伯努瓦·曼德布洛特 Benoit Mandelbrot

来自集智百科 - 复杂系统|人工智能|复杂科学|复杂网络|自组织
Jie（讨论 | 贡献）2020年12月27日 (日) 13:16的版本 （→‎Research career）
跳到导航 跳到搜索

此词条暂由彩云小译翻译，翻译字数共1902，未经人工整理和审校，带来阅读不便，请见谅。

See also: Mandelbrot set

伯努瓦 曼德布洛特
Born1924年11月20日
波兰华沙
Died2010年10月14日
美国马萨诸塞州剑桥
Nationality法国，美国，波兰
Alma mater巴黎综合理工学院，加州理工学院，巴黎大学
Known for曼德布罗集，混沌理论，分形，Zipf –曼德尔布洛特法则
Spouse(s)阿利耶特·卡甘（1955-2010）
Awards法国荣誉军团勋章（1990年侠士·2006年警官），2003日本国际奖，1993沃尔夫奖，1989年哈维奖，1986年富兰克林奖章，1985年巴纳德奖章
Scientific career
Fields数学，空气动力学
Institutions耶鲁大学，IBM公司，太平洋西北国家实验室
Doctoral advisor保罗·皮埃尔·莱维
Doctoral studentsL-E. 卡维，尤金·法玛，肯·穆斯格雷夫，穆拉德·塔克
Influences约翰内斯·开普勒，保罗·列维，索莱姆·曼德布罗特
Influenced纳西姆·尼古拉斯·塔勒布


Benoit B.[n 1] Mandelbrot[n 2] (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of physical phenomena and "the uncontrolled element in life".[3][4][5] He referred to himself as a "fractalist"[6] and is recognized for his contribution to the field of fractal geometry, which included coining the word "fractal", as well as developing a theory of "roughness and self-similarity" in nature.[7]

Benoit B.[n 1] Mandelbrot[n 2] (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of physical phenomena and "the uncontrolled element in life".[5][6][7] He referred to himself as a "fractalist"[8] and is recognized for his contribution to the field of fractal geometry, which included coining the word "fractal", as well as developing a theory of "roughness and self-similarity" in nature.

伯努瓦 曼德布洛特（1924年11月20日至2010年10月14日）是波兰裔法国裔美国数学家和博学家，对实用科学有着广泛的兴趣。他将其称为物理现象的“粗糙艺术”和“生活中不受控制的元素”。他称自己为“分形主义者”，并因其对分形几何学领域的贡献而受到认可，其中包括创造了“分形Fractal”一词，并发展了自然界中的“ 粗糙度Roughness和 自相似性Self-similarity ”理论。


In 1936, while he was a child, Mandelbrot's family emigrated to France from Warsaw, Poland. After World War II ended, Mandelbrot studied mathematics, graduating from universities in Paris and the United States and receiving a master's degree in aeronautics from the California Institute of Technology. He spent most of his career in both the United States and France, having dual French and American citizenship. In 1958, he began a 35-year career at IBM, where he became an IBM Fellow, and periodically took leaves of absence to teach at Harvard University. At Harvard, following the publication of his study of U.S. commodity markets in relation to cotton futures, he taught economics and applied sciences.

In 1936, while he was a child, Mandelbrot's family emigrated to France from Warsaw, Poland. After World War II ended, Mandelbrot studied mathematics, graduating from universities in Paris and the United States and receiving a master's degree in aeronautics from the California Institute of Technology. He spent most of his career in both the United States and France, having dual French and American citizenship. In 1958, he began a 35-year career at IBM, where he became an IBM Fellow, and periodically took leaves of absence to teach at Harvard University. At Harvard, following the publication of his study of U.S. commodity markets in relation to cotton futures, he taught economics and applied sciences.

