Vladimir I. Arnold

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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)
An Abel differential equation y′ = p(x)y + q(x)y 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)
Theorem 1. The nth derivative of the function 1/r along n constant (translation-invariant) vector fields in R coincides on the sphere with a spherical function of degree n. Any nonzero spherical function of degree n can be obtained by this construction from some n-tuple of nonzero vector fields. These n fields are uniquely defined by the function (up to(More)
The topological structures of the generic smooth functions on a smooth manifold belong to the small quantity of the most fundamental objects of study both in pure and applied mathematics. The problem of their study has been formulated by A. Cayley in 1868, who required the classification of the possible configurations of the horizontal lines on the(More)
The multiplication by a constant (say, by 2) acts on the set Z/nZ of residues (mod n) as a dynamical system, whose cycles relatively prime to n all have a common period T (n) and whose orbits consist each of T (n) elements, forming a geometrical progression or residues. The paper provides many new facts on the arithmetical properties of these periods and(More)
We present a Hamiltonian framework for higher-dimensional vortex filaments (or membranes) and vortex sheets as singular 2-forms with support of codimensions 2 and 1, respectively, i.e., singular elements of the dual to the Lie algebra of divergence-free vector fields. It turns out that the localized induction approximation (LIA) of the hydrodynamical Euler(More)
In the 1920’s Marston Morse developed what is now known as Morse theory trying to study the topology of the space of closed curves on S2 ([7], [5]). We propose to attack a very similar problem, which 80 years later remains open, about the topology of the space of closed curves on S2 which are locally convex (i.e., without inflection points). One of the main(More)
We study finite length sequences of numbers which, at the first glance, look like realizations of a random variable (for example, sequences of fractional parts of arithmetic and geometric progressions, last digits of sequences of prime numbers, and incomplete periodic continuous fractions). The degree of randomness of a finite length sequence is measured by(More)