Proof of a Gromov conjecture on the infinitesimal invertibility of the metric inducing operators
We prove a conjecture of Gromov about non-free isometric immersions.
math.DG↗
arXiv subjects
Roberto De Leo
Publications and source records attributed to Roberto De Leo.
On the Numerical Solution of the Far Field Refractor Problem
The far field refractor problem with a discrete target is solved with a numerical scheme that uses and simplify ideas from Caffarelli, Kochengin and Oliker. A numerical implementation is carried out and examples are shown.
math.NA↗
Weak solutions of the cohomological equation on R^2 for regular vector fields
In a recent article, we studied the global solvability of the so-called cohomological equation L_X f=g in C^\infty(\Rt), where X is a regular vector field on the plane and L_X the corresponding Lie derivative. In a joint article with T. Gramchev and A. Kirilov, we studied the existence of global L^1_{loc} weak solutions of the cohomological equation for vector fields depending only on one coordinate. Here we generalize the results of both articles by providing explicit conditions for the existence of global weak solutions to the cohomological equation when X is intrinsically Hamiltonian or of finite type.
math.AP↗
Exponential growth of norms in semigroups of linear automorphisms and Hausdorff dimension of self-projective IFS
Given a finitely generated semigroup S of the (normed) set of linear maps of a vector space V into itself, we find sufficient conditions for the exponential growth of the number N(k) of elements of the semigroup contained in the sphere of radius k as k->infinity. We relate the growth rate lim log N(k)/log k to the exponent of a zeta function naturally defined on the semigroup and, in case S is a semigroup of volume-preserving automorpisms, to the Hausdorff and box dimensions of the limit set of the induced semigroup of automorphisms on the corresponding projective space.
math.DS↗
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 $n$.
math.DG↗
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.
math.DG↗
Partially Isometric Immersions and Free Maps
In this paper we investigate the existence of ``partially'' isometric immersions. These are maps f:M->R^q which, for a given Riemannian manifold M, are isometries on some sub-bundle H of TM. The concept of free maps, which is essential in the Nash--Gromov theory of isometric immersions, is replaced here by that of H-free maps, i.e. maps whose restriction to H is free. We prove, under suitable conditions on the dimension q of the Euclidean space, that H-free maps are generic and we provide, for the smallest possible value of q, explicit expressions for H-free maps in the following three settings: 1-dimensional distributions in R^2, Lagrangian distributions of completely integrable systems, Hamiltonian distributions of a particular kind of Poisson Bracket.
math.DG↗
Global Solvability in Functional Spaces for Smooth Nonsingular Vector Fields in the Plane
We address some global solvability issues for classes of smooth nonsingular vector fields $L$ in the plane related to cohomological equations $Lu=f$ in geometry and dynamical systems. The first main result is that $L$ is not surjective in $C^\infty(\R^2)$ iff the geometrical condition -- the existence of separatrix strips -- holds. Next, for nonsurjective vector fields, we demonstrate that if the RHS $f$ has at most infra-exponential growth in the separatrix strips we can find a global weak solution $L^1_{loc}$ near the boundaries of the separatrix strips. Finally we investigate the global solvability for perturbations with zero order p.d.o. We provide examples showing that our estimates are sharp.
math.AP↗
On a generalized Sierpinski fractal in RP^n
We associate a fractal in $\RPn$ to each vector basis of $\bR^{n+1}$ and we study its measure and asymptotic properties. Then we discuss and study numerically in detail the cases $n=1,2,3$, evaluating in particular their Hausdorff dimension.
math.CA↗
A note on a conjecture of Gromov about non-free isometric immersions
Wextend the results obtained recently by G. D'Ambra and A. Loi towards the proof of a conjecture of M.Gromov on isometric immersions via non-free maps.
math.DG↗
Geometry of plane sections of the infinite regular skew polyhedron {4,6|4}
The asymptotic behavior of open plane sections of triply periodic surfaces is dictated, for an open dense set of plane directions, by an integer second homology class of the three-torus. The dependence of this homology class on the direction can have a rather rich structure, leading in special cases to a fractal. In this paper we present in detail the results for the skew polyhedron {4,6|4} and in particular we show that in this case a fractal arises and that such a fractal can be generated through an elementary algorithm, which in turn allows us to verify for this case a conjecture of S.P.Novikov that such fractals have zero measure.
math.GT↗
Twist solitons in complex macromolecules: from DNA to polyethylene
DNA torsion dynamics is essential in the transcription process; simple models for it have been proposed by several authors, in particular Yakushevich (Y model). These are strongly related to models of DNA separation dynamics such as the one first proposed by Peyrard and Bishop (and developed by Dauxois, Barbi, Cocco and Monasson among others), but support topological solitons. We recently developed a ``composite'' version of the Y model, in which the sugar-phosphate group and the base are described by separate degrees of freedom. This at the same time fits experimental data better than the simple Y model, and shows dynamical phenomena, which are of interest beyond DNA dynamics. Of particular relevance are the mechanism for selecting the speed of solitons by tuning the physical parameters of the non linear medium and the hierarchal separation of the relevant degrees of freedom in ``master'' and ``slave''. These mechanisms apply not only do DNA, but also to more general macromolecules, as we show concretely by considering polyethylene.
q-bio.BM↗
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.
q-bio.BM↗
Topology of plane sections of periodic polyhedra with an application to the Truncated Octahedron
The main results of A. Zorich and I. Dynnikov about plane sections of periodic surfaces are extended to the PL case. As an application, the Stereographic Map of a truncated octahedron, extended to the whole $\Rt$ by periodicity, is analyzed numerically.
math.DG↗
First-principles generation of Stereographic Maps for high-field magnetoresistance in normal metals: an application to Au and Ag
About thirty high-field magnetoresistance Stereographic Maps have been measured for metals between Fifties and Seventies but no way was known till now to compare these complex experimental data with first-principles computations. We present here the method we developed to generate Stereographic Maps directly from a metal's Fermi Surface, based on the Lifshitz model and the recent advances by S.P. Novikov and his pupils. As an application, we test the method with an interesting toy model and then with Au and Ag.
cond-mat.other↗
Characterization of the set of "ergodic directions" in the Novikov's problem of quasi-electrons orbits in normal metals
Novikov's problem of semiclassical orbits of quasi-electrons in a normal metal leads to a correspondance between 3-ply periodic functions in R and fractals in R P^2. These fractals are the complement of infinitely many open sets labeled by integer 2-cycles of T^3. Here we present a characterization of the fractal points in terms of the open sets labels.
math.DS↗
Numerical analysis of the Novikov problem of a normal metal in a strong magnetic field
We present the results of our numerical exploration of the fractal structure found by S.P. Novikov in the problem of the behviour of magnetoresistance in a normal metal under a strong magnetic field. The case we discuss in this paper is the simplest non-trivial one, namely the case of 2 Fermi Surfaces that cut the brillouin zone along of the coordinate axes (i.e. Fermi surfaces have genus 3).
math-ph↗
Existence and measure of ergodic leaves in Novikov's problem on the semiclassical motion of an electron
We show that ``ergodic regime'' appears for generic dispersion relations in the semiclassical motion of electrons in a metal and we prove that, in the fixed energy picture, the measure of the set of such directions is zero.
math-ph↗