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Mathematical Events of the Twentieth Century by Vladimir I. Arnold, et al.

By Vladimir I. Arnold, et al.

This booklet includes a number of 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 position. The articles are written in an off-the-cuff kind, from mathematical philosophy to the outline of the improvement of principles, own stories and provides a distinct account of non-public conferences with well-known representatives of twentieth century arithmetic who exerted nice effect in its improvement. This publication can be of serious curiosity to mathematicians, who will take pleasure in seeing their very own specialities defined with a few historic standpoint. Historians will learn it with an identical rationale, and maybe additionally to pick issues for destiny research.

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Let us return to the question of stability of missile shells. People who constructed these shells have observed that the geometry of the characteristics, which are the asymptotic lines of the shell surface, can present obstacles to inflexibility. The asymptotic lines define a dynamical system similar to that which leads us to the diffeomorphisms of the circle. Consider a hyperbolic cylinder between two horizontal circles (like a section of the Moscow TV tower designed by Shukhov). Moving upward from the lower circle along the asymptotic line (“element” of a hyperboloid of one sheet), we will arrive at a point on the upper circle.

Then I read Poincar´e and observed that for the ellipse, for instance, this transformation is equivalent to a rotation through an angle which depends on the ellipse. This angle is, in general, incommensurable with 2π and hence the mapping T represents an ergodic dynamical system. In the resonance case, when there is a periodic orbit, 8 the periodic points are obstacles to the solution of the initial problem, because the alternating sum of 7 This problem has recently been re-examined by A. B. Skopenkov (1996) who listed all the obstacles to the representation of any continuous function on a plane curve in the form of the sum of two continuous functions of the coordinates.

I thought that the topology of our algebraic function for higher n is complicated and, if there were an expression in terms of functions of fewer variables, then it should be simpler. So I studied the topology of this space — the complement to the discriminant — which is in fact the configuration space of sets of n points in C and the Eilenberg–MacLane space K(π , 1) of the braid group. In one of the first 4 It follows from the Smith theory that the sum of Betti numbers of a real algebraic variety is no greater than the sum of Betti numbers of the complex variety specified by the same equations (all Betti numbers being Z2 -homologies ranks).

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