Projection Methods for some Constrained Systems

This article is concerned with a geometric tool given by a pair of projector operators defined by almost product structures on finite dimensional manifolds, polarized by a distribution of constant rank and also endowed with some geometric structures (Riemann,resp.Poisson,resp.symplectic).The work is motivated by non-holonomic and sub-Riemannian geometry of mechanical systems on finite dimensional manifolds.Two examples are given

A Sampling Theorem for Rotation Numbers of Linear Processes in R2{\R}^{2}

We prove an ergodic theorem for the rotation number of the composition of a sequence os stationary random homeomorphisms in $S^{1}$. In particular, the concept of rotation number of a matrix $g\in Gl^{+}(2,{\R})$ can be generalized to a product of a sequence of stationary random matrices in $Gl^{+}(2,{\R})$. In this particular case this result provides a counter-part of the Osseledec's multiplicative ergodic theorem which guarantees the existence of Lyapunov exponents. A random sampling theorem is then proved to show that the concept we propose is consistent by discretization in time with the rotation number of continuous linear processes on ${\R}^{2}.

Three-loop field renormalization for scalar field theory with Lorentz violation

Applying the counterterm method in minimal subtraction scheme we calculate the three-loop quantum correction to field anomalous dimension in a Lorentz-violating O($N$) self-interacting scalar field theory. We compute the Feynman diagrams using dimensional regularization and $\epsilon$-expansion techniques. As this approximation corresponds to a three-loop term, to our knowledge this is the first time in literature in which such a loop level is attained for a LV theory.Comment: 12 page