1936年，当曼德布罗特还是个孩子时，一家人从波兰华沙移民到了法国。第二次世界大战结束后，曼德布洛特学习了数学，从巴黎和美国的大学毕业，并获得了加州理工学院的航空硕士学位。他的职业生涯大部分时间都是在美国和法国度过，拥有法国和美国双重国籍。1958年，他在IBM开始了35年的职业生涯，并在那里成为了IBM研究员，定期请假到哈佛大学任教。在哈佛大学发表关于棉花期货的美国商品市场研究之后，他开始教授经济学和应用科学。


Because of his access to IBM's computers, Mandelbrot was one of the first to use computer graphics to create and display fractal geometric images, leading to his discovery of the Mandelbrot set in 1980. He showed how visual complexity can be created from simple rules. He said that things typically considered to be "rough", a "mess", or "chaotic", such as clouds or shorelines, actually had a "degree of order".[8] His math and geometry-centered research career included contributions to such fields as statistical physics, meteorology, hydrology, geomorphology, anatomy, taxonomy, neurology, linguistics, information technology, computer graphics, economics, geology, medicine, physical cosmology, engineering, chaos theory, econophysics, metallurgy, and the social sciences.[9]

Because of his access to IBM's computers, Mandelbrot was one of the first to use computer graphics to create and display fractal geometric images, leading to his discovery of the Mandelbrot set in 1980. He showed how visual complexity can be created from simple rules. He said that things typically considered to be "rough", a "mess", or "chaotic", such as clouds or shorelines, actually had a "degree of order".[10] His math and geometry-centered research career included contributions to such fields as statistical physics, meteorology, hydrology, geomorphology, anatomy, taxonomy, neurology, linguistics, information technology, computer graphics, economics, geology, medicine, physical cosmology, engineering, chaos theory, econophysics, metallurgy, and the social sciences.

由于曼德布罗特可以使用IBM的计算机，因此他是最早使用计算机图形来创建和显示分形几何图像的人之一，因此他于1980年发现了曼德布洛特集合。他展示了如何从简单的规则图形创建出视觉复杂性。他认为那些通常被认为是“粗糙”，“杂乱”或“混乱”的事物，例如云层或海岸线，实际上都具有“ 有序度Degree of order ”。他以数学和几何学为中心的延申研究领域包括了统计物理学，气象学，水文学，地貌学，解剖学，分类学，神经学，语言学，信息技术，计算机图形学，经济学，地质学，医学，物理宇宙学，工程学，混沌理论等领域的贡献 ，经济物理学，冶金学和社会科学。


Toward the end of his career, he was Sterling Professor of Mathematical Sciences at Yale University, where he was the oldest professor in Yale's history to receive tenure.[10] Mandelbrot also held positions at the Pacific Northwest National Laboratory, Université Lille Nord de France, Institute for Advanced Study and Centre National de la Recherche Scientifique. During his career, he received over 15 honorary doctorates and served on many science journals, along with winning numerous awards. His autobiography, The Fractalist: Memoir of a Scientific Maverick, was published posthumously in 2012.

Toward the end of his career, he was Sterling Professor of Mathematical Sciences at Yale University, where he was the oldest professor in Yale's history to receive tenure.[12] Mandelbrot also held positions at the Pacific Northwest National Laboratory, Université Lille Nord de France, Institute for Advanced Study and Centre National de la Recherche Scientifique. During his career, he received over 15 honorary doctorates and served on many science journals, along with winning numerous awards. His autobiography, The Fractalist: Memoir of a Scientific Maverick, was published posthumously in 2012.

