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., singularâ€¦ (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 consistâ€¦ (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 problemâ€¦ (More)

In the 1920â€™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, whichâ€¦ (More)

V. I. Arnold reveals some unexpected connections between such apparently unrelated theories as Galois fields, dynamical systems, ergodic theory, statistics, chaos and the geometry of projectiveâ€¦ (More)

A permutation of a set of N elements is decomposing this set into y cycles of lengths xs, defining a partition N = x1 + Â· Â· Â· + xy. The length X1, the height y and the fullness Î» = N/xy of the Youngâ€¦ (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 inâ€¦ (More)