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Fixed points


The importance of infinite variance
Benoît Mandelbrot Mathematician
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I proclaimed the importance of this second variance, infinite variance, very loudly and in general a view of inequality, irregularity, which was very, very unpopular. It's very easy to explain actually and it's very important. Imagine that you take two molecules of a gas, and you find that the total energy is u; then you know perfectly well that each of them has a energy which is u over half, plus or minus a small fluctuation, which actually is Gaussian, more or less. In other words, in classical physics, there is a very profound situation, which is equality plus or minus a small fluctuation. But now imagine on the contrary that you walked down Wall Street and you're allowed to stop two persons more or less blindly and find that the total income for the year 1997 was two million dollars. Now are you going to say that you have found two millionaires? Of course not. Everybody has the absolutely strong conviction, perfectly justified, that most likely one of them is an ordinary wage earner -again distribution - maybe twenty, fifty, maybe a hundred thousand dollars, and the other has all the rest. Therefore the inequality was not something that was in a certain sense quantitative, but qualitative. In one case one half plus or minus, in the other case its two independents, one point which is independent of a sum, and then the other which is the rest. Now, at that point I was very much in a mood already to attribute more importance to qualitative than to quantitative distinctions, because the qualitative one is so sharp it would separate two very different behaviours. If it were just qualitative matter- up to a point it is a liquid and after it's a gas - it wouldn't be right - the critical points would be terrible, they're too complicated.

Benoît Mandelbrot (1924-2010) discovered his ability to think about mathematics in images while working with the French Resistance during the Second World War, and is famous for his work on fractal geometry - the maths of the shapes found in nature.

Listeners: Daniel Zajdenweber Bernard Sapoval

Daniel Zajdenweber is a Professor at the College of Economics, University of Paris.

Bernard Sapoval is Research Director at C.N.R.S. Since 1983 his work has focused on the physics of fractals and irregular systems and structures and properties in general. The main themes are the fractal structure of diffusion fronts, the concept of percolation in a gradient, random walks in a probability gradient as a method to calculate the threshold of percolation in two dimensions, the concept of intercalation and invasion noise, observed, for example, in the absorbance of a liquid in a porous substance, prediction of the fractal dimension of certain corrosion figures, the possibility of increasing sharpness in fuzzy images by a numerical analysis using the concept of percolation in a gradient, calculation of the way a fractal model will respond to external stimulus and the correspondence between the electrochemical response of an irregular electrode and the absorbance of a membrane of the same geometry.

Duration: 2 minutes, 11 seconds

Date story recorded: May 1998

Date story went live: 24 January 2008