在他职业生涯后期，他是耶鲁大学数学科学系的斯特林教授，在那里他被授予了耶鲁历史上最年长的终身教授职位。曼德布罗特还在西北太平洋国家实验室，里尔-北法兰西院校联盟，普林斯顿高等研究院和法国国家科学研究中心担任过职务。他的自传《分形主义者:一个科学特立独行者的回忆录》于2012年死后出版。


Early years 早年生活

家庭背景和早期教育，（4:11）伯努瓦 曼德布洛特访谈，《Web of Stories》 144第1部分

Mandelbrot was born in a Lithuanian Jewish family, in Warsaw during the Second Polish Republic.[11] His father made his living trading clothing; his mother was a dental surgeon. During his first two school years, he was tutored privately by an uncle who despised rote learning: "Most of my time was spent playing chess, reading maps and learning how to open my eyes to everything around me."[12] In 1936, when he was 11, the family emigrated from Poland to France. The move, the war, and his acquaintance with his father's brother, the mathematician Szolem Mandelbrojt (who had moved to Paris around 1920), further prevented a standard education. "The fact that my parents, as economic and political refugees, joined Szolem in France saved our lives," he writes.[6]:17[13]

Mandelbrot was born in a Lithuanian Jewish family, in Warsaw during the Second Polish Republic.[14] His father made his living trading clothing; his mother was a dental surgeon. During his first two school years, he was tutored privately by an uncle who despised rote learning: "Most of my time was spent playing chess, reading maps and learning how to open my eyes to everything around me."[15] In 1936, when he was 11, the family emigrated from Poland to France. The move, the war, and his acquaintance with his father's brother, the mathematician Szolem Mandelbrojt (who had moved to Paris around 1920), further prevented a standard education. "The fact that my parents, as economic and political refugees, joined Szolem in France saved our lives," he writes.

曼德布洛特出生于波兰第二共和国时期，华沙的一个立陶宛犹太家庭。父亲以服装贸易为生，母亲是一名牙科医生。他入学后的前两年，一直由他叔叔私下辅导，这位叔叔尤其鄙视死记硬背的学习方法：“我的大部分时间都花在下棋，阅读地图和学习如何打开我的视角观察周围的一切。”后来1936年，他11岁时，一家人从波兰移民到了法国。这次移民，战争，和他父亲兄弟Szolem Mandelbrojt（数学家斯佐勒姆·曼德尔贝罗亚特，1920年左右移居巴黎）的接触，更进一步阻碍了他受到规范的教育。他曾写道：“我的父母作为经济和政治难民，在法国投靠了佐勒姆，因此而挽救了我们的生命。”


Mandelbrot attended the Lycée Rolin in Paris until the start of World War II, when his family moved to Tulle, France. He was helped by Rabbi David Feuerwerker, the Rabbi of Brive-la-Gaillarde, to continue his studies.[6]:62–63[14] Much of France was occupied by the Nazis at the time, and Mandelbrot recalls this period:

Mandelbrot attended the Lycée Rolin in Paris until the start of World War II, when his family moved to Tulle, France. He was helped by Rabbi David Feuerwerker, the Rabbi of Brive-la-Gaillarde, to continue his studies.[8]:62–63[17] Much of France was occupied by the Nazis at the time, and Mandelbrot recalls this period:

曼德布罗特在巴黎罗兰公立中学学习，一直到第二次世界大战开始，后来他的家人搬到了法国的蒂勒。后来犹太教教士大卫·菲尔韦克Rabbi David Feuerwerker（布里夫拉盖勒德犹太教教士the Rabbi of Brive-la-Gaillarde）帮助他继续学业。当时法国大部分地区都被纳粹占领，曼德布罗特回忆起这段时期：


Our constant fear was that a sufficiently determined foe might report us to an authority and we would be sent to our deaths. This happened to a close friend from Paris, Zina Morhange, a physician in a nearby county seat. Simply to eliminate the competition, another physician denounced her ... We escaped this fate. Who knows why?

Our constant fear was that a sufficiently determined foe might report us to an authority and we would be sent to our deaths. This happened to a close friend from Paris, Zina Morhange, a physician in a nearby county seat. Simply to eliminate the competition, another physician denounced her ... We escaped this fate. Who knows why?

