little material good ... So Tartaglia replied to Ferrari, trying to bring Cardan into the debate. Cardan, however, had no intention of debating with Tartaglia. Ferrari and Tartaglia wrote fruitlessly to each other for about a year, trading the most offensive personal insults but achieving little in the way of resolving the dispute. Suddenly in 1548, Tartaglia received an impressive offer of a lectureship in his home town, Brescia. To clearly establish his credentials for the post, Tartaglia was asked to journey to Milan and take part in the contest with Ferrari. 1548年8月10日,比赛在米兰总督的主持下在米兰的教堂举行,吸引了大量的看客。费拉里带了众多支持者助阵,而塔塔利亚只带了一位同胞兄弟,费拉里可谓占尽了天时地利人和,而且在开场白中就已经表现出他对三次和四次方程的理解要比塔塔利亚透彻。身经百战的塔塔利亚一见大势不妙,在当天晚上悄悄地离开了米兰。 On 10 August 1548, the contest which all Italy wanted to see, for the correspondence between the two antagonists had taken the form of open letters, took place in the Church in the Garden of the Frati Zoccolanti in Milan. A huge crowd had gathered, and the Milanese celebrities came out in force, with Don Ferrante di Gonzaga, governor of Milan, the supreme arbiter. Ferrari ... brought a large crowd of friends and supporters. Alone but for his brother, Tartaglia was a vastly experienced disputant ... By the end of the first day, it was clear that things were not going Tartaglia's way. .... Ferrari clearly understood the cubic and quartic equations more thoroughly than his opponent who decided that he would leave Milan that very night and thus leave the contest unresolved, so victory went to Ferrari. 结果塔塔利亚不仅名誉扫地,而且经济也陷入困境。布雷西亚虽然让他教了一年书,却不支付他的薪水。看来那个时候并没有禁止拖欠教师工资的规定,塔塔利亚打了几场官司也没能把欠薪讨回来,灰溜溜又回到威尼斯继续当他的穷教师。1557年,57岁的塔塔利亚带着对卡当的满腔仇恨,在贫困中死去。 Tartaglia suffered as a result of the contest. After giving his lectures for a year in Brescia, he was informed that his stipend was not going to be honoured. Even after numerous lawsuits, Tartaglia could not get any payment and returned, seriously out of pocket, to his previous job in Venice, nursing a huge resentment of Cardan ... He died in poverty in his house ... 13 Dec 1557 in Venice 费拉里在比赛后名声大震,甚至连皇帝都来请他给太子当老师。但费拉里选择了给米兰总督当估税员发财。1565年,年仅43岁的费拉里已成了富翁,提前退休回博洛尼亚,不幸当年就去世了,据说是被他的妹妹毒死的,为了继承他的财产。 On the strength of this challenge, Ferrari's fame soared and he was inundated with offers of employment, including a request from the emperor himself, who wanted a tutor for his son. Ferrari fancied a more financially rewarding position though, and took up an appointment as tax assessor to the governor of Milan, Ferrando Gonzaga. After transferring to the service of the church, he retired as a young and very rich man. He moved back to his home town of Bologna ... in 1565 but, sadly, Ferrari died later that year. It is claimed that he died of white arsenic poisoning, administered by his own sister. Certainly, according to Cardan, Maddalena refused to grieve at her brother's funeral and, having inherited Ferrari's fortune, she remarried two weeks later. 只有卡当得以长寿,活到了75岁,不过他本来可以活得更长——迷信占星术的卡当预测自己将死于1575年9月21日,为了实现自己的预言,他在那一天自杀。 Cardan is reported to have correctly predicted the exact date of his own death but it has been claimed that he achieved this by committing suicide. 科学研究毕竟是人从事的事业,人性的弱点也会在其中表现出来。做为一项最为看重首创权的工作,因争名夺利结下的种种个人恩怨也就难以避免,有时也难以让人看清其中的是非曲折。虽然根据现代科研的规范和历史资料来看,卡当在这个事件中的所作所为并无过错,他并没有试图去剽窃他人成果,为了公布学术成果与众人分享所作的努力还很值得赞赏,反倒是塔塔利亚死守学术成果的偏执和对卡当的憎恨都有点变态。奇怪的是,在后人的传说中,卡当却成了欺世盗名的骗子,人们对弱者的同情有时会超过了对真相的探求。不过事实的真相毕竟难以掩盖,尤其是在信息发达的今天,更是如此。 2006.9.17. (《经济观察报》2006.9.23,链接: ) 公示链接:
