以下英文部分来自B. B. Mandelbrot, Possible refinement of the lognormal hypothesis concerning the distribution of energy dissipation in intermittent turbulence. Statistical Models Turbulence, Springer, 333-351, 1972. 中文是博主的点评。 Self-similarity and the k^(-5/3) spectrum have not only been observed, but are found to hold beyond their assumed domain of applicability. An unexpected embarrassment of riches, and a puzzle! 大气湍流中能谱的-5/3律延展的范围要远远宽于结构函数2/3律，故不能仅根据能谱来判断惯性子区的范围。 For many scientists, studying turbulence is synonymous with attempting to derive its properties, including those listed above, from the Navier-Stokes equations of fluid mechanics. But one can also follow a different tack and view intermittency and self-similar statistical hierarchies as autonomous phenomena. 湍流表现出很多与复杂系统相似的特征。这些特征可能是更高层次集体行为的涌现。这就好比一个社会系统，本质上都遵循牛顿力学或量子力学和麦克斯韦方程组，但是社会的行为是更高尺度的涌现行为，根本不可能从这些方程组出发来研究。这里的尺度不仅仅表示空间尺度，还有着层次的概念。分子量级是一个层次。分子组成细胞，所有的细胞集团构成一个层次。细胞构成人体器官，所有人体器官是一个层次。个人构成家庭，所有家庭是一个层次。家庭构成国家，所有国家集团是一个层次。。。。不同的层次可以涌现出不同的行为，也可以有相似的行为。回到湍流，湍流在不同尺度上的涌现特征，也完全没必要从微团层次的Navier-Sotkes方程来理解。
复数 M 集 仅有二维形就那么美妙迷人，想必三维数集也会有惊人的分形结构。由二维形寻觅三维形最容易遇到三维 M 集。 复数 x + i y 是二元数，所以复数集在复平面上有二维形。要寻求三维分形数集就得有三元数。这个不难，直接将二元复数再扩展一元成 x + i y + j z ，让 i 2 = j 2 =-1 ， ij =- ji =1 ，就有最简单的三元数。因为三元数三分量 ( x, y, z ) 唯一决定空间一点，所以三元数集在三维空间中有特定的三维形象。 借用 M 集的迭代函数对以上三元数运算取集，便得一个三元数集，称为三维 M 集，形象见图 1 。 三维 M 集乍看大小两球并接，束身扎着粗细不一许多环带，还有一些纤细的正圆环圈绕在身体外围，整体好像一个抖着呼拉圈的宝葫芦。大球后部向里凹陷，小球上顶着更小的圆球，形成分形球串。球串前那根针须不可轻视，一经放大，所见精细结构让人瞠目，原来有无数与整体相似的三维子集蛰伏在针须上。那子集头前还存在更小的相似子集，向细微深处延绵不断，这正是久违的三维自相似现象。 若顺着针须把三维 M 集完整切开，就会看到整个切面是标准的 M 集形，如图2。放大后还能看到那些绕身圆环的切口也是 M 集形。想象出，三维 M 集应该是 M 集绕对称轴旋转一周的痕迹形象，这束身环带应该是 M 集身上的球苞旋转而成，这绕身细环应该是 M 集枝丫上的自相似子集绕转而成。所以 M 集应是三维 M 集的切面子集，三维 M 集也可称 M 旋集。
分形之父Mandelbrot所得部分奖励和荣誉 A partial list of awards received by Mandelbrot 2004 Best Business Book of the Year Award AMS Einstein Lectureship Barnard Medal Caltech Service Casimir Funk Natural Sciences Award Charles Proteus Steinmetz Medal Franklin Medal Harvey Prize Honda Prize Humboldt Preis Fellow, American Geophysical Union IBM Fellowship Japan Prize John Scott Award Lewis Fry Richardson Medal Medaglia della Presidenza della Repubblica Italiana Mdaille de Vermeil de la Ville de Paris Nevada Prize Science for Art Sven Berggren-Priset W?adys?aw Orlicz Prize Wolf Foundation Prize for Physics
据美国新闻媒体报道，美国东部时间10月14日法裔美国数学家Benoit Mandelbrot 85岁（20 November 1924 14 October 2010）在马塞诸塞州剑桥市临终医院因胰腺癌辞世。Mandelbrot是分形之父， 1982年他出版了著名的自然的分形几何著作，标志着分形几何的诞生。他在分形方面的工作成为混沌理论的基础，也是计算机数据压缩和医学图像纹理以及模拟 湍流对飞机机翼造型设计的关键。Benoit Mandelbrot 出生于波兰(父母是犹太人)，孩童时代移居法国，他大部分时间在美国生活和工作，他具有法国和美国双重国籍。 1958-1987年 Mandelbrot 一直在IBM工作，1987-2005在耶鲁大学工作，2005年退休后一直生活在麻州的剑桥市。 Benoit Mandelbrot, a mathematics pioneer and the father of the principle of fractal geometry, has died in the US at the age of 85. The fractal principle uses mathematical fromulas to attempt to understand complexity of natural world In his seminal 1982 work The Fractal Geometry of Nature, Mandelbrot argued that seemingly random patterns could in fact be the same infinitely repeated shape. He once used a cauliflower to describe the mathematical principle, pointing out that the shape of the vegetable was repeated over and over The mathematical principle has been used to measure shapes previously thought unmeasurable, including coastlines and mountains. Mandelbrot also applied the concept to economics, but he was critical of the global financial system, believing it to be too complex to properly function. Fractal geometry can be depicted in intricate and colourful computer designs which have become popular as artworks in their own right. One fractal variation was even named after Mandelbrot. The Mandelbrot Set has had a huge influence on mathematics and culture - examples have even been known to appear as crop formations. Mandelbrot的早年生活 Early years Mandelbrot was born in Warsaw into a Jewish family from Lithuania .He was born into a family with a strong academic traditionhis mother was a medical doctor and he was introduced to mathematics by two uncles, one of whom, Szolem Mandelbrojt , was a Parisian mathematician. However, his father made his living trading clothing. Anticipating the threat posed by Nazi Germany , the family fled from Poland to France in 1936 when he was 11. Mandelbrot attended the Lyce Rolin in Paris until the start of World War II , when his family moved to Tulle . He was helped by Rabbi David Feuerwerker , the Rabbi of Brive-la-Gaillarde , to continue his studies. In 1944 he returned to Paris. He studied at the Lyce du Parc in Lyon and in 1945-47 attended the cole Polytechnique , where he studied under Gaston Julia and Paul Lvy . From 1947 to 1949 he studied at California Institute of Technology , where he earned a master's degree in aeronautics .Returning to France, he obtained a PhD in Mathematical Sciences at the University of Paris in 1952. 