我们一直担心的是，一旦敌人下定决心将我们报告给当局，等待我们的就是死刑。类似的事情就发生在巴黎的一位密友Zina Morhange深上，她曾是附近某个县城的医生。当时只是为了解决竞争对手，另一位医生就举报了她...而我们逃过了一劫。谁知道是为什么？


In 1944, Mandelbrot returned to Paris, studied at the Lycée du Parc in Lyon, and in 1945 to 1947 attended the École Polytechnique, where he studied under Gaston Julia and Paul Lévy. From 1947 to 1949 he studied at California Institute of Technology, where he earned a master's degree in aeronautics.[15] Returning to France, he obtained his PhD degree in Mathematical Sciences at the University of Paris in 1952.[12]

In 1944, Mandelbrot returned to Paris, studied at the Lycée du Parc in Lyon, and in 1945 to 1947 attended the École Polytechnique, where he studied under Gaston Julia and Paul Lévy. From 1947 to 1949 he studied at California Institute of Technology, where he earned a master's degree in aeronautics.[2] Returning to France, he obtained his PhD degree in Mathematical Sciences at the University of Paris in 1952.

1944年，曼德布洛特回到巴黎，在里昂的帕克中学学习，并于1945年至1947年考上了巴黎综合理工学院，在加斯顿·朱莉亚Gaston Julia和保罗·列维Paul Lévy的指导下学习。之后的1947年到1949年，他就读于加利福尼亚理工学院，在那里获得了航空硕士学位。返回法国后，他于1952年在巴黎大学获得数学科学博士学位。

Research career 科研生涯

From 1949 to 1958, Mandelbrot was a staff member at the Centre National de la Recherche Scientifique. During this time he spent a year at the Institute for Advanced Study in Princeton, New Jersey, where he was sponsored by John von Neumann. In 1955 he married Aliette Kagan and moved to Geneva, Switzerland (to collaborate with Jean Piaget at the International Centre for Genetic Epistemology) and later to the Université Lille Nord de France.[16] In 1958 the couple moved to the United States where Mandelbrot joined the research staff at the IBM Thomas J. Watson Research Center in Yorktown Heights, New York.[16] He remained at IBM for 35 years, becoming an IBM Fellow, and later Fellow Emeritus.[12]

From 1949 to 1958, Mandelbrot was a staff member at the Centre National de la Recherche Scientifique. During this time he spent a year at the Institute for Advanced Study in Princeton, New Jersey, where he was sponsored by John von Neumann. In 1955 he married Aliette Kagan and moved to Geneva, Switzerland (to collaborate with Jean Piaget at the International Centre for Genetic Epistemology) and later to the Université Lille Nord de France.[18] In 1958 the couple moved to the United States where Mandelbrot joined the research staff at the IBM Thomas J. Watson Research Center in Yorktown Heights, New York.[18] He remained at IBM for 35 years, becoming an IBM Fellow, and later Fellow Emeritus.

自1949年到1958年，曼德布洛特任职于法国国家科学研究中心。在此期间，他得到了约翰·冯·诺伊曼John von Neumann的赞助，在新泽西州普林斯顿的高级研究学院度过了一年。1955年，他与阿利耶特·卡甘结婚，并搬到瑞士日内瓦（与国际遗传认识论中心的让·皮亚杰Jean Piaget合作），后来又迁往法国里尔大学。到了1958年，这对夫妇搬到了美国，在那里曼德布洛特加入了位于纽约约克敦高地的IBM托马斯·沃森研究中心。他在IBM待了35年，成为IBM院士，后来成为了荣誉退休院士。


From 1951 onward, Mandelbrot worked on problems and published papers not only in mathematics but in applied fields such as information theory, economics, and fluid dynamics.

From 1951 onward, Mandelbrot worked on problems and published papers not only in mathematics but in applied fields such as information theory, economics, and fluid dynamics.