2010年度美国国家科学奖章获得者名单公布 http://news.sciencenet.cn/htmlnews/2010/10/238863.shtm http://scholar.google.com/scholar?hl=enq=D+MumfordbtnG=Searchas_sdt=2000as_ylo=as_vis=0 Optimal approximations by piecewise smooth functions and associated variational problems from uned.es D Mumford , J Shah - Communications on pure and applied , 1989 - interscience.wiley.com 'A preliminary version of this paper was submitted by invitation in 1986 to Computer Vision 1988, L. Erlbaum Press, but it has not appeared! ... Communications on Pure and Applied Mathematics, Vol. XLII 577-685 (1989) Q 1989 John Wiley Sons, Inc. Cited by 2160 - Related articles - All 16 versions Geometric invariant theory D Mumford , J Fogarty, FC Kirwan - 1994 - books.google.com David Mumford Department of Mathematics Harvard University 1 Oxford Street Cambridge, MA 02138, USA John Fogarty Department of Mathematics and Statistics University of Massachusetts Amherst, MA 01003, USA Frances Kirwan Balliol College Oxford OX 13BJ, England ... Cited by 1974 - Related articles - All 5 versions Abelian varieties D Mumford - 1970 - Oxford Univ Press Foundations of Algebraic Analysis Masaki Kashb/ara, Takahiro fawal, and Tatiuo Kimura Translated by Goro too The use of algebraic methods for studying anarysts b an Important theme In modem mathematics. The most sig- nificant devtluumem n this field Is mtootocal analysis, that Is, ... Cited by 1589 - Related articles - All 9 versions Tata lectures on theta I D Mumford - 1983 - books.google.com Modern Birkhauser Classics Many of the original research and survey monographs in pure and applied mathematics published by Birkhauser in recent decades have been groundbreaking and have come to be regarded as foun- dational to the subject. Through the MBC Series, ... Cited by 1283 - Related articles - Find in ChinaCat - All 8 versions The irreducibility of the space of curves of given genus from harvard.edu P Deligne, D Mumford - Publications Mathmatiques de l'IHES, 1969 - Springer Fix an algebraically closed field k. Let Mg be the moduli space of curves of genus g over k. The main result of this note is that Mg is irreducible for every k. Of course, whether or not M s is irreducible depends only on the characteristic of k. When the characteristic is o, we can assume that ... Cited by 1001 - Related articles - All 6 versions Boundary detection by minimizing functionals from northeastern.edu D Mumford , J Shah - Image understanding, 1988 - books.google.com 20 Mumford and Shah their analysis is to model them by smoothly varying functions with explicitly marked discontinuities. This is a mathematical statement of one of the major steps in figure/ground discrimination. This chapter studies a variational approach to this problem. Our approach ... Cited by 584 - Related articles - All 5 versions Lectures on curves on an algebraic surface D Mumford - 1966 - books.google.com ANNALS OF MATHEMATICS STUDIES Edited by Robert C. Gunning, John C. Moore, and Marston Morse 1. Algebraic Theory of Numbers, by Hermann Weyl 3. Consistency of the Continuum Hypothesis, by Kurt Godel 11. Introduction to Nonlinear Mechanics, by N. Kryloff and N. ... Cited by 755 - Related articles - Find in ChinaCat - All 5 versions Algebraic geometry I: Complex projective varieties D Mumford - 1995 - books.google.com OOCIM A.Dold Lectures on Algebraic Topology ISBN 3-540-58660-1 F. Hirzebruch Topological Methods in Algebraic Geometry ISBN 3-540-58663-6 T. Kato Perturbation Theory for Linear Operators ISBN 3-540-5866i-X S. Kobayashi Transformation Groups in Differential Geometry ISBN ... Cited by 527 - Related articles - All 8 versions Creativity syndrome: Integration, application, and innovation MD Mumford , SB Gustafson - Psychological Bulletin, 1988 - Elsevier The present article is concerned with certain conceptual issues embodied in the description and understanding of creative behavior. Initially, we argue that although creativity has been defined in many way, the ultimate concern in studies of creativity is the production of novel, ... Cited by 479 - Related articles - All 7 versions The topology of normal singularities of an algebraic surface and a criterion for simplicity D Mumford - Publications Mathmatiques de l'IHES, 1961 - Springer A normal point P in F is given. A finite sequence