13 research outputs found
Versal deformations of a Dirac type differential operator
If we are given a smooth differential operator in the variable its normal form, as is well known, is the simplest form
obtainable by means of the \mbox{Diff}(S^1)-group action on the space of all
such operators. A versal deformation of this operator is a normal form for some
parametric infinitesimal family including the operator. Our study is devoted to
analysis of versal deformations of a Dirac type differential operator using the
theory of induced \mbox{Diff}(S^1)-actions endowed with centrally extended
Lie-Poisson brackets. After constructing a general expression for tranversal
deformations of a Dirac type differential operator, we interpret it via the
Lie-algebraic theory of induced \mbox{Diff}(S^1)-actions on a special Poisson
manifold and determine its generic moment mapping. Using a Marsden-Weinstein
reduction with respect to certain Casimir generated distributions, we describe
a wide class of versally deformed Dirac type differential operators depending
on complex parameters
On recurrence and ergodicity for geodesic flows on noncompact periodic polygonal surfaces
We study the recurrence and ergodicity for the billiard on noncompact
polygonal surfaces with a free, cocompact action of or . In the
-periodic case, we establish criteria for recurrence. In the more difficult
-periodic case, we establish some general results. For a particular
family of -periodic polygonal surfaces, known in the physics literature
as the wind-tree model, assuming certain restrictions of geometric nature, we
obtain the ergodic decomposition of directional billiard dynamics for a dense,
countable set of directions. This is a consequence of our results on the
ergodicity of \ZZ-valued cocycles over irrational rotations.Comment: 48 pages, 12 figure
Beyond Kinetic Relations
We introduce the concept of kinetic equations representing a natural
extension of the more conventional notion of a kinetic relation. Algebraic
kinetic relations, widely used to model dynamics of dislocations, cracks and
phase boundaries, link the instantaneous value of the velocity of a defect with
an instantaneous value of the driving force. The new approach generalizes
kinetic relations by implying a relation between the velocity and the driving
force which is nonlocal in time. To make this relations explicit one needs to
integrate the system of kinetic equations. We illustrate the difference between
kinetic relation and kinetic equations by working out in full detail a
prototypical model of an overdamped defect in a one-dimensional discrete
lattice. We show that the minimal nonlocal kinetic description containing now
an internal time scale is furnished by a system of two ordinary differential
equations coupling the spatial location of defect with another internal
parameter that describes configuration of the core region.Comment: Revised version, 33 pages, 9 figure
Visualization of a hyperbolic structure in area preserving maps
Consiglio Nazionale delle Ricerche - Biblioteca Centrale - P.le Aldo Moro, 7, Rome / CNR - Consiglio Nazionale delle RichercheSIGLEITItal