从1951年起，曼德布洛特致力于研究相关问题并发表论文，不仅在数学领域，而且在诸如信息论，经济学和流体动力学等应用领域中也发表了论文。


Randomness in financial markets 金融市场的随机性

Mandelbrot saw financial markets as an example of "wild randomness", characterized by concentration and long range dependence. He developed several original approaches for modelling financial fluctuations.[17] In his early work, he found that the price changes in financial markets did not follow a Gaussian distribution, but rather Lévy stable distributions having infinite variance. He found, for example, that cotton prices followed a Lévy stable distribution with parameter α equal to 1.7 rather than 2 as in a Gaussian distribution. "Stable" distributions have the property that the sum of many instances of a random variable follows the same distribution but with a larger scale parameter.[18]

Mandelbrot saw financial markets as an example of "wild randomness", characterized by concentration and long range dependence. He developed several original approaches for modelling financial fluctuations.[19] In his early work, he found that the price changes in financial markets did not follow a Gaussian distribution, but rather Lévy stable distributions having infinite variance. He found, for example, that cotton prices followed a Lévy stable distribution with parameter α equal to 1.7 rather than 2 as in a Gaussian distribution. "Stable" distributions have the property that the sum of many instances of a random variable follows the same distribution but with a larger scale parameter.

曼德布洛特将金融市场视为“野生随机性”的一个例子，其特征是集中性和长相关性。为此他开发了几种可以模拟财务波动的创新方法。在他的早期工作中，他发现金融市场的价格变化并非遵循高斯分布，而是遵循具有无限方差的 列维稳定分布Lévy stable distributions 。他发现，例如棉花价格遵循列维稳定分布，其参数α等于1.7，而不是高斯分布中的2。该“稳定”分布具有以下性质：随机变量的许多实例总和遵循相同的分布，只是比例参数较大。


Developing "fractal geometry" and the Mandelbrot set “分形几何”和曼德布洛特集合的发展

As a visiting professor at Harvard University, Mandelbrot began to study fractals called Julia sets that were invariant under certain transformations of the complex plane. Building on previous work by Gaston Julia and Pierre Fatou, Mandelbrot used a computer to plot images of the Julia sets. While investigating the topology of these Julia sets, he studied the Mandelbrot set which was introduced by him in 1979. In 1982, Mandelbrot expanded and updated his ideas in The Fractal Geometry of Nature.[19] This influential work brought fractals into the mainstream of professional and popular mathematics, as well as silencing critics, who had dismissed fractals as "program artifacts".

As a visiting professor at Harvard University, Mandelbrot began to study fractals called Julia sets that were invariant under certain transformations of the complex plane. Building on previous work by Gaston Julia and Pierre Fatou, Mandelbrot used a computer to plot images of the Julia sets. While investigating the topology of these Julia sets, he studied the Mandelbrot set which was introduced by him in 1979. In 1982, Mandelbrot expanded and updated his ideas in The Fractal Geometry of Nature.[21] This influential work brought fractals into the mainstream of professional and popular mathematics, as well as silencing critics, who had dismissed fractals as "program artifacts".

作为哈佛大学的客座教授，曼德布洛特开始研究名为 朱莉娅集合Julia sets的分形，这些分形在复杂平面的变换下依旧保持不变。在加斯顿·朱莉娅Gaston Julia和皮埃尔·法图Pierre Fatou先前工作的基础上，曼德尔布洛特使用计算机绘制出了朱莉娅集合的图像。在他研究这些朱莉娅集的拓扑时，他研究了他于1979年提出的曼德布洛特集。1982年，他在《大自然的分形几何学》一书中扩展并更新了他的思想。这项颇具影响力的著作将分形技术引入到专业数学和大众数学中，同时也进入到了那些将分形技术仅视为“程序工件”的批评者眼中。

Awards and honors

Mandelbrot's awards include the Wolf Prize for Physics in 1993, the Lewis Fry Richardson Prize of the European Geophysical Society in 2000, the Japan Prize in 2003,[20] and the Einstein Lectureship of the American Mathematical Society in 2006.