of quadratic transformations plus normalizations leads to a non-singular surface F' dominating F . The inverse image of P on F' is the union of a finite sct of curves El, E2, ...,E,. By further quadratic transformations if necessary we ... Cited by 438 - Related articles - All 3 versions Towards an enumerative geometry of the moduli space of curves D Mumford - Selected papers on the classification of varieties , 2004 - books.google.com Towards an Enumerative Geometry of the Moduli Space of Curves David Mumford Dedicated to Igor Shafarevitch on his 60th birthday Introduction The goal of this paper is to formulate and to begin an exploration of the cnumcrative geometry of the set of all curves of arbitrary ... Cited by 410 - Related articles - All 2 versions Stability of projective varieties: lectures... D Mumford - 1977 - L'Enseignement mathmatique Cited by 328 - Related articles The red book of varieties and schemes D Mumford - Lecture notes in mathematics, 1988 - cat.inist.fr ... Titre du document / Document title. The red book of varieties and schemes Auteur(s) / Author(s). MUMFORD D . (1) ; Affiliation(s) du ou des auteurs / Author(s) Affiliation(s). (1) Harvard univ., dep. mathematics, Cambridge MA 02138, ETATS-UNIS Rsum / Abstract. Varits. ... Cited by 359 - Related articles - All 3 versions Filters, random fields and maximum entropy (FRAME): Towards a unified theory for texture modeling from psu.edu SC Zhu, Y Wu, D Mumford - International Journal of Computer Vision, 1998 - Springer ... 112 Zhu, Wu and Mumford we assume that there exists a true joint probability density f (I)over the image spaceL| D | , and f (I)should concentrate on a subspace of L | D | which corresponds to texture images that have perceptually similar texture appearances. ... Cited by 360 - Related articles - BL Direct - All 34 versions Statistics of natural images and models from brown.edu J Huang, D Mumford - cvpr, 1999 - computer.org There has been much attention recently to the statistics of natural images. For example, Ruder- man discusses the approximate scale invariance property of natural images and Field linked the design of the biological vision system to the statis- tics of natural images. ... Cited by 295 - Related articles - BL Direct - All 16 versions An EBV-Genome-Negative Cell Line Established from an American Burkitt Lymphoma; Receptor Characteristics. EBV Infectibility and Permanent Conversion into EBV B Giovanella, A Westman, JS Stehlin, D Mumford - , 1975 - content.karger.com An in vitro line was derived from an American Burkitt lymphoma, designated Ra #1, which produced malignant tumors when inoculated into thymus-deficient nude mice. The cells have B-lymphocyte characteristics, with surface-associated mu and kappa chains and Epstein-Barr virus ... Cited by 328 - Related articles - All 3 versions Minimax entropy principle and its application to texture modeling from psu.edu SC Zhu, YN Wu, D Mumford - Neural Computation, 1997 - MIT Press ... Chun Zhu, Ying Nian Wu, and David Mumford is the partition function, which normalizes p(I; ) into a probability distri- bution. 2.2 Estimation and Computation. Equation 2.3 specifies an exponential family of distributions (Brown, 1986), S = {p(I; ,S) : Rd}, (2.4) where d is the total ... Cited by 318 - Related articles - BL Direct - All 13 versions On the computational architecture of the neocortex from brown.edu D Mumford - Biological cybernetics, 1991 - Springer Abstract. This paper proposes that each area of the cortex carries on its calculations with the active partici- pation of a nucleus in the thalamus with which it is reciprocally and topographically connected. Each corti- cal area is responsible for maintaining and updating the ... Cited by 289 - Related articles - All 7 versions On the Kodaira dimension of the moduli space of curves from psu.edu J Harris, D Mumford - Inventiones mathematicae, 1982 - Springer We consider the closure Dk of D k in 2g and compute the divisor class of O k in terms of the basic divisor classes ... (2k-4) The projectivity of the moduli space of stable curves I: Preliminaries on 'det'and 'Div' F Knudsen, D Mumford - Math. Scand, 1976 Cited by 281 - Related articles On the equations defining abelian varieties. I from kryakin.com D Mumford - Inventiones mathematic, 1966 - Springer My aim is to set up a purely algebraic theory of theta-functions. Actually, since my methods are algebraic and not analytic, the functions themselves will not dominate the picture - although they are there. The basic idea is to construct canonical bases of all linear systems on all ... Cited by 272 - Related articles - All 12 versions Hierarchical Bayesian inference in the visual cortex from psu.edu TS Lee, D Mumford - JOSA A, 2003 - opticsinfobase.org Received October 23, 2002; revised manuscript received February 21, 2003; accepted February 26, 2003 Traditional views of visual processing suggest that early visual neurons in areas V1 and V2 are static spa- tiotemporal filters that extract local features from a visual scene. ... Cited by 266 - Related articles - BL Direct - All 38 versions The role of the primary visual cortex in higher level vision from psu.edu TS Lee, D Mumford , R Romero, VAF Lamme - Vision research, 1998 - Elsevier In the classical feed-forward, modular view of visual processing, the primary visual cortex (area V1) is a module that serves to extract local features such as edges and bars. Representation and recognition of objects are thought to be functions of higher extrastriate cortical areas. ... Cited by 252 - Related articles - All 18 versions Varieties defined by quadratic equations D Mumford - Questions on Algebraic Varieties (CIME, III Ciclo, , 1969 Cited by 250 - Related articles Prym varieties I D Mumford - Contributions to analysis, in A Collection of papers , 1974 Cited by 245 - Related articles Prior learning and Gibbs reaction-diffusion from psu.edu SC Zhu, D Mumford - Pattern Analysis and Machine , 2002 - ieeexplore.ieee.org AbstractThis article addresses two important themes in early visual computation: First, it presents a novel theory for learning the universal statistics of natural imagesa prior model for typical cluttered scenes of the worldfrom a set of natural images, and, second, it proposes a ... Cited by 226 - Related articles - BL Direct - All 19 versions Leading creative people: Orchestrating expertise and relationships MD Mumford , GM Scott, B Gaddis, JM Strange - The Leadership Quarterly, 2002 - Elsevier Global competition, new production techniques, and rapid technological change have placed a premium on creativity and innovation. Although many variables influence creativity and innovation in organizational settings, there is reason to suspect that leaders and their behavior ... Cited by 217 - Related articles - All 5 versions Elastica and computer vision D Mumford , Center for Intelligent Control - 1991 - Center for Intelligent Control Cited by 216 - Related articles Leadership skills for a changing world:: Solving complex social problems MD Mumford , SJ Zaccaro, FD Harding, TO - The Leadership , 2000 - Elsevier Leadership has traditionally been seen as a distinctly interpersonal phenomenon demonstrated in the interactions between leaders and subordinates. The theory of leadership presented in this article proposes that effective leadership behavior fundamentally depends upon the ... Cited by 192 - Related articles - All 5 versions Filtering, segmentation, and depth M Nitzberg, D Mumford , T Shiota - 1993 - books.google.com Series Editors Gerhard Goos Juris Hartmanis Universitiit Karlsruhe Cornell University Postfach 69 80 Department of Computer Science Vincenz-Priessnitz-StraBe 1 4130 Upson Hall W-7500 Karlsruhe. FRG Ithaca, NY 