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第104行： − + − + + + − According to Clarke, "the Mandelbrot set is indeed one of the most astonishing discoveries in the entire history of mathematics. Who could have dreamed that such an incredibly simple equation could have generated images of literally infinite complexity?" Clarke also notes an "odd coincidence<blockquote>the name Mandelbrot, and the word "mandala"—for a religious symbol—which I'm sure is a pure coincidence, but indeed the Mandelbrot set does seem to contain an enormous number of mandalas. He joined the Department of Mathematics at Yale, and obtained his first tenured post in 1999, at the age of 75. At the time of his retirement in 2005, he was Sterling Professor of Mathematical Sciences. − 根据克拉克的说法，“曼德尔布罗特集合确实是整个数学史上最惊人的发现之一。谁能想到这么简单的一个方程竟然能产生无限复杂的图像? ”克拉克还注意到了一个“奇怪的巧合”——曼德尔布洛特的名字，以及“曼荼罗”——意为宗教象征——我确信这纯粹是巧合，但事实上曼德尔布洛特集似乎包含了大量的曼荼罗。他加入了耶鲁大学数学系，并在1999年获得了他的第一个终身职位，那时他75岁。在他2005年退休的时候，他是数学科学的斯特林教席。 − + − + − Mandelbrot created the first-ever "theory of roughness", and he saw "roughness" in the shapes of mountains, coastlines and river basins; the structures of plants, blood vessels and lungs; the clustering of galaxies. His personal quest was to create some mathematical formula to measure the overall "roughness" of such objects in nature.}} − − 曼德布洛特创立了有史以来第一个“粗糙理论” ，他看到了山脉、海岸线和河流盆地形状的“粗糙” ，看到了植物、血管和肺的结构，看到了星系的聚集。他个人的追求是创造一些数学公式来衡量这些物体在自然界中的整体“粗糙度” − − ===Randomness in financial markets=== − − − − In his paper titled How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension published in Science in 1967 Mandelbrot discusses self-similar curves that have Hausdorff dimension that are examples of fractals, although Mandelbrot does not use this term in the paper, as he did not coin it until 1975. The paper is one of Mandelbrot's first publications on the topic of fractals. − − 在他题为《英国海岸有多长？统计自相似性和分维数1967年发表在《科学》杂志上，曼德尔布洛特讨论了自相似曲线，这些曲线都有豪斯多夫维数，是分形的例子，尽管曼德尔布洛特在论文中没有使用这个术语，因为他直到1975年才造出这个术语。这篇论文是曼德布洛特关于分形主题的首批出版物之一。 − − Mandelbrot saw [[financial market]]s as an example of "wild randomness", characterized by concentration and long range dependence. He developed several original approaches for modelling financial fluctuations.<ref>{{cite book |author= Rama Cont |chapter= Mandelbrot, Benoit |journal= Encyclopedia of Quantitative Finance |publisher= Wiley |date= 19 April 2010 |doi= 10.1002/9780470061602.eqf01006 |isbn = 9780470057568|author-link= Rama Cont }}</ref> − − In his early work, he found that the price changes in [[financial market]]s did not follow a [[Gaussian distribution]], but rather [[Paul Lévy (mathematician)|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]].<ref>{{cite web |url=https://www.newscientist.com/article/mg15420784.700-flight-over-wall-st.html |title=''New Scientist'', 19 April 1997 |publisher=Newscientist.com |date=19 April 1997 |accessdate=17 October 2010}}</ref> − − Mandelbrot emphasized the use of fractals as realistic and useful models for describing many "rough" phenomena in the real world. He concluded that "real roughness is often fractal and can be measured." and a maverick. His informal and passionate style of writing and his emphasis on visual and geometric intuition (supported by the inclusion of numerous illustrations) made The Fractal Geometry of Nature accessible to non-specialists. The book sparked widespread popular interest in fractals and contributed to chaos theory and other fields of science and mathematics. − − 曼德布洛特强调用分形作为现实和有用的模型来描述现实世界中的许多“粗糙”现象。他的结论是: “真实的粗糙度通常是分形的，可以测量。”还是个特立独行的人。他非正式和充满激情的写作风格和他对视觉和几何直觉的强调(通过大量插图的支持)使得《自然的分形几何学》对非专业人士来说易于理解。这本书激发了人们对分形的广泛兴趣，并促成了混沌理论和其他领域的科学和数学。 − − − − ===Developing "fractal geometry" and the Mandelbrot set=== − − Mandelbrot also put his ideas to work in cosmology. He offered in 1974 a new explanation of Olbers' paradox (the "dark night sky" riddle), demonstrating the consequences of fractal theory as a sufficient, but not necessary, resolution of the paradox. He postulated that if the stars in the universe were fractally distributed (for example, like Cantor dust), it would not be necessary to rely on the Big Bang theory to explain the paradox. His model would not rule out a Big Bang, but would allow for a dark sky even if the Big Bang had not occurred. − − 曼德布洛特也将他的想法应用于宇宙学。