The small asteroid 27500 Mandelbrot was named in his honor. In November 1990, he was made a Chevalier in France's Legion of Honour. In December 2005, Mandelbrot was appointed to the position of Battelle Fellow at the Pacific Northwest National Laboratory.[21] Mandelbrot was promoted to an Officer of the Legion of Honour in January 2006.[22] An honorary degree from Johns Hopkins University was bestowed on Mandelbrot in the May 2010 commencement exercises.[23]


A partial list of awards received by Mandelbrot:[24]

“ padding-right: 2em”

  • 2004 Best Business Book of the Year Award
  • AMS Einstein Lectureship
  • Barnard Medal
  • Caltech Service
  • Fellow, American Geophysical Union
  • Honda Prize
  • Humboldt Preis
  • IBM Fellowship
  • Japan Prize (2003)
  • Lewis Fry Richardson Medal
  • Medaglia della Presidenza della Repubblica Italiana
  • Médaille de Vermeil de la Ville de Paris
  • Nevada Prize
  • Science for Art
  • Sven Berggren-Priset
  • Władysław Orlicz Prize


Legacy

Mandelbrot died from pancreatic cancer at the age of 85 in a hospice in Cambridge, Massachusetts on 14 October 2010.[28][29] Reacting to news of his death, mathematician Heinz-Otto Peitgen said: "[I]f we talk about impact inside mathematics, and applications in the sciences, he is one of the most important figures of the last fifty years."[28]


Chris Anderson, TED conference curator, described Mandelbrot as "an icon who changed how we see the world".[30] Nicolas Sarkozy, President of France at the time of Mandelbrot's death, said Mandelbrot had "a powerful, original mind that never shied away from innovating and shattering preconceived notions [... h]is work, developed entirely outside mainstream research, led to modern information theory."[31] Mandelbrot's obituary in The Economist points out his fame as "celebrity beyond the academy" and lauds him as the "father of fractal geometry".[32]


Best-selling essayist-author Nassim Nicholas Taleb has remarked that Mandelbrot's book The (Mis)Behavior of Markets is in his opinion "The deepest and most realistic finance book ever published".[7]


Bibliography

in English

  • Fractals: Form, Chance and Dimension, 1977, 2020
  • Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. Selecta Volume E, 1997 by Benoit B. Mandelbrot and R.E. Gomory
  • Fractales, hasard et finance, 1959-1997, 1 November 1998
  • Multifractals and 1/ƒ Noise: Wild Self-Affinity in Physics (1963–1976) (Selecta; V.N) 18 January 1999 by J.M. Berger and Benoit B. Mandelbrot
  • Gaussian Self-Affinity and Fractals: Globality, The Earth, 1/f Noise, and R/S (Selected Works of Benoit B. Mandelbrot) 14 December 2001 by Benoit Mandelbrot and F.J. Damerau
  • Fractals and Chaos: The Mandelbrot Set and Beyond, 9 January 2004
  • The Misbehavior of Markets: A Fractal View of Financial Turbulence, 2006 by Benoit Mandelbrot and Richard L. Hudson
  • The Fractalist: Memoir of a Scientific Maverick, 2014


In French

  • La forme d'une vie. Mémoires (1924-2010) by Benoît Mandelbrot (Author), Johan-Frédérik Hel Guedj (Translator)


References in popular culture

  • In 2004, the American singer-songwriter Jonathan Coulton wrote "Mandelbrot Set". Formerly, it contained the lines "Mandelbrot's in heaven / at least he will be when he's dead / right now he's still alive and teaching math at Yale". Live performances after Mandelbrot's passing in 2010 feature only the first line and a brief rock instrumental.
  • In 2007, the author Laura Ruby published "The Chaos King," which includes a character named Mandelbrot and discussion of chaos theory.
  • In 2017, Liz Ziemska published a novella, Mandelbrot The Magnificent, a fictional account of how Mandelbrot saved his family during WWII