14853, USA Authors Mark Nitzberg David Mumford ... Cited by 208 - Related articles - BL Direct - All 4 versions Theta-characteristics of an algebraic curve from brown.edu D Mumford - Ann. scient. Ec. Norm. Sup, 1971 - archive.numdam.org Gauthier-Villars (ditions scientifiques et mdicales Elsevier), 1971, tous droits rservs. L'accs aux archives de la revue Annales scientifiques de l'.NS (http://www. elsevier.com/locate/ansens), implique l'accord avec les conditions gnrales d 'utilisation ( ... Cited by 170 - Related articles - All 5 versions Curves and their Jacobians D Mumford - 1975 - getcited.org ... Curves and their Jacobians. Post a Comment. CONTRIBUTORS: Author: Mumford , David. PUBLISHER: University of Michigan Press (Ann Arbor). SERIES TITLE: YEAR: 1975. PUB TYPE: Book (ISBN 0472660004 ). VOLUME/EDITION: PAGES (INTRO/BODY): 104 p. ... Cited by 202 - Related articles - Cached - All 2 versions What can be computed in algebraic geometry from arxiv.org D Bayer, D Mumford - Computational algebraic geometry and , 1993 - arxiv.org This paper evolved from a long series of discussions between the two authors, going back to around 1980, on the problems of making effective computations in algebraic geometry, and it took more definite shape in a survey talk given by the second author at a conference on ... Cited by 176 - Related articles - View as HTML - All 16 versions An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equation, Korteweg-deVries equation, and related non- D Mumford - Proc. Int. Symp. on Alg. Geom., Kyoto, 1977 Cited by 133 - Related articles - All 3 versions Some elementary examples of unirational varieties which are not rational from psu.edu M Artin, D Mumford - Proceedings of the London , 1972 - plms.oxfordjournals.org For n\, these are equivalent (Liiroth's theorem). For n = 2 they are equivalent in characteristic 0 (Castelnuovo's theorem) or if the map / in (a) is assumed separable (Zariski's extension of Castelnuovo's theorem). In 1959 ( ), Serre clarified classical work on this problem for n = 3. It has ... Cited by 159 - Related articles - All 9 versions Abelian varieties. Tata Institute of Fundamental Research Studies in Mathematics, 5 D Mumford - Published for the Tata Institute of Fundamental , 1970 Cited by 169 - Related articles A Bayesian treatment of the stereo correspondence problem using half-occluded regions from psu.edu PN Belhumeur, D Mumford - Computer Vision and Pattern , 2002 - ieeexplore.ieee.org A half-occluded region in a stereo pair pixels in one image representing points in is a set of svace visi- ble to that camera or eye only, and not to ihe other. These occur typically as parts of the background imme- diately to the left and right sides of nearby occluding objects, and are ... Cited by 144 - Related articles - All 11 versions Managing creative people: strategies and tactics for innovation MD Mumford - Human Resource Management Review, 2000 - Elsevier With rapid changes in technology, and global competition, the success of many organizations has become progressively more dependent on their ability to bring innovative products to market. Ultimately, however, innovation depends on the generation of creative, new ... Cited by 159 - Related articles - BL Direct - All 4 versions FRAME: Filters, Random fields, and Minimax Entropy--Towards a Unified Theory for Texture Modeling SC Zhu, Y Wu, D Mumford - cvpr, 1996 - computer.org Abstract In this paper, a minimax entropy principle is stud- ied, based on which a novel theory, called FRAME (Filters, Random fields And Minimax Entropy) is pro- posed for texture modeling. FRAME combines attrac- tive aspects of two important themes in texture model- ... Cited by 146 - Related articles - BL Direct - All 6 versions Taxonomic efforts in the description of leader behavior: A synthesis and functional interpretation EA Fleishman, MD