伯努瓦·曼德布洛特 Benoit Mandelbrot

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
Born	1924年11月20日 波兰华沙
Died	2010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
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 students	L-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. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French and 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". He referred to himself as a "fractalist" 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.

{{quote|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? 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.

{引用 | 我们一直担心的是，一个足够坚定的敌人可能会向某个权威举报我们，我们将会被送去送死。来自巴黎的密友吉娜 · 莫汉格就遇到了这种情况，她是附近一个县城的内科医生。仅仅是为了消除竞争，另一个医生谴责她... 我们逃过了这个命运。谁知道为什么呢？1958年，这对夫妇移居美国，在那里曼德尔布洛特加入了位于约克敦海茨的 IBM 汤玛士·Ｊ·华生研究中心的研究人员。

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 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.

在他的早期工作中，他发现金融市场的价格变化并不遵循正态分布，而是具有无限方差的 l é vy 稳定分布。例如，他发现，棉花价格遵循一个 l é vy 稳定分布，参数 α 等于1.7，而不是像正态分布一样遵循2。“稳定”分布具有这样的性质: 一个随机变量的多个实例的和服从相同的分布，但具有更大的尺度参数。

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", like clouds or shorelines, actually had a "degree of order". 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.

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. 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.

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. 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，Mandelbrot 开始研究被称为 Julia 集的分形，它们在复平面的某些变换下是不变的。在加斯顿 · 朱莉娅和皮埃尔 · 法图之前工作的基础上，曼德尔布洛特使用计算机绘制了朱莉娅集合的图像。在研究这些 Julia 集的拓扑结构时，他研究了他在1979年提出的 Mandelbrot 集。1982年，曼德布洛特在《自然的分形几何》一书中扩展和更新了他的观点。这一有影响力的工作带来了分形进入主流的专业和流行数学，以及沉默批评家，谁曾驳回分形作为“程序人工制品”。

Early years

Mandelbrot speaking about the Mandelbrot set, during his acceptance speech for the Légion d'honneur in 2006

曼德布洛特谈到了[2006年他在法国荣誉军团勋章的获奖感言中谈到了曼德布洛特集合]

Mandelbrot was born in a Jewish family, in Warsaw during the Second Polish Republic. 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." Later, the family's move to France, 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.

In 1975, Mandelbrot coined the term fractal to describe these structures and first published his ideas, and later translated, Fractals: Form, Chance and Dimension. 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". 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":

1975年，曼德布洛特创造了分形这个术语来描述这些结构，并首次发表了他的观点，后来被翻译成分形: 形式、机会和维度。计算机科学家、物理学家斯蒂芬•沃尔夫拉姆(Stephen Wolfram)表示，这本书对曼德布洛特来说是一个“突破” ，在此之前，他通常会“将相当简单的数学应用到以前几乎没有认真研究过数学的领域”。沃尔夫勒姆补充说，由于这项新研究的结果，他不再是一个“流浪的科学家” ，后来称他为“分形之父” :

The family emigrated from Poland to France in 1936, when he was 11. "The fact that my parents, as economic and political refugees, joined Szolem in France saved our lives," he writes. 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. Much of France was occupied by the Nazis at the time, and Mandelbrot recalls this period:

/* Styling for Template:Quote */ .templatequote { overflow: hidden; margin: 1em 0; padding: 0 40px; } .templatequote .templatequotecite {

   line-height: 1.5em;
   /* @noflip */
   text-align: left;
   /* @noflip */
   padding-left: 1.6em;
   margin-top: 0;

}

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 markets as an example of "wild randomness", characterized by concentration and long range dependence. He developed several original approaches for modelling financial fluctuations.

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.

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 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. 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".