1974年，他对奥尔伯斯悖论(“黑暗的夜空”之谜)提出了新的解释，证明了分形理论的结果是解决这一悖论的充分而非必要的方法。他假设，如果宇宙中的恒星是分散分布的(例如，康托尔尘埃) ，就没有必要依靠大爆炸理论来解释这一悖论。他的模型不能排除宇宙大爆炸的可能性，但是即使宇宙大爆炸没有发生，也可以考虑到黑暗的天空。 − − − − As a visiting professor at [[Harvard University]], Mandelbrot began to study fractals called [[Julia set]]s 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]]''.<ref>[https://books.google.com/books?id=xJ4qiEBNP4gC&printsec=frontcover ''The Fractal Geometry of Nature''], by Benoît Mandelbrot; W H Freeman & Co, 1982; {{isbn|0-7167-1186-9}}</ref> This influential work brought fractals into the mainstream of professional and popular mathematics, as well as silencing critics, who had dismissed fractals as "[[Artifact (observational)|program artifacts]]". − − − − 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, and the Einstein Lectureship of the American Mathematical Society in 2006. − − 曼德布洛特获得的奖项包括1993年的沃尔夫物理奖、2000年欧洲地球物理学会的刘易斯 · 弗莱 · 理查森奖、2003年的日本奖以及2006年的美国数学学会爱因斯坦讲师奖。 − − [[File:Mandelbrot p1130876.jpg|thumb|right|Mandelbrot speaking about the [[Mandelbrot set]], during his acceptance speech for the [[Légion d'honneur]] 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. Mandelbrot was promoted to an Officer of the Legion of Honour in January 2006. An honorary degree from Johns Hopkins University was bestowed on Mandelbrot in the May 2010 commencement exercises. − − 小行星27500 Mandelbrot 以他的名字命名。1990年11月，他被授予法国法国荣誉军团勋章骑士称号。2005年12月，曼德尔布洛特被任命为太平洋西北国家实验室的巴特尔研究员。2006年1月，曼德尔布洛特被提升为法国荣誉军团勋章安全委员会官员。在2010年5月的毕业典礼上，曼德尔布洛特获得了约翰·霍普金斯大学的荣誉学位。 − − In 1975, Mandelbrot coined the term ''[[fractal]]'' to describe these structures and first published his ideas, and later translated, ''Fractals: Form, Chance and Dimension''.<ref>''Fractals: Form, Chance and Dimension'', by Benoît Mandelbrot; W H Freeman and Co, 1977; {{isbn|0-7167-0473-0}}</ref> According to computer scientist and physicist [[Stephen Wolfram]], the book was a "breakthrough" for Mandelbrot, who until then would typically "apply fairly straightforward mathematics ... to areas that had barely seen the light of serious mathematics before".<ref name=Wolfram>Wolfram, Stephen. [https://www.wsj.com/articles/SB10001424127887324439804578107271772910506 "The Father of Fractals"], ''Wall Street Journal'', 22 November 2012</ref> Wolfram adds that as a result of this new research, he was no longer a "wandering scientist", and later called him "the father of fractals": − − − − A partial list of awards received by Mandelbrot: − − 曼德布洛特获得的部分奖项: − − {{quote|Mandelbrot ended up doing a great piece of science and identifying a much stronger and more fundamental idea—put simply, that there are some geometric shapes, which he called "fractals", that are equally "rough" at all scales. No matter how close you look, they never get simpler, much as the section of a rocky coastline you can see at your feet looks just as jagged as the stretch you can see from space.<ref name=Wolfram />}} − − − − Wolfram briefly describes fractals as a form of geometric repetition, "in which smaller and smaller copies of a pattern are successively nested inside each other, so that the same intricate shapes appear no matter how much you zoom in to the whole. Fern leaves and Romanesco broccoli are two examples from nature."<ref name=Wolfram /> He points out an unexpected conclusion: − − − − {{quote|One might have thought that such a simple and fundamental form of regularity would have been studied for hundreds, if not thousands, of years. But it was not. In fact, it rose to prominence only over the past 30 or so years—almost entirely through the efforts of one man, the mathematician Benoit Mandelbrot.<ref name=Wolfram />}} − − − − Mandelbrot used the term "fractal" as it derived from the Latin word "fractus", defined as broken or shattered glass. Using the newly developed IBM computers at his disposal, Mandelbrot was able to create fractal images using graphic computer code, images that an interviewer described as looking like "the delirious exuberance of the 1960s [[psychedelic art]] with forms hauntingly reminiscent of nature and the human body". He also saw himself as a "would-be Kepler", after the 17th-century scientist [[Johannes Kepler]], who calculated and described the orbits of the planets.