Mumford , Y Zaccaro Kerry, J - The Leadership , 1991 - Elsevier A systematic definition of the behaviors contributing to effective organizational leadership is required for both theory development and the design of training interventions. This article describes an attempt to formulate a general taxonomy capable of describing the functional behavioral ... Cited by 154 - Related articles - All 2 versions Rational equivalence of 0-cycles on surfaces D Mumford - Kyoto Journal of Mathematics, 1969 - projecteuclid.org ... previous :: next. Rational equivalence of 0-cycles on surfaces. D . Mumford . Source: J. Math. Kyoto Univ. Volume 9, Number 2 (1969), 195-204. Full-text: Access by subscription. PDF File (677 KB). Links and Identifiers. Permanent ... Cited by 144 - Related articles - All 4 versions Hirzebruch's proportionality theorem in the non-compact case from uni-due.de D Mumford - Inventiones Mathematicae, 1977 - Springer In this section we will not be concerned specifically with the locally symmetric algebraic varieties D /F, but with general smooth quasi-projective algebraic varieties X. When X is not compact, we want to study the order of poles of differential forms on X at infinity, and when E is ... Cited by 134 - Related articles - All 4 versions 7 Neuronal Architectures for Pattern-theoretic Problems from psu.edu D Mumford - Large-scale neuronal theories of the brain, 1994 - books.google.com 7 Neuronal Architectures for Pattern-theoretic Problems David Mumford As is abundantly clear from the other chapters of this book, there are many levels at which one can attack the problem of modeling the computations of the cortex. For example, at one extreme, one can model ... Cited by 145 - Related articles - All 6 versions A remark on Mahler's compactness theorem from harvard.edu D Mumford - Proceedings of the American Mathematical Society, 1971 - ams.org DAVID MUMFORD Abstract. We prove that if G is a semisimple Lie group without compact factors, then for all open sets UCZG containing the uni- potent elements of G and for all OO, the set of discrete subgroups rCG such that (a) rnU={e\, (b) G/T compact and measure (G/T) ... Cited by 109 - Related articles - All 5 versions A note of Shimura's paper Discontinuous groups and abelian varieties from harvard.edu D Mumford - Mathematische Annalen, 1969 - Springer Recently, there has been considerable discussion of families of abelian varieties parametrized by quotients of bounded symmetric domains by arith- metic subgroups. An exposition of this material can be found in the papers of Shimura, Kuga, Satake and myself in . Subsequently in ... Cited by 112 - Related articles - All 6 versions Stochastic models for generic images from psu.edu D Mumford , B Gidas - Quarterly of applied mathematics, 2001 - Citeseer The idea of using statistical inference for analyzing and understanding images has been used for at least 20 years, going back, for instance, to the work of Grenander Gr] and Cooper Co]. To apply these techniques, one needs, of course, a probabilistic model for some class of ... Cited by 134 - Related articles - View as HTML - BL Direct - All 7 versions Algebraic geometry D Mumford - 1976 - Springer Cited by 140 - Related articles - Find in ChinaCat The rare coagulation disordersreview with guidelines for management from the United Kingdom Haemophilia Centre Doctors' Organisation , PW Collins, S Kitchen, G Dolan, AD Mumford - , 2004 - Wiley Online Library ... Bolton-Maggs, PHB, Perry, DJ, Chalmers, EA, Parapia, LA, Wilde, JT, Williams, MD, Collins, PW, Kitchen, S., Dolan, G. and Mumford , AD (2004), The rare coagulation disorders review with guidelines for management from the United Kingdom Haemophilia Centre Doctors ... Cited by 123 - Related articles - BL Direct - All 4 versions Job Feedback: Giving, Seeking, and Using Feedback for Performance Improvement , RF