See also

模板:External media

Category:1924 births

类别: 1924出生

Category:2010 deaths

分类: 2010年死亡人数

Category:20th-century American mathematicians

范畴: 20世纪美国数学家

Category:21st-century French mathematicians

范畴: 21世纪法国数学家

Category:20th-century American economists

类别: 20世纪美国经济学家

Category:21st-century American economists

类别: 21世纪美国经济学家

Category:Alexander von Humboldt Fellows

类别: 亚历山大·冯·洪堡研究员

Category:California Institute of Technology alumni

类别: 加州理工学院校友

Category:Chaos theorists

范畴: 混沌理论家

Category:Deaths from cancer in Massachusetts

分类: 马萨诸塞州癌症死亡人数

Category:Deaths from pancreatic cancer

分类: 死于胰腺癌

Category:École Polytechnique alumni

类别: 巴黎综合理工学院校友

Category:Fellows of the American Geophysical Union

类别: 美国地球物理联盟研究员


Category:Fellows of the American Statistical Association

类别: 美国统计协会研究员

Notes

Category:Fellows of the Econometric Society

类别: 经济计量学会研究员

  1. ↑ 引用错误：无效<ref>标签；未给name属性为Mandelbrot's_name的引用提供文字
  2. ↑ Pronounced 模板:IPAc-en 模板:Respell in English.[1] When speaking in French, Mandelbrot pronounced his name 模板:IPA-fr.[2]