<ref>Ivry, Benjamin. [http://forward.com/articles/166094/benoit-mandelbrot-influenced-art-and-mathematics/?p=all "Benoit Mandelbrot Influenced Art and Mathematics"], ''Forward'', 17 November 2012</ref> − − − − [[File:Newton-lplane-Mandelbrot.jpg|thumb|A Mandelbrot set]] − − Mandelbrot, however, never felt he was inventing a new idea. He describes his feelings in a documentary with science writer Arthur C. Clarke: − − − − {{quote|Exploring this set I certainly never had the feeling of invention. I never had the feeling that my imagination was rich enough to invent all those extraordinary things on discovering them. They were there, even though nobody had seen them before. It's marvelous, a very simple formula explains all these very complicated things. So the goal of science is starting with a mess, and explaining it with a simple formula, a kind of dream of science.<ref name=Clarke>[https://www.youtube.com/watch?v=Lk6QU94xAb8 "Arthur C Clarke – Fractals – The Colors Of Infinity"], video interviews, 54 min.</ref>}} − − − − According to Clarke, "the [[Mandelbrot set]] is indeed one of the most astonishing discoveries in the entire history of mathematics. Who could have dreamed that such an incredibly simple equation could have generated images of literally ''infinite'' complexity?" Clarke also notes an "odd coincidence<blockquote>the name Mandelbrot, and the word "[[mandala]]"—for a religious symbol—which I'm sure is a pure coincidence, but indeed the Mandelbrot set does seem to contain an enormous number of mandalas.<ref name=Clarke /></blockquote> − − − − Mandelbrot left IBM in 1987, after 35 years and 12 days, when IBM decided to end pure research in his division.<ref name="wos44">{{cite web|url=http://www.webofstories.com/play/10483|title=Web of Stories • Benoît Mandelbrot • IBM: background and policies|last=Mandelbrot|first=Benoît|author2=Bernard Sapoval|author3=Daniel Zajdenweber|date=May 1998|publisher=[[Web of Stories]] | accessdate=17 October 2010}}</ref> He joined the Department of Mathematics at [[Yale]], and obtained his first [[tenure]]d post in 1999, at the age of 75.<ref name="Tenner">{{cite news|url=https://www.theatlantic.com/technology/archive/2010/10/benoit-mandelbrot-the-maverick-1924-2010/64684/|title=Benoît Mandelbrot the Maverick, 1924–2010|last=Tenner|first=Edward|date=16 October 2010|work=[[The Atlantic]] | accessdate=16 October 2010}}</ref> At the time of his retirement in 2005, he was Sterling Professor of Mathematical Sciences. − − − − ===Fractals and the "theory of roughness"=== − − Mandelbrot created the first-ever "theory of roughness", and he saw "roughness" in the shapes of mountains, [[coastline]]s and [[river basin]]s; the structures of plants, [[blood vessel]]s and [[lung]]s; the clustering of [[galaxy|galaxies]]. His personal quest was to create some mathematical formula to measure the overall "roughness" of such objects in nature.<ref name=Mandelbrot />{{rp|xi}} He began by asking himself various kinds of questions related to nature: − − {{quote|Can [[geometry]] deliver what the Greek root of its name [geo-] seemed to promise—truthful measurement, not only of cultivated fields along the Nile River but also of untamed Earth?<ref name=Mandelbrot>Mandelbrot, Benoit (2012). ''The Fractalist: Memoir of a Scientific Maverick'', Pantheon Books. {{isbn|978-0-307-38991-6}}.</ref>{{rp|xii}}}} − − − − In his paper titled [[How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension]] published in [[Science (journal)|''Science'']] in 1967 Mandelbrot discusses [[self-similarity|self-similar]] curves that have [[Hausdorff dimension]] that are examples of ''fractals'', although Mandelbrot does not use this term in the paper, as he did not coin it until 1975. The paper is one of Mandelbrot's first publications on the topic of fractals.<ref>"Dr. Mandelbrot traced his work on fractals to a question he first encountered as a young researcher: how long is the coast of Britain?": Benoit Mandelbrot (1967). "[https://www.nytimes.com/2010/10/17/us/17mandelbrot.html?adxnnl=1&adxnnlx=1332064840-/vD0Sjafcl9t9BNghRf8Qw Benoît Mandelbrot, Novel Mathematician, Dies at 85]", ''The New York Times''.