Morrison, J Adams, MD Mumford , G Stokes, WA - questia.com There is a compelling need for innovative approaches to the solution of many pressing problems involving human relationships in today's society. Such approaches are more likely to be successful when they are based on sound research and applications. This Series in Applied ... Cited by 131 - Related articles An analytic construction of degenerating curves over complete local rings D Mumford - Compositio Math, 1972 - archive.numdam.org Foundation Compositio Mathematica, 1972, tous droits rservs. L'accs aux archives de la revue Compositio Mathematica (http:// http://www.compositio.nl/) implique l'accord avec les conditions gnrales d 'utilisation (http://www.numdam.org/legal.php). Toute utilisation ... Cited by 136 - Related articles - All 5 versions
Izrail M. Gelfand(盖尔芳德)是20世纪最伟大的数学家之一,1978年获得了首届数学沃尔夫奖。生平几乎涉猎数学的所有领域,特别是在泛函分析,群表示论,代数拓扑,微分方程等方面成就卓越。盖尔芳德曾先后于1954年、1962年、1970年三次被邀请在国际数学家大会上作一小时全会报告,这充分说明了他在国际数学界的地位。有人说,I.M. Gelfand的影响在代数几何之外无处不在。 1990年沃尔夫数学奖得主著名数学皮亚捷斯基-夏皮罗称:盖尔芳德、 柯尔莫戈洛夫、沙法列维奇是苏联数学界的三大巨人。盖尔芳德于1989年赴美,在Rutgers大学任教,号称是去改造美国的数学【北大一数学教授如是说】,2009年10月去世,终年96岁。 Izrail M. Gelfand Mikhail S. Gelfand(小盖尔芳德)是老盖尔芳德的孙子,莫斯科大学数学系毕业,从事生命科学领域的研究,曾经两次当选为霍华德休斯医学研究所的研究员(HHMI Investigator)。今天查资料的时候偶尔看到了他的个人主页。在90年代初的时候,俺曾经和他有过较长时间的交往,弹指一挥间,一晃也20年了。 小盖尔芳德 小盖尔芳德的个人主页: http://www.rtcb.iitp.ru/mg_e.htm Research interests : - comparative genomics - metagenomics - metabolic reconstruction and functional annotation of genes and proteins - identification of regulatory signals - evolution of metabolic pathways and regulatory systems - alternative splicing - statistics of DNA sequences Degrees : 2007. Professor of Bioinformatics 1998. Doctor of Science , State Scientific Center for Biotechnology . 1993. Ph.D./Candidate of Science (mathematics / biophysics). Awards : 2010. Academia Europaea, Member 2007 A.A.Baev Prize in Genomics and Genoinformatics (Russian Academy of Sciences) "for work in computer comparative genomics" 2004. "Best Scientist of the Russian Academy of Sciences" award, Fund for Support fo the Russian Science 2000. The President of Russian Federation's Award for Young Doctors of Science. 1999. A.A.Baev Prize from the Russian State Scientific Council "Human Genome" Current grants : 2009-2011. Russian Academy of Sciences, program "Basic Sciences for Medicine", project "Age-related changes in gene expression and alternative splicing" 2009-2011. Russian Academy of Sciences, program "Genetic Diversity", project "Modeling of the interactions of an invading phage and the host's restriction-modification system" 2006-2010. Howard Hughes Medical Institute "Comparative genomics and evolution of regulatory systems" (grant 55005610, PI) 2009-2011. Russian Foundation of Basic Research (09-04-92745) "Regulation and evolution of cellular systems" 2008-2012. Russian Academy of Sciences, program "Molecular and Cellular Biology", project "Comparative genomics, reconstruction and modeling, of metabolism and regulation in bacteria and eukaryotes" 2010-2012. Russian Foundation of Basic Research (10-04-00431) "Co-evolution of DNA-binding proteins and their DNA motifs" Editorial boards : 2009-2011. Member, Editorial Board, Journal of Bacteriology 2006-present. Member, Editorial Board, Molecular Biology (Moscow) 2006-present. Member, Editorial Board, Book Series "Bioinformatics and 2005-present. Member, Editorial Board, Central European Journal of Biology 2005-2011. Member, Editorial Board, Bioinformatics 2005-present. Member, Editorial Board, Biology Direct 2005-present. Member, Editorial Board, BMC Bioinformatics 2004-present. Member, Editorial Board, Journal of Bioinformatics and Computational Biology 2003-present. Member, Editorial Board, PLoS Biology 2003-present. Member, Editorial Board, Lecture Notes in BioInformatics (LNBI), Springer-Verlag. 2002-present. Member, Editorial Review Board, Archaea 1997-present. Associate Editor, Journal of Computational Biology.
