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Benoît Mandelbrot is free a mathematician - American born with Warsaw on November 20th, 1924. He worked at the beginning of his career on original applications of the information theory, then developed a new class of mathematical objects: the fractals objects, or Fractale S.
Biography
Mandelbrot was born with Warsaw in Poland, in a family with a strong academic tradition: his/her mother was doctor and his/her uncle Szolem Mandelbrojt was mathematical professor of to the Collège de France; his/her father, built his life to him on the resale of clothing. Its family left the Poland for Paris in order to flee the threat hitlérienne, because of their Jewish origins. Over there, Mandelbrot was initiated with the Mathématiques by his/her two uncles. After having attended the College Edmond Perrier of Tulle, it continues its studies with the Lycée of the Park to Lyon.
Years of youth: a brilliant departure
After having left the Polytechnic school (X44 promotion) where it followed the courses of Paul Levy, it is interested in the phenomena of Information, ideas of Claude Shannon being then in full rise. Intrigued by the Law of Zipf, empirical and disputed, it poses it in terms of minimization of the use and storage costs of the words by the spirit. By elimination of the variable of cost between the two equations a law appears which does not have this time plus nothing empirical: it is the Loi of Mandelbrot, of which that of Zipf is only one particular case, and who answers better than it the observations (explaining in particular the elbow always observed in the distributions and not explained by the law of Zipf). This work is worth an immediate notoriety to him, in particular thanks to a work of Leon Brillouin: Science and information theory which will have besides a success much larger in its English translation Science and information theory (catastrophic typographical conventions of the French work are not foreign there).
The crossing of the desert
It then leaves France to join the the United States of America attracted by a greater freedom of nonrestricted creativity to only one precise discipline. He works as researcher at IBM on the optimal transmission in the disturbed mediums, while continuing his work on strange objects until neglected enough there by the mathematicians: objects with recursively definite complexity like the curve of Von Koch to which it has a presentiment of a unit. The mathematician Felix Hausdorff prepared the ground besides by defining for these objects a not-whole dimension, the dimension of Hausdorff. As for the mathematician Gaston Julia, it defined objects which have an air of family with the whole.
A new paradigm
It signs in 1973 in a review of economy an article to the quite careful title: new Forms of the chance in sciences . This article indexes the cases where, contrary to the paradigm classically used, the risks are not cancelled, but on the contrary cumulate, and where the traditional statistical prediction does not function any more. It of course quotes examples taken in its field at IBM, the transmission of the signal, but also in unexpected fields: risings of the the Nile, the shape of the clouds, that of the rivers.
It brilliantly concludes that there is not a form of chance, which would always lead to an equalization by the Loi of the great numbers. It is a Illusion due to the fact that we study only these examples by diverting us of the others like badly conditioned , as the mathematicians were diverted curve of von Koch which they regarded as an object monstrous : the spheres or the triangles are regarded as acceptable objects by the mathematicians of the time, but not the clouds nor the trees (at least as geometrical objects). Mathematics of this time remains dumb on the monsters . Not astonishing under these conditions which existing mathematics is regarded as having an immense capacity of explanation of the scientific phenomena, because we consider as scientists only the phenomena that they make it possible to explain ! We are taken in the trap of a circular Argument of which we cannot leave any more.
However, adds Mandelbrot, it is the main part of the phenomena of nature which obey this other type of chance where one cannot apply the law of the great numbers. The standard model makes us pass beside most of reality, and will prevent to us to even see it .
It quotes then as example of this new form of chance studied the example which will become famous coast of Brittany , on which the length depends on the scale to which one measures it, and who has a not-whole dimension of Hausdorff, ranging between 1 and 2: it strictly speaking constitutes neither an object with a dimension, nor an object with two dimensions, and it is by accepting the idea of not-whole size which we will be able to attack these objects which have always escaped with our study: the theory fractale as of this article is semi-officially launched.
The principles will be published by it with very great quantity of examples (hydrology, structure of the lung, granulation of the concretes, Paradoxe of Olbers, turbulences in Mécanique of the fluids, town planning of the cities, and even holes of the cheese of Appenzell) in a work which makes since reference: the fractals Objects - Form, chance and dimension in 1974. It presents to it to the reader objects hitherto little known: curve of Von Koch, sponge of Sierpinski (or sponge of Menger, or Sierpinski-Menger), that the mathematicians kept moderately in their drawers. All these examples in common have what the author names a homothety of scale and which it will indicate a few years later under the name of autosimilarity ( coil-similarity ).
Innovative character of the book (appeared at the beginning in France) in fact an immediate success, world, and which touches this time the general public. It should be noted that the examples of the first edition of this work were all in black and white for reasons of economy and technology of the screens. Thereafter, the fractales appearing an effective tool for the Synthesis of image S complexes, one will not see any more but colors.
Mandelbrot gave its name to a family of fractales (known as of Mandelbrot), manufactured in the complex plan by successive iterations of type Z (new) = Z ² + constant.
Its work on the fractales as a Mathématicien with IBM was worth to him “Emeritus Fellowship” at the research laboratory T.J. Watson. Its work was resumed there by his/her collaborator, Richard Voss. He was prize winner of the Franklin Médaille in 1986.
In addition to discovered Mathematical Fractale S in , it showed the great number of objects described well by Fractale S in the Nature, thus leading to new grounds of Recherche. Fractales are also found in phenomena studied in Théorie of chaos.
It joined the Université Yale in 1987.
In 1991, Mandelbrot (systematically invited with any chance with each bearing congress on the fractales) realized that there had been of it more on planet this year than of days in the year!
November 23rd, 1990 it is made knight of the Légion of honor, and is promoted officer on January 1st, 2006, a distinction which to him is given on September 11th, 2006 by his/her comrade of promotion at the Polytechnic school, the senator Pierre Laffitte.
Finance
Benoît Mandelbrot is also at the origin in 1961 of a model of evolution of the price stock exchange based on the geometry fractale. This financial theory with the advantage of better predicting occurred of the extreme variations, which does not allow the use of the technical analysis based on the Théorie of Dow. Initially recognized relevant, it was then put on side due to complexity, before being re-used since the end of the Années 1990, rich in financial turbulences.
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