</ref><ref name="Mandelbrot_Science_1967">{{cite journal| title=How long is the coast of Britain? Statistical self-similarity and fractional dimension | journal=Science | date=5 May 1967 | author=Mandelbrot, Benoit B. | pages=636–638 | volume=156 | issue=3775 |doi=10.1126/science.156.3775.636 | pmid=17837158|url=http://users.math.yale.edu/~bbm3/web_pdfs/howLongIsTheCoastOfBritain.pdf | bibcode=1967Sci...156..636M | s2cid=15662830 }}</ref> − − − − Mandelbrot emphasized the use of fractals as realistic and useful models for describing many "rough" phenomena in the real world. He concluded that "real roughness is often fractal and can be measured."<ref name=Mandelbrot />{{rp|296}} Although Mandelbrot coined the term "fractal", some of the mathematical objects he presented in ''[[The Fractal Geometry of Nature]]'' had been previously described by other mathematicians. Before Mandelbrot, however, they were regarded as isolated curiosities with unnatural and non-intuitive properties. Mandelbrot brought these objects together for the first time and turned them into essential tools for the long-stalled effort to extend the scope of science to explaining non-smooth, "rough" objects in the real world. His methods of research were both old and new: − − {{quote|The form of geometry I increasingly favored is the oldest, most concrete, and most inclusive, specifically empowered by the eye and helped by the hand and, today, also by the computer ... bringing an element of unity to the worlds of knowing and feeling ... and, unwittingly, as a bonus, for the purpose of creating beauty.<ref name=Mandelbrot />{{rp|292}}}} − − − − Fractals are also found in human pursuits, such as music, painting, architecture, and [[stock market]] prices. Mandelbrot believed that fractals, far from being unnatural, were in many ways more intuitive and natural than the artificially smooth objects of traditional [[Euclidean geometry]]: <blockquote>Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.<br /> —Mandelbrot, in his introduction to ''[[The Fractal Geometry of Nature]]''</blockquote> − − [[File:Mandel zoom 08 satellite antenna.jpg|thumb|right|Section of a Mandelbrot set]] − Mandelbrot has been called an artist, and a visionary<ref name="RLD">{{cite web|author=Devaney, Robert L.|authorlink= Robert L. Devaney |title="Mandelbrot's Vision for Mathematics" in ''Proceedings of Symposia in Pure Mathematics''. Volume 72.1 |publisher=American Mathematical Society |year=2004 |url=http://www.math.yale.edu/mandelbrot/web_pdfs/jubileeletters.pdf |access-date=5 January 2007 |url-status=dead |archive-url=https://web.archive.org/web/20061209093734/http://www.math.yale.edu/mandelbrot/web_pdfs/jubileeletters.pdf |archive-date=9 December 2006 }}</ref> and a maverick.<ref>{{cite web|url=https://www.pbs.org/wgbh/nova/fractals/mandelbrot.html|title=A Radical Mind|last=Jersey|first=Bill|date=24 April 2005|work=Hunting the Hidden Dimension|publisher=NOVA/ PBS|accessdate=20 August 2009}}</ref> His informal and passionate style of writing and his emphasis on visual and geometric intuition (supported by the inclusion of numerous illustrations) made ''The Fractal Geometry of Nature'' accessible to non-specialists. The book sparked widespread popular interest in fractals and contributed to chaos theory and other fields of science and mathematics.+ + + − Mandelbrot died from pancreatic cancer at the age of 85 in a hospice in Cambridge, Massachusetts on 14 October 2010. 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." 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." Mandelbrot's obituary in The Economist points out his fame as "celebrity beyond the academy" and lauds him as the "father of fractal geometry".+ − 2010年10月14日，曼德尔布洛特在马萨诸塞州剑桥的一家临终关怀胰腺癌去世，享年85岁。数学家海因茨-奥托 · 佩特根在听到他去世的消息后说: “如果我们谈论数学内部的影响，以及在科学中的应用，他是过去50年来最重要的人物之一。”曼德布洛特去世时的法国总统尼古拉•萨科齐(Nicolas Sarkozy)表示，曼德布洛特“拥有强大的、独创的头脑，从不回避创新和打破先入为主的观念[ ... ... 他是工作，完全在主流研究之外发展起来，导致了现代信息理论的产生。”曼德布洛特在《经济学人》上发表的讣告指出，他是“学术之外的名人” ，并称赞他是“分形几何之父”。 − Mandelbrot also put his ideas to work in cosmology. He offered in 1974 a new explanation of [[Olbers' paradox]] (the "dark night sky" riddle), demonstrating the consequences of fractal theory as a [[Necessity and sufficiency|sufficient, but not necessary]], resolution of the paradox. He postulated that if the [[star]]s in the universe were fractally distributed (for example, like [[Cantor dust]]), it would not be necessary to rely on the [[Big Bang]] theory to explain the paradox. His model would not rule out a Big Bang, but would allow for a dark sky even if the Big Bang had not occurred.<ref>''Galaxy Map Hints at Fractal Universe'', by Amanda Gefter; New Scientist; 25 June 2008</ref> + + − 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".+ − 最畅销的散文作家兼作家纳西姆·尼可拉斯·塔雷伯 · 曼德布洛特评论说，曼德布洛特的著作《市场的(错误)行为》是他认为“有史以来最深刻、最现实的金融著作”。+
