Thursday, October 21, 2010

Tribute to a Great Mathematician

Benoit Mandelbrot at a mathematics conference in 2008.

The passing of Benoit Mandelbrot, one of the most influential mathematicians of the 20th century, will be remembered and mourned for many years to come. When one sees or hears his name, one of course also conjures up the image of his famous "Mandelbrot set", a fractal mathematical creation bearing an infinite complexity. Indeed, in the case of all fractals (which Mandelbrot helped to originate) one see ever deepening complexity as one descends to ever increasing levels of magnification. It is as if there's an infinite regression capacity within the image itself (readers can glean some idea of this by comparing the two fractal images shown, generated by a fractal computer program).

Perhaps because of his creation of fractals, which had enormously wide applications, Mandelbrot similarly had wide interests and contributed to many disciplines: biology, physics and economics - especially publishing devastating critiques of modern mathematical finance (the same that gave rise to the "quants" who while ensconced in their ivory towers at investment banks - misused the Gaussian Copula formula to create the horrific derivatives known as "credit default swaps" which created the biggest financial meltdown since the Great Depression).

Fractals themselves are basically complex geometric shapes with fragmented forms, whose structures remain the same however much it is blown up or scaled down. (That is, the basic structure, though many more details of the intricate shapes are disclosed as one changes scale via a "zoom" factor). Fractal analysis itself has emerged as a powerful tool in the field of chaotic dynamics, to discern patterns even in emergent biological entities subject to increasing order.

Indeed, in a previous blog I showed as much. This was in conjunction with a polymer evolution model simulating emergence of a very primitive cell (macromolecule) from simpler monomers. The tendency observed (in the simulation) was for smaller length units to evolve to greater length polymers, as if the longer length had been preordained by selection. In the (Juliabrot fractal) model I used, entropy was expressed as a function of length and some partition function Z. Longer lengths prevailed because the difference in free energy was heavily weighted in that direction with entropy taken into account.

In the model (peculiar to most fractal approaches), an iterative mapping was set up with z’ (new value) = z^2 + k, where z = x + iy, and k = a +bi, with i= (-1)^1/2. A conformal mapping was then performed where w = f(x + iy). The entropy S/k = l ln (z),for smaller lengths l much less than L and S/k = (L+ l) ln (z) for larger. The simulation could then explain the basis for a bias in the way amino acids polymerize toward protenoids and peptides. The bias traced to free energy differences.

Of course, there are many species of fractal formats and designs, with the Juliabrot being only one. Overall, dozens exist and the beauty of the program I use ("Winfract") is that you can design any fractal species at will and alter the parameters within it. For example, one first selects the general type from the menu - say the "Orbital fractal" then the specific fractal type from the sub-menu: say the rossler3d. One then chooses the formula to initialize, e.g.

x = y = z = 1

and the steps for iteration, viz.

1) x' = x - (y - x)dt

2) y' = y + (x + ay)dt

3) z' = z + (b + xz - cz)dt

After these preliminaries, one selects the file for the routine, then adjusts the parameters desired, for example the time step, and colors, or any special effects. The program then does the rest via its computations.

Mandelbrot was born in Warsaw on November 20, 1924, then moved (as a child) with his Jewish family to France. There he survived the 2nd World War and began his mathematical career at the Centre National de la Recherche Scientifique in Paris. He emigrated to the U.S. in 1958 where he spent most of his professonal life working for IBM at its main research center in New York.

After leaving IBM, Mandelbrot taught mathematics at Yale University, finally accepting position as a tenured professor in 1999 at the age of 75.

His most famous book, and definitely my favorite is The Fractal Geometry of Nature, published in 1982. His most recent effort (in 2004), co-authored with Richard Hudson, was:The (Mis)Behavior Of Markets: A Fractal View of Risk, Ruin and Reward. This was a devastating attack on the failure of mainstream economists to understand the likelihood of wild swings in prices and the risk of financial disaster. Well, people - even non-mathematicians- understand much more about that since the 2008 meltdown, in many ways similar to a chaotic nonlinear progression such as seen in "catastrophe theory", a related field to which Mandelbrot contributed.

Mandelbrot could certainly be seen or viewed as a "maverick" (moving across many disciplines with little regard to the purist specialists) but it would be a major error to see him as a crank. Instead, he's given us forms of mathematics which can only be used to discern and unravel the maddening complexity of chaotic behavior.

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