好数学家工作范式是如何的呢?这也是我早年感兴趣的问题,因此也看过一些著名数学家的介绍,试图可以发现一些可以借鉴之处。看多之后,自然就想这样的问题:好的数学家有统一的工作范式吗? 好数学家需要多发表论文吗?当然不是。黎曼在解析数论里发表一篇不足十页的论文,即留下了当今数学最重要的难题黎曼假设;在一个就职演说就提出了黎曼几何。黎曼一生发表论文不足20余篇,但没有人怀疑黎曼的伟大( )。然而,倘若我们依此便说好数学家论文应是少而精,则大错特错了。欧拉从19岁开始发表论文,直到76岁,半个多世纪写下了886本书籍和论文,共计七十余卷( )。但是,欧拉是至少不在黎曼之下的伟大数学家。 好数学家需要长时间工作吗?也不是的。英国大数学家Hardy一般每天工作时间不超过4小时,在 里面,是这么描述Hardy一天生活的: In fact for most of his life his day, at least during the cricket season, would consist of breakfast during which he read The Times studying the cricket scores with great interest. After breakfast he would work on his own mathematical researches from 9 o'clock till 1 o'clock. Then, after a light lunch, he would walk down to the university cricket ground to watch a game. In the late afternoon he would walk slowly back to his rooms in College. There he took dinner, which he followed with a glass of wine. 60岁之后,Hardy便基本停止了数学工作。事实上,因为身体和精神上的原因,60岁后,他似乎失去了鲜活的数学创作力。然而,Hardy还是被认为英国继牛顿之后最伟大的数学家。但是,若据此便说好的数学家需要悠闲的工作激发创造力,那也是失之偏颇的。与Hardy相反的是传奇数学家Paul Erdos。在兴奋剂的帮助下,他平均每天工作十几个小时,而且一直工作到80多岁。好在数学研究不同于奥林匹克运动,没有人会说在兴奋剂帮助下做出的结果是无效的。所以,Paul Erdos仍然是个传奇。 那么,好的数学家是需要广泛的研究兴趣还是要坚守一个研究领域呢?20世纪前,数学家大多通晓数学的每一个分支。进入20世纪后,随着数学知识的指数级增长,不再有人通晓数学的所有分支。因此,有人坚持的观点是,若要做一个好的数学家,需浓缩于一个研究方向。这个观点当然是不全面的。正所谓分久必合,合久必分。特别是进入21世纪之后,数学不同分支之间影响、交叉逐渐显现出其重要性。其实,即便是在20世纪,兴趣广泛与坚守阵地也是并存的。仅举我所熟悉的领域为例。I. J. Schoenberg被认为是样条函数的创始人( ),然而,其最初的研究兴趣却是解析数论。他的领路人,也是其后来的岳父,便是大名鼎鼎的数论学家Landau。其后,Schoenberg曾涉足于多个不同数学研究领域,如距离几何、组合等。最后,提出了应用数学中广泛使用的样条函数。Schoenberg在最初的论文里提出了样条函数非常深刻的四种观点,它们并非现在教材所普遍采用的分片光滑多项式定义的方式。这些观点对样条函数的研究产生了极为深刻的影响,这大概与其早年从事纯粹数学研究有关。随后对样条函数产生重要影响的应是Carl de Boor。与Schoenberg不同,de Boor一开始便在逼近论领域工作,其后所发表的论文也仅限于这一领域。但这丝毫不影响de Boor作为一个数学家的重要性。 更多的,有的数学家对纯粹数学及其应用都有兴趣,如欧拉和高斯。而有的数学家则对应用数学深恶痛绝,如Hardy。Hardy对自己在应用方面的贡献评价为从实用的观点看,我的数学生涯的价值为零。 因此,似乎很难找到好数学家的统一工作范式。但是,他们都有一个共同点,就是对数学研究的热爱,和在数学工作中得到的满足和成就感,最为重要的是保持工作的热情与兴趣,这大概是所有好数学家所共同的。至于工作方式,正所谓以无法为有法,一切可全凭自己的喜好,选择自己认为最合适的就可以了。 1. 黎曼的论文可见 http://www.emis.de/classics/Riemann/ 2. http://baike.baidu.com/view/4645.htm 3. http://www-history.mcs.st-and.ac.uk/Biographies/Hardy.html 4. http://www-history.mcs.st-and.ac.uk/Biographies/Schoenberg.html
Niels Henrik Abel 阿贝尔(Niels Henrik Abel,公元1802年─公元1829年)是十九世纪挪威出现的最伟大数学家。他的父亲是挪威克里斯蒂安桑(Kristiansand)主教区芬杜(Findouml;)小村庄的牧师,全家生活在穷困之中。在1815年,当他进入了奥斯陆的一所天主教学校读书,他的数学才华便显露出来。经他的老师霍姆彪(Holmboeuml;)的引导下,他学习了不少当时的名数学家的著作,包括:牛顿(Newton)、欧拉(Euler)、拉格朗日(Lagrange)及高斯(Gauss)等。他不单了解他们的理论,而且可以找出他们一些微小的漏洞。 1820年,阿贝尔的父亲去世,照顾全家七口的重担突然交到他的肩上。虽然如此,1821年阿贝尔透过霍姆彪的补助,仍可进入奥斯陆的克里斯蒂安尼亚大学(University of Christinania),即现在的奥斯陆大学(Universitetet i Oslo)就读,於1822年获大学预颁学位,并由霍姆彪的资助下继续学业。 1823年当阿贝尔的第一篇论文发表後,他的朋友便力请挪威政府资助他到德国及法国进修。当等待政府回覆时,在1824年他发表了他的「一元五次方程没有代数一般解」的论文,可望为他带来肯定地位。他把论文寄了给当时有名的数学家高斯,可惜高斯错过了这篇论文,也不知道这个著名的代数难题已被解破。 1825-26年的冬季,他远赴柏林,并认识了克列尔(Crelle)。克列尔是个土木工程师,而且对数学很有热诚,他跟阿贝尔成为很要好的朋友。1826年,在阿贝尔的鼓励下,克列尔创立了一份纯数学和应用数学杂志(Journal fr die reine und angewandte Mathematik),该杂志的第一期便刊登了阿贝尔在五次方程的工作成果,另外还有方程理论、泛函方程及理论力学等的论文。在柏林,新的数学向导使他继续独立地进行研究工作,後来阿贝尔更到了欧洲不同的地方。 1826年夏天,他在巴黎造访了当时最顶尖的数学家,并且完成了一份有关超越函数的研究报告。这些工作展示出一个代数函数理论,现在称为阿贝尔定理,而这定理也是後期阿贝尔积分及阿贝尔函数的理论基础。他在巴黎被冷落对待,他曾经把他的研究报告寄去科学学院,望可得到好评,但他的努力也是徒然。他在离开巴黎前染顽疾,最初只以为只是感冒,後来才知道是肺结核病。 他辗转回到挪威,但欠下不少钱债。他只好靠教书及收取大学的微薄津贴为生。在1828年,他找到一份代课教师之职来维持生计。但他的穷困及病况并没有减低他对数学的热诚,他在这段期间写了大量的论文,主要是方程理论及椭圆函数,也就是有关阿贝尔方程和阿贝尔群的理论。他比雅可比(Jacobi)更快完全了椭圆函数的理论。此时,阿贝尔的名声经已响遍所有的数学中心,各方面的人也希望为他找到一个适当的教授席位,当中克列尔便希望为他在柏林找得一个教授席位。 在1828年冬天,阿贝尔的病逐渐严重起来。在他圣诞节去芬罗兰(Froland)探他的未婚妻克莱利.肯姆普(Crelly Kemp)期间,病情便更恶化。到1829年1月时,他已知自己寿命不长,出血的症状已无法否认。直至1829年4月6日凌晨,阿贝尔去世了,他的未婚妻坚持不要他人之助照顾阿贝尔,「单独占有这最後的时刻」。 在阿贝尔死後两天,克列尔写信说为阿贝尔成功争取於柏林大学(Freie Universitauml;t Berlin)当数学教授,可惜经已太迟,一代天才数学家经已在收到这消息前去世了。 读读数学家的故事,或许可以给研究许多启示。对于科学家而言,需要知道的历史就是科学史,还有科学家的经历,这些可以为研究提供许多的启迪,虽然目前的研究不同于古人,但是依然不能完全异于古人,否则,难于出现重大的成就,难于出现原创性的成就。而现代科学的发展时候的确让人感到缺乏重大的理论诞生,希望大家能够就此评论。 是我的错觉,还是现在创新的难度加大了,还是因为这个时代是知识爆炸的时代,还是社会太浮躁了?