Category:Fellows of the American Physical Society

类别: 美国物理学会会员


Category:French emigrants to the United States

类别: 移居美国的法国移民

References

Category:French scientists

分类: 法国科学家

  1. ↑ 模板:OED
  2. ↑ Recording of the ceremony on 11 September 2006 at which Mandelbrot received the insignia for an Officer of the Légion d'honneur.
  3. ↑ "Archived copy". Archived from the original on 8 January 2018. Retrieved 8 January 2018.{{cite web}}: CS1 maint: archived copy as title (link)
  4. ↑ Benoit Mandelbrot: Fractals and the art of roughness -{zh-cn:互联网档案馆; zh-tw:網際網路檔案館; zh-hk:互聯網檔案館;}-的存檔，存档日期14 April 2016.. ted.com (February 2010)
  5. ↑ Hudson & Mandelbrot, Prelude, page xviii
  6. ↑ 6.0 6.1 6.2 引用错误：无效<ref>标签；未给name属性为Mandelbrot的引用提供文字
  7. ↑ 7.0 7.1 Gomory, R. (2010). "Benoît Mandelbrot (1924–2010)". Nature. 468 (7322): 378. Bibcode:2010Natur.468..378G. doi:10.1038/468378a. PMID 21085164. S2CID 4393964.
  8. ↑ 引用错误：无效<ref>标签；未给name属性为Wolfram的引用提供文字
  9. ↑ list includes specific sciences mentioned in Hudson & Mandelbrot, the Prelude, p. xvi, and p. 26
  10. ↑ Steve Olson (November–December 2004). "The Genius of the Unpredictable". Yale Alumni Magazine. Archived from the original on 22 October 2014. Retrieved 22 July 2014.
  11. ↑ Hoffman, Jascha (16 October 2010). "Benoît Mandelbrot, Novel Mathematician, Dies at 85 (Published 2010)". The New York Times (in English). ISSN 0362-4331. Archived from the original on 21 January 2017. Retrieved 20 November 2020.
  12. ↑ 12.0 12.1 12.2 Mandelbrot, Benoît (2002). "The Wolf Prizes for Physics, A Maverick's Apprenticeship" (PDF). Imperial College Press. Archived (PDF) from the original on 3 December 2013. Retrieved 23 April 2012.
  13. ↑ "BBC News – 'Fractal' mathematician Benoît Mandelbrot dies aged 85". BBC Online. 17 October 2010. Archived from the original on 18 October 2010. Retrieved 17 October 2010.
  14. ↑ Hemenway P. (2005) Divine proportion: Phi in art, nature and science. Psychology Press.
  15. ↑ 引用错误：无效<ref>标签；未给name属性为guardian_obit的引用提供文字
  16. ↑ 16.0 16.1 Barcellos, Anthony (1984). "Mathematical People, Interview of B. B. Mandelbrot" (PDF). Birkhaüser. Archived (PDF) from the original on 27 April 2015. Retrieved 25 June 2013.
  17. ↑ Rama Cont (19 April 2010). "Mandelbrot, Benoit". Wiley. doi:10.1002/9780470061602.eqf01006. ISBN 9780470057568. 
  18. ↑ "New Scientist, 19 April 1997". Newscientist.com. 19 April 1997. Archived from the original on 21 April 2010. Retrieved 17 October 2010.
  19. ↑ The Fractal Geometry of Nature -{zh-cn:互联网档案馆; zh-tw:網際網路檔案館; zh-hk:互聯網檔案館;}-的存檔，存档日期30 November 2015., by Benoît Mandelbrot; W H Freeman & Co, 1982;
  20. ↑ Laureates of the Japan Prize. japanprize.jp
  21. ↑ "PNNL press release: Mandelbrot joins Pacific Northwest National Laboratory". Pnl.gov. 16 February 2006. Retrieved 17 October 2010.
  22. ↑ "Légion d'honneur announcement of promotion of Mandelbrot to officier" (in français). Legifrance.gouv.fr. Retrieved 17 October 2010.
  23. ↑ "Six granted honorary degrees, Society of Scholars inductees recognized". Gazette.jhu.edu. 7 June 2010. Retrieved 17 October 2010.
  24. ↑ Mandelbrot, Benoit B. (2 February 2006). "Vita and Awards (Word document)". Retrieved 6 January 2007. Retrieved from Internet Archive 15 December 2013.
  25. ↑ View/Search Fellows of the ASA, accessed 20 August 2016.
  26. ↑ "APS Fellow Archive". APS. Retrieved 24 September 2020.
  27. ↑ "Gruppe 1: Matematiske fag" (in Norwegian). Norwegian Academy of Science and Letters. Retrieved 7 October 2010.{{cite web}}: CS1 maint: unrecognized language (link)
  28. ↑ 28.0 28.1 引用错误：无效<ref>标签；未给name属性为nyt_obit的引用提供文字
  29. ↑ "Benoît Mandelbrot, fractals pioneer, dies". United Press International. 16 October 2010. Retrieved 17 October 2010.
  30. ↑ "Mandelbrot, father of fractal geometry, dies". The Gazette. Archived from the original on 19 October 2010. Retrieved 16 October 2010.
  31. ↑ "Sarkozy rend hommage à Mandelbrot" [Sarkozy pays homage to Mandelbrot]. Le Figaro (in French). Retrieved 17 October 2010.{{cite news}}: CS1 maint: unrecognized language (link)
  32. ↑ Benoît Mandelbrot's obituary. The Economist (21 October 2010)
  33. ↑ "Saturday Morning Breakfast Cereal – Mandelbrot".

Category:Harvard University people

类别: 哈佛大学的人


Category:IBM employees

类别: IBM 员工

Bibliography

Category:IBM Fellows

类别: IBM Fellows

Category:IBM Research computer scientists

类别: IBM 研究计算机科学家


Category:Institute for Advanced Study visiting scholars

类别: 高级研究所访问学者

Further reading

Category:Jewish French scientists

类别: 犹太法国科学家

取自“https://wiki.swarma.org/index.php?title=伯努瓦·曼德布洛特_Benoit_Mandelbrot&oldid=20494”