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。

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. 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". 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:

/* Styling for Template:Quote */ .templatequote { overflow: hidden; margin: 1em 0; padding: 0 40px; } .templatequote .templatequotecite {

   line-height: 1.5em;
   /* @noflip */
   text-align: left;
   /* @noflip */
   padding-left: 1.6em;
   margin-top: 0;

}

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." He points out an unexpected conclusion:

/* Styling for Template:Quote */ .templatequote { overflow: hidden; margin: 1em 0; padding: 0 40px; } .templatequote .templatequotecite {

   line-height: 1.5em;
   /* @noflip */
   text-align: left;
   /* @noflip */
   padding-left: 1.6em;
   margin-top: 0;

}

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.

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:

/* Styling for Template:Quote */ .templatequote { overflow: hidden; margin: 1em 0; padding: 0 40px; } .templatequote .templatequotecite {

   line-height: 1.5em;
   /* @noflip */
   text-align: left;
   /* @noflip */
   padding-left: 1.6em;
   margin-top: 0;

}

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

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.

Mandelbrot left IBM in 1987, after 35 years and 12 days, when IBM decided to end pure research in his division. 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.

Fractals and the "theory of roughness"

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. He began by asking himself various kinds of questions related to nature:

/* Styling for Template:Quote */ .templatequote { overflow: hidden; margin: 1em 0; padding: 0 40px; } .templatequote .templatequotecite {

   line-height: 1.5em;
   /* @noflip */
   text-align: left;
   /* @noflip */
   padding-left: 1.6em;
   margin-top: 0;

}

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.

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." 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:

/* Styling for Template:Quote */ .templatequote { overflow: hidden; margin: 1em 0; padding: 0 40px; } .templatequote .templatequotecite {

   line-height: 1.5em;
   /* @noflip */
   text-align: left;
   /* @noflip */
   padding-left: 1.6em;
   margin-top: 0;

}

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:

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.
  —Mandelbrot, in his introduction to The Fractal Geometry of Nature

Section of a Mandelbrot set

Mandelbrot has been called an artist, and a visionary 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.

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 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.

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”。

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, 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. 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.

A partial list of awards received by Mandelbrot:

“ padding-right: 2em” 2004 Best Business Book of the Year Award AMS Einstein Lectureship Barnard Medal Caltech Service Casimir Funk Natural Sciences Award Charles Proteus Steinmetz Medal Fellow, American Geophysical Union Fellow of the American Statistical Association Fellow of the American Physical Society (1987)

Franklin Medal Harvey Prize (1989) Honda Prize Humboldt Preis IBM Fellowship Japan Prize (2003) John Scott Award Légion d'honneur (Legion of Honour)

• Lewis Fry Richardson Medal

• Medaglia della Presidenza della Repubblica Italiana

• Médaille de Vermeil de la Ville de Paris

• Nevada Prize

• Member of the Norwegian Academy of Science and Letters.

• Science for Art

• Sven Berggren-Priset

• Władysław Orlicz Prize

• Wolf Foundation Prize for Physics (1993)

Legacy

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."

Chris Anderson, TED conference curator, described Mandelbrot as "an icon who changed how we see the world". 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".

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".

Bibliography

in English

• Fractals: Form, Chance and Dimension, 1977, 2020

• The Fractal Geometry of Nature, 1982

• 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, Zach Weinersmith' webcomic, Saturday Morning Breakfast Cereal, portrayed Mandelbrot.

• In 2017, Liz Ziemska published a novella, Mandelbrot The Magnificent, a fictional account of how Mandelbrot saved his family during WWII

See also

"How Long is the Coast of Britain?" Louis Bachelier Zipf–Mandelbrot law Seven states of randomness Skewness risk Category:1924 births 类别: 1924出生 Kurtosis risk Category:2010 deaths 分类: 2010年死亡人数 Fractal dimension Category:20th-century American mathematicians 范畴: 20世纪美国数学家 Lacunarity 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 Self-similarity Category:21st-century American economists 类别: 21世纪美国经济学家 Self-affinity Category:Alexander von Humboldt Fellows 类别: 亚历山大·冯·洪堡研究员 Hurst exponent Category:California Institute of Technology alumni 类别: 加州理工学院校友 Fractional Brownian motion Category:Chaos theorists 范畴: 混沌理论家 Multifractal system Category:Deaths from cancer in Massachusetts 分类: 马萨诸塞州癌症死亡人数 1/f noise Category:Deaths from pancreatic cancer 分类: 死于胰腺癌 Mandelbrot Competition 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