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 , and later to the Universit Lille Nord de France . 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 . He remained at IBM for thirty-two years, becoming an IBM Fellow , and later Fellow Emeritus . 学术生涯 Academic career 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 . He became convinced that two key themes, fat tails and self-similar structure, ran through a multitude of problems encountered in those fields. Mandelbrot found that price changes in financial markets did not follow a Gaussian distribution , but rather Lvy stable distributions having theoretically infinite variance . He found, for example, that cotton prices followed a Lvy stable distribution with parameter 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 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. In 1975, Mandelbrot coined the term fractal to describe these structures, and published his ideas in Les objets fractals, forme, hasard et dimension (1975; an English translation Fractals: Form, Chance and Dimension was published in 1977). Mandelbrot developed here ideas from the article Deux types fondamentaux de distribution statistique (1938; an English translation Two Basic Types of Statistical Distribution ) of Czech geographer , demographer and statistician Jaromr Kor?k . While on secondment as Visiting Professor of Mathematics at Harvard University in 1979, 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 of the formula z ² . While investigating how the topology of these Julia sets depended on the complex parameter he studied the Mandelbrot set fractal that is now named after him. (Note that the Mandelbrot set is now usually defined in terms of the formula z ² + c , so Mandelbrot's early plots in terms of the earlier parameter are leftright mirror images of more recent plots in terms of the parameter c .) 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 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. His 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 Knight in the French 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 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. 分形Fractals and regular roughness Although Mandelbrot coined the term fractal , some of the mathematical objects he presented in The Fractal Geometry of Nature had been described by other mathematicians. Before Mandelbrot, they had been 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 non-smooth objects in the real world. He highlighted their common properties, such as self-similarity (linear, non-linear, or statistical), scale invariance , and a (usually) non-integer Hausdorff dimension . He also emphasized the use of fractals as realistic and useful models of many rough phenomena in the real world. Natural fractals include the shapes of mountains , coastlines and river basins ; the structures of plants, blood vessels and lungs ; the clustering of galaxies ; and Brownian motion . Fractals are 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 Mandelbrot has been called 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. When visiting the Museu de la Cincia de Barcelona in 1988, he told its director that the painting The Face of War had given him the intuition about the transcendence of the fractal geometry when making intelligible the omnipresent similitude in the forms of nature. He also said that, fractally, Gaud was superior to Van der Rohe . Death Mandelbrot died in a hospice in Cambridge, Massachusetts , on 14 October 2010 from pancreatic cancer , at the age of 85. Reacting to news of his death, mathematician Heinz-Otto Peitgen said if we talk about impact inside mathematics, and applications in the sciences, he is one of the most important figures of the last 50 years. Chris Anderson described Mandelbrot as an icon who changed how we see the world. French President Nicolas Sarkozy said Mandelbrot had a powerful, original mind that never shied away from innovating and shattering preconceived notions. Sarkozy also added, His work, developed entirely outside mainstream research, led to modern information theory.
标签: Mandelbrot
