By Vladimir I. Arnold, et al.
ISBN-10: 3540232354
ISBN-13: 9783540232353
ISBN-10: 3540294627
ISBN-13: 9783540294627
This ebook includes numerous contributions at the most eminent occasions within the improvement of twentieth century arithmetic, representing a large choice of specialities within which Russian and Soviet mathematicians performed a substantial function. The articles are written in a casual type, from mathematical philosophy to the outline of the improvement of rules, own thoughts and provides a special account of non-public conferences with well-known representatives of twentieth century arithmetic who exerted nice impression in its improvement. This e-book can be of serious curiosity to mathematicians, who will get pleasure from seeing their very own specialities defined with a few old viewpoint. Historians will learn it with a similar rationale, and maybe additionally to choose themes for destiny research.
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I wonder whether Kolmogorov had been involved in the writing of the first paper of I. M. Gel’fand, who was also his student. This is one of the few papers signed by Gel’fand alone, with no collaborators. Gel’fand, whose brilliant papers and highly influential seminar I always admired very much, mastered a special and enviable art of day-to-day collaboration with extremely gifted mathematicians (mostly his former students), resulting in important and beautiful joint publications. I dare to guess that these papers were actually written in most cases by the collaborators.
But surprisingly it turned out that there was no one limit cycle in any of these randomly chosen systems. At the time, examples with three cycles were known, later Chinese mathematicians found examples with four cycles, but, up to now, even the boundedness of the number of cycles (uniform with respect to the coefficients of the polynomials P and Q) conjectured by Hilbert has not been proved. 30 V. I. Arnold In trying to understand this phenomenon, Kolmogorov investigated its abstract variant: a dynamical system on a torus given by a divergence-free field with respect to a volume element.
This happened in 1961–63, and since that time I have been trying to find applications of this philosophy of structural stability. My first idea was to think of the model of hard spheres in statistical mechanics. I speculated that such systems might be considered as the limiting case of geodesic flows on negatively curved manifolds (the curvature being concentrated on the collisions hypersurface). The simplest model of this kind is a system of two elastically colliding disks on the surface of a two-dimensional torus.
Mathematical Events of the Twentieth Century by Vladimir I. Arnold, et al.
by Brian
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