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

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

• Hudson, Richard L.; Mandelbrot, Benoît B. (2004). The (Mis)Behavior of Markets: A Fractal View of Risk, Ruin, and Reward. New York: Basic Books. ISBN 978-0-465-04355-2. https://archive.org/details/misbehaviorofmar00beno. 

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

• Mandelbrot, Benoit B. (2010). The Fractalist, Memoir of a Scientific Maverick. New York: Vintage Books, Division of Random House.

  Category:Members of the Norwegian Academy of Science and Letters

  类别: 挪威科学与文学学会成员

  • Mandelbrot, Benoît B. (1983). The Fractal Geometry of Nature. San Francisco: W.H. Freeman. ISBN 978-0-7167-1186-5. 

  Category:Members of the United States National Academy of Sciences

  类别: 美国国家科学院院士

  • Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe and Cornelia Zahlten: Fractals: An Animated Discussion (63 min video film, interviews with Benoît Mandelbrot and Edward Lorenz, computer animations), W.H. Freeman and Company, 1990.
    (re-published by Films for the Humanities & Sciences,
    )

    Category:Officiers of the Légion d'honneur

    类别: 美国法国荣誉军团勋章协会官员

    • Mandelbrot, Benoit B. (1997) Fractals and Scaling in Finance: Discontinuity, Concentration, Risk, Springer.

    Category:Polish emigrants to France

    类别: 移居法国的波兰移民

    • Mandelbrot, Benoît (February 1999). "A Multifractal Walk down Wall Street". Scientific American. 280 (2): 70. Bibcode:1999SciAm.280b..70M. doi:10.1038/scientificamerican0299-70.

    Category:Polish emigrants to the United States

    类别: 移居美国的波兰移民

    • Mandelbrot, Benoit B., Gaussian Self-Affinity and Fractals, Springer: 2002.

    Category:Polish Jews

    分类: 波兰犹太人

    • Mandelbrot, Benoît; Taleb, Nassim (23 March 2006). "A focus on the exceptions that prove the rule". Financial Times. Archived from the original on 23 October 2010. Retrieved 17 October 2010.

    Category:University of Paris alumni

    类别: 巴黎大学校友

    • "Hunting the Hidden Dimension: mysteriously beautiful fractals are shaking up the world of mathematics and deepening our understanding of nature", NOVA, WGBH Educational Foundation, Boston for PBS, first aired 28 October 2008.

    Category:Wolf Prize in Physics laureates

    类别: 沃尔夫物理学奖获得者

    • Frame, Michael; Cohen, Nathan (2015). Benoit Mandelbrot: A Life in Many Dimensions. Singapore: World Scientific Publishing Company. ISBN 978-981-4366-06-9. 

    Category:Yale University faculty

    类别: 耶鲁学院

    • Mandelbrot, B. (1959) Variables et processus stochastiques de Pareto-Levy, et la repartition des revenus. Comptes rendus de l'Académie des Sciences de Paris, 249, 613–615.

    Category:Yale Sterling Professors

    分类: 耶鲁斯特林教授

    • Mandelbrot, B. (1960) The Pareto-Levy law and the distribution of income. International Economic Review, 1, 79–106.

    Category:20th-century French mathematicians

    范畴: 20世纪法国数学家

    • Mandelbrot, B. (1961) Stable Paretian random functions and the multiplicative variation of income. Econometrica, 29, 517–543.

    Category:21st-century American mathematicians

    范畴: 21世纪美国数学家

    
    

    This page was moved from wikipedia:en:Benoit Mandelbrot. Its edit history can be viewed at 曼德布洛特/edithistory