→Research career
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。
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==
== 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]].<ref name="people">{{Cite journal|last=Barcellos|first=Anthony|title=Mathematical People, ''Interview of B. B. Mandelbrot''|publisher=Birkhaüser|year=1984|url=http://users.math.yale.edu/~bbm3/web_pdfs/inHisOwnWords.pdf}}</ref> In 1958 the couple moved to the United States where Mandelbrot joined the research staff at the [[International Business Machines|IBM]] [[Thomas J. Watson Research Center]] in [[Yorktown Heights, New York]].<ref name="people" /> He remained at IBM for 35 years, becoming an IBM Fellow, and later Fellow [[Emeritus]].<ref name="wolf" />
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]].<ref name="people">{{Cite web|last=Barcellos|first=Anthony|title=Mathematical People, ''Interview of B. B. Mandelbrot''|publisher=Birkhaüser|year=1984|url=http://users.math.yale.edu/~bbm3/web_pdfs/inHisOwnWords.pdf|access-date=25 June 2013|archive-date=27 April 2015|archive-url=https://web.archive.org/web/20150427164851/http://users.math.yale.edu/~bbm3/web_pdfs/inHisOwnWords.pdf|url-status=live}}</ref> In 1958 the couple moved to the United States where Mandelbrot joined the research staff at the [[International Business Machines|IBM]] [[Thomas J. Watson Research Center]] in [[Yorktown Heights, New York]].<ref name="people" /> He remained at IBM for 35 years, becoming an IBM Fellow, and later Fellow [[Emeritus]].<ref name="wolf" />
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。
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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]].
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 market]]s as an example of "wild randomness", characterized by concentration and long range dependence. He developed several original approaches for modelling financial fluctuations.<ref>{{cite book |author= Rama Cont |chapter= Mandelbrot, Benoit |journal= Encyclopedia of Quantitative Finance |publisher= Wiley |date= 19 April 2010 |doi= 10.1002/9780470061602.eqf01006 |isbn = 9780470057568|author-link= Rama Cont }}</ref> In his early work, he found that the price changes in [[financial market]]s did not follow a [[Gaussian distribution]], but rather [[Paul Lévy (mathematician)|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]].<ref>{{cite web |url=https://www.newscientist.com/article/mg15420784.700-flight-over-wall-st.html |title=''New Scientist'', 19 April 1997 |publisher=Newscientist.com |date=19 April 1997 |access-date=17 October 2010 |archive-date=21 April 2010 |archive-url=https://web.archive.org/web/20100421101729/http://www.newscientist.com/article/mg15420784.700-flight-over-wall-st.html |url-status=live }}</ref>
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'''<font color="#ff8000"> 列维稳定分布Lévy stable distributions </font>'''。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 set]]s 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]]''.<ref>[https://books.google.com/books?id=xJ4qiEBNP4gC&printsec=frontcover ''The Fractal Geometry of Nature''] {{Webarchive|url=https://web.archive.org/web/20151130231048/https://books.google.com/books?id=xJ4qiEBNP4gC&printsec=frontcover |date=30 November 2015 }}, by Benoît Mandelbrot; W H Freeman & Co, 1982; {{isbn|0-7167-1186-9}}</ref> This influential work brought fractals into the mainstream of professional and popular mathematics, as well as silencing critics, who had dismissed fractals as "[[Artifact (observational)|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'''<font color="#ff8000"> 朱莉娅集合Julia sets</font>'''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==
==Awards and honors==