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According to a conjecture of E. Witten [21] proved by M. Ko-ntsevich [12], 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)
An Abel differential equation y = p(x)y 2 + q(x)y 3 is said to have a center at a pair of complex numbers (a, b) if y(a) = y(b) for every solution y(x) with the initial value y(a) small enough. This notion is closely related to the classical center-focus problem for plane vector fields. Recently, conditions for the Abel equation to have a center have been(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)
We study Cohen–Macaulay modules over normal surface singularities. Using the method of Kahn and extending it to families of modules, we classify Cohen–Macaulay modules over cusp singularities and prove that a minimally elliptic singularity is Cohen–Macaulay tame if and only if it is either simple elliptic or cusp. As a corollary, we obtain a classification(More)
A spherical function of degree n on the unit sphere in R 3 is the restriction to the sphere of a homogeneous harmonic polynomial of degree n. In the present paper the topological consequences of the following classical fact are discussed. Theorem 1. The nth derivative of the function 1/r along n constant (translation invariant) vector fields in R 3(More)
The Frobenius number of vector a, whose components as are natural numbers (having no common divisor greater than 1), is the minimal integer N (a) which is representable as a sum of the components as with nonnegative multiplicities, together with all the greater integers (like for N (4, 5) = 12). The mean Frobenius number is the arithmetical mean value of(More)