Since the first edition of this book, geometrical methods in the theory of ordinary differential equations have become very popular and some progress has been made partly with the help of computers. Much of this progress...
This is a very unusual book, and I think it will be appropriate to put things into perspective and to provide some background information. V. I. Arnold is a famous mathematician who belongs to the generation of Russian mathematicians, now approaching their 70-th anniversaries. I will not list here Arnold's numerous awards and prizes: such lists are easily… (More)
521 results about them. The function field K(X) is a subfield of K(X /Y); one of the important questions is: when are K(X) and K(X /Y) equal, in other words, when can all formal-rational functions on X /Y be extended to X? Many of the theorems state suitable conditions on X and Y under which K(X) = K(X /Y). The other chapters in the second part deal with… (More)
According to a conjecture of E. Witten  proved by M. Ko-ntsevich , a certain generating function for intersection indices on the Deligne–Mumford moduli spaces of Riemann surfaces coincides with a certain tau-function of the KdV hierarchy. The generating function is naturally generalized under the name the total descendent potential in the theory of… (More)
The higher-dimensional analogue of a continuous fraction is the polyhedral surface, bounding the convex hull of the semigroup of the integer points in a simplicial cone of the euclidian space. The article describes some conjectures and theorems, extending to such higher-dimensional continouos fraction the Lagrange theorem on quadratic irrationals and the… (More)
The critical points of a smooth function are the points where the differential vanishes. A critical point is nondegenerate if the second differential is a nondegener-ate quadratic form. In some neighbourhood of a nondegenerate critical point the function can be represented in the Morse normal form /= ± x\ ± ' • • ± x\ + const using suitable local… (More)
Sturm theory extends the Morse inequality (minorating the number of critical points of functions on a circle) to the higher derivatives. The Legendrian Morse theory (created by Yu. V. Chekanov in 1986) provides the Morse inequality for the multivalued functions (corresponding to the unknoted Legendrian submanifolds of the space of 1-jets of functions). It… (More)
1 Classiication of singularities of fractions In this paper we mean by fraction the ratio (ordered pair) of two germs of (holomorphic or smooth) functions in n variables at a point (which usually will be refered as the origin). For the sake of simplicity mostly the complex case is considered below. The results for the real case are similar, only certain… (More)