Partial immersions and partially free maps

In a recent paper~\cite{DDL10} we studied basic properties of partial immersions and partially free maps, a generalization of free maps introduced first by Gromov in~\cite{Gro70}. In this short note we show how to build partially free maps out of partial immersions and use this fact to prove that the partially free maps in critical dimension introduced in Theorems 1.1-1.3 of~\cite{DDL10} for three important types of distributions can actually be built out of partial immersions. Finally, we show that the canonical contact structure on \bR^{2n+1} admits partial immersions in critical dimension for every $n$.Comment: 8 pages, submitted to the proceedings of the conference DGA201

Solvability of the cohomological equation for regular vector fields on the plane

We consider planar vector field without zeroes X and study the image of the associated Lie derivative operator LX acting on the space of smooth functions. We show that the cokernel of LX is infinite-dimensional as soon as X is not topologically conjugate to a constant vector field and that, if the topology of the integral trajectories of X is ``simple enough'' (e.g. if X is polynomial) then X is transversal to a Hamiltonian foliation. We use this fact to find a large explicit subalgebra of the image of LX and to build an embedding of R^2 into R^4 which rectifies X. Finally we use this embedding to characterize the functions in the image of LX.Comment: 21 pages, 2 figure

Numerical analysis of solitons profiles in a composite model for DNA to rsion dynamics

We present the results of our numerical analysis of a "composite" model of DNA which generalizes a well-known elementary torsional model of Yakushevich by allowing bases to move independently from the backbone. The model shares with the Yakushevich model many features and results but it represents an improvement from both the conceptual and the phenomenological point of view. It provides a more realistic description of DNA and possibly a justification for the use of models which consider the DNA chain as uniform. It shows that the existence of solitons is a generic feature of the underlying nonlinear dynamics and is to a large extent independent of the detailed modelling of DNA. As opposite to the Yakushevich model, where it is needed to use an unphysical value for the torsion in order to induce the correct velocity of sound, the model we consider supports solitonic solutions, qualitatively and quantitatively very similar to the Yakushevich solitons, in a fully realistic range of all the physical parameters characterizing the DNA.Comment: 16 pages, 9 figure